![Integration of GRACE Data
for Improvement of Hydrological Models
Chandan Banerjee and D. Nagesh Kumar
1 Introduction
Observation is the first and most primary step in various disciplines of geosciences
such as hydrology, meteorology, oceanography, geology, glaciology, and other plan-
etary sciences. Hydrology or hydrological sciences which essentially deals with the
question “What happens to the rain?” largely depends on gauge observations, which
have been the longest running bastion furnishing long time series of datasets. Hydro-
logical studies require datasets of both meteorological and hydrological variables
such as temperature, humidity, precipitation, streamflow, etc. to monitor, understand,
and model the complex physical processes which convert precipitation to surface
water, soil moisture, groundwater, or streamflow. For a long time, hydrological stud-
ies were completely driven by datasets produced only by gauge measurements and
to some extent field surveys. Although gauge measurements and field datasets are
indispensable tools to understand the natural processes even today, they suffer from
several limitations [6] such as
(i) localized nature of the gauges provides information only for a particular loca-
tion;
(ii) gauges cannot provide data at locations inaccessible to humans;
(iii) data procured by gauges are not easily available due to political control over
data sharing policies; and
(iv) management and maintenance of gauges are big challenges faced by concerned
authorities.
Moreover, from hydrological modeling perspective, gauge datasets are quite limit-
ing and do not help incorporate mathematical modeling of many physical processes.
Consequently, development of a systematic framework that provides us with obser-
vational datasets of the Earth having the desired properties such as global coverage,
C. Banerjee · D. N. Kumar (B)
Department of Civil Engineering, Indian Institute of Science, Bengaluru, India
e-mail: nagesh@iisc.ac.in
© Springer Nature Switzerland AG 2019
S. K. Singh and C. T. Dhanya (eds.), Hydrology in a Changing World, Springer Water,
https://doi.org/10.1007/978-3-030-02197-9_1
1](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-12-320.jpg)
![2 C. Banerjee and D. N. Kumar
continuously available in time, and accessible across political boundaries, specif-
ically of hydrological and meteorological variables, was necessary. As a result,
Earth-observing satellite remote sensing has been developed to complement the
gauge-based observations and enhance our knowledge and understanding of vari-
ous physical processes [52, 55, 57]. This has not only enhanced our ability to model
complex hydrological processes [36, 53, 86] to a large extent but also improved our
capabilities to predict and forecast hydrological extremes which have now reached
new levels.
The journey of remote sensing observations started in 1972 with Earth Resources
Technology Satellite (ERTS) 1 [14] launched by National Aeronautics and Space
Administration (NASA), USA, which later came to be known as Landsat 1. It car-
ried a multispectral scanner (MSS) recording data in four spectral bands, viz., red,
green, and two infrared bands. Since then the technology used for remote sensing
has grown by leaps and bounds. Remote sensing satellites now record not only in
the optical and near-infrared bands but also in thermal and microwave bands. The
spatial, spectral, temporal as well as radiometric resolutions have improved with
each new satellite. New data acquisition techniques are being developed such as the
synthetic-aperture radar (SAR) used to procure terrain and land cover information
[44], satellite altimeters used to measure depth of seabed, radiometers used to esti-
mate surface soil moisture [63], and hyperspectral imagers having a very high spectral
resolution are used for various applications in the fields of agriculture, mineralogy,
and environmental sciences [46].
The Terrestrial Water Storage (TWS) estimate derived from the Gravity Recovery
and Climate Experiment (GRACE) satellite data is a remarkable addition to the vast
set of remote sensing observations [76, 83]. Compared to the previous satellites,
GRACE uses a completely different technique of data acquisition. While most of the
previous satellites can observe only surface features of the land, GRACE satellites
are able to acquire information about water storages in any form at any depth. TWS
refers to the total water storage in a column of land present in the form, be it snow,
ice, surface water, soil moisture, and groundwater. Although the spatial and temporal
resolution of the GRACE data is coarse as compared to many other satellites, the
unique nature of the data makes it an invaluable tool to observe terrestrial hydrolog-
ical processes [79, 93]. The water storage that is the most difficult to observe and
monitor is groundwater and in situ well observations were the only way to mon-
itor them until the advent of GRACE. Well observations suffer from the obvious
limitations of consistency, unavailability of data for the required period, inadequate
spatial distribution of observation wells, and above all, political control over data for
transboundary aquifers. GRACE on the other hand provides a global observational
dataset periodically for the past 15 years. Using GRACE-derived datasets, scientists
have identified depleting groundwater levels in different parts of the world such as
Sacramento and San Joaquin River basins, California’s Central Valley, and High
Plains aquifer in USA [12, 23, 64, 65], Bengal Basin of Bangladesh [68] and Gan-
ga–Brahmaputra–Meghna River basin [34] in South Asia, transboundary river basins
in the Middle East [32, 88], Northern China [31] and Southern Murray Darling River
basin [15] in Australia.](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-13-320.jpg)
![Integration of GRACE Data for Improvement of Hydrological Models 3
GRACE data is being used to solve a host of scientific problems other than the
numerous studies related to groundwater. GRACE-derived TWS, also known as TWS
Anomaly (TWSA) and its derivative TWS Change (TWSC), are used to study the
dynamics of the terrestrial part of the hydrologic cycle and unravel its complex nature
[5, 27, 45, 80]. It is used to understand water budget at the spatial scale of large river
basins or continents [38, 79]. Terrestrial water budget, atmospheric water budget, or
a coupling of the two is used to estimate evapotranspiration or river discharge [60,
59, 70, 79, 78]. Evapotranspiration is an important part of the terrestrial hydrolog-
ical cycle as it is the terrestrial feedback to the atmosphere and affects the climate.
However, it is a complex hydrological variable which is difficult to estimate by the
various energy balance and aerodynamic methods as they are highly data intensive.
GRACE provides a rather simple method of its estimation. River discharge is an
equally important parameter affecting the seas and oceans, determining the fresh-
water input to the system. Ocean salinity, sea surface temperature, and various other
parameters are dependent on the amount of freshwater that comes into the oceans in
the form of river discharge. The GRACE-based method of river discharge estimation
is specifically helpful for large rivers which do not have a defined stream but forms a
large delta system as it meets the ocean, as in case of the rivers Ganga–Brahmaputra,
Indus, Irrawaddy, Mekong, and Yangtze. GRACE also finds application in drought-
related studies [29, 84]. Precipitation is typically used for drought identification,
monitoring, and management. Recently, a few studies have also used soil moisture
to monitor droughts. However, TWS data which is the total of all the water storages
helps improve the impact assessment of a drought by providing a holistic estimate
of the total amount of water lost during a drought and the time taken to regain.
The area of application of GRACE data which would be of interest for the present
discussion is its integration into hydrological models, which are sophisticated tools
used for prediction of various hydrological parameters. Prediction of river discharge
has been the sole objective of hydrologic models for a long time due to the limited
number of hydrological variables observed (as discussed earlier). However, with the
increasing number of observations, specifically satellite-based observations and huge
improvement in the computational capabilities, the structure and functions of hydro-
logical models have also evolved. They now represent more complex processes at
finer spatial and temporal scales and predict various hydrological parameters along
with streamflow [17]. Integration of GRACE data into a hydrological model should
further improve the representation of the physical processes and prediction of com-
plex parameters such as evapotranspiration, soil moisture, and snow accumulation.
In this chapter, we discuss in detail the various ways of integrating GRACE data into
a hydrological model. We elaborate on the physics behind acquisition of TWS data
through GRACE satellites and the available data products. We also review various
hydrological models used for GRACE-based studies discussing models which are
more often chosen over the others.](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-14-320.jpg)
![4 C. Banerjee and D. N. Kumar
2 GRACE Data and Gravity Recovery
Before divulging into the details of integrating GRACE data with hydrological mod-
els, it is important to understand the science of gravity recovery. The GRACE satellite
mission is a joint venture by the US and German space agencies, NASA and DLR
(DeutschesZentrum fur“
r Luft-und Raumfahrt), respectively, under the NASA Earth
System Science Pathfinder Program. The mission which was launched on March
17, 2002 consists of a pair of small and identical satellites (Fig. 1) orbiting at an
altitude of 500 km from the Earth’s surface with a separation between them of about
220 km along track. The satellites are connected by a highly accurate inter-satellite
microwave K band ranging system constantly measuring the minute changes in the
inter-satellite distance/range of the order of 10 µm. The distance between the two
satellites changes due to the changes in earth surface features, which vary in den-
sity. Higher density relates to high mass, thus culminating into greater gravitational
force and vice versa. If the Earth was homogenous in nature, the range between
the two satellites would remain constant. However, the mass distribution is highly
heterogeneous as well as constantly changing in time. The most dynamic constituent
of the planet is water that circulates through the oceans, atmosphere, lithosphere,
cryosphere, and biosphere. As a result, the time variable gravity signal acquired by
the GRACE satellites through the measurement of the inter-satellite range rate mainly
consists of the temporal variations of water as it moves from one storage compartment
to another. After removing the fluctuations in the mass of the atmosphere and oceans,
also known as Atmosphere and Ocean De-aliasing (AOD) from the total gravity sig-
nal, the seasonal and inter-annual fluctuations in TWS are obtained, expressed as
centimeters of Equivalent Water Thickness (EWT) [73, 75, 76, 89].
The inter-satellite range rate, the primary variable observed by the GRACE satel-
lites, must go through a long course of data processing to be converted to TWS.
There are three primary centers constituting the Science Data System (SDS) which
perform the processing of the Level 1 dataset to provide Level 2 and Level 3 datasets.
These centers are the Center for Space Research (CSR) at the University of Texas at
Austin, Jet Propulsion Laboratory (JPL), NASA and the German Research Center for
Geosciences (GFZ) Helmholtz Center, Potsdam. The Level 1 data from GRACE con-
sists of the inter-satellite range, range rate, range acceleration, and non-gravitational
accelerations from each satellite. The Level 2 data product is the monthly gravity field
estimates available in the form of spherical harmonic coefficients, whereas the Level
3 dataset is mass anomaly expressed in terms of EWT of TWS [39, 77]. The three
data processing centers use different data processing techniques which include dis-
tinct static gravity models, different de-aliasing schemes and varied order and degree
of the spherical harmonic coefficients to produce three separate datasets commonly
known as CSR, JPL, and GFZ datasets. However, there are other research groups
which also use other varieties of processing techniques to produce Level 2 and Level
3 data products such as the Delft Mass Transport (DMT) model of Delft University
of Technology (TU Delft) [35, 43], ITG-Grace2010 of Bonn University, and a host of
datasets produced by NASA’s Goddard Space Flight Center (GSFC). JPL’s TELLUS](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-15-320.jpg)
![Integration of GRACE Data for Improvement of Hydrological Models 5
Fig. 1 Illustration of the twin satellites of the GRACE mission, connected by the along-track K-
band microwave ranging system (Credits: NASA/JPL-Caltech)
website provides Level 3 monthly gridded as well as mascon products of GRACE
TWS estimate derived from Level 2 dataset of the three primary data centers viz
CSR, GFZ, and JPL. These products are easily accessible, available along with the
error estimates and are ready to use for hydrologists [39].
3 Large-Scale Hydrological Models
Mathematical models of the hydrological processes, commonly known as hydrolog-
ical models, have a long history as they evolved from simple lumped models with a
single output to much more sophisticated stochastic distributed hydrological mod-
els which use several input variables and estimate wide range hydrologic responses
[69]. Most of the primitive models are known as rainfall–runoff models which take
rainfall and very primary land surface characteristics to estimate runoff. However,
these models were an improvement over statistical models used for the prediction
of runoff because the former contains representation of some physical processes
and their usability in real time when forced with real-time precipitation [11]. With
advancement in computational capabilities and proliferation of remotely sensed data,
the hydrological models have hugely improved in terms of the simulation time steps,
number of climatological forcings and land surface characteristics, spatial resolu-
tion, and number of output variables. These developments finally culminated to an](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-16-320.jpg)
![6 C. Banerjee and D. N. Kumar
increased number of physical processes represented within the models as well as the
accuracy with which they are represented. Thus, the hydrological models of the new
generation are of great use for prediction of various hydrologic variables such as
runoff, streamflow, evapotranspiration, etc. which help for water resources manage-
ment [8, 42, 71]. Moreover, they also provide a robust framework to run numerical
experiments to understand the effects of various natural and anthropogenic changes
in the land surface properties and climate such as deforestation, urbanization, expan-
sion of agricultural land, global warming, increasing extreme rainfall, etc. [7, 24].
Another aspect of hydrological models that has changed with improvement in
various technologies as well as the urge to improve the accuracy in prediction of
large-scale hydrological processes is the expanse of the land surface modeled within
a single framework. Most hydrological models refer to catchment or river basin
scale modeling where the primary output variable of interest is the streamflow at
the mouth of the river basin. However, these models are calibrated for a single
catchment such that the model parameters are tuned to represent hydrologic and
climatic processes occurring only within that catchment. A new variety of models
are the Land Surface Models (LSMs) which are included within the atmospheric
General Circulation Models (GCMs) to represent the interaction of the atmosphere
with the land surface in the form of mass and energy exchange [10, 21].
As discussed earlier, the river discharges from the land surface into the oceans alter
several of its physical properties which in turn affect the climate. As a result, these
LSMs are coupled with a River Routing Model (RRM) to convert the runoff produced
by the LSM to streamflow and finally the river discharge into the oceans. LSMs have
evolved greatly over the past few decades to accurately represent the partitioning of
the incoming net radiative energy into latent and sensible heat fluxes and the parti-
tioning of precipitation into runoff, evaporation, and water storage. One such LSM
is the Community Land Model (CLM), part of the Community Earth System Model
(CESM) of the National Center for Atmospheric Research (NCAR), USA [10]. The
hydrologic processes represented in the model (shown in Fig. 2) include interception
of precipitation by canopy, throughfall, transpiration, soil evaporation, canopy evap-
oration, infiltration, runoff, soil moisture, aquifer recharge, snow accumulation, melt,
and sublimation. Other than the hydrologic cycle, the model includes other physi-
cal processes such as land biogeophysics, biogeochemistry, ecosystem dynamics,
and anthropogenic interventions (Fig. 2). A similar framework is the Noah-Multi-
parameterization Land Surface Model (Noah-MPLSM) which includes detailed veg-
etation dynamics including canopy shading and under-canopy snow dynamics along
with the capability to differentiate between C3 and C4 pathways of photosynthesis
[51, 92]. The Noah-MP LSM version 1.6 was implemented in Weather Research
and Forecasting (WRF) Model version 3.6. WRF is a numerical weather predic-
tion model developed mainly by NCAR and National Centers for Environmental
Prediction (NCEP).
LSMs coupled within a GCM framework are not the only large-scale hydrological
models simulating the water and energy cycles along with geochemical processes and
vegetation dynamics. There are many large-scale uncoupled or stand-alone LSMs,
sometimes also known as the Global Hydrological Models (GHMs) simulating phys-](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-17-320.jpg)
![Integration of GRACE Data for Improvement of Hydrological Models 7
Fig. 2 A schematic diagram showing the energy cycle, hydrological cycle, biogeochemical, veg-
etation dynamics, and land use change represented within the Community Land Model (CLM)
(Credits: http://www.cesm.ucar.edu/models/clm/)
ical processes at global scale such as the WaterGAP model (Water—Global Analy-
sis and Prognosis model) [2], developed by University of Kassel and University of
Frankfurt,Germany,PCR-GLOBWB(PCRasterGLOBalWaterBalancemodel)[96]
conceived by Utrecht University, The Netherland, ISBA-TRIP (Interactions between
Soil, Biosphere, Atmosphere—Total Runoff Integrating Pathways) [3, 19] created by
Centre National de Recherchés Météorologiques, France and the Global Land Data
Assimilation System (GLDAS) framework [61], developed by GSFC and NCEP. The
WaterGAP model is designed for the assessment of macro-scale processes of the ter-
restrial hydrological cycle, taking into consideration anthropogenic component to
simulate freshwater availability and irrigation water use. PCR-GLOBWB includes
subgrid schemes for partitioning of rainfall into runoff, infiltration, interflow, ground-
water recharge, and baseflow, as well as routing of the generated runoff. The model
includes detailed anthropogenic effects to the extent that it includes more than 6000
manmade reservoirs. Thus, the human water use is completely integrated into the
hydrological model at time step, calculating water demand, surface and groundwater
abstraction, consumptive water use, and return flow. ISBA is a relatively simple LSM
calculating variability in energy and water budgets with a saturation excess overland
flow approach to simulate runoff based on TOPMODEL hydrological model [9].
This is coupled with TRIP, a simple RRM which converts runoff simulated by ISBA](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-18-320.jpg)
![8 C. Banerjee and D. N. Kumar
into river discharge for a global river network. The GLDAS framework consists of
four different LSMs, viz., Mosaic, CLM, Noah, and Variable Infiltration Capacity
(VIC) models forced by a single forcing dataset. These four models differ mainly in
the depth of soil considered for the simulation of soil water interaction and storage
as well as the number of layers into which the total depth is divided. It should be
noted that both the ISBA-TRIP model and the GLDAS set of models do not include
any anthropogenic effects of water storage and water use. Although the LSMs are of
great use, a major challenge lies in the calibration of these models which can only be
carried out using vegetation indices or streamflow for large river basins by routing
the simulated runoff. GRACE provides a very useful data first for the evaluation of
such LSMs and eventually can be assimilated into the models for better estimation
of various hydrologic variables. In the following sections, these two approaches of
integration of GRACE data are discussed in detail.
4 Evaluation of Model Simulations Using GRACE Data
As discussed in the previous section, coupled and uncoupled LSMs try to simu-
late several hydrological variables by incorporating most complex of the physical
processes, mimicking them to the maximum extent possible. Scientists are contin-
uously trying to improve these models by better parameterization and adding more
and more hydrological, geophysical, and biophysical processes into these models. A
handy dataset for quick evaluation of the hydrological fields simulated by these mod-
els is the GRACE dataset. Due to its continuous global coverage, GRACE data could
be used for evaluation of LSMs in cold regions affected by snow, arid, and semi-arid
regions characterized by very low or no soil moisture conditions as well as areas char-
acterized by a heavy to very heavy monsoonal rainfall. Table 1 provides a detailed
chronological list of various studies carried out at global, continental, regional, and
river basin scales to compare and evaluate various LSMs to estimate the accuracy
with which it simulates various water storages and physical processes affecting them.
Some studies tried to improve the estimation of certain variables by incorporating
better and more detailed process representation and validated the improvements by
comparison with GRACE data. The CLM LSM of NCAR is one such model which
has seen continual efforts of betterment and corresponding evaluation using GRACE
data. One of the limitations observed with CLM 2.0 was its representation of frozen
soil which included completely frozen soil in areas with temperature below 0 °C,
resulting in higher and earlier than expected runoff caused by spring season rainfall.
The modifications suggested were allowance for the coexistence of ice and water in
soil, the concept of a fractional permeable area, and considering both liquid water
and ice together as soil moisture for calculating hydraulic conductivity. These mod-
ifications improved both the surface runoff and the soil water storage estimates of
CLM when the simulations were evaluated for river basins in the cold regions, viz.,
Lena, Yenisei, Mackenzie, Ob, Churchill-Nelson, and Amur using both streamflow
and GRACE data [48, 49]. Another deficiency of the CLM model was its inability](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-19-320.jpg)
![Integration of GRACE Data for Improvement of Hydrological Models 9
to model the groundwater dynamics as the column of soil considered extends only
to 3.8 m below the surface. In another attempt to improve the CLM model, a Sim-
ple Groundwater Model (SIMGM) which represents an unconfined aquifer along
with the recharge and discharge processes was included within the framework [50].
Although the modification worked out well for all the 12 river basins considered in
the study, it is not expected to do well in cold regions where the water table is exposed
to freezing conditions due to the obvious differences in the physical processes. In a
more recent attempt to accurately represent groundwater dynamics within the CLM
model version 4.5, it was found that addition of a no-flux boundary condition at
the base of the soil layer improved the estimate. As a result, these simulations from
the improved CLM models were found to agree well with GRACE-derived TWS
observations [72].
A few studies also tried to improve the ISBA-TRIP hydrological model by com-
paring modified versions of the model with GRACE data. Initial comparisons of the
model with GRACE data outlined some model deficiencies such as the high storage in
the form of surface water within the river channel as a part of the routing scheme over-
estimated the maximum and underestimated the minimum TWS values mainly in the
tropical region. Other deficiencies within the ISBA-TRIP model were the calculation
of evaporation and snow accumulation. However, the major limitation was identified
as the oversimplified routing model and the absence of anthropogenic effects within
the model [3, 19, 54, 87]. Although human impact was not included in the modified
version, improvements were suggested for TRIP—the routing model which included
a simple groundwater reservoir and a variable streamflow velocity calculation. Sev-
eral other LSMs were evaluated globally or regionally using GRACE data. Inclusion
of a water exchange scheme between continents and oceans included in the Organis-
ing Carbon and Hydrology in Dynamic Ecosystems (ORCHIDEE) LSM resulted in
better simulation of land water storage [Ngo-Duc et al., 2007]. GRACE data when
compared to the Australian Water Resource Assessment (AWRA) model suggested
a need for improvement in representations of diffuse groundwater discharge pro-
cesses and interaction between surface and groundwater [van Dijk et al., 2011]. Doll
et al. [2014] found that the WaterGAP model version 2.2 underestimates TWS as
compared to GRACE with a phase lag of a month observed between the two. Evalu-
ation of the GLDAS framework-versions 1 & 2 carried out for China by Wang et al.
[2016] showed inconsistency in the rate of change of TWS. The four land surface
models (Noah, SAC-Sacramento Soil Moisture Accounting Model, (VIC) Variable
Infiltration Capacity Model, and Mosaic) applied in the newly implemented National
Centers for Environmental Prediction (NCEP) operational and research versions of
the North American Land Data Assimilation System version 2 (NLDAS-2) were also
evaluated using GRACE data [Xia et al., 2016]. A common source of inconsistency
ob-served between the GRACE observation and model simulation was attributed
to the error and uncertainty present in the precipitation dataset which is a primary
forcing for all hydrologic models.](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-20-320.jpg)
![10 C. Banerjee and D. N. Kumar
Table 1 Comparative list of studies evaluating and comparing hydrological models with GRACE
data
Authors
(Year)
GRACE data Model Input data Study region Study period
Niu and Yang
[49]
Chen et al.
[16], Seo and
Wilson [67]
CLM 2.0 with
SIMTOP
GLDAS 1-degree
3-hourly data
(2002–2004)
Lena, Yenisei,
Mackenzie,
Ob, Churchill-
Nelson and
Amur
August
2002–July
2004
Niu and Yang
[48]
Chen et al.
[16], Seo and
Wilson [67]
CLM and
Modified
CLM
GLDAS 1-degree
3-hourly data
(2002–2004)
Global and
Ob, Yangtze,
Amazon, Taz
and Ural
River Basin
August
2002–July
2004
Swenson and
Wahr [74]
Swenson and
Wahr [73]
Atmospheric
and Terrestrial
Water Balance
model
GCM output and
NCEP/DOE R-2 for
atmospheric water
balance and
GLDAS/Noah LSM for
terrestrial water budget
Mississippi
and Ohio-
Tennessee
River basins
June
2002–April
2004
Ngo-Duc
et al. [47]
Ramillien
et al. [56]
ORCHIDEE
modified to
include a
routing
scheme
P: 6-hourly NCEP/NCAR
constrained by
monthly CMAP;
Others: 6-hourly NCC
(NCEP/NCAR)
corrected by CRU
atmospheric forcing
Global and
Amazon,
Congo, Niger,
Mississippi,
Yangtze,
Ganges,
Brahmaputra,
Mekong
May
2002–Decem-
ber
2003
Niu et al. [50] Chen et al.
[16], Seo and
Wilson [67]
Modified
CLM with
SIMTOP and
SIMGM
1-degree 3-hourly
GLDAS dataset
(2002–2004)
12 Global
river basins
not affected
by snow or ice
August
2002–Decem-
ber
2004
Alkama et al.
[3]
CSR-RL04,
JPL-RL 4.1,
GFZ-RL04
estimates
ISBA-TRIP 3-hourly 1-degree
Princeton University data
Global and 33
large river
basins
Aug
2002–Dec
2006
Decharme
et al. [19]
CSR-RL04,
JPL-RL4.1,
GFZ-RL04
estimates
TRIP with
groundwater
storage and
variable flow
velocity
Runoff simulated by
ISBA of Alkama et al. [3]
Global and 12
large river
basins
Aug
2002–Dec
2006
van Dijk et al.
[97]
1-degree
gridded TWS
estimates
from CSR
Australian
Water
Resource
Assessment
(AWRA)
0.05-degree gridded
meteorological forcings
obtained by interpolation
of Station data
Continental
Australia
January
2003–Decem-
ber
2010
Grippa et al.
[27]
RL04 of CSR,
JPL and GFZ,
DEOSDMT,
GRGS-
EIGEN-GL04
and 10 day, 4°
GSFC
HTESSEL,
ORCHIDEE-
CWRR,
ISBA,
JULES,
SETHYS,
NOAH,
CLSM, SSiB,
SWAP
Rainfall: TRMM 3B42,
Atmospheric forcings:
ECMWF short-term
forecast data Downwell
Radiative fluxes: mix of
ECMWF and Land
Surface Analysis Satellite
Applications Facility
West Africa Jan 2003–Dec
2007
(continued)](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-21-320.jpg)
![Integration of GRACE Data for Improvement of Hydrological Models 11
Table 1 (continued)
Authors
(Year)
GRACE data Model Input data Study region Study period
Pedinotti et al.
[54]
CSR-RL04,
JPL-RL4.1,
GFZ-RL04
estimates
ISBA-TRIP TRMM-3B42 and
RFE-Hybrid rainfall for
ISBA-TRIP CHS, other
atmospheric forcings
from ECMWF
Niger River
Basin
Jan 2003–Dec
2007
Vergnes and
Decharme
[87]
CSR-RL04,
JPL-RL 4.1,
GFZ-RL04
estimates
TRIP Total runoff from ISBA
simulation by Alkama
et al. [4]
Global and 12
large river
basins
August
2002–August
2008
Rosenberg
et al. [62]
I-degree
gridded CSR
dataset
VIC modified
to include
SIMGM
1/8th-degree Gridded
from precipitation and
maximum/minimum
temperature data from
NOAA Cooperative
Observer stations and
wind data from
NCEP-NCAR reanalysis
Colorado
River Basin
2002–2010
Cai et al. [13] 1-degree
gridded TWS
estimates
from CSR
RL4.0
Noah-MP NLDAS Phase 2
atmospheric forcing at
1/8° resolution
Mississippi
River Basin
2003–2009
Doll et al. [20] 0.5-degree
gridded
GFZ-RL05,
CSR-RL05
and ITG-
Grace2010
WaterGAP
2.2
Daily climate dataset
WFD (WATCH Forcing
Data)/WFDEI (Watch
Forcing Data
ERA-Interim)
Global 2003–2009
Swenson and
Lawrence [72]
CSR RL05 CLM version
4.5 with
modification
1.25 longitude × 0.9
latitude ECMWF
ERA-Interim Reanalysis
data
Lower
Colorado
River basin, in
the
southwestern
United States,
and a region
in
northeastern
Australia
2002–2014
Ahmed et al.
[1]
1-degree
gridded TWS
estimates
from CSR
RL05
CLM4.5-SP
and
GLDAS-Noah
GLDAS: NOAA and
CPC/CMAP and CLM:
CRU/CRUNCEP
Continental
Africa (Niger,
Zambezi,
Okavanko,
Limpopo)
2003–2010
Wang et al.
[90]
GRACE
Tellus RL05
CSR, JPL,
GFZ
GLDAS1
(Noah, CLM,
Mosaic, VIC)
GLDAS2
(Noah 3.3)
ECMWF &
NCEP–NCAR reanalyses
data, NOAA/GDAS and
Princeton University
atmospheric fields,
AGRMET radiation
fields,
China 2002–2010
(continued)](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-22-320.jpg)
![12 C. Banerjee and D. N. Kumar
Table 1 (continued)
Authors
(Year)
GRACE data Model Input data Study region Study period
Xia et al. [91] GRACE
Tellus RL05
CSR, JPL,
GFZ average
NLDAS-2
operation
(Mosaic and
Noah) and
research
(SAC-Clim
and VIC4.0.5)
CPC, PRISM & NARR
precipitation data and 2-m
air temperature from
NARR
USA 2003–2014
Zhang et al.
[95]
GRACE
RL05 Level-2
products from
GFZ
LSDM,
WGHM,
JSBACH,
MPI-HM
WFDEI dataset based on
ERA-Interim reanalysis
data
31 largest
river basins
2003–2012
5 GRACE Data Assimilation
Data assimilation is a statistical technique of combining the simulations or forecasts
from a prediction model with measurements from an observing system to produce
improved estimates. Evaluation of LSMs has been one of the most explored tech-
niques of utilizing GRACE data for the improvement of model physics and simu-
lation accuracies. However, it is an indirect method where model deficiencies are
figured out by comparing model outputs with GRACE observations followed by
improving model physics solely based on our understanding of the intricate details
of hydrological processes. This to some extent is limiting since the knowledge and
understanding of the hydrological processes are itself limited and the large infor-
mation hidden within the GRACE observations may be completely overlooked. As
an alternate method of data integration, GRACE data assimilation techniques were
explored where the observational dataset is directly utilized to improve the model
simulation at each time step. Although it apparently does not improve model physics
or our understanding of hydrological processes, GRACE data assimilation improves
model simulations to a great extent, also facilitating spatial and temporal disaggre-
gation of GRACE data as a byproduct. Table 2 gives a detailed chronological list of
studies performed in this field of research.
The assimilation of GRACE data into LSMs has two major challenges. The typical
temporal and spatial resolution of the GRACE observation is much coarse as com-
pared to the LSMs. The GRACE data provided by NASA JPL’s TELLUS website
has a spatial resolution of ~100 km (1 degree) and a temporal resolution of a month.
On the contrary, most LSMs are run at a daily or sub-daily scale, with the spatial res-
olution varying from 5 km (0.05°) to a maximum of 50 km (0.5°). Hence, the process
of data assimilation invariably includes a spatial and temporal disaggregation tech-
nique. Consequently, a widely used and efficient data assimilation technique, known
as the Ensemble Kalman Filter (EnKF) [22], is used in most of the previous literature
(Table 2). The EnKF is a variant of a statistical technique known as the Kalman filter
and is used for large problems. It has the inherent assumptions that the probability](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-23-320.jpg)
![Integration of GRACE Data for Improvement of Hydrological Models 13
Table
2
Comparative
list
of
GRACE
data
assimilation
studies
Authors
(Year)
GRACE
data
Model
Input
data
Study
region
Study
period
Assimilation
method
Zaitchik
et
al.
[94]
CSR
(RL01),
GFZ
(RL03),
JPL
(RL02)
CLSM
GLDAS
forcing
database
Mississippi
(4
sub-
catchments)
January
2003–May
2006
Ensemble
Kalman
Smoother
Houborg
et
al.
[29]
CSR-RL04
CLSM
NLDAS-2
ad
GLDAS
data
for
study
period
model
run
and
Princeton
University
data
for
long-term
simulation
North
America
August
2002–July
2009
Ensemble
Kalman
Smoother
Li
et
al.
[41]
CSR-RL04
CLSM
GLDAS
forcing
database
Western
and
Central
Europe
August
2002–July
2009
Ensemble
Kalman
Smoother
Huang
et
al.
[30]
CSR-RL05
Noah-MP
0.1-degree,
3-hourly,
near-surface
meteorological
dataset
produced
by
the
ITPCAS
Yangtze
River
basin
Jan
2003–Dec
2010
Proposed
framework
Reager
et
al.
[58]
CSR-RL05
CLSM
Same
as
Zaitchik
et
al.
[94]
Mississippi
river
basin
April
2002—
Dec
2014
Ensemble
Kalman
Smoother
Tangdamrongsub
et
al.
[82]
CSR-RL05
OpenStreams
wflow_hbv
model
(HBV-96)
European
Climate
Assessment
&
Dataset
(ECA
&
D),
ENSEMBLES
project
and
Princeton
University
Dataset
Rhine
river
basin
Dec
2003—Oct
2007
Ensemble
Kalman
Filter
Girotto
et
al.
[25]
Gridded
CSR-RL05
CLSM
MERRA
USA
Jan
2003—Dec
2013
Sequential
Kalman
filtering
technique
(continued)](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-24-320.jpg)
![14 C. Banerjee and D. N. Kumar
Table
2
(continued)
Authors
(Year)
GRACE
data
Model
Input
data
Study
region
Study
period
Assimilation
method
Schumacher
et
al.
[66]
TWS
values
using
WGHM,
ITG-
GRACE2010
error
covariance
WGHM
CRU
TS
3.2,
GPCC,
WFDEI
Mississippi
river
basin
August
2003
EnKF,
SQRA,
SEIK
Girotto
et
al.
[26]
Gridded
CSR-RL05
CLSM
MERRA
India
Jan
2003–Aug
2015
3D
Ensemble
Kalman
Filter
Khaki
et
al.
[33]
ITSG-
Grace2014
W3RA
Princeton
University
forcing
dataset
Australia
Feb
2002–Dec
2012
(stochastic)
EnKF,
ETKF,
SQRA,
DEnKF,
EnSRF,
EnOI
and
PF
with
Multinomial
(PFMR)
and
Systematic
(PFSR)
Resampling
Tangdamrongsub
et
al.
[81]
CSR-RL05
PCR-
GLOBWB
ECMWF
Era-Interim,
TRMM,
CRU,
Princeton
and
China
Daily
Ground
Climate
Dataset
Hexi
corridor
in
Northern
China
April
2002–Dec
2010
EnKF
with
and
without
errors
Tian
et
al.
[85]
JPL-RL05
M,
3-degree
mascon
product
W3
Model
WFDEI
forcing
data,
Global
Tree
cover
fraction
map
and
MODIS
white-sky
albedo
Australia
Jan
2010–Dec
2013
EnKF
and
EnKS](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-25-320.jpg)
![Integration of GRACE Data for Improvement of Hydrological Models 15
Fig. 3 A schematic diagram showing the concept of a typical Kalman Filter (Credit: Melda Ulusoy,
MathWorks)
distributions are all Gaussian and the predictive model is linear. The Kalman filter
(Fig. 3) is a recursive filtering mechanism which combines the simulation of a model
and a noisy measurement, both of which are assumed to be Gaussian distributions to
estimate the most likely state variables. The model estimate is generally less prob-
able and contains more uncertainty than the measurement. However, the use of the
EnKF provides an optimal estimate of the state variable which is much more proba-
ble and contains less uncertainty as compared to both the model prediction and the
measurement, as shown in Fig. 3.
The second challenge is the hydrological variable of interest. GRACE obser-
vations result into TWS data which, as discussed earlier, is the aggregation of all
the surface and subsurface water storages. To assimilate GRACE TWS data, there
needs to be a hydrological variable within the model to which it can be mapped.
The problem in this case is that all hydrological models have separate surface and
subsurface storages modeled as different processes. Even if all the storages are added
up to create a hydrological variable to be mapped against GRACE TWS data, it falls
short due to the absence of groundwater storage. Most of the hydrological models
incorporate groundwater dynamics as a boundary condition at the bottom of the
soil column considered which is typically 2–4 m in depth from the ground surface.
To resolve this issue, the catchment land surface model is the most preferred LSM
used for assimilation as it contains an unconfined groundwater reservoir. Several
studies have assimilated the GRACE TWS data with one of the primary objectives
being improvement of groundwater estimation. Zaitchik et al. [2008] assimilated
GRACE data into the CLSM using an ensemble Kalman smoother. Results indi-
cated an improved correlation between observed ground-water and data assimilated
simulated groundwater. In a similar effort, GRACE data was assimilated into the
OpenStreams wflow_hbv model using an ensemble Kalman filter for the Rhine river
basins. Results show increase in correlation between observed and simulated ground-
water from 0.6 to 0.7 and 15% reduction in RMSE as a result of this data assimilation
[Tangdamrongsub et al., 2015]. In both the cases, slight improvement in streamflow
simulation was also observed. Tangdamrongsub et al. [2017] showed that assimi-](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-26-320.jpg)
![16 C. Banerjee and D. N. Kumar
lation of GRACE data increased the accuracy of groundwater estimate, simulated
for a semi-arid region in northern China by PCR-GLOBWB by 25%. GRACE data
assimilation was also carried out with the objective of drought assessment because
most frameworks lack information of groundwater and soil moisture of deeper lay-
ers. Houborg et al. [2012] and Li and Rodell [2015] assimilated GRACE data into
CLSM model to derive drought indicators for North America and conterminous US
respectively. A similar exercise was carried out for western and central Europe by Li
et al. [2012]. These efforts disaggregated GRACE data in both spatial and temporal
dimension. GRACE data assimilation was also carried out to estimate human induced
changes in TWS and assess regional flood potential [Y Huang et al., 2015a; Reager
et al., 2015]. Further studies concentrated on improving the data assimilation using
better variants of the ensemble Kalman Filter and other hydrologic dataset such as
the soil moisture from Soil Moisture and Ocean Salinity (SMOS) mission [Girotto
et al., 2016; Girotto et al., 2017; Khaki et al., 2017; Schumacher et al., 2016; Tian
et al., 2017].
6 Conclusions
The hydrological models altogether have improved from the simple lumped models
and now include not only hydrological processes but all such physical, chemical,
and biological processes that affect or is affected by water (a typical example of
which is shown in Fig. 2). Integration of GRACE data into hydrological models
has improved their model physics and prediction capabilities. Such models now
represent better dynamics of frozen soil, dry soil in arid climate, groundwater, and
vegetation. This also improved the estimation of various hydrological and vegetation
parameters. Further improvements were achieved by GRACE data assimilation into
hydrological models with the added advantage of disaggregation of GRACE TWS
observations. Moreover, the GRACE data processing techniques have also improved
with the most recent studies using Release 05 dataset which has a much higher
accuracy as compared to the initial releases. The GRACE Follow-On (GRACE-FO)
missionis scheduledtobelaunchedin2018whichis expectednot onlytocontinuethe
unique GRACE observations but also to have some improvements as compared to its
forerunner [18]. Meanwhile, scientists are still working on the processing techniques
of the GRACE data and the new Release 06 of the GRACE dataset having better
accuracy is available for use [28]. Thus, there are numerous avenues in which further
improvement is possible that will unravel new vistas of knowledge in future.
References
1. Ahmed M, Sultan M, Yan E, Wahr J (2016) Assessing and improving land surface model outputs
over africa using GRACE, field, and remote sensing data. Surv Geophys 37(3):529–556. https://](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-27-320.jpg)
![An Analysis of Spatio-Temporal Changes
in Drought Characteristics over India
Ganeshchandra Mallya, Shivam Tripathi and Rao S. Govindaraju
1 Introduction
1.1 Introduction to Droughts
Droughts are among the world’s costliest disasters with an annual cost estimated
in the range of $6–$8 billion [20]. Unlike other natural disasters such as floods and
earthquakes, droughts manifest slowly and are already a serious threat before they are
detected. Droughts have major impacts on agriculture, natural habitats and ecosys-
tems, and economies of affected regions. Modeled precipitation and temperature
results from different climate change scenarios indicate that droughts are likely to
intensify over many parts of the world in the next 20–50 years [13], suggesting the
need to assess drought impacts more accurately and develop appropriate mitigation
strategies.
When a drought event occurs, moisture deficits are identified from many hydro-
logic variables such as precipitation, streamflow, soil moisture, snowpack, ground-
water levels, and reservoir storage [81]. Because drought impacts are experienced
differently across the world, no universally accepted definition of drought exists.
However, three types of droughts are commonly featured in the scientific literature
[17]:
a. Meteorological droughts result from deficits in precipitation amounts when com-
pared to the long-term average for a region. This shortage in precipitation can
develop quickly and also end abruptly.
G. Mallya · R. S. Govindaraju (B)
Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA
e-mail: govind@purdue.edu
S. Tripathi
Department of Civil Engineering, Indian Institute of Technology, Kanpur,
UP 208016, India
© Springer Nature Switzerland AG 2019
S. K. Singh and C. T. Dhanya (eds.), Hydrology in a Changing World, Springer Water,
https://doi.org/10.1007/978-3-030-02197-9_2
23](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-34-320.jpg)
![24 G. Mallya et al.
b. Agricultural drought conditions prevail when available soil moisture is insuffi-
cient to replace evapotranspiration losses in the root zone [83]. The timing of
soil moisture deficit plays a critical role, because deficiencies during the grow-
ing season can adversely impact crop yields. During droughts, plants are under
stress and cannot fight off pests. Fertilizers and pesticides are also not effective
in the absence of moisture resulting in failure of crops. The onset of agricul-
tural droughts depends on antecedent soil moisture conditions and usually lags
meteorological droughts.
c. Hydrologic droughts reflect shortages in water supply mainly in the form of
reduced streamflows, reservoir and lake levels, and groundwater levels. Hydro-
logic droughts persist for longer durations when compared to meteorological and
agricultural droughts because precipitation deficits translate to deficit in other
hydrologic variables with significant time lags in some instances.
In this chapter, we focus on meteorological droughts. Precipitation deficits are not
the only cause of droughts. Industrial and agricultural water demands have increased
exponentially over the last few decades leading to water scarcity. With the increase in
the emission of greenhouse gases, a steady rise in temperature has been observed over
many parts of the globe. Increasing temperatures have affected the global hydrologic
cycle leading to spatiotemporal variability of precipitation at different scales [54,
74], thereby affecting drought characteristics.
1.2 Drought Characterization and Monitoring
Drought indicators or indices are commonly used to characterize and monitor
droughts and their impacts. Generally, all drought indicators use some measure of
water deficit for analysis. Multiple hydrometeorological variables can also be used in
a single drought indicator to capture the complex interactions that lead to droughts.
Some of the desirable properties of a drought index are: (1) it should be sensitive to
the timescale appropriate for the problem at hand; (2) the index should be able to
capture the characteristics of both shorter and longer duration droughts; (3) it should
be applicable to the problem being studied; (4) it should be possible to identify his-
torical droughts; (5) the index should be capable of monitoring droughts on a near
real-time basis [21, 32]; and (6) it should have drought forecasting capability.
Several studies provide comprehensive review of drought indices [32, 55]. Palmer
drought severity index (PDSI; Palmer [61]) is a popular meteorological drought index
that uses precipitation and temperature for estimating demand and supply of soil
moisture within a two-layer water balance model. PDSI provides outlooks of mois-
ture conditions that are comparable across regions and over different months. PDSI
values typically vary from −4.0 to +4.0, negative values indicating drought condi-
tions, while positive values indicate wet conditions. Another drought index that is
popular because of its computational simplicity and forecasting ability at different
time scales is the standardized precipitation index (SPI; McKee et al. [51]). The SPI](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-35-320.jpg)
![An Analysis of Spatio-Temporal Changes … 25
is recommended by the World Meteorological Organization as a standard meteoro-
logical drought-monitoring index [30]. The SPI first fits a probability distribution
to historic precipitation time series data, and then normalizes the fitted distribution
using the standard inverse Gaussian function to compute the drought index. SPI val-
ues are dimensionless with negative values indicating drought conditions, and the
magnitudes of their departures from zero indicating the severity of the drought.
The crop moisture index (CMI; Palmer [60]) that monitors short-term moisture
supply was developed to monitor agricultural droughts. With the improvements in
satellite remote sensing, monitoring crop and vegetation health over large spatial
extents have become routine. For example, vegetation condition index (VCI; Liu and
Kogan [40]) uses the advanced very high-resolution radiometer radiance (AVHRR)
data to study drought characteristics (early onset, intensity, frequency, and duration)
and vegetation health. Along similar lines, the normalized difference water index
(NDWI; Gao [23]) uses near-infrared (NIR) and short-wave infrared (SWIR) chan-
nels to study the variation of moisture content and spongy mesophyll in vegetation
canopies.
Monthly non-exceedance probability computed by compiling weighted values of
variables such as reservoir storage, stream flow, snowpack, and precipitation resulted
in the development of a hydrologic drought index called the surface water supply
index (SWSI; Shafer and Dezman [71]). Other popular hydrologic drought indices
are standardized streamflow index (SSI) and standardized runoff index (SRI; Shukla
and Wood [73]). The SRI is computed and interpreted along similar lines as SPI.
Drought indices have been used for identifying droughts and their triggers [76],
assessing drought status [35], forecasting droughts [1], performing drought risk anal-
ysis [31], and studying relationship of droughts with local-scale regional hydrolog-
ical variables such as water quality [75] and large-scale climate patterns like El
Niño–Southern Oscillation [11, 41, 69]. Drought indices are also used for classify-
ing droughts and quantifying their temporal trends. These two applications of drought
indices are reviewed in the following paragraphs.
1.3 Drought Classification
Drought classification schemes typically classify droughts based on their severity or
intensity, and are often based on drought indices that measure degree of departure
of hydrometeorological variables, such as precipitation and streamflow, from their
long-term averages. Water resource planners rely on drought classification to select
drought mitigation strategies. Hence, weather agencies throughout the world rou-
tinely issue drought classification bulletins. For example, the US Drought Monitor
releases a weekly update of drought status in USA by classifying droughts into five
classes—D0 to D4 with the latter representing exceptional drought. Likewise, India
Meteorological Department (IMD) issues drought bulletins classifying droughts into
three categories, namely, mild, moderate, and severe.](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-36-320.jpg)
![26 G. Mallya et al.
Common quantitative drought classification schemes work in two steps—first, by
defining a drought index using hydrometeorological observations of typically 30-
year period to establish normal conditions [33] and next, by categorizing droughts
based on predefined thresholds on the index value. Examples include IMD classifica-
tion that uses departure of rainfall from its long-term average as a drought index, and
US Drought Monitor classification that, along with other indices, uses standardized
precipitation index (SPI) as a drought index. Among several drought classification
schemes [13, 32, 55], the scheme based on standardized precipitation index (SPI;
McKee et al. [51]) is very popular because of its computational simplicity and versa-
tility in comparing different hydrometeorological variables at different time scales.
In SPI, historical observations are used to compute the probability distribution of the
monthly and seasonal (4, 6, and 12 months) precipitation totals. The fitted probabil-
ity distributions are then normalized using the standard inverse Gaussian function
to calculate SPI values. A negative value of SPI indicates precipitation less than the
median rainfall, and the magnitude of departure from zero represents the severity of
a drought.
Standard SPI-based drought classification, though popular, has many weaknesses
[48]. It provides discrete classification and ignores uncertainties arising from data
errors, model assumptions, and parameter estimates. Thus, users are not aware of
inherent uncertainties in drought classification often required for making informed
decisions. Further, in the context of SPI, there is an ongoing debate on the selection
of the parametric distribution for fitting the data. McKee et al. [50] in their original
paper on SPI recommend a gamma distribution. Lloyd-Huges and Saunders [42]
found gamma distribution to be an appropriate model for Europe. Guttman [27] sug-
gested Pearson-III distribution as the best universal model for SPI because it provides
more flexibility than the gamma distribution. Rossi and Cancelliere [67] found nor-
mal, lognormal, and gamma distributions to be suitable for different datasets in their
study.LoukasandVasiliades[43]investigateddifferenttheoreticaldistributionsusing
Kolmogorov–Smirnov (K–S) test and chi-squared test and found extreme value-I dis-
tribution to be the most suitable for studying droughts over Thessaly, Greece. Mishra
et al. [53] argue that different distributions may be appropriate for different drought
durations (window size), and recommend the K–S test for choosing an appropri-
ate distribution. Bonaccorso et al. [7] used Lilliefors test to choose among normal,
lognormal, and gamma distributions while Russo et al. [68] used three parameters
generalized extreme value (GEV) distribution for SPI analysis. Thus, there is no
consensus on the choice of distribution for SPI analysis.
Mallya et al. [48] used hidden Markov model (HMM) for drought classification by
conceptualizing hidden states in the model to represent drought states. Their model
avoided the need for specifying thresholds for drought classification and provided
probabilistic drought classification by accounting for model uncertainties; however,
the number of hidden states (drought classes) was prespecified. To facilitate com-
parison of HMM drought index (HMM-DI) classification with standard methods,
they specified 11 hidden states. Since the number of states is imposed on the model,
it is possible that for datasets with short record length the model suffers from an
overspecification problem, i.e., the model structure is more complicated than sup-](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-37-320.jpg)
![An Analysis of Spatio-Temporal Changes … 27
ported by the dataset. Specifically, in the HMM context, overspecification would
occur if the number of specified hidden states is more than that needed to model
the data. Overspecification can result in parameter identification problems leading
to unreliable results.
Mallya et al. [47] proposed a method that adapts SPI drought classification
methodology by employing gamma mixture model (Gamma-MM) in a Bayesian
framework. The method alleviates the problem of selecting suitable distribution for
SPI analysis, quantifies modeling uncertainties, and propagates them for probabilistic
drought classification. Further, it avoids overspecification using a Bayesian approach
for optimally selecting the number of hidden states in the model.
1.4 Temporal Trends in Droughts
Temporal trends in droughts are identified by determining changes or sudden shifts
in the distributional properties of the underlying hydrological variable. Classical
drought indices such as SPI, or even probabilistic drought indices such as HMM-DI or
probabilistic SPI, make several model assumptions about the hydrological variables
used in their construct. Among them, the most important assumption is that the time
series of hydrological variable is stationary, i.e., its distributional properties used to
define droughts do not change over time. Thus, temporal trends in droughts cannot
be estimated using classical drought indices. Nevertheless, hydrological time series
may exhibit nonstationarity due to changes in climate and land use or due to natural
cycles that operate over a period of several years to decades, and hence it is important
to study temporal trends of droughts.
Several studies in the literature have proposed methods to perform drought analy-
sis under nonstationary conditions. Mishra and Desai [52] used autoregressive inte-
grated moving average (ARIMA) models and variants of artificial neural networks
(ANNs), namely, recursive multistep neural network approach (RMSNN) and direct
multistep neural network approach (DMSNN) for drought forecasting in presence of
nonstationarity. Coulibaly and Baldwin [12] proposed the use of dynamic recurrent
neural network (RNN) to model and forecast nonstationary hydrologic time series.
Belayneh et al. [4] used wavelet analysis to first denoise the series, and then train
ANNs or support vector regressors on the decomposed signals to perform drought
forecasting in arid regions of Ethiopia. Türkeş and Tatlı [77] studied droughts in non-
stationary precipitation series by modifying the classical SPI using the concepts of
empirical mode decomposition. Unlike the classical SPI, the modified SPI accounts
for local or higher order statistics in the precipitation time series. Han et al. [29] pro-
posed the use of ARIMA models on potentially nonstationary remote sensing data to
predict vegetation temperature condition index for drought forecasting. Verdon-Kidd
and Kiem [80] emphasized the need to evaluate risk to water resources systems during
drought under a nonstationary climate over Australia. Mitra and Srivastava [56] use](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-38-320.jpg)
![28 G. Mallya et al.
the modified Mann–Kendall test on SPI and SPEI series to study the spatiotemporal
variability of meteorological droughts in southeast USA.
The literature review suggests that in the context of SPI, temporal changes in
droughts are mostly studied using one of the following two approaches. The first
approach divides the study period into smaller intervals or epochs (~30 years) where
the underlying rainfall series is assumed to be stationary, and computes relative SPI
for each epoch [18], and then compares drought characteristics between different
epochs. The second approach allows the distribution of the hydrological variable to
change with time, but the parameters of the distribution can follow only a prespecified
temporal pattern. For example, the standardized nonstationary precipitation index
(SnsPI) proposed by Russo et al. [68] assumes that the scale parameter of the Gamma
distribution for rainfall varies linearly with time.
The main objectives of this chapter are as follows:
(a) To investigate drought characteristics in India using a probabilistic drought clas-
sification approach that adapts SPI methodology by employing gamma mixture
model (Gamma-MM) in a Bayesian framework [47], and to compare the results
with classical SPI.
(b) To use an alternate methodology for studying temporal changes in droughts [46]
that does not require—(i) making stationarity assumption about the precipitation
time series and (ii) prespecifying the nature of temporal trend in the precipitation
series.
(c) TostudytemporalchangesinthedroughtsinIndiathroughthisalternatemethod-
ology, and compare results with existing methods.
The remainder of the chapter is structured as follows. The next section describes
the study area, India, and provides an account of its historical droughts. The pre-
cipitation datasets available for the study area are described in Sect. 2. Section 3
presents the mathematical formulation of the methods used for classifying droughts
and quantifying temporal variation. These methods will be applied to precipitation
data over India and the results will be presented and discussed in Sect. 4. The chapter
ends with a set of concluding remarks.
2 Study Area and Dataset
The study area, India, receives 80% of its annual precipitation during 4-months
long southwest summer monsoon [3, 62]. The monsoon precipitation makes landfall
around the first week of June near Kerala, India, and moves northeast toward the
Himalayas. By the first week of July, almost the entire country typically receives
some precipitation that continues until the end of September [9]. From beginning of
October to December, cool and dry winds from Central Asia cross India diagonally
from northeast to southwest. These winds humidify the air as they blow over the Bay
of Bengal, resulting in northeast monsoon precipitation predominantly over the state
of Tamil Nadu, and partly over other states of Odisha, Andhra Pradesh, Karnataka,](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-39-320.jpg)
![An Analysis of Spatio-Temporal Changes … 29
and Kerala [36]. Though the Indian monsoon is believed to be one of the most stable
monsoon systems [34, 49, 65], it has large inter- and intraseasonal variability that
can sometimes result in weak monsoon or droughts over India [37, 59]. Since the
country’s gross domestic product (GDP), particularly food and power production, is
closelylinkedtomonsoonrains,variousstrategieshavebeendevelopedovertheyears
to mitigate the effects of droughts (e.g., drought-prone areas programme (DPAP), and
desert development programme (DDP)). Implementing effective drought mitigation
strategies requires real-time reliable classification of droughts.
2.1 Droughts over India and Their Consequences
Each type of drought has its own consequences, and the effects are felt by the general
population. Since India is mainly an agricultural economy, droughts have histori-
cally had major impact on farmers. Agricultural droughts result in low crop yield
and sometimes a complete failure of the crop. The agricultural fields can quickly
turn into large dust bowls thereby leading to topsoil loss. This, in turn, causes stress
in maintaining healthy livestock. Scarcity of water also leads to unhygienic con-
ditions—leading to faster spread of diseases among the population. The economic
consequences during/following a prolonged drought event can be detrimental to the
poor farming community in India. Lack of crop insurance and inadequate financial
support from government-backed banks often forces farmers to borrow money from
private lenders. This leads to social disputes and mass migration from villages to
cities in search of alternate employment opportunities [15, 64]. The non-farming
community living in towns and cities face consequences of droughts in the form
of shortage of water supply for household and industrial use. Droughts often lead
to rise in commodity and fuel prices, thus causing economic stress for lower and
middle-income families within the affected region.
In an effort to build a resilient society, the Central and State Governments in
India have developed drought mitigation programs such as drought-prone areas pro-
gramme (DPAP), desert development programme (DDP), and national watershed
development programme in rainfed areas (NWDPRA) that provide material, educa-
tional, and financial support for the following:
i. Lake restoration and capacity building of existing reservoirs.
ii. Rainwater harvesting, cloud seeding to trigger rains.
iii. Large-scale desalination plants in major coastal cities to decrease reliance on
groundwater and river water for domestic and industrial supply.
iv. Low-interest agricultural loans to farmers, and guaranteed employment for at
least 100 days in a year under National Rural Employment Guarantee scheme.](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-40-320.jpg)
![30 G. Mallya et al.
2.2 Recent Drought Literature over India
Niranjan Kumar et al. [58] used SPEI to study the variability of monsoon droughts
over India, and found that El Nino/Southern Oscillation (ENSO) as the most influenc-
ing factor. They implicate the warming of the equatorial Indian Ocean to the increased
droughts over India in the recent decades. Mahajan and Dodamani [44] performed
trend analysis of drought events over Upper Krishna Basin in Maharashtra. Kumar
et al. [39] used historical rainfall and sea surface temperature (SST) records to show
that warmest SST anomalies in the central equatorial Pacific are better indicators of
severe droughts over India. Varikoden et al. [78] showed that droughts associated
with El Nino are very intense in most parts of the subcontinent, when compared to
droughts during non-El Nino years. Mallya et al. [45] used SPI, SPEI, HMM-based
drought index, and Gaussian mixture models and found that irrespective of the pre-
cipitation dataset or the choice of drought index, the drought severity and frequency
over India increased significantly during recent decades. Their study also found that
droughts are becoming more regional and are showing a general shift to the agricul-
turally important coastal South India, central Maharashtra, and Indo-Gangetic plains.
Zhang et al. [86] found that the soil moisture and vegetation drought indices were
best suited to study the impact on wheat production in India. Naresh Kumar et al.
[57] studied the spatiotemporal patterns of droughts over India using SPI and found
that area under moderate droughts have increased in recent decades.
2.3 India Meteorological Department (IMD) Precipitation
Dataset
To analyze meteorological droughts over India, long-term precipitation data are
required. Daily rainfall data at a spatial resolution of 1° for both latitude and longi-
tude were obtained from India Meteorological Department (IMD) and are based on
a total 1803 stations distributed over India that have at least 90% availability for the
period 1901–2004 [63]. The gridded data, consisting of 357 grid points, have been
obtained by interpolating rain gage data. The IMD datasets are standard datasets
widely used in monsoon-related studies over India [25]. Figure 1 shows the study
area along with the grid locations for which rainfall data were available as circular
markers. The grids where results are discussed in subsequent sections are denoted
as square red-colored markers. Because of its large geographical extent, the study
area consists of several streams and rivers (some of which are perennial). Of these
streams, some drain into the Arabian Sea or the Bay of Bengal, while few rivers cross
international borders into neighboring countries. The main networks of some of the
major rivers of India are shown in Fig. 1.](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-41-320.jpg)
![An Analysis of Spatio-Temporal Changes … 33
of 208 cm with a standard deviation of 21 cm), while region 5 (Northwest monsoon
region spanning over Gujarat and Rajasthan) is the driest (mean precipitation of
52 cm with a standard deviation of 13 cm).
An analysis of previous water year drought events based on SPI values (see Fig. 4)
computed using an average representative cumulative precipitation time series over
India (without considering grids located over hilly regions) suggested that the most
severe conditions persisted during the periods early 1900s, 1918, 1951–52, 1965–68,
1972, late 1980s (Bengal drought), and early 2000s. The most recent drought condi-
tions over India occurred in 2012–2013, with the West Central region—specifically
the state of Maharashtra—being severely impacted by the drought.
Figure 5 shows time series of percentage area under drought computed using his-
torical SPI values for water year beginning in June of each year over India. According
to this figure, approximately 72% of the grids were under drought during 2002, fol-
lowed by 69% of the grids in 1972, 68% in 1918, 62% in 1965, and 60% in 1960.
Therefore, the combination of Figs. 4 and 5 suggests that 2002 drought was the
most severe on record in terms of severity and extent. However, the actual damage
caused due to droughts in recent years are much lower compared to some of the
previous droughts due to improved drought mitigation programs [14]. The result of
Mann–Kendall trend test on the time series of area under drought suggests that the
trend is positive (Sen’s slope 0.01, shown as red-dashed line in Fig. 5), although
not statistically significant (at α 0.05).
3 Methodology
The mathematical formulations of the two drought classification methods, namely,
SPI and Gamma-MM, are presented in Sects. 3.1 and 3.2, respectively. Next, the
methods used for studying temporal changes in droughts over India are described.
3.1 Standardized Precipitation Index (SPI)
The method involves the following steps:
1. Decide a drought duration (time window) and estimate cumulative precipitation
during that period. For example, to estimate droughts during a summer mon-
soon season, estimate cumulative precipitation during 4 months of the summer
monsoon season (JJAS) for each year. This will yield an annual time series of
cumulative precipitation. Likewise, for analyzing water year droughts obtain an
annual time series of cumulative precipitation for 12 months starting on June 1
and ending on May 31 of following year.
2. Fit a gamma distribution to the cumulative precipitation series. The cumulative
distribution function (CDF) of the gamma distribution is standardized using the](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-44-320.jpg)
![36 G. Mallya et al.
standard inverse Gaussian function to compute the SPI drought index. As stated
earlier, a negative value of SPI indicates drought conditions and the magnitude
of its departure from zero indicates the severity of a drought.
3. Decide a threshold on CDF to determine drought class. To draw parallels with
the US Drought Monitor, we have used the same thresholds as used by them for
SPI drought classification (Table 1).
3.2 Gamma Mixture Model (Gamma-MM)
As discussed in the Introduction section, there is an ongoing debate on the choice
of a suitable distribution for fitting data in SPI analysis. Mallya et al. [47] addressed
this problem using the gamma mixture model (Gamma-MM). Given sufficient num-
ber of components in the mixture, the Gamma-MM is proven to provide arbitrarily
close approximation to any general continuous distribution in the range (0, ∞) (see,
DeVore and Lorentz [16]).
The use of Gamma-MM is not new in hydrology. To model data with multiple
modes and different types of skewness, Evin et al. [19] proposed the use of Gamma-
MMforstrictlypositivehydrologicaldata.Intheassessmentofhydrologicaldroughts
for Yellow River in China, Shiau et al. [72] first fitted mixtures of exponential and
gamma distributions to drought duration and drought severity, respectively, and then
used the copula method to construct a bivariate drought distribution. While the mix-
tures help represent the subpopulations within an overall population, the copula
method describes the dependence between variables of interest. In the following, we
provide a brief description of the Gamma-MM. The readers are referred to Wiper
et al. [82] and Richardson and Green [66] for details on mixture models. A summary
of the mathematical details of the Gamma-MM as described in Mallya et al. [47] is
presented below.
Table 1 US Drought
Monitor classification
scheme. SPI ranges are
prescribed for the inverse of
the normal distribution.
Corresponding thresholds on
CDF are given in the last
column
Category Description SPI range Threshold on
CDF
D0 Abnormally
dry
−0.5 to −0.8 0.212–0.309
D1 Moderate
drought
−0.8 to −1.3 0.097–0.212
D2 Severe
drought
−1.3 to −1.6 0.055–0.097
D3 Extreme
drought
−1.6 to −1.9 0.023–0.055
D4 Exceptional
drought
−2.0 or less 0.023 or less](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-47-320.jpg)
![An Analysis of Spatio-Temporal Changes … 37
Let the cumulative rainfall at time t be denoted by xt , t
1, . . . , N
xt ∈ R and X [x1, . . . , xN ]T
. If the total number of compo-
nents of Gamma-MM, M, is known a priori, then the weighted sum of M mixtures
of gamma is given by the following equation:
P(xt |λ)
M
i1
wi G
xt |vi ,
vi
μi
(1)
where wi are the mixture weights or mixing ratios, and G
xt | νi , νi
μi
are the com-
ponents of gamma densities of the form,
G
xt | νi ,
νi
μi
νi
μi
νi
(νi )
x(vi −1)
t exp
−
νi
μi
xt
, (2)
with mean μi and shape parameter νi . Further, the mixture weights satisfy the con-
straint M
i1 wi 1. The parameter set is represented as λ {w, μ, v}
where w [w1, w2, . . . , wM ]T
, μ [μ1, μ2, . . . , μM ]T
and v [ν1, ν2, . . . , νM ]T
.
In the Bayesian framework, the model parameters are obtained by specifying
prior distributions to model parameters. Parameter estimation is accomplished by
introducing a latent variable Z [z1, . . . , zN ]T
for each time step. The variable
zt is an M-dimensional binary random variable, zt [zt1, . . . , zt M ]T
, in which a
particular element is equal to 1 and all other elements are zero, i.e., M
i1 zti 1
and zti ∈ {0, 1}. The variable zt denotes the component to which the data xt belongs,
and hence it is also called an indicator variable. The conditional distribution of xt
given zt is
P(xt |zti 1) ∼ G
xt |νi ,
νi
μi
(3)
The posterior probability of the model parameters and latent variables is obtained
by applying Bayes’ rule as
P(λ|X) ∝ P(X|λ)P(λ) (4)
where the parameter set λ includes the latent variable as well. The likelihood function
given the latent variable is
P(X|λ) P(X|Z, μ,ν)
N
t1
M
i1
G
xt |νi ,
νi
μi
zti
(5)
Following Wiper et al. [82] the prior distribution over the model parameter is
given as P(λ) P(Z|w)P(w)P(μ)P(ν) with](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-48-320.jpg)
![38 G. Mallya et al.
P(Z|w)
N
t1
M
i1
w
zti
i ,
P(w) Dir(w|) C()
M
i1
w
φi −1
i , [φ1, . . . , φM ]T,
P(ν) Exp(ν|θ)
M
i1
1
θi
exp(−θi νi ), θ [θ1, . . . , θM ]T, and
P(μ) GI(μ|α,β)
M
i1
β
αi
i
(αi )
μ
−αi −1
i exp
−
βi
μi
, α [α1, . . . , αM ]T and β [β1, . . . , βM ]T
where Dir, Exp, and GI represent Dirichlet, Exponential, and Inverted gamma dis-
tributions, respectively, and C() is a normalizing constant. The prior distribution
is made non-informative by assigning following values to the hyperparameters.
φi 1; θi 0.01; αi βi 1 for i 1, . . . , M.
The posterior distribution P(λ|X) does not have a closed form and has to be
estimated by either deterministic approximation (variational Bayes’ methods) or
stochastic approximation (MCMC; Markov chain Monte Carlo methods). In this
study, the posterior distribution is estimated using stochastic approximation by sam-
pling the posterior distribution with Gibbs sampler, an MCMC algorithm [24].
The Gibbs sampling algorithm samples posterior distribution of the parameters
by sequentially sampling from the conditional distribution of a parameter given all
other parameters. The sampling starts with an initial value and proceeds as follows:
1. Set iteration number j 0, and parameters to their initial value λ(0)
w(0)
, μ(0)
, ν(0)
. The initial value is obtained by randomly sampling from the
prior distribution of the parameters.
2. Sample from P
z
( j+1)
t |X, w( j)
, μ( j)
, ν( j)
∼ Multinomial(zt |rt )
where rt [rt1, . . . ,rt M ]T
, rti sti
M
i1 sti
and sti wi G
xt |νi , νi
μi
and
Multinomial represents multinomial distribution.
3. Sample from P w( j+1)
|X, Z( j+1)
, μ( j)
, ν( j)
∼ Dir
w| ˆ
where ˆ
[φi + ni , . . . , φM + nM ]T
and ni
N
t1
zti .
4. Sample from P μ( j+1)
|X, Z( j+1)
, w( j+1)
, ν( j)
∼ GI
μ|α̂, β̂
where α̂ [αi + ni νi , . . . , αM + nM νM ]T
and β̂
βi + νi
M
t1
xt zti , . . . , βM + νM
M
t1
xt zt M
T
.
5. Sample from P v( j+1)
|X, Z( j+1)
, w( j+1)
, μ( j+1)
. This conditional distribu-
tion does not have a closed form. Hence, samples are generated using
Metropolis–Hastings algorithm from a proposal distribution P(ṽi |vi ) ∼
G(h, h|vi ) and are accepted with a probability](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-49-320.jpg)
![An Analysis of Spatio-Temporal Changes … 39
min
1,
f (ṽi )P(vi |ṽi )
f (vi )P(ṽi |vi )
,
where f (vi ) ∝
v
ni νi
i
(vi )ni exp
−νi
θi + t xt zti
μi
+ ni log μi − log
N
t1;zti 1 xt
.
If the new sample ṽi is rejected, the current value of vi is retained. The above
procedure is repeated to sample vi for all components i 1, . . . , M. In this
study, the parameter of the proposal distribution, h, is set to 2.
6. Set j j + 1 and go to Step 2 until convergence. In this study, 15,000 samples
were generated after ignoring initial 500 samples (burn-in period). Trace plots
of the samples were monitored for convergence.
To keep the notations uncluttered, the iteration number is omitted from the param-
eters of the conditional distributions.
In the above formulation of Gamma-MM, we have assumed that the number of
mixture components, M, is known. However, in a general context, M is not known
and should be estimated from data. One approach for estimating M is to consider it as
a model parameter, assign prior distribution to it and estimate posterior distribution
by MCMC method. Since changing M will result in a different model structure, usual
MCMC algorithms such as Gibbs sampler cannot be applied. Instead, reversible jump
MCMC (RJMCMC; Green [26] and Richardson and Green [66]) may be used. In
this study, we implemented RJMCMC for Gamma-MM as described by Richardson
and Green [66] and Wiper et al. [82]. The results suggested that RJMCMC algorithm
requires significantly higher number of iterations for convergence compared to a
model where M is specified. We found that if we start with a model having sufficiently
large number of components, M, the Bayesian algorithm automatically prunes the
components that are not relevant by making the mixing ratio (w) very small, thereby
determining optimum number of components. We recommend the latter approach
for hydrological applications where the number of components is usually limited to
2 or 3.
In the Bayesian framework, mixture models have the identifiability problem, i.e.,
a M component mixture model will have a total of M! equivalent solutions. The
problem can be avoided by introducing asymmetricity in the likelihood function. For
example, in the context of Gamma-MM, Wiper et al. [82] recommended the following
restrictiononthemeans of themixturecomponents,μ1 μ2 · · · μM .However,
for finding a good density model, as required in the present application, the problem
of identifiability is not relevant because any of the equivalent solutions is as good as
another [6].](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-50-320.jpg)
![40 G. Mallya et al.
3.3 Temporal Trends in Droughts
Evaluation of temporal trends associated with retrospective drought events provides
a basis to understand regional patterns of severity and duration of droughts. It also
provides insight into the nature of possible future droughts and potential vulnerabil-
ities over the study region. In this study, we first investigated trends in precipitation
series at each IMD grid and also at each homogeneous monsoon region. A modified
Mann–Kendall trend test that accounts for autocorrelation in time series [28, 38] was
used to detect trends in summer monsoon and water year precipitation. The trends
were tested at 5% significance level (α). The effect of spatial correlations in the data
[8, 85] on the trend results was accounted using false discovery rate (FDR) [5, 79].
If the precipitation series exhibits a trend, drought classification methods such as
SPI and Gamma-MM are not applicable for drought analysis because they assume the
precipitation time series to be stationary. In this study, we apply an alternate method
for drought analysis that overcomes this problem by explicitly removing trends from
the precipitation series. The method is compared with relative SPI [18] and SnsPI [68]
for performing drought analysis of a nonstationary precipitation series. The following
paragraphs describe this alternate method and provide a summary of relative SPI and
SnsPI methods.
3.3.1 An Alternate Method for Drought Analysis of a Nonstationary
Precipitation Series
A precipitation series may exhibit nonstationarity because of any of several reasons
[22]—(a) the mean is a function of time, (b) the variance or other higher order
moments are functions of time, and (c) the stochastic mechanism generating time
series is nonstationary. In this study, we consider nonstationarity arising from first
kind. As proposed by Mallya et al. [46], we consider a trend stationary process in
which the mean trend is deterministic. Once the trend is estimated and removed from
the data, the residual series is a stationary stochastic process. A trend stationarity
process, yt , is expressed as
yt f (t) + zt (6)
where t represents time, zt denotes a zero-mean stationary process, and f (t) is
a function of time representing trend of the process at time t. The trend can be
determined either extrinsically by specifying a linear or nonlinear functional form for
f (t) [e.g., regression models] or intrinsically by using the data without prespecifying
a functional form (e.g., empirical mode decomposition [84]).
Drought classification for nonstationary time series can be approached in one of
the following two ways—(a) assuming that the “normal” conditions for a station are
evolving, and hence the drought thresholds for different categories are changing with
time (similar to SnsPI) and (b) assuming that the “normal” conditions for a station](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-51-320.jpg)
![42 G. Mallya et al.
4 Results and Discussion
This section is divided into three subsections. Section 4.1 presents the results for
drought classification using SPI and Gamma-MM drought indices. Section 4.2
describes the results for precipitation trend analysis. Section 4.3 presents the results
of the proposed methodology for the drought classification of nonstationary precip-
itation series (detrended-SPI) and compares them with the results of classical SPI,
relative SPI, and SnsPI for synthetic and real-world precipitation series.
4.1 Drought Classification
The drought indices described in Sect. 3 are applied to study seasonal (4-month
time window) and water year (12-month time window) droughts in India. India has
three seasons each spanning about 4 months: Winter (October to January), Summer
(February to May), and Summer Monsoon (June to September). The water year in
India extends from June to May of the following year. For example, 1999 water year
starts on June 1, 1999 and ends on May 31, 2000. First, an annual time series of
cumulative precipitation during any chosen season or water year is computed. Next,
droughts are classified using SPI and Gamma-MM methods. The latter method is
also referred to as probabilistic SPI. Both the methods assume that the cumulative
precipitation time series is stationary, and consists of independent and identically
distributed samples. In the following paragraphs, summer monsoon and water year
droughts from the two methods are presented for a selected IMD grid over India.
4.1.1 Summer Monsoon Droughts
The results are presented for IMD grid 251 located in northeast India and are among
thehighestrainfallreceivingregionsoftheworld.Figure6showstheempiricalcumu-
lative distribution function (CDF) obtained using Weibull plotting position formula
[10] along with CDFs of fitted gamma distribution (fitted using maximum likelihood
approach) and gamma mixture model (Gamma-MM) for summer monsoon precipi-
tation (June–September). The CDF of Gamma-MM is closer to empirical CDF than
the CDF of gamma distribution, particularly, for the smaller rainfall values [F(X)
0.25], which are critical for drought classification. The Gamma-MM owes its bet-
ter fit to the large number of tuning parameters (3M − 1, where M is number of
components in Gamma-MM) compared to two-parameter gamma distributions.
Increasing the number of mixture components (M) in Gamma-MM method
ensures that the model provides a better fit to the data. However, it may also result in
overfitting. The Gamma-MM model addresses this problem using a Bayesian frame-
work that avoids overfitting by marginalizing over the model parameters instead of
makingpointestimates.Figure7showsthemixingratioofafive-componentGamma-](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-53-320.jpg)
![and the hope seems reasonable; though the slavery question will
probably be the point on which the next election will turn, which
may again be to the disadvantage of the colonel. If he should
become president, he will be as strange-looking a potentate as
ever ruled. His countenance is wild, though with much cleverness
in it; his hair wanders all abroad, and he wears no cravat. But
there is no telling how he might look if dressed like other people.
I was fortunate enough once to catch a glimpse of the invisible
Amos Kendall, one of the most remarkable men in America. He is
supposed to be the moving spring of the whole administration; the
thinker, planner, and doer; but it is all in the dark. Documents are
issued of an excellence which prevents their being attributed to
persons who take the responsibility of them; a correspondence is
kept up all over the country for which no one seems to be
answerable; work is done, of goblin extent and with goblin speed,
which makes men look about them with a superstitious wonder;
and the invisible Amos Kendall has the credit of it all. President
Jackson's Letters to his Cabinet are said to be Kendall's; the
Report on Sunday Mails is attributed to Kendall; the letters sent
from Washington to appear in remote country newspapers,
whence they are collected and published in the Globe as
demonstrations of public opinion, are pronounced to be written by
Kendall. Every mysterious paragraph in opposition newspapers
relates to Kendall; and it is some relief to the timid that his having
now the office of postmaster-general affords opportunity for open
attacks upon this twilight personage; who is proved, by the faults
in the postoffice administration, not to be able to do quite
everything well. But he is undoubtedly a great genius. He unites
with his great talent for silence a splendid audacity. One proof of
this I have given elsewhere, in the account of the bold stroke by
which he obtained the sanction of the Senate to his appointment
as postmaster-general.
[11]](https://image.slidesharecdn.com/4980990-250525131347-c2ee6395/85/Hydrology-In-A-Changing-World-Challenges-In-Modeling-1st-Ed-Shailesh-Kumar-Singh-61-320.jpg)
Hydrology In A Changing World Challenges In Modeling 1st Ed Shailesh Kumar Singh
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- 5. SpringerWater Shailesh Kumar Singh C. T. Dhanya Editors Hydrology in a Changing World Challenges in Modeling
- 7. The book series Springer Water comprises a broad portfolio of multi- and interdisciplinary scientific books, aiming at researchers, students, and everyone interested in water-related science. The series includes peer-reviewed monographs, edited volumes, textbooks, and conference proceedings. Its volumes combine all kinds of water-related research areas, such as: the movement, distribution and quality of freshwater; water resources; the quality and pollution of water and its influence on health; the water industry including drinking water, wastewater, and desalination services and technologies; water history; as well as water management and the governmental, political, developmental, and ethical aspects of water. More information about this series at http://www.springer.com/series/13419
- 8. Shailesh Kumar Singh • C. T. Dhanya Editors Hydrology in a Changing World Challenges in Modeling 123
- 9. Editors Shailesh Kumar Singh Hydrological Processes National Institute of Water and Atmospheric Research Christchurch, New Zealand C. T. Dhanya Department of Civil Engineering Indian Institute of Technology (IIT) Delhi New Delhi, India ISSN 2364-6934 ISSN 2364-8198 (electronic) Springer Water ISBN 978-3-030-02196-2 ISBN 978-3-030-02197-9 (eBook) https://doi.org/10.1007/978-3-030-02197-9 Library of Congress Control Number: 2018966838 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 10. Contents Integration of GRACE Data for Improvement of Hydrological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chandan Banerjee and D. Nagesh Kumar An Analysis of Spatio-Temporal Changes in Drought Characteristics over India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Ganeshchandra Mallya, Shivam Tripathi and Rao S. Govindaraju Urban Hydrology in a Changing World . . . . . . . . . . . . . . . . . . . . . . . . 73 James A. Griffiths and Shailesh Kumar Singh Uncertainty in Calibration of Variable Infiltration Capacity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Ankita Pradhan and J. Indu Predictability of Hydrological Systems Using the Wavelet Transformation: Application to Drought Prediction . . . . . . . . . . . . . . . 109 Rajib Maity and Mayank Suman Land–Atmosphere Interactions in Indian Monsoon at Sub-seasonal to Seasonal Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Amey Pathak and Subimal Ghosh Assessment of Climate Change Impacts on IDF Curves in Qatar Using Ensemble Climate Modeling Approach . . . . . . . . . . . . . . . . . . . 153 Abdullah Al Mamoon, Ataur Rahman and Niels E. Joergensen River Water Temperature Modelling Under Climate Change Using Support Vector Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Shaik Rehana Assessing the Impact of Climate Change on Water Resources: The Challenge Posed by a Multitude of Options . . . . . . . . . . . . . . . . . 185 Riddhi Singh and Basudev Biswal v
- 11. Streamflow Connectivity in a Large-Scale River Basin . . . . . . . . . . . . 205 Koren Fang, Bellie Sivakumar, Fitsum M. Woldemeskel and Vinayakam Jothiprakash Climate Change Impacts on Four Agricultural, Headwater Watersheds from Varying Climatic Regions of New Zealand. . . . . . . . 225 M. S. Srinivasan, Shailesh Kumar Singh and R. J. Wilcock vi Contents
- 12. Integration of GRACE Data for Improvement of Hydrological Models Chandan Banerjee and D. Nagesh Kumar 1 Introduction Observation is the first and most primary step in various disciplines of geosciences such as hydrology, meteorology, oceanography, geology, glaciology, and other plan- etary sciences. Hydrology or hydrological sciences which essentially deals with the question “What happens to the rain?” largely depends on gauge observations, which have been the longest running bastion furnishing long time series of datasets. Hydro- logical studies require datasets of both meteorological and hydrological variables such as temperature, humidity, precipitation, streamflow, etc. to monitor, understand, and model the complex physical processes which convert precipitation to surface water, soil moisture, groundwater, or streamflow. For a long time, hydrological stud- ies were completely driven by datasets produced only by gauge measurements and to some extent field surveys. Although gauge measurements and field datasets are indispensable tools to understand the natural processes even today, they suffer from several limitations [6] such as (i) localized nature of the gauges provides information only for a particular loca- tion; (ii) gauges cannot provide data at locations inaccessible to humans; (iii) data procured by gauges are not easily available due to political control over data sharing policies; and (iv) management and maintenance of gauges are big challenges faced by concerned authorities. Moreover, from hydrological modeling perspective, gauge datasets are quite limit- ing and do not help incorporate mathematical modeling of many physical processes. Consequently, development of a systematic framework that provides us with obser- vational datasets of the Earth having the desired properties such as global coverage, C. Banerjee · D. N. Kumar (B) Department of Civil Engineering, Indian Institute of Science, Bengaluru, India e-mail: nagesh@iisc.ac.in © Springer Nature Switzerland AG 2019 S. K. Singh and C. T. Dhanya (eds.), Hydrology in a Changing World, Springer Water, https://doi.org/10.1007/978-3-030-02197-9_1 1
- 13. 2 C. Banerjee and D. N. Kumar continuously available in time, and accessible across political boundaries, specif- ically of hydrological and meteorological variables, was necessary. As a result, Earth-observing satellite remote sensing has been developed to complement the gauge-based observations and enhance our knowledge and understanding of vari- ous physical processes [52, 55, 57]. This has not only enhanced our ability to model complex hydrological processes [36, 53, 86] to a large extent but also improved our capabilities to predict and forecast hydrological extremes which have now reached new levels. The journey of remote sensing observations started in 1972 with Earth Resources Technology Satellite (ERTS) 1 [14] launched by National Aeronautics and Space Administration (NASA), USA, which later came to be known as Landsat 1. It car- ried a multispectral scanner (MSS) recording data in four spectral bands, viz., red, green, and two infrared bands. Since then the technology used for remote sensing has grown by leaps and bounds. Remote sensing satellites now record not only in the optical and near-infrared bands but also in thermal and microwave bands. The spatial, spectral, temporal as well as radiometric resolutions have improved with each new satellite. New data acquisition techniques are being developed such as the synthetic-aperture radar (SAR) used to procure terrain and land cover information [44], satellite altimeters used to measure depth of seabed, radiometers used to esti- mate surface soil moisture [63], and hyperspectral imagers having a very high spectral resolution are used for various applications in the fields of agriculture, mineralogy, and environmental sciences [46]. The Terrestrial Water Storage (TWS) estimate derived from the Gravity Recovery and Climate Experiment (GRACE) satellite data is a remarkable addition to the vast set of remote sensing observations [76, 83]. Compared to the previous satellites, GRACE uses a completely different technique of data acquisition. While most of the previous satellites can observe only surface features of the land, GRACE satellites are able to acquire information about water storages in any form at any depth. TWS refers to the total water storage in a column of land present in the form, be it snow, ice, surface water, soil moisture, and groundwater. Although the spatial and temporal resolution of the GRACE data is coarse as compared to many other satellites, the unique nature of the data makes it an invaluable tool to observe terrestrial hydrolog- ical processes [79, 93]. The water storage that is the most difficult to observe and monitor is groundwater and in situ well observations were the only way to mon- itor them until the advent of GRACE. Well observations suffer from the obvious limitations of consistency, unavailability of data for the required period, inadequate spatial distribution of observation wells, and above all, political control over data for transboundary aquifers. GRACE on the other hand provides a global observational dataset periodically for the past 15 years. Using GRACE-derived datasets, scientists have identified depleting groundwater levels in different parts of the world such as Sacramento and San Joaquin River basins, California’s Central Valley, and High Plains aquifer in USA [12, 23, 64, 65], Bengal Basin of Bangladesh [68] and Gan- ga–Brahmaputra–Meghna River basin [34] in South Asia, transboundary river basins in the Middle East [32, 88], Northern China [31] and Southern Murray Darling River basin [15] in Australia.
- 14. Integration of GRACE Data for Improvement of Hydrological Models 3 GRACE data is being used to solve a host of scientific problems other than the numerous studies related to groundwater. GRACE-derived TWS, also known as TWS Anomaly (TWSA) and its derivative TWS Change (TWSC), are used to study the dynamics of the terrestrial part of the hydrologic cycle and unravel its complex nature [5, 27, 45, 80]. It is used to understand water budget at the spatial scale of large river basins or continents [38, 79]. Terrestrial water budget, atmospheric water budget, or a coupling of the two is used to estimate evapotranspiration or river discharge [60, 59, 70, 79, 78]. Evapotranspiration is an important part of the terrestrial hydrolog- ical cycle as it is the terrestrial feedback to the atmosphere and affects the climate. However, it is a complex hydrological variable which is difficult to estimate by the various energy balance and aerodynamic methods as they are highly data intensive. GRACE provides a rather simple method of its estimation. River discharge is an equally important parameter affecting the seas and oceans, determining the fresh- water input to the system. Ocean salinity, sea surface temperature, and various other parameters are dependent on the amount of freshwater that comes into the oceans in the form of river discharge. The GRACE-based method of river discharge estimation is specifically helpful for large rivers which do not have a defined stream but forms a large delta system as it meets the ocean, as in case of the rivers Ganga–Brahmaputra, Indus, Irrawaddy, Mekong, and Yangtze. GRACE also finds application in drought- related studies [29, 84]. Precipitation is typically used for drought identification, monitoring, and management. Recently, a few studies have also used soil moisture to monitor droughts. However, TWS data which is the total of all the water storages helps improve the impact assessment of a drought by providing a holistic estimate of the total amount of water lost during a drought and the time taken to regain. The area of application of GRACE data which would be of interest for the present discussion is its integration into hydrological models, which are sophisticated tools used for prediction of various hydrological parameters. Prediction of river discharge has been the sole objective of hydrologic models for a long time due to the limited number of hydrological variables observed (as discussed earlier). However, with the increasing number of observations, specifically satellite-based observations and huge improvement in the computational capabilities, the structure and functions of hydro- logical models have also evolved. They now represent more complex processes at finer spatial and temporal scales and predict various hydrological parameters along with streamflow [17]. Integration of GRACE data into a hydrological model should further improve the representation of the physical processes and prediction of com- plex parameters such as evapotranspiration, soil moisture, and snow accumulation. In this chapter, we discuss in detail the various ways of integrating GRACE data into a hydrological model. We elaborate on the physics behind acquisition of TWS data through GRACE satellites and the available data products. We also review various hydrological models used for GRACE-based studies discussing models which are more often chosen over the others.
- 15. 4 C. Banerjee and D. N. Kumar 2 GRACE Data and Gravity Recovery Before divulging into the details of integrating GRACE data with hydrological mod- els, it is important to understand the science of gravity recovery. The GRACE satellite mission is a joint venture by the US and German space agencies, NASA and DLR (DeutschesZentrum fur“ r Luft-und Raumfahrt), respectively, under the NASA Earth System Science Pathfinder Program. The mission which was launched on March 17, 2002 consists of a pair of small and identical satellites (Fig. 1) orbiting at an altitude of 500 km from the Earth’s surface with a separation between them of about 220 km along track. The satellites are connected by a highly accurate inter-satellite microwave K band ranging system constantly measuring the minute changes in the inter-satellite distance/range of the order of 10 µm. The distance between the two satellites changes due to the changes in earth surface features, which vary in den- sity. Higher density relates to high mass, thus culminating into greater gravitational force and vice versa. If the Earth was homogenous in nature, the range between the two satellites would remain constant. However, the mass distribution is highly heterogeneous as well as constantly changing in time. The most dynamic constituent of the planet is water that circulates through the oceans, atmosphere, lithosphere, cryosphere, and biosphere. As a result, the time variable gravity signal acquired by the GRACE satellites through the measurement of the inter-satellite range rate mainly consists of the temporal variations of water as it moves from one storage compartment to another. After removing the fluctuations in the mass of the atmosphere and oceans, also known as Atmosphere and Ocean De-aliasing (AOD) from the total gravity sig- nal, the seasonal and inter-annual fluctuations in TWS are obtained, expressed as centimeters of Equivalent Water Thickness (EWT) [73, 75, 76, 89]. The inter-satellite range rate, the primary variable observed by the GRACE satel- lites, must go through a long course of data processing to be converted to TWS. There are three primary centers constituting the Science Data System (SDS) which perform the processing of the Level 1 dataset to provide Level 2 and Level 3 datasets. These centers are the Center for Space Research (CSR) at the University of Texas at Austin, Jet Propulsion Laboratory (JPL), NASA and the German Research Center for Geosciences (GFZ) Helmholtz Center, Potsdam. The Level 1 data from GRACE con- sists of the inter-satellite range, range rate, range acceleration, and non-gravitational accelerations from each satellite. The Level 2 data product is the monthly gravity field estimates available in the form of spherical harmonic coefficients, whereas the Level 3 dataset is mass anomaly expressed in terms of EWT of TWS [39, 77]. The three data processing centers use different data processing techniques which include dis- tinct static gravity models, different de-aliasing schemes and varied order and degree of the spherical harmonic coefficients to produce three separate datasets commonly known as CSR, JPL, and GFZ datasets. However, there are other research groups which also use other varieties of processing techniques to produce Level 2 and Level 3 data products such as the Delft Mass Transport (DMT) model of Delft University of Technology (TU Delft) [35, 43], ITG-Grace2010 of Bonn University, and a host of datasets produced by NASA’s Goddard Space Flight Center (GSFC). JPL’s TELLUS
- 16. Integration of GRACE Data for Improvement of Hydrological Models 5 Fig. 1 Illustration of the twin satellites of the GRACE mission, connected by the along-track K- band microwave ranging system (Credits: NASA/JPL-Caltech) website provides Level 3 monthly gridded as well as mascon products of GRACE TWS estimate derived from Level 2 dataset of the three primary data centers viz CSR, GFZ, and JPL. These products are easily accessible, available along with the error estimates and are ready to use for hydrologists [39]. 3 Large-Scale Hydrological Models Mathematical models of the hydrological processes, commonly known as hydrolog- ical models, have a long history as they evolved from simple lumped models with a single output to much more sophisticated stochastic distributed hydrological mod- els which use several input variables and estimate wide range hydrologic responses [69]. Most of the primitive models are known as rainfall–runoff models which take rainfall and very primary land surface characteristics to estimate runoff. However, these models were an improvement over statistical models used for the prediction of runoff because the former contains representation of some physical processes and their usability in real time when forced with real-time precipitation [11]. With advancement in computational capabilities and proliferation of remotely sensed data, the hydrological models have hugely improved in terms of the simulation time steps, number of climatological forcings and land surface characteristics, spatial resolu- tion, and number of output variables. These developments finally culminated to an
- 17. 6 C. Banerjee and D. N. Kumar increased number of physical processes represented within the models as well as the accuracy with which they are represented. Thus, the hydrological models of the new generation are of great use for prediction of various hydrologic variables such as runoff, streamflow, evapotranspiration, etc. which help for water resources manage- ment [8, 42, 71]. Moreover, they also provide a robust framework to run numerical experiments to understand the effects of various natural and anthropogenic changes in the land surface properties and climate such as deforestation, urbanization, expan- sion of agricultural land, global warming, increasing extreme rainfall, etc. [7, 24]. Another aspect of hydrological models that has changed with improvement in various technologies as well as the urge to improve the accuracy in prediction of large-scale hydrological processes is the expanse of the land surface modeled within a single framework. Most hydrological models refer to catchment or river basin scale modeling where the primary output variable of interest is the streamflow at the mouth of the river basin. However, these models are calibrated for a single catchment such that the model parameters are tuned to represent hydrologic and climatic processes occurring only within that catchment. A new variety of models are the Land Surface Models (LSMs) which are included within the atmospheric General Circulation Models (GCMs) to represent the interaction of the atmosphere with the land surface in the form of mass and energy exchange [10, 21]. As discussed earlier, the river discharges from the land surface into the oceans alter several of its physical properties which in turn affect the climate. As a result, these LSMs are coupled with a River Routing Model (RRM) to convert the runoff produced by the LSM to streamflow and finally the river discharge into the oceans. LSMs have evolved greatly over the past few decades to accurately represent the partitioning of the incoming net radiative energy into latent and sensible heat fluxes and the parti- tioning of precipitation into runoff, evaporation, and water storage. One such LSM is the Community Land Model (CLM), part of the Community Earth System Model (CESM) of the National Center for Atmospheric Research (NCAR), USA [10]. The hydrologic processes represented in the model (shown in Fig. 2) include interception of precipitation by canopy, throughfall, transpiration, soil evaporation, canopy evap- oration, infiltration, runoff, soil moisture, aquifer recharge, snow accumulation, melt, and sublimation. Other than the hydrologic cycle, the model includes other physi- cal processes such as land biogeophysics, biogeochemistry, ecosystem dynamics, and anthropogenic interventions (Fig. 2). A similar framework is the Noah-Multi- parameterization Land Surface Model (Noah-MPLSM) which includes detailed veg- etation dynamics including canopy shading and under-canopy snow dynamics along with the capability to differentiate between C3 and C4 pathways of photosynthesis [51, 92]. The Noah-MP LSM version 1.6 was implemented in Weather Research and Forecasting (WRF) Model version 3.6. WRF is a numerical weather predic- tion model developed mainly by NCAR and National Centers for Environmental Prediction (NCEP). LSMs coupled within a GCM framework are not the only large-scale hydrological models simulating the water and energy cycles along with geochemical processes and vegetation dynamics. There are many large-scale uncoupled or stand-alone LSMs, sometimes also known as the Global Hydrological Models (GHMs) simulating phys-
- 18. Integration of GRACE Data for Improvement of Hydrological Models 7 Fig. 2 A schematic diagram showing the energy cycle, hydrological cycle, biogeochemical, veg- etation dynamics, and land use change represented within the Community Land Model (CLM) (Credits: http://www.cesm.ucar.edu/models/clm/) ical processes at global scale such as the WaterGAP model (Water—Global Analy- sis and Prognosis model) [2], developed by University of Kassel and University of Frankfurt,Germany,PCR-GLOBWB(PCRasterGLOBalWaterBalancemodel)[96] conceived by Utrecht University, The Netherland, ISBA-TRIP (Interactions between Soil, Biosphere, Atmosphere—Total Runoff Integrating Pathways) [3, 19] created by Centre National de Recherchés Météorologiques, France and the Global Land Data Assimilation System (GLDAS) framework [61], developed by GSFC and NCEP. The WaterGAP model is designed for the assessment of macro-scale processes of the ter- restrial hydrological cycle, taking into consideration anthropogenic component to simulate freshwater availability and irrigation water use. PCR-GLOBWB includes subgrid schemes for partitioning of rainfall into runoff, infiltration, interflow, ground- water recharge, and baseflow, as well as routing of the generated runoff. The model includes detailed anthropogenic effects to the extent that it includes more than 6000 manmade reservoirs. Thus, the human water use is completely integrated into the hydrological model at time step, calculating water demand, surface and groundwater abstraction, consumptive water use, and return flow. ISBA is a relatively simple LSM calculating variability in energy and water budgets with a saturation excess overland flow approach to simulate runoff based on TOPMODEL hydrological model [9]. This is coupled with TRIP, a simple RRM which converts runoff simulated by ISBA
- 19. 8 C. Banerjee and D. N. Kumar into river discharge for a global river network. The GLDAS framework consists of four different LSMs, viz., Mosaic, CLM, Noah, and Variable Infiltration Capacity (VIC) models forced by a single forcing dataset. These four models differ mainly in the depth of soil considered for the simulation of soil water interaction and storage as well as the number of layers into which the total depth is divided. It should be noted that both the ISBA-TRIP model and the GLDAS set of models do not include any anthropogenic effects of water storage and water use. Although the LSMs are of great use, a major challenge lies in the calibration of these models which can only be carried out using vegetation indices or streamflow for large river basins by routing the simulated runoff. GRACE provides a very useful data first for the evaluation of such LSMs and eventually can be assimilated into the models for better estimation of various hydrologic variables. In the following sections, these two approaches of integration of GRACE data are discussed in detail. 4 Evaluation of Model Simulations Using GRACE Data As discussed in the previous section, coupled and uncoupled LSMs try to simu- late several hydrological variables by incorporating most complex of the physical processes, mimicking them to the maximum extent possible. Scientists are contin- uously trying to improve these models by better parameterization and adding more and more hydrological, geophysical, and biophysical processes into these models. A handy dataset for quick evaluation of the hydrological fields simulated by these mod- els is the GRACE dataset. Due to its continuous global coverage, GRACE data could be used for evaluation of LSMs in cold regions affected by snow, arid, and semi-arid regions characterized by very low or no soil moisture conditions as well as areas char- acterized by a heavy to very heavy monsoonal rainfall. Table 1 provides a detailed chronological list of various studies carried out at global, continental, regional, and river basin scales to compare and evaluate various LSMs to estimate the accuracy with which it simulates various water storages and physical processes affecting them. Some studies tried to improve the estimation of certain variables by incorporating better and more detailed process representation and validated the improvements by comparison with GRACE data. The CLM LSM of NCAR is one such model which has seen continual efforts of betterment and corresponding evaluation using GRACE data. One of the limitations observed with CLM 2.0 was its representation of frozen soil which included completely frozen soil in areas with temperature below 0 °C, resulting in higher and earlier than expected runoff caused by spring season rainfall. The modifications suggested were allowance for the coexistence of ice and water in soil, the concept of a fractional permeable area, and considering both liquid water and ice together as soil moisture for calculating hydraulic conductivity. These mod- ifications improved both the surface runoff and the soil water storage estimates of CLM when the simulations were evaluated for river basins in the cold regions, viz., Lena, Yenisei, Mackenzie, Ob, Churchill-Nelson, and Amur using both streamflow and GRACE data [48, 49]. Another deficiency of the CLM model was its inability
- 20. Integration of GRACE Data for Improvement of Hydrological Models 9 to model the groundwater dynamics as the column of soil considered extends only to 3.8 m below the surface. In another attempt to improve the CLM model, a Sim- ple Groundwater Model (SIMGM) which represents an unconfined aquifer along with the recharge and discharge processes was included within the framework [50]. Although the modification worked out well for all the 12 river basins considered in the study, it is not expected to do well in cold regions where the water table is exposed to freezing conditions due to the obvious differences in the physical processes. In a more recent attempt to accurately represent groundwater dynamics within the CLM model version 4.5, it was found that addition of a no-flux boundary condition at the base of the soil layer improved the estimate. As a result, these simulations from the improved CLM models were found to agree well with GRACE-derived TWS observations [72]. A few studies also tried to improve the ISBA-TRIP hydrological model by com- paring modified versions of the model with GRACE data. Initial comparisons of the model with GRACE data outlined some model deficiencies such as the high storage in the form of surface water within the river channel as a part of the routing scheme over- estimated the maximum and underestimated the minimum TWS values mainly in the tropical region. Other deficiencies within the ISBA-TRIP model were the calculation of evaporation and snow accumulation. However, the major limitation was identified as the oversimplified routing model and the absence of anthropogenic effects within the model [3, 19, 54, 87]. Although human impact was not included in the modified version, improvements were suggested for TRIP—the routing model which included a simple groundwater reservoir and a variable streamflow velocity calculation. Sev- eral other LSMs were evaluated globally or regionally using GRACE data. Inclusion of a water exchange scheme between continents and oceans included in the Organis- ing Carbon and Hydrology in Dynamic Ecosystems (ORCHIDEE) LSM resulted in better simulation of land water storage [Ngo-Duc et al., 2007]. GRACE data when compared to the Australian Water Resource Assessment (AWRA) model suggested a need for improvement in representations of diffuse groundwater discharge pro- cesses and interaction between surface and groundwater [van Dijk et al., 2011]. Doll et al. [2014] found that the WaterGAP model version 2.2 underestimates TWS as compared to GRACE with a phase lag of a month observed between the two. Evalu- ation of the GLDAS framework-versions 1 & 2 carried out for China by Wang et al. [2016] showed inconsistency in the rate of change of TWS. The four land surface models (Noah, SAC-Sacramento Soil Moisture Accounting Model, (VIC) Variable Infiltration Capacity Model, and Mosaic) applied in the newly implemented National Centers for Environmental Prediction (NCEP) operational and research versions of the North American Land Data Assimilation System version 2 (NLDAS-2) were also evaluated using GRACE data [Xia et al., 2016]. A common source of inconsistency ob-served between the GRACE observation and model simulation was attributed to the error and uncertainty present in the precipitation dataset which is a primary forcing for all hydrologic models.
- 21. 10 C. Banerjee and D. N. Kumar Table 1 Comparative list of studies evaluating and comparing hydrological models with GRACE data Authors (Year) GRACE data Model Input data Study region Study period Niu and Yang [49] Chen et al. [16], Seo and Wilson [67] CLM 2.0 with SIMTOP GLDAS 1-degree 3-hourly data (2002–2004) Lena, Yenisei, Mackenzie, Ob, Churchill- Nelson and Amur August 2002–July 2004 Niu and Yang [48] Chen et al. [16], Seo and Wilson [67] CLM and Modified CLM GLDAS 1-degree 3-hourly data (2002–2004) Global and Ob, Yangtze, Amazon, Taz and Ural River Basin August 2002–July 2004 Swenson and Wahr [74] Swenson and Wahr [73] Atmospheric and Terrestrial Water Balance model GCM output and NCEP/DOE R-2 for atmospheric water balance and GLDAS/Noah LSM for terrestrial water budget Mississippi and Ohio- Tennessee River basins June 2002–April 2004 Ngo-Duc et al. [47] Ramillien et al. [56] ORCHIDEE modified to include a routing scheme P: 6-hourly NCEP/NCAR constrained by monthly CMAP; Others: 6-hourly NCC (NCEP/NCAR) corrected by CRU atmospheric forcing Global and Amazon, Congo, Niger, Mississippi, Yangtze, Ganges, Brahmaputra, Mekong May 2002–Decem- ber 2003 Niu et al. [50] Chen et al. [16], Seo and Wilson [67] Modified CLM with SIMTOP and SIMGM 1-degree 3-hourly GLDAS dataset (2002–2004) 12 Global river basins not affected by snow or ice August 2002–Decem- ber 2004 Alkama et al. [3] CSR-RL04, JPL-RL 4.1, GFZ-RL04 estimates ISBA-TRIP 3-hourly 1-degree Princeton University data Global and 33 large river basins Aug 2002–Dec 2006 Decharme et al. [19] CSR-RL04, JPL-RL4.1, GFZ-RL04 estimates TRIP with groundwater storage and variable flow velocity Runoff simulated by ISBA of Alkama et al. [3] Global and 12 large river basins Aug 2002–Dec 2006 van Dijk et al. [97] 1-degree gridded TWS estimates from CSR Australian Water Resource Assessment (AWRA) 0.05-degree gridded meteorological forcings obtained by interpolation of Station data Continental Australia January 2003–Decem- ber 2010 Grippa et al. [27] RL04 of CSR, JPL and GFZ, DEOSDMT, GRGS- EIGEN-GL04 and 10 day, 4° GSFC HTESSEL, ORCHIDEE- CWRR, ISBA, JULES, SETHYS, NOAH, CLSM, SSiB, SWAP Rainfall: TRMM 3B42, Atmospheric forcings: ECMWF short-term forecast data Downwell Radiative fluxes: mix of ECMWF and Land Surface Analysis Satellite Applications Facility West Africa Jan 2003–Dec 2007 (continued)
- 22. Integration of GRACE Data for Improvement of Hydrological Models 11 Table 1 (continued) Authors (Year) GRACE data Model Input data Study region Study period Pedinotti et al. [54] CSR-RL04, JPL-RL4.1, GFZ-RL04 estimates ISBA-TRIP TRMM-3B42 and RFE-Hybrid rainfall for ISBA-TRIP CHS, other atmospheric forcings from ECMWF Niger River Basin Jan 2003–Dec 2007 Vergnes and Decharme [87] CSR-RL04, JPL-RL 4.1, GFZ-RL04 estimates TRIP Total runoff from ISBA simulation by Alkama et al. [4] Global and 12 large river basins August 2002–August 2008 Rosenberg et al. [62] I-degree gridded CSR dataset VIC modified to include SIMGM 1/8th-degree Gridded from precipitation and maximum/minimum temperature data from NOAA Cooperative Observer stations and wind data from NCEP-NCAR reanalysis Colorado River Basin 2002–2010 Cai et al. [13] 1-degree gridded TWS estimates from CSR RL4.0 Noah-MP NLDAS Phase 2 atmospheric forcing at 1/8° resolution Mississippi River Basin 2003–2009 Doll et al. [20] 0.5-degree gridded GFZ-RL05, CSR-RL05 and ITG- Grace2010 WaterGAP 2.2 Daily climate dataset WFD (WATCH Forcing Data)/WFDEI (Watch Forcing Data ERA-Interim) Global 2003–2009 Swenson and Lawrence [72] CSR RL05 CLM version 4.5 with modification 1.25 longitude × 0.9 latitude ECMWF ERA-Interim Reanalysis data Lower Colorado River basin, in the southwestern United States, and a region in northeastern Australia 2002–2014 Ahmed et al. [1] 1-degree gridded TWS estimates from CSR RL05 CLM4.5-SP and GLDAS-Noah GLDAS: NOAA and CPC/CMAP and CLM: CRU/CRUNCEP Continental Africa (Niger, Zambezi, Okavanko, Limpopo) 2003–2010 Wang et al. [90] GRACE Tellus RL05 CSR, JPL, GFZ GLDAS1 (Noah, CLM, Mosaic, VIC) GLDAS2 (Noah 3.3) ECMWF & NCEP–NCAR reanalyses data, NOAA/GDAS and Princeton University atmospheric fields, AGRMET radiation fields, China 2002–2010 (continued)
- 23. 12 C. Banerjee and D. N. Kumar Table 1 (continued) Authors (Year) GRACE data Model Input data Study region Study period Xia et al. [91] GRACE Tellus RL05 CSR, JPL, GFZ average NLDAS-2 operation (Mosaic and Noah) and research (SAC-Clim and VIC4.0.5) CPC, PRISM & NARR precipitation data and 2-m air temperature from NARR USA 2003–2014 Zhang et al. [95] GRACE RL05 Level-2 products from GFZ LSDM, WGHM, JSBACH, MPI-HM WFDEI dataset based on ERA-Interim reanalysis data 31 largest river basins 2003–2012 5 GRACE Data Assimilation Data assimilation is a statistical technique of combining the simulations or forecasts from a prediction model with measurements from an observing system to produce improved estimates. Evaluation of LSMs has been one of the most explored tech- niques of utilizing GRACE data for the improvement of model physics and simu- lation accuracies. However, it is an indirect method where model deficiencies are figured out by comparing model outputs with GRACE observations followed by improving model physics solely based on our understanding of the intricate details of hydrological processes. This to some extent is limiting since the knowledge and understanding of the hydrological processes are itself limited and the large infor- mation hidden within the GRACE observations may be completely overlooked. As an alternate method of data integration, GRACE data assimilation techniques were explored where the observational dataset is directly utilized to improve the model simulation at each time step. Although it apparently does not improve model physics or our understanding of hydrological processes, GRACE data assimilation improves model simulations to a great extent, also facilitating spatial and temporal disaggre- gation of GRACE data as a byproduct. Table 2 gives a detailed chronological list of studies performed in this field of research. The assimilation of GRACE data into LSMs has two major challenges. The typical temporal and spatial resolution of the GRACE observation is much coarse as com- pared to the LSMs. The GRACE data provided by NASA JPL’s TELLUS website has a spatial resolution of ~100 km (1 degree) and a temporal resolution of a month. On the contrary, most LSMs are run at a daily or sub-daily scale, with the spatial res- olution varying from 5 km (0.05°) to a maximum of 50 km (0.5°). Hence, the process of data assimilation invariably includes a spatial and temporal disaggregation tech- nique. Consequently, a widely used and efficient data assimilation technique, known as the Ensemble Kalman Filter (EnKF) [22], is used in most of the previous literature (Table 2). The EnKF is a variant of a statistical technique known as the Kalman filter and is used for large problems. It has the inherent assumptions that the probability
- 24. Integration of GRACE Data for Improvement of Hydrological Models 13 Table 2 Comparative list of GRACE data assimilation studies Authors (Year) GRACE data Model Input data Study region Study period Assimilation method Zaitchik et al. [94] CSR (RL01), GFZ (RL03), JPL (RL02) CLSM GLDAS forcing database Mississippi (4 sub- catchments) January 2003–May 2006 Ensemble Kalman Smoother Houborg et al. [29] CSR-RL04 CLSM NLDAS-2 ad GLDAS data for study period model run and Princeton University data for long-term simulation North America August 2002–July 2009 Ensemble Kalman Smoother Li et al. [41] CSR-RL04 CLSM GLDAS forcing database Western and Central Europe August 2002–July 2009 Ensemble Kalman Smoother Huang et al. [30] CSR-RL05 Noah-MP 0.1-degree, 3-hourly, near-surface meteorological dataset produced by the ITPCAS Yangtze River basin Jan 2003–Dec 2010 Proposed framework Reager et al. [58] CSR-RL05 CLSM Same as Zaitchik et al. [94] Mississippi river basin April 2002— Dec 2014 Ensemble Kalman Smoother Tangdamrongsub et al. [82] CSR-RL05 OpenStreams wflow_hbv model (HBV-96) European Climate Assessment & Dataset (ECA & D), ENSEMBLES project and Princeton University Dataset Rhine river basin Dec 2003—Oct 2007 Ensemble Kalman Filter Girotto et al. [25] Gridded CSR-RL05 CLSM MERRA USA Jan 2003—Dec 2013 Sequential Kalman filtering technique (continued)
- 25. 14 C. Banerjee and D. N. Kumar Table 2 (continued) Authors (Year) GRACE data Model Input data Study region Study period Assimilation method Schumacher et al. [66] TWS values using WGHM, ITG- GRACE2010 error covariance WGHM CRU TS 3.2, GPCC, WFDEI Mississippi river basin August 2003 EnKF, SQRA, SEIK Girotto et al. [26] Gridded CSR-RL05 CLSM MERRA India Jan 2003–Aug 2015 3D Ensemble Kalman Filter Khaki et al. [33] ITSG- Grace2014 W3RA Princeton University forcing dataset Australia Feb 2002–Dec 2012 (stochastic) EnKF, ETKF, SQRA, DEnKF, EnSRF, EnOI and PF with Multinomial (PFMR) and Systematic (PFSR) Resampling Tangdamrongsub et al. [81] CSR-RL05 PCR- GLOBWB ECMWF Era-Interim, TRMM, CRU, Princeton and China Daily Ground Climate Dataset Hexi corridor in Northern China April 2002–Dec 2010 EnKF with and without errors Tian et al. [85] JPL-RL05 M, 3-degree mascon product W3 Model WFDEI forcing data, Global Tree cover fraction map and MODIS white-sky albedo Australia Jan 2010–Dec 2013 EnKF and EnKS
- 26. Integration of GRACE Data for Improvement of Hydrological Models 15 Fig. 3 A schematic diagram showing the concept of a typical Kalman Filter (Credit: Melda Ulusoy, MathWorks) distributions are all Gaussian and the predictive model is linear. The Kalman filter (Fig. 3) is a recursive filtering mechanism which combines the simulation of a model and a noisy measurement, both of which are assumed to be Gaussian distributions to estimate the most likely state variables. The model estimate is generally less prob- able and contains more uncertainty than the measurement. However, the use of the EnKF provides an optimal estimate of the state variable which is much more proba- ble and contains less uncertainty as compared to both the model prediction and the measurement, as shown in Fig. 3. The second challenge is the hydrological variable of interest. GRACE obser- vations result into TWS data which, as discussed earlier, is the aggregation of all the surface and subsurface water storages. To assimilate GRACE TWS data, there needs to be a hydrological variable within the model to which it can be mapped. The problem in this case is that all hydrological models have separate surface and subsurface storages modeled as different processes. Even if all the storages are added up to create a hydrological variable to be mapped against GRACE TWS data, it falls short due to the absence of groundwater storage. Most of the hydrological models incorporate groundwater dynamics as a boundary condition at the bottom of the soil column considered which is typically 2–4 m in depth from the ground surface. To resolve this issue, the catchment land surface model is the most preferred LSM used for assimilation as it contains an unconfined groundwater reservoir. Several studies have assimilated the GRACE TWS data with one of the primary objectives being improvement of groundwater estimation. Zaitchik et al. [2008] assimilated GRACE data into the CLSM using an ensemble Kalman smoother. Results indi- cated an improved correlation between observed ground-water and data assimilated simulated groundwater. In a similar effort, GRACE data was assimilated into the OpenStreams wflow_hbv model using an ensemble Kalman filter for the Rhine river basins. Results show increase in correlation between observed and simulated ground- water from 0.6 to 0.7 and 15% reduction in RMSE as a result of this data assimilation [Tangdamrongsub et al., 2015]. In both the cases, slight improvement in streamflow simulation was also observed. Tangdamrongsub et al. [2017] showed that assimi-
- 27. 16 C. Banerjee and D. N. Kumar lation of GRACE data increased the accuracy of groundwater estimate, simulated for a semi-arid region in northern China by PCR-GLOBWB by 25%. GRACE data assimilation was also carried out with the objective of drought assessment because most frameworks lack information of groundwater and soil moisture of deeper lay- ers. Houborg et al. [2012] and Li and Rodell [2015] assimilated GRACE data into CLSM model to derive drought indicators for North America and conterminous US respectively. A similar exercise was carried out for western and central Europe by Li et al. [2012]. These efforts disaggregated GRACE data in both spatial and temporal dimension. GRACE data assimilation was also carried out to estimate human induced changes in TWS and assess regional flood potential [Y Huang et al., 2015a; Reager et al., 2015]. Further studies concentrated on improving the data assimilation using better variants of the ensemble Kalman Filter and other hydrologic dataset such as the soil moisture from Soil Moisture and Ocean Salinity (SMOS) mission [Girotto et al., 2016; Girotto et al., 2017; Khaki et al., 2017; Schumacher et al., 2016; Tian et al., 2017]. 6 Conclusions The hydrological models altogether have improved from the simple lumped models and now include not only hydrological processes but all such physical, chemical, and biological processes that affect or is affected by water (a typical example of which is shown in Fig. 2). Integration of GRACE data into hydrological models has improved their model physics and prediction capabilities. Such models now represent better dynamics of frozen soil, dry soil in arid climate, groundwater, and vegetation. This also improved the estimation of various hydrological and vegetation parameters. Further improvements were achieved by GRACE data assimilation into hydrological models with the added advantage of disaggregation of GRACE TWS observations. Moreover, the GRACE data processing techniques have also improved with the most recent studies using Release 05 dataset which has a much higher accuracy as compared to the initial releases. The GRACE Follow-On (GRACE-FO) missionis scheduledtobelaunchedin2018whichis expectednot onlytocontinuethe unique GRACE observations but also to have some improvements as compared to its forerunner [18]. Meanwhile, scientists are still working on the processing techniques of the GRACE data and the new Release 06 of the GRACE dataset having better accuracy is available for use [28]. Thus, there are numerous avenues in which further improvement is possible that will unravel new vistas of knowledge in future. References 1. Ahmed M, Sultan M, Yan E, Wahr J (2016) Assessing and improving land surface model outputs over africa using GRACE, field, and remote sensing data. Surv Geophys 37(3):529–556. https://
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- 34. An Analysis of Spatio-Temporal Changes in Drought Characteristics over India Ganeshchandra Mallya, Shivam Tripathi and Rao S. Govindaraju 1 Introduction 1.1 Introduction to Droughts Droughts are among the world’s costliest disasters with an annual cost estimated in the range of $6–$8 billion [20]. Unlike other natural disasters such as floods and earthquakes, droughts manifest slowly and are already a serious threat before they are detected. Droughts have major impacts on agriculture, natural habitats and ecosys- tems, and economies of affected regions. Modeled precipitation and temperature results from different climate change scenarios indicate that droughts are likely to intensify over many parts of the world in the next 20–50 years [13], suggesting the need to assess drought impacts more accurately and develop appropriate mitigation strategies. When a drought event occurs, moisture deficits are identified from many hydro- logic variables such as precipitation, streamflow, soil moisture, snowpack, ground- water levels, and reservoir storage [81]. Because drought impacts are experienced differently across the world, no universally accepted definition of drought exists. However, three types of droughts are commonly featured in the scientific literature [17]: a. Meteorological droughts result from deficits in precipitation amounts when com- pared to the long-term average for a region. This shortage in precipitation can develop quickly and also end abruptly. G. Mallya · R. S. Govindaraju (B) Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA e-mail: govind@purdue.edu S. Tripathi Department of Civil Engineering, Indian Institute of Technology, Kanpur, UP 208016, India © Springer Nature Switzerland AG 2019 S. K. Singh and C. T. Dhanya (eds.), Hydrology in a Changing World, Springer Water, https://doi.org/10.1007/978-3-030-02197-9_2 23
- 35. 24 G. Mallya et al. b. Agricultural drought conditions prevail when available soil moisture is insuffi- cient to replace evapotranspiration losses in the root zone [83]. The timing of soil moisture deficit plays a critical role, because deficiencies during the grow- ing season can adversely impact crop yields. During droughts, plants are under stress and cannot fight off pests. Fertilizers and pesticides are also not effective in the absence of moisture resulting in failure of crops. The onset of agricul- tural droughts depends on antecedent soil moisture conditions and usually lags meteorological droughts. c. Hydrologic droughts reflect shortages in water supply mainly in the form of reduced streamflows, reservoir and lake levels, and groundwater levels. Hydro- logic droughts persist for longer durations when compared to meteorological and agricultural droughts because precipitation deficits translate to deficit in other hydrologic variables with significant time lags in some instances. In this chapter, we focus on meteorological droughts. Precipitation deficits are not the only cause of droughts. Industrial and agricultural water demands have increased exponentially over the last few decades leading to water scarcity. With the increase in the emission of greenhouse gases, a steady rise in temperature has been observed over many parts of the globe. Increasing temperatures have affected the global hydrologic cycle leading to spatiotemporal variability of precipitation at different scales [54, 74], thereby affecting drought characteristics. 1.2 Drought Characterization and Monitoring Drought indicators or indices are commonly used to characterize and monitor droughts and their impacts. Generally, all drought indicators use some measure of water deficit for analysis. Multiple hydrometeorological variables can also be used in a single drought indicator to capture the complex interactions that lead to droughts. Some of the desirable properties of a drought index are: (1) it should be sensitive to the timescale appropriate for the problem at hand; (2) the index should be able to capture the characteristics of both shorter and longer duration droughts; (3) it should be applicable to the problem being studied; (4) it should be possible to identify his- torical droughts; (5) the index should be capable of monitoring droughts on a near real-time basis [21, 32]; and (6) it should have drought forecasting capability. Several studies provide comprehensive review of drought indices [32, 55]. Palmer drought severity index (PDSI; Palmer [61]) is a popular meteorological drought index that uses precipitation and temperature for estimating demand and supply of soil moisture within a two-layer water balance model. PDSI provides outlooks of mois- ture conditions that are comparable across regions and over different months. PDSI values typically vary from −4.0 to +4.0, negative values indicating drought condi- tions, while positive values indicate wet conditions. Another drought index that is popular because of its computational simplicity and forecasting ability at different time scales is the standardized precipitation index (SPI; McKee et al. [51]). The SPI
- 36. An Analysis of Spatio-Temporal Changes … 25 is recommended by the World Meteorological Organization as a standard meteoro- logical drought-monitoring index [30]. The SPI first fits a probability distribution to historic precipitation time series data, and then normalizes the fitted distribution using the standard inverse Gaussian function to compute the drought index. SPI val- ues are dimensionless with negative values indicating drought conditions, and the magnitudes of their departures from zero indicating the severity of the drought. The crop moisture index (CMI; Palmer [60]) that monitors short-term moisture supply was developed to monitor agricultural droughts. With the improvements in satellite remote sensing, monitoring crop and vegetation health over large spatial extents have become routine. For example, vegetation condition index (VCI; Liu and Kogan [40]) uses the advanced very high-resolution radiometer radiance (AVHRR) data to study drought characteristics (early onset, intensity, frequency, and duration) and vegetation health. Along similar lines, the normalized difference water index (NDWI; Gao [23]) uses near-infrared (NIR) and short-wave infrared (SWIR) chan- nels to study the variation of moisture content and spongy mesophyll in vegetation canopies. Monthly non-exceedance probability computed by compiling weighted values of variables such as reservoir storage, stream flow, snowpack, and precipitation resulted in the development of a hydrologic drought index called the surface water supply index (SWSI; Shafer and Dezman [71]). Other popular hydrologic drought indices are standardized streamflow index (SSI) and standardized runoff index (SRI; Shukla and Wood [73]). The SRI is computed and interpreted along similar lines as SPI. Drought indices have been used for identifying droughts and their triggers [76], assessing drought status [35], forecasting droughts [1], performing drought risk anal- ysis [31], and studying relationship of droughts with local-scale regional hydrolog- ical variables such as water quality [75] and large-scale climate patterns like El Niño–Southern Oscillation [11, 41, 69]. Drought indices are also used for classify- ing droughts and quantifying their temporal trends. These two applications of drought indices are reviewed in the following paragraphs. 1.3 Drought Classification Drought classification schemes typically classify droughts based on their severity or intensity, and are often based on drought indices that measure degree of departure of hydrometeorological variables, such as precipitation and streamflow, from their long-term averages. Water resource planners rely on drought classification to select drought mitigation strategies. Hence, weather agencies throughout the world rou- tinely issue drought classification bulletins. For example, the US Drought Monitor releases a weekly update of drought status in USA by classifying droughts into five classes—D0 to D4 with the latter representing exceptional drought. Likewise, India Meteorological Department (IMD) issues drought bulletins classifying droughts into three categories, namely, mild, moderate, and severe.
- 37. 26 G. Mallya et al. Common quantitative drought classification schemes work in two steps—first, by defining a drought index using hydrometeorological observations of typically 30- year period to establish normal conditions [33] and next, by categorizing droughts based on predefined thresholds on the index value. Examples include IMD classifica- tion that uses departure of rainfall from its long-term average as a drought index, and US Drought Monitor classification that, along with other indices, uses standardized precipitation index (SPI) as a drought index. Among several drought classification schemes [13, 32, 55], the scheme based on standardized precipitation index (SPI; McKee et al. [51]) is very popular because of its computational simplicity and versa- tility in comparing different hydrometeorological variables at different time scales. In SPI, historical observations are used to compute the probability distribution of the monthly and seasonal (4, 6, and 12 months) precipitation totals. The fitted probabil- ity distributions are then normalized using the standard inverse Gaussian function to calculate SPI values. A negative value of SPI indicates precipitation less than the median rainfall, and the magnitude of departure from zero represents the severity of a drought. Standard SPI-based drought classification, though popular, has many weaknesses [48]. It provides discrete classification and ignores uncertainties arising from data errors, model assumptions, and parameter estimates. Thus, users are not aware of inherent uncertainties in drought classification often required for making informed decisions. Further, in the context of SPI, there is an ongoing debate on the selection of the parametric distribution for fitting the data. McKee et al. [50] in their original paper on SPI recommend a gamma distribution. Lloyd-Huges and Saunders [42] found gamma distribution to be an appropriate model for Europe. Guttman [27] sug- gested Pearson-III distribution as the best universal model for SPI because it provides more flexibility than the gamma distribution. Rossi and Cancelliere [67] found nor- mal, lognormal, and gamma distributions to be suitable for different datasets in their study.LoukasandVasiliades[43]investigateddifferenttheoreticaldistributionsusing Kolmogorov–Smirnov (K–S) test and chi-squared test and found extreme value-I dis- tribution to be the most suitable for studying droughts over Thessaly, Greece. Mishra et al. [53] argue that different distributions may be appropriate for different drought durations (window size), and recommend the K–S test for choosing an appropri- ate distribution. Bonaccorso et al. [7] used Lilliefors test to choose among normal, lognormal, and gamma distributions while Russo et al. [68] used three parameters generalized extreme value (GEV) distribution for SPI analysis. Thus, there is no consensus on the choice of distribution for SPI analysis. Mallya et al. [48] used hidden Markov model (HMM) for drought classification by conceptualizing hidden states in the model to represent drought states. Their model avoided the need for specifying thresholds for drought classification and provided probabilistic drought classification by accounting for model uncertainties; however, the number of hidden states (drought classes) was prespecified. To facilitate com- parison of HMM drought index (HMM-DI) classification with standard methods, they specified 11 hidden states. Since the number of states is imposed on the model, it is possible that for datasets with short record length the model suffers from an overspecification problem, i.e., the model structure is more complicated than sup-
- 38. An Analysis of Spatio-Temporal Changes … 27 ported by the dataset. Specifically, in the HMM context, overspecification would occur if the number of specified hidden states is more than that needed to model the data. Overspecification can result in parameter identification problems leading to unreliable results. Mallya et al. [47] proposed a method that adapts SPI drought classification methodology by employing gamma mixture model (Gamma-MM) in a Bayesian framework. The method alleviates the problem of selecting suitable distribution for SPI analysis, quantifies modeling uncertainties, and propagates them for probabilistic drought classification. Further, it avoids overspecification using a Bayesian approach for optimally selecting the number of hidden states in the model. 1.4 Temporal Trends in Droughts Temporal trends in droughts are identified by determining changes or sudden shifts in the distributional properties of the underlying hydrological variable. Classical drought indices such as SPI, or even probabilistic drought indices such as HMM-DI or probabilistic SPI, make several model assumptions about the hydrological variables used in their construct. Among them, the most important assumption is that the time series of hydrological variable is stationary, i.e., its distributional properties used to define droughts do not change over time. Thus, temporal trends in droughts cannot be estimated using classical drought indices. Nevertheless, hydrological time series may exhibit nonstationarity due to changes in climate and land use or due to natural cycles that operate over a period of several years to decades, and hence it is important to study temporal trends of droughts. Several studies in the literature have proposed methods to perform drought analy- sis under nonstationary conditions. Mishra and Desai [52] used autoregressive inte- grated moving average (ARIMA) models and variants of artificial neural networks (ANNs), namely, recursive multistep neural network approach (RMSNN) and direct multistep neural network approach (DMSNN) for drought forecasting in presence of nonstationarity. Coulibaly and Baldwin [12] proposed the use of dynamic recurrent neural network (RNN) to model and forecast nonstationary hydrologic time series. Belayneh et al. [4] used wavelet analysis to first denoise the series, and then train ANNs or support vector regressors on the decomposed signals to perform drought forecasting in arid regions of Ethiopia. Türkeş and Tatlı [77] studied droughts in non- stationary precipitation series by modifying the classical SPI using the concepts of empirical mode decomposition. Unlike the classical SPI, the modified SPI accounts for local or higher order statistics in the precipitation time series. Han et al. [29] pro- posed the use of ARIMA models on potentially nonstationary remote sensing data to predict vegetation temperature condition index for drought forecasting. Verdon-Kidd and Kiem [80] emphasized the need to evaluate risk to water resources systems during drought under a nonstationary climate over Australia. Mitra and Srivastava [56] use
- 39. 28 G. Mallya et al. the modified Mann–Kendall test on SPI and SPEI series to study the spatiotemporal variability of meteorological droughts in southeast USA. The literature review suggests that in the context of SPI, temporal changes in droughts are mostly studied using one of the following two approaches. The first approach divides the study period into smaller intervals or epochs (~30 years) where the underlying rainfall series is assumed to be stationary, and computes relative SPI for each epoch [18], and then compares drought characteristics between different epochs. The second approach allows the distribution of the hydrological variable to change with time, but the parameters of the distribution can follow only a prespecified temporal pattern. For example, the standardized nonstationary precipitation index (SnsPI) proposed by Russo et al. [68] assumes that the scale parameter of the Gamma distribution for rainfall varies linearly with time. The main objectives of this chapter are as follows: (a) To investigate drought characteristics in India using a probabilistic drought clas- sification approach that adapts SPI methodology by employing gamma mixture model (Gamma-MM) in a Bayesian framework [47], and to compare the results with classical SPI. (b) To use an alternate methodology for studying temporal changes in droughts [46] that does not require—(i) making stationarity assumption about the precipitation time series and (ii) prespecifying the nature of temporal trend in the precipitation series. (c) TostudytemporalchangesinthedroughtsinIndiathroughthisalternatemethod- ology, and compare results with existing methods. The remainder of the chapter is structured as follows. The next section describes the study area, India, and provides an account of its historical droughts. The pre- cipitation datasets available for the study area are described in Sect. 2. Section 3 presents the mathematical formulation of the methods used for classifying droughts and quantifying temporal variation. These methods will be applied to precipitation data over India and the results will be presented and discussed in Sect. 4. The chapter ends with a set of concluding remarks. 2 Study Area and Dataset The study area, India, receives 80% of its annual precipitation during 4-months long southwest summer monsoon [3, 62]. The monsoon precipitation makes landfall around the first week of June near Kerala, India, and moves northeast toward the Himalayas. By the first week of July, almost the entire country typically receives some precipitation that continues until the end of September [9]. From beginning of October to December, cool and dry winds from Central Asia cross India diagonally from northeast to southwest. These winds humidify the air as they blow over the Bay of Bengal, resulting in northeast monsoon precipitation predominantly over the state of Tamil Nadu, and partly over other states of Odisha, Andhra Pradesh, Karnataka,
- 40. An Analysis of Spatio-Temporal Changes … 29 and Kerala [36]. Though the Indian monsoon is believed to be one of the most stable monsoon systems [34, 49, 65], it has large inter- and intraseasonal variability that can sometimes result in weak monsoon or droughts over India [37, 59]. Since the country’s gross domestic product (GDP), particularly food and power production, is closelylinkedtomonsoonrains,variousstrategieshavebeendevelopedovertheyears to mitigate the effects of droughts (e.g., drought-prone areas programme (DPAP), and desert development programme (DDP)). Implementing effective drought mitigation strategies requires real-time reliable classification of droughts. 2.1 Droughts over India and Their Consequences Each type of drought has its own consequences, and the effects are felt by the general population. Since India is mainly an agricultural economy, droughts have histori- cally had major impact on farmers. Agricultural droughts result in low crop yield and sometimes a complete failure of the crop. The agricultural fields can quickly turn into large dust bowls thereby leading to topsoil loss. This, in turn, causes stress in maintaining healthy livestock. Scarcity of water also leads to unhygienic con- ditions—leading to faster spread of diseases among the population. The economic consequences during/following a prolonged drought event can be detrimental to the poor farming community in India. Lack of crop insurance and inadequate financial support from government-backed banks often forces farmers to borrow money from private lenders. This leads to social disputes and mass migration from villages to cities in search of alternate employment opportunities [15, 64]. The non-farming community living in towns and cities face consequences of droughts in the form of shortage of water supply for household and industrial use. Droughts often lead to rise in commodity and fuel prices, thus causing economic stress for lower and middle-income families within the affected region. In an effort to build a resilient society, the Central and State Governments in India have developed drought mitigation programs such as drought-prone areas pro- gramme (DPAP), desert development programme (DDP), and national watershed development programme in rainfed areas (NWDPRA) that provide material, educa- tional, and financial support for the following: i. Lake restoration and capacity building of existing reservoirs. ii. Rainwater harvesting, cloud seeding to trigger rains. iii. Large-scale desalination plants in major coastal cities to decrease reliance on groundwater and river water for domestic and industrial supply. iv. Low-interest agricultural loans to farmers, and guaranteed employment for at least 100 days in a year under National Rural Employment Guarantee scheme.
- 41. 30 G. Mallya et al. 2.2 Recent Drought Literature over India Niranjan Kumar et al. [58] used SPEI to study the variability of monsoon droughts over India, and found that El Nino/Southern Oscillation (ENSO) as the most influenc- ing factor. They implicate the warming of the equatorial Indian Ocean to the increased droughts over India in the recent decades. Mahajan and Dodamani [44] performed trend analysis of drought events over Upper Krishna Basin in Maharashtra. Kumar et al. [39] used historical rainfall and sea surface temperature (SST) records to show that warmest SST anomalies in the central equatorial Pacific are better indicators of severe droughts over India. Varikoden et al. [78] showed that droughts associated with El Nino are very intense in most parts of the subcontinent, when compared to droughts during non-El Nino years. Mallya et al. [45] used SPI, SPEI, HMM-based drought index, and Gaussian mixture models and found that irrespective of the pre- cipitation dataset or the choice of drought index, the drought severity and frequency over India increased significantly during recent decades. Their study also found that droughts are becoming more regional and are showing a general shift to the agricul- turally important coastal South India, central Maharashtra, and Indo-Gangetic plains. Zhang et al. [86] found that the soil moisture and vegetation drought indices were best suited to study the impact on wheat production in India. Naresh Kumar et al. [57] studied the spatiotemporal patterns of droughts over India using SPI and found that area under moderate droughts have increased in recent decades. 2.3 India Meteorological Department (IMD) Precipitation Dataset To analyze meteorological droughts over India, long-term precipitation data are required. Daily rainfall data at a spatial resolution of 1° for both latitude and longi- tude were obtained from India Meteorological Department (IMD) and are based on a total 1803 stations distributed over India that have at least 90% availability for the period 1901–2004 [63]. The gridded data, consisting of 357 grid points, have been obtained by interpolating rain gage data. The IMD datasets are standard datasets widely used in monsoon-related studies over India [25]. Figure 1 shows the study area along with the grid locations for which rainfall data were available as circular markers. The grids where results are discussed in subsequent sections are denoted as square red-colored markers. Because of its large geographical extent, the study area consists of several streams and rivers (some of which are perennial). Of these streams, some drain into the Arabian Sea or the Bay of Bengal, while few rivers cross international borders into neighboring countries. The main networks of some of the major rivers of India are shown in Fig. 1.
- 42. An Analysis of Spatio-Temporal Changes … 31 Grid 40 Grid 125 Grid 169 Grid 251 Fig. 1 Map showing the study area along with the location of 1° × 1° grids of India Meteorological Department (IMD) precipitation dataset. Red square markers denote the stations where results are discussed in subsequent sections. The location of major rivers in India are also shown in the map 2.4 Homogenous Monsoon Regions Based on rainfall characteristics, the Indian Institute of Tropical Meteorology has divided the study area into six homogenous monsoon regions. The geographical extent of each of these regions is shown in Fig. 2. Dividing the entire study area into smaller regions, instead of working with a single representative average precipitation time series for the entire country, is needed to account for large spatiotemporal variability of precipitation across the country. Of the six regions, the hilly region
- 43. 32 G. Mallya et al. Fig. 2 Map showing homogenous monsoon regions of India Modified from Indian Institute of Tropical Meteorology (labeled as region 6 in Fig. 2) consists of grids located at high altitudes and often has poor precipitation estimates. The grids belonging to this region are not included in this study, especially when computing or reporting regional or all-India metrics of precipitation or droughts. The cumulative precipitation time series for different seasons and the water year (June to May of following year) were computed for each of the six regions using the average of precipitation time series recorded at all grids within a region. Figure 3 shows the histogram of water year precipitation series computed over each region. Each panel within Fig. 3 also contains the mean and standard deviation of the water yearprecipitationtimeseries.Thevaluesofmeanandstandarddeviationsuggeststhat region 1 (Northeast monsoon region) is the wettest among the six (mean precipitation
- 44. An Analysis of Spatio-Temporal Changes … 33 of 208 cm with a standard deviation of 21 cm), while region 5 (Northwest monsoon region spanning over Gujarat and Rajasthan) is the driest (mean precipitation of 52 cm with a standard deviation of 13 cm). An analysis of previous water year drought events based on SPI values (see Fig. 4) computed using an average representative cumulative precipitation time series over India (without considering grids located over hilly regions) suggested that the most severe conditions persisted during the periods early 1900s, 1918, 1951–52, 1965–68, 1972, late 1980s (Bengal drought), and early 2000s. The most recent drought condi- tions over India occurred in 2012–2013, with the West Central region—specifically the state of Maharashtra—being severely impacted by the drought. Figure 5 shows time series of percentage area under drought computed using his- torical SPI values for water year beginning in June of each year over India. According to this figure, approximately 72% of the grids were under drought during 2002, fol- lowed by 69% of the grids in 1972, 68% in 1918, 62% in 1965, and 60% in 1960. Therefore, the combination of Figs. 4 and 5 suggests that 2002 drought was the most severe on record in terms of severity and extent. However, the actual damage caused due to droughts in recent years are much lower compared to some of the previous droughts due to improved drought mitigation programs [14]. The result of Mann–Kendall trend test on the time series of area under drought suggests that the trend is positive (Sen’s slope 0.01, shown as red-dashed line in Fig. 5), although not statistically significant (at α 0.05). 3 Methodology The mathematical formulations of the two drought classification methods, namely, SPI and Gamma-MM, are presented in Sects. 3.1 and 3.2, respectively. Next, the methods used for studying temporal changes in droughts over India are described. 3.1 Standardized Precipitation Index (SPI) The method involves the following steps: 1. Decide a drought duration (time window) and estimate cumulative precipitation during that period. For example, to estimate droughts during a summer mon- soon season, estimate cumulative precipitation during 4 months of the summer monsoon season (JJAS) for each year. This will yield an annual time series of cumulative precipitation. Likewise, for analyzing water year droughts obtain an annual time series of cumulative precipitation for 12 months starting on June 1 and ending on May 31 of following year. 2. Fit a gamma distribution to the cumulative precipitation series. The cumulative distribution function (CDF) of the gamma distribution is standardized using the
- 45. 34 G. Mallya et al. 150 200 250 300 0 5 10 15 20 25 30 Region : 1 Bins: Precipitation (cm) Frequency Mean : 208.4cm Std. Dev. : 20.9cm 80 100 120 140 160 180 0 5 10 15 20 25 Region : 2 Bins: Precipitation (cm) Frequency Mean : 135.4cm Std. Dev. : 15.1cm 60 80 100 120 140 160 0 5 10 15 20 25 Region : 3 Bins: Precipitation (cm) Frequency Mean : 112.6cm Std. Dev. : 15.6cm 80 100 120 140 160 180 0 10 20 30 40 Region : 4 Bins: Precipitation (cm) Frequency Mean : 116.6cm Std. Dev. : 12.9cm 20 40 60 80 100 0 5 10 15 20 25 Region : 5 Bins: Precipitation (cm) Frequency Mean : 51.8cm Std. Dev. : 13.4cm 100 150 200 250 0 5 10 15 20 25 30 Region : 6 Bins: Precipitation (cm) Frequency Mean : 175.2cm Std. Dev. : 25.7cm Fig. 3 Histogram of cumulative water year (June to May) precipitation at each homogenous monsoon region over India
- 46. An Analysis of Spatio-Temporal Changes … 35 Fig. 4 SPI time series corresponding to water year (June to May) over the Indian monsoon region (IMR, excluding grids over hilly regions shown in Fig. 2) 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 10 20 30 40 50 60 70 SPI: Time window - 12month(s), Sen's slope: 0.01 Water Year % Area under drought Fig. 5 Time series of percentage area under drought during the period 1901–2003 over India/IMR. The results correspond to water year droughts (12-month time window, June to May) computed using standard SPI. The red-dashed line indicates the Sen’s slope
- 47. 36 G. Mallya et al. standard inverse Gaussian function to compute the SPI drought index. As stated earlier, a negative value of SPI indicates drought conditions and the magnitude of its departure from zero indicates the severity of a drought. 3. Decide a threshold on CDF to determine drought class. To draw parallels with the US Drought Monitor, we have used the same thresholds as used by them for SPI drought classification (Table 1). 3.2 Gamma Mixture Model (Gamma-MM) As discussed in the Introduction section, there is an ongoing debate on the choice of a suitable distribution for fitting data in SPI analysis. Mallya et al. [47] addressed this problem using the gamma mixture model (Gamma-MM). Given sufficient num- ber of components in the mixture, the Gamma-MM is proven to provide arbitrarily close approximation to any general continuous distribution in the range (0, ∞) (see, DeVore and Lorentz [16]). The use of Gamma-MM is not new in hydrology. To model data with multiple modes and different types of skewness, Evin et al. [19] proposed the use of Gamma- MMforstrictlypositivehydrologicaldata.Intheassessmentofhydrologicaldroughts for Yellow River in China, Shiau et al. [72] first fitted mixtures of exponential and gamma distributions to drought duration and drought severity, respectively, and then used the copula method to construct a bivariate drought distribution. While the mix- tures help represent the subpopulations within an overall population, the copula method describes the dependence between variables of interest. In the following, we provide a brief description of the Gamma-MM. The readers are referred to Wiper et al. [82] and Richardson and Green [66] for details on mixture models. A summary of the mathematical details of the Gamma-MM as described in Mallya et al. [47] is presented below. Table 1 US Drought Monitor classification scheme. SPI ranges are prescribed for the inverse of the normal distribution. Corresponding thresholds on CDF are given in the last column Category Description SPI range Threshold on CDF D0 Abnormally dry −0.5 to −0.8 0.212–0.309 D1 Moderate drought −0.8 to −1.3 0.097–0.212 D2 Severe drought −1.3 to −1.6 0.055–0.097 D3 Extreme drought −1.6 to −1.9 0.023–0.055 D4 Exceptional drought −2.0 or less 0.023 or less
- 48. An Analysis of Spatio-Temporal Changes … 37 Let the cumulative rainfall at time t be denoted by xt , t 1, . . . , N xt ∈ R and X [x1, . . . , xN ]T . If the total number of compo- nents of Gamma-MM, M, is known a priori, then the weighted sum of M mixtures of gamma is given by the following equation: P(xt |λ) M i1 wi G xt |vi , vi μi (1) where wi are the mixture weights or mixing ratios, and G xt | νi , νi μi are the com- ponents of gamma densities of the form, G xt | νi , νi μi νi μi νi (νi ) x(vi −1) t exp − νi μi xt , (2) with mean μi and shape parameter νi . Further, the mixture weights satisfy the con- straint M i1 wi 1. The parameter set is represented as λ {w, μ, v} where w [w1, w2, . . . , wM ]T , μ [μ1, μ2, . . . , μM ]T and v [ν1, ν2, . . . , νM ]T . In the Bayesian framework, the model parameters are obtained by specifying prior distributions to model parameters. Parameter estimation is accomplished by introducing a latent variable Z [z1, . . . , zN ]T for each time step. The variable zt is an M-dimensional binary random variable, zt [zt1, . . . , zt M ]T , in which a particular element is equal to 1 and all other elements are zero, i.e., M i1 zti 1 and zti ∈ {0, 1}. The variable zt denotes the component to which the data xt belongs, and hence it is also called an indicator variable. The conditional distribution of xt given zt is P(xt |zti 1) ∼ G xt |νi , νi μi (3) The posterior probability of the model parameters and latent variables is obtained by applying Bayes’ rule as P(λ|X) ∝ P(X|λ)P(λ) (4) where the parameter set λ includes the latent variable as well. The likelihood function given the latent variable is P(X|λ) P(X|Z, μ,ν) N t1 M i1 G xt |νi , νi μi zti (5) Following Wiper et al. [82] the prior distribution over the model parameter is given as P(λ) P(Z|w)P(w)P(μ)P(ν) with
- 49. 38 G. Mallya et al. P(Z|w) N t1 M i1 w zti i , P(w) Dir(w|) C() M i1 w φi −1 i , [φ1, . . . , φM ]T, P(ν) Exp(ν|θ) M i1 1 θi exp(−θi νi ), θ [θ1, . . . , θM ]T, and P(μ) GI(μ|α,β) M i1 β αi i (αi ) μ −αi −1 i exp − βi μi , α [α1, . . . , αM ]T and β [β1, . . . , βM ]T where Dir, Exp, and GI represent Dirichlet, Exponential, and Inverted gamma dis- tributions, respectively, and C() is a normalizing constant. The prior distribution is made non-informative by assigning following values to the hyperparameters. φi 1; θi 0.01; αi βi 1 for i 1, . . . , M. The posterior distribution P(λ|X) does not have a closed form and has to be estimated by either deterministic approximation (variational Bayes’ methods) or stochastic approximation (MCMC; Markov chain Monte Carlo methods). In this study, the posterior distribution is estimated using stochastic approximation by sam- pling the posterior distribution with Gibbs sampler, an MCMC algorithm [24]. The Gibbs sampling algorithm samples posterior distribution of the parameters by sequentially sampling from the conditional distribution of a parameter given all other parameters. The sampling starts with an initial value and proceeds as follows: 1. Set iteration number j 0, and parameters to their initial value λ(0) w(0) , μ(0) , ν(0) . The initial value is obtained by randomly sampling from the prior distribution of the parameters. 2. Sample from P z ( j+1) t |X, w( j) , μ( j) , ν( j) ∼ Multinomial(zt |rt ) where rt [rt1, . . . ,rt M ]T , rti sti M i1 sti and sti wi G xt |νi , νi μi and Multinomial represents multinomial distribution. 3. Sample from P w( j+1) |X, Z( j+1) , μ( j) , ν( j) ∼ Dir w| ˆ where ˆ [φi + ni , . . . , φM + nM ]T and ni N t1 zti . 4. Sample from P μ( j+1) |X, Z( j+1) , w( j+1) , ν( j) ∼ GI μ|α̂, β̂ where α̂ [αi + ni νi , . . . , αM + nM νM ]T and β̂ βi + νi M t1 xt zti , . . . , βM + νM M t1 xt zt M T . 5. Sample from P v( j+1) |X, Z( j+1) , w( j+1) , μ( j+1) . This conditional distribu- tion does not have a closed form. Hence, samples are generated using Metropolis–Hastings algorithm from a proposal distribution P(ṽi |vi ) ∼ G(h, h|vi ) and are accepted with a probability
- 50. An Analysis of Spatio-Temporal Changes … 39 min 1, f (ṽi )P(vi |ṽi ) f (vi )P(ṽi |vi ) , where f (vi ) ∝ v ni νi i (vi )ni exp −νi θi + t xt zti μi + ni log μi − log N t1;zti 1 xt . If the new sample ṽi is rejected, the current value of vi is retained. The above procedure is repeated to sample vi for all components i 1, . . . , M. In this study, the parameter of the proposal distribution, h, is set to 2. 6. Set j j + 1 and go to Step 2 until convergence. In this study, 15,000 samples were generated after ignoring initial 500 samples (burn-in period). Trace plots of the samples were monitored for convergence. To keep the notations uncluttered, the iteration number is omitted from the param- eters of the conditional distributions. In the above formulation of Gamma-MM, we have assumed that the number of mixture components, M, is known. However, in a general context, M is not known and should be estimated from data. One approach for estimating M is to consider it as a model parameter, assign prior distribution to it and estimate posterior distribution by MCMC method. Since changing M will result in a different model structure, usual MCMC algorithms such as Gibbs sampler cannot be applied. Instead, reversible jump MCMC (RJMCMC; Green [26] and Richardson and Green [66]) may be used. In this study, we implemented RJMCMC for Gamma-MM as described by Richardson and Green [66] and Wiper et al. [82]. The results suggested that RJMCMC algorithm requires significantly higher number of iterations for convergence compared to a model where M is specified. We found that if we start with a model having sufficiently large number of components, M, the Bayesian algorithm automatically prunes the components that are not relevant by making the mixing ratio (w) very small, thereby determining optimum number of components. We recommend the latter approach for hydrological applications where the number of components is usually limited to 2 or 3. In the Bayesian framework, mixture models have the identifiability problem, i.e., a M component mixture model will have a total of M! equivalent solutions. The problem can be avoided by introducing asymmetricity in the likelihood function. For example, in the context of Gamma-MM, Wiper et al. [82] recommended the following restrictiononthemeans of themixturecomponents,μ1 μ2 · · · μM .However, for finding a good density model, as required in the present application, the problem of identifiability is not relevant because any of the equivalent solutions is as good as another [6].
- 51. 40 G. Mallya et al. 3.3 Temporal Trends in Droughts Evaluation of temporal trends associated with retrospective drought events provides a basis to understand regional patterns of severity and duration of droughts. It also provides insight into the nature of possible future droughts and potential vulnerabil- ities over the study region. In this study, we first investigated trends in precipitation series at each IMD grid and also at each homogeneous monsoon region. A modified Mann–Kendall trend test that accounts for autocorrelation in time series [28, 38] was used to detect trends in summer monsoon and water year precipitation. The trends were tested at 5% significance level (α). The effect of spatial correlations in the data [8, 85] on the trend results was accounted using false discovery rate (FDR) [5, 79]. If the precipitation series exhibits a trend, drought classification methods such as SPI and Gamma-MM are not applicable for drought analysis because they assume the precipitation time series to be stationary. In this study, we apply an alternate method for drought analysis that overcomes this problem by explicitly removing trends from the precipitation series. The method is compared with relative SPI [18] and SnsPI [68] for performing drought analysis of a nonstationary precipitation series. The following paragraphs describe this alternate method and provide a summary of relative SPI and SnsPI methods. 3.3.1 An Alternate Method for Drought Analysis of a Nonstationary Precipitation Series A precipitation series may exhibit nonstationarity because of any of several reasons [22]—(a) the mean is a function of time, (b) the variance or other higher order moments are functions of time, and (c) the stochastic mechanism generating time series is nonstationary. In this study, we consider nonstationarity arising from first kind. As proposed by Mallya et al. [46], we consider a trend stationary process in which the mean trend is deterministic. Once the trend is estimated and removed from the data, the residual series is a stationary stochastic process. A trend stationarity process, yt , is expressed as yt f (t) + zt (6) where t represents time, zt denotes a zero-mean stationary process, and f (t) is a function of time representing trend of the process at time t. The trend can be determined either extrinsically by specifying a linear or nonlinear functional form for f (t) [e.g., regression models] or intrinsically by using the data without prespecifying a functional form (e.g., empirical mode decomposition [84]). Drought classification for nonstationary time series can be approached in one of the following two ways—(a) assuming that the “normal” conditions for a station are evolving, and hence the drought thresholds for different categories are changing with time (similar to SnsPI) and (b) assuming that the “normal” conditions for a station
- 52. An Analysis of Spatio-Temporal Changes … 41 are fixed (with respect to a reference period, and hence the thresholds for drought classification are fixed), but the frequency of droughts are changing with time (as in relative SPI). The proposed method classifies droughts for nonstationary time series using both the approaches. The steps of the proposed methods are as follows: (a) Identify trend in the time series, f (t) using either an extrinsic approach (regres- sion model) or an intrinsic approach (empirical mode decomposition). (b) Determine zt byremovingtrendfromthetimeseriesandfitasuitabledistribution to obtain the cumulative distribution function FZ (zt ) P(Z ≤ zt ), where Z is a random variable belonging to a family of stationary stochastic process (for example—normal, lognormal, generalized extreme value, etc.). (c) Determine the CDF of the rainfall time series at time t as FYt (yt ) P(Yt ≤ yt ) P(Z ≤ yt − f (t)) FZ (yt − f (t)). (7) Estimate drought thresholds at each time step using FYt (yt ) and the SPI drought definitions given in Table 1. Select a reference year (t0), and determine FYt0 and corresponding drought thresh- olds. Estimate drought class at time t based on these fixed thresholds for the reference year (t0). 3.3.2 Standardized Nonstationary Precipitation Index (SnsPI) The SnsPI method fits a nonstationary model to the precipitation data by linearly varying the scale parameter (st ) of the gamma distribution with time. Following the notations used in Sect. 3.2, the gamma distribution is represented as G(xt |ν, st ) with scale parameter st μt ν and E(xt ) μt b1+b2t, where b1 and b2 are constants. In this study, the parameters ν, b1, and b2 are estimated by the maximum likelihood method. 3.3.3 Relative SPI The relative SPI is defined with respect to a reference period in which precipitation time series is assumed to be stationary. For estimating relative SPI, the gamma distribution is first fitted to the reference period and to the period for which the temporal changes have to be analyzed. The two distributions are then compared to determine changes in the concerned period with respect to the reference period.
- 53. 42 G. Mallya et al. 4 Results and Discussion This section is divided into three subsections. Section 4.1 presents the results for drought classification using SPI and Gamma-MM drought indices. Section 4.2 describes the results for precipitation trend analysis. Section 4.3 presents the results of the proposed methodology for the drought classification of nonstationary precip- itation series (detrended-SPI) and compares them with the results of classical SPI, relative SPI, and SnsPI for synthetic and real-world precipitation series. 4.1 Drought Classification The drought indices described in Sect. 3 are applied to study seasonal (4-month time window) and water year (12-month time window) droughts in India. India has three seasons each spanning about 4 months: Winter (October to January), Summer (February to May), and Summer Monsoon (June to September). The water year in India extends from June to May of the following year. For example, 1999 water year starts on June 1, 1999 and ends on May 31, 2000. First, an annual time series of cumulative precipitation during any chosen season or water year is computed. Next, droughts are classified using SPI and Gamma-MM methods. The latter method is also referred to as probabilistic SPI. Both the methods assume that the cumulative precipitation time series is stationary, and consists of independent and identically distributed samples. In the following paragraphs, summer monsoon and water year droughts from the two methods are presented for a selected IMD grid over India. 4.1.1 Summer Monsoon Droughts The results are presented for IMD grid 251 located in northeast India and are among thehighestrainfallreceivingregionsoftheworld.Figure6showstheempiricalcumu- lative distribution function (CDF) obtained using Weibull plotting position formula [10] along with CDFs of fitted gamma distribution (fitted using maximum likelihood approach) and gamma mixture model (Gamma-MM) for summer monsoon precipi- tation (June–September). The CDF of Gamma-MM is closer to empirical CDF than the CDF of gamma distribution, particularly, for the smaller rainfall values [F(X) 0.25], which are critical for drought classification. The Gamma-MM owes its bet- ter fit to the large number of tuning parameters (3M − 1, where M is number of components in Gamma-MM) compared to two-parameter gamma distributions. Increasing the number of mixture components (M) in Gamma-MM method ensures that the model provides a better fit to the data. However, it may also result in overfitting. The Gamma-MM model addresses this problem using a Bayesian frame- work that avoids overfitting by marginalizing over the model parameters instead of makingpointestimates.Figure7showsthemixingratioofafive-componentGamma-
- 54. An Analysis of Spatio-Temporal Changes … 43 Fig. 6 Empirical CDF along with CDFs obtained by fitting gamma distribution (Gamma CDF) and gamma mixture model (Gamma-MM CDF) to the 4-month cumulative precipitation during summer monsoon (June to September) at IMD grid 251. The gray band shows 5 and 95 percentile of the Gamma-MM CDF and the green-dotted line shows width of its credible interval 1000 2000 3000 4000 5000 6000 7000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 X (Cumulative Precipitation in mm) F(X) Empirical CDF Gamma CDF Gamma-MM CDF 5-95 %tile Credible interval Fig. 7 Mixing ratios of the components of the Bayesian Gamma-MM. Two components are identified as significant for characterizing summer monsoon (June to September) droughts at IMD grid 251 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Components Mixing ratio (w) MM fitted to cumulative winter precipitation at this station. The model identifies that three of the five components have negligible contribution and are effectively pruned from the model. Thus, the Bayesian framework identifies optimal number of mixture components needed to fit the data. The Bayesian framework also allows quantification of model uncertainties and their propagation to model estimates. In the context of Gamma-MM, the posterior distribution of model parameters is estimated from which the CDF is obtained. Unlike maximum likelihood approach that yields a point estimate of CDF, the Bayesian approach treats CDF as a random variable and yields distribution of CDFs for a given value of precipitation. The gray shaded band in Fig. 6 represents 90% credible interval (5 and 95 percentile). The width of the credible interval is not constant but varies with the magnitude of precipitation. It has a maximum value of 0.145 near the
- 55. Exploring the Variety of Random Documents with Different Content
- 56. lamented the existence of much fanaticism in the United States; but he saw the evils of an establishment the more clearly, not the less, from being aware of the faults in the administration of religion at home. The most animated moment of our conversation was when I told him I was going to visit Mr. Madison on leaving Washington. He instantly sat upright in his chair, and with beaming eyes began to praise Mr. Madison. Madison received the mention of Marshall's name in just the same manner; yet these men were strongly opposed in politics, and their magnanimous appreciation of each other underwent no slight or brief trial. Judge Porter sometimes came, a hearty friend, and much like a fellow-countryman, though he was a senator of the United States, and had previously been, for fourteen years, Judge of the Supreme Court of Louisiana. He was Irish by birth. His father was vindictively executed, with cruel haste, under martial law, in the Irish rebellion; and the sons were sent by their noble-minded mother to America, where Alexander, the eldest, has thus raised himself into a station of high honour. Judge Porter's warmth, sincerity, generosity, knowledge, and wit are the pride of his constituents, and very ornamental to the Senate. What their charm is by the fireside may be imagined. Such are only a few among a multitude whose conversation filled up the few evenings we spent at home. Among the pleasantest visits we paid were dinners at the president's, at the houses of heads of departments, at the British legation, and at the Southern members' congressional mess. We highly enjoyed our dinings at the British legation, where we felt ourselves at home among our countrymen. Once, indeed, we were invited to help to do the honours as English ladies to the seven Judges of the Supreme Court, and seven great lawyers besides, when we had the merriest day that could well be. Mr. Webster fell chiefly to my share, and there is no merrier man than he; and Judge Story would enliven a dinner-table at Pekin. One laughable peculiarity at the British legation was the confusion of tongues among the servants, who
- 57. ask you to take fish, flesh, and fowl in Spanish, Italian, German, Dutch, Irish, or French. The foreign ambassadors are terribly plagued about servants. No American will wear livery, and there is no reason why any American should. But the British ambassador must have livery servants. He makes what compromise he can, allowing his people to appear without livery out of doors except on state occasions; but yet he is obliged to pick up his domestics from among foreigners who are in want of a subsistence for a short time, and are sure to go away as soon as they can find any employment in which the wearing a livery is not requisite. The woes of this state of things, however, were the portion of the host, not of his guests; and the hearty hospitality with which we were ever greeted by the minister and his attachés, combined with the attractions of the society they brought together, made our visits to them some of the pleasantest hours we passed in Washington. Slight incidents were perpetually showing, in an amusing way, the village-like character of some of the arrangements at Washington. I remember that some of our party went one day to dine at Mr. Secretary Cass's, and the rest of us at Mr. Secretary Woodburys'. The next morning a lady of the Cass party asked me whether we had candied oranges at the Woodburys'. No. Then, said she, they had candied oranges at the attorney-general's. How do you know? Oh, as we were on the way, I saw a dish carried; and as we had none at the Cass's, I knew they must either be for the Woodburys or the attorney-general. There were candied oranges at the attorney-general's. When we became intimate some time afterward with some Southern friends, with whom we now dined at their congressional mess, they gave us an amusing account of the preparations for our dinner. They boarded (from a really self-denying kindness) at a house where the arrangements were of a very inferior kind. Two sessions previous to our being there they had invited a large party of eminent persons to dinner, and had committed the ordering of the arrangements to a gentleman of their mess, advising him to
- 58. engage a French cook in order to ensure a good dinner. The gentleman engaged a Frenchman, concluding he must be a cook, which, however, he was not; and the dinner turned out so unfortunately, that the mess determined to ask no more dinner- company while they remained in that house. When we arrived, however, it was thought necessary to ask us to dinner. There was little hope that all would go rightly; and the two senators of the mess were laughingly requested, in case of any blunder, to talk nullification as fast as possible to us ladies. This was done so efficaciously, that, when dinner was over, I could not have told a single dish that was on the table, except that a ham stood before me, which we were too full of nullification to attack. Our hosts informed us, long afterward, that it was a bad dinner badly served; but it was no matter. At the president's I met a very large party, among whom there was more stiffness than I saw in any other society in America. It was not the fault of the president or his family, but of the way in which the company was unavoidably brought together. With the exception of my party, the name of everybody present began with J, K, or L; that is to say, it consisted of members of Congress, who are invited alphabetically, to ensure none being left out. This principle of selection is not, perhaps, the best for the promotion of ease and sociability; and well as I liked the day, I doubt whether many others could say they enjoyed it. When we went in the president was standing in the middle of the room to receive his guests. After speaking a few words with me, he gave me into the charge of Major Donelson, his secretary, who seated me, and brought up for introduction each guest as he passed from before the president. A congressional friend of mine (whose name began with a J) stationed himself behind my chair, and gave me an account of each gentleman who was introduced to me; where he came from, what his politics were, and how, if at all, he had distinguished himself. All this was highly amusing. At dinner the president was quite disposed for conversation. Indeed, he did nothing but talk. His health is poor, and his diet of the sparest. We
- 59. both talked freely of the governments of England and France; I, novice in American politics as I was, entirely forgetting that the great French question was pending, and that the president and the King of the French were then bandying very hard words. I was most struck and surprised with the president's complaints of the American Senate, in which there was at that time a small majority against the administration. He told me that I must not judge of the body by what I saw it then, and that after the 4th of March I should behold a Senate more worthy of the country. After the 4th of March there was, if I remember rightly, a majority of two in favour of the government. The ground of his complaint was, that the senators had sacrificed their dignity by disregarding the wishes of their constituents. The other side of the question is, that the dignity of the Senate is best consulted by its members following their own convictions, declining instructions for the term for which they are elected. It is a serious difficulty, originating in the very construction of the body, and not to be settled by dispute. The president offered me bonbons for a child belonging to our party at home, and told me how many children (of his nephew's and his adopted son's) he had about him, with a mildness and kindliness which contrasted well with his tone upon some public occasions. He did the honours of his house with gentleness and politeness to myself, and, as far as I saw, to every one else. About an hour after dinner he rose, and we led the way into the drawing- room, where the whole company, gentlemen as well as ladies, followed to take coffee; after which every one departed, some homeward, some to make evening calls, and others, among whom were ourselves, to a splendid ball at the other extremity of the city. General Jackson is extremely tall and thin, with a slight stoop, betokening more weakness than naturally belongs to his years. He has a profusion of stiff gray hair, which gives to his appearance whatever there is of formidable in it. His countenance bears commonly an expression of melancholy gravity; though, when
- 60. roused, the fire of passion flashes from his eyes, and his whole person looks then formidable enough. His mode of speech is slow and quiet, and his phraseology sufficiently betokens that his time has not been passed among books. When I was at Washington albums were the fashion and the plague of the day. I scarcely ever came home but I found an album on my table or requests for autographs; but some ladies went much further than petitioning a foreigner who might be supposed to have leisure. I have actually seen them stand at the door of the Senate Chamber, and send the doorkeeper with an album, and a request to write in it, to Mr. Webster and other eminent members. I have seen them do worse; stand at the door of the Supreme Court, and send in their albums to Chief-justice Marshall while he was on the bench hearing pleadings. The poor president was terribly persecuted; and to him it was a real nuisance, as he had no poetical resource but Watts's hymns. I have seen verses and stanzas of a most ominous purport from Watts, in the president's very conspicuous handwriting, standing in the midst of the crowquill compliments and translucent charades which are the staple of albums. Nothing was done to repress this atrocious impertinence of the ladies. I always declined writing more than name and date; but senators, judges, and statesmen submitted to write gallant nonsense at the request of any woman who would stoop to desire it. Colonel Johnson, now Vice-president of the United States, sat opposite to me at the president's dinner-table. This is the gentleman once believed to have killed Tecumseh, and to have written the Report on Sunday Mails, which has been the admiration of society ever since it appeared; but I believe Colonel Johnson is no longer supposed to be the author of either of these deeds. General Mason spoke of him to me at New-York with much friendship, and with strong hope of his becoming president. I heard the idea so ridiculed by members of the federal party afterward, that I concluded General Mason to be in the same case with hundreds more who believe their intimate friends sure of being president. But Colonel Johnson is actually vice-president,
- 61. and the hope seems reasonable; though the slavery question will probably be the point on which the next election will turn, which may again be to the disadvantage of the colonel. If he should become president, he will be as strange-looking a potentate as ever ruled. His countenance is wild, though with much cleverness in it; his hair wanders all abroad, and he wears no cravat. But there is no telling how he might look if dressed like other people. I was fortunate enough once to catch a glimpse of the invisible Amos Kendall, one of the most remarkable men in America. He is supposed to be the moving spring of the whole administration; the thinker, planner, and doer; but it is all in the dark. Documents are issued of an excellence which prevents their being attributed to persons who take the responsibility of them; a correspondence is kept up all over the country for which no one seems to be answerable; work is done, of goblin extent and with goblin speed, which makes men look about them with a superstitious wonder; and the invisible Amos Kendall has the credit of it all. President Jackson's Letters to his Cabinet are said to be Kendall's; the Report on Sunday Mails is attributed to Kendall; the letters sent from Washington to appear in remote country newspapers, whence they are collected and published in the Globe as demonstrations of public opinion, are pronounced to be written by Kendall. Every mysterious paragraph in opposition newspapers relates to Kendall; and it is some relief to the timid that his having now the office of postmaster-general affords opportunity for open attacks upon this twilight personage; who is proved, by the faults in the postoffice administration, not to be able to do quite everything well. But he is undoubtedly a great genius. He unites with his great talent for silence a splendid audacity. One proof of this I have given elsewhere, in the account of the bold stroke by which he obtained the sanction of the Senate to his appointment as postmaster-general. [11]
- 62. It is clear that he could not do the work he does (incredible enough in amount any way) if he went into society like other men. He did, however, one evening; I think it was at the attorney- general's. The moment I went in, intimations reached me from all quarters, amid nods and winks, Kendall is here: That is he. I saw at once that his plea for seclusion (bad health) is no false one. The extreme sallowness of his complexion, and hair of such perfect whiteness as is rarely seen in a man of middle age, testified to disease. His countenance does not help the superstitious to throw off their dread of him. He probably does not desire this superstition to melt away; for there is no calculating how much influence was given to Jackson's administration by the universal belief that there was a concealed eye and hand behind the machinery of government, by which everything could be foreseen, and the hardest deeds done. A member of Congress told me this night that he had watched through four sessions for a sight of Kendall, and had never obtained it till now. Kendall was leaning on a chair, with head bent down, and eye glancing up at a member of Congress with whom he was in earnest conversation, and in a few minutes he was gone. Neither Mr. Clay nor any of his family ever spoke a word to me of Kendall except in his public capacity; but I heard elsewhere and repeatedly the well-known story of the connexion of the two men early in Kendall's life. Tidings reached Mr. and Mrs. Clay one evening, many years ago, at their house in the neighbourhood of Lexington, Kentucky, that a young man, solitary and poor, lay ill of a fever in the noisy hotel in the town. Mrs. Clay went down in the carriage without delay, and brought the sufferer home to her house, where she nursed him with her own hands till he recovered. Mr. Clay was struck with the talents and knowledge of the young man (Kendall), and retained him as tutor to his sons, heaping benefits upon him with characteristic bounty. Thus far is notorious fact. As to the causes of their separation and enmity, I have not heard Kendall's side of the question, and therefore say nothing; but go on to the other notorious facts, that Amos Kendall
- 63. left Mr. Clay's political party some time after Adams had been, by Mr. Clay's influence, seated in the presidential chair, and went over to Jackson; since which time he has never ceased his persecutions of Mr. Clay through the newspapers. It was extensively believed, on Mr. Van Buren's accession, that Kendall would be dismissed from office altogether; and there was much speculation about how the administration would get on without him. But he appears to be still there. Whether he goes or stays, it will probably be soon apparent how much of the conduct of Jackson's government is attributable to Kendall's influence over the mind of the late president, as he is hardly likely to stand in the same relation to the present. I was more vividly impressed with the past and present state of Ireland while I was in America than ever I was at home. Besides being frequently questioned as to what was likely to be done for the relief of her suffering millions—suffering to a degree that it is inconceivable to Americans that freeborn whites should ever be—I met from time to time with refugee Irish gentry, still burning with the injuries they or their fathers sustained in the time of the rebellion. The subject first came up with Judge Porter; and I soon afterward saw, at a country-house where I was calling, the widow of Theobald Wolfe Tone. The poor lady is still full of feelings which amazed me by their bitterness and strength, but which have, indeed, nothing surprising in them to those who know the whole truth of the story of Ireland in those dreadful days. The descendants of the rebels cannot be comforted with tidings of anything to be done for their country. Naturally believing that nothing good can come out of England—nothing good for Ireland —they passionately ask that their country shall be left to govern herself. With tears and scornful laughter they beg that nothing may be done for her by hands that have ravaged her with gibbet, fire, and sword, but that she may be left to whatever hopefulness may yet be smouldering under the ashes of her despair. Such is the representation of Ireland to American minds. It may be imagined what a monument of idiotcy the forcible
- 64. maintenance of the Church of England in Ireland must appear to American statesmen. I do not understand this Lord John Russell of yours, said one of the most sagacious of them. Is he serious in supposing that he can allow a penny of the revenues, a plait of the lawn-sleeves of that Irish Church to be touched, and keep the whole from coming down, in Ireland first, and in England afterward? We fully agree in the difficulty of supposing Lord John Russell serious. The comparison of various, but, I believe, pretty extensive American opinions about the Church of England yields rather a curious result. No one dreams of the establishment being necessary or being designed for the maintenance of religion; it is seen by Chief-justice Marshall and a host of others to be an institution turned to political purposes. Mr. Van Buren, among many, considers that the church has supported the state for many years. Mr. Clay, and a multitude with him, anticipates the speedy fall of the establishment. The result yielded by all this is a persuasion not very favourable (to use the American phrase) to the permanence of our institutions. Among our casual visiters at Washington was a gentleman who little thought, as he sat by our fireside, what an adventure was awaiting him among the Virginia woods. If there could have been any anticipation of it, I should have taken more notice of him than I did; as it is, I have a very slight recollection of him. He came from Maine, and intended before his return to visit the springs of Virginia, which he did the next summer. It seems that he talked in the stages rashly, and somewhat in a bragging style—in a style, at least, which he was not prepared to support by a harder testimony —about abolitionism. He declared that abolitionism was not so dangerous as people thought; that he avowed it without any fear; that he had frequently attended abolition meetings in the North, and was none the worse for it in the slave states, c. He finished his visit at the Springs prosperously enough; but, on his return, when he and a companion were in the stage in the midst of the forest, they met at a crossroad—Judge Lynch; that is, a mob with hints of cowhide and tar and feathers. The mob stopped the
- 65. stage, and asked for the gentleman by name. It was useless to deny his name, but he denied everything else. He denied his being an abolitionist; he denied his having ever attended abolition meetings, and harangued against abolitionism from the door of the stage with so much effect, that the mob allowed the steps to be put up, and the vehicle to drive off, which it did at full speed. It was not long before the mob became again persuaded that this gentleman was a fit object of vengeance, and pursued him; but he was gone as fast as horses could carry him. He did not relax his speed even when out of danger, but fled all the way into Maine. It was not on the shrinking at the moment that one would animadvert so much as on the previous bragging. I have seen and felt enough of what peril from popular hatred is, in this martyr age of the United States, to find it easier to venerate those who can endure than to despise those who flinch from the ultimate trial of their principles; but every instance of the infliction of Lynch punishment should be a lesson to the sincerest and securest to profess no more than they are ready to perform. One of our mornings was devoted to an examination of the library and curiosities of the State Department, which we found extremely interesting. Our imaginations were whirled over the globe at an extraordinary rate. There were many volumes of original letters of Washington's and other revolutionary leaders bound up, and ordered to be printed, for security, lest these materials of history should be destroyed by fire or other accident. There were British parliamentary documents. There was a series of the Moniteur complete, wherein we found the black list of executions during the reign of terror growing longer every day; also the first mention of Napoleon; the tidings of his escape from Elba; the misty days immediately succeeding, when no telegraphic communication could be made; his arrival at Lyons, and the subsequent silence till the announcement became necessary that the king and princes had departed during the night, and that his majesty the emperor had arrived at his palace of the Tuileries at eight o'clock the next evening. Next we turned to Algerine (French) gazettes, publishing
- 66. that Mustaphas and such people were made colonels and adjutants. Then we lighted upon the journals of Arnold during the Revolutionary war, and read the postscript of his last letter previous to the accomplishment of his treason, in which he asks for hard cash, on pretence that the French had suffered so much by paper money that he was unwilling to offer them any more. Then we viewed the signatures of treaties, and decreed Metternich's to be the best; Don Pedro's the worst for flourish, and Napoleon's for illegibility. The extraordinary fact was then and there communicated to us that the Americans are fond of Miguel from their dislike of Pedro, but that they hope to get along very well with the Queen of Portugal. The treaties with oriental potentates are very magnificent, shining, and unintelligible to the eyes of novices. The presents from potentates to American ambassadors are laid up here; gold snuffboxes set in diamonds, and a glittering array of swords and cimeters. There was one fine Damascus blade, but it seemed too blunt to do any harm. Then we lost ourselves in a large collection of medals and coins—Roman gold coins, with fat old Vespasian and others—from which we were recalled to find ourselves in the extremely modern and democratic United States! It was a very interesting morning. We took advantage of a mild day to ascend to the skylight of the dome of the Capitol, in order to obtain a view of the surrounding country. The ascent was rather fatiguing, but perfectly safe. The residents at Washington declare the environs to be beautiful in all seasons but early winter, the meadows being gay with a profusion of wild flowers; even as early as February with several kinds of heart's-ease. It was a particularly cold season when I was there; but on the day of my departure, in the middle of February, the streets were one sheet of ice, and I remember we made a long slide from the steps of our boarding-house to those of the stage. But I believe that that winter was no rule for others. From the summit of the Capitol we saw plainly marked out the basin in which Washington stands, surrounded by hills except where the Potomac spreads its waters. The city was intended to occupy the
- 67. whole of this basin, and its seven theoretical avenues may be traced; but all except Pennsylvania Avenue are bare and forlorn. A few mean houses dotted about, the sheds of a navy-yard on one bank of the Potomac, and three or four villas on the other, are all the objects that relieve the eye in this space intended to be so busy and magnificent. The city is a grand mistake. Its only attraction is its being the seat of government, and it is thought that it will not long continue to be so. The far-western states begin to demand a more central seat for Congress, and the Cincinnati people are already speculating upon which of their hills or tablelands is to be the site of the new Capitol. Whenever this change takes place all will be over with Washington; thorns shall come up in her palaces, and the owl and the raven shall dwell in it, while her sister cities of the east will be still spreading as fast as hands can be found to build them. There was a funeral of a member of Congress on the 30th of January; the interment of the representative from South Carolina, whose death I mentioned in connexion with Mr. Calhoun. We were glad that we were at Washington at the time, as a congressional funeral is a remarkable spectacle. We went to the Capitol at about half an hour before noon, and found many ladies already seated in the gallery of the Hall of Representatives. I chanced to be placed at the precise point of the gallery where the sounds from every part of the house are concentred; so that I heard the whole service, while I was at such a distance as to command a view of the entire scene. In the chair were the President of the Senate and the Speaker of the Representatives. Below them sat the officiating clergyman; immediately opposite to whom were the president and the heads of departments on one side the coffin, and the judges of the Supreme Court and members of the Senate on the other. The representatives sat in rows behind, each with crape round the left arm; some in black; many in blue coats with bright buttons. Some of the fiercest political foes in the country; some who never meet on any other occasion—the president and the South Carolina senators, for instance—now sat knee to knee, necessarily looking
- 68. into each others' faces. With a coffin beside them, and such an event awaiting their exit, how out of place was hatred here! After prayers there was a sermon, in which warning of death was brought home to all, and particularly to the aged; and the vanity of all disturbances of human passion when in view of the grave was dwelt upon. There sat the gray-headed old president, at that time feeble, and looking scarcely able to go through this ceremonial. I saw him apparently listening to the discourse; I saw him rise when it was over, and follow the coffin in his turn, somewhat feebly; I saw him disappear in the doorway, and immediately descended with my party to the Rotundo, in order to behold the departure of the procession for the grave. At the bottom of the stairs a member of Congress met us, pale and trembling, with the news that the president had been twice fired at with a pistol by an assassin who had waylaid him in the portico, but that both pistols had missed fire. At this moment the assassin rushed into the Rotundo where we were standing, pursued and instantly surrounded by a crowd. I saw his hands and half-bare arms struggling above the heads of the crowd in resistance to being handcuffed. He was presently overpowered, conveyed to a carriage, and taken before a magistrate. The attack threw the old soldier into a tremendous passion. He fears nothing, but his temper is not equal to his courage. Instead of his putting the event calmly aside, and proceeding with the business of the hour, it was found necessary to put him into his carriage and take him home. We feared what the consequences would be. We had little doubt that the assassin Lawrence was mad; and as little that, before the day was out, we should hear the crime imputed to more than one political party or individual. And so it was. Before two hours were over, the name of almost every eminent politician was mixed up with that of the poor maniac who caused the uproar. The president's misconduct on the occasion was the most virulent and protracted. A deadly enmity had long subsisted between General
- 69. Jackson and Mr. Poindexter, a senator of the United States, which had been much aggravated since General Jackson's accession by some unwarrantable language which he had publicly used in relation to Mr. Poindexter's private affairs. There was a prevalent expectation of a duel as soon as the expiration of the president's term of office should enable his foe to send him a challenge. Under these circumstances the president thought proper to charge Mr. Poindexter with being the instigator of Lawrence's attempt. He did this in conversation so frequently and openly, that Mr. Poindexter wrote a letter, brief and manly, stating that he understood this charge was made against him, but that he would not believe it till it was confirmed by the president himself; his not replying to this letter being understood to be such a confirmation. The president showed this letter to visiters at the White House, and did not answer it. He went further; obtaining affidavits (tending to implicate Poindexter) from weak and vile persons whose evidence utterly failed; having personal interviews with these creatures, and openly showing a disposition to hunt his foe to destruction at all hazards. The issue was, that Lawrence was proved to have acted from sheer insanity; Poindexter made a sort of triumphal progress through the states, and an irretrievable stain was left upon President Jackson's name. Every one was anxiously anticipating the fierce meeting of these foes on the president's retirement from office, when Mr. Poindexter last year, in a fit either of somnambulism or of delirium from illness, walked out of a chamber window in the middle of the night, and was so much injured that he soon died. It so happened that we were engaged to a party at Mr. Poindexter's the very evening of this attack upon the president. There was so tremendous a thunder-storm that our host and hostess were disappointed of almost all their guests except ourselves, and we had difficulty in merely crossing the street, being obliged to have planks laid across the flood which gushed between the carriage and the steps of the door. The conversation
- 70. naturally turned on the event of the morning. I knew little of the quarrel which was now to be so dreadfully aggravated; but the more I afterward heard, the more I admired the moderation with which Mr. Poindexter spoke of his foe that night, and as often as I subsequently met him. I had intended to visit the president the day after the funeral; but I heard so much of his determination to consider the attack a political affair, and I had so little wish to hear it thus treated, against the better knowledge of all the world, that I stayed away as long as I could. Before I went I was positively assured of Lawrence's insanity by one of the physicians who were appointed to visit him. One of the poor creature's complaints was, that General Jackson deprived him of the British crown, to which he was heir. When I did go to the White House, I took the briefest possible notice to the president of the insane attempt of Lawrence; but the word roused his ire. He protested, in the presence of many strangers, that there was no insanity in the case. I was silent, of course. He protested that there was a plot, and that the man was a tool, and at length quoted the attorney- general as his authority. It was painful to hear a chief ruler publicly trying to persuade a foreigner that any of his constituents hated him to the death; and I took the liberty of changing the subject as soon as I could. The next evening I was at the attorney-general's, and I asked him how he could let himself be quoted as saying that Lawrence was not mad. He excused himself by saying that he meant general insanity. He believed Lawrence insane in one direction; that it was a sort of Ravaillac case. I besought him to impress the president with this view of the case as soon as might be. It would be amusing, if it were possible to furnish a complete set of the rumours, injurious (if they had not been too absurd) to all parties in turn, upon this single and very common act of a madman. One would have thought that no maniac had ever before attacked a chief magistrate. The act might so easily have remained
- 71. fruitless! but it was made to bear a full and poisonous crop of folly, wickedness, and wo. I feared on the instant how it would be, and felt that, though the president was safe, it was very bad news. When will it come to be thought possible for politicians to have faith in one another, though they may differ, and to be jealous for their rivals rather than for themselves?
- 72. THE CAPITOL. ... You have unto the support of a true and natural aristocracy the deepest root of a democracy that hath been planted. Wherefore there is nothing in art or nature better qualified for the result than this assembly.— Harrington's Oceana. The places of resort for the stranger in the Capitol are the Library, the Supreme Court, the Senate Chamber, and the Hall of Representatives. The former library of Congress was burnt by the British in their atrocious attack upon Washington in 1814. Jefferson then offered his, and it was purchased by the nation. It is perpetually increased by annual appropriations. We did not go to the library to read, but amused ourselves for many pleasant hours with the prints and with the fine medals which we found there. I was never tired of the cabinet of Napoleon medals; the most beautifully composed piece of history that I ever studied. There is a cup carved by Benvenuto Cellini, preserved among the curiosities of the Capitol, which might be studied for a week before all the mysteries of its design are apprehended. How it found its way to so remote a resting-place I do not remember. Judge Story was kind enough to send us notice when any cause was to be argued in the Supreme Court which it was probable we might be able to understand, and we passed a few mornings
- 73. there. The apartment is less fitted for its purposes than any other in the building, the court being badly lighted and ventilated. The windows are at the back of the judges, whose countenances are therefore indistinctly seen, and who sit in their own light. Visiters are usually placed behind the counsel and opposite the judges, or on seats on each side. I was kindly offered the reporter's chair, in a snug corner, under the judges, and facing the counsel; and there I was able to hear much of the pleadings and to see the remarkable countenances of the attorney-general, Clay, Webster, Porter, and others, in the fullest light that could be had in this dim chamber. At some moments this court presents a singular spectacle. I have watched the assemblage while the chief-justice was delivering a judgment; the three judges on either hand gazing at him more like learners than associates; Webster standing firm as a rock, his large, deep-set eyes wide awake, his lips compressed, and his whole countenance in that intent stillness which easily fixes the eye of the stranger; Clay leaning against the desk in an attitude whose grace contrasts strangely with the slovenly make of his dress, his snuffbox for the moment unopened in his hand, his small gray eye and placid half-smile conveying an expression of pleasure which redeems his face from its usual unaccountable commonness; the attorney-general, his fingers playing among his papers, his quick black eye, and thin tremulous lips for once fixed, his small face, pale with thought, contrasting remarkably with the other two; these men, absorbed in what they are listening to, thinking neither of themselves nor of each other, while they are watched by the groups of idlers and listeners around them; the newspaper corps, the dark Cherokee chiefs, the stragglers from the Far West, the gay ladies in their waving plumes, and the members of either house that have stepped in to listen; all these I have seen at one moment constitute one silent assemblage, while the mild voice of the aged chief-justice sounded through the court.
- 74. Every one is aware that the wigs and gowns of counsel are not to be seen in the United States. There was no knowing, when Webster sauntered in, threw himself down, and leaned back against the table, his dreamy eyes seeming to see nothing about him, whether he would by-and-by take up his hat and go away, or whether he would rouse himself suddenly, and stand up to address the judges. For the generality there was no knowing; and to us, who were forewarned, it was amusing to see how the court would fill after the entrance of Webster, and empty when he had gone back to the Senate Chamber. The chief interest to me in Webster's pleading, and also in his speaking in the Senate, was from seeing one so dreamy and nonchalant roused into strong excitement. It seemed like having a curtain lifted up through which it was impossible to pry; like hearing auto-biographical secrets. Webster is a lover of ease and pleasure, and has an air of the most unaffected indolence and careless self-sufficiency. It is something to see him moved with anxiety and the toil of intellectual conflict; to see his lips tremble, his nostrils expand, the perspiration start upon his brow; to hear his voice vary with emotion, and to watch the expression of laborious thought while he pauses, for minutes together, to consider his notes, and decide upon the arrangement of his argument. These are the moments when it becomes clear that this pleasure-loving man works for his honours and his gains. He seems to have the desire which other remarkable men have shown, to conceal the extent of his toils, and his wish has been favoured by some accidents; some sudden, unexpected call upon him for a display of knowledge and power which has electrified the beholders. But on such occasions he has been able to bring into use acquisitions and exercises intended for other occasions, on which they may or may not have been wanted. No one will suppose that this is said in disparagement of Mr. Webster. It is only saying that he owes to his own industry what he must otherwise owe to miracle. What his capacity for toil is was shown, in one instance among many, in an affair of great interest to his own state. On the 7th of
- 75. April, 1830, the town of Salem, Massachusetts, was thrown into a state of consternation by the announcement of a horrible murder. Mr. White, a respectable and wealthy citizen of Salem, about eighty years of age, was found murdered in his bed. The circumstances were such as to indicate that the murder was not for common purposes of plunder, and suspicions arose which made every citizen shudder at the idea of the community in which he lived containing the monsters who would perpetrate such a deed. A patrol of the citizens was proposed and organized, and none were more zealous in propositions and in patrolling than Joseph and John Knapp, relatives of the murdered man. The conduct of these young men on the occasion exposed them to dislike before any one breathed suspicion. Several acquaintances of the family paid visits of condolence before the funeral. One of these told me, still with a feeling of horror, how one of the Knapps pulled his sleeve, and asked, in an awkward whisper, whether he would go up stairs and see the old devil. The old gentleman's housekeeper had slept out of the house that particular night; a back window had been left unfastened, with a plank placed against it on the outside; and a will of the old gentleman's (happily a superseded one) was missing. Suspicious circumstances like these were found soon to have accumulated so as to justify the arrest of the two Knapps, and of two brothers of the name of Crowninshield. A lawyer was ready with testimony that Joseph Knapp, who had married a grand-niece of Mr. White, had obtained legal information, that if Mr. White died intestate, Knapp's mother- in-law would succeed to half the property. Joseph Knapp confessed the whole in prison, and Richard Crowninshield, doubtless the principal assassin, destroyed himself. The state prosecutors were in a great difficulty. Without the confession, the evidence was scarcely sufficient; and though Joseph Knapp was promised favour from government if he would repeat his evidence on the side of the prosecution in court, it was not safe, as the event proved, to rely upon this in a case otherwise doubtful. The attorney and solicitor-general of the state were both aged and feeble men; and, as the day of trial drew on, it became more and
- 76. more doubtful whether they would be equal to the occasion, and whether these ruffians, well understood to be the murderers, would not be let loose upon society again, from bad management of the prosecution. The prosecuting officers of the government were prevailed upon, within three days of the trial, to send to seek out Mr. Webster and request his assistance. A citizen of Salem, a friend of mine, was deputed to carry the request. He went to Boston: Mr. Webster was not there, but at his farm by the seashore. Thither, in tremendous weather, my friend followed him. Mr. Webster was playing checkers with his boy. The old farmer sat by the fire, his wife and two young women were sewing and knitting coarse stockings; one of these last, however, being no farmer's daughter, but Mr. Webster's bride, for this was shortly after his second marriage. My friend was first dried and refreshed, and then lost no time in mentioning business. Mr. Webster writhed at the word, saying that he came down hither to get out of hearing of it. He next declared that his undertaking anything more was entirely out of the question, and pointed, in evidence, to his swollen bag of briefs lying in a corner. However, upon a little further explanation and meditation, he agreed to the request with the same good grace with which he afterward went through with his task. He made himself master of all that my friend could communicate, and before daybreak was off through the woods, in the unabated storm, no doubt meditating his speech by the way. He needed all the assistance that could be given him, of course; and my friend constituted himself Mr. Webster's fetcher and carrier of facts for these two days. He says he was never under orders before since his childish days; but in this emergency he was a willing servant, obeying such laconic instructions as Go there; Learn this and that; Now go away; and so forth. At the appointed hour Mr. Webster was completely ready. His argument is thought one of the finest, in every respect, that he has produced. I read it before I knew anything of the circumstances which I have related; and I was made acquainted
- 77. with them in consequence of my inquiry how a man could be hanged on evidence so apparently insufficient as that adduced by the prosecution. Mr. Webster had made all that could be made of it; his argument was ingenious and close, and imbued with moral beauty; but the fact was, as I was assured, the prisoners were convicted on the ground of the confession of the criminal more than on the evidence adduced by the prosecutors; though the confession could not, after all, be made open use of. The prisoners had such an opinion of the weakness of the case, that Joseph, who had been offered favour by government, refused to testify, and the pledge of the government was withdrawn. Both the Knapps were hanged. The clearness with which, in this case, a multitude of minute facts is arranged, and the ingenuity with which a long chain of circumstantial evidence is drawn out, can be understood only through a reading of the entire argument. Even these are less remarkable than the sympathy by which the pleader seems to have possessed himself of the emotions, the peculiar moral experience, of the quiet, good people of Salem, when thunderstruck with this event. While shut up at his task, Mr. Webster found means to see into the hearts which were throbbing in all the homes about him. One thing more, said he to my friend, who was taking his leave of him on the eve of the trial. Do you know of anything remarkable about any of the jury? My friend had nothing to say, unless it was that the foreman was a man of a remarkably tender conscience. To this we doubtless owe the concluding passage of the argument, delivered, as I was told, in a voice and manner less solemn than easy and tranquil. Gentlemen—Your whole concern should be to do your duty, and leave consequences to take care of themselves. You will receive the law from the court. Your verdict, it is true, may endanger the prisoner's life; but, then, it is to save other lives. If the prisoner's guilt has been shown and proved beyond all reasonable doubt, you will convict him. If such reasonable doubts still remain, you
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