TIME INTEGRATION OF EVAPOTRANSPIRATION USING A TWO SOURCE SURFACE ENERGY BALANCE MODEL USING NARR REANALYSIS WEATHER DATA AND SATELLITE BASED METRIC DATA :: Electronic Theses and Dissertations

TIME INTEGRATION OF EVAPOTRANSPIRATION USING A TWO SOURCE
SURFACE ENERGY BALANCE MODEL USING NARR REANALYSIS
WEATHER DATA AND SATELLITE BASED METRIC DATA
A Dissertation
Presented in Partial Fulfillment of the Requirements for the
Degree of Doctorate of Philosophy
with a
Major in Civil Engineering
in the
College of Graduate Studies
University of Idaho
by
Ramesh Dhungel
May 2014
Major Professor: Richard G. Allen, Ph.D., P.E.

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Authorization to Submit Dissertation
This dissertation of Ramesh Dhungel, submitted for the degree of Doctorate of Philosophy
with a Major in Civil Engineering and titled “Time integration of evapotranspiration using a
two source surface energy balance model using NARR reanalysis weather data and satellite
based METRIC data” has been reviewed in final form. Permission, as indicated by the
signatures and dates given below, is now granted to submit final copies to the College of
Graduate Studies for approval.
Major Professor: Date:
Richard G. Allen, Ph.D.
Committee
Members: Date:
Fritz R. Fiedler, Ph.D.
Date:
Karen Humes, Ph.D.
Date:
Ricardo Trezza, Ph.D.
Department
Administrator: Date:
Richard Nielsen, Ph.D.
Discipline’s
College Dean: Date:
Larry Stauffer, Ph.D.
Final Approval and Acceptance
Dean of the College
of Graduate Studies: Date:
Jie Chen, Ph.D.

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Abstract
A backward averaged two source accelerated numerical solution (BATANS) of surface
energy fluxes was developed that utilizes satellite-based Mapping Evapotranspiration at High
Resolution using Internalized Calibration (METRIC) data and NARR reanalysis weather data.
BATANS simulates surface energy fluxes and surface temperature for surface energy balance
when no thermal based surface temperature is available. BATANS was able to partition both
satellite based Metric ET at the time of satellite overpass and time-integrate ET between the
satellite overpass dates. In addition, BATANS was able to time integrate ET every three hours
with the partitioned surface energy fluxes. Partitioned ET on the satellite overpass day was
used to estimate soil surface moisture and root zone moisture based on the soil and canopy
resistances. Estimated soil moisture at the start of the satellite overpass was used in soil water
balance while time integrating ET. A Jarvis type canopy resistance model was utilized by
BATANS to estimate the canopy transpiration. An irrigation module was developed to
simulate irrigation in agricultural lands when soil moisture falls below a set threshold. The
time integrated ET was calibrated and validated using METRIC ET at the start of simulation
period and at the next satellite overpass date. Any difference between the simulated and
METRIC ET at the end of the simulation period was adjusted linearly assuming that error
grows at the same rate over the time. Result showed good agreement between simulated and
Metric derived surface energy fluxes and surface temperature in most of the agricultural
lands. However, the agreement in the surface temperature was not as good in desert and dry
areas. Daily time integrated ET from BATANS between the satellite overpass was also
compared to reference ET from AgriMet weather station which showed a reasonable trend.
The simple FAO-56 soil water balance model was compared to the advance numerical

iv
solution based Hydrus-1D model to validate the soil water balance between the satellite
overpasses. Results showed that both models estimated similar soil water balance.

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Acknowledgements
I would like to acknowledge my advisor Dr. Richard G. Allen for his generous support
and providing me this opportunity. This work would not have been possible without his help
and guidance. I want to acknowledge Dr. Ricardo Trezza and Mr. Clarence Robison for their
valuable suggestions. I would like to thank Dr. Fritz Fiedler and Dr. Karen Humes for
continuous support and critical review of this thesis. I want to thank Dr. Timothy Link, Mr.
Jeremy Greth, Mr. John Stewart, Mr. Carlos Kelly and Mr. Wesley Kelly for their
suggestions. I am thankful to Water Resources Department at Kimbelry R & E center and
Department of Civil Engineering, University of Idaho. I am also thankful to ARS, Kimberly
for providing me working space.
I am grateful to my entire family members, especially my mother Geeta and my late
father Chandra Raj for their support and inspiration, sister Rama and her husband Kailash,
Rachana and her husband Bikash, and my brother Ranjan and his wife Sweta for their help.
Last but not the least; I am thankful to my wife Bibha and my loving son Ayush for
supporting me in the entire study period.

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Dedication
This dissertation is dedicated to my parents, Geeta and the late Chandra Raj

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Table of Contents
Authorization to Submit Dissertation......................................................................................... ii
Abstract .....................................................................................................................................iii
Acknowledgements.................................................................................................................... v
Dedication ................................................................................................................................. vi
Table of Contents..................................................................................................................... vii
List of Figures ............................................................................................................................ x
List of Tables............................................................................................................................ xv
Chapter 1: Parameterization of Moisture at the Soil Surface and Root Zone with a Two
Source Surface Energy Balance Model using NARR Reanalysis Weather and Satellite
Based METRIC Data Set at a Satellite Overpass Time......................................................... 1
Abstract .............................................................................................................................. 1
1.0 Overview...................................................................................................................... 2
2.0 Methodology................................................................................................................ 5
2.1 Model Description........................................................................................................ 5
2.2 Convergence of Surface Energy Balance................................................................... 10
2.3 Model Implementation ............................................................................................... 12
3.0 Application................................................................................................................. 19
3.1 Data Requirements..................................................................................................... 19
3.2 Study Area.................................................................................................................. 20
4.0 Results and Discussions............................................................................................. 21
4.1 Combined Result for 05/17/2008 satellite overpass .................................................. 23
4.1.1 Sensible Heat Flux (H)............................................................................................ 23
4.1.2 Ground Heat Flux (G)............................................................................................. 26
4.1.3 Bulk Surface Temperature (Tb) ............................................................................... 28
4.2 Partitioned Surface Energy Balance Fluxes for 05/17/2008..................................... 32
4.2.1 Soil Surface Evaporation (Ess) and Canopy Transpiration (T)............................... 33
4.2.2 Soil Surface Resistance and Canopy Resistance (rss and rsc).................................. 33
4.2.3 Soil Surface and Root Zone Soil Moisture (θsur and θroot)....................................... 34
5.0 Conclusions................................................................................................................ 36
References........................................................................................................................ 38

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Chapter 2: Time Integration of Evapotranspiration Using a Two Source Surface Energy
Balance Model Using NARR Reanalysis Weather Data and Satellite Based Metric Data . 45
Abstract ............................................................................................................................ 45
1.0 Overview.................................................................................................................... 46
2.0 Methodology .............................................................................................................. 48
2.1 Surface Energy Balance Model.................................................................................. 48
2.2 Soil Water Balance Sub-model .................................................................................. 54
2.2.1 Soil Water Balance of Soil Surface Layer: Surface to dsur (100 mm) layer............ 57
2.2.2 Soil Water Balance of Root Zone: Surface to droot (1-2 m) layer............................ 58
2.3 Irrigation Sub-model and Components...................................................................... 59
2.4 Soil Surface Resistance (rss) and Canopy Resistance (rsc) Computation Procedure . 62
3.0 Application................................................................................................................. 63
3.1 Data Requirements and Study Area........................................................................... 63
4.0 Results and Discussions............................................................................................. 67
4.1 Satellite Overpass Time ............................................................................................. 67
4.1.1 Extrapolation Model ............................................................................................... 72
4.1.2 Inversion Model ...................................................................................................... 76
4.2 Three Hourly Results (Extrapolation Model) ............................................................ 79
4.2.1 Irrigated Agricultural Pixel A1 (Coordinate: 2600407, 1328596)......................... 79
4.2.2 Desert Pixel D1 (Coordinates: 2601776, 1325307) ............................................... 83
4.2.3 Irrigated Agricultural Pixel A2 (Coordinate: 2612319, 1332522)......................... 88
4.2.4 Degree of Agreement and Distribution of Error..................................................... 90
4.3 Correction of Hourly and Daily Evapotranspiration (ET) ........................................ 92
5.0 Conclusions................................................................................................................ 97
References........................................................................................................................ 99
Chapter 3: Comparisons between the FAO-56 Soil Water Evaporation Model and
HYDRUS-1D Evaporation Model over a Range of Soil Types ........................................ 104
Abstract .......................................................................................................................... 104
1.0 Overview.................................................................................................................. 105
1.1 FAO-56 with Skin Evaporation Enhancement......................................................... 106
1.2 Hydrus-1D Model..................................................................................................... 111

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1.2.1 Initial and Boundary Conditions of Hydrus-1D ................................................... 112
2.0 Materials and Methods............................................................................................. 114
2.1 Hydrus-1D and FAO-56 Model Setup...................................................................... 114
3.0 Results and Discussions........................................................................................... 116
3.1 Comparison of FAO-56 Original, FAO-56 Enhanced, and Hydrus-1D Models to
Lysimeter........................................................................................................................ 116
3.2 Comparison of Results between FAO-56 and Hydrus-1D Models with Daily Data for
Theoretical Soils ............................................................................................................ 123
3.2.1 FAO-56 vs. Hydrus-1D model with Initial Pressure Head of -3 m....................... 124
3.2.2 FAO-56 vs. Hydrus-1D Model with Initial Pressure Head of -0.5 m and -10 m.. 129
3.3 Comparison of Results between FAO-56 and Hydrus-1D Model with Hourly
Timesteps........................................................................................................................ 130
3.4 Sensitivity Analyses on REW and Soil Type............................................................. 132
3.4.1 Daily Calculation Timesteps................................................................................. 132
3.4.2 Hourly Calculation Timesteps .............................................................................. 134
4.0 Conclusions.............................................................................................................. 136
References...................................................................................................................... 138
Appendix-A: Parameters and Variables in BATANS........................................................ 140
Parameters and Variables............................................................................................... 141
References.......................................................................................................................... 150

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List of Figures
Fig. 1.1. Two source model for sensible heat flux (H) with major components of surface
energy balance and resistances................................................................................................... 8
Fig. 1.2. Two source model for latent heat flux (LE) with major components of surface energy
balance and resistances .............................................................................................................. 9
Fig. 1.3. Flowchart of BATANS convergence at satellite overpass time using NARR
reanalysis weather data and satellite based METRIC data ...................................................... 11
Fig. 1.4. Flowchart of computation soil moisture at root zone (θroot) from Jarvis-type model 12
Fig. 1.5. F4 function for Jarvis-type model developed from available water fraction (AWF)
where θwp = 0.12 m3
/m3
and θfc = 0.36 m3
/m3
......................................................................... 17
Fig. 1.6. Relationship between soil surface resistance (rss) and soil surface moisture (θsur) for
different soil types.................................................................................................................... 19
Fig. 1.7. Study area near American Falls, ID overlaying NARR pixel and a Landsat path 39
image for May 17, 2008........................................................................................................... 21
Fig. 1.8. METRIC generated instantaneous ET (mm/hr), NDVI, and fc on 05/17/2008 in
southern Idaho, near American Falls ....................................................................................... 23
Fig. 1.9. Comparison of METRIC to simulated bulk sensible heat flux (W/m2
) on 05/17/2008
in southern Idaho, near American Falls ................................................................................... 24
Fig. 1.10. Scatter plots of METRIC and simulated sensible heat flux (W/m2
) for URA and
BLA AOIs on 05/17/2008 and 06/18/2008 in southern Idaho, near American Falls .............. 26
Fig. 1.11. Comparison of METRIC to simulated ground heat flux (W/m2
) on 05/17/2008 in
southern Idaho, near American Falls ....................................................................................... 27
Fig. 1.12. Comparison of METRIC to simulated bulk surface temperature (K) on 05/17/2008
in southern Idaho, near American Falls ................................................................................... 29
Fig. 1.13. Scatter plot of METRIC and simulated bulk surface temperature (Tb) for URA and
BLA AOIs on 05/17/2008 and 06/18/2008 in southern Idaho, near American Falls .............. 30
Fig. 1.14. Estimated soil evaporation (mm/hr) and canopy transpiration (mm/hr) after
adjusting for fc on 05/17/2008 in southern Idaho, near American Falls .................................. 33
Fig. 1.15. Estimated soil surface resistance (s/m) from Step 2 and canopy resistance (s/m)
from Step 1 on 05/17/2008 in southern Idaho, near American Falls ....................................... 34
Fig. 1.16. Estimated soil moisture at root zone (m3
/m3
) from Step 1 and soil surface moisture

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(m3
/m3
) from Step 2 on 05/17/2008 in southern Idaho, near American Falls.......................... 36
energy balance and resistances................................................................................................. 51
Fig. 2.2. Flowchart of BATANS convergence using NARR reanalysis weather data and
satellite based METRIC data ................................................................................................... 53
Fig. 2.3. An illustration of the partitioning of ET at satellite overpass dates and extrapolation
of ET between satellite overpass dates..................................................................................... 54
Fig. 2.4. A representative sketch of soil surface and root zone control volume for soil water
balance...................................................................................................................................... 56
Fig. 2.5. A representative sketch of different soil moistures and parameters used in irrigation
sub-model................................................................................................................................. 60
Fig. 2.6. NARR reanalysis data in three hour time intervals during the simulation period from
05/17/2008 to 06/18/2008 ........................................................................................................ 65
Fig. 2.7. Study area near American Falls, ID overlaying NARR reanalysis pixel and a Landsat
path 39 image for May, 17, 2008............................................................................................. 66
Fig. 2.8. Visual comparison of METRIC ET at satellite overpass time to simulated ET three
hours after satellite overpass and NDVI at satellite overpass on 05/17/2008 in southern Idaho,
near American Falls ................................................................................................................. 68
Fig. 2.9. Comparison of simulated ET to METRIC ET and NDVI at following satellite
overpass date i.e. 06/18/2008 in southern Idaho, near American Falls.................................... 70
Fig. 2.10. Comparisons between simulated ET to METRIC ET and NDVI inside upper right
agricultural (URA) AOI at next satellite overpass date i.e. 06/18/2008 in southern Idaho, near
American Falls ......................................................................................................................... 72
Fig. 2.11. Scatter plot between simulated and METRIC ET at next satellite overpass date for
bottom left agricultural (BLA) AOI on 06/18/2008 in southern Idaho, near American Falls . 74
Fig. 2.12. Scatter plot of METRIC ET vs. METRIC NDVI and simulated ET vs. METRIC
NDVI for bottom left agricultural (BLA) AOI on 06/18/2008 in southern Idaho, near
American Falls ......................................................................................................................... 75
Fig. 2.13. Simulation results of daily ET for agricultural pixels A1 (ET_A1), A2 (ET_A2) and
desert pixel D1 (ET_D1) compared to reference ET (ETr) from Aberdeen AgriMet station
between satellite overpass dates............................................................................................... 76

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Fig. 2.14. Simulation results of combined ET (ET_simulated), soil evaporation (Ess) , canopy
transpiration (T), and precipitation (P) for an irrigated agricultural pixel A1 (Land use 82,
NDVI: 0.71 to 0.83 and fc: 0.86 to 1) from 05/17/2008 to 06/18/2008 ................................... 80
Fig. 2.15. Simulation results of bulk surface temperature (Tb) and air temperature (Ta) from
NARR reanalysis for an irrigated agricultural pixel A1 (Land use 82, NDVI: 0.71 to 0.83 and
fc: 0.86 to 1) from 05/17/2008 to 06/18/2008........................................................................... 81
Fig. 2.16. Cumulative results of precipitation (Cum_P), ET (Cum_ET), irrigation (Cum_Irri),
and reference ET (Cum_ETr) for an irrigated agricultural pixel A1 (Land use: 82, NDVI: 0.71
to 0.83 and fc: 0.86 to 1) from 05/17/2008 to 06/18/2008........................................................ 81
Fig. 2.17. Simulation results of soil moisture (θsur) at surface and root zone (θroot) for an
agricultural pixel A1 (Land use 82, NDVI: 0.71 to 0.83 and fc: 0.86 to 1) from 05/17/2008 to
06/18/2008 ............................................................................................................................... 82
Fig. 2.18. Simulation results of soil surface resistance (rss) and canopy resistance (rsc) for an
irrigated agricultural pixel A1 (Land use 82, NDVI: 0.71 to 0.83 and fc: 0.86 to 1) from
05/17/2008 to 06/18/2008 ........................................................................................................ 83
Fig. 2.19. Simulation results of combined ET (ET_simulated), soil evaporation (Ess) and
canopy transpiration (T) for a desert pixel D1 (Land use 52, NDVI = 0.2 to 0.17) from
05/17/2008 to 06/18/2008 ........................................................................................................ 84
Fig. 2.20. Simulation results of bulk surface temperature (Tb) and air temperature from NARR
reanalysis (Ta) for desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17) from 05/17/2008 to
06/18/2008 ............................................................................................................................... 85
Fig. 2.21. Cumulative results of precipitation (Cum_P), ET (Cum_ET) and reference ET
(Cum_ETr) for a desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17) from 05/17/2008 to
06/18/2008 ............................................................................................................................... 86
Fig. 2.22. Simulation results for soil moisture at the surface (θsur) and root zone (θroot) for a
desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17) from 05/17/2008 to 06/18/2008 ............... 87
Fig. 2.23. Simulation results of soil surface resistance (rss) and canopy resistance (rsc) for a
desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17) from 05/17/2008 to 06/18/2008 ............... 88
canopy transpiration (T) for an irrigated agricultural pixel A2 (Land use 82, NDVI : 0.12 to
0.32 and fc : 0.05 to 0.27) from 05/17/2008 to 06/18/2008...................................................... 89

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fc: 0.05 to 0.27) from 05/17/2008 to 06/18/2008...................................................................... 89
and reference ET (Cum_ETr) for an irrigated agricultural pixel A2 (Land use 82, NDVI: 0.12
to 0.32 and fc: 0.05 to 0.27) from 05/17/2008 to 06/18/2008................................................... 90
Fig. 2.27. Error distribution using triangular interpolation method......................................... 91
Fig. 2.28. Error map (mm/hr) at the end of simulation period i.e. 06/18/2008 approximately at
11 am in southern Idaho, near American Falls ........................................................................ 92
Fig. 2.29. Simulation results of combined ET before correction (ET_sim) and after correction
(ET_cor) for an irrigated agricultural pixel A1 (Land use 82, NDVI: 0.71 to 0.83 and fc: 0.86
to 1) from 05/17/2008 to 06/18/2008....................................................................................... 93
(ET_cor) for a desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17, fc: 0.28) from 05/17/2008 to
06/18/2008 ............................................................................................................................... 94
to 0.27) from 05/17/2008 to 06/18/2008.................................................................................. 95
Fig. 2.32. Simulation results of daily ETrF before correction (ETrF_sim) and after correction
(ETrF_cor) and daily METRIC ETrF (ETrF_METRIC) for an irrigated agricultural pixel A1
(Land use 82, NDVI: 0.71 to 0.83 and fc: 0.86 to 1) from 05/17/2008 to 06/18/2008 ............ 96
(Land use 82, NDVI: 0.12 to 0.32 and fc: 0.05 to 0.27) from 05/17/2008 to 06/18/2008 ....... 97
Fig. 3.1. Soil water evaporation from Stage 1 and Stage 2 in the FAO-56 model (Allen et al.,
1998) ...................................................................................................................................... 108
Fig. 3.2. Daily average Ker based on the alfalfa reference from (a) the original FAO-56 model
and (b) the enhanced (skin evaporation) FAO-56 and from the Hydrus-1D for silt loam soil
and sandy clay loam (hini = -3 m and -10 m respectively) over a one-year period at Kimberly,
Idaho with daily calculation timesteps................................................................................... 119
Fig. 3.3. Daily average evaporation from (a) the original FAO-56 model and (b) the enhanced

xiv
(skin-evaporation) FAO-56 model and from the Hydrus-1D for silt loam soil and sandy clay
loam (hini = -3 m and -10 m respectively) over a one-year period at Kimberly, Idaho with
daily input data....................................................................................................................... 120
Fig. 3.4. Scatter plot of daily evaporation from (a) the original FAO-56 model and (b)
enhanced (skin-evaporation) FAO-56 model compared against lysimeter measurements for a
bare silt loam soil near Kimberly, Idaho with daily data during August and September 1977
................................................................................................................................................ 121
Fig. 3.5. Scatter plot of daily evaporation from Hydrus-1D for (a) silt loam soil (b) sandy
clay loam soil for hini = -3 m vs. lysimeter measurements near Kimberly, Idaho with daily
input data during August........................................................................................................ 122
Fig. 3.6. Daily average Ker based on the alfalfa reference from (a) the original FAO-56 model
and (b) the enhanced (skin evaporation) FAO-56 and from the Hydrus-1D for silt loam and
sandy clay loam textures using hini = -3 m over a one-year period at Kimberly, Idaho with
daily precipitation and weather data inputs............................................................................ 126
Fig. 3.7. Daily evaporation based on the alfalfa reference from (a) the original FAO-56
model and (b) the enhanced (skin evaporation) FAO-56 and from the Hydrus-1D for silt loam
and sandy clay loam textures using hini = -3 m over a one-year period at Kimberly, Idaho with
daily precipitation and weather data inputs............................................................................ 127
Fig. 3.8. Commutative evaporation based on the alfalfa reference from the FAO-56 model
with the skin evaporation enhancement and from the Hydrus-1D for silt loam and sandy clay
loam soil (hini = - 3 m) over a one-year period at Kimberly, Idaho with daily calculation
timesteps................................................................................................................................. 128
enhanced (skin-evaporation) FAO-56 model vs. daily evaporation estimated by the Hydrus-
1D using an hini = -3 m for a bare silt loam soil over a one-year period at Kimberly, Idaho
with a daily precipitation and weather data inputs................................................................. 129
Fig. 3.10. Scatter plot of daily evaporation from the enhanced FAO-56 model with hourly
calculation timesteps summed daily and daily calculation timesteps for hini = - 3 m for a bare
silt loam soil over a one-year period at Kimberly, Idaho....................................................... 136

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List of Tables
Table 1.1. Fluxes, parameters, variables, and boundary conditions used in BATANS ............ 7
Table 1.2. Data from NARR and METRIC at satellite overpass date 05/17/2008 and
06/18/2008 ............................................................................................................................... 20
Table 1.3. Surface energy fluxes for different land use classes and conditions on 05/17/2008
.................................................................................................................................................. 25
Table 1.4. Statistics of combined surface energy balance fluxes on 05/17/2008.................... 32
Table 2.1. Fluxes, parameters, variables, and boundary condition used in BATANS............ 49
Table 2.2. Data from NARR reanalysis for the study area...................................................... 67
Table 2.3. Statistics of combined surface energy balance fluxes from extrapolation and
inversion model compared with METRIC estimations............................................................ 78
Table 3.1. Standard soil hydraulic parameters for Hydrus-1D simulations .......................... 115
Table 3.2. Standard input data for a silt loam soil for the FAO-56 model............................ 116
Table 3.3. Summary of comparisons between the FAO-56 models, Hydrus-1D and lysimeter
for daily precipitation and reference evaporation inputs for the Kimberly, Idaho data......... 123
Table 3.4. Summary of comparisons between the FAO-56 models and Hydrus-1D model for
daily precipitation and weather data from Kimberly Idaho for year 2002 and a silt loam soil
texture where regressions are expressed against Hydrus-1D for three initial pressure heads 130
hourly precipitation and weather data and hourly calculation time steps for a silt loam soil
texture at Kimberly, Idaho during year 2002......................................................................... 132
Table 3.6. Summary of sensitivity analyses on impacts of soil texture in Hydrus-1D
simulations with comparisons against the enhanced FAO-56 model using daily weather and
precipitation inputs................................................................................................................. 134
Table 3.7. Summary of sensitivity analyses on reducing REW in the enhanced FAO-56 model
for hourly simulations for a silt loam soil texture vs the Hydrus-1D model for weather data
from Kimberly, ID during 2002............................................................................................. 135

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Chapter 1: Parameterization of Moisture at the Soil Surface and Root Zone with a Two
Source Surface Energy Balance Model using NARR Reanalysis Weather and Satellite
Based METRIC Data Set at a Satellite Overpass Time
By
Ramesh Dhungel and Richard G. Allen
Abstract
A backward averaged two source accelerated numerical solution (BATANS) of
surface energy balance model was developed to partition evaporation and transpiration
coupled with the soil surface and root zone soil moisture for two 2008 satellite overpass dates
in southern Idaho. The model was applied to estimate soil surface and canopy temperatures
for nearly 500,000 pixels using available climatic NARR reanalysis data and METRIC ET
images produced with Landsat. An iterative procedure was used for convergence of the
surface energy balance flux with an aerodynamic equation for latent and sensible heat flux.
An innovative inversion technique was developed to compute soil moisture at the surface and
root zone. This technique utilizes soil surface and canopy resistances that were derived by
inverting the two-source model. The model is able to predict the expected small canopy
resistances (60-120 s/m) in well-watered, fully-transpiring agricultural fields. The predicted
soil moisture was about 0.18-0.22 m3
/m3
in the root zone for agricultural lands and sagebrush
deserts for the May 17, 2008 overpass. This value reflects that the sagebrush desert and
grassland were not fully dry on that date. The sensible heat flux between the simulated and
METRIC model had R2
values of about 0.76-0.95 across the 30,000 pixels for the AOIs
compared. The combined surface temperature (Tb) had R2
values of about 0.69-0.89 for
different AOIs. The maximum RMSE values for sensible heat flux and temperature were

2
about 71 W/m2
and 13 K respectively across the 30,000 pixels. The large temperature
difference between the surface of the sagebrush desert and the air above resulted in a sharp
temperature profile. This fact, coupled with difficulties in the parameterization of the Jarvis-
type (Jarvis, 1976) model for sagebrush desert, caused the simulated combined temperatures
for sagebrush desert to be about 10 K lower than the thermal-band-based surface temperature
from Landsat.
1.0 Overview
SEBAL (Bastiaanssen et al., 1998), SEBS (Su, 2002), ALEXI (Norman et al., 2003)
and METRIC (Allen et al., 2007) are some of the most widely used evapotranspiration (ET)
models that utilize remote sensing. These models use the surface energy balance method to
compute ET from satellite images taken on overpass dates. Surface temperature is the most
important parameter used to complete the surface energy balance. SEBAL and METRIC use
the surface temperature derived from the thermal band of Landsat images to compute ET
using the surface energy balance method. The traditional SEBAL and METRIC models are
single source model. They compute ET from a parameter that includes the bulk contribution
of soil and vegetation parameter without partitioning surface energy fluxes. Along with
satellite images, SEBAL and METRIC need different roughness parameters and vegetation
indices to compute combined ET. These models are able to compute relatively accurate ET
values using satellite images from satellite overpass dates without quantifying other complex
hydrological parameters, such as soil moisture. But when satellite images are not available,
these methods face difficulties.
The main objective of this paper is to estimate soil surface moisture (θsur) and root
zone moisture (θroot) at satellite overpass dates. These soil moisture values are needed to

3
compute the soil-water balance while extrapolating ET between satellite overpass dates
(Dhungel and Allen, 2014b). To accomplish this objective, a two source surface energy
balance method is adopted, rather than a single source method. A two source surface energy
balance method partitions surface energy fluxes as well as surface temperature into soil and
canopy values separately. The success of partitioning surface energy fluxes into soil and
canopy values ultimately helps to estimate θsur and θroot. The challenge is to conduct a
complex two source surface energy balance without using thermal band based surface
temperature. The primary reason for not using the thermal band based surface temperature at
satellite overpass dates is to ascertain whether surface temperature can be computed
accurately enough inside the surface energy balance with an iterative process. Secondly,
surface temperature from the satellite based thermal band is bulk surface temperature, and
difficulties arise when partitioning this bulk surface temperature into soil and canopy portions.
Finally, the success of estimating surface temperature inside the surface energy balance is
important when extrapolating ET between the satellite overpass dates, when no satellite
images and thermal based surface temperatures are available.
Some of the earlier studies that used two source surface energy balances were
Raupach, 1989, McNaughton and Van den Hurk, 1995, Shuttleworth and Wallace, 1985,
Choudhury and Monteith, 1988, Norman et al., 1995, Li et al., 2005 and Colaizzi et al., 2012.
BATANS estimates soil surface temperature (Ts) and canopy temperature (Tc) iteratively
using the aerodynamic equation of the sensible heat fluxes for soil and canopy portions,
respectively. A two-source surface energy balance aerodynamic equation is adopted, as
opposed to the widely used Penman-Monteith ET equation. This is because the Penman-
Monteith equation collapses to surface energy balance when the slope of saturated vapor

4
pressure (Δ) is computed using actual surface temperatures. The aerodynamic method requires
resistances and temperatures of different sources and sinks to compute latent and sensible heat
fluxes. In this study, relatively accurate METRIC ET data are utilized as a boundary condition
of ET for partitioning surface energy fluxes at satellite overpass. To complete the surface
energy balance, North American Regional Reanalysis (NARR) weather based meteorological
data is used along with METRIC ET.
Soil moisture can be measured directly in the field, but difficulties arise when these
kinds of data are needed in greater spatial and temporal resolution. Limited research had been
done to estimate θroot with remote sensing technology. Current remote sensing technology
does not have the capability to measure soil moisture directly at the rooting zone. Scott et al.,
2003 used evaporative fraction to estimate θroot using the SEBAL model in different parts of
the world. Das et al., 2006 assimilated aircraft-based remotely-sensed surface soil moisture
into a distributed Soil–Water–Atmosphere–Plant (SWAP) model to compute θroot. Albergel et
al., 2008 used near-surface soil moisture to estimate θroot using an exponential filter.
Microwave technology is most commonly used to estimate soil moisture for a few centimeters
below the ground surface. Most of these above mentioned methods have different limitations;
a new innovative method was adopted to accomplish the objective of this study.
Soil moisture at surface (θsur) and θroot are measures of resistances at the soil surface
and canopy respectively, which are needed while computing ET with the aerodynamic
method. In this developed procedure, θsur and θroot are computed from a two-source surface
energy balance by an indirect inversion of resistances. In general, soil moisture is used to
estimate rss and rsc because soil moisture can be measured directly and quantified easily. In
this approach, resistances are initially estimated based on partitioned METRIC ET, and soil

5
moistures are back calculated based on these partitioned resistances.
2.0 Methodology
This procedure is accomplished in three phases.
1) Phase 1 Inversion: METRIC generated ET is partitioned into soil surface evaporation
(Ess) and transpiration (T) to calculate soil surface moisture (θsur) and root zone moisture
(θroot) at the satellite overpass time. The partitioning of ET and calculation of θsur and θroot
at the satellite overpass time is achieved by a two-step procedure.
a) Step 1: Estimate T and θroot by assuming the soil is completely dry and letting the
canopy transpire fully, if the conditions allow, using surface energy balance. In this
process, a minimum value of rsc limits the maximum T while inverting Jarvis-type
function.
b) Step 2: Back calculate T, Ess and θsur with an estimated rsc that defines T of Step 1.
2) Phase 2 Interpolation: Estimate θsur and θroot every three hours, using the θsur and θroot
values of Phase 1 as an initial starting point of the simulation of soil-water balance, in
order to extrapolate ET between two satellites overpass dates with METRIC and NARR
(Mesinger, F., and Coauthors, 2006) reanalysis data sets.
3) Phase 3 Correction: Correct simulated ET using METRIC ET at the next satellite passing
date.
This paper discusses Phase 1. Phase 2 and 3, which involve the extrapolation and correction
of ET, will be discussed in Dhungel and Allen, 2014b. There are two models (Step 1 and Step
2) associated with Phase 1.
2.1 Model Description
BATANS applies the Monin-Obukhov similarity theory from canopy height to

6
blending height over all contained 30 m pixels. Meteorological conditions, such as surface
roughness, atmospheric stability and heterogeneity length scales, govern the blending height
(Essery et al., 2003), which often varies between 50 m and 100 m above the surface
(Wieringa, 1986; Mason, 1988 ; Raupach and Finnigan, 1995). SEBAL (Bastiaanssen et al.,
1998) and METRIC (Allen et al., 2007) use 200 m as the blending height, while ALEXI
(Norman et al., 2003) uses 50 m. Suñén et al., 2012 tested a 30-60 m blending height. This
study uses a blending height of 30 m because of the availability of different meteorological
data from NARR reanalysis.
Aerodynamic resistance from substrate to canopy height (ras) is computed for the soil
portion and aerodynamic resistance from canopy height to blending height (rah) is computed
for the soil and canopy portion combined (fig. 1.1). The H is computed separately for the soil
and canopy portions and blended together at the d + zom height (fig. 1.1). This model is similar
to a parallel or patch model proposed by Norman et al, 1995, as no intermediate temperature
is computed at height d + zom. In this study, both Ts and Tc are computed iteratively within the
surface energy balance, so introduction of another intermediate temperature, as in a series
model, creates difficulties in the solution and convergence of the two source surface energy
balance. It is assumed that use of bulk sensible heat flux for computing rah with a combined
stability correction will carry the properties of the soil and canopy portions. Table 1.1 shows
the fluxes, parameters, and boundary conditions that are applied in the model.

7
Table 1.1. Fluxes, parameters, variables, and boundary conditions used in BATANS
Parameters Symbol Min Max Units
Incoming solar radiation RS↓ - - W/m2
Incoming longwave radiation RL↓ - - W/m2
Net Radiation Rn - - W/m2
Measurement height (Blending height) Z - - m
Stability correction parameter ψ - - -
Soil surface temperature Ts 265 350 K
Canopy temperature Tc 265 350 K
Combined temperature Tb 265 350 K
Air Temperature at blending height Ta - - K
Wind speed at blending height uz - - m/s
Specific humidity at blending height qa - - kg/kg
Soil surface evaporation Ess 0.0001 1.4 Ref_ET mm/hr
Canopy transpiration T 0.0001 1.4 Ref_ET mm/hr
Combined (bulk) Sensible heat flux H -200 500 W/m2
Sensible heat flux for soil portion Hs -200 500 W/m2
Sensible heat flux for canopy portion Hc -200 500 W/m2
Ground heat flux G -200 700 W/m2
Latent heat flux for soil (LEs) - - W/m2
Latent heat flux for canopy (LEc) - - W/m2
Friction velocity u* 0.01 500 m/s
Aerodynamic resistance from canopy height to blending height rah 1 500 s/m
Normalized difference vegetation Index (NDVI) for fraction of
cover
NDVI 0.15 0.8 -
Albedo soil αs 0.15 0.28 -
Albedo canopy αc 0.15 0.24 -
Leaf area index LAI fc LAI METRIC -
Single area leaf equivalent bulk stomatal resistance rl 80 5000 s/m
Fraction of vegetation cover fc 0.05 1 -
Mean boundary layer resistance per unit area of vegetation rb 0 - s/m
Roughness length of momentum zom 0.01 - m
Roughness length of heat zoh - - m
Minimum roughness length zos 0.01 - m
Bulk boundary layer resistance of the vegetative elements in the
canopy
rac 0 5000 s/m
Canopy resistance rsc 0 5000 s/m
Soil surface resistance rss 35 5000 s/m
Aerodynamic resistance between the substrate and canopy height
(d + zom)
ras - - s/m
Height of canopy hc - - m
Manageable allowable depletion MAD 0 0.95 -
Relative Evaporative fraction (ETrF) Rel_ETrF 0.55 1 -
Soil moisture at surface θsur - - m3
/m3
Soil moisture at root zone θroot - - m3
/m3
Available water fraction AWF 0 1 -
Soil moisture at wilting point θwilt - - m3
/m3
Soil moisture at field capacity θfc - - m3
/m3
Figure 1.1 shows the model for H with the parameters of surface energy balance
fluxes.

8
energy balance and resistances
Equations 1 and 2 show the aerodynamic equations for sensible heat flux of the soil
(Hs) and canopy (Hc) portions, respectively.
(1)
(2)
where Ta is the air temperature at blending height (K), ρa is the atmospheric density (kg/m3
),
and cp is the specific heat capacity of moist air (J/kg K). ras is computed according to the
Shuttleworth and Wallace, 1985 procedure shown in the appendix-A. The value of rac is small

9
compared to the other resistances, whose computation procedures are shown in the appendix-
A. zom is the roughness length of momentum (m) and zos is the minimum roughness length
above the surface which is assigned as 0.01 m for land surface. Figure 1.2 shows the two
source model for latent heat flux (LE). Equations (3), (4), and (5) show the latent heat flux
for soil (LEs), canopy (LEc), and water (LEw) respectively.
Fig. 1.2. Two source model for latent heat flux (LE) with major components of surface energy
balance and resistances
( ) (3)

10
( ) (4)
( )
(5)
where eo
s is the saturation vapor pressure at the soil surface (kPa), eo
c is the saturation vapor
pressure of the canopy (kPa), ea is the actual vapor pressure of air (kPa), qo
w is the specific
humidity of the air at saturation (kg/kg), qa is the actual specific humidity (kg/kg), γ is the
psychrometric constant (kPa/o
C), and λ is latent heat of vaporization (J/kg).
2.2 Convergence of Surface Energy Balance
The complexity of the solution and convergence of the surface energy balance
increases as greater numbers of the parameters and fluxes are uncertain and need to be
computed iteratively. Figure 1.3 shows backward averaged two source accelerated numerical
solutions (BATANS) of the surface energy balance at satellite overpass time. Separate surface
energy balance models, which differ in their method of separating Ess and T from METRIC
ET, are used for Step 1 and Step 2. Figure 1.3 is the representative surface energy balance for
both steps. Section 2.3 discusses the procedure of partitioning Ess and T in Step 1 and Step 2.
Initial Ts and Tc are estimated by inverting equations (1) and (2), assuming neutral conditions.
An initial value of H is taken from METRIC ET to start the surface energy balance. In the
iterative process, friction velocity (u*), ground heat flux (G), LE, and H are backward
averaged and updated during each iteration as a nested loop. Friction velocity (u*), G, LE, and
H are assumed to be the most significant parameters for the purpose of convergence. These
parameters are monitored during the convergence of surface energy balance and Monin-
Obukhov similarity correction. Aerodynamic resistance (rah) is updated by with new values in
each iteration. The convergence criterion for the entire grid is taken to be when 99% of the

11
total pixels rah are converged within +1 s/m.
Fig. 1.3. Flowchart of BATANS convergence at satellite overpass time using NARR
reanalysis weather data and satellite based METRIC data
After the convergence of the surface energy balance fluxes, θroot is computed outside

12
the energy balance loop using a Jarvis-type equation (eqn. (12)). Soil moisture at root zone
(θroot) represents the soil moisture from the soil surface to the rooting zone which is 2 m for
the big tree forest and sage brush desert, and 1 m for rest of the vegetation. Figure 1.4 shows
the procedure of inverting a Jarvis-type model to compute θroot. A detailed description of this
process is described in Section 2.3.
Fig. 1.4. Flowchart of computation soil moisture at root zone (θroot) from Jarvis-type model
2.3 Model Implementation
Step 1: Estimation of T
Soil moisture prior to the satellite overpass date and at the satellite overpass time is not

13
always available in a large spatial and temporal resolution. In this study, soil surface
evaporation (Ess) at the satellite overpass date is computed using eqn. (3), assuming the soil is
completely dry. With this assumption, the model can be applied in study areas where no
information of the soil moisture is available. Soil surface resistance (rss) is set at 3000 s/m in
Step 1 to make Ess very small. Transpiration (T) is partitioned by subtracting Ess from bulk
METRIC ET (ETMETRIC) from weighting by fc (eqn. 6). Transpiration (T) is updated as the
fluxes of the surface energy balance are updated and solved iteratively. The initial estimate of
T is computed from Equation 6.
(6)
1. (a) Estimation of rsc
Once rah is converged from BATANS (fig. 1.3), rsc is computed by inverting aerodynamic
equation of LEc (eqn. 4).
(7)
Saturation vapor pressure of the canopy (eo
c) is computed during the iteration process using
iteratively computed Tc (eqn. 8).
( ) (8)
Canopy resistance (rsc) is converted to rl to check if the minimum value of rl is smaller than
rlmin i.e. 80 s/m. Effective Leaf area index (LAIeff) is computed by using a modified version of
Ben-Mehrez et al. (1992) equation of partial canopy. The Ben-Mehrez et al. (1992) is
modified to concentrate LAI in a vegetation portion and to separate the soil portion from the
vegetation portion (eqn. (9)). While partitioning ET into soil and canopy portion, LAI is
weighted by fc in all the equations that contain LAI.

14
(9)
rl is computed using LAIeff as according to Allen et al., 1998 (eqn. (10)) which limits the
maximum T in a pixel.
{
(10)
Canopy resistance (rsc) is corrected with rl if rl is below rl_min using eqn. (11).
[ ]
(11)
1. (b) Estimation of θroot
Once, rsc_cor is estimated, θroot is computed by inverting a Jarvis-type equation for
canopy resistance (eqn. (12), fig. 1.4)). Impacts of different environmental factors (F1, F2, F3,
F4) are considered while computing canopy resistance using a Jarvis-type function. In
Equation 12, the impacts of environmental factors associated with solar radiation (F1) and soil
moisture at the root zone (F4) are solely considered. The influence of other widely used
environmental factors like temperature and humidity are not considered, as the temperature is
computed iteratively and these two parameters are considered to be relatively consistent
between two satellite overpass dates. The weighting coefficients of F1 and F4 are limited to 1.
(12)
Equation 13 shows the computation procedure for F1.
(13)
rlmax and rlmin are maximum and minimum values for rl respectively (s/m). rlmax and rlmin are
taken as 5000 s/m (Kumar et al., 2010) and 80 s/m respectively (Allen et al., 1998). rlmin is

15
assigned as 80 s/m for all land use classes except for the sagebrush desert and grassland,
where a value of 300 s/m is used. This assumes that desert vegetation has higher resistance to
transpiration. Parameter f in Jarvis-type model is computed using eqn. (14), where Rgl is
minimum solar radiation necessary for photosynthesis (T) to occur, and Rg is incident solar
radiation.
(
( )
)
(14)
The typical values of Rgl for irrigated cropland and barren/sparsely vegetated area are
100 and 999 W/m2
respectively (Kumar et al., 2010). There is a larger uncertainty in the Rgl in
sagebrush desert. The majority of rsc values computed from the Jarvis-type equation are for
the agricultural land and forest environment, and very little research had been done in the
desert environment. For agricultural land, minimum solar radiation necessary for
photosynthesis (Rgl) is assigned as 100 W/m2
(Stewart, 1988; Alfieri, 2008). A sensitivity
analysis is carried out with different set data of rlmin, Rgl, fc and LAI to produce a reasonable
value of θroot for May, 2008. Sensitivity analysis concluded that a Rgl of 100 W/m2
can be used
for all land use classes. Along with other parameters, there is also a greater uncertainty in LAI
in sagebrush desert and grasslands. METRIC estimates very small values of LAI (0.05 to
0.11) in sagebrush desert and grassland, while a study conducted in Wyoming showed that
LAI is about 0.7 for sagebrush desert (Miller, 1987). Clark and Seyfried, 2001 had also
compiled LAI of sagebrush desert, which are from 0.13 to 1.1 at the Reynolds Creek
Experimental Watershed in Idaho. After finding the values of Rgl, fc, and LAI, F4 is computed
with an inverted Jarvis-type model (eqn. (12)) using Equation 15.

16
(15)
Available water fraction (AWF) is calculated from θroot, soil moisture at field capacity
(θfc), and soil moisture at wilting point (θwp) (eqn. 18). Soil moisture at field capacity (θfc) and
θwp depend on the soil types. Their typical values are 0.36 m3
/m3
and 0.12 m3
/m3
(Kumar et
al., 2010; Allen et al., 1998) respectively, for a silt loam soil. Studies frequently use the AWF
function to represent the F4 function, creating a very sharp decline of soil moisture (dashed
line, fig. 1.5) when sufficient soil moisture is available for transpiration. The F4 curve acquires
a concave down shape to mimic soil moisture at the root zone, as well as the stress level of
vegetation (fig. 1.5). The F4 function decreases slowly within moisture ranges from θfc and
certain threshold moisture content (θt) and decreases sharply from θt to θwp. Vegetation has
very little to no stress in the soil moisture range from θfc to certain threshold moisture content
(θt). The plant can transpire at a reference rate in these ranges of soil moisture if atmospheric
conditions are favorable. When θroot decreases to θt, vegetation is no longer able to transpire in
a reference rate. A detailed discussion of the computational procedure of θt is carried out in
next paper (paper 2).
To compute the F4 function for a Jarvis-type equation, a logistic growth model is
frequently used in agricultural modeling (France and Thornley, 1984; Anderson et al., 2007).
This relationship preserves the characteristics of T with respect to θroot (fig. 1.5), assuming it
is independent of the soil types and textures. Using Equations 15 and 16, θroot is computed by
the following procedure:
[ ] (16)

17
Fig. 1.5. F4 function for Jarvis-type model developed from available water fraction (AWF)
where θwp = 0.12 m3
/m3
and θfc = 0.36 m3
/m3
After computing F4 (eqn. 15) from the Jarvis-type equation, AWF is computed by inverting
the logistic growth model (eqn. 17).
[ ]
(17)
Finally, θroot (eqn. 19) is computed by inverting the AWF equation (eqn. 18).
(18)
( ) (19)
Step 2: Estimation of Ess_cor
In Step 2, an iterative process is carried out for the convergence of rah using different
set of the surface energy balance model. In this step, transpiration (Tcor) is refined by rsc_cor
from Step 1 using the aerodynamic equation for vegetation (eqn. 4). With a better estimation
of Tcor, soil surface evaporation is corrected (Ess_cor) simultaneously in this step (eqn. 20).
0.0
0.2
0.4
0.6
0.8
1.0
0.08 0.13 0.18 0.23 0.28 0.33
Weightingfactor
Soil Moisture
AWF F4

18
(20)
2. (a) Estimation of rss
Once rah has converged from the surface energy balance in Step 2, rss_cor (eqn. 21) is
estimated by inverting the aerodynamic equation for LEs (eqn. 3).
(21)
Saturation vapor pressure at the soil surface (eo
s) is computed using Ts iteratively inside the
surface energy balance (eqn. 22).
( )
(22)
2. (b) Estimation of θsur
In this modeling approach, soil moisture at the surface (θsur) is estimated by inverting
the rss equation as proposed by Sun, 1982 (eqn. 23). Soil moisture at surface (θsur) represents
the evaporation layer of soil, from the surface to a depth of 10 cm. Equation 23 was developed
for a loam soil, which has been extensively used (Mahfouf and Noilhan, 1991, Ács, F., 2003,
Yang and Qin, 2009 etc.). Limited research has been done into developing a relationship
between θsur and rss for all soil types and characteristics. Figure 1.6 shows some of the
previously published relationships between θsur and rss for different soil types. Soil moisture at
the surface (θsur) may be calculated by inverting Equation 23, but only if the value of rss is
greater than 33.5 s/m. If not, θsur will become indefinite. Therefore, a minimum value of 35
s/m for rss is used in the model.
( )
(23)

19
Fig. 1.6. Relationship between soil surface resistance (rss) and soil surface moisture (θsur) for
different soil types
( )
(24)
The estimated θsur in Step 2 and θroot from Step 1 are used in soil water balance while
extrapolating ET in Phase 2.
3.0 Application
3.1 Data Requirements
A Python based ArcGIS scripting code is used to develop BATANS. Meteorological
data needed for BATANS are taken from NARR reanalysis. The NARR reanalysis is a
valuable source of climatic gridded data with a high temporal resolution. Data are collected
every 3 hours, but many data are not usable because of their coarse spatial resolution. The
data acquired from NARR reanalysis are air temperature (Ta) at 30 m, wind speed (uz) at 30
m, specific humidity (qa) at 30 m, incoming shortwave (Rs↓), incoming long wave radiation
(RL↓), surface runoff (Srun), and precipitation (P). Another set of vegetation indices and
roughness data are taken from METRIC (Allen et al., 2007). The data from the METRIC
0
1000
2000
3000
4000
0 0.1 0.2 0.3 0.4
Soilsurfaceresistance(s/m)
Soil Moisture (m3/m3)
Sun, 1982 Camillo and Gurney, 1986
Passerat, 1986 Kondo et al., 1990
Sellers et al., 1986 Walker, 1996

20
model are 30 m specific estimates for surface albedo (α), roughness length of momentum
transfer (zom), broadband emissivity (εo), leaf area index (LAI), and instantaneous hourly ET
(ETins). Broadband emissivity (εo) is partitioned into soil and canopy portions. Surface albedo
(α) is partitioned into soil and canopy portions based on θsur. The METRIC model is
processed for Landsat image path 39 on May 18, 2000 and June 18, 2008 in southern Idaho.
The National Land Cover Database (NLCD) is used to determine the land use classes. The
NARR reanalysis data sets have a 32 km grid size, while a METRIC product has a 30 m grid
size. The NARR reanalysis values in Table 1.2 correspond to the 32 km grid size that covers
the entire study area used in the developed model. Table 1.2 shows data from both NARR
reanalysis and METRIC.
Table 1.2. Data from NARR and METRIC at satellite overpass date 05/17/2008 and
06/18/2008
Parameters 051/17/2008 06/18/2008 METRIC Data
Wind speed at 30 m (uz) 1.2 m/s 3.1 m/s Surface albedo (α)
Air temperature at 30 m (Ta) 295.4 K 294.8 K Roughness length of momentum
transfer (zom)
Incoming shortwave radiation (Rs↓) 974 W/m2
988 W/m2
Broadband emissivity (εo)
Incoming longwave radiation (RL↓) 303 W/m2
310 W/m2
Leaf area index (LAI)
Precipitation (P) 0 mm/3 hr 0 mm/3 hr Instantaneous hourly ET (ETins)
Surface runoff (Srun) 0 mm/3 hr 0 mm/3 hr
Specific humidity at 30 m (qa) 0.0041 kg/kg 0.0052 kg/kg
Most of the soil in the area of interest consists of silt loam and loam soil (Westermann
and Tindall, 1995; City of American Falls, 2009). The Soil Survey Geographic (SSURGO)
database is used to acquire information about this soil.
3.2 Study Area
The test project area is in southern Idaho, near American Falls, and includes irrigated
agriculture, desert, and water bodies. In this study, a small 500 km2
AOI is chosen for

21
convergence and computational efficiency (fig. 1.7). The blue square box in fig. 1.7 is a
NARR pixel which overlays the entire AOI. Landsat path 39 images from May, 17, 2008
overlay the NARR pixel. A single pixel of NARR has an area of 1024 km2
, which is an area
equivalent to that of about 1 million Landsat image pixels.
Fig. 1.7. Study area near American Falls, ID overlaying NARR pixel and a Landsat path 39
image for May 17, 2008
4.0 Results and Discussions
In this section, discussion is carried out for combined and partitioned surface energy
fluxes separately. Surface energy fluxes computed from Step 2 are final flux values of the
Phase 1 inversion process. Likewise, canopy resistance (rsc) and θroot from Step 1 are final
values of the Phase 1 inversion process. Some of the statistical parameters suggested by

22
Legates and McCabe, 1999 are used to evaluate model performance. These statistical
parameters are coefficients of determination (R2
), root mean square error (RMSE), mean
absolute error (MAE), modified index of agreement (di), and coefficient of efficiency (E).
These parameters are summarized in table 1.4.
Before analyzing θsur and θroot, it is important to evaluate the surface energy balance
fluxes and parameters. Simulated results from BATANS are compared to METRIC results
after adjusting and combining fc values. To evaluate model performance, three areas of
interest (AOI) are chosen in different land use classes. They are the upper right agricultural
land (URA AOI), bottom left agricultural land (BLA AOI) and middle sagebrush desert and
grassland (MD AOI), as shown in Figure 1.8a. Figures 1.8a, 1.8b and 1.8c show ET, NDVI
and fc respectively at one satellite overpass date (05/17/2008). Dark blue colors (NDVI from
0.68 to 0.8) in fig. 1.8a indicate that the center pivots are transpiring nearly at reference rate,
while sagebrush desert and grassland (NDVI from 0.19 to 0.27) have ET from about 0.1 to
0.25 mm/hr.

23
a) ETins b) NDVI c) fc
Fig. 1.8. METRIC generated instantaneous ET (mm/hr), NDVI, and fc on 05/17/2008 in
southern Idaho, near American Falls
4.1 Combined Result for 05/17/2008 satellite overpass
In this section, combined surface energy fluxes and results are discussed.
4.1.1 Sensible Heat Flux (H)
Figure 1.9 shows METRIC and simulated H at the satellite overpass time. Simulated
H results match closely with METRIC H in visual inspection. The mean of H for the
METRIC data is about 159 W/m2
with a standard deviation of 43 W/m2
, while simulated H is
about 183 W/m2
with a standard deviation of 53.4 W/m2
for URA AOI. The difference in the
mean is about 25 W/m2
, which is less than 1 standard deviation from METRIC H. For BLA
AOI, the mean is about 167 W/m2
, while simulated H is
212 W/m2
. In BLA AOI, the difference in mean
increased compared to URA AOI, and is about 45 W/m2
. For MD AOI, which consists of
sagebrush desert and grassland, the METRIC mean is about 269 W/m2
with a standard

24
deviation of 15 W/m2
, while simulated H is 323 W/m2
.
Table 1.4 compiles the statistical parameters between simulated H and METRIC H values.
Simulated H is systemically greater than the METRIC results in all three AOIs. It is
difficult to determine a consistent primary cause of these differences, as simulated H is from a
combination of soil and canopy portions. One of the reasons of systemically greater H from
BATANS is that G is consistently smaller. There are biases in the computation of fluxes in the
surface energy balance approach, as LE is only as accurate as the combined estimates of Rn,
G, and H (Allen et al., 2013). These flux biases are one possible reason for the differences
between simulated and METRIC H. Use of the standard aerodynamic equation in the sparse
canopy, where the surface-air temperature difference is made proportional to the sensible heat
flux by means of an aerodynamic resistance, can produce a 50% error in surface energy
balance (Choudhury et al., 1986, Stewart et al., 1989).
a) Sensible heat flux (METRIC) b) Sensible heat flux (Simulated)
Fig. 1.9. Comparison of METRIC to simulated bulk sensible heat flux (W/m2
) on 05/17/2008
in southern Idaho, near American Falls
Table 1.3 shows the results of the surface energy balance fluxes of a particular pixel

25
for different land use classes and conditions. The simulated H matches closely to METRIC H
in all four agricultural pixels except for desert and grasslands, where simulated H is
noticeably larger than METRIC H (table 1.3).
Table 1.3. Surface energy fluxes for different land use classes and conditions on 05/17/2008
Models Coordinates
(m)
NLCD
Land use classes
fc Tb
(K)
H
(W/m2
)
G
(W/m2
)
Rn
(W/m2
)
LE
(W/m2
)
METRIC 2612097,
1330202
82 0.063 305 119 96 641 426
Simulated 305 92 84 603 427
METRIC 2606520,
1327977
52 0.28 321 278 111 519 130
Simulated 309 332 77 543 131
METRIC 2604335,
1326667
71 0.28 324 275 110 546 114
Simulated 312 331 76 526 115
METRIC 2600245,
1328521
82 0.85 301 98 43 600 459
Simulated 305 130 11 599 458
METRIC 2609171,
1333273
82 0.05 320 250 100 445 95
Simulated 314 284 105 485 95
METRIC 2612312,
1329483
82 0.24 309 155 90 603 358
Simulated 305 161 64 585 359
Figure 1.10 shows a scatter plot of about 30,000 METRIC H and simulated H pixels
sampled on two different dates. The coefficients of determination (R2
) between the METRIC
H and simulated H are about 0.79 and 0.77 for URA AOI and BLA AOI respectively (table
1.4) for 05/17/2008. Most of the points in both scatter plots are above the 1:1 line, indicating
that simulated H is larger than METRIC H (fig. 1.10a and 1.10b). Root mean square errors
(RMSEs) are about 34 W/m2
, 54 W/m2
and 55 W/m2
for URA, BLA, and MD AOIs,
respectively, for 05/17/2008. On 06/18/2008, R2
increased to 0.95 and 0.90 for URA and BLA
AOIs respectively (fig. 1.10c and 1.10d). The increased R2
on 06/18/2008 is probably because
of increased convergence of surface energy fluxes with higher wind speed.

26
a) URA 05/17/2008 c) URA 06/18/2008
b) BLA 05/17/2008 d) BLA 06/18/2008
Fig. 1.10. Scatter plots of METRIC and simulated sensible heat flux (W/m2
) for URA and
BLA AOIs on 05/17/2008 and 06/18/2008 in southern Idaho, near American Falls
4.1.2 Ground Heat Flux (G)
Figure 1.11 shows comparisons between METRIC and simulated G. BATANS
y = 1.1x + 5.8
R² = 0.79
50
100
150
200
250
300
350
400
50 100 150 200 250 300 350 400
Simulatedsensibleheatflux(W/m2)
METRIC sensible heat flux (W/m2)
y = 0.94x + 65.1
R² = 0.95
50
100
150
200
250
300
350
400
0 50 100 150 200 250 300 350 400
Simulatedsensibleheatflux(W/m2) METRIC sensible heat flux (W/m2)
y = 1.12x + 22.6
R² = 0.77
0
50
100
150
200
250
300
350
400
0 50 100 150 200 250 300 350 400
y = 1.04x + 59.8
R² = 0.90
0
50
100
150
200
250
300
350
400
450
500
0 100 200 300 400 500

27
assumes that there is no G in the vegetated portion of the surface energy balance, making G
close to zero as fc approaches 1. The mean G of METRIC data is about 77 W/m2
with a
standard deviation of 17 W/m2
, while for simulated URA AOI data it is 67 W/m2
with a
standard deviation of 27.4 W/m2
(Table 1.4). For BLA AOI, the mean is about 90 W/m2
with
a standard deviation of 42.3 W/m2
for METRIC, while for simulated G data it is 60 W/m2
. For MD AOI, which is comprised of sagebrush desert
and grassland, the mean is about 108 W/m2
for
METRIC data, and for simulated G data it is 77 W/m2
.
Results show that METRIC G and simulated G matched closely with a maximum difference
of 30 W/m2
in all AOIs. The RMSEs are 18 W/m2
, 46 W/m2
, and 30 W/m2
for URA, BLA,
and MD AOIs, respectively. METRIC has larger G (yellowish green) in woody lands (NLCD
land class 90), compared to simulation results.
a) Ground heat flux (METRIC) b) Ground heat flux (Simulated)
Fig. 1.11. Comparison of METRIC to simulated ground heat flux (W/m2
) on 05/17/2008 in
southern Idaho, near American Falls

28
4.1.3 Bulk Surface Temperature (Tb)
Figure 1.12 shows simulated bulk surface temperature (Tb) from the BATANS and
METRIC models. METRIC temperature is a radiometric surface temperature computed from
the thermal band of a Landsat, while simulated bulk surface temperature is the combination of
Tc and Ts. For a URA AOI, the mean Tb of METRIC data is about 310 K with a standard
deviation of 5.4 K, while the simulated mean is 307 K with a standard deviation of 3.2 K
(table 1.4). For BLA AOI, the METRIC mean is 310 K with a standard deviation of 6.1 K,
while the simulated mean is 307 K with a standard deviation of 3.1 K. BATANS is able to
simulate surface temperature in a manner similar to METRIC in URA and BLA AOIs. For
MD AOI, the mean of METRIC is about 322 K with a standard deviation of 1.1 K, while the
simulated mean is 311 K with a standard deviation of 1.2 K. MD AOI has a larger difference
in Tb compared to other two AOIs.
Table 1.3 showed the temperature comparison of individual pixels for different land
use classes. BATANS is able to simulate surface temperatures close to those of METRIC
when surface temperature is low, but when surface temperature is high, BATANS has
difficulty simulating the resultant steep temperature profile. Numerous factors affect the
simulation of surface temperature from the surface energy balance. As mentioned earlier, the
surface temperature from METRIC is radiometric temperature, while the simulated
temperature is an aerodynamic temperature. These temperatures can differ largely in sparse
canopy and semi-arid areas without any bias (Stewart et al., 1993; Colaizzi, et al., 2004;
Kustas, 2007). Stewart et al., 1993 discussed the large difference between the radiometric and
aerodynamic temperatures in semi-arid areas.

29
a) METRIC surface temperature b) Bulk surface temperature (Simulated)
Fig. 1.12. Comparison of METRIC to simulated bulk surface temperature (K) on 05/17/2008
in southern Idaho, near American Falls
The R2
between METRIC and simulated Tb is about 0.74 for the URA AOI and 0.69
for BLA AOI (fig. 1.13) for 05/17/2008 data. The RMSE is about 3.8 K for URA AOI and 5.0
K for BLA AOI for 05/17/2008 data. The majority of points lie below the 1:1 line in both
URA AOI and BLA AOI, showing that simulated Tb is smaller than METRIC Tb. For the date
06/18/2008, R2
increased to 0.89 and 0.81 for URA and BLA AOIs respectively as of H (fig.
1.10, fig 1.13).
Results showed lower temperature in the sagebrush desert and grassland from
BATANS, evidencing the difficulty involving in the simulation of temperature in the sparsely
vegetated desert using the 32 km NARR reanalysis for 30 m data. Air temperature (Ta) at 30
m is assumed to be same for agricultural land and desert if both lie in same NARR pixel, but
the desert environment has a larger temperature difference between surface and air than
agricultural land. Thus, the desert has a steeper temperature profile than agricultural land at 30

30
m, which is the blending height. A number of efforts have been made to match simulated
surface temperature to Landsat surface temperature. Sensitivity analysis is conducted by
varying attenuation coefficient (n) (Appendix-A, eqn. 6) and exponent (m) on fc (Appendix-
A, eqn. 14b), but none of them are sufficient to increase surface temperature in desert and
grasslands.
a) URA 05/17/2008 c) URA 06/18/2008
b) BLA 05/17/2008 d) BLA 06/18/2008
Fig. 1.13. Scatter plot of METRIC and simulated bulk surface temperature (Tb) for URA and
BLA AOIs on 05/17/2008 and 06/18/2008 in southern Idaho, near American Falls
y = 0.49x + 153.4
R² = 0.74
295
300
305
310
315
320
325
330
295 300 305 310 315 320 325 330
Simulatedsurfacetemperature(K)
METRIC surface temperature(K)
y = 0.45x + 164.6
R² = 0.89
295
300
305
310
315
320
325
330
295 300 305 310 315 320 325 330
y = 0.41x + 180.3
R² = 0.69
295
300
305
310
315
320
325
330
295 300 305 310 315 320 325 330
y = 0.41x + 177.5
R² = 0.80
290
295
300
305
310
315
320
325
290 295 300 305 310 315 320 325

31
Table 1.4 shows the comparison between the METRIC and simulated results for
different statistical measures for 05/17/2008. The means of the METRIC and simulation
results are similar in most cases, except for some differences in H and Tb in sagebrush desert
and grassland. The R2
is about 0.78 for G in URA AOI and 0.32 for BLA AOI. The R2
for Tb
for MD AOI is 0.23, showing a weak coefficient of determination. The R2
values for Rn are
0.88, 0.75 and 0.61 for URA, BLA, and MD AOIs respectively. The RMSEs of Rn are 30
W/m2
, 34 W/m2
, and 17 W/m2
for URA, BLA, and MD AOIs respectively. The coefficient of
efficiency (E) has a negative sign in H for BLA and MD AOI; in G for URA AOI, BLA AOI,
and MD AOI; in Tb for MD AOI; and in Rn for MD AOI, indicating that the observed mean is
a better predictor than the model (Wilcox et al., 1990). Modified index of agreement (di) is
also very weak in Tb for MAD AOI.

32
Table 1.4. Statistics of combined surface energy balance fluxes on 05/17/2008
Fluxes METRIC Simulated R2
Slope Intercept MAE RMSE
(W/m2
)
E di
Combined
H
Mean
(W/m2
)
Standard
deviation
(W/m2
)
Mean
(W/m2
)
Standard
deviation
(W/m2
)
URA AOI 159 43.2 183 53.4 0.78 1.1 5.8 30.2 33.8 0.38 0.64
BLA AOI 167 48.8 212 62 0.76 1.12 22.63 46.4 53.6 -0.20 0.55
MD AOI 269 15.1 323 20.0 0.72 1.06 35.8 54.3 55.4 -11.1 0.19
Combined
G
Mean
(W/m2
)
Standard
deviation
(W/m2
)
Mean
(W/m2
)
Standard
deviation
(W/m2
)
R2
(W/m2
)
E di
URA AOI 77 16.9 67 27.43 0.78 1.4 -44.4 14.2 18.2 -0.16 0.55
BLA AOI 90 42.3 60 33.0 0.32 0.44 20.3 32.0 46.5 -0.2 0.49
MD AOI 108 6.4 77 6.1 0.45 0.61 13.05 29.5 29.9 -21.1 0.15
Combined
Temp.
(Tb)
Mean
(K)
Standard
deviation
(K)
Mean
(K)
Standard
deviation
(K)
R2
(K)
E di
URA AOI 310 5.4 307 3.2 0.74 0.49 154.9 3.1 3.8 0.49 0.60
BLA AOI 310 6.1 307 3.1 0.69 0.41 180.4 4.2 5.05 0.32 0.52
MD AOI 322 1.1 311 1.2 0.23 0.49 150 11.3 11.42 -112.2 0.07
Combined
Rn
Mean
(W/m2
)
Standard
deviation
(W/m2
)
Mean
(W/m2
)
Standard
deviation
(W/m2
)
R2
(W/m2
)
E di
URA AOI 544 62.6 558 44.0 0.88 0.62 215.5 25.4 30.4 0.76 0.72
BLA AOI 549 58.2 566 43.2 0.75 0.62 222.8 27.4 33.9 0.66 0.67
MD AOI 502 16.3 531 9.0 0.61 1.5 -282.6 13.2 17.2 -3.15 0.41
A - Upper right agricultural, BLA-Bottom left agricultural land, MD- Middle Desert
4.2 Partitioned Surface Energy Balance Fluxes for 05/17/2008
In the following section, results of the partitioned soil and canopy surface energy
balance fluxes and parameters are discussed. Surface energy balance is carried out separately
for both soil and canopy portions when computing fluxes and parameters. These partitioned
fluxes are later converted to bulk equivalents adjusting the fc derived from NDVI (Section

33
4.1).
4.2.1 Soil Surface Evaporation (Ess) and Canopy Transpiration (T)
Figure 1.14 shows the actual contribution of LEs and LEc to total LE in terms of
mm/hr, after adjusting fc. Soil surface evaporation (Ess) is low for the agricultural land, where
fc is very high, as its contribution to total ET is negligible. Soil surface evaporation (Ess) is
significant where fc is small and the soil is wet because of recent irrigation or precipitation
events. Canopy transpiration (T) is very high where fc and METRIC ET are high, showing
that the crop is fully transpiring. Results confirmed that BATANS is able to partition and
simulate reasonable value of Ess and T at satellite overpass.
a) Soil evaporation (Ess) b) Canopy transpiration (T)
Fig. 1.14. Estimated soil evaporation (mm/hr) and canopy transpiration (mm/hr) after
adjusting for fc on 05/17/2008 in southern Idaho, near American Falls
4.2.2 Soil Surface Resistance and Canopy Resistance (rss and rsc)
As mentioned in earlier sections, soil surface resistance (rss) and rsc are computed by
inverting the aerodynamic equations of LEs and LEc, respectively. Canopy resistance (rsc) has
low values in fully covered agricultural lands and higher values where fc is small, in

34
sagebrush desert and grassland. Canopy resistance (rsc) in land use classes 81 and 82, which
are irrigated agricultural lands, is about 55-120 s/m. The model is able to reproduce a realistic
value of rsc in irrigated agricultural land because rsc should be low in irrigated agricultural
areas. Likewise, rsc in sagebrush desert and grassland is about 150 to 5000 s/m (fig. 1.15). As
discussed earlier, it is assumed that the desert vegetation faces more resistance while
transpiring than agricultural land vegetation. Soil surface resistance (rss) in agricultural land is
about 60-5000 s/m, while rss for sagebrush desert and grasslands is about 500-1000 s/m. When
fc is near 1, rss is elevated very highly in agricultural land, and Ess makes a very small
contribution to total ET.
a) Soil surface resistance (rss) b) Canopy resistance (rsc)
Fig. 1.15. Estimated soil surface resistance (s/m) from Step 2 and canopy resistance (s/m)
from Step 1 on 05/17/2008 in southern Idaho, near American Falls
4.2.3 Soil Surface and Root Zone Soil Moisture (θsur and θroot)
The complexity of using the Jarvis-type equation in desert increases as Rgl, fc, rsc, rlmin,
LAI, and physiological characteristics of the plants are more difficult to quantify precisely

35
than in agricultural land. The developed model estimated θsur from 0.02 to 0.3 m3
/m3
in an
agricultural land (fig. 1.16). Soil moisture at the surface (θsur) is high in some areas where fc is
low, indicating recent irrigation or precipitation events. For instance, the developed model is
able to capture the recent irrigation or precipitation event with θsur as 0.23 m3
/m3
for an
agricultural pixel (coordinate of 2602197, 1332720 m, fc ~ 0.3) with an ET of about 0.65
mm/hr. Soil surface moisture (θsur) is low in the agricultural land where rss is high. Inverted
θsur for sagebrush desert is small, i.e. 0.01-0.05 m3
/m3
(fig. 1.16).
The θroot of agricultural land is slightly greater than that of sage brush desert and
grassland. The θroot of sagebrush desert is about 0.18-0.20 m3
/m3
, while in agricultural land it
is about 0.18-0.22 m3
/m3
. This indicates that soil moisture from the winter snow is still
present in the desert. It is important to establish reasonable initial moisture content at the root
zone in the desert to preserve the accurate stress level of the desert vegetation, even though
desert environments have lower ET than agricultural areas.

36
a) Soil surface moisture (θsur) b) Root zone soil moisture (θroot)
Fig. 1.16. Estimated soil moisture at root zone (m3
/m3
) from Step 1 and soil surface moisture
(m3
/m3
) from Step 2 on 05/17/2008 in southern Idaho, near American Falls
5.0 Conclusions
BATANS has been developed and applied in southern Idaho to estimate soil moisture
of the soil surface layer (θsur) and root zone (θroot). BATANS was able to simulate the surface
temperature and soil moisture at satellite overpass time for two different dates. These data are
needed to initialize the soil water balance and plant parameters while extrapolating ET
between the two satellite overpass dates. Within the various uncertainties, the simulated and
METRIC H, G, and Tb matched closely for agricultural fields but less closely for sagebrush
desert and grassland. The coefficient of determination (R2
) between the simulated and
METRIC was about 0.51-0.95 for sensible heat flux for different sampled AOI’s in the
agricultural areas. Root mean square error (RMSE) of sensible heat flux was about 34-71
W/m2
. The error could have been due to a bias in METRIC H, differences in spatial resolution
between METRIC and NARR reanalysis data, assumptions in the calculation of G, the
computation procedure for fc, and the use of a standard aerodynamic equation for computing

37
the aerodynamic temperature. The simulated G was systemically smaller than METRIC G,
which could be due to the assumption that no G occurred under vegetation. The R2
between
the simulated and METRIC Tb was about 0.69-0.89 for agricultural land, and the maximum
RMSE was about 4 K. There was less variation in data when temperature was low. As H, G,
and Tb are simultaneously calculated iteratively, any error within the surface energy
parameters was, in essence, distributed among all fluxes. At dry areas and desert, the standard
aerodynamic equation with an aerodynamic resistance was not able to explain the high
radiometric temperature from Landsat, where the simulated surface temperatures were about
10 K lower than those from METRIC. Total surface energy fluxes replicated partitioned
surface energy fluxes weighted by fc. Partitioning of ET not only helped to explicate the
surface energy balance fluxes for soil and canopy, but also facilitated the computation of θsur
and θroot. The model was able to simulate high rss (up to 5000 s/m) in dry areas and low rss (~
35 s/m) in wet areas. Canopy resistance (rsc) in well watered transpiring agricultural land was
about 40-120 s/m, and it increased up to 5000 s/m in dry areas. The simulated θsur and θroot
had reasonable values for the month of May, 2008, though these values need to be calibrated
and validated independently in the field. Both agricultural land and desert had θroot values of
about 018 - 0.22 m3
/m3
, showing that some moisture remained in the desert in May. Soil
moisture at the surface (θsur) varied according to soil surface resistance (rss) at the surface. As
this model had been tested within a specific area of interest, future work could be done to
expand the concept and apply it to more areas.

38
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soils. In T. A. Tindall and D.T. Westermann (eds.). Proceedings of the Western

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Nutrient Management Conference, Vol. 2:172-181. Salt Lake City, UT, March 6-7.
Potash and Phosphate Institute, Manhattan, KS.
Wieringa, J., 1986. Roughness-dependent geographical interpolation of surface wind speed
averages. Quart. J. Roy. Meteor. Soc., 112, 867–889.
Wilcox, B. P., Rawls, W. J., Brakensiek, D. L., and Wight, J. R., 1990. Predicting runoff from
rangeland catchments: A comparison of two models, Water Resour. Res., 26, 2401-
2410, 1990.
Yang, K., Chen, Y. Y., and Qin, J., 2009. Some practical notes on the land surface modeling
in the Tibetan Plateau, Hydrol. Earth Syst. Sci., 13, 687-701, doi:10.5194/hess-13-
687-2009, 2009.

45
Chapter 2: Time Integration of Evapotranspiration Using a Two Source Surface Energy
Balance Model Using NARR Reanalysis Weather Data and Satellite Based Metric Data
By
Abstract
A backward averaged two source accelerated numerical solution of the surface energy
balance model (BATANS) was developed to extrapolate evapotranspiration (ET) between
Landsat satellite overpass dates that were used for the time-integration of METRIC model
derived ET images estimations. BATANS was demonstrated with data from southern Idaho
over the one month period from 05/17/2008 to 06/18/2008. NARR reanalysis meteorological
data were used to calculate surface energy balance fluxes for 3-hour time steps, and the
METRIC data set was used to define initial surface characteristics and soil water conditions.
Fraction of vegetation cover (fc) was used to partition surface energy balance fluxes into soil
and canopy components, as defined by the normalized difference vegetation index (NDVI).
BATANS was able to estimate the surface energy balance fluxes for the 3 hour period
between satellite overpasses. This was accomplished by using surface temperature
approximated from the energy balance, which utilized NARR weather data. Surface
temperature was iteratively computed within the surface energy balance using air temperature
at 30 m and an aerodynamic equation for sensible heat flux. Soil surface resistance (rss) and
Jarvis-model-based canopy resistance (rsc) were used to calculate latent heat flux (LE) using
an aerodynamic expression. METRIC generated ET was used to initialize the soil water
content for the surface and root zone layers at the start of the simulation period, validate the

46
simulated results from the model at the next satellite overpass date, and make adjustments.
The soil moisture sub-model was used to track soil moisture at the surface and root zone,
which defined the evaporation and transpiration portions respectively. An irrigation sub-
model was developed to factor in irrigation for known irrigated agricultural fields, which is
critical when computing ET for heavily irrigated areas. Any mismatch between the simulated
and METRIC ET at the following Landsat image date was adjusted over the simulation period
with a time-based linear correction to increase the accuracy and reduce computation time. The
developed extrapolation model was able to reasonably predict ET from irrigated agricultural
lands and desert at the end of the simulation period. Due to the possible mismatch in timing of
irrigation events between simulated and actual conditions, BATANS produced lower ET than
METRIC in some cases when the NDVI was low. In these cases, potential for evaporation
effects following irrigation was high when compared to METRIC estimations at the end of the
simulation period. The average simulated and METRIC ET matched closely in all sampled
areas of interest (AOI). The R2
for ET for about 30,000 pixels was about 057 to 0.68 for
various AOIs and the RMSE was about 0.03 to 0.17 mm/hour.
1.0 Overview
The motivation for this study was the need for ET maps having high temporal and
spatial resolution and availability of weather based gridded data to calculate surface energy
fluxes. A two source surface energy balance model has been developed to partition ET which
is needed to accurately estimate ET. This allows one to understand the behavior of the land
surface processes and the various resistances associated with them. The main objective of the
study is to extrapolate ET between the satellite overpass dates using the two source surface
energy balance model based on NARR reanalysis weather data and METRIC estimations. The

47
result of this study can be used to analyze hourly, monthly and seasonal trends in ET,
recharge patterns, and other hydrological and crop production aspects. This allows users,
ranging from farmers to state agencies, to help manage water rights and drought conditions.
The first part of this study (Dhungel and Allen, 2014a) estimated soil moisture and resistances
at the satellite overpass time to parameterize the model and to set initial boundary conditions
for the second part. In this section (second part), ET is extrapolated between the satellite
overpass dates using the soil moisture at the previous satellite overpass date as an initial
starting point of the soil water balance. This procedure estimates soil and canopy temperature
iteratively, as no gridded thermal sensor based temperature data or satellite images were
available.
Past research has explored methods for partitioning ET into soil and canopy portions
using a variety of approaches which are generally based on the surface energy balance. In
general, these researchers have been limited to point based ET (weather station based
meteorological data) or single event based ET (satellite overpass date) because the
partitioning of ET requires large quantities of data. Surface energy balance models like
traditional SEBAL (Bastiaanssen et al., 1998) and METRIC (Allen et al., 2007) are single
source and single event (satellite overpass date) based models which calculate relatively
accurate ET at satellite overpass dates. METRIC interpolates daily ET between the satellite
overpass dates using a mathematical spline of the fraction of reference evapotranspiration
(ETrF).
Some of the earlier studies are based on two source surface energy balance by
Raupach, 1989, McNaughton and Van den Hurk, 1995, Shuttleworth and Wallace, 1985,
Choudhury and Monteith, 1988, Norman et al., 1995, Li et al., 2005, and Colaizzi, 2012.

48
Application of a two source model to partition ET to a larger area such as going from regional
to global scales or larger to smaller time steps is more challenging. Methods used to partition
ET for gridded data generally include two source surface energy balance models with
aerodynamic equations (eqn. 2, 3, 4 and 5) or separate Penman-Monteith type equations for
the canopy and soil. Soil and canopy temperature and various resistances from sources and
sinks are needed while partitioning ET with these models. Shuttleworth and Wallace, 1985,
proposed a two source surface energy balance model, in which Penman Monteith based ET
equation, were written separately for soil and vegetation portions with detailed resistances.
This method was a cornerstone for two source ET models. Later, Norman et al., 1995
proposed a two source model using the radiometric temperature and the Priestly Taylor
equation of ET to partition soil and vegetation. These approaches have been widely used in
the remote sensing community. The problem with the Shuttleworth and Wallace procedure for
estimating ET is that large quantities of data need to be measured and parameters need to be
fitted, which might not be possible in larger spatial applications. The Norman et al., 1995
method has been widely accepted by the remote sensing community, although it also faces
many challenges because of its use of the relatively less accurate Priestley Taylor equation.
The need for thermal based radiometric surface temperature also creates more requirements
that may not always be available.
2.0 Methodology
2.1 Surface Energy Balance Model
A detailed discussion about the two source model implemented in this study has been
presented in the first paper (Dhungel and Allen, 2014a). This paper elaborates on the
procedure to apply the model to extrapolate ET between the satellite image dates and is good

49
background for this publication. Table 2.1 shows the fluxes, parameters and boundary
conditions of BATANS.
Table 2.1. Fluxes, parameters, variables, and boundary condition used in BATANS
Parameters Symbol Min Max Units
Incoming solar radiation RS↓ - - W/m2
Incoming longwave radiation RL↓ - - W/m2
Net Radiation Rn - - W/m2
Measurement height (Blending height) Z - - m
Stability correction parameter ψ - - -
Soil surface temperature Ts 265 350 K
Canopy temperature Tc 265 350 K
Combined temperature Tb 265 350 K
Air Temperature at blending height Ta - - K
Wind speed at blending height uz - - m/s
Specific humidity at blending height qa - - kg/kg
Soil surface evaporation Ess 0.0001 1.4 Ref_ET mm/hr
Canopy transpiration T 0.0001 1.4 Ref_ET mm/hr
Combined (bulk) Sensible heat flux H -200 500 W/m2
Sensible heat flux for soil portion Hs -200 500 W/m2
Sensible heat flux for canopy portion Hc -200 500 W/m2
Ground heat flux G -200 700 W/m2
Latent heat flux for soil (LEs) - - W/m2
Latent heat flux for canopy (LEc) - - W/m2
Friction velocity u* 0.01 500 m/s
Aerodynamic resistance from canopy height to blending height rah 1 500 s/m
Normalized difference vegetation Index (NDVI) for fraction of
cover
NDVI 0.15 0.8 -
Albedo soil αs 0.15 0.28 -
Albedo canopy αc 0.15 0.24 -
Leaf area index LAI fc LAI METRIC -
Single area leaf equivalent bulk stomatal resistance rl 80 5000 s/m
Fraction of vegetation cover fc 0.05 1 -
Mean boundary layer resistance per unit area of vegetation rb 0 - s/m
Roughness length of momentum zom 0.01 - m
Roughness length of heat zoh - - m
Minimum roughness length zos 0.01 - m
Bulk boundary layer resistance of the vegetative elements in the
canopy
rac 0 5000 s/m
Canopy resistance rsc 0 5000 s/m
Soil surface resistance rss 35 5000 s/m
Aerodynamic resistance between the substrate and canopy height
(d + zom)
ras - - s/m
Height of canopy hc - - m
Manageable allowable depletion MAD 0 0.95 -
Relative Evaporative fraction (ETrF) Rel_ETrF 0.55 1 -
Soil moisture at surface θsur - - m3
/m3
Soil moisture at root zone θroot - - m3
/m3
Available water fraction AWF 0 1 -
Soil moisture at wilting point θwilt - - m3
/m3
Soil moisture at field capacity θfc - - m3
/m3

50
The sensible heat flux (H) is estimated separately for the soil and canopy portions and
then blended together at canopy height as shown in eqn. (1).
(1)
where fc is the fraction of vegetation cover, Hc and Hs are sensible heat from vegetation and
soil portions (W/m2
) respectively. Equations of sensible heat flux for the soil and canopy are
shown in eqn. (2) and eqn. (3) respectively. Soil surface temperature (Ts) and canopy
temperature (Tc) are computed by inverting eqn. (2) and (3) respectively.
(2)
(3)
where ρa is the atmospheric density (kg/m3
), cp is the specific heat capacity of moist air (J/kg
K), Ta is the air temperature at 30 m taken from NARR reanalysis (K), ras is the aerodynamic
resistance between the substrate and canopy height (d + zom) (s/m), d is zero plane
displacement (m), zom is the roughness length of momentum (m), rah is the aerodynamic
resistance between d + zom and the 30 m blending height for both soil and canopy (s/m), and
rac (s/m) is the bulk boundary layer resistance of the vegetative elements in the canopy (s/m).
ras is computed in accord with the Shuttleworth and Wallace, 1985 procedure (Appendix-A),
and is assumed to be zero for fully covered agricultural land (fc =1). Equations (4) and (5)
show the latent heat flux equations for the soil and canopy portions respectively.
( )
(4)
( )
(5)
where LEs and LEc are sensible heat flux for the soil and canopy portions respectively

51
(W/m2
), eo
s is the saturation vapor pressure at the soil surface (kPa), eo
c is the saturation vapor
of the canopy (kPa), ea is the actual vapor pressure of the air (kPa), rss is soil surface
resistance (s/m), and rsc is canopy resistance (s/m). Equation (6) is total latent heat flux (LE)
after adjusting fc for soil and canopy portions.
(6)
Figure 2.1 shows the model for H, including the surface energy balance fluxes and
parameters to calculate these fluxes. The parameters and fluxes of fig. 2.1 are described in
Table 2.1.
energy balance and resistances
Figure 2.2 is a flowchart of the complete surface energy balance model (BATANS)
and a solution technique adopted for the Monin-Obukhov stability correction. The Monin-

52
Obukhov stability correction is applied to the combined sensible heat flux (H) from canopy
height (d + zom) to the blending height (z). It is assumed that the combined stability correction
maintains the characteristics of the soil and canopy fluxes. A detailed formulation of stability
correction is shown in the appendix-A. A blending height of 30 m is used in this study
because of the availability of NARR reanalysis weather data at this height. To start the surface
energy balance, initial estimates of Ts and Tc are calculated from Hs and Hc from the earlier
time steps. Aerodynamic resistance (rah) is initially computed as neutral atmospheric
conditions, and updated during the iteration process. For each time step, a new set of NARR
reanalysis weather data and METRIC interpolated data are used to conduct the surface energy
balance (fig. 2.2). Soil moisture at the surface (θsur) and root zone (θroot) are tracked using the
soil water balance at each time step. Initial estimates of the surface energy fluxes are updated
with new values in each iteration until convergence is reached. A backward averaging of H,
G, LE and u* is done to expedite the convergence of the surface energy balance variables.
Once rah is converged within + 1 s/m for each pixel and 99% of the total pixels are converged,
the model advances to the new time step.
Without incoming shortwave radiation at night, the surface energy balance fluxes, i.e.
net radiation (Rn), ground heat flux (G) and H, become negative and stable near the surface
boundary layer. This stability causes difficulty in iteratively determining surface temperature
due to numerical instability. Therefore, no iteration for convergence is done during night,
where a neutral condition is utilized. Situations in which the model converges with negative
sensible heat flux during daytime are indicative of advection of energy.

53
Fig. 2.2. Flowchart of BATANS convergence using NARR reanalysis weather data and
satellite based METRIC data
Figure 2.3 shows the process of extrapolation of ET between the satellite overpass
dates. ET METRIC at the start of the satellite overpass is used to initialize the soil moisture

54
estimations at the surface and root zone. ET METRIC at the next satellite overpass date is
used to calibrate and validate the model. The ET energy balance is simulated every 3 hours
using BATANS.
Fig. 2.3. An illustration of the partitioning of ET at satellite overpass dates and extrapolation
of ET between satellite overpass dates
In this section, a detailed discussion is carried out on the soil water balance model
adopted in this study. Soil water balance is a key element of the land surface model as it
helps to track soil moisture at surface and root zone, which is necessary to estimate
resistances to energy. These resistances are measure of ET in aerodynamic equations.
2.2 Soil Water Balance Sub-model
In general, three kinds of soil water balance methods are widely used in land surface,
irrigation and hydrological modeling. The first method involves the physical mass balance of
soil water content (Budyko, 1956 and Manabe, 1969), and it has been extensively used in

55
different forms for different numbers of soil layers (Milly, 1992, Schaake et al., 1996 and
Allen et al., 1998). Depending on the complexity of the modeling approach, some important
soil water movement processes, such as capillary rise, gravitational processes, and root uptake
water, may be ignored. Another widely used soil water balance model is the two layer force
restore model (Deardorff, 1977, Deardorff, 1978, Sellers, 1986, Noilhan and Mahfouf, 1996
and WEB-SVAT modeling). The diffusive flux of water between the surface and bulk root
layer may be estimated by the function of soil moisture at the root zone (θroot) and force
restore coefficients. Finally, a multi-layered soil water balance method with a one
dimensional Richards’ equation may be used for the prediction of soil moisture (Lee and
Abriola, 1999 and Irannejad and Shao, 1998). These water balance approaches to land surface
modeling have been discussed in detail by Shao, 1998, Peng et al., 2002 and Barrgaoui, 2012.
This study implements is a simple physical water mass balance type soil water balance
model. This is an advancement of the first modeling approach described in the previous
section, in which soil evaporation (Ess) and canopy transpiration (T) are computed separately,
in contrast to a conventional single source model. This method incorporates soil moisture
content through soil surface resistance in the soil surface layer and canopy resistances at the
rooting depth layer. In this implemented soil water sub-model, the soil profile is divided into
three layers (fig. 2.4). The 1st
(top) layer is the soil surface layer, where evaporation takes
place, and it is 100 mm in depth from the surface. The 2nd
layer represents the rooting depth
(droot) where transpiration takes place and includes the 1st
(evaporation) layer as a subset. The
2nd
layer is extended from the surface to 1-2 m depth depending on the type of the vegetation.
The rooting depth is set at 1 m for all vegetated areas, except for forest and sagebrush desert,
for which it is set at 2 m. This is because the root zones of mature forests and certain other

56
kinds of vegetation like sage brush desert can go up to 2 m depth (Richards and Caldwell,
1987 and Robson and Kingery, 2006). The rooting depth of the cultivated agricultural
vegetation is generally around 0.2 – 2.0 m (NRCS handbook). These droot values can be
refined according to the land use type and crop type to match observed soil moisture in the
root zone. The 3rd
layer is the deep percolation layer that is used to keep track of fluxes
outside of the root zone. The implemented soil water balance focuses on the first two layers.
The 1st
layer is a subset of the 2nd
layer. This allows for estimation of the evaporation from
exposed soil and quantification of the impact of this evaporation on depletion of water
available for transpiration in the root zone. Similar method are implemented by Schaake et al.,
1996, FAO 56 (Allen et al., 1998), Hunsaker et al., 2003, and Er-Raki, 2008.
Fig. 2.4. A representative sketch of soil surface and root zone control volume for soil water
balance

57
The blue block in fig. 2.4 shows the control volume of the soil surface layer, and the
green block shows the control volume of the bulk root zone layer. In this model, no particular
information regarding the soil moisture is measured by any other means. The actual
distribution of water in the vertical profile with respect to time is not known. A soil water
balance of the surface layer was conducted first to determine Ess. Soil surface evaporation
(Ess) is subtracted from a soil water balance of root zone in later calculations to calculate final
soil moisture at root zone. The soil water balance is updated every three hours to compute a
new set of θsur and θroot.
In the following section, the detailed description of soil water balance of the soil
surface layer and root zone layer is carried out.
2.2.1 Soil Water Balance of Soil Surface Layer: Surface to dsur (100 mm) layer
Soil moisture of the upper dsur at the current time step is computed by using soil
moisture from the previous time step (θsur (i-1)) and balancing the soil water fluxes of the
current time step takes into account precipitation (P(i)), soil surface evaporation (Ess(i)), surface
runoff (Srun(i)), irrigation (Irr(i)) and deep percolation (DPe(i)). The depth involving Ess, i.e. dsur,
is generally taken as 100 mm (Dickinson, 1984). Soil moisture at the satellite overpass time is
taken as the starting point, and it is estimated by an inversion model (Dhungel and Allen,
2014a). The soil water balance of the upper soil portion is shown eqn. (7).
( ) (7)
where i is current time step, θsur(i) is volumetric water content (VWC) for the current time step
(m3
/m3
), θsur(i-1) is the VWC of the surface layer for the previous time steps (m3
/m3
), Ess(i) is
soil surface evaporation (mm), P(i) is precipitation (mm), Srun(i) is surface runoff (mm), Irr(i) is
irrigated water (mm), dsur is soil surface depth (mm), DPe(i) is deep percolated water from the

58
upper soil layer to the root zone (m3
/m3
), CRe(i) is the capillary rise from the root zone into the
1st
soil surface (m3
/m3
), and Te(i) is transpiration from the soil surface layer (mm). CRe(i) and
Te(i) are neglected to simplify the soil water balance in this sub-model. The θsur(i ) is in units of
m3
/m3
while the parameters P(i), Irr(i), Irr(i), Srun(i), Ess(i) and Te(i) are in units of mm of water per
unit area.
In soil water balance, the unit millimeters of water is initially converted to m3
of water
distributed over the depth (dsur or droot), and later converted to a m3
/m3
equivalent by dividing
the depth (eqn. 8 – 10). Ess(i) is computed iteratively in this process and is updated according to
θsur(i) with each iteration. In the implemented soil moisture sub-model, the soil water balance
of the surface layer at the current time step is written as the following conditional equation,
assuming CRe(i) and Te(i) as zero (eqn. (8)).
{
( ) (8)
When θsur(i) is greater than field capacity (θfc), soil water from the surface layer is deep
percolated to the 2nd
root zone layer (droot); therefore VWC of the upper surface layer is
limited to θfc.
2.2.2 Soil Water Balance of Root Zone: Surface to droot (1-2 m) layer
The soil water balance of the rooting depth (droot) is computed from eqn. (9).
( ) (9)
where θroot(i) is the VWC at the root zone for the current time step (m3
/m3
), θroot(i-1) is the VWC
of the root zone from the previous time steps (m3
/m3
), T(i) is transpiration from the root zone
(mm), DP(i) is deep percolation below the root zone (m3
/m3
), and CR(i) is the capillary rise

59
from 3rd
layer to the root zone (m3
/m3
). In this sub-model, the soil water balance of the root
zone layer for the current time step is written using the following conditional equation, which
assumes CR (i) and DP (i) to be zero.
{
( ) (10)
When the soil water content at root zone (θroot) is greater than the field capacity (θfc), it will be
deep percolated (DP(i)) below to the bulk layer. Ti is computed iteratively and any change in
θroot (i) will update Ti with each iteration. Capillary rise (CR (i)) from the 3rd
layer into the bulk
layer (2nd
) is neglected.
So as to accurately estimate ET in irrigated agricultural land, it is important to account
irrigation events between the satellite overpass dates. BATANS incorporates an irrigation
sub-model to account irrigation events. The following section discusses the developed
irrigation sub-model.
2.3 Irrigation Sub-model and Components
Irrigation (Irr) is assumed to be applied to irrigated agricultural lands in a semiarid
climate when soil moisture in the root zone is below the threshold moisture content (θt). The
amount of depleted soil moisture at droot is added as irrigation to recharge the soil moisture to
the field capacity (θfc). Threshold moisture content (θt) is defined as the moisture content at
which plants start getting stressed and no longer transpire at reference rates. At the time of
irrigation, both surface and root zone moisture content is within the field capacity (θfc) using
irrigation sub-model. As mentioned earlier, when the moisture content is greater than field
capacity (θfc), water is released from the root zone as a deep percolation (DP). Equation (11)
shows the conditional equation for an irrigation sub-model used in the developed model. The

60
total amount of the applied irrigation water depends on soil moisture below the θt and root
depth (droot).
{ (11)
Figure 2.5 shows a representative sketch of different soil moisture contents and available
water for vegetation in the irrigation sub-model.
Fig. 2.5. A representative sketch of different soil moistures and parameters used in irrigation
sub-model
Total available water (TAW) in fig. 2.5 is water available from soil moisture at field
capacity (θfc) to wilting point (θwp). The threshold moisture content (θt) is computed using
readily available water (RAW) and θfc (eqn. (12)).
(12)
Readily available water (RAW) is computed from the manageable allowable depletion
(MAD) and total available water (TAW) as according to eqn. (13).
(13)

61
Manageable allowable depletion (MAD) is determined according to the derived
relative ETrF (Ref_ETrF) at the time of irrigation. Ref_ETrF is used to distribute the plant
stress by MAD in between the satellite overpass dates. A regression equation (eqn. (14)) has
been developed for MAD to account for different levels of plant stress for different kinds of
vegetation and conditions (Allen, 2013). MAD is limited to a maximum value of 0.95.
√ (14)
where p is the fraction of depletion of available water before stress, which is generally taken
as 0.5. Ref_ETrF was computed from eqn. (15), and it is limited to a range from 0.55 to 1.
MAD and Rel_ETrF are used for irrigated agricultural lands, where actual irrigation is
applied, and it is not used for other land use classes.
(15)
where ETrFpot(ave) is the average potential ETrF for two satellite overpass dates, and ETrFpot(act)
is the actual ETrF for two satellite overpass dates. ETrFpot(ave) and ETrFact(ave) are calculated
from eqn. (16a) and (16b) respectively.
( ) (16a)
(16b)
where ETrFact(S) is actual ETrF of the current satellite overpass date, ETrFact(E) is actual ETrF of
the next satellite overpass date, NDVI(S) is NDVI of the current overpass date, and NDVI (E) is
the NDVI of the next satellite overpass date, from the METRIC model.
As mentioned earlier, resistances to energy fluxes are measure of ET estimations. In
this section, a brief description of the formulation and computational procedure of soil surface

62
resistance (rss) and canopy resistance (rsc) is discussed. A detailed discussion of these
resistances is carried out in Dhungel and Allen, 2014a.
2.4 Soil Surface Resistance (rss) and Canopy Resistance (rsc) Computation Procedure
To compute Ess and T from the aerodynamic equation for the latent heat flux of the
soil and canopy, rss and rsc is needed. After calculating θsur from the soil water balance (eqn.
(8)), rss is computed using an equation proposed by Sun (1982) (eqn. (17)). rss depends on the
amount of soil moisture on the soil surface i.e. top layer.
( )
(17)
where θsat is soil moisture at saturation (m3
/m3
).
While computing rsc using a Jarvis-type equation (eqn. (20)), different environmental
weighting functions (F1, F2, F3, F4) were used to because the effect of plant stresses is needed.
Some of the commonly used weighing functions for plant stress are soil moisture, solar
radiation, temperature and humidity. In this study, weighting functions representing the
environmental effects of plant stress due to photosynthetically active radiation (F1) and soil
moisture (F4) are solely considered. As discussed in Dhungel and Allen, 2014a, environmental
factors like temperature and humidity are not considered in this study, because temperature
and humidity are considered to be relatively consistent between two satellite image dates. The
formulation of F1 is discussed in Dhungel and Allen, 2014a. While computing F4 (eqn. (19)),
logistic growth curve of available water fraction (AWF) developed in Dhungel and Allen,
2014a is used to compute F4 (eqn. (18)). Soil moisture at field capacity (θfc) and soil moisture
at wilting point (θwp) depend on the soil types. The typical values of θfc and θwp for silt loam
soil are 0.36 m3
/m3
and 0.12 m3
/m3
(Kumar et al., 2010, Allen, 1998) respectively, and they
are used in the development of this model. This is due to match the study area’s soil type (silt

63
loam).
(18)
[ ] (19)
The weighting coefficients F1 and F4 are limited to 1. Weighting coefficients near or equal to
1 show that there is no stress in vegetation associated with the environmental factor. When
these weighing coefficients are small, rsc becomes large, evidencing a very high stress level in
the vegetation.
(20)
where rl is single area leaf equivalent bulk stomatal resistance (s/m).
3.0 Application
3.1 Data Requirements and Study Area
The model was coded in a Python scripting with ArcGIS modules and functions. The
model runs every three hours for an entire month with a full surface energy balance (fig. 2.2).
So, it is important to have an efficient platform to run the model since it is computing
resource intensive. Python and GIS handle time series calculations efficiently compared to
other image processing software like ERDAS Imagine. When the model is implemented for
larger areas, the model should run faster with the use of servers or computing clouds.
Data acquired from the METRIC model has a 30 m resolution while NARR reanalysis
weather data has 32 km resolution. Even though weather data acquired from the NARR
reanalysis has coarser resolution (32 km), it is assumed that these weather data have a low
variability within each 32 km pixel size. NARR reanalysis is a valuable source of climate data

64
with high temporal resolution; data is collected every 3 hours. Data acquired from the NARR
reanalysis are air temperature (Ta) at 30 m, wind speed at 30 m (uz), specific humidity (qa) at
30 m, surface runoff (Srun), precipitation (P), incoming shortwave at surface (Rs↓), and
incoming long wave radiation at the surface (RL↓). Since 30 m is assumed to be the blending
height over all half million 30 m pixels contained in the 32 km cell, Ta, uz and qa taken at 30
m. Data acquired from the METRIC model were surface albedo (α), roughness length of
momentum transfer (zom), broadband emissivity (εo), leaf area index (LAI), and
reference ET fraction (ETrF) for two satellite overpass dates. Albedo (α), zom, εo and LAI were
linearly interpolated between the satellite overpass dates. It is because these parameters vary
little between the satellite overpass, and it’s difficult to acquire higher temporal and spatial
resolution.
The extrapolation model was run from 05/17/2008 to 06/18/2008 for the test case. The
model was ran for all of the 3 hours’ time steps between the satellite overpass dates with a
simulation period of about a month for an area near American Falls in southern Idaho. Figure
2.6 shows the P, Srun, Rs↓, RL↓, uz, and Ta from NARR reanalysis for every three hours for the
entire month. The maximum precipitation (P) is about 2.5 mm/3 hr. Air temperature (Ta) is
below 300 K for the entire simulation period. The maximum wind speed is about 14 m/s and
the average is 5.2 m/s.

65
Fig. 2.6. NARR reanalysis data in three hour time intervals during the simulation period from
05/17/2008 to 06/18/2008
Figure 2.7 shows the study area location as a 500 km2
area of interest that is contained
within one NARR reanalysis pixel (blue box). It also shows the Landsat path 39 image for
May 17, 2008. The smaller area of interest with different land use classes is chosen for
computational efficiency and convergence purposes. Use of a single pixel of NARR
reanalysis data can create biases in calculation, but the data has been carefully scrutinized and
compared with ground data. No bias has been found between the ground meteorological data
and the NARR reanalysis. NARR reanalysis data was compared to the Aberdeen AgriMet
weather station (lat. =42.95 N, long. =112.83 W and elevation = 1341.46 m) data, and the two
sets are correlated well.
0
2
4
0
250
500
750
1000
1
31
61
91
121
151
181
211
241
Precipitation/Runoff
(mm/3hr)
Radiation(W/m2)
Index number (every 3 hours )
P Incoming shortwave Incoming Longwave Srun
0
3
6
9
12
15270
280
290
300
310
1
31
61
91
121
151
181
211
241
Windspeed(m/s)
Temperature(K)
Air temperature Wind speed

66
Fig. 2.7. Study area near American Falls, ID overlaying NARR reanalysis pixel and a Landsat
path 39 image for May, 17, 2008
The soil in the study area is comprised of silt loam and loam. The Soil Survey
Geographic (SSURGO) database has been used to acquire soil moisture data, including soil
moisture at wilting point (θwp), field capacity (θfc) and saturation (θsat) for different types of
soil. To identify land use classes, National Land Cover Database (NLCD, 2006) land use
classes are used. Not much land use expansion and agricultural development happened in this
region from 2006 to 2008 so it was still a valid land use classification to use for that time
period on such a small region. Parameters like rooting depth (droot), minimum solar radiation
necessary for photosynthesis to occur (Rgl), roughness length of momentum (zom), height of
vegetation (hc), irrigation, and minimum value single area leaf equivalent bulk stomatal

67
resistance (rlmin) differ by land use class. Table 2.2 shows values of data acquired from NARR
reanalysis on 05/17/2008 at 11 am (start of simulation), on 05/17/2008 at 2 pm (three hours
after the first satellite overpass) and on 06/18/2008 at 11 am (end of the simulation).
Table 2.2. Data from NARR reanalysis for the study area
Variables 05/17/2008
11 am
05/17/2008
2 pm
06/18/2008
11 am
Wind speed at 30 m (uz) 1.2 m/s 3 m/s 3.1 m/s
Air Temperature at 30 m (Ta) 295.4 K 299 K 294.8 K
Specific humidity at 30 m (qa) 0.0041 kg/kg 0.004 kg/kg 0.0052 kg/kg
Incoming shortwave radiation (Rs↓) 974 W/m2
967 W/m2
988 W/m2
Incoming longwave radiation (RL↓) 303 W/m2
320 W/m2
310 W/m2
Surface runoff (Srun) 0 mm/3hr 0 mm/3hr 0 mm/3hr
Precipitation (P) 0 mm/3hr 0 mm/3hr 0 mm/3hr
Three different sub-areas of interest (sub-AOIs) are chosen inside the main AOI to
evaluate model performance using different statistical measurements at the end of a satellite
overpass date. They are upper right agricultural land (URA AOI), middle desert (MD AOI)
and bottom left agricultural land (BLA AOI) (fig. 2.8a).
4.1 Satellite Overpass Time
Figure 2.8 shows the METRIC ET and NDVI at the satellite overpass date on
05/17/2008 approximately at 11 am and simulated ET for the next three hours (05/17/2008 at
2 pm). The maximum ET from METRIC at the satellite overpass on 05/17/2008 is about 0.94
mm/hr. It is intended to examine how well the model is able to simulate ET after three hours
with similar meteorological conditions. This analysis is important for the validation of the
extrapolation and inversion models. Partitioning of the surface energy balance fluxes on the
satellite overpass date is done by the inversion model (Dhungel and Allen, 2014a), while
extrapolation of ET for every three hours is done by the extrapolation model as discussed in

68
this paper. Three hours after the satellite overpass time, Ta is increased by 4 K and the rest of
the parameters are similar to the satellite overpass time (Table 2.2). Visually, the simulation
results of ET from the extrapolation model show a very similar trend three hours after the
satellite overpass time when compared to the inversion model.
a) METRIC ET (mm/hr) b) Simulated ET (mm/hr) c) NDVI
(05/17/2008, 11 am) (05/17/2008, 2 pm) (05/17/2008, 11 am)
Fig. 2.8. Visual comparison of METRIC ET at satellite overpass time to simulated ET three
hours after satellite overpass and NDVI at satellite overpass on 05/17/2008 in southern Idaho,
near American Falls
Figure 2.9 shows the simulated ET, METRIC ET and the NDVI at the end of the
simulation period i.e. 06/18/2008, which was 32 days after the starting date. From a visual
inspection, it can be ascertained that the model was able to simulate a pattern of ET similar to
METRIC ET. Statistics confirmed that the simulated ET mimicked METRIC results in
irrigated agricultural lands having high NDVI. Desert and grassland regions also showed
similar results between the developed model and METRIC outputs. Most of the larger

69
discrepancies occurred in the irrigated agricultural pixels where NDVI is low. In agricultural
lands where NDVI is low, ET is generally dominated by Ess. Soil surface evaporation (Ess)
increases when irrigation or precipitation occurs and decreases rapidly when the soil drying.
To have high ET in irrigated agricultural lands where NDVI is low, there must be a recent
precipitation or irrigation event. There was no precipitation during the week prior to the
satellite overpass date, so the higher values of ET in low NDVI areas probably resulted from
an actual irrigation event in the field. In this case, the irrigation sub-model is not able to
capture this actual event in the field probably because θroot is still above threshold moisture
content (θt). This process can easily create a mismatch between METRIC and simulated ET at
the next satellite overpasses date. Even though the difference in ET at satellite overpass can
be large due to this mismatch, the actual discrepancy may be smaller if the model just missed
the irrigation time before satellite overpass date. Further discussion related to irrigation
assumptions is presented in Section 4.2.
In addition to the irrigation scheduling time, numerous other factors may create
mismatches between the simulated ET and METRIC ET. These include incorrect initial
moisture content and inaccurate parameters such as droot, rss, and rsc, etc. Some of the aspects
of these variables and conditions are discussed in the later sections. While calculating H and
LE, different sets of resistances are acting in combination within the aerodynamic equation.
Formulations of these resistances directly affect the final value of simulated ET, and
resistances may need to be adjusted so that the simulated ET matches METRIC ET at next
satellite overpass date. This is because resistances and related fluxes are highly
interdependent.
For the land use classes 52 and 72 (sagebrush desert and grassland), BATANS is able

70
to simulate ET similar to METRIC (fig. 2.9). Even if there is no precipitation in the week
prior to the second satellite overpass date, both models are able to capture ET for the second
overpass date, residual ET is possibly due to transpiration supplied by the stored soil moisture
in the root zone. The simulation was conducted for the period from May to June, 2008; the
root zone soil layer was still wet because of winter snow in this period. ET simulated from the
extrapolation model and METRIC is fairly constant in desert and grassland for these satellite
overpass dates (Fig 2.9).
Simulated ET (mm/hr) METRIC ET (mm/hr) NDVI
Fig. 2.9. Comparison of simulated ET to METRIC ET and NDVI at following satellite
overpass date i.e. 06/18/2008 in southern Idaho, near American Falls
Figure 2.10 shows a comparison between the simulated ET, METRIC ET and NDVI
for a small area inside the URA AOI. The suspected mismatch in irrigation time between the
simulated and actual events, METRIC produced high ET in a right portion of central pivot C-
1 area, where the simulation model was not able to capture the effects of that wetting event.
As NDVI is low in C-1 area (0.10 - 0.31), soil water evaporated relatively quickly after

71
irrigation. For instance, in one of the irrigated agricultural pixel A2 (defined later) that lies in
C-1 area where METRIC estimated high ET, irrigation sub-model applied irrigation once at
05/19/2008 for the entire simulation period. This observation clearly shows that one of the
primary reasons for the mismatch in ET in low NDVI areas may be due to a discrepancy in
irrigation timing.
Since the rooting depth is considered a constant 1 m for the irrigated agricultural land,
the model may have over-predicted soil moisture in the root zone within low NDVI areas,
which reduced the frequency of irrigation. But in reality, rooting depths may vary with the
stage of plant growth and types of the vegetation. Frequent irrigation is required at the initial
stage of vegetation growth, compared to the fully grown vegetation, because droot is smaller
compared to mature vegetation. So to increase accuracy of the simulated ET, where the NDVI
is low, it is important to have an appropriate rooting depth according to the vegetation type
and growth stage. Table 2.3 shows the statistical values of the fluxes from both the METRIC
and the extrapolation model at the next satellite overpass date, which is 32 days after the start
of the first satellite overpass date. Table 2.3 incorporates both the extrapolation model
(column 1-12) and inversion model (13-14) results.

72
Simulated ET (mm/hr) METRIC ET (mm/hr) NDVI
Fig. 2.10. Comparisons between simulated ET to METRIC ET and NDVI inside upper right
agricultural (URA) AOI at next satellite overpass date i.e. 06/18/2008 in southern Idaho, near
American Falls
The following section discusses the results of BATANS in between the satellite
overpass every three hours.
4.1.1 Extrapolation Model
The simulated mean ET from the extrapolation model and METRIC model matches
very closely with a maximum difference of 0.1 mm/hr in all AOIs with an R2
of about 057 -
0.68 (Table 2.3). It shows that the model is able to predict a similar mean ET after a month of
simulation with a reasonable accuracy. Even though the simulated mean ET was consistent to
METRIC estimations, individual variations were seen due to parameter uncertainty and
irrigation mismatch. Simulated bulk surface temperature (Tb) in URA and BLA AOIs matches
closely to the METRIC model, with a difference of 1 K. Mid desert (MD) AOI has a lower Tb
than METRIC, which may be because of the steep temperature profile in the desert compared
to irrigated agricultural areas. As discussed in Dhungel and Allen, 2014a, BATANS has
difficulty simulating the higher temperature in the desert using aerodynamic type equation in

73
surface energy balance. Simulated mean G is slightly lower than METRIC G which is
possibly because of the assumption that there is no G in the vegetation portion. The maximum
difference between the mean G values is about 27 W/m2
, which occurred in URA AOI. The
largest difference occurs in mean H occurred in BLA AOI, and it was about 80 W/m2
. The
maximum difference in mean Rn was about 44 W/m2
, and it occurred in MD AOI.
Some other statistical parameters that were used to evaluate the degree of agreement
between the simulated and METRIC models includes the mean absolute error (MAE), root
mean square error (RMSE), coefficient of efficiency (E), and the index of agreement (di). The
coefficient of determination (R2
) of the fluxes varied over a wide range for different fluxes
and parameters (Table 2.3). The index of agreement (di) varies from 0 to 1, like R2
, and higher
values show better agreement between the simulated and METRIC data. In most cases, R2
and
index of agreement (di) have similar values, which verify the statistical accuracy of these
parameters. Coefficient of efficiency (E) has a negative sign in some surface energy balance
flux parameters, showing that the observed mean is a better predictor than the simulated
model (Wilcox et al., 1990). Higher values of E between the simulated and METRIC results
indicate better agreement, and E can vary from minus infinity to 1. MAE and RMSE are
larger in H, with a maximum value of 108 W/m2
, showing greater variance in the individual
errors in the simulated H. Even if some of the simulated surface energy fluxes vary from the
METRIC estimations, final ET values have less variation, as the bias of the surface energy
fluxes are possibly neutralized. Thermal based surface temperature is one of the dominating
factors when computing ET from surface energy balance models. Simulated surface energy
fluxes from BATANS are based on the iteratively computed surface temperature, a method

74
that has more uncertainty when calculating fluxes than the METRIC model. So, some
variations between the simulated and METRIC estimations are anticipated.
Figure 2.11a shows a hourly scatter plot of simulated ET and METRIC ET of about
30,000 sample pixels at the end of the simulation period (at 06/18/2008) for BLA AOI. The
coefficient of determination (R2
) is about 0.67 and the RMSE is about 0.17 mm/hr (fig.
2.11a). The hourly scatter plot can have more variability than daily ET, so it is equally
important to compare the daily scatter plot of ET. The daily ET scatter plot is also computed
by letting the extrapolation model run beyond the satellite overpass time for a full day,
without interruption at the satellite overpass time. The scatter plot shows that daily ET and
hourly ET have similar trends (fig. 2.11a and 2.11b). The R2
of the daily ET is about 0.65 (fig.
2.11b).
a) Hourly ET (mm/hr) b) Daily ET (mm/day)
Fig. 2.11. Scatter plot between simulated and METRIC ET at next satellite overpass date for
bottom left agricultural (BLA) AOI on 06/18/2008 in southern Idaho, near American Falls
y = 1.04x - 1.12
R² = 0.65
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
SimulatedET(mm/day)
METRIC ET (mm/day)
y = 1.22x - 0.2
R² = 0.67
0
0.2
0.4
0.6
0.8
1
0 0.3 0.6 0.9
SimulatedET(mm/hr)
METRIC ET (mm/hr)

75
Figure 2.12 shows the hourly scatter plots between METRIC ET vs. NDVI and
simulated ET vs. NDVI for BLA AOI. In the simulated ET vs. NDVI plot, the data tends to
approach the 1:1 line because of a possible mismatch in irrigation events, and some influence
of NDVI on the simulated ET, since ET is weighted based on NDVI.
a) METRIC ET vs. NDVI (mm/hr) b) Simulated ET vs. NDVI (mm/day)
Fig. 2.12. Scatter plot of METRIC ET vs. METRIC NDVI and simulated ET vs. METRIC
NDVI for bottom left agricultural (BLA) AOI on 06/18/2008 in southern Idaho, near
American Falls
Figure 2.13 shows the comparison between the daily ET from BATANS and reference
ET (ETr) from the Aberdeen weather station for different selected pixels during the simulation
period. Pixel A1 is an agricultural pixel with high NDVI (0.71 to 0.83), A2 is agricultural
pixel with low NDVI (0.12 to 0.32), and D1 is a desert pixel with an NDVI of about 0.2. Pixel
A1 closely followed ETr, as it is near fully covered with vegetation. Pixels A2 and D1 have
lower daily ET compared to A1, because fc is small in these pixels and precipitation was low
in this period. These results confirmed that BATANS is able to simulate a realistic trend of
ET throughout a simulation period between two images. A detailed discussion of these
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
METRICET(mm/hr)
METRIC NDVI
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
SimulatedET(mm/hr)
METRIC NDVI

76
individual pixels is carried out in the section 4.2. The average daily ET are about 4.6 mm/day,
2.2 mm/day and 1.47 mm/day for A1, A2 and D1 pixels respectively.
Fig. 2.13. Simulation results of daily ET for agricultural pixels A1 (ET_A1), A2 (ET_A2) and
desert pixel D1 (ET_D1) compared to reference ET (ETr) from Aberdeen AgriMet station
between satellite overpass dates
4.1.2 Inversion Model
While inverting METRIC ET using the inversion model, the simulated model cannot
replicate H, G, Ts and Rn exactly like the METRIC results. So, the surface energy balance
fluxes can have different results, depending upon the degree of error when calculating surface
temperature and other fluxes. It is important to understand that the METRIC and developed
inversion model use different methods to calculate final ET, though both use surface energy
balance approaches. So, the results of these two models are expected to vary to some extent
because of different approaches. As mentioned earlier, the inversion model only uses
METRIC ET and surface roughness parameters, and the rest of the surface energy fluxes are
computed iteratively using NARR reanalysis data. Another interesting observation is the
inversion model at 06/18/2008 (col. 13-14) has results similar to the interpolation model
results (col. 2-3) at the end of the simulation period. The mean G from the inversion model is
less than 10 W/m2
different from the interpolation model. Mean H from the inversion model
0
2
4
6
8
10
12
136 141 146 151 156 161 166 171
ET(mm/day)
Day of the year
ET_A1 ET_A2 ET_D1 ETr

77
matches very closely to interpolated H; it is within a 20 W/m2
difference. Difference in Rn is
also within 20 W/m2
. The results indicate that both simulated and inversion model fluxes
match very well with all parameters. The major objective of this study is to extrapolate ET
between the satellite overpass, the major focus is to simulate realistic ET using BATANS
which was ultimately obtained. This allows for the consideration of using BATANS to
extrapolate ET every 3 hours in between satellite images with the utilization of NARR data.
However, other surface energy fluxes can have biases and can vary according to the
iteratively calculated surface temperature. It is highly beneficial to simulate relatively
accurate ET every three hours based on weather data from NARR reanalysis to estimate ET
between images, even though surface energy fluxes can have some variation.

78
Table 2.3. Statistics of combined surface energy balance fluxes from extrapolation and inversion model compared with METRIC
estimations
URA-Upper right agricultural, BLA-Bottom left agricultural land, MD- Middle Desert
Fluxes METRIC
Results
(06/18/2008 11am.)
Simulated Results
(Extrapolation model)
(06/18/2008 11am.)
Simulated Results
(Inversion model)
(06/18/2008 11 am.)
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Statistical
measures
Mean Standard
deviation
Mean Standard
Deviation
R2
Slope Intercept MAE RMSE E di Mean Standard
Deviation
ET (mm/hr) mm/hr) (mm/hr) (mm/hr) % (mm/hr) (mm/hr) (mm/hr) (mm/hr) (mm/hr) (mm/hr)
URA AOI 0.62 0.20 0.60 0.27 0.64 1.09 -0.08 0.11 0.17 0.29 0.71 0.62 0.2
BLA AOI 0.59 0.19 0.52 0.28 0.68 1.2 -0.18 0.13 0.17 0.17 0.68 0.58 0.19
MD AOI 0.16 0.04 0.18 0.04 0.57 0.56 0.08 0.02 0.03 0.49 0.64 0.16 0.05
Temperature
(Tb)
K K K K % K K K K K K
URA AOI 303.1 6.5 304.2 4.1 0.55 0.46 163.18 3.8 4.5 0.51 0.67 303.5 3.2
BLA AOI 304.0 5.8 304.9 3.9 0.51 0.48 157.5 3.4 4.19 0.48 0.60 304 2.6
MD AOI 322.1 0.87 309.1 0.47 0.0 -0.03 319.6 12.9 12.9 -217.1 0.05 309.1 0.73
Ground heat
flux
(G)
W/m2
W/m2
W/m2
W/m2
% W/m2
W/m2
W/m2
W/m2
W/m2
W/m2
URA AOI 57.3 24.51 40.0 45.6 0.86 1.73 -59.2 27.3 30.2 -0.52 0.58 31.0 36.05
BLA AOI 70.4 44.0 50.0 48.4 0.38 0.68 1.9 34.07 45.3 -0.06 0.54 36.1 35.2
MD AOI 115.2 11.2 99.4 4.7 0.31 0.23 72.3 15.8 18.4 -1.6 0.37 88.2 8.3
Sensible heat
flux (H)
W/m2
W/m2
W/m2
W/m2
% W/m2
W/m2
W/m2
W/m2
W/m2
W/m2
URA AOI 109.4 73.9 172.0 96.93 0.50 0.93 69.7 70.1 92.5 -0.56 0.49 168.7 71.5
BLA AOI 125.6 71.3 205.1 106.6 0.54 0.4 36.3 90.2 108.7 -1.3 0.49 191.1 78.1
MD AOI 288.0 28.1 333.5 20.5 0.56 0.54 175.1 48.2 49.0 -2.0 0.28 355.9 13.6
Net radiation
(Rn)
W/m2
W/m2
W/m2
W/m2
% W/m2
W/m2
W/m2
W/m2
W/m2
W/m2
URA AOI 590.51 49.7 613.7 50.7 0.54 0.75 168.79 34.64 43.07 0.25 0.61 624.1 40.6
BLA AOI 592.0 45.6 608.14 50.6 0.49 0.78 144.6 32.03 40.5 0.21 0.61 622.8 63.17
MD AOI 508.7 13.6 552.38 7.75 0.0 0.03 534.9 43.8 46.2 -10.55 0.19 562.3 9.6

79
4.2 Three Hourly Results (Extrapolation Model)
4.2.1 Irrigated Agricultural Pixel A1 (Coordinate: 2600407, 1328596)
In this section, the behavior of individual pixels is discussed for the entire simulation
period. Figure 2.14 shows the comparison between the simulated ET, soil surface
evaporation (Ess), and transpiration (T) for an irrigated agricultural pixel A1 that is inside one
of the central pivots. In this pixel, NDVI varies from 0.71 to 0.83 at the start and end of the
simulation period respectively. ET from the METRIC model, at the start and end of the
satellite overpass, are 0.67 mm/hr and 0.7 mm/hr respectively. ET from the simulated model
is about 0.79 mm/hr at the end of the simulation period. Reference ET (ETr) from the
Aberdeen AgriMet weather station is used as an upper boundary condition to limit Ess and T,
which is about 0.9 mm/hr at 06/18/2008. There is very low precipitation during this
simulation period, less than 3 mm. Figure 2.14 shows that the simulated Ess is very small
compared to T as the pixel is near fully covered with vegetation. Therefore, a majority of ET
is the contribution of T, and small precipitation events are not able to elevate Ess when fc is
near to 1. ET varied according to the weather conditions and the influence of NDVI is not
dominant because NDVI is relatively high throughout the simulation period. Figure 2.14 also
includes the linearly interpolated NDVI. The results confirmed that the BATANS model was
able to simulate realistic ET every three hours and at the end of the simulation period.

80
Fig. 2.14. Simulation results of combined ET (ET_simulated), soil evaporation (Ess) , canopy
transpiration (T), and precipitation (P) for an irrigated agricultural pixel A1 (Land use 82,
NDVI: 0.71 to 0.83 and fc: 0.86 to 1) from 05/17/2008 to 06/18/2008
Figure 2.15 shows the comparison between simulated bulk surface temperature (Tb)
and air temperature (Ts) at 30 m. The differences between Tb and Ts are small in most of the
time steps because the pixel is fully vegetated irrigated agricultural land, creating nearly
neutral conditions. The discussion above also confirmed that the simulated surface
temperature and thermal temperature are statistically consistent. The maximum simulated
surface temperature is about 305 K. This analysis confirmed that BATANS is able to
simulate realistic surface temperature at fully vegetated agricultural pixel.
0
1
2
3
4
50.0
0.2
0.4
0.6
0.8
1.0
1.2
1
31
61
91
121
151
181
211
241
P(mm/3hr)
ET(mm/hr)
Index number (every 3 hours)
P ET_simulated Ess T NDVI

81
fc: 0.86 to 1) from 05/17/2008 to 06/18/2008
Figure 2.16 shows the cumulative values of ETr, simulated ET, precipitation and
irrigation. Cumulative ETr is about 35 mm higher than cumulative ET. The total ET for the
simulation period was about 185 mm. A total applied irrigation and precipitation match
closely to the cumulative ET, which is about 204 mm. Irrigation was applied once at
05/25/2008 within the simulation period. Cumulative P is about 35 mm.
and reference ET (Cum_ETr) for an irrigated agricultural pixel A1 (Land use: 82, NDVI:
0.71 to 0.83 and fc: 0.86 to 1) from 05/17/2008 to 06/18/2008
Figure 2.17 shows θsur and θroot with the precipitation events. θroot decreased gradually
265
273
281
289
297
305
313
1
31
61
91
121
151
181
211
241
Temperature(K)
Tb Ta
0
80
160
240
1
31
61
91
121
151
181
211
241
Cumulativewater(mm/hr)
Cum_P Cum_ET Cum_Irri Cum_ETr

82
after the irrigation event, and was about 0.2 m3
/m3
at the end of the simulation period. Soil
moisture at root (θroot) responded gently to the low precipitation because the addition of water
to θroot was small compared to total water available at root zone. Soil moisture at surface
(θsur) decreased rapidly after the irrigation event and dried out at the end of the simulation
period, as there was no precipitation prior to the satellite overpass. Soil moisture at surface
(θsur) responded quickly in the precipitation and irrigation events.
Fig. 2.17. Simulation results of soil moisture (θsur) at surface and root zone (θroot) for an
agricultural pixel A1 (Land use 82, NDVI: 0.71 to 0.83 and fc: 0.86 to 1) from 05/17/2008 to
06/18/2008
Figure 2.18 shows the simulation results of rss and rsc. Soil surface resistance (rss)
decreased rapidly at the time of precipitation and irrigation. The soil surface resistance
reached a maximum value of 5000 s/m at the start of the simulated period (05/17/2008) and
stayed low during the precipitation events. At the end of the simulation period, rss again
elevated to the maximum allowed value of 5000 s/m, as there was no precipitation and
irrigation. Canopy resistance (rsc) has small values (~35 s/m) in the day time that increase at
night because the F1 function in the Jarvis-type equation becomes very small without solar
radiation input. This creates a very large rsc at night time. The maximum value of rsc at night
time gradually decreases and becomes fairly constant with an increase in LAI within the
0
1
2
3
4
50.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
1
31
61
91
121
151
181
211
241
P(mm/3hr)
Moisturecontent(m3/m3)
P θsur θroot

83
pixel.
Fig. 2.18. Simulation results of soil surface resistance (rss) and canopy resistance (rsc) for an
irrigated agricultural pixel A1 (Land use 82, NDVI: 0.71 to 0.83 and fc: 0.86 to 1) from
05/17/2008 to 06/18/2008
4.2.2 Desert Pixel D1 (Coordinates: 2601776, 1325307)
Figure 2.19 shows a comparison between simulated ET, Ess and T for a desert pixel
with NDVI of about 0.2. The simulated ET in the desert is low, naturally, (less than 0.3
mm/day) throughout the simulation period, as desert ET is more limited by precipitation than
agricultural lands. Soil surface evaporation (Ess) is high after light precipitation events in
comparison to T, because of the low fc in the desert. The contribution of T in the desert is
fairly consistent within the simulation period; water that is transpiring comes from the droot.
After the precipitation events, ET rises about 0.3 mm/hr, becoming the maximum value for
this simulation period. Simulated ET closely matches METRIC ET at the end of the
simulation period.
0
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3
4
50
800
1600
2400
3200
4000
4800
1
31
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P(mm/3hr)
Resistances(s/m)
P rss rsc

84
canopy transpiration (T) for a desert pixel D1 (Land use 52, NDVI = 0.2 to 0.17) from
05/17/2008 to 06/18/2008
Figure 2.20 shows the comparison between the simulated bulk surface temperature
(Tb) and Ta, where Tb is larger than Ta during the day-time and smaller at night. Table 2.3
shows that the mean of Tb that is systematically smaller than METRIC thermal based surface
temperature at the satellite overpass time. The maximum value of Tb is about 314 K in the
simulation period, which is about 8 K less than thermal based radiometric temperature. Tb
averaged about 309 K, which is smaller than the thermal based surface temperature at desert.
This analysis also confirmed that BATANS is able to produce higher surface temperature in
desert compared to air temperature and surface temperature of fully vegetated agricultural
pixel in daytime.
0
1
2
3
4
50.00
0.10
0.20
0.30
1
31
61
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121
151
181
211
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P(mm/3hr)
ET(mm/hr)
P ET_simulated Ess T

85
Fig. 2.20. Simulation results of bulk surface temperature (Tb) and air temperature from
NARR reanalysis (Ta) for desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17) from 05/17/2008
to 06/18/2008
Figure 2.21 shows the cumulative precipitation, ET and ETr, for desert pixel D1.
Cumulative ET was about 19 mm higher than cumulative precipitation in this simulation
period, probably because T is supplied by θroot in the root zone.
260
270
280
290
300
310
320
1
31
61
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121
151
181
211
241
Temperature(K)
Tb Ta

86
Fig. 2.21. Cumulative results of precipitation (Cum_P), ET (Cum_ET) and reference ET
(Cum_ETr) for a desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17) from 05/17/2008 to
06/18/2008
Figure 2.22 shows θsur and θroot for the desert pixel D1 during the entire simulation
period. Soil moisture at the surface (θsur) abruptly rose and fell in response to precipitation
events. Soil moisture at the root (θroot) responded to precipitation much less quickly as the
addition when the soil moisture at 2 m rooting depth was small. Soil moisture at the 2 m
rooting depth steadily decreased from 0.204 m3
/m3
to 0.19 m3
/m3
.
0
60
120
180
240
1
31
61
91
121
151
181
211
241
Cumulativewater(mm)
Cum_P Cum_ET Cum_ETr

87
Fig. 2.22. Simulation results for soil moisture at the surface (θsur) and root zone (θroot) for a
desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17) from 05/17/2008 to 06/18/2008
Figure 2.23 shows the comparison between rss and rsc for pixel D1. Soil surface
resistance (rss) decreased rapidly at the time of precipitation and irrigation and it reached a
minimum value of 35 s/m. At the start and end of the simulation period, when there was no
precipitation, rss reached a maximum allowed value of about 5000 s/m. Canopy resistance
(rsc) had very small values during the day time, but it increased at night because of the F1
function at the Jarvis-type equation. The canopy resistance (rsc) of pixel D1 during the day
was found to be greater than that of pixel A1 because of the formulation of the Jarvis
function. The desert vegetation may face greater transpiration resistance than cultivated,
irrigated vegetation.
0
1
2
3
4
50.19
0.20
0.21
0.22
1
31
61
91
121
151
181
211
241
Moisturecontentatrootzone(m3/m3)
P θroot
0
1
2
3
4
50.00
0.05
0.10
0.15
0.20
1
31
61
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151
181
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P(mm/3hr)
Moisturecontentatsurface(m3/m3)
P θsur

88
Fig. 2.23. Simulation results of soil surface resistance (rss) and canopy resistance (rsc) for a
desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17) from 05/17/2008 to 06/18/2008
4.2.3 Irrigated Agricultural Pixel A2 (Coordinate: 2612319, 1332522)
Another simulation was carried out for different agricultural pixel, A2, which had
lower NDVI than pixel A1. Figure 2.24 shows a comparison between the simulated ET, Ess,
and T with NDVI from 0.12 to 0.32 in the area where simulated ET is significantly lower
than METRIC ET. The simulated ET of this pixel is smaller than that of pixel A1 because fc
is significantly smaller in A2. Soil surface evaporation (Ess) was higher than T at the start of
the simulation period because the irrigation was applied at 05/19/2008. When the soil became
dry at the end of the simulation period, Ess became very small and T gradually increased with
an increase of fc. The increase in T was not able to match METRIC ET because fc was still
very low and there were no irrigation or precipitation events at the end of the simulation
period. METRIC ET at the end of the simulation period was about 0.72 mm/hr, while
simulated ET was 0.21 mm/hr.
0
1
2
3
4
50
800
1600
2400
3200
4000
4800
1
31
61
91
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151
181
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241
P(mm/3hr)
Resistance(s/m)
P rss rsc

89
canopy transpiration (T) for an irrigated agricultural pixel A2 (Land use 82, NDVI : 0.12 to
0.32 and fc : 0.05 to 0.27) from 05/17/2008 to 06/18/2008
Bulk surface temperature (Tb) was higher during the day and lower at night than Ta
(fig. 2.25) because the soil surface of pixel A2 is largely exposed; Tb rose up to 315 K in the
dry environment during the day time. The trend of the Tb in pixel A2 is similar to that of
pixel D1, as both pixels have very small fc.
fc: 0.05 to 0.27) from 05/17/2008 to 06/18/2008
0
1
2
3
4
50.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
31
61
91
121
151
181
211
241
P(mm/3hr)
ET(mm/hr)
P ET_simulated Ess T NDVI
265
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285
295
305
315
325
1
31
61
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121
151
181
211
241
Temperature(K)
Tb Ta

90
In pixel A2, cumulative ETr has a greater difference than the cumulative ET in pixel
A1 because ET is low throughout the simulation period (fig. 2.26). In pixel A2, irrigation
was applied once at 05/19/2008, three days after the first satellite overpass. The total ET of
the simulation period was about 88 mm.
and reference ET (Cum_ETr) for an irrigated agricultural pixel A2 (Land use 82, NDVI: 0.12
to 0.32 and fc: 0.05 to 0.27) from 05/17/2008 to 06/18/2008
4.2.4 Degree of Agreement and Distribution of Error
After ET was extrapolated between the two satellite overpass dates, a comparison
between simulated and METRIC ET was conducted. The degree of agreement between the
results is evaluated using different statistical techniques which were discussed in section 4.1.
This was done to validate the model and to reduce the error between simulated and METRIC
ET, it is necessary to analyze the effects of individual variables and conditions on the
simulation process. Assuming the trend in ET from BATANS is correct, a triangular
interpolation method may be used to adjust these errors to all points within the simulation
period. Figure 2.27 shows the triangular method of error distribution, assuming that there is
0
80
160
240
1
31
61
91
121
151
181
211
241
Cumulativewater(mm/hr)
Cum_P Cum_ET Cum_Irri Cum_ETr

91
no error at the beginning of the simulation and maximum error at the end of simulation, the
next satellite overpass date.
Fig. 2.27. Error distribution using triangular interpolation method
where ETM(S) is the METRIC ET at the start of the simulation (mm/hr), ETC (i) is corrected ET
for every three hour time step (i) (mm/hr), ETS (i) is simulated ET for every three hours’ time
step (i) (mm/hr), i(S) is the start index of current satellite overpass date, and i(E) is the end
index of next satellite overpass date.
In this process, error (Err) between the simulated ET (ETS (E)) and METRIC ET
(ETM(E)) is computed for the next satellite overpass time in mm/hr (06/18/2008). Once a
corrected ET at the next satellite overpass time is obtained, ET is adjusted linearly. The
assumption was made that the error grows at a constant rate. Index number (i) is a measure of
the distance between the current and the next satellite overpass date for every three hours.
Simulated ET (ETs) for every three hour time step is corrected using the second term of eqn.
(21), where height is associated with error in ET (mm/hr).
( )
(21)
Ess and T are written in separate equations so that their errors may be computed and
corrected. After correction, METRIC and the corrected ET have identical values with an R2
value of 1 at the end of the simulation (06/18/2008). Figure 2.28 shows the error map at the

92
end of the simulation period. As per assumption, the maximum error occurs on the end of the
simulation period (06/18/2008), and it is simply the difference between the simulated and
METRIC ET.
Fig. 2.28. Error map (mm/hr) at the end of simulation period i.e. 06/18/2008 approximately
at 11 am in southern Idaho, near American Falls
4.3 Correction of Hourly and Daily Evapotranspiration (ET)
In the following section, the adjusted ET from the above procedure is examined.
Figure 2.29 shows the comparison of simulated ET to the corrected ET for the irrigated
agricultural pixel A1. The simulated ET needs very little correction in pixel A1, as BATANS
is able to simulate relatively accurate ET in higher NDVI irrigated agricultural land although
it struggles with low NDVI pixels.

93
to 1) from 05/17/2008 to 06/18/2008
Figure 2.30 shows the desert pixel D1, which also needed a small correction to match
METRIC ET in most of the time steps within the simulation period. BATANS was able to
simulate relatively accurate ET in the sage brush desert and grassland regions, even if the
temperature differences in these land use classes were high.
0
1
2
3
4
50.0
0.2
0.4
0.6
0.8
1.0
1.2
1
31
61
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151
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P(mm/3hr)
ET(mm/hr)
P ET_sim ET_cor

94
(ET_cor) for a desert pixel D1 (Land use 52, NDVI: 0.2 to 0.17, fc: 0.28) from 05/17/2008 to
06/18/2008
Irrigated agricultural Pixel A2 needs substantial correction because the simulated
model has lower values of ET than METRIC ET. Figure 2.31 shows that the simulated ET
elevated about 0.8 mm/hr after the irrigation and ET increased rapidly after precipitation
events even though the fc in this pixel is low. This indicates that the influence of NDVI by fc
in the simulation result is not the cause of low ET in this pixel at the end of the simulation
period. Simulated ET is lower than METRIC ET at the end of the simulation period because
of the possible mismatch in irrigation, as there is no irrigation and precipitation at the end of
the simulation period.
0
1
2
3
4
50.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
1
31
61
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121
151
181
211
241
P(mm/3hr)
ET(mm/hr)
P ET_sim ET_cor

95
to 0.27) from 05/17/2008 to 06/18/2008
As indicated earlier section, the main objective of BATANS is to extrapolate ET in
between the satellite overpass dates. The METRIC model computes daily ET in between the
satellite overpass dates with the mathematical spline interpolation of ETrF. Figure 2.32 shows
the daily ETrF between the satellite overpass dates before and after the correction and the
METRIC interpolated ETrF for agricultural pixel A1. Simulated ETrF is able to incorporate
the variations in the meteorological conditions compared to METRIC ETrF. At the same
time, the correction of ETrF is achieved using the previously discussed error distribution
procedure.
0
1
2
3
4
50.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
31
61
91
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151
181
211
241
P(mm/3hr)
ET(mm/hr)
P ET_sim ET_cor

96
(Land use 82, NDVI: 0.71 to 0.83 and fc: 0.86 to 1) from 05/17/2008 to 06/18/2008
Figure 2.33 shows the simulated ETrF, corrected ETrF and METRIC ETrF for the
irrigated agricultural pixel A2. After the correction, simulated ETrF is similar to METRIC
ETrF at the end of the simulation period. These inconsistencies are possibly due to the
mismatch in irrigation time, as discussed in an earlier section, daily simulated ETrF at the end
of the simulation period is low. However, ETrF from METRIC shows higher values at the
start and end of the simulation period.
0
0.2
0.4
0.6
0.8
1
1.2
136 146 156 166 176
ETrF
Day of the year
ETrF_sim ETrF_METRIC ETrF_cor

97
(Land use 82, NDVI: 0.12 to 0.32 and fc: 0.05 to 0.27) from 05/17/2008 to 06/18/2008
5.0 Conclusions
A two source surface energy balance model was developed to extrapolate ET between
the two satellite overpass dates, when no Landsat image or thermal based surface temperature
map was available. The extrapolation is driven by NARR reanalysis data sets and
parameterized and initialized using the Landsat based METRIC ET data set. The developed
model was tested in southern Idaho from 05/17/2008 to 06/18/2008. It was able to simulate
ET every three hours for the simulation period and produce reasonable values. A scatter plot
of surface energy fluxes between the simulation and METRIC showed some variation in
data, but the means of the fluxes were similar; within one standard deviation for most fluxes.
The irrigation model supplied water to the known irrigated agricultural lands when soil
moisture was below a threshold. However, the pattern of irrigation and actual irrigation time
in the field was stochastic in nature. A mismatch in the irrigation timing caused by the
assumption of constant rooting depth possibly created differences between the simulated and
METRIC ET when NDVI was low and the opportunity for differences in evaporation due to

98
differences in irrigation timing were high. The Jarvis-type function was able to simulate
lower values of rsc in irrigated agricultural areas.
The soil water balance model was able to track soil moisture at the root zone and
surface. The simulated surface temperature matched closely to the thermal based temperature
in agricultural areas. Simulated surface temperature from the aerodynamic equation with
aerodynamic resistance was not able to explain the high temperatures in the desert. METRIC
ET was the target ET at the end of the simulation period and the distribution of error was
done according to the difference from METRIC ET. Any mismatch between simulated ET
and METRIC ET at the end of the simulation period was corrected for each time step,
preserving the pattern of ET with a linear time-based-proportioning method.

99
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Chapter 3: Comparisons between the FAO-56 Soil Water Evaporation Model and
HYDRUS-1D Evaporation Model over a Range of Soil Types
By
Abstract
A simple ‘slab’ evaporation model by the United Nations Food and Agriculture
Organization (FAO-56) was compared against the more sophisticated finite element Hydrus-
1D model to determine the relative performance of the simple FAO-56 model over a range of
soil types, time series and wetting conditions. The FAO-56 model was applied with and
without a skin evaporation enhancement. Soil water balances and estimated evaporation rates
from the FAO-56 evaporation model with the skin evaporation enhancement matched closely
with those simulated by the Hydrus-1D model for several soil types. In a separate test,
simulations from both FAO-56 and Hydrus-1D models had good agreement with evaporation
measured from a weighing lysimeter. Both hourly and daily timesteps were evaluated with
the FAO-56 model. Both timesteps sufficiently explained the observed evaporation rates and
estimations by the Hydrus-1D model that employs a timestep-optimization strategy. The root
mean square error of daily average evaporation estimates by the original and enhanced FAO-
56 models were 0.67 mm/day and 0.27 mm/day respectively, when compared to Hydrus-1D
for a silt loam soil and an hini (hini) of -3 m. The R2
of the FAO-56 model with and without
skin evaporation enhancement vs. the Hydrus-1D model was about 0.88 and 0.22
respectively when applied to daily calculation timestamps for an hini of -3 m. The R2
for
hourly data summed to daily averages varied from 0.82 to 0.90 for the silt loam soil over a

105
range of estimates of readily evaporable water, REW. In general, the FAO-56 model with
skin enhancement and implicit stage 1 – stage 2 transition produces estimates of daily E that
are similar to those of Hydrus-1D using both daily and hourly calculation timesteps for the
FAO-56 model. This is a useful finding and indicates that models incorporating the FAO-56
procedure should generally provide accurate accounting for E.
1.0 Overview
Water balance is important in many hydrologic studies for quantifying the amount of
surface water or ground water that is evaporated, transpired or contributes to ground water
recharge. Water balance computations are important in agricultural areas to understand and
quantify the water available for crop production in both temporal and spatial dimensions.
Efficient use of water in agricultural fields is enhanced by the accurate calculation of
evapotranspiration (ET). Evaporation of water from bare soil is an important, but challenging
component of ET estimations, and is routinely estimated in operational models.
The FAO-56 soil water balance model Allen et al., 1998, 2005 has been widely used
as an approximate estimate of evaporation (Hunsaker et al., 2002, Howell et al., 1995, Burt et
al., 2002, Mutiziger et al., 2005, Allen, 2011). The FAO-56 model computes a soil water
balance of the upper soil profile (0.10 to 0.15 m), under the assumption that the soil surface
layer functions as an evaporative slab that gains and loses water from precipitation,
infiltration, and evaporation events. The slab model functions as a fully mixed layer so that
small additions of precipitation (P) get mixed completely into the slab. The complete mixing
can incorrectly dampen evaporation spikes following small wetting amounts as compared to
measured values (Allen, 2011). Therefore, Allen (2011) extended the basic FAO-56 slab
model to incorporate an additional shallow skin layer at the soil surface that can hold small

106
amounts of precipitation near the surface that can release the water quickly as stage 1
evaporation.
Comparisons between the original and extended FAO-56 model outputs with
lysimeter measurements and against using the more theoretical and rigorous Hydrus-1D
model of Simunek et al. (2008) provides a useful insight into the behavior of the FAO-56
model and extension for both small and extended evaporation events. The Hydrus-1D model
computes soil water flow by solving the Richards Equation (eqn. (12)) with specified
boundary conditions. The main objective is to compare the relatively simple FAO-56 model,
enhanced with skin evaporation, to the Hydrus-1D model outputs under different
environmental conditions and soil types. The following section describes the FAO-56 model
with the skin evaporation enhancement and the Hydrus-1D model.
1.1 FAO-56 with Skin Evaporation Enhancement
The FAO-56 evaporation model is a simple slab model intended for use in routine
operational models used in irrigation scheduling or for determining crop water requirements
(Allen et al. 1998, 2005). The model has low data requirements, requiring only field capacity
and wilting point of the surface soil layer and weather-based reference ET. The model can
be operated on hourly or daily time steps. The evaporation rate is expressed as a function of
the water content, or depletion from the soil slab that is generally assumed to be 0.10 to 0.15
m in depth. The amount of water stored in the slab is the sum of the water stored at the end
of the previous period plus any rainfall or irrigation inputs and less evaporation. Water stored
in the soil slab is constrained between water content at air dry and field capacity (θfc). The air
dry content is generally approximated as one-half of the water content at the wilting point
(θwp) (Allen et al., 1998). The maximum storage is termed total evaporable water (TEW, eqn.

107
(5)).
The FAO-56 model follows a two stage approach where stage 1 is defined as an
energy limited stage where evaporation is at a maximum rate limited by energy availability at
the soil surface, and stage 2 is a falling rate stage that occurs when the cumulative depth of
evaporation (depletion) from the soil surface layer exceeds the maximum evaporation depth
occurring during stage 1. That maximum evaporation depth (mm) during stage 1 is known as
readily evaporable water (REW) and is the maximum depth of water that can be evaporated
from the soil surface layer without hydraulic restriction. Stage 1 evaporation generally
follows an irrigation or rainfall event. During stage 1, where depletion from the soil surface
layer (De) is less than REW:
(1)
where E1 is the evaporation rate during stage 1, and Kemax describes the potential rate of
evaporation, relative to the reference ET. ETr is reference evapotranspiration (ET) which
changes hourly or daily according to weather conditions. Reference ET is defined as the ET
rate from an extensive surface of uniform, well watered vegetation and represents a near
maximum, surface-energy-controlled ET rate. Values for Kemax range from 1.05 to 1.2 when
applied with a clipped grass reference, ETo, and 0.8 to 1.0 when applied to the taller alfalfa
reference, ETr, (Allen et al., 2005). The higher values for Kemax apply when a dry soil slab has
been heated by solar radiation prior to the wetting event.
In stage 2 drying, where De > REW, the evaporation rate is assumed in the FAO-56
model to decrease in proportion to the water remaining in the top soil layer:
(2)
(3)

108
[ ] (4)
where Kr is a soil evaporation reduction coefficient that is multiplied by the potential
evaporation rate. De (i-1) is the cumulative depth of evaporation (depletion) from the soil
surface layer at the end of time step (i-1). The minimum of the two values separated by the
comma in the ‘min [ ]’ function restricts Kr to less than or equal to 1.0. Figure 3.1 shows the
soil water evaporation of stage 1 and stage 2 of the FAO-56 model expressed as Kr. Units for
TEW, REW and De (i-1) are in mm.
Fig. 3.1. Soil water evaporation from Stage 1 and Stage 2 in the FAO-56 model (Allen et al.,
1998)
The FAO-56 estimates TEW from the soil slab using data for θfc and θwp, soil
parameters that are commonly available:
[√ ] (5)
where Ze is the assumed effective depth of the surface soil layer that dries completely by
evaporation, m. The ‘min [ ]’ function selects the minimum of the two values separated by

109
the comma. In Allen et al., (1998, 2005), the value of TEW during winter or other cool
periods was reduced by the (ETo/5)0.5
multiplier to approximate reductions in potential drying
depth during those periods when less radiation energy is available for heating the soil surface
layer and supplying energy for evaporation. Consequently, the effective depth of drying
decreases.
Ritchie et al. (1989) proposed potential values for REW based on soil texture, with
values ranging from 4 to 10 mm. Allen and Robison (2007) proposed estimating REW from
TEW as:
[ ] (6)
where TEW and REW are in mm.
The FAO slab model is designed for application in a dual evaporation (E) +
transpiration (T) process where calculation of T is determined separately. However, because
evaporation is the focus of this paper, the following equations governing E assumed that the
soil surface is completely bare and that T is zero. In the soil water balance used to estimate
Kr, cumulative depletion depth (Dei) , mm, at the end of timestep i , applied to hourly or daily
timesteps is (Allen 2011):
[ ] (7)
where De i-1 is cumulative depth of evaporation (depletion) from the soil surface layer at the
end of timestep i-1, Pi is precipitation at the surface during time step i (mm), ROi is surface
runoff (mm), Ii is irrigation depth that, in eq. (7) is assumed to infiltrate over the entire soil
surface. Parameter fb is the fraction of the P and I occurring during a time step that
contributes to E during the same time step (fb = 0 to 1), and 1- fb is the fraction of P and I
whose wetting effect does not impact E until the next time step. Ei is the evaporation rate for

110
timestep i computed from eqn. (1) or (2). Infiltration in excess of De i, is passed to below the
slab layer into the soil profile below. No return of that water is permitted. Allen et al. (1998,
2005) and Allen (2011) give modifications to eqn. (7) for situations where vegetation is
present.
Evaporation rates associated with light wetting events (less than 5-10 mm) are treated
by the FAO-56 model skin evaporation enhancement of Allen (2011). That enhancement
accounts separately for surface wetness of the skin of the soil surface where the stage 1
evaporation is assumed to be stored. The depletion of REW is accounted for by an additional
state variable DREW that represents depletion from the skin layer. The water balance for the
skin layer is computed similar to eqn. (7) as:
[ ] (8)
Most variables in eqn. (8) are the same as for eqn. (7). DREWi is constrained between 0
and REW. Infiltration in excess of DREW is passed below the skin layer and mixed into the
evaporation slab. The spatial infiltration efficiency factor (Ceff) represents the effectiveness
of the skin layer in capturing and retaining P and I additions without any bypass flow. In
most applications, the value for Ceff is set to 1.0 assuming that water flows deeper in soil only
after fully recharging the skin layer.
Allen (2011) modified eqn. (3) when applied to daily timesteps to improve the
estimated transition from stage 1 to stage 2 drying and the impact on the value for Ke on the
day of the transition:
[ ] (9)

111
where Ft is the fraction of the timestep interval that resides in stage 1:
[ ] (10)
Ft ranges from 0 to 1.
Ke is modified by the skin evaporation enhancement of Allen (2011) as:
[
[ ]
] (11)
The use of eqn. (10) and (11) simplifies the coding and calculation of E and makes the
procedure continuous through the transition from stage 1 to stage 2 drying.
1.2 Hydrus-1D Model
Hydrus-1D is a numerical model that solves a modified Richard’s equation using a
finite element method for both saturated and unsaturated water flow in one dimension in a
partially saturated rigid porous media. The Modified Richards’ assumes that the air phase is
not significant in liquid flow and that water flow due to thermal gradients can be neglected
(Šimůnek, J., 2008). Equation 12 describes the Richards’ equation for partially rigid porous
media.
[ ( )] (12)
where θ is the volumetric water content, t is time, h is water pressure head (soil water head),
x is the spatial coordinate (positive upward) i.e. x = L at soil surface and x = 0 at the bottom
of the soil profile, S is a sink (root extraction) term, γ is the angle between the flow direction
and the vertical axis (i.e., γ = 00
for vertical flow, 900
for horizontal flow), and K(h) is the
unsaturated hydraulic conductivity function (Šimůnek, J., 2008).

112
The van Genuchten equation is a commonly used parametric model for relating
volumetric water content (θv) to the matric potential (ψm). A statistical pore-size distribution
model of Mualem [1976] was used to develop the van Genuchten [1980] equation. The van
Genuchten equation is shown in eqn. (13).
[
| |
] (13)
(14)
[
⁄
] (15)
⁄ (16)
where θr and θs are the residual and saturated water contents, respectively, h is the pressure
head at the soil surface and α, n and m are parameters directly dependent on the shape of the
water retention curve θ (ψ) curve for a specific soil type and condition, Se is effective
saturation, l is a pore-connectivity parameter, and m is the parameter for the soil water
retention function.
1.2.1 Initial and Boundary Conditions of Hydrus-1D
Hydrus-1D (Šimůnek, 2008) uses an initial boundary condition for the solution of
Richards’s equation:
(17)
where hi is initial water pressure head at x at the beginning of the simulation (t0). The actual
evaporative flux is computed by Richards’s equation using system dependent boundary

113
condition. Actual evaporative flux is assumed equal to the potential evaporative flux if the
pressure head is greater than some minimum pressure at the soil surface (hCritA) that is
defined from equilibrium conditions between soil water and atmospheric vapor. Neumann
boundary condition is applied to compute surface pressure at this condition. This limits the
absolute value of the flux by satisfying the following two conditions (Šimůnek, J., 2008):
| ( )| (18)
(19)
where Emax is the maximum potential rate of infiltration or evaporation under the current
atmospheric conditions (Emax = E1 from eqn. (1)) and hA and hS are the minimum and
maximum pressure heads allowed under the prevailing soil conditions, L is the z-coordinate
(depth of the soil profile) of the soil surface above a certain reference plane. The minimum
pressure head (hA) can be calculated from the relative humidity (Hr). Generally, the
maximum value of the head allowed (hs) is set at zero.
(20)
(21)
where M is the molecular weight of water (M = 0.018015 kg mol-1
), g is the gravitational
acceleration (9.807 m s-2
), and R is the gas constant (R = (=8.314 J mol-1
K-1
). If hs is positive
during rainfall, the model creates a layer of water ponded on the surface. Excess water on the
surface is immediately removed, which is one of the assumptions in the Hydrus-1D model.
In Hydrus-1D model, one of the system independent boundary must be specified at the soil

114
surface (x=L) or at the bottom of the profile (x=0) (Šimůnek, 2008).
(22)
( )
(23)
( ) (24)
where h0 is the surface boundary condition for the pressure head (L) and q0 is water flux
boundary condition at the bottom of the soil profile (L/T).
2.0 Materials and Methods
Comparisons between the original and enhanced FAO-56 evaporation models and the
Hydrus-1D is done in two parts. Section 1 compares model estimates against evaporation
measured by weighing lysimeter for a silt loam soil in southern Idaho. Section 2 compares
the models and conducts a sensitivity analysis for theoretical soil types. Daily and hourly
data sets were used to run the Hydrus-1D and FAO-56 models to evaluate the impact of
timestep length on evaporation estimates.
2.1 Hydrus-1D and FAO-56 Model Setup
Soil hydraulic parameters recommended by the Hydrus-1D manual for its generalized
silt loam, silt and sandy clay soils (Šimůnek, J., 2008) were used in the simulations. As
summarized in Table 3.1, a 3 m soil depth was used for simulations with Hydrus-1D to insure
that the upper soil profile was not impacted by the lower boundary condition. The 3 m profile
was divided into 1001 vertical nodes representing 3 mm layers, to reduce numerical
linearization error. The single porosity van Genuchten Mualem model without hysteresis was
used for simulation with a single soil material. The atmospheric boundary layer with surface
layer option was used as an upper boundary condition which permits water to build up on the

115
surface. A -150 m minimum pressure head was permitted at the soil surface. The lower
boundary condition was chosen as a free drainage condition. Transpiration was assumed to
be zero as all simulations were conducted for a bare soil condition. The computed reference
ETo representing a grass reference was used to approximate a potential evaporation rate in
the Hydrus-1D model, using Kemax = 1.2 in Section 1 of the study. In section 2, a reference
ETr representing the tall reference crop of ASCE-EWRI (2005) was utilized, using Kemax = 1
as recommended by Allen (2011).
Tables 3.1 and 3.2 summarize the soil hydraulic properties for the three soil types
used in the Hydrus-1D model and parameters used in the FAO-56 for the silt loam soil. The
FAO-56 model is recommended to be applied using a 0.1 to 0.15 m depth of surface soil
water (Ze) subjected to drying by evaporation (Allen et al., 1998, 2011) while the Hydrus-1D
model simulates a soil water balance for a deeper depth and considers upward flow toward
the surface. In this study, Ze was taken as 0.1 m for the FAO-56 model.
Table 3.1. Standard soil hydraulic parameters for Hydrus-1D simulations
Soil Properties Symbol Units
Sandy
Clay Loam
Silt
Loam
Silt
Residual soil water content θr
m3
/m3
0.1 0.067 0.034
Saturated soil water content θs m3
/m3
0.39 0.45 0.46
Parameter α in the soil water retention function
[L-1
]
α
mm-1
0.0059 0.002 0.0016
Parameter n in the soil water retention function n 1.48 1.41 1.37
Saturated hydraulic conductivity, Ks [LT-1
] Ks mm/day 314.5 108 60
Tortuosity parameter in the conductivity
function
Tr
0.5 0.5 0.5
(Source: Hydrus-1D Šimůnek, J., 2008)

116
Table 3.2. Standard input data for a silt loam soil for the FAO-56 model
Soil Properties Symbol Units Silt Loam
Field Capacity water content θfc m3
/m3
0.36
Wilting Point water content θwp m3
/m3
0.22
Depth of Surface Soil Layer subjected to Drying by Evaporation Ze m 0.1
Total Evaporable Water (calculated) TEW mm 25
Readily Evaporable Water REW mm 8
(Source: FAO-56, Allen et. al, 1998.)
3.1 Comparison of FAO-56 Original, FAO-56 Enhanced, and Hydrus-1D Models to
Lysimeter
This section involved the comparison of the FAO-56 models and the Hydrus-1D
model using evaporation data measured by a weighing lysimeter under a bare soil condition.
The lysimeter was managed by Dr. J.L. Wright of the USDA-ARS (ret.) and was located near
Kimberly, Idaho (latitude 42.54o
, longitude -114.35o
, elevation 1195 m). Data for a bare soil
surface condition were collected from August 1st
to September 24th
, 1977. The weighing
lysimeter was 1.83 m on a side and 1.2 m deep and was filled with repacked Portneuf silt
loam soil during its construction in 1968. Details on the lysimeter installation and
management and soil characteristics are given in Wright (1968, 1981). Total precipitation
(precip.) plus irrigation (Irri.) during the observation period was 241 mm and total reference
evapotranspiration was 239 mm. A daily evaporation coefficient (Ke) was calculated by
dividing actual evaporation (E) over daily time periods by the associated reference
evapotranspiration (ETo). Kemax in the FAO-56 models was set to 1.2 for stage 1 evaporation
estimation to be congruent with the use of the grass ETo following recommendations by
Allen et al., (1998) and Allen (2011). Weather data were collected at the USDA-ARS
research center near Kimberly over a grassed surface (Wright 1981). TEW and REW were

117
set at 25 mm and 8 mm respectively for the silt loam soil texture and Ze was set at 0.1 m for
the FAO-56 model. The parameters of the FAO-56 were held constant for the silt loam soil
setting, as recommended values for REW and TEW fall in the range for the silt loam soil for
wide variety of soils. Ke max ETo was used to represent the potential E in Hydrus to be
consistent with the maximum value set for FAO-56 simulations.
Figure 3.2 shows Ke estimated from the original FAO-56, enhanced FAO-56 and the
Hydrus-1D models compared against lysimeter measurements for daily timesteps where the
Hydrus-1D model was applied for both silt loam and sandy clay loam soils to serve as a
means of bracketing the specific Portneuf silt loam of the lysimeter. Two different hini s of -3
m and -10 m were also used to bracket the likely hini of the lysimeter at the beginning of the
drying events. The sandy clay loam of Hydrus has a slightly larger soil water retention
parameter α as compared to the silt loam soil while the saturated hydraulic conductivity is 50
% larger than that of silt loam. The Hydrus-1D manual recommended 0.1 m3
/m3
residual soil
water content for a sandy clay loam texture and 0.067 m3
/m3
for silt loam soil.
Figure 3.2 shows that the Hydrus-1D runs simulated relatively large evaporation
spikes (Ke reaching 1.2) following small precipitation events that were on the order of < 5 to
10 mm whereas the original FAO-56 slab model did not show a substantial increase in Ke for
those same precipitation events. This observation is most obvious during the period of
09/14/1977 to 09/24/1977 with small precipitation events of less than 5 mm. In contrast, the
FAO-56 model enhanced with skin evaporation produced substantially increased estimates
for Ke following the small wetting events, with behavior similar to Ke from Hydrus-1D. On
the 09/16/1977 date having a light precipitation events of only a few mm, the lysimeter
observed Ke = 0.24 while the original FAO-56 estimated Ke = 0.05, and the enhanced FAO-

118
56 model and Hydrus-1D, for all soil types, estimated 1.2. This occurred because of timing
assumptions made for the daily precipitation additions.
On 08/26/1977, Ke based on lysimeter measurements was about 1.7, which is
uncharacteristically high. The cause of that value, which substantially exceeds the value
imposed in the models for Ke max, is uncertain. It was likely caused by a combination of
random lysimeter error, an increase in evaporation on the first day of drying due to the
presence of additional sensible heat energy available from the hot soil slab for evaporation
and potentially by an underestimation of the reference ET for that day. The values for Ke
from Hydrus-1D and the enhanced FAO-56 were at the upper limit of 1.2.
Ke from the Hydrus silt loam run with hini of -3 m tended to Ke during drying events.
When hini was lowered to -10 m for the silt loam soil, Ke decreased during drying event but
was still slightly larger than lysimeter measurements and slightly above estimates by the
enhanced FAO-56 model (fig. 3.2 and 3.3). Hydrus runs for the sandy clay loam soil with hini
of -3 m followed lysimeter measurements during drying events relatively closely, with slight
underestimation at times. Ke computed from Hydrus-1D for the two silt loam simulations ran
greater than that produced by the original FAO-56 slab model over most of the simulation
period and Ke computed for the sandy clay loam simulation was lower than that by the
original FAO-56 model.

119
Fig. 3.2. Daily average Ker based on the alfalfa reference from (a) the original FAO-56
soil and sandy clay loam (hini = -3 m and -10 m respectively) over a one-year period at
Kimberly, Idaho with daily calculation timesteps
Figure 3.3 shows the data of fig. 3.2 plotted in terms of daily average evaporation rate
where Hydrus was operated using an hini = -3 m. As with the Ke comparisons, all models
matched relatively closely to weighing lysimeter data, although the FAO method tended to
reduce more quickly to lower evaporation values late in drying periods as compared to the
lysimeter. This effect could be corrected by using a larger value for TEW, for example, 30
mm rather than 25 mm, as was shown by Allen (2011). Evaporation estimation improved
with the skin enhancement where the original FAO-56 model estimated smaller values for E
following light wetting events due to assumed mixing of the evaporation layer. The skin
enhancement, with the more shallow 'reservoir' at the surface, made the small water additions
immediately available for evaporation at higher rates. The Hydrus-1D simulations for the silt
0
25
50
75
1000.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
8/5/1977
8/7/1977
8/9/1977
8/11/1977
8/13/1977
8/15/1977
8/17/1977
8/19/1977
8/21/1977
8/23/1977
8/25/1977
8/27/1977
8/29/1977
8/31/1977
9/2/1977
9/4/1977
9/6/1977
9/8/1977
9/10/1977
9/12/1977
9/14/1977
9/16/1977
9/18/1977
9/20/1977
9/22/1977
9/24/1977
Precipitation(mm)
Ke
Precip (mm) FAO 56-skin Lysimeter Silt loam(-3m)
Silt Loam(-10m) Sandy Clay loam(-3m) FAO-56

120
loam soil slightly over-predicted evaporation during drying events as with the Ke
calculations. Simulations for the sandy clay loam with its higher saturated hydraulic
conductivity (314 mm/day) caused Hydrus-1D to estimate less evaporation in drying period
compared to the silt loam soil. It is because soil with higher saturated hydraulic conductivity
tends to have faster infiltration at drying period.
Fig. 3.3. Daily average evaporation from (a) the original FAO-56 model and (b) the
enhanced (skin-evaporation) FAO-56 model and from the Hydrus-1D for silt loam soil and
sandy clay loam (hini = -3 m and -10 m respectively) over a one-year period at Kimberly,
Idaho with daily input data
Figure 3.4 shows the results of the previous simulations expressed as a scatter plot of
daily evaporation between the FAO-56 model vs. the lysimeter data before and after
application of skin evaporation enhancement. The R2
was 0.83 before the enhancement while
it increased to 0.88 after enhancement. The root mean square error decreased from 1.1
mm/day to 0.85 mm/day. A majority of evaporation points fell below the 1:1 line with the
0
25
50
75
1000.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
8/5/1977
8/7/1977
8/9/1977
8/11/1977
8/13/1977
8/15/1977
8/17/1977
8/19/1977
8/21/1977
8/23/1977
8/25/1977
8/27/1977
8/29/1977
8/31/1977
9/2/1977
9/4/1977
9/6/1977
9/8/1977
9/10/1977
9/12/1977
9/14/1977
9/16/1977
9/18/1977
9/20/1977
9/22/1977
9/24/1977
Precipi+Irri.(mm)
Evaporation(mm)
Precip (mm) FAO 56-skin Silt Loam(-3m) Lysimeter
Silt loam(-10m) Sandy Clay loam (-3m) FAO-56

121
original FAO-56 model, whereas, after enhancement, these points approached the 1:1 line,
with the increased R2
.
a) b)
enhanced (skin-evaporation) FAO-56 model compared against lysimeter measurements for a
bare silt loam soil near Kimberly, Idaho with daily data during August and September 1977
Figure 3.5 shows a scatter plot of daily evaporation from Hydrus-1D model vs. the
lysimeter data for silt loam and sandy clay loam soils. The R2
was 0.78 and 0.81 for silt loam
and sandy clay loam soil respectively using -3 m hini at all nodes. The RMSE was 1.1
mm/day and 0.93 mm/day for the silt loam and sandy clay loam. The scatterplot shows the
slight overestimation of evaporation from Hydrus-1D for the silt loam soil and conversely the
underestimation for the sandy loam soil. Hydrus-1D simulations for the silt loam tended to
estimate higher than the lysimeter during dry down periods as previously noted. Some of the
overestimation may be due to using a relatively 'wet' hini for the entire soil profile which
enabled an extended period of upward flux of water to be simulated during the
surface drying.
y = 0.97x - 0.57
R² = 0.83
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Evaporation(mm)FAO56
Evaporation (mm) Lysimeter
y = 1.03x - 0.38
R² = 0.88
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Evaporation (mm) Lysimeter

122
a) b)
Fig. 3.5. Scatter plot of daily evaporation from Hydrus-1D for (a) silt loam soil (b) sandy
clay loam soil for hini = -3 m vs. lysimeter measurements near Kimberly, Idaho with daily
input data during August
Table 3.3 summarizes statistics for the comparison of original FAO-56; enhanced
FAO-56 and Hydrus-1D models vs. Lysimeter, along with cumulative evaporation rate. The
lowest cumulative evaporation estimation came from the original FAO-56 model (106 mm)
while the highest value came from Hydrus-1D for the silt loam soil (173 mm) with the initial
pressure head of -3 m. Cumulative evaporation from the enhanced FAO-56 model was about
123 mm, and the lysimeter measured 139 mm. Hydrus-1D for sandy clay loam estimated
about 127 mm.
y = 0.84x + 1.0
R² = 0.78
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Evaporation(mm)Hydrus-
SiltLoam(-3m)
Evaporation (mm) Lysimter
y = 0.90x + 0.03
R² = 0.81
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Evaporation(mm)Hydrus-
SandyClayLoam(-3m)
Evaporation (mm) Lysimter

123
Table 3.3. Summary of comparisons between the FAO-56 models, Hydrus-1D and lysimeter
for daily precipitation and reference evaporation inputs for the Kimberly, Idaho data
FAO-56 vs. Lysimeter
Model Parameters
Param.
Values
Cumulative
ET Slope Intercept R2 RMSE
mm mm/day
Original
FAO-56
REW 8 mm 106 0.97 -0.57 0.83 1.1
Enhanced
FAO-56
TEW 25 mm 123 1.03 -0.38 0.88 0.85
Lysimeter 139
Hydrus-1D vs. Lysimeter
Model Parameters
Param.
Values
Cumulative
ET, mm
Slope Intercept R2
RMSE
Hydrus-Silt
Loam(-3 m)
Initial Pressure
head
-3 m 173 0.84 1.09 0.78 1.18
Hydrus-
Sandy Clay
Loam(-3 m)
-3 m 127 0.90 0.03 0.81 0.93
3.2 Comparison of Results between FAO-56 and Hydrus-1D Models with Daily Data for
Theoretical Soils
In the previous section, Hydrus-1D simulations for the silt loam soil slightly over
predicted ET during drying events and slightly under predicted for the sandy clay loam when
compared to lysimeter measurements. In this section, simulations are described where both
FAO-56 and the Hydrus-1D models are applied using specific soil parameters for three
common soils in Hydrus-1D model runs and for a silt loam soil for FAO-56 model runs. The
most common soil type in southern Idaho is silt loam (Westermann and Tindall, 1995, City
of American Falls, 2009). The intent of this exercise was to evaluate the sensitivity of E from
Hydrus-1D to soil texture and behavior of the FAO model over an extended period of four
seasons, as well as impact of the skin enhancement. The one-year period included a range of
evaporative demands and wetting sequences. The product Kemax ETr was used to represent an
upper evaporation limit for both the FAO-56 and the Hydrus-1D models. Reference
evaporation was calculated using the ASCE Penman-Monteith equation for the tall (alfalfa)

124
reference for both the FAO-56 and the Hydrus-1D model runs. Calculations were made
using the RefET model (Allen, 2010) using daily and hourly weather data. Weather and
precipitation data were acquired from the Twin Falls AgriMet weather station (lat. =
42.54611 N, long. = 114.34527 W, elev. = 1195 m) for year 2002. The annual cumulative
reference evapotranspiration was 1560 mm and 1470 mm for daily and hourly calculation
time steps respectively, with annual precipitation of 154 mm.
Model run scenarios were conducted using the Hydrus-1D model for three hini
profiles. Simulations were made with an hini of -3.0 m representing approximate field
capacity, -10.0 m (-1.0 atmosphere) and -0.5 m (representing near saturation). The FAO-56
recommends a REW value for silt loam soil of 8 -11 mm, with 8 mm used in these
simulations, and TEW as 25 mm when Ze = 0.1m. The computed evaporation coefficient
(Ker) was based on the alfalfa reference, with a Kemax = 1.0 representing a maximum
evaporation rate. Hourly simulation results from the Hydrus-1D model were summed to daily
values prior to making comparisons with FAO-based simulations and prior to computing Ker
by dividing E by ETr to avoid numerical division errors while computing Ker caused by small
or zero values for evaporation and reference ET during night time.
3.2.1 FAO-56 vs. Hydrus-1D model with Initial Pressure Head of -3 m
This scenario compared simulated Ke from the original and enhanced FAO-56 models
with simulations by the Hydrus-1D model for silt loam and sandy clay loam soil for hini of -3
m, representing the field capacity water content. Results presented in fig. 3.6a show the daily
Ke time series during the simulation year. Ke is expressed as Ker in the figures, representing
the reference ETr basis. As previously demonstrated, the original FAO-56 model with the
mixed evaporation slab did not produce the characteristic short-duration evaporation spikes

125
associated with small precipitation events that were simulated by the enhanced FAO-56 and
Hydrus-1D model (fig. 3.6a ad 3.6b). For example, on 05/21/2002, following a 2.5 mm
precipitation, the Ker value computed by the original FAO-56 remained near zero, while the
Hydrus-1D model for a silt loam soil texture estimated Ke =1.0 which is the upper limit
specified for the model. With the skin evaporation enhancement, the FAO-56 model also
estimated Ker = 1.0 (fig. 3.6b). Likewise, on 09/17/2202, the original FAO-56 model
estimated Ker = 0.03 while the Hydrus-1D model and enhanced FAO-56 model both
estimated Ker = 0.8. In nearly all cases, the FAO-56 model with skin-evaporation
enhancement and Hydrus-1D model predicted similar Ker following precipitation events, and
with similar decay of Ker with time of drying. Residual Ker during long drying periods during
late November and December was estimated higher with the Hydrus-1D model when
potential evaporation rates were low and upward flux from the wet soil profile was simulated
to support a Ker averaging about 0.15. As in the lysimeter comparison, the Ker from the
Hydrus-1D model for sandy clay loam soil was smaller during the winter months compared
to the silt loam soil texture (fig. 3.6b) while matching simulations for the silt loam texture
during the rest of the year (fig. 3.6b).
The FAO-56 model has no provision for simulating upward flux from below the
evaporation slab. Instead, the average impact of upward flux is accounted for by specifying a
larger depth for Ze, based on field observation of total evaporation between wetting events.
Comparisons with Hydrus-1D results for an initial pressure condition near field capacity
suggests that an upward flux provision may not be necessary for estimating evaporation from
the soil surface, provided an appropriate value for Ze (for example, Ze = 0.1 – 0.15 m) is used
in the FAO-56 model to represent the effective depth of the evaporative layer.

126
a)
b)
Fig. 3.6. Daily average Ker based on the alfalfa reference from (a) the original FAO-56
and sandy clay loam textures using hini = -3 m over a one-year period at Kimberly, Idaho
with daily precipitation and weather data inputs
Figure 3.7 shows daily evaporation rate from the models associated with the Ker
shown in fig. 3.6, with hini = -3.0 m in the Hydrus-1D run. As with the Ker comparison, the
original FAO-56 (fig. 3.7a) did not produce the same increase in evaporation rate following
relatively small precipitation events (5 to 10 mm) as did the enhanced FAO-56 and Hydrus-
1D models (fig. 3.7b). On a number of days following precipitation events, estimated E was
0
5
10
15
200.0
0.2
0.4
0.6
0.8
1.0
1.2
Precipitation(mm)
Ker
Precip (mm) FAO 56 Hydrus- Silt Loam (-3m)
0
5
10
15
200.0
0.2
0.4
0.6
0.8
1.0
1.2
1/1/2002
1/16/2002
1/31/2002
2/15/2002
3/2/2002
3/17/2002
4/1/2002
4/16/2002
5/1/2002
5/16/2002
5/31/2002
6/15/2002
6/30/2002
7/15/2002
7/30/2002
8/14/2002
8/29/2002
9/13/2002
9/28/2002
10/13/2002
10/28/2002
11/12/2002
11/27/2002
12/12/2002
12/27/2002
Precipitation(mm)
Ker
Precip (mm) FAO 56 - skin Hydrus-Silt Loam (-3m) Hydrus-Sandy Clay Loam (-3m)

127
nearly identical between the Hydrus-1D and enhanced FAO-56 model due to the use of the
same value for Kemax and as an upper limit on potential E (fig. 3.7b).
a)
b)
Fig. 3.7. Daily evaporation based on the alfalfa reference from (a) the original FAO-56
and sandy clay loam textures using hini = -3 m over a one-year period at Kimberly, Idaho
with daily precipitation and weather data inputs
Cumulative evaporation matched closely among the models for the silt loam soil
throughout the year, where cumulative evaporation over the year totaled 166 mm and 175
mm for the enhanced FAO-56 and the Hydrus-1D models respectively (fig. 3.8). The
0
5
10
15
200.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Precipitation(mm)
Evaporation(mm)
Precip (mm) FAO 56 Hydrus-Silt Loam (-3m)
0
5
10
15
200.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
1/1/2002
1/16/2002
1/31/2002
2/15/2002
3/2/2002
3/17/2002
4/1/2002
4/16/2002
5/1/2002
5/16/2002
5/31/2002
6/15/2002
6/30/2002
7/15/2002
7/30/2002
8/14/2002
8/29/2002
9/13/2002
9/28/2002
10/13/2002
10/28/2002
11/12/2002
11/27/2002
12/12/2002
12/27/2002
Precipitation(mm)
Evaporation(mm)
Precip (mm) FAO 56 - skin Hydrus-Silt Loam(-3m) Hydrus-Sandy Clay Loam(-3m)

128
cumulative evaporation from Hydrus-1D for sandy clay loam was smaller i.e. 142 mm. The
enhanced FAO-56 and Hydrus-1D model for the silt loam texture exceeded the 154 mm of
recorded precipitation due to evaporation of assumed initial water stored in the soil profilem.
Fig. 3.8. Commutative evaporation based on the alfalfa reference from the FAO-56 model
with the skin evaporation enhancement and from the Hydrus-1D for silt loam and sandy clay
loam soil (hini = - 3 m) over a one-year period at Kimberly, Idaho with daily calculation
timesteps
Figures 3.9a and 3.9b show scatter plots of daily averaged evaporation amounts
estimated by the original FAO-56 (fig. 3.9a) and by the enhanced FAO-56 model (fig. 3.9b)
vs. daily averaged evaporation amounts from the Hydrus-1D model with hini = -3 m for the
silt loam texture. There was significant improvement in the agreement between E estimated
from the FAO-56 with the skin evaporation enhancement and Hydrus-1D (fig. 3.9b) where
the coefficient of determination (R2
) increased from 22% to 88%. With the enhancement,
some of the evaporation estimates fell above the 1:1 line, but with a majority of estimates
were located near the 1:1 line. The RMSE between the original FAO-56 and the Hydrus-1D
for the silt loam texture with hini = -3 m was about 0.67 mm/day, whereas after the
enhancement, the RMSE decreased by more than 50% to 0.27 mm/day. There are different
0
5
10
15
200.0
50.0
100.0
150.0
200.0
250.0
300.0 1/1/2002
1/16/2002
1/31/2002
2/15/2002
3/2/2002
3/17/2002
4/1/2002
4/16/2002
5/1/2002
5/16/2002
5/31/2002
6/15/2002
6/30/2002
7/15/2002
7/30/2002
8/14/2002
8/29/2002
9/13/2002
9/28/2002
10/13/2002
10/28/2002
11/12/2002
11/27/2002
12/12/2002
12/27/2002
Precipitation(mm)
Cum.Evap.(mm)
Precip (mm) FAO 56 - skin HYDRUS 1D -3m Hydrus- Sandy Clay Loam(-3m)

129
parameters in the Hydrus-1D model as well as the FAO-56 that can create variability in the
evaporation computation. However, standard values were used with both models for this
exercise.
a) b)
enhanced (skin-evaporation) FAO-56 model vs. daily evaporation estimated by the Hydrus-
1D using an hini = -3 m for a bare silt loam soil over a one-year period at Kimberly, Idaho
with a daily precipitation and weather data inputs
3.2.2 FAO-56 vs. Hydrus-1D Model with Initial Pressure Head of -0.5 m and -10 m
In this scenario, simulations were conducted for initial pressure heads of -0.5 m (near
saturation) and -10 m (1 atmosphere). Because of the relatively wet soil water profile
assumed by the hini of -0.5 m for all profile nodes, the Hydrus-1D model estimated slightly
larger values for evaporation for the silt loam texture throughout the simulation period as
compared to both the original and enhanced FAO-56 models. The higher estimation by
Hydrus-1D was due to upward flow from the wet soil profile and higher initial water storage
coupled with relatively higher hydraulic conductivity. The total cumulative evaporation from
the Hydrus-1D model decreased when hini was reduced to -3.0 m (Table 3.4).
The second Hydrus-1D run using an hini of -10 m at all nodes explored the influence
of a relatively dry initial soil profile on simulated evaporation in the dry Idaho climate. In
y = 0.43x + 0.22
R² = 0.22
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Evaporation (mm) Hydrus-Slit Loam
FAO-56 FAO-56(skin)

130
this scenario, Hydrus-1D produced smaller Ker during the initial months (January to
February) as compared to the FAO-56 model and as compared to previous Hydrus-1D runs
using -0.5 m and -3 m. The Ker from Hydrus-1D model decayed more quickly following
wetting events due to the dry subsoil and less water storage. The results matched closely
with those from the FAO-56 and with Hydrus-1D model simulations for all hini except for the
winter months. The results of simulations by Hydrus-1D model for hini = -10 m are
summarized in table 3.4.
daily precipitation and weather data from Kimberly Idaho for year 2002 and a silt loam soil
texture where regressions are expressed against Hydrus-1D for three initial pressure heads
Model Parameters Values
Cumulative
ET Slope Intercept R2 RMSD
mm mm/day
Original
FAO-56
REW 8 mm 158 0.43 0.22 0.22 0.67
Enhanced
FAO-56
TEW 25 mm 166
1.08 -0.06 0.88 0.27
Hydrus-Silt
Loam Soil
Initial
Pressure head
-3 m 175
Original
FAO-56
REW 8 mm 158 0.45 0.16 0.26 0.68
Enhanced
FAO-56
TEW 25 mm 166
1.05 -0.17 0.88 0.30
Hydrus-Silt
Loam
Initial
Pressure head
-0.5 m 220
Enhanced
FAO-56
REW 8 mm
166
1.06 0.03 0.83 0.33TEW 25 mm
Hydrus- Silt
Loam
Initial
Pressure head
-10 m 146
Annual measured precipitation = 154 mm
3.3 Comparison of Results between FAO-56 and Hydrus-1D Model with Hourly Timesteps
A second set of analyses was conducted using hourly precipitation and weather data
and hourly calculation timesteps with the FAO-56 models. The Hydrus-1D utilizes its own
automated determination of calculation timestep depending on the rate of change of state
variables. The same generalized silt loam soil texture was utilized for the models, with a

131
TEW of 25 mm and REW of 8 mm used in FAO-56 simulations. The hourly estimates for E
from both the Hydrus-1D model and the FAO-56 model were summed to daily values prior
to making comparisons to avoid calculating large ratios for Ker during periods of near zero
ETr during nighttime and early morning periods, the Hydrus-1D model was run using hini = -
0.5 m and -3.0 m.
The hourly results followed similar trends observed for the daily data and calculation
timesteps for both initial pressure heads. For the hini -3.0 m condition, cumulative annual
evaporation from the enhanced FAO-56 model and the Hydrus-1D models totaled 162 mm
and 169 mm respectively over the annual period. The R2
between the original and enhanced
FAO-56 model and Hydrus-1D models was 0.54 and 0.83 respectively (Table 3.5). Even
though the R2
increased significantly, the RMSD did not change noticeably following the
skin evaporation enhancement for hini of -3.0 m (Table 3.5), where the RMSD between the
original FAO-56 with hourly calculation timesteps and the Hydrus-1D was about 0.38
mm/day and 0.378 mm/day after the enhancement. For a hini of -0.5 m in Hydrus-1D, the
FAO-56 model after enhancement produced cumulative evaporation of 162 mm, showing a
slight increase compared to the original FAO-56 model (Table 3.5). The R2
between the
FAO-56 models and Hydrus-1D run prior to and following enhancement was 0.58 and 0.82
respectively. Total cumulative annual evaporation estimated by Hydrus-1D increased by
about 40 mm when hini was increased from -3.0 to -0.5 m. The hini = -0.5 m represents a very
wet condition near saturation.

132
hourly precipitation and weather data and hourly calculation time steps for a silt loam soil
texture at Kimberly, Idaho during year 2002
Cumulative
ET Slope Intercept R2 RMSD
mm mm/day
Original FAO-56 REW 8 mm 157 0.85 0.03 0.54 0.38
Enhanced FAO-56 TEW 25 mm 162
1.41 -0.206 0.83 0.378
Hydrus-Silt Loam
Initial Pressure
head
-3 m 169
Original FAO-56 REW 8 mm 157 0.83 -0.054 0.58 0.40
Enhanced FAO-56 TEW 25 mm 162
1.33 -0.33 0.82 0.39
Hydrus-Silt Loam
Initial Pressure
head
-0.5 m 212
3.4 Sensitivity Analyses on REW and Soil Type
3.4.1 Daily Calculation Timesteps
A sensitivity analysis was conducted to evaluate the influence of simulating different
soil types in the Hydrus-1D model. As in section 3.1 with the lysimeter comparisons,
Hydrus-1D simulations for a sandy clay loam soil with initial pressure head of -3 m had less
evaporation compared to the silt loam soil. The Hydrus-1D model with hini = -3.0 m
produced an R2
= 0.89 and 0.81 for the silt and sandy clay loam soils respectively, when
compared to the enhanced FAO-56 model. The RMSD value for the silt soil was about 0.26
mm/day between the enhanced FAO-56 model and the Hydrus-1D model with -3 m hini. For
the sandy clay loam soil, RMSD increased to 0.34 mm/day for hini = -3.0 m showing
increased departure between the two model types.
Sensitivity analyses were carried out on the enhanced FAO-56 model to assess the
sensitivity to changes in the model parameter REW describing the depth of evaporation
associated with stage 1 drying. REW is often estimated from observations but can be a
somewhat subjective estimate. A second set of sensitivity analyses was conducted for the silt

133
loam soil by changing values for both REW and TEW in the enhanced FAO-56 model. When
REW was decreased to 5 mm, the agreement with Hydrus-1D for -3.0 m hini was slightly
improved (R2
= 0.90). RMSD was also the lowest among the other daily simulations with a
value of 0.23 mm/day. Cumulative evaporation from the enhanced FAO-56 model decreased
by only 2 mm since the value for TEW remained unchanged, and the FAO-56 is constrained
by a conservation water balance. Decreasing REW for the silt loam soil decreased the
differences between the model and Hydrus-1D run and increased R2
. Some of the
evaporation estimates that fell above 1:1 line earlier came closer to the 1:1 line with the
smaller value for REW. When REW was increased to 11 mm, the R2
value decreased slightly
to 0.85 and the RMSD value increased to 0.32 mm/day.
Field capacity and wilting point soil water contents were changed to vary the values
for TEW for the silt loam soil but there were no significant changes in results, primarily due
to the relatively small depths of wetting events that tended to capture and hold most of the
precipitation for all levels of TEW tested. In regions having relatively large depths of
precipitation per wetting event, the sensitivity of estimated cumulative evaporation to TEW
is expected to be much greater. R2
values ranged from 0.8 to 0.9 and RMSD ranged from
0.23 mm/day to 0.34 mm/day between the enhanced FAO-56 model and the Hydrus-1D
model over a wide range of the simulations with different soil types and ranges of REW and
TEW. With daily calculation time steps, the cumulative ET from the enhanced FAO-56
model was 159 mm and R2
was about 0.85 compared to Hydrus-1D when TEW was
decreased about 25% (i.e. 18 mm) with REW set at 8 mm. The agreement between the
models was slightly reduced and RMSD was increased to 0.32 mm/day. The R2
and RMSD
slightly improved for daily calculation timesteps with decreased REW.

134
Sensitivity was also explored for the impact of using smaller vertical nodes in the
Hydrus-1D model, where a grid size of half millimeter was specified. Differences in results
from runs employing the 3 mm node distance were very small. Table 3.6 summarizes the
results of the sensitivity analyses.
Table 3.6. Summary of sensitivity analyses on impacts of soil texture in Hydrus-1D
simulations with comparisons against the enhanced FAO-56 model using daily weather and
precipitation inputs
Cumulative
ET Slope Intercept R2
RMSD
mm/day
mm
Silt Soil
Enhanced
FAO-56
REW 8 mm
166
1.08 -0.08 0.89 0.26TEW 25 mm
Hydrus-1D Initial Pressure head -3 m 182
Sandy Clay loam Soil
Enhanced
FAO-56
REW 8 mm
166
1.057 0.044 0.81 0.34TEW 25 mm
Silt Loam Soil
Enhanced
FAO-56
REW 5 mm
164
1.001 -0.029 0.90 0.23TEW 25 mm
Enhanced
FAO-56
REW 8 mm
159
1.119 -0.099 0.85 0.32TEW 18 mm
Enhanced
FAO-56
REW 11 mm
166
1.105 -0.073 0.85 0.32TEW 25 mm
3.4.2 Hourly Calculation Timesteps
Sensitivity analyses were also carried out on REW in the enhanced FAO-56 model

135
for hourly precipitation data and calculation timesteps. The REW was lowered to a value of 3
mm, which represents a very quick drying surface and short duration of stage 1 drying (when
Ker = 1). R2
improved to 0.89 as compared to R2
= 0.82 for REW = 8 mm when compared to
the Hydrus-1D model with an hini of -0.5 m. Similarly, when compared to the Hydrus-1D run
for an hini of -3.0 m, the R2
increased from 0.83 to 0.90 when lowering REW to 3 mm. The
RMSD decreased to the value of 0.26 mm/day and 0.22 mm/day for hini of -0.5 m and -3 m
respectively. Decreasing REW seems to have a greater impact for hourly timestep calculation
as compared to using daily timesteps due to finer resolution in simulating the duration of
stage 1 drying and the particular shape of the drydown curve vs. time. For example, on the 4th
of August, evaporation computed from the enhanced FAO-56 reduced from 6.6 mm/day
when REW was 8 mm to 3.6 mm/day with REW of 3 mm/day. The evaporation estimated by
Hydrus-1D was 2.8 mm/day for -0.5 m hini and 2.7 mm/day for -3.0 m hini. Table 3.7
summarizes the results of hourly timesteps data for the reduced REW.
Table 3.7. Summary of sensitivity analyses on reducing REW in the enhanced FAO-56
model for hourly simulations for a silt loam soil texture vs the Hydrus-1D model for weather
data from Kimberly, ID during 2002
Cumulative
ET Slope Intercept R2
RMSD
mm mm/day
FAO-56
REW 3 mm
161
-0.121 1.219 0.90 0.22TEW 25 mm
Hydrus-Silt
Loam
Initial Pressure
head
-3.0 m 169
FAO-56
REW 3 mm
161
-0.225 1.146 0.89 0.26
TEW 25 mm
Hydrus-Silt
Loam
Initial Pressure
head
-0.5 m 212
Figure 3.10 shows a scatterplot of daily evaporation computed using daily timesteps
and vs. using hourly timesteps data summed to daily values for enhanced FAO-56 model for

136
hini -3.0 m for silt loam soil. The results showed that daily timesteps evaporation generated
small variations to hourly timesteps values with R2
of 0.86 and RMSD of 0.28 mm/day.
Fig. 3.10. Scatter plot of daily evaporation from the enhanced FAO-56 model with hourly
calculation timesteps summed daily and daily calculation timesteps for hini = - 3 m for a bare
silt loam soil over a one-year period at Kimberly, Idaho
4.0 Conclusions
The study illustrated the benefits of the skin layer evaporation enhancement of Allen
(2011) incorporated into the FAO-56 model in simulating E during the hours or days
immediately following shallow wetting events. The improved behavior was confirmed by
comparison of model results to field lysimeter measurements for bare soil conditions and
against the Hydrus-1D model. The simple FAO-56 model enhanced with skin evaporation
layer was relatively successful in producing short-lived evaporation spikes associated with
small precipitation events when compared to the more complex numerical solutions used in
the Hydrus-1D model and with the lysimeter data. Sensitivity analyses indicated that overall
accuracy of the evaporation coefficient Ker for smaller precipitation events in the FAO-56
model are somewhat sensitive to the values for TEW and REW terms. The coefficient of
y = 0.90x + 0.034
R² = 0.86
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Evaporation(mm)FAO-56hourly
timesteps
Evaporation (mm) FAO-56 Daily timesteps

137
agreement for Ker was slightly higher for daily timesteps as compared to hourly calculation
time steps with the FAO-56 model. Otherwise, simulation results were similar in all
applications, indicating that daily calculation timesteps are sufficient when using the skin
evaporation enhancement that implicitly identifies the transition time between stage 1 and
stage 2 drying. The evaporation of initial water stored in the soil profile and initial water
potential hini of the soil played an important role in total cumulative E estimated by Hydrus-
1D and the duration of drydown following wetting events. The simulation results showed that
Hydrus-1D simulations to be relatively sensitive to saturated hydraulic conductivity. In
general, the FAO-56 model with skin enhancement and implicit stage 1 – stage 2 transition
estimation produces estimates of daily E that are similar to those of Hydrus-1D using daily or
hourly calculation timesteps for the FAO-56 model. This is a useful finding and indicates
that models incorporating the FAO-56 procedure will generally provide accurate accounting
for E.

138
References
Allen R.G., 2011. Skin layer evaporation to account for small precipitation events-An
enhancement to the FAO-56 evaporation model. Agricultural Water Management.
Allen, R. G., Pereira L. S., Raes, D., and Smith, M., 1998. Crop evapotranspiration:
Guidelines for computing crop requirements.” Irrigation and Drainage Paper 56,
FAO, Rome, Italy.
Allen, R.G., Pereira, L.S., Smith, M., Raes, D. and Wright, J. L., 2005. FAO-56 dual crop
coefﬁcient method for estimating evaporation from soil and application extensions. J.
Irrig. Drain Eng. ASCE 131 (1), 2–13.
Burt, C. M., Mutziger, A. J., Howes, D. J., and Solomon, K. H., 2002. Evaporation from
irrigated agriculture land in California. A study funded by the CALFED and the
California State University! Agricultural Research Initiative. Available on the web
page (http:// www.itrc.orglreports!reportsindex.html). Irrigation Training and Re-
search Center. California Polytechnic State Univ. San Luis Obispo, Calif
FAO-56 modeled evaporation from bare soil. J. Irrig. Drain. Engrg. ASCE 131 (1),
59–72.
Howell, T. A., Steiner, J. L., Schneider, A. D., and Evett, S. R., 1995. Evapotranspiration of
irrigated winter wheat-Southern Plains. Trans. ASAE, 38(3), 745-759.Trans. ASAE,
38(3), 745~759.
http://websoilsurvey.nrcs.usda.gov/app/WebSoilSurvey.aspx) <accessed 2012>.
http://www.groundwatersoftware.com/v9_n5_hydrus.htm <accessed 2010>.
http://www.pc-rogress.com/en/Default.aspx?HYDRUS 1D-faq-01-10 <accessed 2010>.

139
Hunsaker, D. J., Pinter, P. J. and Cai, H., 2002. Alfalfa basal crop coefficients for FAO-56
procedures in the desert regions of the southwestern U.S.Transactions of the ASAE,
45 (6) (2002), pp. 1799–1815.
Mutziger, A. J., Burt, C. M., Howes, D. J. and Allen, R. G., 2005. Comparison of measured
and FAO-56 Modeled Evaporation from Bare Soil.
Šimůnek, J., Šejna, M., Saito, H., M. Sakai, and van Genuchten, M. T., 2008. The HYDRUS-
1D Software Package for Simulating the One-Dimensional Movement of Water,
Heat, and Multiple Solutes in Variably-Saturated Media. Department of
environmental sciences, University of California Riverside, Riverside, California.
Weeks B., and Wilson, G. W., 2006. Prediction of evaporation from soil slopes. Canadian
Geotech J. 43: 815-829.
Wright, J. L., 1982.
Wright, J. L., 1991.

140
Appendix-A: Parameters and Variables in BATANS

141
Parameters and Variables
Fraction of vegetation cover is computed assuming a linear relationship between
normalized difference vegetation index (NDVI) and fraction of vegetation cover using eqn.
(1).
(1)
where fc is fraction of vegetation cover, NDVI is normalized difference vegetation index,
NDVImax is the maximum value of NDVI which is taken as 0.80 and NDVImin is the
minimum value of NDVI which is 0.15.
Actual vapor pressure is computed from pressure and specific humidity using eqn.
(2).
(2)
where ea is actual vapor pressure (kPa), P is pressure (kPa) and qa is specific humidity
(kg/kg).
Height of vegetation is computed according leaf area index suggested by Allen et al.,
2010 (eqn. (3)).
{
(3)
where hc is height of vegetation (m) and LAI is leaf area index.
Zero plane displacement is computed from eqn. (4) (Choudhury and Monteith, 1988).
(4)
where d is zero plane displacement (m) and X is parameter for calculating zero plane
displacement.

142
X parameter of the zero plane displacement is calculated using eqn. (5).
AI (5)
where cd is mean drag coefficient for individual leaves which is used as a 0.2 (Shaw and
Pereira, 1982; Choudhury and Monteith, 1988). Shuttleworth and Gurney, 1990 recommends
the value of cd as 0.07.
Attenuation coefficient is computed using eqn. (6) (Choudhury and Monteith, 1988).
{
(6)
where n is attenuation coefficient.
Rooting depth which is approximately used as 1 or 2 m according to land use classes
according to eqn. (7).
{ (7)
where droot is rooting depth (m).
Soil Portion Surface Energy Balance
Outgoing long wave radiation of soil portion is computed using eqn. (8).
(8)
where RL_s↑ is outgoing longwave radiation of soil portion (W/m2
), Ts is surface temperature
of soil portion (K), εo_s is broad band emissivity of soil portion, and σ is Stefan-Boltzmann
constant (W/m2
/K4
).
Surface temperature of soil portion is computed inverting sensible heat flux
equation of soil portion as an initial guess (eqn. (9)).
(9)

143
where Hs is sensible heat flux of soil portion (W/m2
), rah is aerodynamic resistance from
vegetation height to blending height (s/m), ras is aerodynamic resistance between the
substrate and vegetation height (s/m), cp is the specific heat capacity of air (J/kg K), ρa is the
atmospheric density (kg/m3
), and Ta is air temperature at blending height (K).
Soil surface net radiation of soil portion is computed from eqn. (10).
( ) (10)
where Rn_s is net radiation of soil portion (W/m2
), Rs↓ is incoming shortwave radiation
(W/m2
), RL↓ is incoming longwave radiation (W/m2
), RL_s↑ is outgoing longwave radiation of
soil portion (W/m2
), and αs is soil portion albedo.
When the surface is fully covered with vegetation, the understory aerodynamic
resistance (ras (fc =1)) between zos and d + zom is computed as according to Choudhury and
Monteith (1988) using eqn. (11).
[ ( ) [ [ ]]]
(11)
where ras (fc=1) is aerodynamic resistance between the substrate and vegetation height (s/m) of
fully vegetated portion (fc =1), zom is roughness length of momentum (m), and K (h) is eddy
diffusion coefficient of momentum correction.
The eddy diffusion coefficient of momentum correction (K (h)) from zom to hc has
been computed using eqn. (12) (Choudhury and Monteith, 1988).
( )
(12)
where K(h) is eddy diffusion coefficient of momentum correction, ψm_30m is stability
correction of momentum for blending height, z is height of measurements of wind speed and

144
air temperature at blending height (m), and uz is wind speed at blending height (m/s).
When the surface is completely bare, the aerodynamic resistance (ras (fc =0)) from zos to
d + zom can be written using eqn. (13).
[ ( ) ] [ ( ) ]
(13)
where ras (fc=1) is aerodynamic resistance between the substrate and vegetation height (s/m) for
soil portion (fc =0), ψhd+zom is stability correction of heat from zom to d + zom, and zos is
minimum roughness length (m).
Finally, aerodynamic resistance for a partial surface is computed by weighting by fc
in a parallel combination of resistances (eqn. (14)).
(14a)
(14b)
where ras (int) is aerodynamic resistance between the substrate and vegetation height (s/m) of
partial surface, m is an exponent on fc where m is 0.5 or 0.2 to give more weight to the ras
from the vegetation component to increase rah and in turn increase Tc.
Ground heat flux of soil portion is estimated either from Hs or Rn_s using eqn. (15)
(Allen et al., 2012). The comma in the ‘max ()’ function selects the maximum of the two
values.
(15)
where Gs is ground heat flux of soil portion (W/m2
).

145
The heat flux of water is computed using eqn. (16) (Allen et al., 2012). Surface
energy balance of water portion is carried out in the soil portion of surface energy balance
replacing LEs to LEw equation. ras is assumed to be zero and rah is computed from zom to
blending height. The calculation procedure of rah of water portion is shown in table A.1.
(16)
where Gw is heat flux of water (W/m2
).
Finally, the sensible heat flux of soil portion is computed as a residual from the
surface energy balance of soil portion using eqn. (17).
(17)
where LEs is latent heat flux of soil portion (W/m2
).
Vegetation Portion Surface Energy Balance
Outgoing long wave radiation of vegetation portion is computed using eqn. (18).
(18)
where RL_c↑ is outgoing long wave radiation of vegetation portion (W/m2
), Tc is surface
temperature of vegetation portion (K), and εo_c is broad band emissivity of vegetation portion.
Surface temperature of vegetation portion is computed from eqn. (19).
(19)
where Hc is sensible heat flux of vegetation portion (W/m2
) and rac is bulk boundary layer
resistance of the vegetative elements in the vegetation (s/m).
Net radiation of vegetation portion (Rn_c) is computed using eqn. (20).
( ) (20)
where αc is albedo of vegetation portion and RL_c↑ is outgoing longwave radiation of
vegetation portion (W/m2
).

146
Sensible heat flux of vegetation portion is computed and updated as a residual using
eqn. (21). It is assumed that there is no ground heat flux in the vegetated portion.
(21)
where λc is latent heat flux of vaporization of vegetation portion (J/kg) and T is transpiration
(mm/hr).
The bulk boundary layer resistance of the vegetative elements in the vegetation is
computed using eqn. (22).
(22)
where rac is bulk boundary layer resistance of the vegetative elements in the vegetation (s/m)
and rb is mean boundary layer resistance per unit area of vegetation (s/m).
Mean boundary layer resistance per unit area of vegetation (rb) is taken as 25 s/m
(Denmead 1976, Uchijima 1976 and Shuttleworth and Wallace, 1985). rb/ 2 is mean leaf
boundary layer resistance of amphistomatous leaves per unit surface vegetation (Ahuja et al,
2000). The value of rac in close vegetation is less than 10 s/m (Ahuja et al, 2000). Sensitivity
showed that the surface energy flux parameters are less sensitive with rb. rb varied from 1 to
18 s/m with seasonal mean of 10 s/m for long leaf width corn (Farahani and Baush, 1995).
Combined Section
Monin Obukhov length is computed from the combined value of sensible heat flux
(H) using eqn. (23). H is computed combining soil and vegetation portion with adjusted fc.
(23)
where L is Monin Obukhov length (m), H is combined sensible heat flux (W/m2
), u* is

147
friction velocity (m/s), g is acceleration due to gravity i.e. 9.81 (m/s2
), k is von Karman
constant i.e. 0.41.
For computing rah from d + zom to z, a standard equation of aerodynamic equation
eqn. (24) is used and modified for different scenarios as in table A.1.
[ ( ) ] [ ( ) ( )] (24)
where ψh_30m is stability correction of heat for blending height and Z1 is integration constant
(m).
For the smooth transition of rah, integration constant (Z1) was interpolated based on
fraction of cover of vegetation (fc). When fc is equal to 1, Z1 is Z1_full i.e. 0.1 * zom. For the
partial covered area with fc < 0.6, Z1 is Z1_par i.e. hc – d. For smooth transition, when fc is in
between 0.6 to 1, linear interpolation is done between Z1_par and Z1_full.
{
( )
(25)
In a fully vegetated surface scenario, the transfer begins at the top of the vegetation
and integration constant Z1 is zoh i.e 0.1 zom. Stability correction of heat integration constant
i.e height (ψh (zoh/L)) is neglected as it is small compared to other integration constants
(scenario 1, table A.1). But in partially covered vegetated surface scenario, integration
constant Z1 becomes (hc - d) which is significant i.e. ψh {(hc – d)/L} and it has been included
in rah equation (scenario 2, table A.1). For water bodies, d and (ψh (zoh/L) is considered as
zero (scenario 3, table A.1) as integration is done from zom to blending height.

148
Table A.1. Computation strategy for aerodynamic resistance (rah) from vegetation height to
blending height
Scenario Cover type d zom Z1 Height of
measurement
( )
1 Fully covered
with vegetation
d zom zoh d + zom to z Neglected
2 Partially covered
with vegetation
d zom hc - d d + zom to z ( )
3 Water Bodies 0 zom zom zom to z 0
Aerodynamic resistance (rah) for fully vegetated and partially vegetated conditions is
shown in eqn. (26).
{
[ ( ) ] [ ( ) ]
[ ( ) ] [ ( ) ]
(26)
Parameter X of stability correction for 30 m is computed from eqn. (27).
[ ]
(27)
Correction of momentum for 30 m (ψm_30m) is computed from eqn. (28).
{
{[ ]} [
( )
]
( )
(28)
Correction of heat for 30 m (ψh_30m) is computed from eqn. (29).
{
( )
( )
(29)
Parameter X for d + zom is computed from eqn. (30).

149
[ ]
(30)
Correction of heat for d + zom (ψhd+zom) is computed from eqn. (31).
{
( )
( )
(31)
Parameter X for hc – d is computed from eqn. (32).
[ ]
(32)
Correction of heat for hc – d (ψhhc-d) is computed from eqn. (33).
{
( )
( )
(33)

150
References
Ahuja, L., Rojas, K. W. and Hanson, J. D., 2000. Root Zone Water Quality
Model: Modelling Management Effects on Water Quality and Crop Production.
Water Resources Publication.
Farahani, H.J., & Bausch, W.C., 1995. Performance of Evapotranspiration models for maize
–bare soil to closed canopy. Transactions of the ASAE, 38(4), 1049-1059.
Denmead, O.T., 1976. Temperate Cereals. In: J.L. Monteith (Editor), Vegetation and the
Atmosphere, Vol. 2. Academic Press, London, pp. 1—33.
Uchijima, Z., 1976. Maize and rice. Vegetation and the Atmosphere Vol. 2, J. L. Monteith
(Eds.), 33-64.

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