Hydrology_Lecture note on chapter two about precipitation occurrence

2. Precipitation
2.1 Precipitation mechanism
2.2 Precipitation measurement
2.3 Precipitation Data Analysis
2.4 Basin average rainfall
2.5 Depth area duration analysis
2.6 Intensity – duration – frequency analysis (IDF)

2. Precipitation
Why do we study precipitation?
 Precipitation can be in the form of rainfall, snowfall or in other forms.
 Precipitation is the source of all waters which enters the land and flows as
overland flow. Overland flow discharges into the streams and then lakes or
ocean.
Hydrologists need to understand how the amount, rate, duration, and quality of precipitation
are distributed in space and time in order to assess, predict, and forecast hydrologic
responses of a catchment.

Why do we study precipitation?

2.1 Precipitation mechanism
 Three mechanisms are needed for formation of precipitation
 Types of Precipitation
a) Lifting and Cooling: Lifting of air mass to higher altitudes causes cooling of air
b) Condensation: Conversion of water vapor into liquid droplets
c) Droplet Formation: Growth of droplets is required if the liquid water present in
a cloud is to reach ground against the lifting mechanism of air
 Depending upon the way in which the air is lifted and cooled so as to cause
precipitation, we have three types of precipitation, as given below:
a) Cyclonic Precipitation
b) Convective Precipitation
c) Orographic Precipitation

a) Cyclonic Precipitation
 Cyclonic precipitation is caused by lifting of an air mass due to the pressure difference.
Cyclonic precipitation may be either frontal or non-frontal cyclonic precipitation.
i) Frontal precipitation
 It results from the lifting of warm and moist
air on one side of a frontal surface over a
wedge of colder denser air.
 A front may be warm front or cold front
depending upon whether there is active or
passive accent of warm air mass over cold
air mass.

 Cold front:
o A colder, denser air mass lifts the warm, moist air ahead of it.
o As the air rises, it cools and its moisture condenses to produce clouds and
precipitation.
o Due to the steep slope of a cold front, forceful rising motion is often produced,
leading to the development of showers and occasionally severe thunderstorms.

 Warm front:
o The warm, less dense air rises up over the colder air ahead of the front.
o The air cools as it rises and its moisture condenses to produce clouds and
precipitation
o Warm fronts move more slowly than cold fronts, so the rising motion along
warm fronts is much more gradual.
o Warm front precipitation is generally light to moderate.

ii) Non-frontal cyclonic precipitation
 If low pressure occurs in an area (called cyclone),
air will flow horizontally from the surrounding
area (high pressure), causing the air in the low-
pressure area to lift.
 When the lifted warm-air cools down at higher
attitude, non-frontal cyclonic precipitation will
occur.

b) Convectional precipitation
 Convectional or convective rainfall results when heating of the ground surface by
the sun causes warming of the air, and locally strong vertical air motions occur.
 If the air is thermally unstable, it continues to rise and the resulting cooling,
condensation and cloud formation may lead to short-term and locally intense
precipitation.
 In tropical areas, due to the greater heating,
the resulting precipitation may be much more
intense, and short–lived.
 Such rainfall is dependent on heating, and
moistening of the air from below, and is most
common in tropical regions.

c) Orographic precipitation
 Orographic precipitation is caused by air masses which strike some natural
topographic barriers like mountains, and cannot move forward and hence rise up over
barriers such as mountain ranges or islands in oceans, and is analogous to warm air
being forced upward at a cold front, causing condensation and precipitation.
 Typically more rain falls on windward than leeward slopes, since as the air descends it
warms and the cloud and rain reduces.
 The intensity of orographic precipitation tends to increase with the depth of the
uplifted layer of moist air. All the precipitation we have in Himalayan region is because
of this nature.

 Southern slope of the Himalayas is a
good example of this kind.
 Similarly, winds coming from ocean
strike the western slopes of coastal
ranges causing heavy rains.

2.2 Precipitation measurement
 Rain gauges for precipitation measurement are of two types
 Key parameters in precipitation measurement:
1) Non-recording rain gauges (cumulative)
2) Recording rain gauges (tipping bucket)
 Amount (mm)
 Intensity (mm/hr)
 Duration (minutes, hours)

 Non-recording rain gauges are commonly used
 They do not record the data
 They collect rain and this collected rain is then
measured in a graduated cylinder
 This type of gauge measures precipitation for
only a specified period
1) Non-recording rain gauges:
𝐷𝑒𝑝𝑡ℎ 𝑜𝑓 𝑟𝑎𝑖𝑛 =
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑟𝑎𝑖𝑛 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑐𝑚3
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒 𝑜𝑓 𝑔𝑎𝑢𝑔𝑒𝑠 𝑖𝑛 𝑐𝑚2

 As long as the gauge is > 3 cm diameter, any size
gauge can work for non-recording rain
measurement
 US standard – 20 cm diameter
 Read once daily normally
 Problems associated with non-recording gauges:
 Place as close to ground as possible – level ground
 Place away from trees and buildings,
 It should not be too exposed to wind (low brush helps),
 Sensitive to wind for light rain

 The instrument records the graphical variation of the rainfall, the total collected
quantity in a certain time interval and the intensity of the rainfall (mm/hour).
 They automatically record rainfall without any bottle reading so that it allows
continuous measurement of the rainfall
 The rainfall is recorded automatically on a graph paper with mechanical
arrangements
 A graph of total rainfall vs time, which is known as mass curve of rainfall is plotted
by the gauges
2) Recording gauges
 Three types of commonly used recording gauges:
1) Tipping bucket gauges
2) Weighing type gauges
3) Float recording gauges

 The collector is funneled into two
compartment buckets
 When one compartment of bucket is filled
with rain water, it becomes over balanced
and tips such that the other compartment
takes its place beneath the funnel.
 As the bucket is tipped, it automatically
activates an electronic circuit.
1) Tipping bucket gauges

 It consists of a storage bin, which is weighed to record the mass.
 It weighs rain or snow which falls into a bucket, set on a platform with a spring
or lever balance.
 The increasing weight of the bucket and its contents are recorded on a chart.
 The record shows accumulation of precipitation.
 Storage gauges are used in remote areas where frequent servicing is not
possible. Weighing type storage gauges operate for 1 or 2 months without any
servicing required.
 This type of rain gauges are designed to operate for entire season without
attention.
2) Weighing type storage gauges

 Generally a daily chart is used for the diagrams. For remote locations weekly charts
can also be used (revolving drum completes one revolution in seven days). Weekly
charts do not have the same detail as daily charts.
Recorded diagram
Recording rain gauge (weighing type)

 Hyetograph from a recorded diagram
Recorded diagram
Time
(min)
 Depth
(mm)
t (min) d (mm) Intensity
(mm/hr)
𝑡0 𝑑0
𝑡1 𝑑1 ∆𝑡1 = 𝑡1 − 𝑡0 ∆𝑑1 = 𝑑1 − 𝑑0
𝑖1 =
∆𝑑1
∆𝑡1
1
60
𝑡2 𝑑2 ∆𝑡2 = 𝑡2 − 𝑡1 ∆𝑑2 = 𝑑2 − 𝑑1
𝑖2 =
∆𝑑2
∆𝑡2
1
60
⋮ ⋮ ⋮ ⋮ ⋮
𝑡𝑛−1 𝑑𝑛−1
𝑡𝑛 𝑑𝑛 ∆𝑡𝑛 = 𝑡𝑛 − 𝑡𝑛−1 ∆𝑑𝑛
= 𝑑𝑛 − 𝑑𝑛−1
𝑖𝑛 =
∆𝑑𝑛
∆𝑡𝑛
1
60 Hyetograph

 The rainfall collected in the funnel
shaped collector is led into a float
chamber, causing the float to rise.
 As the float rises, a pen attached to the
float through a lever system records
the rainfall on a rotating drum driven
by a clockwork mechanism.
 A syphon arrangement empties the
float chamber when the float has
reached a preset maximum level.
3) Float recording gauges

 Presentation of rainfall data
 Hyetograph
 Plot of rainfall intensity against time,
where rainfall intensity is depth of
rainfall per unit time
 Mass curve of rainfall
 Plot of accumulated precipitation
against time, plotted in chronological
order.
 Point rainfall
 It is also known as station rainfall . It
refers to the rainfall data of a station

o The major problem is under-catch due to wind turbulence around the gauge.
o Particular measurement challenges are faced when measuring rainfall in forested
areas, in very steep terrain and during very intense rainstorms.
o Thus, a raingauge site should not be over–exposed and subject to strong winds, nor
should it be unduly sheltered by nearby obstacles.
o As a general rule the gauge should be at a distance of at least twice (and preferably
four times) the height of any obstacle.
 Problems in the collection of accurate rainfall data

 Even if raingauges provide accurate point measurements, they are only
representative of a limited spatial extent.
 Hydrologists often need to estimate the volume of rain-fall over a catchment
area and require an adequate number of measurements in order to assess the
spatial variation.
 This may be achieved with a network of raingauges alone, or by using additional
information from remote sensing by weather radar or satellites.
 Areal rainfall

 Design of raingauge networks
 The accuracy of areal precipitation estimates will increase as the gauging network
density increases. But a dense network is difficult and expensive to maintain, and so a
number of general guidelines for gauge density have been produced (e.g., WMO).
 The UK has one of the highest densities of raingauges in the world with an average of
one gauge per 80 km2 (Allott, 2010).
 There has been a general reduction in hydrometric networks in recent years (Mishra
and Coulibaly, 2009), due partly to save costs and partly in response to an increase in
methods of remote sensing.

 The World Meteorological Organization (Perks et al.,1996) evaluated the adequacy of
hydrological Networks and gave the following broad guidelines for the minimum
gauge density of precipitation networks in various geographical regions:
 One raingauge per 25 km2 for small mountainous islands with irregular
precipitation;
 250 km2 per gauge for mountainous areas;
 575 km2 elsewhere in temperate, Mediterranean and tropical climates, and
 10,000 km2 for arid and polar climates.

a) Check for Continuity: Estimation of missing data
o P1, P2, P3,…, and Pm are annual precipitation at neighboring M stations
of 1, 2, 3,…, and M, respectively.
o Px is the missing annual precipitation at station X
o N1, N2, N3,…, Nm and Nx are the normal annual precipitation at all M
stations and at X, respectively
 Adjustment of precipitation data
 Check for continuity and consistency of rainfall records
o Normal rainfall as standard of comparison
o Normal rainfall: Average value of rainfall at a particular date, month or year
over a specified 30-year period.
2.3 Processing and Analysis of Hydrological Data

1) Arithmetic Average Method
 In Station Average Method, the missing record is computed as the simple average
of the values at the nearby gauges
 This method is used when normal annual precipitations at various stations show
variation within 10% with respect to station X
 Several approaches are commonly used to estimate the missing values:
1) Station Average: Arithmetic average
2) Normal Ratio
3) Inverse Distance Weighting
4) Regression methods
𝑃𝑥 =
1
𝑀
𝑃1 + 𝑃2 + 𝑃3 + ⋯ + 𝑃𝑀
where Px is the missing precipitation record; P1, P2 , …, Pm are the precipitation
records at the neighboring stations; and M is the number of neighboring stations.

2) Normal Ratio Method
 Used when normal annual precipitations at various stations show variation >10%
with respect to station X
 If the annual precipitations vary considerably by more than 10 %, the missing record
is estimated by the Normal Ratio Method
 It can be estimated by weighing the precipitation at the neighboring stations by the
ratios of normal annual precipitations.
where Nx is the annual-average precipitation at the gauge with missing values;
N1 , N2 , …, Nm are the annual average precipitation at neighboring gauges
𝑃𝑥 =
𝑁𝑥
𝑀
𝑃1
𝑁1
+
𝑃2
𝑁2
+
𝑃3
𝑁3
+ ⋯ +
𝑃𝑚
𝑁𝑚

3) Inverse Distance Method (IDM)
 The Inverse Distance Method (IDM) weights the annual average values only by
their distances, di, from the gauge with the missing data and so does not require
information about average annual precipitation at the gauges.
𝑃𝑥 =
σ𝑖=1
𝑁 𝑃𝑖
𝑑𝑖
𝑏
σ𝑖=1
𝑁 1
𝑑𝑖
𝑏
The value of b:
 It can be 1 if the weights are inversely
proportional to distance or
 It can be 2 if the weights are proportional to
distance squared.
𝑃𝑥 =
10
252 +
20
152 +
30
102
1
252 +
1
152 +
1
102
= 25.24 𝑚𝑚
4) Regression Method
 The regression method can be used to estimate the missing precipitation value If
relatively few values are missing at the gauge of interest.

 Causes of inconsistency in records:
 Shifting of rain-gauge to a new location
 Change in the ecosystem due to damages, deforestation, obstruction, etc.
 Occurrence of observational error from a certain date (both personal and
instrumental)
 The most common method of checking for inconsistency of a record is the
Double-Mass Curve analysis (DMC).
b) Check for Consistency

 Double-Mass Curve analysis (DMC)
o The curve is a plot on arithmetic graph paper, of cumulative precipitation
collected at a gauge where measurement conditions may have changed
significantly against the average of the cumulative precipitation for the
same period of record collected at several gauges in the same region
o The data is arranged in the reverse order, i.e., the latest record as the first
entry and the oldest record as the last entry in the list
o A change in proportionality between the measurements at the suspect
station and those in the region is reflected in a change in the slope of the
trend of the plotted points

o If a Double Mass Curve reveals a change in slope that is significant and is due to
changed measurement conditions at a particular station, the values of the earlier
period of the record should be adjusted to be consistent with latter period records
before computation of areal averages.
o The adjustment is done by applying a correction factor K, on the records before the
slope change given by the following relationship.
𝐾 =
𝑆𝑙𝑜𝑝𝑒 𝑓𝑜𝑟 𝑝𝑒𝑟𝑖𝑜𝑑 𝑎𝑓𝑡𝑒𝑟 𝑠𝑙𝑜𝑝𝑒 𝑐ℎ𝑎𝑛𝑔𝑒
𝑆𝑙𝑜𝑝𝑒 𝑓𝑜𝑟 𝑝𝑒𝑟𝑖𝑜𝑑 𝑏𝑒𝑓𝑜𝑟𝑒 𝑠𝑙𝑜𝑝𝑒 𝑐ℎ𝑎𝑛𝑔𝑒

 When each recorded data comes from the same
parent population, they are consistent.
 Break in the year: 1987
 Correction Ratio
𝑀𝑐
𝑀𝑎
=
𝑐
𝑎
= 𝑘
𝑃𝑐𝑥 = 𝑃𝑥 ×
𝑀𝑐
𝑀𝑎
where
Pcx: corrected precipitation at any time period t1 at station X
Px: Original recorded precipitation at time period t1 at station X
Mc: corrected slope of the double mass curve
Ma: original slope of the mass curve

Example
 The annual records of five
precipitation stations are
given in Table. Check the
consistency of station A.
Adjust the record if it is
inconsistent.

1) The mean of a group of stations (B, C, D, and E) is computed in column 7.
2) The accumulated values for station A and the group of stations are given in
columns 8 and 9, respectively.
3) Column 8 is plotted against column 9. The breakpoint is observed at 1999.
4) The ratio of recent to past slope = 1.06/0.78 = 1.36.
5) The data prior to the breakpoint (1995–1998) are corrected by the factor 1.36,
as indicated in Table.
 Solution

2.4 Basin average precipitation
o Why do we want a basin-average precipitation?
 Methods for computing the areal average precipitation.
1) Arithmetic average
2) Thiessen-weighted average
3) Isohyetal method
4) Inverse distance weighting
 Estimation of areal precipitation from point measurements

 The arithmetic mean method uses the mean of precipitation record from all
gauges in a catchment – considers the stations inside the basin.
 This method can be used if gages distributed uniformly over watershed and
rainfall does not vary much in space.
1) Arithmetic average
ത
𝑃 =
1
𝑁
෍
𝑖=1
𝑁
𝑃𝑖
where
Pi is the rainfall at the ith rain-gauge station
N is the total number of rain-gauge stations

 The rainfall measured at each station is given a weightage on the basis of an area
closest to the station.
 The average rainfall over the basin is computed by considering the precipitation
from each station multiplied by the percentage of basin area enclosed by the
Thiessen polygon.
 The total average areal rainfall is the summation averages from all the stations.
 The rain gauge stations outside the basin area can be considered effectively by
Thiessen polygon method.
 Assumes rain at any point in watershed equal to rainfall at nearest station
2) Thiessen polygon average

 Steps of Thiessen polygon weighted average rainfall estimation:
a) Draw lines joining adjacent gages
b) Draw perpendicular bisectors to the lines created in step a)
c) Extend the lines created in step b) in both directions to form irregular polygon
areas that are representative areas for gages
d) Compute representative area for each gage
e) Compute the areal average using the following
ത
𝑃 =
1
𝐴
෍
𝑖=1
𝑁
𝐴𝑖𝑃𝑖
where
Pi is the rainfall at the ith rain-gauge station
Ai is the representative area for the ith rain-gauge station
A is the total area of the basin

3) Isohyetal method
 The Isohyetal method is the most recommended method of estimating areal rainfall.
 It is considered to be the most accurate method, if there is a sufficiently dense gage
network to construct an accurate Isohyetal map
 The method requires the preparation of the Isohyetal map of the catchment from a
network of gauging stations
 Areas between the Isohyets and the catchment boundary are measured
 The areal rainfall is calculated from the product of the inter-Isohyetal areas and the
corresponding mean rainfall between the Isohyets divided by the total catchment
area.

 Steps of Isohyetal method areal rainfall estimation:
 Plot gauge locations on a map
 Interpolate between rain amounts between
gauges at a selected interval
 Connect points of equal rain depth to produce
lines of equal rainfall amounts (isohyets)
 Compute aerial rain using:
where
Pi is the values of the Isohyets
ai is the inter-isohyet area between each pair of adjacent isohyets

Example: P1 = 10 mm, P2 = 20 mm, and P3 = 30 mm. Total area = 50 Km2
ത
𝑃 =
5 × 5 + 20 × 15 + 15 × 25 + 10 × 35
50
= 21 𝑚𝑚
 Arithmetic average:
ത
𝑃 =
10 + 20 + 30
3
= 20 𝑚𝑚
 Thiessen polygon weighted average:
A1 = 15 Km2, A2 = 15 Km2, and A3 = 20 Km2
ത
𝑃 =
10 × 15 + 20 × 15 + 30 × 20
50
= 21 𝑚𝑚
 Isohyetal method

4) Inverse distance weighting (IDW) method
 Prediction at a point is more influenced by nearby
measurements than that by distant measurements.
 The prediction at an ungauged point is inversely
proportional to the distance to the measurement points.
 The IDW method is usually used to fill missing data or to
predict a value at the ungauged site, but it can be also
used for areal estimation if the computation is made at the
basin center point
ത
𝑃 =
σ𝑖=1
𝑁 𝑃𝑖
𝑑𝑖
2
σ𝑖=1
𝑁 1
𝑑𝑖
2
where
Pi is the precipitation at the ith rain-gauge station
di is the distance between the site of interest and the ith rain-gauge station
N is the total number of rain-gauge stations

2.5 Depth-Area-Duration relationships
 It indicates the areal distribution characteristic of a storm of given duration
 Depth-Area relationship
 For a rainfall of given duration, the average depth decreases with the area in an
exponential fashion given by:
ത
𝑃 = 𝑃0𝑒𝑥𝑝 −𝐾𝐴𝑛
where
ത
𝑃: average depth in cm over area (A, km2),
P0: highest amount of rainfall in cm at the
storm center
K and n: constants for a given region
Typical DAD curves

2.6 Intensity-Duration-Frequency (IDF) curves
 IDF curves describe the relationship between rainfall intensity, rainfall duration, and
return period.
 IDF curves are commonly used in the design of hydrologic, hydraulic, and water
resource systems. Example: erosion control, highway construction, culvert design etc.
 IDF curves are obtained through frequency analysis of rainfall observations.
 IDF curves can be used in case of many design problems such as runoff disposal,
erosion control, highway construction, culvert design etc.
 The relation can be expressed in general form as:
𝑖 =
𝑘𝑇𝑥
𝐷 + 𝑎 𝑛
where
i: Intensity (cm/hr), T: Return period
D: Duration (hours)
K, x, a, and n: are constants for a given catchment

 Intensity-Depth relationship
Example: 12 hour duration, 100-year frequency,
depth is 70 mm
average intensity is 70mm/12hr = 5.8 mm/hr
Example:
20 min duration, 5-year frequency
intensity is 5.5 in/hr
 Intensity-Duration-Frequency (IDF)

1) Data:
 From rainfall measurements, for every year of record, determine the annual
maximum rainfall intensity for specific durations (or the annual maximum rainfall
depth over the specific durations).
 Common durations for design applications are: 5-min, 10-min, 15-min, 30-min, 1-hr,
2-hr, 6-hr, 12-hr, and 24-hr
 How to construct IDF
 The development of IDF curves requires that a frequency analysis be performed
for each set of annual maxima, one each associated with each rain duration.
 The basic objective of each frequency analysis is to determine the exceedance
probability distribution function of rain intensity for each duration.
2) Frequency Analysis:

 Two options for this frequency analysis:
1) Use an empirical plotting position approach to estimate the exceedance
probabilities based on the observations.
2) Fit a theoretical extreme value distribution (e.g., Gumbel Type I) to the
observations and then use the theoretical distribution to estimate the
rainfall events associated with given exceedance probabilities.

a) Rank the observations in descending order (Table 2, Column 1)
b) Compute the exceedance probability associated with each rainfall volume using the
following expression (Table 2, Column 4):
c) Transform the volume data into rainfall intensity by dividing volume by the
corresponding duration (Table 2, Column 6).
d) Plot empirical distribution of rainfall intensity (Columns 5 and 6 in Figure 1).
e) Repeat this procedure for each of the desired durations.
1) Empirical Plotting Position Approach
 Select for example the 30-min duration data from Table 1 and proceed as follows:
where m is the number of observations, p is the exceedance probability
and T is the corresponding return period (Table 2, Column 5).

Table 1. Maximum annual rainfall intensity for the shown duration

Table 2. 30-min rainfall – Frequency Analysis
 i(mm/hr) = 15.9/0.5 = 31.8 mm/hr
Frequency analysis 30-min rain Intensity

2) Theoretical extreme value (EV) distribution approach
 Select the Gumbel (Type I) distribution for our example as EV distribution.
 The Gumbel Type I distribution is given as:
where µ is the location parameter and
 is the scale parameter.
 It can be shown that the value of the random variable XT associated with a given
return period, T, may be obtained from the following expression,
where X (overbar) is the mean of the observations, and
S is the standard deviation of the observations.
 The frequency factor associated with return period T, KT is given by

 The above equations are applied to each set of annual maxima corresponding to
each duration, as follows:
1) Compute the frequency factors (KT) associated with the desired return
periods (e.g., 2, 5, 10, 25, 50, 100, 1000).
2) For each duration (e.g., 5-min, 10-min, …etc.), compute the sample mean and
sample standard deviations of the series of annual maxima, (x1,…….,xm) (see
Table 1).

Precipitation intensity associated with each return period.
3) Compute the precipitation intensity associated with each return period using the
following equation:

4) Plot the results.
Figure. Intensity-Duration-Frequency curves

Hydrology_Lecture note on chapter two about precipitation occurrence

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Hydrology_Lecture note on chapter two about precipitation occurrence