Precipitation analysis - Methods and solved example

1. 1
Precipitation Analysis
 Areal Precipitation Estimation
 Depth - Area Analysis
 Depth - Duration Analysis
 Intensity - Duration Analysis
 Intensity-Duration-Frequency Analysis

2. 2
Quantitative Description of Rainfall
• A rainfall event, or storm, describes a period of time having
measurable and significant rainfall,
• preceded and followed by periods with no measurable rainfall.
• The time elapsed from start to end of the rainfall event is the
rainfall duration.
For small catchment rainfall duration is measured in minutes,
while for very large catchments it may be measured in days.
Rainfall durations of 1, 2, 3, 6, 12 and 14 h are common in
hydrologic analysis and design.
Rainfall depth is measured in mm, cm, or in and considered to
be uniformly distributed over the catchment area. For instance, a
60-mm, 6-h rainfall event produces 60 mm of depth over a 6-h
period.

3. 3
• Rainfall depth and duration tend to vary widely, depending on
geographic location, climate, and time of the year.
• Other things being equal, larger rainfall depths tend to occur
more infrequently than smaller rainfall depths.
Average rainfall intensity is the ratio of rainfall depth to rainfall
duration. For example, a rainfall event producing 60 mm in 6 h
represents an average rainfall intensity of 10 mm/h.
Rainfall intensity varies widely in space and time. Typically,
rainfall intensities are in the range 0.1 - 30.0 mm/h, but can be as
large as 150 to 350 mm/h in extreme cases.

4. 4
Rainfall Frequency refers to the average time elapsed between
occurrences of two rainfall events of same depth and duration.
The actual elapsed time varies widely and can therefore be
interpreted only in a statistical sense.
For instance, if at a certain location a 100-mm rainfall event
lasting 6-h occurs on the average once every 50 y, the 100-mm,
6-h rainfall frequency for this location would be 1 in 50 years,
1/50, or 0.02.
The reciprocal of rainfall frequency is referred to as return
period or recurrence interval. In the case of the previous
example, the return period corresponding to a frequency of
0.02 is 50 y.
Rainfall Frequency

5. 5
• Generally, larger rainfall depths tend to be associated with longer
return period.
• The longer the return period, the longer the historical record needed
to find out the statistical properties of the distribution of annual
maximum rainfall.
Due to the shortage of long rainfall records, extrapolations are usually
necessary to estimate rainfall depths associated with long return
periods.
The first step in designing a water-control works is to determine the
probable recurrence of rainfall of different intensity and duration so
that an economical size of structure can be provided.
Where human life is endangered, however, the design should handle
runoff from storms even grater than have been recorded.
Rainfall Frequency

6. 6
Temporal Rainfall Distribution
The temporal rainfall distribution represents the variation of rainfall depth
within a storm duration. It can be expressed in either discrete or continuous
form. The discrete form is referred to as hyetograph, a histogram of
rainfall depth (or rainfall intensity) with time increments as abscissas and
rainfall depth (or rainfall intensity) as ordinates.
Time
depth
or
intensity
Hyetograph
Time (%)
Depth
(%)
The continuous form in the temporal rainfall distribution, a function
describing the rate of rainfall accumulation with time. Rainfall duration
(abscissas) and rainfall depth (ordinates) can be expressed in percentage of
total value, the result is dimensionless temporal rainfall distribution.

7. 7
Spatial Rainfall Distribution
Rainfall varies spatially, i.e., the same amount of rain does not fall
uniformly over the entire catchment. Isohyets are used to depict the spatial
variation of rainfall. An isohyte is a contour line showing the loci of equal
rainfall depth.
3
4
5
4
Isohyets

8. 8
Depth-Area Duration Analysis
• In designing hydraulic structures for controlling river flow, an
engineer needs to know the areal rainfall of the area draining to the
control point.
• The techniques of relating areal rainfall depths to area by analyzing
several storms gives depth-area relationships for different specific
durations.
• Generally, the greater the catchment area, the smaller the averaged
storm depth i.e. P  1/A (Example on Page 46)

9. 9
Depth – Area- Duration Analysis
• The purpose: To determine the maximum precipitation amounts over various area
sizes during the passage of storms of various durations for the computation of
probable maximum precipitation (PMP) estimates.
• The y-axis is the area size and X-axis Maximum average areal precipitation.
• Example: During a 36-hour storm event, the heaviest 12-hour 100 sq-mi
precipitation was about 11 inches.

10. 10
Depth - Duration Analysis
Storm depth (h) and duration (t) are directly related, i.e., storm depth increases
with duration. An equation relating storm depth and duration is
h = c tn (1)
where c is a coefficient and n is an exponent (a positive real number less than 1).
Typically n varies between 0.2 and 0.5, indicating that storm depth increases at a
lesser rate than storm duration.
The applicability of such an equation, however, is limited to the regional or local
conditions for which it is derived.
Time
Depth
Lin-Lin
Time
Depth
Log-Log

11. 11
Storm intensity (i) and duration are inversely related. Differentiating
Equation 1 with respect to duration (time), we have;
Intensity - Duration Analysis
1
)
( 

 n
t
n
c
dt
n
ct
d
dt
dh
m
t
a
i 
where a = cn, and m = 1-n. Since n is less than 1, it follows that m is also less than 1.
Equation 2 is used for duration’s exceeding two hours.
Where a and b are constants to be determined by regression analysis.
(2)
Another intensity-duration model (for shorter duration less than 2 hours) is
the following:
b
t
a
i

 (3)
n
X
d
-
Y
=
c


X)
( 2
-
X2
n
Y
X
-
XY
n
=
d





Y = c + dt
Simple Linear Model
A. N Talbot’s formula

12. 12
A general intensity-duration model combining the features of Equations 2
and 3 is
 m
b
t
a
i


Intensity - Duration Analysis
(4)
Taking log
Log(i) = log (a) - m log (t+b)
Therefore various values of b may be assumed and
graphs plotted till a straight-line is obtained. Time
Intensit
y

13. 13
Example
Determine the equation relating rainfall intensity and duration for the
following 10-y frequency rainfall data.
Duration (min) 5 10 15 30 60 120 180
Intensity (cm/h) 8.0 5.0 4.0 2.5 1.5 1.0 0.8
Solution
The data suggest that the relation is of hyperbolic type, with greater
intensities associated with shorter durations. Therefore, an equation of the
following type is applicable:
b
t
a
i


This equation can be linearalized in the following way:
a
t
a
b
i


1
t
d
c
y 

y = 1/i
c = b/a
d = 1/a
A. N Talbot’s formula

14. 14
t i
(min) (cm/hr)
5 8.0
10 5.0
15 4.0
30 2.5
60 1.5
120 1.0
180 0.8
Total =
X Y X2 XY
t 1/i t2 t (1/i)
5 0.1250 25 0.625
10 0.2000 100 2
15 0.2500 225 3.75
30 0.4000 900 12
60 0.6667 3600 40
120 1.0000 14400 120
180 1.2500 32400 225
420 4 51650 403
i = a/(t+b)
(cm/hr)
4.9330
4.2584
3.7462
2.7527
1.7987
1.0624
0.7538
Model
Calculations
c = 0.006422495
d = 0.170602665
a = 1/d = 155.70
b = a c = 26.56

15. 15
Figure: Intensity-duration-frequency curve.
0
1
2
3
4
5
6
7
8
9
0 20 40 60 80 100 120 140 160 180 200
Rainfall duration t (min)
Rainfall
intensity
i
(cm/h)
Data
Model
i = 155.7 / (26.56 + t)

16. 16
Model
Calculations
6503
.
0
6586
.
22
t
i 
t i
x = log(t) y = log(i) x2 x y
5 8 0.6990 0.9031 0.48856 0.63123
10 5 1.0000 0.6990 1.00000 0.69897
15 4 1.1761 0.6021 1.38319 0.70808
30 2.5 1.4771 0.3979 2.18189 0.58781
60 1.5 1.7782 0.1761 3.16182 0.31312
120 1 2.0792 0.0000 4.32299 0.00000
180 0.8 2.2553 -0.0969 5.08625 -0.21856
7 Total 10.4648 2.6812 17.6247 2.7206
a' = 1.3552
a = 10^a' 22.6586
b = -0.6503
m = - b 0.6503
x
b
a
y
t
m
a
i




'
)
log(
)
log(
)
log(
m
t
a
i 

17. 17
Intensity-Duration and Frequency of Rainfall
In hydrologic design projects, the use of intensity-duration-frequency
relationships is recommended. Intensity refers to rainfall intensity, duration
refers to rainfall duration and frequency refers to Return Period, which is
expected value of the recurrence interval (time between occurrence).
“It is defined as the average period of time (years) within which the depth of
rainfall for a given duration will be equalled or exceeded once on the average.”
The IDF relationships are also referred to as IDF curves.
A general expression for IDF can be obtained by assuming that the constant a in
Equation 4 [i = a / (t+b)m] is related to return period as: a = K Tx
Where k is a coefficient, T is return period and x is an exponent. Therefore
 m
b
t
x
T
K
i

 Where t in min, T in years and I in cm/hour.
The values of k, b, m, and x are evaluated from
measured data.
Sherman Equation

18. 18
IDF Curves of Maximum rainfall in Chicago, USA
(from Chow, Maidment and Mays)