Lesson 7 Estimation of Missing Rainfall Data
7.1 Estimating Missing Data
The point observation from a precipitation gage may have a
short break in the record because of instrument failure or
absence of the observer. Thus, it is often necessary to
estimate the missing record using data from the neighboring
station. The following methods are most commonly used for
estimating the missing records.
1.
Simple Arithmetic Method
2.
Normal Ratio Method
3.
Modified normal ratio method
4.
Inverse distance method
5.
Linear programming method
For m stations, 1, 2, 3, …,m, the annual precipitation values
are P1, P2, P3, …, Pm, respectively. At station x (not
included in the above m stations), the missing annual
precipitation (Px) should be found out. The normal annual
precipitation N1, N2, N3, …,Ni at each of the above (m+1)
stations including the station x is known.
7.1.1 Normal Precipitation - It is the average value of
precipitation at a particular date, month or year over a
specified 30 year period. Thus, the term normal annual
precipitation at station A means the average annual
precipitation at A based on a specified 30 year of record.
7.1.2 Simple Arithmetic Average - The missing
precipitation Px can be determined using simple arithmetic
average, if the normal annual precipitation at various
stations are within 10% of the normal precipitation at
station, x, as follows:
(
7.1)
7.1.3 Normal Ratio Method - If the normal precipitations
vary considerably then Px is estimated by weighting the
precipitation at various stations by the ratios of normal
annual precipitation. The normal ration method gives Px
as:
(7.2)
This method is based selecting m (m is usually 3) stations
that are near and approximately evenly spaced around the
station with the missing record.
Example 7.1 The normal annual rainfall at stations A, B, C
and D in a basin are 80.97, 67.59, 76.28, and 92.01 cm,
respectively. In the year 1975, the station D was
inoperative and the stations A, B, and C recorded annual
rainfall of 91.11, 72.23, and 79.89 cm,
respectively. Estimate the rainfall at station D in that year.
Solution: As the normal rainfall values vary by more than
10%, the ration method is adopted.
7.1.4 Modified Normal Ratio Method
Normal ratio method is modified to incorporate the effect
of distance in the estimation of missing rainfall.
(7.3)
Where is normal rainfall, is the distance between the index
station i and the gauge station with missing data or ungaged
station, n is the number of index stations and b is the
constant by which the distance is weighted (normally 1.52.0) commonly used D0.5
7.1.5Inverse Distance Method
The inverse distance method has been advocated to be the
most accurate method as compare to other two methods
discussed above.
Amount of rainfall to be estimated at a location is a
function of;
1.
2.
rainfall measured at the surrounding index stations
distance to each index station from the ungauged
location
Rainfall rx at station x is given by;
(7.4)
b = 2 is commonly used.
As in inverse distance method the weighting is strictly
based on distance, hence this method is not satisfactory for
hilly regions.
Example 7.2 Data for the base station and 5 surrounding
stations are tabulated below. Find missing data at ‘A’ using
(i) modified normal ratio method and (ii) inverse distance
method.
Station
Distance (D) from
the base station
(km)
Rainfall
(cm|)
Normal
Rainfall
(cm)
A (base
station)
-
?
102
B
1.5
2.5
114
C
1.21
3.4
122
D
0.85
1.5
95
E
1.3
2.2
106
F
2.11
1.8
104
Solution: (i) Using modified normal ratio method
Weight calculation for different stations other than the base
station is shown in following table, using b = 2
Station
Weight (a)
B
1.22
C
1.1
D
0.92
E
1.14
F
1.45
Sum
5.83
1.0
Hence,
= 2.12 cm
(ii) Using Inverse Distance Method
Station
Weight (a)
B
0.44
C
0.68
D
1.38
E
0.59
F
0.22
Sum
3.31
1.0
Hence,
7.1.6Linear Programming Method
Linear programming (LP) method selects a base station and
several surrounding index stations and determines optimal
weighting factor by minimizing the deviation between
observed and computed rainfall at a base station for a
number of rainfall events.
Thus it determines optimal weighting factors for the base
station and associated index stations.
This method can be formulated as,Objective is
to
minimize sum of deviation for k events i.e.,
Minimize
Subjected to
(Non-negativity constraints)
Where,
i = index for “index station”
j = index for rainfall events
= observed rainfall at base station ‘b’ for event
‘j’
= computed rainfall at base station for event
‘j’
For any event, computed rain – observed rain = deviation
Deviation could be either positive or negative value
(unrestricted in sign), such variables are replaced by the
difference of two non-negative variables (LP requirement)
i.e., U.
Solved Example
Assume that rainfall is not known at the station D. The
normal precipitations of the three neighbouring gauging
stations are as follows:
Station
Station coordinate
Normal Annual
Precipitation (cm)
Precipitation
(cm)
A
(1,2.5)
28
25
B
(4,1)
15
10
C
(3,5)
30
25
D
(3,3)
25
?
Compute the rainfall at this point using
a. Simple Arithmetic Method
b. Normal Ratio Method
c. Modified Normal Ratio Method
d. Inverse Distance Method
Answer
a. Simple Arithmetic Method
b. Normal Ratio Method
c. Modified Normal Ratio Method
d. Inverse Distance Method
References
Singh, V. P. (1994). Elementary Hydrology.Prentice Hall
of India Private Limited,New Delhi.
Suggested Reading
Subramanya, K. (1994). Engineering Hydrology.Third
edition, Tata McGraw Hill, New Delhi.