Estimation of Missing Rainfall Data

Estimation of Missing Rainfall Data
Rainfall data are generally collected at point locations (mainly at meteorological stations)
However, rainfall data might be incomplete. Missing data therefore could be attributed to:
Malfunctioning التقصير
Mismanagementسوء لادارة
Inability to take the measurement عدم القدرة لاخذ القياس
Vandalism تخريب متعمد
Therefore, when part of rainfall data is missing, then estimation of missing data should be made

1- Station – Average Method
Consider n rain gages present in a region with measured data for a given storm event
The data at station X are missing for a storm event
Then
P_x= 1/n ?_(i=1)^n?(P_i )
Use this method when the annual rainfall of any station is within the 10% of the average annual rainfall from the gages

Example: Find the missing rainfall data at station D for the following storm events given the average annual rainfall data,




Solution: The average annual rainfall at the four gages is 40.7 in and thus all the annual readings are within 10% of the average
10% of the average annual rainfall from the gages = 40.7+ (40.7 * 10/100) = 44.77
Annual rainfall < 44.77

P_D= 1/3 x (2.6 + 3.1 + 2.3) = 2.67 in

2- Normal – Ratio Method
In regions where the annual average rainfall differs considerably between locations, the normal – ratio method is preferred
P_x= ?_(i=1)^n??A_x/(nA_i ) (P_i ) ?
Px: missing rainfall data at x
Ax: annual rainfall at station x
Ai: annual rainfall at station i
n: No. of stations (known precipitation)
Example : Find the missing rainfall data at station D for the following storm events given the average annual rainfall data




P_x= ?_(i=1)^n??A_x/(nA_i ) (P_i ) ?
P_D= 40/3x41 x 2.4 + 40/3x37 x 2.3 + 40/3x46 x 3.1 = 2.51 in
Use the flowing equation when the annual rainfall of any station is greater than the 10% of the average annual rainfall from the gages
PX= NX/3*[PA/ NA+ PB/ NB+ PC/ NC]

where:
A,B,C = rainfall for each station is know
X= missing data station
N= average annual rainfall for station
P = rainfall for station during limit time
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Example: One of four monthly reading gages on a catchment area develops a fault in a month when the other three gages record, respectively (122, 89, 107) mm. If the average annual precipitation of these three gages are (935, 1120, 979) mm, respectively and (1200mm) of the broken gage. Estimate the missing monthly precipitation at the later station?
Solution:
(NA-NX)/NX*100 = (935-1200)/1200*100= 22%
(NB-NX)/NX*100 = (1120-1200)/1200*100= 7% <10%
(NC-NX)/NX*100 = (979-1200)/1200*100= 18% > 10%
PX = NX/3*[PA/ NA+ PB/ NB+ PC/ NC]
= 1200/3[122/935+89/1120+107/979]
=127.7 mm
Testing of Consistency of precipitation Records.
Numerous factors could affect the consistency of the record at a given station.
damage and replacement of a rain gage
change in measurement procedure, or
human, mechanical, or electrical error in taking readings
Changes in type, location, and/or environment of the gage are common
Trees may grow up or buildings may be constructed around a gage
So it is important for a hydrologist to determine if the precipitation record is affected by such artificial alterations of measurement conditions and to correct them if they are present
The most common technique for detecting and correcting the inconsistent precipitation data is the double-mass curve
The double-mass curve is a plot of the successive cumulative annual precipitation collected at a gage where measurement conditions may have changed significantly versus the successive cumulative of average annual precipitation for the same period of years collected at several gages in the same region
A change in the proportionality between the measurements at the suspect station and those of the region is reflected in a change in the slope of the trend of the plotted points










Correction Factor
K = slope (after) / slope (before)




EX:
The problem is in station E
Find the average for stations A to D and then the cumulative
Find the cumulative for station E

















Apparently, there is a slope difference
The slope before the change is 0.77 while after the change it is 1.05
To reflect the conditions that exist before the break then multiply by 0.77/1.05 all the records after change
To reflect the conditions that exist after the break then multiply by 1.05/0.77 all the records before the change













Double Mass Curve Analysis (Consistency Test):DMC
DMC: is a plot of accumulative totals for each year at the gage
against the sum of accumulative totals for the same years at a number of adjacent gages.
DMC: is used to check if trended in rainfall data of a certain gage is due to Meteorological condition only.

Example: Annual precipitation at rain gage (X) and the annual precipitation at (15) surrounding rain gages are listed in the following table:
a. Examine the consistency of st. (X) data;
b. When did a change in regime occur?
c. Adjust the data and determine what difference this makes to the 33 year annual average precipitation at st. (X).
Annual precipitation (mm) Year
(15) Sts. St.(X)
13.9 13.4 1938
9.9 10.7 1939
10.1 10.9 1940
13.7 12 1941
13.1 13.3 1942
13.2 14.6 1943
10.9 9.0 1944
11.4 11.8 1945
10.2 9.7 1946
13.9 15.4 1947
13.0 12.5 1948
13.1 11.5 1949
13.1 11.5 1950
10.9 13.9 1951
13.2 14.1 1952
10.0 10.4 1953
8.8 7.9 1954
9.6 13.3 1955
10.2 16.3 1956
15.9 22.7 1957
10.9 13.9 1958
10.2 14.7 1959
10.3 14 1960
10.2 11.4 1961
11.8 13.8 1962
9.2 10.0 1963
10.2 10.5 1964
14.0 16.7 1965
8.4 9.3 1966
11.5 18.4 1967
9.0 14.1 1968
13.0 19.8 1969
13.1 17.1 1970
10.7 16.0 1971
Rainfall Classification
Very light — < 0.25 mm/hr
Light — 0.25 mm/hr - 1.0 mm/hr
Moderate — 1.0 mm/hr - 4.0 mm/hr
Heavy — 4.0 mm/hr - 16.0 mm/hr
Very heavy — 16.0 mm/hr – 50 mm/hr
Extreme — > 50.0 mm/hr