**Raingage Network Density Requirement for Hydrologic Modeling**
By
**John C. Schaake, Jr.**
July 6, 1981
The number, N, of rain gages required to measure mean areal precipitation (MAP) amounts
over an area of A square miles is
If these gages are uniformly distributed over the areas, the maximum MAP error will be
less than 20 percent during 80 percent of all storms. The duration of the period of
rainfall accumulation used to derive this relation is 1/4 of the basin response lag time.
The exponent, 0.3, of Eq (1) implies that the required number of gages doubles as the
catchment are changes by a factor of 10. The coefficient, 0.8, of Eq (1) scales the
requirement and implies that a density of 0.8 gages per square mile is needed to measure
rainfall over a one square mile catchment. Table 1 was derived from Eq (1):
Table 1. The required number of gages for given area sizes
Catchment Area (sq. mile) |
Number of Gages, N |
Required Gage Density |
gages/sq. mile |
sq. mile/gage |
1 |
0.8 |
0.8 |
1.3 |
10 |
1.6 |
0.16 |
6.3 |
100 |
3.2 |
0.032 |
31.0 |
1000 |
6.4 |
0.0064 |
156.0 |
10000 |
12.8 |
0.0013 |
781.0 |
Eq (1) was derived from the following considerations.
A regression analysis of precipitation measurement errors for periods of 6 hours in the
Muskingum Basin (Office of Hydrologic Director, 1947) produced the following relation for
the median value of the coefficient of variation of the measurement errors
where G is the gage density in square miles per gage (i.e., G =
A/N). Values of for individual storms are distributed about this
median value so that 80 percent have a value of (call this ) less
than 1.57 . Therefore
According to similar studies of precipitation measurement errors on networks
operated by Illinois State Water Survey, Huff found that the value
of varied with the -0.22 power of the period of rainfall accumulation,
t. Substituting this into Eq (2) while assuring that the equation
would apply for t = 6 hours, gives
and
In order to have sufficient precipitation information as input to a hydrologic model,
the precipitation must be accumulated for periods of t that are not longer that 1/4 of the
basin response lag time,* T*_{l} .
Substituting this requirement and the fact that G = A/N into Eqs (4) and (5) gives :
and
If the value of *T*_{l} for an individual basin is known, then
Eq (7) gives the following gage requirements to assure = 0.20:
At the planning stage of a river forecasting system, the value of *T*_{l}
may not always be known. The values of *T*_{l} varies with catchment area,
slope and other factors. But the main factor affecting *T*_{l} is A. A
study of values of *T*_{l} for catchment areas ranging from the entire
Mississippi River Basin down to a small parking lot gave the following relation for the
median value of *T*_{l} as a function of A:
and the standard deviation of values of log_{10} *T*_{l} for
individual catchments of area A was found to be 0.15. This means relatively fast
responding basins, one standard deviation below the median, would have :
whereas relatively slow responding basins, one standard deviation above the median,
would have :
Because relatively more rapidly responding basins require slightly more dense networks,
Eq (10) was substituted into Eq (8) to derive Eq (1). At the other extreme, for relatively
slowly responding basins, if Eq (11) were substituted into Eq (8), only 77 percent as many
gages would be required as are needed for relatively fast basins. In view of the other
uncertainties at the planning stage Eq (1) should give adequate, slightly conservative,
results for planning purposes.
__References__:
Office of Hydrologic Director, 1947, __Thunderstorm Rainfall__,
Hydrometeorological Report No. 5, Weather Bureau, The Hydrometeorological Section, US
Department of Commerce, Vicksburg, Mississippi. |