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

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 log₁₀ 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.