Basin average rainfall and its sampling error

[1] The sampling error from the randomly located rain gauge network has a significant meaning, especially for hydrologic applications where the basin average rainfall estimated using point measurements is used as an input for various rainfall-runoff analyses. In this paper we derived an equation for the sampling error from the randomly located rain gauge network of a basin and compared it with that from the evenly spaced rain gauge network derived by North and Nakamoto [1989], also implemented by Yoo [2000]. To make the analytical derivation of the sampling error equation available, we assumed that the locations of the rain gauges planted through years in a given basin follow the uniform distribution and the rainfall field is weakly statistically homogeneous in space and time. The derived equation is neatly factored with the rainfall spectrum and the design filter, which considers the second-order statistics of a rainfall field and sampling design characteristics such as the basin size, the basin shape, and the number of rain gauges within the basin. The derived equation is also flexible enough to consider various shapes of the basins such as circles, ellipses, and rectangles as well. We applied the two equations for the sampling errors from evenly spaced and randomly located rain gauge network cases to the GATE data field using the rainfall model of North and Nakamoto [1989] and then compared the results. As a result of the study, we found that the sampling errors from the randomly located and evenly spaced rain gauge networks are more or less the same as the standard deviation of the m gauge averages until some threshold number of rain gauges is reached. With more rain gauges the sampling error from the evenly spaced rain gauge network decreases much faster, while that from the randomly located rain gauge networks stagnates. However, if considering the typical rain gauge density in a natural basin, we may conclude that the sampling error involved in the estimation of basin average rainfall using point measurements is almost the same as the standard deviation of the m gauge averages.