Analytical and practical examples of estimating the average nearest-neighbor distance in a rain gauge network

Abstract

This paper deals with estimating the average n-th neighbor distance in a rain gauge network using analytical expressions. Research was carried out to find the relationship between the total number of rain gauges, the area size, and the shape of a domain on one side and the average distance to the n-th neighbor on the other side. Authors such as HERTZ (1909) and CHANDRASEKHAR (1943) have shown that when a random distribution of rain gauges and an infinite domain are assumed, a simple analytical formula can be obtained. For a circular domain, one gets a first neighbor formula, given by an integral which has to be integrated numerically. For neighbors located further away and other domain shapes, meaning not infinite and not circular, the best way to estimate the average n-th neighbor distance seems to be a computer program which uses a random number generator to distribute the rain gauges in a domain. After the process of random distribution is complete, then the distance to the n-th neighbor is found for each rain gauge. These distances are then averaged. We performed tests using seven domain shapes and the results show that the average distance to the nearest neighbor is always larger in a limited domain rather than in an infinite domain with the same rain gauge density. The difference between the average distance in an infinite and in a particular domain depends on the shape of the domain and the total number of rain gauges within it. The length of a boundary does not inherently influence the magnitude of the difference. However, the difference is small if the total number of rain gauges is large and a simple infinite domain equation can usually be used. For any tested domain the difference is less than 10 % if the number of rain gauges is larger than 100. The assumption of randomly distributed rain gauges limits the usefulness of the method. There are cases, such as smaller countries or country sub-regions, where the rain gauges might be more or less evenly scattered in the domain. In these cases the method is still useful and the distance to the n-th neighbor could be estimated using a simple analytical expression.