NOTE ON MAPPING 0 E EXTREME VALUES—ITS PROCEDURES AND PROBLEMS—

Abstract

A hydrological map is one of the most effective tools to express hydrological phenomena, especially spatial distribution of hydrological variables. Since different expressing or processing procedures for one statistic or variable might give different information, we have to consider how to express concerned hydrological elements. In this paper, the author tries to show differences of information in three different expressions for one statistic and to analyze them from a statistical point of view. As a statistic, taken are the minimum values of annual precipitation at 85 stations over Japan (in Fig. 1); the minimum values are expressed by three different ways, that is, observed values xmin, normalized values zmin, and modular coefficients kmin. The observed values are minimums during observation periods, which are different but longer than 40 years as shown in Table 1. The normalized values are defined by Eq. (1). As long as annual precipitation is normally distributed, these values can be compared under the absolutely common standard. The third values, modular coefficients are defined by Eq. (2). These three values are generally used to express hydrological variables. Distribution maps of minimum values of annual precipitation by the three ways are shown in Fig. 2, which shows different distribution patterns. This means that these three values for one statistic might not be equivalent in information content. Relationships between z and k, which are function of coefficient of variation (Eq. (3)), are discussed statistically. Under normal distributions, z and k are statistically equivalent only when mi=mj and si=sj, or si/mi=sj/mi, where i and j denote stations. For the former case, xi=xj, but for the latter, xi≠xj. Otherwise, they are not in one-to-one correspondence as shown in Fig. 3. This means that isolines of modular coefficients are not always significant statistically. Therefore, distribution map by modular coefficients in Fig. 2 has many closed areas, which notifies spatial discontinuity among individual values. Under non-normal distributions, when si/mi=sj/mj, z and k are just equivalent in one-to-one correspondence. However, they are not compared under the absolutely common standard. As a conclusion, we have to consider well whether or not isolines are meaningful for the concerned variables as well as extreme values, because even equal values are not always equivalent in statistical means.