On the Threshold Method for Rainfall Estimation: Choosing the Optimal Threshold Level
Kedem, Benjamin, Pavlopoulos, Harry, Journal of the American Statistical Association
The purpose of this article is to present a statistical solution to a problem of scientific interest in the field of meteorology. The problem is to choose an optimal threshold level needed for the so-called threshold method for measuring rainfall from space via satellite. The threshold method calls for the estimation of the instantaneous area average rain rate from the fraction of the area above which the instantaneous rain rate exceeds a fixed threshold. We suggest two criteria for deriving optimal thresholds. In the first, the optimal level minimizes a certain variance. In the second, the optimal level maximizes a certain correlation. The two schemes provide reasonably close threshold levels across a wide range of rain types.
By rain rate we mean instantaneous rain intensity measured in mm/hr at points in space. By a snapshot we mean the map of rain rate over a given region at a fixed instant. The area refers to the area of a given region. From now on we shall refer to the area average rain rate and the fraction of the area covered by rain rate exceeding a given threshold as the area average and fractional area, respectively. In what follows we assume that rain rate possesses all moments, and thus we exclude "fat-tailed" distributions, conditional on rain, from our discussion. The threshold level is denoted by [tau]; clearly, [tau][greater than or equal to]0.
It has recently been observed (Chiu 1988; Doneaud, Niscov, Priegnitz, and Smith 1984; Rosenfeld, Atlas, and Short 1990; Short, Wolff, Rosenfeld, and Atlas 1989) that the area average and fractional area are highly correlated. This observation has been confirmed from quite a few data sets from different parts of the world by showing that the sample correlation can easily exceed 95% and in some cases can even exceed 99% (Short et al. 1989). In particular, for the well-known and widely used data set referred to as GATE, to be described later, it has been found empirically by Chiu (1988) that the threshold level [tau] = 5 leads to a squared correlation of 98% and drops dramatically for other levels. For level [tau] = 0 the squared correlation drops to below 80%. The drop in squared correlation from a maximum of 98% is shown in Figure 1.
[FIGURE 1 OMITTED]
We shall provide, under some assumptions inspired by the GATE data set, a theoretical explanation for the empiricism observed in Chiu (1988). In particular, by means of our optimality criteria, we shall show why threshold levels around [tau] = 5 mm/hr are optimal for GATE-like rain.
1.1 The Need for the Threshold Method
The National Aeronautics and Space Administration (NASA) is contemplating at present the Tropical Rainfall Measuring Mission (TRMM), a satellite program for measuring rain rate from space over the tropics and subtropics. TRMM will provide monthly and seasonal rainfall estimates averaged over areas of about [1.sup.5] [km.sup.2]. The idea is to equip a satellite at a low altitude orbit with a precipitation radar, the first ever in space, and passive microwave radiometers operating at several frequencies. Rain rate is to be recovered from radar reflectivity and microwave temperature. The measurements are needed for a better understanding of the variability of the Earth's climate and a description of the global hydrological cycle. An extensive description of TRMM and its scientific goals is given in Simpson 1988 and in Simpson, Adler, and North 1988.
From a statistical point of view, the mission poses great challenge, because rain rate will be measured indirectly through covariates whose precise relationship to rain rate is not entirely clear. For example, microwave temperature is related to rain rate nonlinearly and the relationship itself is empirical (Wilheit 1986). Thus converting microwave temperature to rain rate is quite tricky. Likewise, the relationship between rain rate and radar reflectivity, commonly referred to as the Z-R relationship, is also empirical and is given by the model Z = [aR. …