# A New Spatial Distance Metric for Verification of Precipitation
## 3 Results
### Analysis of Idealized Examples
Since the fields are non-binary, only the PSD and CDST provide results. The low and high-intensity regions' magnitude (the sum of values) is comparable (the ratio is about 7.5:10). The positioning of the high-intensity regions is different for each comparison. In comparison #13 (Figure 3a), the centers of high and low-intensity areas are collocated, and both the PSD and CDST give a good estimate of the actual displacement. In contrast, in #14 and #15 (Figure 3b,c), the higher intensity areas are displaced either toward the center or to the sides. Depending on the overlap and distance between the low- and high-intensity regions, the resulting values can noticeably decrease or increase. Figure 2: Same as Figure 1 but for the six comparisons that use a pair of either binary or Gaussian events of elliptical shape. Since both fields in Comparison #14 are vertically symmetrical around the middle, the center of mass is located on the middle line. As \(d\) changes, the blue event is transposed in the left-right direction, and the distance between the centers of mass will change linearly--thus, the CDST will also change linearly with \(d\). At approximately \(d=30\), the centers of mass of the two events are collocated, and \(CDST=0\). The behavior of PSD is roughly similar, with two local maximums at \(d=0\) and 100 grid points and one minimum in the middle. The minimum of PSD is near \(d=55\) when the two high-intensity regions have considerable overlap, while at the same time, there is also some overlap between the low-intensity regions. Comparison #16 (Figure 3d) is similar to previous comparisons, but the ellipses are vertically aligned with the circular areas diagonally displaced. Comparison #17 (Figure 3e) is similar to #9 (Figure 2c), but the collocated higher intensity regions cause a slight reduction of the PSD value at small values of \(d\). It is worthwhile mentioning what would happen if comparisons #13 to #15 were analyzed using a metric that requires thresholding. If a single threshold value would be used that was smaller than the low-intensity precipitation denoted by the lighter colors, then comparisons #13 to #15 would all provide identical results. This happens since the binary fields for these three comparisons would be identical and all elliptical events inside  
Figure 3: Same as Figure 1 but for the six comparisons that use a set of displaced double-level events. The area with low-intensity values is elliptical but contains a smaller circular area with high-intensity values. the fields would be identical, regardless of the actually intensity of precipitation inside them. If the threshold would be set to an intermediate value (precipitation intensity that would correspond between the low- and high-intensities shown by lighter and darker colors), then the resulting binary events would represent only the small circular high-intensity regions while the low-intensity regions would be ignored. Of course, if the chosen threshold was higher than the high-intensity precipitation denoted by the darker colors, then there would be no events left after the thresholding. As PSD does not require thresholding it takes into account all the precipitation at the same time, while also taking note of the differences in its intensity. Figure 4 shows the comparisons with noisy, scattered, and two-set event fields. Comparison #18 (Figure 4a) is the same as #1 (Figure 1a) but with a small amount of randomly generated noise in both fields. The results of the PSD, CDST, and dFSS for the noisy situation are almost the same as for the non-noisy situation, indicating that these metrics are not overly sensitive to noise. On the other hand, the noise greatly influences the MED and BDEL, with significantly smaller displacement estimates. This behavior indicates that the MED and BDEL are quite sensitive to small changes in the fields. The behaviour can be linked to the use of nearest-neighbor methodology they are based on.