# A New Spatial Distance Metric for Verification of Precipitation
## 3 Results
### Analysis of Idealized Examples
Comparison #22 (Figure 4e) is similar to #21, but the events in the bottom set are larger (the magnitude of events in the bottom set is twice the magnitude of the events in the top set). A real world analogy would be a two pairs of stronger and weaker precipitation events. At \(d=0\) the PSD, dFSS and CDST are close to 33 grid points, which is the magnitude weighted average displacement of the events in the two sets (i.e., \(50\times 2/3+0\times 1/3=33.3\)). As \(d\) increases, the PSD and dFSS also increase more or less linearly to about 50 grid points at \(d=50\)--thus, they approximate well the magnitude weighted average displacement of the events in the two sets. On the other hand, the CDST again decreases with \(d\), and the BDEL and MED values tend to be smaller than PSD and dFSS. Surprisingly, the BDEL values are almost identical to #21, which indicates that the magnitude of events does not influence the results. This behavior can likely be attributed to the nearest-neighbor distances being evaluated over the whole domain instead of only the areas with events. The larger size of the events probably does not significantly affect the distance maps in most parts of the domain, and the BDEL value remains almost the same. In comparison #23 (Figure 4f), the events in each set are biased--there is a larger and a smaller event in each set. Again BDEL values are almost identical to #21 for the same reasons as for #22. Surprisingly, MED values are also practically identical to #21. This behavior can be attributed to the nearest-neighbor methodology used by MED. Only the distance to the nearest event is evaluated, regardless of how big or small that event might be. On the other hand, PSD, dFSS, and CDST are markedly larger compared to #21. This behavior makes sense--since each set is biased, one could expect the displacement to be larger since the events of each set do not balance each other out perfectly. The behavior of CDST is nearly linear with respect to \(d\), but its value does not change much. The center of mass of one field is located more towards the top, while the center of mass of the other field is more towards the bottom. The vertical distance between the centers of mass is larger than the horizontal (which depends on \(d\)) so the CDST does not change much with \(d\). The behavior of PSD and dFSS is more variable and complex, with PSD consistently larger compared to dFSS. They both have a global minimum at approximately \(d=22\) grid points and are larger for other values of \(d\). Figure 5 shows the results of six additional non-spatial measures for comparison #1 with the displaced circular events. Figure 5 can be compared with the bottom panel of Figure 1a, which shows the results of the distance metrics. The non-spatial measures are the root-mean-square-error (RMSE, [13]) and five measures based on contingency tables: the probability of detection (POD, also called hit rate), false alarm ratio (FAR), critical success index (CSI, also called threat score), Heidke skill score (HSS) and Gilbert skill score (GSS, also called equitable threat score). The description of the five contingency-table-based measures is available in [14]. If the forecast is perfect (i.e., \(d=0\)) the values of the non-spatial measures are either 0 (RMSE, FAR) or 1 (POD, CSI, HSS, GSS). As \(d\) increases, the overlap becomes smaller, and the values of RMSE and FAR increase, while the values of other measures decrease, which is as expected. However, once \(d\) reaches 40, their values do not change anymore (for example, FAR starts at 0 and increases towards the worst possible value of 1 at \(d=40\), but does not change when the displacement is further increased). This happens since the overlap becomes zero at \(d=40\) and remains zero for all larger values of \(d\). Thus, as long as there is no overlap, the non-spatial measures will always give identical results regardless of the actual displacement. Clearly, this behavior does not make much sense since the forecast is evidently better if \(d=40\) instead of \(d=100\).