# A New Spatial Distance Metric for Verification of Precipitation
## 3 Results
### Analysis of Idealized Examples
Namely, since the noisy events are more or less evenly spread out throughout the domain, it is never far to the nearest event, which results in a severely reduced estimate of the displacement. Moreover, BDEL has almost a constant value of about 11 grid points, which can be attributed to the nearest-neighbour distances being evaluated over the whole domain (which includes the regions without events). Figure 4: Same as Figure 1 but for noisy, scattered and two-set event fields. Comparison #19 (Figure 4b) uses the so-called scattered events. Both fields contain small-scale single-grid-point events randomly scattered inside a square-shaped envelope. With respect to \(d\), the two envelopes can be overlapping or are far apart. The real-world analogy is a region of small-scale convective systems, with the region being either correctly or incorrectly positioned in the forecast. For larger displacements, when the two envelopes do not overlap (e.g., \(d>40\) grid points), the results are quite similar to comparison #3 (Figure 1c), which uses the fully-filled square-shaped events. However, when the two envelopes overlap, the results diverge from #3, with PSD and dFSS providing smaller displacement estimates. This behavior can be interpreted as measures transitioning from estimating the large-scale envelopes' displacement toward the displacement of individual small-scale events in the overlapping portions of the envelopes. At \(d=0\), the envelopes overlap fully, and all metrics provide an estimate between 2 and 3 grid points, which corresponds quite well to the average nearest-neighbor distance between the single-grid-point events. Even CDST is not zero at \(d=0\), which might be surprising since the centers of the two envelopes are collocated. The reason is that the small-scale events in the two envelopes are randomly generated, resulting in the centers of mass of the two fields not necessarily being collocated. Comparison #20 (Figure 4c) is a mix of comparisons #3 and #19. One field contains the square-shaped envelope with small-scale scattered events, while the other contains a fully-filled square-shaped event of identical size. While the dFSS cannot provide a result (due to the bias being too large), the results of other methods are similar to comparison #19. However, there are some differences at smaller values of \(d\) when the two square-shaped regions overlap. For example, at \(d=0\), PSD is 2.7, CDST 0.3, BDEL 4.5 and MED is 1.5 grid points. Comparisons #21 to #23 (Figure 4d-f) use two sets of events. The sets are located near the top or the bottom of the domain. The \(d\) reflects the displacement of the events in the top set only, while the displacement in the bottom set is fixed to 50 grid points. These setups are interesting as they indicate how the metrics behave in a multi-event situation. In #21 (Figure 4d) all the events are identical. At \(d=0\) most of the metrics provide results of about 25 grid points (except MED, which is about 20 grid points). A displacement of approximately 25 grid points makes sense since this is the average displacement of the events in the two sets--at \(d=0\), the displacement of the top set is zero, and the displacement of the bottom set is always 50 grid points. As \(d\) increases, so do the values of all measures (except CDST). PSD and dFSS increase more or less linearly with \(d\) and are close to 50 grid points at \(d=50\)--thus, they both approximate well the average displacement of the events in the two sets. The values of BDEL and MED tend to be somewhat smaller and behave less linearly with respect to \(d\). Contrary to other measures, CDST decreases with \(d\) and reaches 0 at \(d=50\). This is again linked to the centers of mass of the two fields being collocated. Namely, since all the events are identical, the center of mass of each field will always be located on the middle line between the bottom and top domain borders. At \(d=50\), the centers of mass of both fields are collocated in the exact center of the domain, resulting in CDST = 0.