# A New Spatial Distance Metric for Verification of Precipitation
## 3 Results
### Analysis of Idealized Examples  
The behavior of the new method was analyzed by applying the new measure along with four other measures to a set of 23 situations with idealized geometric fields. The situations and the corresponding results are shown in Figures 1-4. All comparisons are based on fields constructed on a 200 \(\times\) 200 regular grid. Some of the situations are based on the cases from [3,12] while others are new. Some of the fields are binary, while others are multi-level or continuous. They are described in detail in the supplementary material. Each situation also depends on a variable parameter \(d\), reflecting the displacement of one of the events. The results are always provided for a range of \(d\) values, starting from 0 and going to either 50 or 100 grid points. All the input fields that were used for the idealizedsituations are freely available on github (for details please refer to the Data Availability Statement at the end of the article). Figure 1 shows the first six comparisons denoted by indices #1 to #6. Comparisons #1-#3 (Figure 1a-c) use a single event of either circular, Gaussian, or square shape, with an identical but displaced event in the other field. The PSD and CDST give a near-perfect estimate of the actual displacement for the whole range of \(d\) values for the circular and square-shaped events. All other measures underestimate the displacement with the smallest underestimation provided by dPSS. For the Gaussian events, due to the events being non-binary, only PSD and CDST provide a result, with both giving a near-perfect estimate of the actual displacement for the whole range of \(d\) values. Figure 1: The subfigures (**a–f**) show six idealized situations that were used to analyze the behavior of the distance measures. The top panel in each subfigure shows the layout of the idealized situation. The blue color corresponds to the values in the first field, and the red color corresponds to the values in the second field. The higher the value, the darker the respective color. The arrow marked with \(d\) shows the displacement of the event it is pointing to. For example, for \(d=0\), the blue circular event in (**a**) will completely overlap the red event, while for \(d>0\), the blue event is moved toward the right. The bottom panel in each subfigure shows the results of the distance metrics. Since the results depend on the \(d\), they are provided for values ranging from 0 to 100 grid-points. If the value of a certain metric is undefined, its results are not shown (e.g., the input fields are non-binary, but the metric can only be used with binary fields). For the dFSS, the underestimation is quite well documented and discussed in [3]. For example, Figure 3c in [3] shows that the relative underestimation is the largest when the displacement is equal to the diameter of the circular events. For comparisons #1-#3 the underestimation additionally increases for large displacements when \(d\) approaches 100. This happens due to the so-called border effect, which can result in dFSS underestimating the displacement if the events are located relatively close to the domain borders (this is also mentioned [3]). The border effect can be mitigated by increasing the domain size (this was not done here). Regarding MED, the likely cause of the underestimation is its reliance on distance maps generated via the nearest-neighbor methodology. When the MED between two non-overlapping events is calculated, the distance between the grid points of one event to the closest part of the second event is considered, which results in an underestimation. BDEL is also based on distance maps. Still, since the nearest-neighbor distances are evaluated over the whole domain, including the non-event regions, its behavior is more complicated and more difficult to interpret.