# A New Spatial Distance Metric for Verification of Precipitation
## 3 Results
### Analysis of Idealized Examples
For example, for comparison #1 at \(d=100\), reducing the domain size (thereby reducing the size of non-event regions that surround the central region with the two events) will decrease the BDEL value, while increasing the domain size will increase the value. Comparison #4 (Figure 1d) is similar to #1 but uses circular events with different sizes (diameters 20 and 10 grid points). For \(d>10\), both PSD and CDST give a near-perfect estimate of the actual displacement, while the other measures underestimate it. Due to the two fields having a significant bias, the dFSS cannot provide a result. At \(d<10\), the CDST approaches zero as \(d\to 0\), while the decrease of PSD slows and is about 6 grid points at \(d=0\). The CDST being zero at \(d=0\) is a consequence of the centers-of-mass of the two circular events being collocated. However, since the two fields are not identical, having a displacement of zero (which indicates a perfect forecast) does not make much sense. Since the overlap of the two events is not perfect, and parts of the larger event stick out from the area of the smaller event, it makes more sense that the displacement is larger than zero. Since the larger event sticks out from the smaller event for 0 to 10 grid points, with the middle being five grid points, the PSD result of 6 grid points seems quite reasonable. Comparison #5 (Figure 1e) uses a circular and a displaced Gaussian event. Similarly to #2, only PSD and CDST provide a result, with both giving a near-perfect estimate of the actual displacement for \(d>20\). Similarly to Comparison #04, the CDST approaches zero as \(d\to 0\), while the decrease of PSD slows and is about ten grid points at \(d=0\). Comparison #6 (Figure 1f) uses a linear gradient event, with values increasing from 0 to 1 from right to left, and a circular event. Again, since the gradient event is non-binary, only PSD and CDST provide results. Since the magnitude of the gradient event is the largest on the left side of the domain, the resulting estimates of displacements are largest when the circular event is on the right side (i.e., if \(d\) is large). At \(d=100\), both measures provide a similar result of \(\approx 80\) grid points, but the difference between the two measures starts to increase when \(d\) decreases. At approximately \(d=13\), the PSD and CDST values are the smallest, with PSD being 31 grid points while CDST is zero. Again, CDST being zero does not make much sense since the two fields are, in fact, very different, regardless of the value of \(d\). Figure 2 shows the six comparisons that use a pair of either binary or Gaussian events of elliptical shape in various combinations of displacement and orientation. Comparisons #7 and #8 (Figure 2a,b) use a pair of identically oriented binary elliptically events. The PSD and CDST give a near-perfect estimate of the actual displacement for the whole range of \(d\) values. All other measures underestimate the displacement with the smallest underestimation provided by dFSS. The underestimation is especially large for #8 in the case of MED and BDEL, which is again linked to the nearest-neighbor methodology they are based on (due to the horizontal orientation, one end of the ellipse will always stick quite far out towards the center of the other ellipse). In comparison #9 (Figures 2c) the two elliptical events are perpendicular, with all the measures, except CDST, providing a larger than zero result for \(d=0\). At \(d>20\), the PSD and CDST give an excellent estimate of the actual  
displacement, while the other measures underestimate it. For the comparisons #10-#12 (Figure 2d-f), with the non-binary Gaussian elliptical events, only PSD and CDST can provide results, which are very similar to the ones for binary elliptical events (comparisons #7-#9). Comparisons #13 to #17 (Figure 3) use a set of displaced double-level events in various combinations of displacement and orientation. The area with low-intensity values is elliptical but contains a smaller circular area with high-intensity values.