# A New Spatial Distance Metric for Verification of Precipitation
## 3 Results
### Analysis of Real-World Examples  
While the analysis of idealized examples provides numerous valuable insights into the properties of the new metric, it is also important to demonstrate its behavior in real-world examples. Figure 6 shows some examples of 6-hourly precipitation forecasts over Europe and North Africa (domain size \(360\times 360\) grid points and \(0.125^{\circ}\) resolution). The examples are taken from the study done by [PERSON] and [PERSON] [15]. The forecasts were calculated by the European Centre for Medium-range Weather Forecasting (ECMWF) model, with 6-hourly analysis precipitation (0000-0600 UTC precipitation accumulations) compared with the corresponding precipitation from the 1-, 5- and 9-day forecast. We only studied the PSD since we wanted to analyze the full continuous fields and did not want to use thresholding (the MED and BDEL can only provide results for binary input fields). While the CDST can be used with non-binary fields, we nevertheless decided not to use it here since the analysis of idealized cases showed that it often does not provide meaningful results. Figure 6a shows a 1-day forecast, which is very similar to the analysis. The majority of precipitation is overlapping (Q = 0.25). Thus, the resulting PSD value is small (3 grid points). Figure 6b shows a 9-day forecast with most of the precipitation concentrated in either one or two dominating events. The single dominating event in the forecast is located between the two events in the analysis. There is very little overlapping precipitation (Q=0.85), which results in a medium PSD value of 47 grid points. Figures 6c,d show situations with multiple events spread out over a large portion of the domain. In both cases, about 30% of precipitation is overlapping. At the same time, the regions of precipitation in Figure 6c are less displaced, resulting in a lower PSD value (27 grid points), compared to Figure 6c, where the PSD value is larger (71 grid points). Figures 6e,f show 9-day forecasts that are very different from the analysis. There is less than 20% of overlapping precipitation, and the displacement is large. In Figure 6e, most of the precipitation in the forecast can be found over Western Europe, with the largest amount of precipitation in the analysis located over the western Balkans, which results in a rather large PSD value of 85 grid points. The displacement is even larger in Figure 6f, with most precipitation located either in the NW part of the domain or over North Africa, resulting in an even larger PSD value of 132 points. Finally, Figure 7 shows the analysis of time-series of ECMWF precipitation forecasts for the same domain as before. It shows the monthly and yearly averages of PSD values for the 6-h precipitation accumulations (0000-0600 UTC) in the period 2007-2014. There is a clear difference between the 1-, 5-, and 9-day forecasts, with the average PSD values being approximately 5, 25, and 49 grid points, respectively. The corresponding values of standard deviations of PSD are 2, 13, and 24 grid points. The relatively large standard deviations indicate a large spread of the PSD values that presumably reflect various weather situations that can be present in the analyzed domain. Nevertheless, the series of yearly averages of PSD values has a statistically significant decreasing trend for the 1-day forecast (\(-0.29\) grid points/year, significant at \(p>0.007\)) and the 5-day forecast (\(-0.52\) grid points/year, \(p>0.07\)). The 9-day forecast does not have a significant trend. A modified Mann-Kendall test that utilizes a trend-free pre-whitening method [16] was used to determine the statistical significance of a trend. The decreasing and significant trends of the 1- and 5-day forecasts  
Figure 6: PSD for selected examples of 6-hourly operational ECMWF precipitation forecasts over Europe and North Africa. The domain size is \(360\times 360\) grid points. The left panel of each figure shows the analysis precipitation, while the right panel shows the forecast for the same 6-h period.