# A New Spatial Distance Metric for Verification of Precipitation
###### Abstract  
Precipitation is an essential meteorological variable affecting the biosphere and human societies. At the same time, precipitation is notoriously difficult to predict and verify. A new spatial distance metric for verification of precipitation is presented. It is called the Precipitation Smoothing Distance (PSD). The aim was to develop a measure that would provide a good and meaningful approximation of the displacement of precipitation events in the two fields. An estimate of spatial displacement is very appealing for forecast interpretation because it is easy to understand and mimics how humans tend to judge fields by eye. Contrary to most other distance metrics, the new metric does not require thresholding and can thus be used to analyze binary and non-binary fields (e.g., continuous or multi-level). The analysis of idealized situations showed that the new metric provides a meaningful approximation of the displacement. Typically the estimate of displacement provided by PSD was better than the results provided by most other metrics. The measure is also not overly sensitive to noise, its results are directly related to the actual displacements of precipitation events, and the events with a larger magnitude have a bigger influence on the resulting value. The analysis of ECMWF precipitation forecasts over Europe and North Africa confirmed that the new metric provides a meaningful approximation of the displacement even in more complex real-world situations. precipitation; verification; spatial displacement; PSD metric +
Footnote †: journal: Applied Sciences  
13 April 202216  
skill score displacement [3]. All these methods provide a global estimate of spatial distance, a single value related to the spatial distribution of all events present in the two fields (as opposed to providing a different value for each individual event). The behavior of each metric is different, and each has its unique properties. Most of these metrics also require binary fields as input. The binary fields denote the spatial extent of the precipitation events (regions with non-zero values). The binary event fields can usually be obtained from the original continuous fields by using a threshold. For example, one can define a physical precipitation threshold and set all the values that are the same or larger than the threshold to one, and all the remaining value to zero. An alternative option is to use a so-called frequency (percentile) threshold ([7; 8; 9]). The frequency threshold corresponds to a variable physical precipitation threshold where the value of the physical threshold is adjusted so that in each field a predefined portion of the domain area exceeds the threshold [3]. Using a frequency threshold is an effective method for constructing pairs of a binary fields that are unbiased (having a small bias is a requirement for some metrics, for example the dFSS) and to allow a focus on the spatial error. The use of frequency threshold does not work in all situations, however. For example, if precipitation in one field covers 10% of the domain area, while in the other field the precipitation covers just 1% of domain area, it is not possible to obtain unbiased binary fields for frequency thresholds larger than 1%. This is quite a problem since in real-world situations there can oftentimes be a large mismatch in the spatial extent of precipitation in the observed and forecast fields. Regardless of weather a physical or a frequency threshold is used, the conversion from continuous to binary fields inevitably removes some information from the fields, which might be important for a subjective analysis that a human performed. For example, all the non-zero grid points are treated equally in a binary-based approach. If the value in both fields is above the threshold, the magnitude of the actual difference between the values in the two fields does not matter. This can lead to results that are not in line with a subjective forecast evaluation.