## 2 Methods
### The QI-Based Estimation of Precipitation  
The design and development of the _qPrec_ precipitation estimation software was started from the scratch. The governing idea was to combine probabilistic and variational approaches. The schematic of the processing chain of the _qPrec_ software is sketched in Figure 2.  
The _qPrec_ processing chain (Figure 2) consists of four blocks. The first one is the calibration step. At the very beginning, the precipitation-related variables are selected. This may be, for instance, the radar reflectivity in terms of the CMAX (column max, i.e., the highest reflectivity in the corresponding column above the Earth's surface), CAPPI (constant altitude PPI, i.e., a slice through a multitude of different PPI scans at a predefined altitude above the Earth's surface), etc. ; further _qi_, H-SAF (EUMETSAT Satellite Application Facility on Support to Operational Hydrology and Water Management [29]; for instance, the H-SAF product P-IN-SEVIRI (H03B)--instantaneous maps by infrared images from operational geostationary satellites \"calibrated\" by precipitation measurements from satellite sensors in sun-synchronous orbits); or any other variable that is directly associated with the precipitation, such as the lightning density. Let us denote the selected precipitation-related variables as \(y_{k}\), \(k=1,\ldots\)\(K\), where \(K=N+M\), \(N\) is the total number of radars at the given point, and \(M\) is the number of additional, rainfall-related fields (such as an H-SAF product as mentioned above). It is not advised to select different radar reflectivity products at the same time (such as CAPPI and CMAX), due to their cross-correlation. Figure 2: The processing chain of the _qPrec_ software. “dBz” indicates the radar reflectivity products, _qi_ stands for quality indices, and H-SAF is the EUMETSAT Satellite Application Facility on Support to Operational Hydrology and Water Management. \(K\)—the total number of the rainfall-related products, \(P\)—the probability of precipitation, \(RR\)—the rainfall rate, \(\sigma^{2}\)—the variance of the rainfall rate. The input fields of \(y_{k}\) are calibrated by the rain-gauge data to obtain an unbiased estimate of the rainfall-rate derived from the given input variable. The calibration is carried out on the basis of pairs of the rain gauge measurements and the corresponding radar-based rainfall estimation within the range of the given radar (usually approximately for 100-200 rain gauges) and for a long-term period lasting for several weeks (which, for instance, allows retaining the seasonal or intra-annual variability of the precipitation). The outputs of the first processing block are four functions that describe the actual relationship between the input field \(y_{k}\) on one hand, and the probability of precipitation \(P\), the mean rainfall rate \(RR\), the variance of the rainfall rate \(\sigma^{2}\), and the mean quality index \(\overline{q! }\) of the inputs on the other (\(f_{P}\), \(f_{RR}\), \(f_{or}\) and \(f_{qi}\), respectively). Figure 3 presents an example of these four relationships for CMAX as the precipitation-related input field. The second processing block is the conversion. In this step, the input maps are converted by the calibration functions to maps of the mean rainfall rate, variance of the rainfall rate, probability of precipitation, and quality index. The processing Blocks #1 and #2 are repeated \(K\) times with each circle corresponding to a different input field \(y_{k}\), \(k=1,\ldots,K\). The third block is the combination of the \(K\) converted input maps into a common map of the mean rainfall rate, variance of the rainfall rate, probability of precipitation, and quality index.