# Western Mediterranean Precipitation Extremes, the Result of Quasi-Resonant Sea-Atmosphere Feedbacks
## 2 Materials and Methods
### Decomposition of SST and Precipitation Depth in Period Ranges  
The coevolution of atmospheric and marine processes is highlighted from the cross-wavelet analysis of precipitation height and SST and a common time reference [27; 28]. When the latter is chosen where the ultimate concentration of precipitation occurs, the phases of both precipitation height and SST anomalies are expressed in relation to the date of occurrence of the western Mediterranean precipitation extreme. SST anomalies referring to the Mediterranean and peripheral seas, as well as the Atlantic Ocean, according to 12 regions are represented in Figure 1.  
The cross-wavelet spectrum of both precipitation height and SST and the time reference is performed within a period range characteristic of harmonic modes of the declination of the Sun. Each time series, whether it is SST or rainfall height, is first represented as a time-period spectrum. Then, this spectrum is sampled so as to characterize each series by an amplitude and a phase, both in a predetermined period range and for a specific date. To obtain that, the spectrum is scale-averaged within the prescribed period range and time-averaged within a time interval framing the date. Instead of a wavelet spectrum, a cross-wavelet spectrum is preferred here so that the series to be processed is \"compared\" to a reference series. This reference series is one of the most representative rainfall series of EREs, where precipitation is torrential. The time interval is defined in such a way as to frame the date of appearance of the ERE, its width being equal to the period in order to optimize the representativeness of the results. All of this is to ensure that the phase represents the elapsed time relative to the date of occurrence of the ERE. The same reference is used for all regionalized time series, whether they are SST or rainfall height. SST data are centered in order to express SST anomalies.  
Three period ranges are used, 0.71 to 1.43 days, 1.43 to 2.9 days, and 11.4 to 45.7 days whose mean periods within the logarithmic scales are 0.95, 1.9, and 21.5 days, respectively; the low and high limits of the period ranges are equal to 3/4 and 3/2 times the mean period, respectively. The first two period ranges refer to the 1/384 and 1/192 harmonic modes to display the depth of precipitation over 1 and 2 days, respectively. The third period range reflects the merging of 1/24 and 1/12 harmonic modes characteristic of both the clustering of lows and harmonization of SST anomalies in the present case of Mediterranean events [19]. This considers the possible overlap of the two characteristic period ranges of the two phenomena.  
Mediterranean rains are characterized by precipitation amounts that can exceed 200 mm in a few hours, over areas of only a few km\({}^{2}\). The problem of representativeness and accuracy of their measurement therefore arises. However, these problems are bypassed here by choosing the upper class of precipitation amounts greater than 60 mm per day. As will be shown, precipitation anomalies higher than this threshold form compact spots. The compactness of the amplitude of precipitations, which is observed for all classes, shows that the temporal and spatial resolution of the data, as well as their accuracy, fulfill the requirements of the present study. Otherwise, anomalies would be scattered. On the other hand, any biases resulting from the calibration of satellite measurements in relation to terrestrial gauges are of little importance in the quantitative representation of torrential rains at daily intervals whose height is well above this threshold.