Probabilistic approach in the heavy rainfall modelling – is model verification needed?

The aim of this paper is an attempt to answer the question whether it is necessary to update the probabilistic model of heavy rainfall with the passage of time. Based on IMWM-NRI measurement data in Wroclaw, probabilistic models for data from 1960–2009 with data up to 2017 were compared. The analysis showed that intense rainfall recorded in recent years can significantly affect the course of the depthduration-frequency (DDF) model.


Introduction
Secure sewerage systems designing is intended to ensure a required water drainage standard, defined as the adaptation of the drainage system to receiving the maximum -forecasted rainfall water streams occurring with a frequency equal to the allowable -socially acceptable -frequency of occurrence of a terrain surface flooding ( Table 1). The current European standard EN 752 allows the frequency of sewerage system overflows to the rare repeatability of their occurrence: one time per 10 years -in case of rural areas and one time per 20, 30 or 50 years for urban areas -respective to the spatial development type. The quoted European standard recommends, for the designing of drainage areas, the following rainfall occurrence frequencies: once a year in case of rural areas, and once in 2, 5 or 10 years for municipal areas, whereby there can be no overload in the operation of the gravity systems -work under pressure. It, therefore, follows that, among others, the need to select the sewers for incomplete filling, namely with a capacity reserve in case of less frequent rainfall [1,2].
The sizing of drainage or combined sewage systems in Poland presents difficulties resulting from the lack of a reliable precipitation model. Most often used model of Błaszczyk from 1954 lowers calculation results for rainfall intensities by 40%, as shown in the paper, on the example of precipitation measured on the meteorological station IMWM-NRI (the Institute of Meteorology and Water Management -National Research Institute) in Wroclaw from the time span 1960-2009. This has consequences when sizing drainage areas according to the recommendations of European standard EN 752, directly affecting the higher frequency of occurrence of these unfavourable phenomena in Poland [3,4]. The aim of the work is to verify the depth-durationfrequency (DDF) model of Kotowski and Kaźmierczak for Wroclaw (developed on the basis of measurement data from 1960-2009 [5]), extending the scope of measurement data to 2017, in which several rainfall events with extremely high rainfall interval values occurred [6]. In the light of the increasingly frequent and intensified storms observed in recent years in the area of Wroclaw and Poland, the authors ask whether the probabilistic model needs updating.

Materials and methods
The meteorological station of the Institute of Meteorology and Water Management -National Research Institute in Wroclaw (southern west part of Poland) is a part of national measurement and observation network at hydrological and meteorological service. The station coordinates: 51-06 N, 16-54 E; terrain altitude: about 120 m above sea level. In this study, precipitation records from this IMWM-NRI meteorological station from 1960-2017 long-term period were used as a research material. To implement the measurement program, the station in Wroclaw uses standard equipment, typical for Polish National Hydrological and Meteorological Service synoptic stations, i.e. meteorological instruments connected to the MAWS -meteorological automatic weather station. The atmospheric precipitation measurement is carried out in parallel with the automatic SEBA rain gauge, which records the 1-minute rainwater and with the participation of a meteorological observer, which records 6 hour checksums and a daily sum using Hellmann's rain gauge. The collected measurement data is compared and verified after each reading [5,7,8].
In the first stage of the research, an update of the sequence of the largest rainfall data in the assumed intervals of rainfall duration (from 5 to 4320 minutes) necessary for the construction of the probabilistic model was carried out.   The research in question was based on the methodology employed by Kotowski with the team, which assumes the following procedure for determining maximum rainfall with a specific duration and probability of occurrence [7][8][9][10][11]:  isolation of the largest precipitation sums in intervals of 5 to 4320 minutes in an amount equal to the number of years of the data stream under investigation,  in the resulting distribution series, non-binding assignment, the probability of empirical crossing,  determining the estimators of parameters (shape, scale, location) of selected probability distributions 2 ITM Web of Conferences 23, 00036 (2018) https://doi.org/10.1051/itmconf/20182300036 XLVIII Seminar of Applied Mathematics with the maximum likelihood method (MLM) by numerically maximizing of the reliability function,  analysis of matching theoretical distributions to measurement data using the following indicators: BIC -Bayesian Schwartz information criterion, AIC -Akaike information criterion, using the Anderson-Darling test and the relative mean square residual error (RRMSE),  the probabilistic model is constructed by the dependence of the probability distribution parameters as a function of time for the quantile of the random variable of the best fit distribution.
Using the above procedure, the generalized exponential distribution (GED) parameters were determined with the use of a maximum likelihood method [12] for the data from 1960-2009 and 1968-2017. Then, the theoretical precipitation heights with the specified duration and frequency of occurrence were calculated using the calculated parameters and the formula for this purpose [7,12]: The final step was to compare the results of 2 heavy rainfall probabilistic models for selected rainfall durations and also for selected empirical probabilities (C-year return period).

Results
Based on rainfall data from both periods (1960-2009 and 1968-2017), the GED distribution parameters were calculated for all analyzed rainfall durations. The calculation results are shown in Table 2. The parameters were determined by the maximum likelihood method by numerically maximizing of the reliability function.  By substituting the parameters from Table 2 to Formula (1), the theoretical precipitation heights with given duration (from t = 5 to t = 4320 minutes) and frequency of occurrence (from C = 1 to C = 50 years) were calculated. The calculation results are presented in Table 3 for the data from 1960-2009 and in Table 4 for the data from 1968-2017. In order to compare the obtained results, the percentage differences (Table 5) between rainfall amounts from Table 3 and 4 were calculated. In order to increase the transparency of the results, differences equal to or greater than 5% are marked in green.
In the case of 22 results (out of 122), there was an increase in rainfall amounts by at least 5%, with as much as 18 times for the frequency of occurrence of C = 10 years and higher. The highest increase was reported for short-term rainfall (t = 5, t = 10 and t = 15 minutes) and lasting 12 and 18 hours.   The conducted comparative analysis showed no case of a significant (greater than 5%) decrease in precipitation amounts.

Conclusions
The aim of this paper is an attempt to answer the question whether it is necessary to update the probabilistic model of heavy rainfall with the passage of time.
Based on the IMWM-NRI measurement data, the probabilistic models of maximum rainfall were developed. The models were based on the generalized exponential distribution (GED), whose parameters were determined by the maximum likelihood method. The models were developed on the basis of 50-year measurement series from 1960-2009 and 1968-2017. The developed models allow determining the forecasted maximum rainfall amount with given durations from t = 5 to t = 4320 minutes and frequencies of occurrence from C = 1 to C = 50 years.
The analysis showed that heavy rainfall recorded in recent years can significantly affect the depth-durationfrequency (DDF) model. In the case of 18% of results, there was an increase in precipitation amounts by at least 5% -especially for the frequency of occurrence C = 10 years and higher. The highest increase was recorded for short-term rainfall (t = 5, t = 10 and t = 15 minutes) and lasting 12 and 18 hours.
The conducted comparative analysis showed no case of a significant (greater than 5%) decrease in precipitation amounts for the analyzed durations from t = 5 to t = 4320 minutes and frequencies of occurrence from C = 1 to C = 50 years.
The obtained results confirm the observations in the literature on increasing the intensity of rainfall over the last decades. Therefore, there are premises for updating the maximum precipitation models developed in the past, as well as attempt to develop prognostic models taking into account the described trends of changes.