- freely available
Hydrology 2019, 6(4), 89; https://doi.org/10.3390/hydrology6040089
- The sample size of continuous rainfall series with a high resolution plays a crucial role for a robust calibration of SRGs. In many locations, this sample size is very short and/or only AMR series are available.
- The basic versions of SRGs usually underestimate extreme values at fine scales. Many attempts at improving the modeling were carried out, but they implied an unsuitable over parameterization for short sample sizes or an inability to reconstruct other features, such as dry/wet ratios.
- Is it possible to calibrate a SRG by only using AMR series, which are usually more and more longer than continuous data with a high-resolution for many locations? Even, AMR series are the unique available data set for some locations.
- If so, for a specific SRG are there some parameters which mostly influence the extreme value distributions?
- In Section 2, the authors provided a brief overview of the NSRP model and described the adopted calibration and validation procedures, in order to derive possible relationships among extreme value distributions and NSRP parameters.
- The obtained results, from calibration and validations procedures, are illustrated in Section 3.
- Section 4 regards discussion about results and future developments.
2. Materials and Methods
2.1. Brief Theoretical Description of NSRP Model
- It is assumed that the number of storms is a homogeneous Poisson random variable. This means that the inter-arrivals, , between the origins of two consecutive storms are independent and identically distributed, and they follow an exponential distribution:
- A number, , of rain cells (also named bursts or pulses) is associated with each storm origin; is usually considered as geometric or Poisson distributed. In this work, a geometric distribution is assumed and, with the aim of having at least one burst for each storm, the random variable is used, with , so that and:
- With respect to the associated storm origin, each burst origin occurs after a waiting time, W, which is assumed as an exponentially distributed variable with parameter and :
- Each burst has a rectangular shape, with an intensity I and a duration D which are both considered as exponential distributed, with parameters and , respectively, and , :
- The total precipitation intensity, at time t is then calculated as the sum of the intensities which are related to the active bursts at time t:
2.2. Calibration Procedure
- Then, 1000 five-parameter sets were generated with a Monte Carlo approach , and, for each set, a 500-year continuous time series of 1 min rainfall heights was simulated, from which AMR series, related to 5, 15, 30, and 60 min, were extracted. The choice of generating only one long realization is according to the ergodicity property for a stationary stochastic process [43,44].
- Although analytical expressions of probability distributions do not exist for AMR series from the basic version of a NSRP model, they can be approximated with an EV1 distribution :The parameters and are related to the theoretical mean, , and standard deviation, , of H in the following way :
- The final step regarded the investigation of possible relationships among each NSRP parameter from the set with the sets , estimated into Step 3 for the considered durations (5, 15, 30, and 60 min).
2.3. Validation Procedure
- Estimation of EV1 parameters (with the ML technique or others).
- Generation of NSRP parameters, by considering the investigated relationships. If no relationship is found for one (or more) NSRP parameter, this is considered as a random uniform variable, with the variation range in Table 1, and then, for each (virtual or real) site, 1000 NSRP five-parameter sets are generated, in which only the parameters with a robust relationship with EV1 distribution present the same value, while the others vary in a uniform way.
- Reproduction of a 500-year continuous 1 min data series, for each generated NSRP five-parameter set and consequent determination of the associated AMR series.
- Evaluation of frequency distributions for each EV1 parameter, from the whole ensemble of 1000 NSRP generations, and comparison with the corresponding ML estimates of the starting sample. Analogous comparisons can be carried out by considering mean and standard deviation values.
- It is expected that the waiting time between two consecutive storm occurrences could imply effects on monthly and annual scale, but not at finer resolutions (in particular, sub-hourly). In fact, a heavy rainfall event can fully develop itself, without being affected by a large or short waiting time with the previous and the successive storms. Greater values for could determine large dry periods and consequently low aggregated rainfall values only at monthly and yearly scales.
- Similar considerations can be made for and . In fact, the number of bursts (and consequently their occurrences) for each storm could mainly influence the aggregated process (see Equation (7) and Figure 1) at coarser resolutions (from daily or multi-daily), but not at finer scales (hourly and sub-hourly), especially if durations of the pulses are always sub-hourly on average, as assumed in this work (see Table 1).
- observed 15-min and 60-min AMR series, related to Cosenza rain gauge (Southern Italy), having a sample size M = 29 years for 15-min AMR and M = 63 years for 60-min AMR, and characterized by a clear EV1 behavior, as shown in EV1 probabilistic plots (Figure 11).
- For each generated NSRP five-parameter set, a 500-year continuous 1 min data series was reproduced, from which the associated 5, 15, 30, and 60-min AMR series were extracted, together with the correspondent EV1 parameter estimations.
- From the previous step, frequency distributions for , were obtained and compared with the ML estimates related to the specific (virtual or real) starting data series (Figure 7 and Figure 8). Comparisons were also carried out in EV1 probabilistic plots (Figure 9 and Figure 10), among starting AMR series, their EV1 theoretical curves, and EV1 curves associated with (i) median values for and (ii) values which were calculated with the relationships reported in Figure 2.
Conflicts of Interest
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|NSRP Parameter||Min. Value||Max. Value|
|Synthetic sample 1||240||6||8||8||0.25|
|Synthetic sample 2||240||6||8||12||0.25|
|5 min||15 min||30 min||60 min|
|Synthetic sample 1||1.30||3.65||0.45||9.68||0.25||16.21||0.15||22.85|
|Synthetic sample 2||0.85||5.40||0.30||14.40||0.17||24.06||0.10||33.85|
|15 min||60 min|
|Cosenza rain gauge||0.27||13||0.14||17.2|
|Synthetic sample 1||[120; 360]||[2; 10]||[5; 24]||8.43||0.26|
|Synthetic sample 2||[120; 360]||[2; 10]||[5; 24]||12.10||0.27|
|Cosenza rain gauge||[120; 360]||[2; 10]||[5; 24]||13.15||0.10|
|50% EV1_NSRP||ML Parameter Estimation from Data|
|Performance Measures||5 min||15 min||0 min||60 min||5 min||15 min||30 min||60 min|
|Synthetic sample 1||0.017||0.042||0.037||0.019||0.010||0.024||0.018||0.018|
|Synthetic sample 2||0.041||0.023||0.036||0.069||0.006||0.008||0.033||0.018|
|Cosenza data series||0.143||0.041||0.090||0.049|
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