Simulating rainfall time-series: how to account for statistical variability at multiple scales?

Abstract

Daily rainfall is a complex signal exhibiting alternation of dry and wet states, seasonal fluctuations and an irregular behavior at multiple scales that cannot be preserved by stationary stochastic simulation models. In this paper, we try to investigate some of the strategies devoted to preserve these features by comparing two recent algorithms for stochastic rainfall simulation: the first one is the modified Markov model, belonging to the family of Markov-chain based techniques, which introduces non-stationarity in the chain parameters to preserve the long-term behavior of rainfall. The second technique is direct sampling, based on multiple-point statistics, which aims at simulating a complex statistical structure by reproducing the same data patterns found in a training data set. The two techniques are compared by first simulating a synthetic daily rainfall time-series showing a highly irregular alternation of two regimes and then a real rainfall data set. This comparison allows analyzing the efficiency of different elements characterizing the two techniques, such as the application of a variable time dependence, the adaptive kernel smoothing or the use of low-frequency rainfall covariates. The results suggest, under different data availability scenarios, which of these elements are more appropriate to represent the rainfall amount probability distribution at different scales, the annual seasonality, the dry-wet temporal pattern, and the persistence of the rainfall events.

Appendix: Summary of the test on synthetic data

As shown in Sect. 3, the two considered algorithms present a different behavior with respect to various characteristics of the signal and training data amounts considered. The relative error ${\varDelta }=(s-r)/r$ (r = reference, s = simulations median) is calculated on a selection of indicators (Table 4), to summarize the average performance of the two techniques. Positive values indicate overestimation and negative ones underestimation: for example ${\varDelta }\hbox {Q} 95=-0.50$ indicates that the 95-th percentile has been underestimated by 50%. The chosen error indicators mainly regard the error in the tail of the considered probability distributions, since the central and lower part are generally preserved by both algorithms.

Table 4 Selection of indicators summarizing the average performance of the techniques. The relative error of the simulations median is considered for: the i-th quantile of the distribution (${\varDelta }Qi$), the i-th lag of the autocorrelation function (ACF, ${\varDelta }$lagi) and the minimum moving average using a n-months-long moving window (MMA, ${\varDelta }n$m). For each error indicator, a couple of values referring to the two algorithms is given. Bullets indicate a superior performance by MMM and asterisks a superior performance by DS

In accordance with the results shown in previous publications (Mehrotra and Sharma 2007a, b; Oriani et al. 2014), it is shown here that both techniques can generate replicates of the same size as the training data set preserving the rainfall variability at multiple scales. The error on the tail of the distribution (${\varDelta }Q99$ and ${\varDelta }Q100$) is in fact very low for the daily rainfall amount up to the decennial scale in the 1-million simulation group. Reducing the available amount of data, MMM can extrapolate extremes by using a conditional kernel smoothing technique, while DS remains limited to the range of data found in the TI. At higher scales, both techniques can preserve an unbiased distribution even when using a small amount of daily data. Nevertheless, the uncertainty shown by the 05–95 percentile boundaries of the realizations (Fig. 3) suggests that, for a reliable simulation of the considered reference signal, a 10,000-day training data set should at least be used. This principle is confirmed by all the results shown in the previous sections.

Both techniques have a comparable performance regarding the regime A and B spell length distribution: the reference model presents an extremely variable regime duration and a highly skewed distribution which can be preserved when using large training data sets only. In addition, MMM shows a considerable error in the 1000-day group: this may indicate that the model based on the 30- and 365-day wetness indexes needs a larger data set to be calibrated.

The time dependence structure of the total signal is simulated quite accurately by DS as confirmed by a small ACF error on all relevant lags. Conversely, MMM is structured to accurately preserve the lag-1 autocorrelation. To avoid the underestimation of persistence using MMM it is therefore necessary to include the appropriate information in the time dependence structure of the model. This is not needed using DS since it can automatically simulate complex time dependence by generating multiscale patterns similar to the ones found in the training data set. Large errors shown by both techniques in the ACF of the separate regimes are due to their inability to capture the non-stationarity of the two-regime alternation in absence of prior information about it.

Finally, the error on the dry/wet spell length distributions and on the minimum moving average confirms the same tendency: we observe a better performance of DS when sufficient training data are available. MMM is more reliable in case of scarce data availability.