Using the normality assumption to calculate probability‐based standardized drought indices: selection criteria with emphases on typical events

Abstract

Enhancing the capability of both standardized precipitation index (SPI) and standardized precipitation evapotranspiration index (SPEI) for quantifying wet and dry events under distinct climate conditions is of paramount importance. The different recommendations of recent studies regarding the best distribution to calculate the SPEI and the lack of studies addressing the effect of different parameters estimation methods on the SPI motivated us to apply and adapt distinct testing methodologies to select candidate models for calculating these standardized drought indices (SDI). The study is based on two data sets. The first represents a tropical–subtropical region of Brazil. The second comprises the same weather stations that were used for developing the original version of the SPEI. The study also emphasized the performance of the models within the range of typical SDI values [−2.0 : 2.0]. Along with goodness‐of‐fit tests, we calculated the mean absolute errors between the indices values estimated from the candidate distributions, and their corresponding theoretical values derived from the standard normal distribution. The two‐parameter gamma and the generalized extreme value distributions are, respectively, recommended for general use in SPI and SPEI algorithms (1–12‐month timescales). The unbiased probability weighted moments are recommended to estimate the distributions parameters. The study also described a trade‐off between choosing the best model for the central part and for the tails of the distributions. This trade‐off suggests that the methodologies used to select models for the SDI algorithms may have to decide which part of the distribution (central or tails) should be emphasized. The behaviour of the errors among different wet/dry categories showed that both indices were only capable of representing drought and floods in a similar probabilistic way within the range [−2.0 : 2.0]. This feature supports our decision to emphasize model performances within such range.

Global and regional performances of SPI candidate distribution functions in observations and simulations

“…In contrast, the index's main disadvantage is the mean by which its standardization is realized and concerns the identification of a suitable theoretical distribution function to describe and normalize highly non-normal precipitation distributions (Lloyd-Hughes and Saunders, 2002). The choice of that suitable theoretical distribution function is a key decision in the index's algorithm (Blain et al, 2018;Stagge et al, 2015;Sienz et al, 2012). This study illuminates reasons for a missing consensus on this choice and attempts to establish such a consensus for both simulations and observations.…”

“…A biased drought description would result from an inadequacy of the fitted distribution function to describe precipitation. Such an inadequacy has been identified for the gamma (Guenang et al, 2019;Blain et al, 2018;Blain and Meschiatti, 2015;Stagge et al, 2015;Sienz et al, 2012;Touma et al, 2015;Naresh Kumar et al, 2009;Lloyd-Hughes and Saunders, 2002) as well as the Pearson type III distribution (Blain et al, 2018;Blain and Meschiatti, 2015;Stagge et al, 2015) in many parts of the world. This lead to the request for further investigations of candidate distribution functions (Blain et al, 2018;Blain and Meschiatti, 2015;Stagge et al, 2015;Touma et al, 2015;Sienz et al, 2012;Lloyd-Hughes and Saunders, 2002;Guttman, 1999).…”

“…As a result, many studies that use SPI directly fit the gamma (Mo and Lyon, 2015;Ma et al, 2015;Yuan and Wood, 2013;Quan et al, 2012;Yoon et al, 2012) or the Pearson type III distribution (Ribeiro and Pires, 2016) without assessing the normality of SPI's resulting distribution with goodness-of-fit tests or other statistical analyses beforehand. The selected PDF, however, is of critical importance because the choice of this PDF is the key decision involved in the calculation of SPI and indeed many authors have urged to investigate the adequacy of distribution functions for new data-sets and regions before applying them (Blain et al, 2018;Stagge et al, 2015;Touma et al, 2015;Sienz et al, 2012). Such a negligence has potentially far-reaching consequences in terms of a biased drought description (Guenang et al, 2019;Sienz et al, 2012).…”

“…Several studies have investigated the adequacy of PDFs fitted onto observed precipitation while focusing on different candidate distribution functions (Blain and Meschiatti, 2015), different parameter estimation methods in the fitting procedure (Blain et al, 2018), different SPI time scales (Guenang et al, 2019), general drought climatology (Lloyd-Hughes and Saunders, 2002), and even the most appropriate methodology to test different candidate distribution functions (Stagge et al, 2015).…”

“…Nevertheless, some common conclusions can be drawn. Most investigations only analyzed 2-parameter distribution functions (Guenang et al, 2019;Blain et al, 2018;Stagge et al, 2015;Lloyd-Hughes and Saunders, 2002). Among those, they agreed depending on the accumulation period and/or the location either on the Weibull or the gamma distribution to be best suited in most cases.…”