Non-stationary modeling of the meteorological drought index SPIt using generalized additive models for location, scale and shape

Introduction

Traditional hydrological time series analyses often assume stationarity, particularly in the estimation of drought indices such as the Standardized Precipitation Index (SPI). However, increasing climate variability and anthropogenic influences have introduced significant non-stationarity into hydrological processes. This challenges the reliability of stationary-based assessments and highlights the need for models that can adapt to changing conditions. Generalized Additive Models for Location, Scale, and Shape (GAMLSS) offer a flexible framework for modeling such dynamics by allowing distribution parameters to vary over time or in relation to covariates. Recent studies suggest that non-stationary modeling improves drought characterization, particularly at longer time scales. Yet, findings remain mixed: while some report better accuracy with non-stationary approaches, others find stationary models still perform adequately, depending on regional and climatic factors. Given these variations, regional validation becomes essential. This study evaluates the performance of a non-stationary SPI-based index, referred to as SPIt, in comparison with the traditional stationary SPI. The case study is the Karkheh river basin in western Iran—a region with distinct climatic conditions compared to where SPIt was originally developed. Drought monitoring is conducted using monthly precipitation data from five stations, focusing on seasonal periods outside the dry summer months due to limited rainfall during that time. By comparing the two indices, the study aims to assess whether accounting for non-stationarity leads to more accurate drought representation in semi-arid climates.

Materials and Methods

The study focuses on five meteorological stations: Pol-Zal, Bostan, Pol-Kohneh, Noorabad and Halilan, spanning a historical data period ranging from 1971 to 2022 (C. E.) depending on rain gauge station availability. These stations were selected due to their diverse altitudes, geographical spread within the Karkheh basin (The latitude range is 47 to 48 degrees north and longitude 31 to 34 degrees east), and sufficiently long precipitation records, especially for the winter season (December to February), which accounts for the bulk of annual rainfall in the region. Precipitation data were analyzed for stationarity using the non-parametric Mann-Kendall trend test. Stations exhibiting significant trends were modeled using non-stationary GAMLSS, where the shape parameter of the gamma distribution was allowed to vary with time using polynomial functions optimized through the Akaike Information Criterion (AIC). The study employed a two-parameter gamma distribution to model winter precipitation in both stationary and non-stationary conditions. For drought assessment, two indices were used. SPI that Assumes stationary gamma-distributed precipitation, standardized to a normal distribution and SPIt that extends SPI by allowing the shape parameter of the gamma distribution to vary over time, thereby accommodating non-stationarity.

Results and Discussion

The Mann-Kendall test revealed significant decreasing trends in precipitation at Polzal and Bostan stations at 95% confidence level, and a similar albeit weaker trend at Norabad. No significant trends were detected at Pol-Kohneh and Holilan. The goodness-of-fit tests (Kolmogorov-Smirnov and Chi-square) confirmed that the gamma distribution was appropriate for all stations. GAMLSS modeling showed that non-stationary models outperformed stationary ones at stations with evident trends. For instance, AIC values were reduced by 5, 6, and 1 unit at Pol-Zal, Bostan, and Norabad, respectively, indicating better fit under non-stationary conditions. Time series analysis of the shape parameter in GAMLSS revealed temporal variability at all stations, supporting the hypothesis of non-stationarity. Worm plots for residual analysis confirmed model adequacy in both conditions, but improvements in model residuals under non-stationarity were evident at trend-affected stations. A comparison of SPI and SPIt indices indicated substantial differences in drought classification over time. At Polzal, years such as 1352 and 1354 showed no drought under SPI but were classified as moderate drought (D1) under SPIt. Similarly, years with similar rainfall amounts (e.g., 127 mm in 1971 vs. 125 mm in 2010) were categorized differently in SPIt, highlighting the model's sensitivity to underlying non-stationarity. At Holilan station, where no significant trend was observed, SPI and SPIt provided nearly identical results, reaffirming the utility of SPIt in trend-sensitive environments. A station-wise drought frequency comparison between SPI and SPIt further revealed that non-stationary modeling generally results in higher estimated drought frequencies at trend-affected stations. For example, the frequency of droughts at Polzal increased from 48% (SPI) to 52% (SPIt). Similar increases were noted at Bostan and Noorabad. Conversely, at Holilan and Polkohneh, where no significant trends were detected, the drought frequency remained the same or slightly decreased under SPIt. Moreover, the frequency of severe droughts (D4) decreased under the non-stationary model, with D4 events dropping from 2% to 0% at Pol-Zal, from 3% to 0% at Bostan, and from 6% to 4% at Noorabad. This suggests that the SPI may overestimate drought severity when stationarity is incorrectly assumed.

Conclusion

Long-term drought monitoring at various rain gauge stations highlights the importance of considering changes in precipitation when making decisions and setting policies in watersheds. When significant trends are present, drought analysis can be performed under either stationary or non-stationary assumptions, depending on the objective. If the primary concern is drought frequency, non-stationary analysis is strongly recommended. Results showed that at stations with trends such as Polzal, Bostan, and Norabad the frequency of droughts was underestimated under stationary analysis compared to non-stationary models. For example, frequencies increased from 48, 48, and 43 (stationary) to 52, 52, and 47 (non-stationary), respectively. However, in non-trending stations, stationary models may still provide reliable results for frequency estimation. In contrast, if the focus is on severe droughts, stationary models may outperform non-stationary ones at trend-affected stations. Non-stationary analysis yielded zero severe drought events, while stationary models identified 2, 3, and 6 cases in Palzal, Bostan, and Norabad, respectively. In non-trend stations like Polkohneh and Holilan, non-stationary analysis was more effective in detecting severe events. These findings align with previous research suggesting that while non-stationary models, such as those using GAMLSS, offer better parameter estimation, stationary models may sometimes better reflect reality in future projections. Therefore, although non-stationary modeling is essential under climate variability, the choice of model should depend on the monitoring goal. It is also recommended to incorporate time-varying variance and alternative probability distributions for better drought characterization under potential extreme rainfall events.