Uncertainty analysis of sample size and time scale in estimation of SPI using bootstrap resampling method

Drought is a complicated natural phenomenon that occurs basically due to the lack of precipitation over a time period. And its occurrence usually results in great costs on various parts of the natural and the society. Among the various indices in climate drought monitoring, the SPI index (McKee et al., 1993) is the most well-known index, in terms of easy access to its data (rainfall); It is also possible to calculate it in any time window; Ability to calculate magnitude, frequency and continuity; The possibility of quick detection of soil moisture and the possibility of showing the spatial distribution of drought-dominated areas are widely used all over the world (Mishra and Desai, 2005). Standardized Precipitation Index (SPI) can be calculated for any location based on long-term recorded precipitation data. Calculating the SPI drought index in each of the time scales can be considered as one of the advantages of this index. The length of the rainfall data recording period as well as the nature of the probability distributions play an important role in calculating the SPI drought index and these factors are among its limitations.

Methods and materials:

In the present research, the uncertainty of SPI estimation in choosing the length of the statistical period and time scale was investigated using monthly rainfall based on the gamma distribution function at Bandar Abbas synoptic station in two periods of 31 and 64 years. Therefore, the rainfall for the time scales of 3, 6, 12, 24 and 48 months in the two mentioned time periods was calculated and then using the bootstrap method for each rainfall event in each time scale, 1000 random data were generated and the SPI confidence interval was about 95% confidence was calculated. The size of the confidence interval (the difference between the upper band and the lower band) was considered as uncertainty and the absolute ratio error (ARE) between the estimated and observed values was calculated.

ARE and uncertainty due to bootstrap estimation were estimated in two time periods and scales from 3 to 48. The greater the difference between the bands, the lower the certainty and the greater the uncertainty (Vergeni et al., 2015). Therefore, a longer period of time (large number of samplings) has less uncertainty, since the number of samples decreases as the time scale increases, so the uncertainty bandwidth increases. For example, in the 3 and 6-month scale in the 31-year period, the estimation error is 0.09, 0.11 and in the 66-year period is 0.03, 0.05, and the uncertainty has decreased from 0.051, 0.63 and 0.38, 0.41, respectively. In the 12-month scale, the error rate in the short and long term is 0.12 and 0.01, respectively, and the uncertainty is 0.81 and 0.55, respectively. In the 24-month scale, the error in the 31-year period is 0.31, which decreases to 0.1 in the long-term period of 66 years, and in 48 months, it decreases from 0.45 to 0.27, and in the same way, the uncertainty level in the long-term period is Scales 24 and 48 are reduced.Based on the results of two short (31 years) and long (66 years) samples, it showed that there was more uncertainty and error in the samples with a small number, and considering the overestimation or underestimation caused by the length of the short period in the diagnosis of drought classes Historical events have been misjudged and lead to inappropriate drought mitigation measures, as classifying a high-grade drought event (Class 9) to a lower grade (Class 8 or lower) can lead to misleading decision-making.

SPI was able to successfully detect historical droughts in the two investigated periods. The increase of records from 31 to 66 years led to the emergence of one of the most unique features of SPI, i.e. repeatability, reversibility and predictability. The confidence interval resulting from the difference between the upper and lower bands estimated by the bootstrap resampling method indicates uncertainty, the smaller the bandwidth or the result of the difference, the lower the uncertainty and the more reliable it is. In this research, the error caused by bootstrap estimation in 1000 sampling times was also estimated. Based on the obtained results, increasing the number of samples from 31 years to 66 years causes a decrease in uncertainty and error, and as a result, increasing the time scale due to the decrease in the number of samples due to data averaging causes an increase in uncertainty and estimation error. The studied area has It has had long dry and wetter periods in the past and has experienced numerous historical events. Therefore, it is necessary to pay attention to the possible judgments caused by wrong estimation and uncertainty in the observational data, and in this way, seek to reduce the possible damages caused by overestimating or underestimating drought.