2.2.3. Trend Analysis

The methods applied in climatic and hydrological trend analysis are typically classified into two types: parametric and non-parametric [65,66]. The latter normally requires fewer assumptions (e.g., normality of study data) compared to the former. In reality, the assumptions on data distribution are difficult to satisfy. Therefore, the parametric methods are considered less robust than the non-parametric methods [66]. Among all non-parametric methods, the Mann–Kendall test (MKT) [67,68] has been applied extensively in the field of climatology and hydrology [14–16,45,69,70]. The approach first identifies the sign of each possible pair of data in the study time series, followed by the determination of the corresponding test statistic z. The null hypothesis (H0) assumes no significant monotonic trend in the time series while the alternative hypothesis suggests otherwise. The null hypothesis is rejected when |Z| > Z1−<sup>α</sup>/2, where Z1−α/2 is the probability of the standard normal distribution at a significance level of α. This study employed the MKT in assessing the significance of a trend and uses 0.05 as the significance level.

**Figure 3.** SPEI-12 of: (**a**) San Francisco Bay region; (**b**) Sacramento River region; and (**c**) Colorado River region during the historical period (1951–2013). Blue color indicates wet conditions; red color designates drought conditions. The purple line is the threshold below which extreme drought conditions exist.

This study further applied the non-parametric Theil–Sen approach (TSA) [71,72] to identify the slope of significant trends determined via the MKT. In this approach, the slope values (vector *TS*) of all data pairs are first calculated:

$$TS = \frac{V\_i - V\_j}{i - j} \text{ i } = 1, 2, \dots, n; j = 1, 2, \dots, n; i > j \tag{2}$$

where *n* is the length of study record period; and *Vi* and *Vj* are time series values at time *i* and *j*, respectively (*i* > *j*). The median of *TS* is then used as overall slope of the trend identified for the study time series. A positive (negative) slope value represents an increasing (decreasing) trend. In this study, trend analysis is conducted in both historical (1951–2013) and future periods (2020–2099).
