4.1. Case Study
In order to quantitatively assess the changes in extreme precipitation from different perspectives, a number indicators focusing on various aspects of extreme precipitation have been proposed. According to the recommendations of the World Meteorological Organization, other relevant studies [
14,
15], four extreme precipitation indicators are used in this study to describe extreme precipitation features in Loess Plateau, including, the amount (P95), the number of days (D95), the intensity (I95), and the proportion (R95) of extreme precipitation. Detailed definitions of these indicators are shown in
Table 1.
Six two-dimensional and three three-dimensional combination schemes of the four indicators can be formulated. The indicator combination schemes are shown in
Table 2, and the joint return period of “AND”, “OR”, and Kendall for these combinations are characterized. Taking “AND” return periods of the combination events of {D95, P95} and {D95, P95, R95} as examples, the combination {D95, P95} indicates D95 and P95 simultaneously exceed their thresholds (Equation (10)), which means that an extreme precipitation event with long time and large amount would happen and severe flood may occur. The combination {D95, P95, R95} indicates that an extreme precipitation event with long duration and large amount occurs, and the proportion of this event to annual precipitation also exceeds the threshold (Equation (11)). For the reason that the annual rainfall is largely completed by extreme precipitation in this situation, adaptation strategies should be devised to achieve reasonable storage of water resources in the flood season and allocation of water resources throughout the year. The meaning of the other types of return periods for each joint event can be inferred from the definitions in
Section 3.2.2.
The spatial risk characteristics for single indicators and multi-indicator combinations throughout the Loess Plateau can be obtained from the results over all stations in the region. In order to explore the trend in different extreme precipitation indicators and their joint risks, a 30-year moving window approach is applied. From the beginning of 1959–1988 to the end of 1989–2018, 31 groups of results for all stations were obtained. For each group, every indicator is fitted with marginal distribution (the GMM) and copulas are fitted for all two and three-dimensional combinations. The initial group (from 1959 to 1988) and the final group (from 1989 to 2018) of the 31 results belong to two independent time periods. The comparison of the results for these two groups can directly reflect the changes of extreme precipitation in the past two representative periods. By evaluating the outcomes of 31 groups in time series, the trend for both the univariate and multivariate risks can then be obtained. The Mann–Kendall (MK) test [
40] is applied to evaluate whether the trend is significant, and the linear regression method is used to quantitative assess the degree of change. The annual extreme precipitation indicators are used in this study, and the Mann–Kendall test is basically not affected by serial correlation.
Taking Yulin station (Station 34 in
Figure 1) as an example, the whole calculation process for the three indicators of P95, D95, and R95 during the period 1959–1988 is illustrated. GMM is used to construct the marginal distributions for the three indicators, and the test results of KS, RMSE, and AIC are shown in
Table 3.
Table S1 in the Supplementary Materials also shows the statistical test comparisons between GMM and the best fitted ones of five other commonly used parametric distributions (Gamma, Pearson type III (P III), lognormal (LN), Log Pearson type III (LP III), and generalized extreme value (GEV)) [
23]. The
p-values of KS tests are all greater and the AIC values are all smaller for GMM than other parametric distributions.
Figure S1 shows the PDF fitting results of GMM with other parameter distributions in
Table S1. It can be seen that even in the case of non-multimodal PDF, the performance of GMM can be similar to the optimal parameter distribution (
Figure S1c) or even better (
Figure S1b). Candidate copulas of Frank Gumbel, Gaussian, and t copulas (at two- or three-dimensional forms) are applied to build the joint probability distributions of multi-indicator combinations. The optimal copulas selected according to the KS, RMSE and AIC tests are shown in
Table 3, and the corresponding test values are also displayed. It can be seen that the p-values of KS test for all indicators and multi-indicator combinations are all greater than 0.05, which means that the fitted distributions can effectively describe the probability characteristics of the indicators in various dimensions. The test results of the other copulas such as Frank and Gumbel are shown in
Table S2 in the Supplementary Materials.
Figure 3a–f further shows the goodness-of-fit by comparing the empirical PDFs and CDFs with those from GMM distributions of the three indicators, and the theoretical distributions show good agreements with the empirical distributions.
Figure 3g–i shows the bivariate CDFs for three two-dimensional indicator combinations.
Given the 10-year univariate return period of three indicators, joint return periods of “AND”, “OR”, and Kendall for two- and three-dimensional cases are then evaluated. It can be seen from
Table 4 that, for the same indicator combination scheme, the ranking order of the three return periods is T
or < T
k (Kendall return period) < T
and. This can also be inferred from the corresponding probabilities of various return periods in
Figure 2, that is, P(and) < P(k) < P(or). Compared to the two-dimensional cases, it can be found that the T
or of the three-dimensional return period is smaller. The maximum return period (17.45 years) can be found when the three indicators occur simultaneously.
4.2. Changes of Different Schemes in Two 30-Year Stages
Based on the fitted GMM distributions, for a given return period, the values of the corresponding indicators can be obtained through the univariate return period formula (Equation (9)). In order to conduct a more illustrative comparison of the 10-year return period values between different regions, the values of four indicators at all stations under two 30-year periods are interpolated to the whole region by using the Kriging method (
Figure 4).
Figure 4a–d provides the comparisons of four indicator values between the time period of 1959–1988 (T1 period, left ones) and time period of 1989–2018 (T2 period, right ones).
The results show that the univariate return period values of the 10-year stage for all groups are similar in spatial distribution between the two 30-year stages over the entire region. The minimum values of D95 and P95 appear in the northwest of Loess Plateau, and there is a stepwise increasing trend from northwest to southeast for these two indicators. In the southeastern area, it is more prone to heavy rainfall for a long time (D95), which brings greater amounts of precipitation (P95). The values of I95 exhibit obvious increasing direction from west to east, while R95 presents a clear North–South gap. The mean values (averaged over the stations) of D95, P95, and I95 in Region 1 are 0.67 days, 78.2 mm, and 15.6 mm/day smaller than those in Region 2 for T1 period. Also, these indicators in Region 1 are 0.63 days, 81.2 mm, and 16.7 mm/day smaller than those in Region 2 for T2 period. The mean value of R95 in Region 1 is slightly higher than that in Region 2, and the values of R95 could be above 0.5 in the north part of Region 1. This indicates that, with the least annual precipitation in the northwest, nearly half of the annual precipitation would fall by extreme heavy precipitation.
The spatial distributions of the indicator values behave differently between the two time periods. The 10-year return period values of the four indicators in the T2 period (1989–2018), when compared with those in T1 period (1959–1988), exhibit the following changes. The value of D95 decreases slightly throughout the region, with an average decrease of 0.26 days. The station with the largest decrease of D95 appears in the northeast with a decrease of 1.95 days, whereas the station with the largest growth of D95 appears in the southwest region with an increase of 1.2 days. Similar to the changes of D95, P95 decreases significantly in north and southeast, with an average decrease of 13.13 mm. I95 increases significantly in some stations in the southwest and southeast regions with the largest increase of 9.96 and 25.4 mm/day, respectively. The spatial difference of R95 in different time periods is not obvious, and the change of annual precipitation does not bring obvious change of R95.
Table 5 displays the average changes of four indicators in different regions and the number of stations which show growth changes by comparing with that in T1 period.
Under the condition that the indicator values at a 10-year return period, the joint return periods of T
and for two-dimensional or three-dimensional indicator combinations can then be obtained.
Figure 5 shows T
and at each station in T1 period (i.e., 1959–1988) (sub-graphs on the left) of six two-dimensional indicator combinations, and the spatial changes in T2 (i.e., 1989–2018) compared with that in the T1 period (right ones of each group). The red areas in the right sub-graphs indicate decreasing T
and in past 30 years, which signify that the regions are more prone to joint risk than the T1 period.
The averaged Tand of three two-dimensional combination events {D95, P95}, {D95, R95}, and {P95, R95} in the T1 period are 13.05, 14.68, and 13.35 years, respectively. The Tand values are only slightly longer than the return period of a single event (10 years), and the minimum values of the three combined events are 11.03, 12.25, and 11.31 years, respectively. It is proved that these three groups of combination events are extremely easy to happen, and the events of different indicators in each combination at a 10-year return period level usually occur together. There is no obvious regularity in the spatial distributions of Tand for these three combinations. The spatial distributions of the changes in Tand for the three groups in T2 period by comparing with that in T1 are obviously different from each other. It is worth noticing that in the central region of the Loess Plateau, the return periods of the three indicator combinations are all decreased, indicating that increased chances for occurrence of the three joint events in this region.
The mean Tand values of the other three two-dimensional combination events of {D95, I95}, {P95, I95}, and {I95, R95} are 32.65, 22.37, and 22.45, respectively. The occurrence probability is relatively small compared with the above three combinations, but the severity of the events may be far greater than the former ones. For example, {D95, I95} represents a long-time heavy rainfall event with high precipitation intensity. For these three combination events, their spatial distributions of Tand are basically the same (T1 period), and their spatial distributions of changing trend in the past 30 years are also similar to each other. The three combination events share a common feature that all of them contain the I95 index. The above results may be because the joint distributions are significantly affected by the I95 indicator, which means that the contribution of I95 to the joint return period is far greater than other indicators. The Tand of the three combined events in the past 30 years have decreased greatly in the northern and central regions, which implies the occurrence probability of extreme joint events has increased significantly. A comprehensive evaluation of Tand of the six two-dimensional indicator combinations shows that the probability of all the six joint extreme events in the central part of the region has increased.
Figure 6 shows the T
and values of three three-dimensional indicator combinations at each station in 1959–1988, and also the spatial changes in 1989–2018 compared with that in the T1 stage. The T
and values of {D95, P95, R95} are significantly smaller than those for the other two groups, with an average of only 15.53 years in 1959–1988. For the two three-dimensional events that contain the precipitation intensity indicator, the T
and values are generally greater than that of the rest one. The mean T
and of {D95, I95, R95} and {P95, I95, R95} from 1959 to 1988 is 31.2 and 24.8 years, respectively. The spatial distribution and the change direction of joint return periods of the two events are similar with each other and are also similar to those for the two-dimensional events in
Figure 5b,d,e. In T1 stage, the T
and values of {D95, I95, R95} and {P95, I95, R95} in Region 1 are generally smaller than those in Region 2.
Figure 6b,c shows that the change direction of T
and in T2 stage compared with T1 stage has decreased in the northern and central regions, and increased in other regions. This may also be because the T
and values of different combination events are highly affected by the precipitation intensity indicator. In the north part of Region 1, where extreme precipitation occurs frequently in T1 stage, the three-dimensional T
and get even smaller in T2 stage.
Figures S2–S5 (in the Supplementary Materials) provide the T
k and T
or values of two- and three-dimensional indicator combinations at each station in 1959–1988, and the spatial changes in 1989–2018 compared with those in the T1 stage. In two-dimensional cases at T1 stage, the T
k values are all smaller than T
and over all stations (see explanation in
Section 4.1) and are similar to T
and in magnitude distribution and change direction. The average T
k values corresponding to
Figure S2a–f are 1.55, 10.09, 2.37, 5.66, 1.72, and 5.73 years smaller than those of T
and, respectively. The magnitude distribution of T
or values in T1 stage are opposite to T
and, as well as the change direction in T2 stage. This is because, given the CDF values of two indicators, the value of T
and is only related to bivariate CDF like T
or (Equations (10) and (12)). With the change of one bivariate CDF, the change directions of the two return periods are opposite with each other. In three-dimensional cases at T1 stage, the average T
K values corresponding to
Figure S3a–c are, respectively, 2.82, 8.80, and 6.83 years smaller than T
and. The average T
or values in
Figure S5a–c are only 6.98, 5.50, and 5.82 years, respectively.
4.3. The Trend of Return Period of Multidimensional Moving Window Series
Figure 7 shows the changing trends in 10-year return period values for the four individual indicators calculated from the 30-year moving time series over all stations. The significance of changing trend and magnitude are also displayed in
Figure 7.
Table 6 provides statistics on the proportion of stations with significant increasing/decreasing trends, and average changes in the two regions. The results show that in the semi-arid area of the Loess Plateau (Region 1), the changing trends of D95 and P95 are similar in significance and direction. The 10-year return period values exhibit a significant decrease trend for D95 and P95 at 46% and 38% of the stations, respectively. However, some stations in the southwest area show a significant increasing trend. For these two indicators in Region 2, 42% and 50% of the stations display a significant decreasing trend, while only 16% and 24% of the stations show a significant increasing trend. Forty-two percent of the stations in Region 2 show a significant decreasing trend for I95, with a decrease ranging in magnitude from 2.2 to 31.6 mm/day. The southern region generally exhibits a significant increase in extreme precipitation intensity, which may increase the risk of flooding in that region. The distribution of stations with significant decreasing trends of R95 is consistent with that of P95. However, more stations in the southern part of Region 1 show significant increasing trends. The proportion of extreme precipitation in annual precipitation increases. This means that the precipitation concentration degree increases, which will aggravate the spatial–temporal distribution of precipitation
Figure 8 displays the significance and magnitude of the changing trends for T
and of the six two-dimensional indicator combinations based on the 30-year moving time series over all stations.
Figure 8b,d,f shows that the joint return periods of three two-dimensional events {D95, I95}, {P95, I95}, and {I95, R95} in the whole region present a relatively consistent trend. The joint return periods of these three events in the entire region show significant increasing trends at 30, 25, and 29 stations, respectively. Moreover, 19 of them are shared stations, and these stations are mainly located in the south of Region 1. There are 17, 16, and 19 stations showing significant decreasing trends, 14 of which are shared stations, which are mainly located in the middle and north parts of the Loess Plateau. The average reductions in the T
and return period are 26.8, 9.25, and 9.39 years for the three events, respectively. These results suggest that the possibility of flood events with long time and high intensity in the middle of the Loess Plateau would increase. The decrease in T
and indicates that extreme precipitation events with the same degree occur more frequently. For the other three two-dimensional combined events, it can be seen that the return period of T
and for {P95, R95} is most significantly reduced in the entire region, with a total number of 24 stations showing a significant decreasing trend.
Figure 9 shows the significance and magnitude of the trends in joint return periods of T
and for three-dimensional indicator combinations over all stations. It can be seen that some stations in the central region of Loess Plateau exhibit significant downward trends for all three-dimensional events. The changing trend of T
and for {D95, P95, R95} in the middle of Region 2 shows a considerable differentiation state: The changing trend for stations in the western region decreased significantly, whereas those in the eastern region increased significantly. The stations with significant decreasing trend in T
and for {D95, P95, R95} are mainly located in the sub-humid regions, with an average reduction of 20.5 years. However, the stations with significantly decreasing return periods for {P95, I95, R95} are located near the boundary of 400 mm rainfall, with an average decrease of 13.5 years. This means that the combination event of {P95, I95, R95} in the semi-arid and semi-humid junction area tends to be more frequent.
Figures S6–S9 in the Supplementary Materials show the significance and magnitude of the trend changes for T
k and T
or for the two- and three-dimensional situations over all stations. In two-dimensional cases, the T
or and T
and basically show opposite changing trends, whereas T
k and T
and show similar changing trends and directions, with only differences in magnitude. For example, for {D95, I95}, {P95, I95} and {I95, R95}, the number of stations showing an increasing trend in T
k are 30, 23, and 25, respectively, in which 29, 22, and 24 stations are shared with the T
and cases. For the events of {D95, I95, R95} and {P95, I95, R95}, the stations with significant decreasing trends in T
k (mainly located in the central Loess Plateau) decreased by an average of 10.61 and 5.05 years, respectively. The sites with significant decrease trends in T
or are mainly located in the southwest and northeast parts of Loess Plateau.