# Temporal and Spatial Characteristics of Multidimensional Extreme Precipitation Indicators: A Case Study in the Loess Plateau, China

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## Abstract

**:**

## 1. Introduction

## 2. Overview of the study area and data

^{2}, the Loess Plateau, which is located in the north of central China, is selected as the study area (Figure 1). The altitude of the Loess Plateau mainly ranges between 900 to 2200 m, and the terrain is characterized by high-west and low-east patterns. In the Loess Plateau, there are some mountains up to 3500 m above sea level, as well as plains and basins at an altitude of only 400 to 500 m. The Loess Plateau is also the largest area covered by loess in the world, presenting loose soil structure and fragile eco-environmental systems.

## 3. Method

#### 3.1. Gaussian Mixture Model

_{i}(i = 1, 2,…, n) are the non-negative mixture weights which satisfy $\sum _{i=1}^{n}{\alpha}_{i}=1$, G

_{i}(x; μ

_{i}, σ

_{i}) represents the ith Gaussian density component, and can be expressed as

_{i}, and σ

_{i}represent the mean and standard deviation of the ith Gaussian model, respectively. Given an appropriate number of components, the GMM can fully approximate a wide variety of distributions encountered in practical applications [22,32].

_{i}= (α

_{i}, μ

_{i}, σ

_{i}), Θ = (θ

_{1}, θ

_{2}, …, θ

_{n}), and the indicator x has a sample size of m, the likelihood function Լ of the model with n components can be expressed as

^{−5}. The goodness-of-fit for GMM is examined using the root mean square error (RMSE), Akaike’s information criteria (AIC [27]), and the Kolomogorov–Smirnov (KS) test (at the 5% significance level of the p-value) [30].

#### 3.2. Joint Probability Distribution and Return Period

#### 3.2.1. Joint Probability Distribution for Extreme Precipitation Indicators

_{1}, X

_{2}, …, X

_{d}) be a d-dimensional random vector formed by d extreme precipitation indicators. Based on the theory of copula proposed by Sklar [34], the d-dimensional distribution function F

_{X}can be expressed by the Copula function as

^{d}→ [0,1]; ${F}_{{X}_{1}}({x}_{1}),\text{\hspace{0.17em}}{F}_{{X}_{2}}({x}_{2}),\cdot \cdot \cdot ,{F}_{{X}_{d}}({x}_{d})$ denote continuous marginal distribution functions of the d-dimensional random vector X.

#### 3.2.2. Return Periods for Extreme Precipitation Indicators and Their Combinations

_{1}and X

_{2}(F(X

_{1}) = u, F(X

_{2}) = v), respectively. C(u

_{i}, v

_{i}) = t is the joint probability for a given (u

_{i}, v

_{i}). The three return periods are defined based on three different types of probability expressions. The space formed by the dotted lines, u = 1 and v = 1 in the upper right corner indicates the probability that both the two indicators exceed their threshold values P{u> u

_{i}∩ v> v

_{i}}. The return period ${T}_{{X}_{1},{X}_{2}}^{and}$ corresponding to the simultaneous occurrence of two events can be expressed as

_{1}, X

_{2}, and X

_{3}represent extreme indicators; x

_{1}, x

_{2}, and x

_{3}are the certain thresholds of the indicators; F(x), F(x

_{1},x

_{2}), …, F(x

_{1},x

_{2},x

_{3}) denote the cumulative probability distributions (CDFs) or joint CDFs corresponding to the thresholds; u, v, and w represent the marginal distributions of the indicators; and C(u,v), …, C(u,v,w) stand for copula functions describing joint distributions of extreme precipitation indicators.

_{i}∪ v > v

_{i}}). The corresponding return period expressed by ${T}_{{X}_{1},{X}_{2}}^{or}$ can be calculated as

- Simulate a sample u
_{1}, ..., u_{m}from the d-dimensional copula C; - For i = 1 ,..., m calculate w
_{i}= C(u);_{i} - Estimate Kc: ${\widehat{K}}_{C}(t)=\frac{1}{m}{\displaystyle {\sum}_{i=1}^{m}1({w}_{i}\le t)}$.

## 4. Case Study and Results Analysis

#### 4.1. Case Study

_{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

_{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.

_{and}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 T

_{and}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 T

_{and}for these three combinations. The spatial distributions of the changes in T

_{and}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.

_{and}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 T

_{and}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 T

_{and}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 T

_{and}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.

_{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.

_{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

_{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.

_{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.

_{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.

## 5. Discussions and Conclusions

_{and}values of all joint extremes are decreased. T

_{k}values are smaller than T

_{and}over all stations and are similar to T

_{and}in magnitude distribution and change direction. The magnitude distribution of T

_{or}values in T1 stage are opposite to T

_{and}, as well as the change direction in T2 stage.

## Supplementary Materials

**a**) PDF for D95, (

**b**) PDF for P95, (

**c**) PDF for R95, Figure S2: Joint return periods (T

_{k}) at each station in 1959-1988 (left ones) and spatial changes in 1989-2018 (right ones) for six two-dimensional indicator combinations: (

**a**) {D95, P95}, (

**b**) {D95, I95}, (

**c**) {D95, R95}, (

**d**) {P95, I95}, (

**e**) {P95, R95}, and (

**f**) {I95, R95}, Figure S3: Joint return periods (T

_{k}) at each station in 1959-1988 (left ones) and spatial changes in 1989-2018 (right ones) for three three-dimensional indicator combinations: (

**a**) {D95, P95, R95}, (

**b**) {D95, I95, R95}, and (

**c**) {P95, I95, R95}, Figure S4: Joint return periods (T

_{or}) at each station in 1959-1988 (left ones) and spatial changes in 1989-2018 (right ones) for six two-dimensional indicator combinations: (

**a**) {D95, P95}, (

**b**) {D95, I95}, (

**c**) {D95, R95}, (

**d**) {P95, I95}, (

**e**) {P95, R95}, and (

**f**) {I95, R95}, Figure S5: Joint return periods (T

_{or}) at each station in 1959-1988 (left ones) and spatial changes in 1989-2018 (right ones) for three three-dimensional indicator combinations: (

**a**) {D95, P95, R95}, (

**b**) {D95, I95, R95}, and (

**c**) {P95, I95, R95}, Figure S6: The significance and magnitude of the trends in joint return periods (T

_{k}) over 30-year moving window series for six two-dimensional indicator combinations: (

**a**) {D95, P95}, (

**b**) {D95, I95}, (

**c**) {D95, R95}, (

**d**) {P95, I95}, (

**e**) {P95, R95}, and (

**f**) {I95, R95}, Figure S7: The significance and magnitude of the trends in joint return periods (T

_{k}) over 30-year moving window series for three three-dimensional indicator combinations: (

**a**) {D95, P95, R95}, (

**b**) {D95, I95, R95}, and (

**c**) {P95, I95, R95}, Figure S8: The significance and magnitude of the trends in joint return periods (T

_{or}) over 30-year moving window series for six two-dimensional indicator combinations: (

**a**) {D95, P95}, (

**b**) {D95, I95}, (

**c**) {D95, R95}, (

**d**) {P95, I95}, (

**e**) {P95, R95}, and (

**f**) {I95, R95}, Figure S9: The significance and magnitude of the trends in joint return periods (T

_{or}) over 30-year moving window series for three three-dimensional indicator combinations: (

**a**) {D95, P95, R95}, (

**b**) {D95, I95, R95}, and (

**c**) {P95, I95, R95}., Table S1: Statistical test comparisons between GMM and the best fitted ones of five other distributions (Gamma, Pearson type III (P III), lognormal (LN), Log Pearson type III (LP III), generalized extreme value (GEV)) at Yulin Station, Table S2: Statistical test results for copulas at Yulin Station.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Location of the Loess Plateau with 63 stations: (

**a**) location of the Loess Plateau, (

**b**) stations.

**Figure 3.**The probability density functions (PDFs) and cumulative probability distributions (CDFs) for three precipitation indicators and bivariate CDFs for three indicator combinations at Yulin Station: (

**a**) PDF for D95, (

**b**) PDF for P95, (

**c**) PDF for R95, (

**d**) CDF for D95, (

**e**) CDF for P95, (

**f**) CDF for R95, (

**g**) bivariate CDF for {P95, D95}, (

**h**) bivariate CDF for {D95, R95}, and (

**i**) bivariate CDF for {P95, R95}.

**Figure 4.**The values of four extreme precipitation indicators at 10-year return period for time periods of 1959–1988 (left ones) and 1989–2018 (right ones): (

**a**) D95, (

**b**) P95, (

**c**) I95, and (

**d**) R95.

**Figure 5.**Joint return periods (T

_{and}) at each station in 1959–1988 (left ones) and spatial changes in 1989–2018 (right ones) for six two-dimensional indicator combinations: (

**a**) {D95, P95}, (

**b**) {D95, I95}, (

**c**) {D95, R95}, (

**d**) {P95, I95}, (

**e**) {P95, R95}, and (

**f**) {I95, R95}.

**Figure 6.**Joint return periods (T

_{and}) at each station in 1959–1988 (left ones) and spatial changes in 1989–2018 (right ones) for three three-dimensional indicator combinations: (

**a**) {D95, P95, R95}, (

**b**) {D95, I95, R95}, and (

**c**) {P95, I95, R95}.

**Figure 7.**Changing trends in 10-year return period values over 30-year moving window series for four indicators: (

**a**) D95, (

**b**) P95, (

**c**) I95, and (

**d**) R95. Blue dots indicate significant decreasing trends, red dots denote significant increasing trends, and the crosses indicate non-significant trend.

**Figure 8.**The significance and magnitude of the trends in joint return periods (T

_{and}) over 30-year moving window series for six two-dimensional indicator combinations: (

**a**) {D95, P95}, (

**b**) {D95, I95}, (

**c**) {D95, R95}, (

**d**) {P95, I95}, (

**e**) {P95, R95}, and (

**f**) {I95, R95}.

**Figure 9.**The significance and magnitude of the trends in joint return periods (T

_{and}) over 30-year moving window series for three three-dimensional indicator combinations: (

**a**) {D95, P95, R95}, (

**b**) {D95, I95, R95}, and (

**c**) {P95, I95, R95}.

Indices | Abbreviations | Definitions | Unit |
---|---|---|---|

Number of extreme precipitation days | D95 | Number of days with P > 95th percentile (daily precipitation exceeding the 95th percentile of precipitation series during 1971–2000). | days |

The amount of extreme heavy precipitation | P95 | Annual total amount of precipitation with P > 95th percentile | mm |

The intensity of extreme precipitation | I95 | Average daily precipitation intensity of extreme precipitation | mm/day |

Ratio of extreme precipitation | R95 | Ratio of annual total precipitation due to events exceeding the 95th percentile | - |

ID | Combinations | ID | Combinations | ID | Combinations |
---|---|---|---|---|---|

1 | {D95, P95} | 4 | {P95, I95} | 7 | {D95, P95, R95} |

2 | {D95, I95} | 5 | {P95, R95} | 8 | {D95, I95, R95} |

3 | {D95, R95} | 6 | {I95, R95} | 9 | {P95, I95, R95} |

**Table 3.**Statistical test results for Gaussian mixture model (GMM) and best-fitted copulas at Yulin Station.

Scheme | Distribution | KS Test | RMSE | AIC | |
---|---|---|---|---|---|

$\mathit{T}$ | p-Value | ||||

D95 | GMM | 0.143 | 0.568 | 0.055 | −170.089 |

P95 | GMM | 0.089 | 0.972 | 0.029 | −208.096 |

R95 | GMM | 0.082 | 0.987 | 0.032 | −202.164 |

{D95, P95} | Gaussian | 0.022 | 0.615 | 0.027 | −215.108 |

{D95, R95} | t | 0.024 | 0.697 | 0.032 | −203.671 |

{P95, R95} | Gaussian | 0.028 | 0.790 | 0.030 | −208.887 |

{D95, P95, R95} | t | 0.032 | 0.648 | 0.021 | −228.451 |

**Table 4.**Joint returned periods of “AND” (T

_{and}), “OR” (T

_{or}), and Kendall (T

_{k}) for two and three-dimensional cases.

Return Periods | {D95, P95} | {D95, R95} | {P95, R95} | {D95, P95, R95} |
---|---|---|---|---|

T_{or} | 8.09 | 7.28 | 7.79 | 7.01 |

T_{k} | 11.66 | 13.50 | 11.77 | 12.66 |

T_{and} | 13.09 | 15.99 | 13.96 | 17.45 |

Region | Number of Stations | D95 | P95 | I95 | R95 | ||||
---|---|---|---|---|---|---|---|---|---|

A | N | A | N | A | N | A | N | ||

1 | 25 | −0.24 | 7 | −11.29 | 11 | 0.36 | 14 | −0.013 | 11 |

2 | 38 | −0.27 | 14 | −8.27 | 14 | 1.44 | 24 | −0.004 | 16 |

**Table 6.**Statistical information on the increase and decrease trend of the four indicators in two regions: The proportion of stations with significant increasing trend (PI) and the total growth over the moving window series (TG); the proportion of stations with significant decreasing trend (PD) and total reduction over the moving window series (TD).

Indices | Region 1 | Region 2 | ||
---|---|---|---|---|

PI/TG | PD/TD | PI/TG | PD/TD | |

D95 | 0.31/0.54 | 0.46/−0.63 | 0.16/0.53 | 0.42/−0.95 |

P95 | 0.19/22.04 | 0.38/−35.7 | 0.24/39.40 | 0.50/−37.01 |

I95 | 0.27/6.79 | 0.38/−6.80 | 0.42/12.17 | 0.29/−11.63 |

R95 | 0.38/0.05 | 0.46/−0.06 | 0.34/0.04 | 0.34/−0.05 |

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**MDPI and ACS Style**

Sun, C.; Huang, G.; Fan, Y.
Temporal and Spatial Characteristics of Multidimensional Extreme Precipitation Indicators: A Case Study in the Loess Plateau, China. *Water* **2020**, *12*, 1217.
https://doi.org/10.3390/w12041217

**AMA Style**

Sun C, Huang G, Fan Y.
Temporal and Spatial Characteristics of Multidimensional Extreme Precipitation Indicators: A Case Study in the Loess Plateau, China. *Water*. 2020; 12(4):1217.
https://doi.org/10.3390/w12041217

**Chicago/Turabian Style**

Sun, Chaoxing, Guohe Huang, and Yurui Fan.
2020. "Temporal and Spatial Characteristics of Multidimensional Extreme Precipitation Indicators: A Case Study in the Loess Plateau, China" *Water* 12, no. 4: 1217.
https://doi.org/10.3390/w12041217