Using Copulas to Evaluate Rationality of Rainfall Spatial Distribution in a Design Storm

In the absence of long-series streamflow records in plain areas, design storm, which serves as the most important input in a hydrologic model, plays an important role in flood control and water resources management. For a large drainage basin, design storm may be estimated for sub-basins separately; thus the spatial distribution of design storm needs to be carefully treated. However, few studies have been carried out to evaluate the rationality of the spatial distribution in a design storm, which means the storm over space should be in accordance with actual needs or its distributing patterns. Taking the Tai Lake Basin (TLB), 3-d Copula-based models combining extreme rainfall of different sub-basins were built using long-term rainfall data, and conditional probabilities of sub-basins encountering certain amounts of rainfall were investigated to evaluate the rationality of the design storm. Results show that the spatial distribution of the design storm based on a typical year is hardly rational, in which rainfall of the northeastern part of the basin is suggested to be weakened while in the southwest to be strengthened; after the rainfall is redistributed based on long-term information, it shows a better rationality of spatial distribution. Such information provides valuable significance in guiding flood control of TLB, and the considered evaluating method can be used for similar basins in plain areas.


Introduction
Flood control planning is used strategically in dealing with flood events in a basin or river.In flood control planning, design flood, which is usually derived from long-series streamflow data, provides the basis for flood control standards (usually described as a return period).In a way, it serves as macro guidance of the construction and operation of hydraulic structures.However, this method is practically impossible for plain areas where it is ungauged or streamflow is difficult to measure because it can flow back and forth.Under this circumstance, an alternative method came into being with the rapid development of hydrologic models [1][2][3], through which design flood can be derived using design storm as a model input.Hence, in such areas design storm becomes important since it can have an indirect but far-reaching influence on flood control planning of the whole basin.There are basins all over the world using design storm as a model input (as long as the long-term rainfall data is available) to get design flood [4,5].Researchers have been applying this idea to estimate design flood in different

Methods
In the flood control planning text of TLB, administrators are designing a storm in which rainfall values of the whole basin and the prioritized sub-basin are fixed (assume that return periods of both are 50/100 years), while, rainfall values over the non-prioritized sub-basins are spatially distributed based on a typical year.Since rainfall of the whole basin and prioritized area is settled, the problem lies in how the remaining rainfall is spatially distributed over the non-prioritized sub-basins.
3-d Copulas are utilized to establish joint distributions among three extreme rainfall variables.Figure 1 shows the flow chart of how to build joint distributions among the three variables.The first and the second variables are rainfall of the whole basin and prioritized sub-basin, respectively, and are regarded as conditional ones.Then, each one of the non-prioritized sub-basins is taken out as the third variable to build a 3-d joint distribution along with the two conditional variables.Afterwards, the probability of each non-prioritized sub-basin encountering a certain amount of rainfall is derived under the two conditional variables.Based on the results, analysis and evaluation is carried out accordingly.This method ensures that the potential correlations among rainfall of different sub-basins and the intentions of administrators are all taken into consideration.
Water 2018, 10, x FOR PEER REVIEW 3 of 14 Copulas.Part 2 serves as an overview of the study domain and basic data.Part 3 establishes and selects the optimal marginal and joint distributions for the 3-d relationship, performs calculation on coincidence and conditional probabilities of rainfall combinations, and evaluates the distribution rationality of two distributing schemes based on the derived conditional probability equations.

Methods
In the flood control planning text of TLB, administrators are designing a storm in which rainfall values of the whole basin and the prioritized sub-basin are fixed (assume that return periods of both are 50/100 years), while, rainfall values over the non-prioritized sub-basins are spatially distributed based on a typical year.Since rainfall of the whole basin and prioritized area is settled, the problem lies in how the remaining rainfall is spatially distributed over the non-prioritized sub-basins.
3-d Copulas are utilized to establish joint distributions among three extreme rainfall variables.Figure 1 shows the flow chart of how to build joint distributions among the three variables.The first and the second variables are rainfall of the whole basin and prioritized sub-basin, respectively, and are regarded as conditional ones.Then, each one of the non-prioritized sub-basins is taken out as the third variable to build a 3-d joint distribution along with the two conditional variables.Afterwards, the probability of each non-prioritized sub-basin encountering a certain amount of rainfall is derived under the two conditional variables.Based on the results, analysis and evaluation is carried out accordingly.This method ensures that the potential correlations among rainfall of different subbasins and the intentions of administrators are all taken into consideration.

Figure 1.
Flow chart of how spatial distribution rationality is evaluated using conditional probability analysis.

Distributing Schemes
Two schemes of rainfall distribution in non-prioritized sub-basins need to be evaluated, one is the so-called "typical year scheme" which is currently used in the TLB flood control, and derives rainfall from an outstanding historical rainfall year.In consideration of the potential problem in the current scheme, an alternative which is based on the average rainfall information in the past decades is proposed (we may name it "average scheme").Equation (1) shows how rainfall distribution in different non-prioritized sub-basins is derived.

(
) where XD represents the design rainfall we try to derive, XB and XN are the design rainfall of the whole basin and northern part of the basin, respectively, F and FN are the area of the whole basin and northern part of the basin respectively, n is the number of non-prioritized sub-basins, i is the ordinal

Distributing Schemes
Two schemes of rainfall distribution in non-prioritized sub-basins need to be evaluated, one is the so-called "typical year scheme" which is currently used in the TLB flood control, and derives rainfall from an outstanding historical rainfall year.In consideration of the potential problem in the current scheme, an alternative which is based on the average rainfall information in the past decades is proposed (we may name it "average scheme").Equation (1) shows how rainfall distribution in different non-prioritized sub-basins is derived.
Water 2018, 10, 758 4 of 14 where X D represents the design rainfall we try to derive, X B and X N are the design rainfall of the whole basin and northern part of the basin, respectively, F and F N are the area of the whole basin and northern part of the basin respectively, n is the number of non-prioritized sub-basins, i is the ordinal of the non-prioritized sub-basins, x i and f i are the typical rainfall and area of each non-prioritized sub-basins, X M represents the typical rainfall of the non-prioritized sub-basins.
It is noteworthy that the difference between the "typical year scheme" and "average scheme" lies in the value of X M , that is, if the formal scheme is chosen, X M is derived using the rainfall information of a typical outstanding year, if the latter, an average information of rainfall over the past decades is applied to X M .

Maginal and Joint Distribution
The first step involved in building Copulas is to decide the optimal marginal distribution.
Researchers have pointed out that some certain distributions may not be proposed as general models [28], and one recent study [29] has presented a framework which attempts to propose a single approach for stochastic modelling of any hydroclimatic process.However, it is hardly our main goal here to discuss the best marginal distribution.In this study, four types of commonly used distributions in hydrology are adopted as candidates, namely Pearson III type distribution (P-III), Logistic distribution (LOG), Log-normal distribution (LOGN) and Weibull distribution (WEI).For the sake of achieving better accuracy, this study selects the optimal one for every rainfall variable.
Afterwards, the second step is the joint distribution in which Copula functions are utilized.The Copula function is a multi-variable joint distribution with a domain between 0 and 1, this offers us convenient access to combine two or more marginal distributions into a probabilistic model.Taking a 2-d combination, assume that F X (x) = u and F Y (y) = v represent cumulative distribution functions of variable X and Y, respectively, their Copula function can be described as follows: where F(x, y) serves as the joint distribution function of stochastic variable X and Y, θ and C θ represent the parameter of the Copula function and the Copula function itself.

2-d
Copulas are also involved when dealing with conditional probabilities of 3-d Copulas, thus relevant joint distribution calculation involves both 2-d and 3-d ones.The Archimedean Copula family, which is commonly used in hydrology [19][20][21][22][23][24][25][26][27], is adopted as candidates of joint distributions in this study.This Copula family includes three main members, namely Clayton, Gumbel and Frank; eventually, an optimal one will be selected for probability calculation for the sake of accuracy.All of these Copulas are single-parameter functions which are easy and flexible to build and solve.The expressions of these Copulas are shown in Table 1.

Goodness of Fit
Criteria are necessary to assess the fit quality of marginal and joint distributions.In terms of marginal distributions, this study utilizes the Kolmogorov-Smimov (K-S) test and least Akaike information criterion (AIC) to select the optimal ones; when it comes to joint distributions, we use AIC and the criteria of ordinary least square (OLS) to assess the fitting quality between theoretical and empirical distributions of samples.The OLS and AIC are specifically the following equations: where F emp (x i1 , x i2 , . . ., x im ), C( u i1 , u i2 , . . ., u im ) are empirical and theoretical cumulative probabilities respectively; m is the dimension of the function; n is the number of observations; k is the number of parameters; MSE is the root mean square error.The goodness of fit is believed to be better when a smaller AIC or OLS value is obtained.For 3-d observations (p i , q j , z k ), the empirical frequency can be calculated using the equation below: where F emp (p i , q j , z k ) is the empirical frequency, N is the number of observations, n ijk represents the times when measured data is less than (p i , q j , z k ) simultaneously.

Conditional Probability
A 3-d conditional probability of the joint distribution can be derived using expressions of 2-d and 3-d Copulas.According to the requirements of calculation, for a data group (X 1 , X 2 , X 3 ), the probability of X 1 ≤ x 1 under condition of X 2 = x 2 , X 3 = x 3 can be derived using Equation (7). where On the basis of Equation ( 7), Equations ( 8)- (11) show the derivation process of probability The following derivations are based on Clayton Copula because it was proven to outperform the other two types, Frank and Gumbel Copula in this study (see Section 4.2 for discussion).
First, the joint distribution of the 3-d Clayton Copula can be described as the following: The partial derivative of C 3 over u 1 , u 2 is derived: Water 2018, 10, 758 6 of 14 The partial derivative of C 2 over u 1 , u 2 is derived: According to Equation ( 7), the conditional probability of Clayton Copula C 3 (u 3 |u 1 , u 2 ) can be derived using the following:

Study Domain
The TLB, located in the southern part of Yangtze River Delta, China (within 119 • 3 1"-121 • 54 26" E, 30 • 7 19"-32 • 14 56" N), has a typical humid sub-tropical monsoon climate, with the annual mean rainfall of about 1177 mm.The area of the basin is around 36,500 km 2 , which is definitely a rather large one.Figure 2 depicts the relative geographical location of TLB and its seven sub-basins which are divided according to terrain and flood control needs.The area of North, YCDM, SH, HJH, HQ and ZX is 11,510, 4314, 4466, 7480, 3192 and 5930 km 2 , respectively.As is revealed from the elevation, the TLB shows a high altitude in the West mountainous area while the rest is basically plain areas.

Study Domain
The TLB, located in the southern part of Yangtze River Delta, China (within 119°3′1″-121°54′26″ E, 30°7′19″-32°14′56″ N), has a typical humid sub-tropical monsoon climate, with the annual mean rainfall of about 1177 mm.The area of the basin is around 36,500 km 2 , which is definitely a rather large one.Figure 2 depicts the relative geographical location of TLB and its seven sub-basins which are divided according to terrain and flood control needs.The area of North, YCDM, SH, HJH, HQ and ZX is 11,510, 4314, 4466, 7480, 3192 and 5930 km 2 , respectively.As is revealed from the elevation, the TLB shows a high altitude in the West mountainous area while the rest is basically plain areas.As one of the most developed regions of China, the TLB has witnessed impressive economic achievements in the past decades.With the degree of urbanization of 80% in 2015 [30], the basin supports 4.8% of the nation's population (1.39 billion) and produces 11.6% of the gross domestic product (GDP), with 0.38% of the land area of China [31].Also, flood disasters have been taking place frequently in the basin due to the long duration of rainfall in summer [32].According to the most outstanding flood disasters of TLB in recent decades, the northern part of the basin (namely sub-basin HX and WCXY) has been suffering more rainfall compared with the rest of the sub-basins.With several big cities (such as the city of Wuxi and Changzhou) located within the North, this area is suffering heavy losses in the economy and has drawn attention from administrators.This explains As one of the most developed regions of China, the TLB has witnessed impressive economic achievements in the past decades.With the degree of urbanization of 80% in 2015 [30], the basin supports 4.8% of the nation's population (1.39 billion) and produces 11.6% of the gross domestic product (GDP), with 0.38% of the land area of China [31].Also, flood disasters have been taking place frequently in the basin due to the long duration of rainfall in summer [32].According to the most outstanding flood disasters of TLB in recent decades, the northern part of the basin (namely sub-basin HX and WCXY) has been suffering more rainfall compared with the rest of the sub-basins.With several big cities (such as the city of Wuxi and Changzhou) located within the North, this area is suffering heavy losses in the economy and has drawn attention from administrators.This explains why the northern part of the basin is considered to be the so called "prioritized area"; accordingly, the rest of the sub-basins (namely YCDM, SH, HQ, HJH, and ZX) are the non-prioritized ones.
According to the flood control planning of TLB, when a 50 years return period storm is designed for the whole basin, it is suggested that rainfall in the prioritized area remains a relatively high return period (50 years also), due to frequent disasters and heavy economic losses; meanwhile, rainfall of the non-prioritized sub-basins is spatially distributed based on the measurement of a typical year (namely 1991).Nevertheless, it turned out the real rainfall of year 1991 varies significantly in space over the rest of the sub-basins, that is to say, its spatial distribution needs to be evaluated and furtherly replaced with an alternative if necessary.

Data Collection
High-density rainfall gauges (see Figure 1) have been established in the basin, thus, the long-time series rainfall data is collected.According to hydrological yearbooks and the latest monitoring data from the administration, the Arithmetical Average Method is adopted to derive daily rainfall series between 1951~2014 as the research data; the Annual Maximum method is used to extract the maximum 30-day, 60-day and 90-day rainfall of the whole basin or different sub-basins, which will be used for building marginal and joint distribution models.
Figure 3 provides daily rainfall of TLB between 1951~2014, the grey dotted lines represent daily rainfall of all the years except 1991, the dotted blue one is the average of them, and the solid black one represents 1991.Compared with the average, the year 1991 saw a significantly outstanding rainfall during flood season in the historical records; especially, there were two tight rainfall periods in the middle of June and beginning of July, respectively, which were rather harsh for the flood control, it explains why rainfall of 1991 is widely used as a model input in previous studies of TLB [33], and of course, it is picked as "typical year" in this study.
Water 2018, 10, x FOR PEER REVIEW 7 of 14 why the northern part of the basin is considered to be the so called "prioritized area"; accordingly, the rest of the sub-basins (namely YCDM, SH, HQ, HJH, and ZX) are the non-prioritized ones.
According to the flood control planning of TLB, when a 50 years return period storm is designed for the whole basin, it is suggested that rainfall in the prioritized area remains a relatively high return period (50 years also), due to frequent disasters and heavy economic losses; meanwhile, rainfall of the non-prioritized sub-basins is spatially distributed based on the measurement of a typical year (namely 1991).Nevertheless, it turned out the real rainfall of year 1991 varies significantly in space over the rest of the sub-basins, that is to say, its spatial distribution needs to be evaluated and furtherly replaced with an alternative if necessary.

Data Collection
High-density rainfall gauges (see Figure 1) have been established in the basin, thus, the longtime series rainfall data is collected.According to hydrological yearbooks and the latest monitoring data from the administration, the Arithmetical Average Method is adopted to derive daily rainfall series between 1951~2014 as the research data; the Annual Maximum method is used to extract the maximum 30-day, 60-day and 90-day rainfall of the whole basin or different sub-basins, which will be used for building marginal and joint distribution models.
Figure 3 provides daily rainfall of TLB between 1951~2014, the grey dotted lines represent daily rainfall of all the years except 1991, the dotted blue one is the average of them, and the solid black one represents 1991.Compared with the average, the year 1991 saw a significantly outstanding rainfall during flood season in the historical records; especially, there were two tight rainfall periods in the middle of June and beginning of July, respectively, which were rather harsh for the flood control, it explains why rainfall of 1991 is widely used as a model input in previous studies of TLB [33], and of course, it is picked as "typical year" in this study.The rainfall data from stations have undergone quality control, including analytical processing to ensure their consistency, reliability and representativeness.

Marginal Distribution
Table 2 shows the test results for four marginal distributions of extreme rainfall of different subbasins (including the whole basin and North part of the basin), the bold numbers, which are smallest among the four candidate distributions, represent the best fitting according to the AIC.Results The rainfall data from stations have undergone quality control, including analytical processing to ensure their consistency, reliability and representativeness.least errors.To be more detailed, Clayton Copula fits the joint distributions best in all types, except for the AIC value of Max 90-d HJH, where Frank Copula is slightly better.Frank Copula is the second best in terms of AIC values, while regarding OLS values, Frank and Gumbel are similar and both are worse than Clayton.Therefore, Clayton Copula should be selected as the optimal type in fitting the joint distributions, which consists of the generally accepted conclusion that Clayton may perform better than others when applied to positive dependence [34].

Conditional Probability
Two schemes of design storm (namely "typical year scheme" and "average scheme") remain to be evaluated.Based on the selected 3-d joint distributions, conditional probabilities of the five nonprioritized sub-basins encountering a corresponding rainfall are calculated using Equation (11).Rationality of spatial distribution is evaluated and analyzed afterwards.Before analysis, there is one thing we should highlight: In previous studies (for instance [35]), researches usually get conditional probabilities of U2 ≤ u2 under the condition of U1 > u1, while in this study, we derived the conditional probability of U3 ≤ u3 under the condition of (U1 = u1, U2 = u2) instead.The equation has to be derived in this way because it needs to keep consistent with the rainfall we are studying.
In scheme 1, conditional probabilities of the five non-prioritized sub-basins are shown in Table 3.It is notable that only a max-90-day rainfall result with 100 years return period is provided, since administrators did not derive the other two durations with 100 years return period in the manual.Taking the 50-year return-period and max-30-day rainfall for the example, conditional probabilities for YCDM, SH and ZX encountering a rainfall which exceeds 601.0, 532.5, and 456.3 mm are 1.42%,

Conditional Probability
Two schemes of design storm (namely "typical year scheme" and "average scheme") remain to be evaluated.Based on the selected 3-d joint distributions, conditional probabilities of the five nonprioritized sub-basins encountering a corresponding rainfall are calculated using Equation (11).Rationality of spatial distribution is evaluated and analyzed afterwards.Before analysis, there is one thing we should highlight: In previous studies (for instance [35]), researches usually get conditional probabilities of U2 ≤ u2 under the condition of U1 > u1, while in this study, we derived the conditional probability of U3 ≤ u3 under the condition of (U1 = u1, U2 = u2) instead.The equation has to be derived in this way because it needs to keep consistent with the rainfall we are studying.
In scheme 1, conditional probabilities of the five non-prioritized sub-basins are shown in Table

Conditional Probability
Two schemes of design storm (namely "typical year scheme" and "average scheme") remain to be evaluated.Based on the selected 3-d joint distributions, conditional probabilities of the five non-prioritized sub-basins encountering a corresponding rainfall are calculated using Equation (11).
Water 2018, 10, 758 10 of 14 Rationality of spatial distribution is evaluated and analyzed afterwards.Before analysis, there is one thing we should highlight: In previous studies (for instance [35]), researches usually get conditional probabilities of U 2 ≤ u 2 under the condition of U 1 > u 1 , while in this study, we derived the conditional probability of U 3 ≤ u 3 under the condition of (U 1 = u 1 , U 2 = u 2 ) instead.The equation has to be derived in this way because it needs to keep consistent with the rainfall we are studying.
In scheme 1, conditional probabilities of the five non-prioritized sub-basins are shown in Table 3.It is notable that only a max-90-day rainfall result with 100 years return period is provided, since administrators did not derive the other two durations with 100 years return period in the manual.Taking the 50-year return-period and max-30-day rainfall for the example, conditional probabilities for YCDM, SH and ZX encountering a rainfall which exceeds 601.0, 532.5, and 456.3 mm are 1.42%, 2.73%, and 44.32%, respectively.It means when a max-30-day rainfall of an exact 50 years return period takes place in the whole basin and in the North simultaneously, there exists huge differences among the probabilities of YCDM, SH and ZX encountering the corresponding rainfall.Obviously, if we take the rainfall of the whole basin and North as a premise, the spatial distribution of max-30-day rainfall over the non-prioritized sub-basins is considered to be quite uneven, which we believe is too big for YCDM and SH while too small for ZX (no problem with this expression because the total rainfall of the non-prioritized sub-basins is fixed).The analysis with max-60-day and max-90-day rainfall shows a similar situation but less remarkable, one possible reason is that max-30-day rainfall of 1991 is more unevenly distributed than the other two durations over the non-prioritized sub-basins, it illustrates the fact that deriving a storm from only one typical year is in a way arbitrary and can be problematic.For rainfall of a 100 years return period, there still exists pretty huge gaps among probabilities, the probabilities of HJH and ZX are 12.29% and 12.96%, while YCDM and HQ are 2.26% and 2.04%.Moreover, it is seen that the probabilities decrease in general from 50 years return period to 100 years return period, this is comprehensible since it always becomes less probable to encounter a larger quantity of rainfall.In order to make a comparison straightforwardly among the possibilities, radar plots are shown in Figure 7.It is clearly seen that the shapes of radar plots are quite irregular.
Hence, considering the five non-prioritized sub-basins, the spatial distribution of design storm in the flood control planning (scheme 1) is not quite reasonable.Extreme rainfall would be too large in YCDM and SH, but too small in ZX.It shows a different degree of reasonability when it comes to different durations or different return periods.
Similarly, conditional probabilities in non-prioritized sub-basins of scheme 2 are shown in Table 4, and accordingly, radar plots are also provided in Figure 8.For extreme rainfall of 50 years return period, conditional probabilities of the five sub-basins are between 9.0%~13.7%,9.0%~16.3%,and 10.5%~17.7%,respectively, for max-30-day, max-60-day and max-90-day rainfall.For max-90-day rainfall of 100 years return period, probabilities are between 5.4%~9.5%.It is obvious that compared with scheme 1, the values of probabilities are all bounded to a much smaller range, also, radar plots also show a much more regular shape with different duration periods.In other words, conditional probabilities of the non-prioritized sub-basins become more uniform after the rainfall is redistributed over the space based on a long-term information of rainfall data.Thus, scheme 2 can show a more reasonable spatial distribution in terms of different durations and return periods.For rainfall of a 100 years return period, there still exists pretty huge gaps among probabilities, the probabilities of HJH and ZX are 12.29% and 12.96%, while YCDM and HQ are 2.26% and 2.04%.Moreover, it is seen that the probabilities decrease in general from 50 years return period to 100 years return period, this is comprehensible since it always becomes less probable to encounter a larger quantity of rainfall.In order to make a comparison straightforwardly among the possibilities, radar plots are shown in Figure 7.It is clearly seen that the shapes of radar plots are quite irregular.Hence, considering the five non-prioritized sub-basins, the spatial distribution of design storm in the flood control planning (scheme 1) is not quite reasonable.Extreme rainfall would be too large in YCDM and SH, but too small in ZX.It shows a different degree of reasonability when it comes to different durations or different return periods.
Similarly, conditional probabilities in non-prioritized sub-basins of scheme 2 are shown in Table 4, and accordingly, radar plots are also provided in Figure 8.For extreme rainfall of 50 years return period, conditional probabilities of the five sub-basins are between 9.0%~13.7%,9.0%~16.3%,and 10.5%~17.7%,respectively, for max-30-day, max-60-day and max-90-day rainfall.For max-90-day rainfall of 100 years return period, probabilities are between 5.4%~9.5%.It is obvious that compared with scheme 1, the values of probabilities are all bounded to a much smaller range, also, radar plots  also show a much more regular shape with different duration periods.In other words, conditional probabilities of the non-prioritized sub-basins become more uniform after the rainfall is redistributed over the space based on a long-term information of rainfall data.Thus, scheme 2 can show a more reasonable spatial distribution in terms of different durations and return periods.Based on the analysis and evaluation above, it is proven that the "typical year scheme" for distributing the storm over the non-prioritized sub-basins is hardly reasonable, and it can be problematic if applied to guiding flood control or construction of hydraulic structures in the TLB.
To be specific, if we adopt the "typical year scheme" as the default storm of the TLB, for subbasin YCDM and SH, flood risk can be overestimated by hydrologic models, accordingly, this might Based on the analysis and evaluation above, it is proven that the "typical year scheme" for distributing the storm over the non-prioritized sub-basins is hardly reasonable, and it can be problematic if applied to guiding flood control or construction of hydraulic structures in the TLB.
To be specific, if we adopt the "typical year scheme" as the default storm of the TLB, for sub-basin YCDM and SH, flood risk can be overestimated by hydrologic models, accordingly, this might lead to unnecessary public spending on flood control in these sub-basins.Moreover, higher water levels in these areas can further influence the water yield distribution in the whole basin.Meanwhile, for sub-basin ZX, it leads to a severe underestimation of flood risk which may expose the sub-basin itself to a higher flood risk than expected.With this sub-basin located in the mountainous area of TLB (see Figure 2), it can be problematic since ZX is frequently suffering from torrential flood hazards during the flood season.On the contrary, if the "average scheme" is adopted instead, flood risks taken by the non-prioritized sub-basins can be more uniform, which is better for the flood control of the TLB as a whole.Of course, this method can be applied to other similar basins if administrators have similar "requests" for the storm of the whole basin and "prioritized sub-basin"; moreover, flexibility is also a highlight since it provides a new perspective, for instance, a 2-d copula can be applied when we only consider the "prioritized sub-basin" as our "pre-condition".

Conclusions
Aimed at tackling the spatial distribution problem of a design storm over non-prioritized sub-basins, this study takes the TLB as an example, it proposes 3-d Copula-based models which can build joint distributions among extreme rainfall of different sub-basins, and investigates and derives conditional probability solutions using mathematical methods.Afterwards, with the calculated conditional probabilities, the rationality of two distributing schemes is analyzed and evaluated.The following conclusions have been reached: 1.
For the "typical year scheme", the probability analysis of extreme rainfall shows huge gaps among non-prioritized sub-basins, especially for the max-30-day rainfall of 50 years return period.If this scheme is applied to hydrologic modelling for TLB, sub-basin YCDM and SH are considered to take a higher flood risk than they should, which can lead to unnecessary public spending on hydraulic structures in these sub-basins.Meanwhile, sub-basin ZX is taking a lower flood risk than it should, and it can be dangerous since this sub-basin is suffering from torrential flood disasters in reality.Generally, the scheme shows a different degree of rationality when it comes to different durations or different return periods, thus it is in urgent need to be redistributed spatially over the five non-prioritized sub-basins.

2.
Conditional probabilities of the non-prioritized sub-basins can be more uniform after rainfall is redistributed based on long-term information of data.This new scheme shows a more rational spatial distribution, in which flood risks taken by different non-prioritized sub-basins are much more uniform and are better for the flood prevention of the TLB as a whole.

3.
The proposed 3-d Copula-based method is proven to be very useful to evaluate spatial distribution of design storm in a large-scale drainage basin where sub-basins need to be considered separately for design storm.Further studies are needed to propose a more reasonable scheme for design storm of sub-basins where rainfall events are not independent.

Figure 1 .
Figure 1.Flow chart of how spatial distribution rationality is evaluated using conditional probability analysis.

Figure 2 .
Figure 2. Elevation of the Tai Lake Basin and its 7 sub-basins.

Figure 2 .
Figure 2. Elevation of the Tai Lake Basin and its 7 sub-basins.

Figure 3 .
Figure 3.Comparison of daily rainfall during flood season in 1991 and other years.

Figure 3 .
Figure 3.Comparison of daily rainfall during flood season in 1991 and other years.

Figure 7 .
Figure 7. Radar plots of conditional probabilities of the five non-prioritized sub-basins encountering a corresponding rainfall (scheme 1 "typical year scheme").(a) 50 years return period; (b) 100 years return period.

Figure 7 .
Figure 7. Radar plots of conditional probabilities of the five non-prioritized sub-basins encountering a corresponding rainfall (scheme 1 "typical year scheme").(a) 50 years return period; (b) 100 years return period.

Figure 8 .
Figure 8. Radar plots of conditional probabilities of the five non-prioritized sub-basins encountering a corresponding rainfall (scheme 2 "average scheme").(a) 50 years return period; (b) 100 years return period.

Figure 8 .
Figure 8. Radar plots of conditional probabilities of the five non-prioritized sub-basins encountering a corresponding rainfall (scheme 2 "average scheme").(a) 50 years return period; (b) 100 years return period.

Table 3 .
Conditional probabilities of the five non-prioritized sub-basins encountering corresponding rainfall in scheme 1 ("typical year scheme").

Table 4 .
Conditional probabilities of the five non-prioritized sub-basins encountering corresponding rainfall in scheme 2 ("average scheme").

Period Duration of Extreme Rainfall Conditional Rainy Zone (mm) Rainfall (mm) of Non-Prioritized Sub-Basins and Corresponding Conditional Probabilities
Water 2018, 10, x FOR PEER REVIEW 11 of 14

Table 4 .
Conditional probabilities of the five non-prioritized sub-basins encountering corresponding rainfall in scheme 2 ("average scheme").