CN102508997B - Underground water model output uncertainty analysis method - Google Patents

Abstract

The invention discloses an underground water model output uncertainty analysis method in combination of a frequency analysis method and a sensitivity analysis method. The frequency analysis process comprises a parameter estimating process and a supposed checking process, wherein a typical distribution function in 7 is selected as an alternative probability density function, the supposed checking is carried out, and a proper probability density function is selected for the underground water level sequence. The sensitivity analysis method comprises stepwise regression analysis and mutual entropy analysis; the stepwise regression analysis can be used for analyzing the trend of the uncertain importance of an input variable, and the mutual entropy analysis can be used for well identifying the uncertainty factor. The method can be used for making up the deficiency in research content of the traditional uncertainty analysis method, analyzing the probability distribution characteristic and identifying the key uncertainty factor influencing the variable probability distribution, thus the production and source of the underground water model uncertainty can be well understood, feedback is provided for the data acquiring work of the underground water model, and the uncertainty of the model is reduced.

Description

A kind of groundwater model exports Uncertainty Analysis Method

Technical field

The present invention relates to a kind of model and export Uncertainty Analysis Method, be specifically related to a kind of to groundwater model output Uncertainty Analysis Method.

Background technology

Along with the development of method for numerical simulation and computer technology, groundwater model has become the basic tool of Groundwater Resource Management and planning.Ground water regime is a complexity, open huge system, and it is subject to the impact of the hydrology, meteorology, geologic condition and mankind's activity.Due to the restriction of observational data, when utilizing groundwater model to carry out numerical simulation, often there is deviation with actual observation in analog result, i.e. the uncertainty of groundwater model, thus affect the reliability of groundwater simulation.Therefore, in order to reduce the risk of groundwater simulation and prediction, the uncertainty analysis of groundwater model becomes the focus that Present Domestic is studied outward.

At present, the method of groundwater model uncertainty analysis mainly contains: 1. parameter uncertainty analytical approach, to by model parameter, the uncertainty that boundary condition etc. cause is analyzed, inverting is carried out to the Posterior probability distribution of mode input parameter, and (Beven is predicted to the output of model, K., Freer, J., 2001. Equifinality, data assimilation, and uncertainty estimation in mechanistic modeling of complex environmental systems using the GLUE methodology. J Hydrol, 249 (1-4): 11-29, Hassan, A.E., et al, 2008. Uncertainty assessment of a stochastic groundwater flow model using GLUE analysis. J Hydrol, 362 (1-2): 89-109), 2. conceptual model Uncertainty Analysis Method, consider the uncertainty of model structure, the result of comprehensive multiple conceptual model carries out uncertainty prediction (Rojas, R., et al, 2008, Conceptual model uncertainty in groundwater modeling:Combining generalized likelihood uncertainty estimation and Bayesian model averaging, Water Resour Res, 44 (12), Ye, M., et al, 2010, A Model-Averaging Method for Assessing Groundwater Conceptual Model Uncertainty, Ground Water, 48 (5), 716-728), 3. the Sensitivity Analysis of model output, sensitivity analysis is carried out to the output (as water level, concentration) of model, identify the uncertain factor (Mishra of groundwater model, S., et al, 2009, Global Sensitivity Analysis Techniques for Probabilistic Ground Water Modeling. Ground Water, 47 (5): 730-747).These methods are widely used in groundwater Numerical Simulation field, and are subject to the approval of hydrologists.However, these methods can not be analyzed the Probability Characteristics of groundwater model output and factor of influence thereof.The Probability Characteristics that model exports, as the final form of expression of model uncertainty, determines accuracy and the accuracy of modeling and prediction.Further, the influence factor of model output probability distribution better can understand the probabilistic generation of groundwater simulation and source.

Summary of the invention

Goal of the invention: the object of the invention is to for the deficiencies in the prior art, a kind of source can understanding model uncertainty is better provided, identify the uncertain factor of the key of groundwater model, thus the groundwater model controlling uncertain generation exports Uncertainty Analysis Method.

Technical scheme: groundwater model of the present invention exports Uncertainty Analysis Method, comprises the following steps:

Frequency analysis:

(1) in the frequency analysis process of groundwater model output variable, normal state, lognormality, gamma-2 is selected, logarithm gamma-2, p-III, logarithm p-III and be uniformly distributed as alternative probability density function;

(2) according to principle of maximum entropy (Principle of Maximum Entropy, POME), to normal state, lognormality, gamma-2, logarithm gamma-2, p-III, logarithm p-III and uniformly distributed function carry out parameter estimation;

(3) Chi-squared inspection is selected, for groundwater model output variable selects suitable probability density function;

Sensitivity analysis:

(4) select stepwise regression analysis method, Linear correlative analysis is carried out to the Probability Characteristics of groundwater model output variable and mode input variable;

(5) select cross-entropy analytical approach, the Probability Characteristics of lower water model output variable and the correlationship of mode input variable are analyzed.

Above-mentioned steps 1) in, selection 7 in probability distribution function there is different distribution characteristicss, representative and popularity.Therefore, it is possible to prevent because alternative distribution function not fully causes the deviation to groundwater model output variable Distribution estimation.

Above-mentioned steps 4) in, adopt the stepwise regression analysis based on linear model, trend analysis can be carried out to the correlationship of variable, and contrast with the result of cross-entropy analysis, the uncertain factor of model of cognition output variable probability distribution.

Above-mentioned steps 5) in, cross-entropy analysis, based on information entropy theory, can overcome the restriction of linear model in correlation analysis, carries out good identification to the nonlinear relationship of amount complicated and changeable.

The present invention obtains the Probability Characteristics of groundwater model output by frequency analysis method, adopts the Key Influential Factors of stepwise regression analysis and the distribution of cross-entropy analytical approach model of cognition output probability.The probability distribution of model output variable is as the probabilistic direct embodiment of groundwater model, and it is the final goal of uncertainty study.The probability distribution scope of variable, shape and position are determined by the probability density function of this variable.Therefore, the factor of influence of output variable probability distribution can export uncertain source and forming process at fundamentally Controlling model.The sensitivity analysis of groundwater model output variable probability distribution better can understand the source of model uncertainty, identifies the uncertain factor of the key of groundwater model, thus is conducive to controlling probabilistic generation.

Frequency analysis method combines with Sensitivity Analysis by the present invention, sets up a kind of model and exports Uncertainty Analysis Method.The present invention has following beneficial effect relative to existing technology:

(1) frequency analysis is used in the uncertainty analysis of groundwater model, the probability density function that underground water output variable is obeyed can be obtained, the Probability Characteristics of output variable is quantitatively described;

(2) the groundwater model sensitivity analysis from conventional is different, this sensitivity analysis to as if the probability distribution of groundwater model output variable, can from the input of more deep layer understanding model on the impact of output variable;

(3) adopt cross-entropy analytical approach, can than classic method more effectively, the uncertain factor of the key of model of cognition output variable probability distribution accurately;

(4) by identifying the uncertain factor of groundwater model output variable probability distribution, better can export probabilistic source by interpretation model, for the data collection effort of actual groundwater simulation provides feedback, being conducive to the uncertainty reducing model.

Accompanying drawing explanation

Fig. 1 is groundwater model schematic diagram in the embodiment of the present invention.

Fig. 2 is the result of the stepwise regression analysis of groundwater level average and variance in the embodiment of the present invention, (a), and (b), (c) represents output variable respectively yfor inspection well 1, the average of the water level sequence of 2,3, (d), (e), (f) represents output variable respectively yfor inspection well 1, the variance of the water level sequence of 2,3.

Fig. 3 is the cross-entropy analysis result of groundwater level average and variance in the embodiment of the present invention, and (a), (b), (c) represents output variable respectively yfor inspection well 1, the average of the water level sequence of 2,3, (d), (e), (f) represents output variable respectively yfor inspection well 1, the variance of the water level sequence of 2,3.

Embodiment

The present invention is based on a desirable groundwater modeling, the probability distribution of model output variable is analyzed.The output variable of groundwater model is groundwater level.Obtained the Probability Characteristics of groundwater level by frequency analysis method, then sensitivity analysis is carried out to it, identify the crucial uncertain factor.

the foundation of desirable groundwater model

As shown in Figure 1, this ideal three-dimensional groundwater model is a rectangle in the plane, long 3600m, wide 1800m, and subdivision is the grid cell of 20 m* 20m.Model extends 53m on vertical, is divided into three layers, is followed successively by from top to bottom: table water aquifer (1-30m), aquitard (31-33m), confined aquifer (34-53m), each water-bearing zone occurrence level.Suppose that water-bearing media is heterogeneous body, infiltration coefficient field meets stationary distribution in layer.Infiltration coefficient is expressed by space random function theory kspatial Variability, adopt isotropic index covariance function to describe ln kdistribution in each layer.Table 1 is ln in each layer of model kthe parameter of covariance function.The specific yield of each layer of model ( sy) and coefficient of storage ( s) be all assumed to be homogeneous distribution.

Infiltration coefficient in each water-bearing zone of table 1 groundwater model kspatial distributed parameters

As shown in Figure 1, the north of model and south are given flow border, are respectively and become a mandarin and Outlet boundary.Model western part is general head boundary (general head boundary).East is the river of a wide 20m, and riverbed thickness is 2m, and riverbed Bottom Altitude is 45m.Model top evenly accepts precipitation recharges.A bite pumped well and three observation stations are set in model confined aquifer.Table 2 is the interval of model parameter, the setting of model boundary condition as table 3, shown in table 4.

The interval of table 2 groundwater model parameter

The interval of table 3 groundwater model boundary parameter

Parameter	Interval
		Riverbed infiltration coefficient ( m/ d)	0.1-10.0
General head boundary transmissibility	50.0-5000.0

The interval of table 4 groundwater model boundary condition and variation range

Border	Interval	Variation range
			Precipitation rate ( mm/ month)	5.0-50.0	[-100%, 100%]
To draw water rate (10 4 m 3/ month)	0.2-2.0	[-100%, 100%]
			Intake rate (10 4 m 3/ month)	0.1-1.0	[-100%, 100%]
Handling rate (10 4 m 3/ month)	0.1-1.0	[-100%, 100%]
			River level ( m)	40.0-44.0	[-1.0m, 1.0m]
General border head ( m)	46.0-50.0	[-1.0m, 1.0m]

1.2 frequency analysis method

Frequency analysis is widely used a kind of analytical approach in hydrologic forecast.The content of frequency analysis be for observation data series ( x ₁, x ₂..., x _n ) select suitable probability density function, thus the distribution characteristics of situational variables and forecasting.The key step of frequency analysis has: (1) selects suitable probability density function; (2) according to principle of maximum entropy, based on observation data sequence, parameter estimation is carried out to alternative probability density function; (3) uncertainty forecast is carried out to variable.

The parameter estimation procedure of alternative probability density function is as follows:

(1) be uniformly distributed

Probability density function is: (1)

Parameter estimation is: (2)

θ ₁, θ ₂represent equally distributed upper and lower border, ( x ₁..., x _n ) be DS, nfor sample size, x _max , x _min be respectively the maxima and minima of DS;

(2) normal distribution

Probability density function is: (3)

Parameter estimation is: , μrepresent the average of sample, σ ²represent the variance of sample; ( x ₁..., x _n ) be DS, nfor sample size;

(3) gamma-2 distribution

Probability density function is: (4)

Based on principle of maximum entropy (Principle of Maximum Entropy, POME), gamma-2 parameter estimation is:

(5)

α, βrepresent form parameter and the scale parameter of gamma-2 distribution function respectively;

(4) P-III distribution

Probability density function is: (6)

Parameter estimation based on POME, P-III distribution is:

(7)

α, β, crepresent the form parameter of P-III distribution function, scale parameter and location parameter respectively, σ _x for sample canonical variance, ( x ₁..., x _n ) be DS, nfor sample size, c _s for the coefficient of skew, c _v for coefficient of dispersion;

(E) lognormality, logarithm gamma-2, logarithm p-III distribute

First Logarithm conversion is carried out to DS, more corresponding parameter estimation is carried out to lognormality, logarithm gamma-2 and logarithm P-III distribution.

Carry out Chi-squared inspection to alternative probability density function, its step is as follows:

(1) use knumber axis is divided into by-1 number kindividual interval (-∞, t ₁], ( t ₁, t ₂] ..., ( t _k-2, t _k-1], ( t _k-1,+∞], k≈ 1.87 ( n-1) ^0.4, nfor sample size;

(2) calculate sample sequence ( x ₁, x ₂..., x _n ) fall into quantity in each interval n _i , i=1,2 ..., k, and calculate for selecting probability density function f ₀ ( x) probability in each interval:

(8)

(3) Chi-squared statistic is calculated χ ²:

(9)

(4) confidence level is specified αif, p(χ ²>=χ 2 1-α)>= α, think that sample sequence is obeyed for selecting probability density function, otherwise refusal.

sensitivity Analysis

Stepwise regression analysis is one linear regression method simply and easily.Its hypothesis input variable and output variable meets linear relationship:

(10)

y' be the output variable of matching, x _j for input variable, b _j for regression coefficient, j=1 .., k, represent the quantity of regression variable.Regression variable is introduced by stepwise regression analysis one by one, and each step all will be carried out test and be guaranteed that each regression coefficient has conspicuousness, under certain confidence level, carries out tdistribution tests.The foundation of regression model can be divided into two steps: (1) constantly will have the variable import regression model of maximum partial correlation coefficient; (2) carry out significance test to variablees all in model, get rid of the inapparent variable of partial correlation coefficient, until all variablees all have remarkable importance in model, and the partial correlation coefficient not entering the variable of model is all not remarkable.

Cross-entropy analysis is a kind of Sensitivity Analysis based on information entropy theory.By set up contingency table (contingency tables) determine data ( x, y) between correlationship.By input variable xspan be divided into iindividual interval, output variable yspan be divided into jindividual interval.Set up ioK jrow contingency table, statistical sample data fall into each list cell ( i, j) quantity n _ij , p _ij represent sample data fall into interval ( x _i , y _j ) probability, p _ij = n _ij / n, nfor sample size. p _i. , p _.j represent that sample falls into interval respectively x _i , y _j probability, p _i. = n _i. / n, p _.j = n _.j / n, n _i. , n _.j represent that sample falls into interval respectively x _i , y _j quantity.

Based on information theory, variable x, yinformation entropy h( x), h( y) be:

(11)

Variable x, ysimultaneous information entropy h( x, y) be:

(12)

In information theory, the cross-entropy between variable i( x, y) represent their degree that interdepends. x, ycross-entropy represent due to x(or y) information and cause y(or x) probabilistic minimizing.That is:

(13)

In cross-entropy is analyzed, input variable xto with output variable yuncertain importance (uncertainty importance) have following two kinds of index expressions:

(1) coefficient of uncertainty u( x, y)

(14)

(2) rstatistic

(15)

This two indices all changes between [0,1], represents xwith yindependent or completely relevant.

modeling scheme designs

Based on Monte Carlo analogue technique, frequency analysis is carried out to the groundwater level obtained, and sensitivity analysis is carried out to its probability distribution, identify the crucial uncertain factor.Monte Carlo simulation steps is as follows:

(1) spatial spreading and time discrete are carried out to groundwater model.The simulation phase is set to 100 years, and within one month, be a stress phase, each stress phase is divided into 6 time periods.Export the mean value that water level is 6 mimic water-depths in each stress phase.In 1-100, the water level sequence of average in certain specific month is as research object.

(2) model parameter is set, comprises the hydraulic conductivity of infiltration coefficient, specific yield, coefficient of storage, riverbed infiltration coefficient, general head boundary.According to the infiltration coefficient Characteristics of spatial variability (table 1) in model water-bearing zone, permeability coefficient random field is generated by direct Fourier's changing method (direct Fourier transform).Remaining mode input parameter is (table 2, table 3) evenly random sampling generation in corresponding span.

(3) model boundary condition is set, comprises the rate of drawing water, border influx, boundary current output, river water level, general border head.The scope (table 4) arranged by model boundary, the assignment of boundary condition is divided into two steps: 1. in the interval of corresponding boundary condition, for monthly (the 1-12 month) evenly randomly draws an average; 2. based on this average, corresponding variation range all with sampling 100 times, as the boundary condition value in this month in the simulation phase.

(4) set up groundwater Numerical Simulation model based on Modflow-2005, moving model also carries out frequency analysis to the groundwater level of each observation station.Based on groundwater level, carry out parameter estimation for each for selecting probability density function, and carry out Chi-Square inspection.Remarkable water level is set αbeing 0.05, standby selecting function all not by Chi-Square inspection if all, is then unknown probability distribution by this water level sequence mark.

(5) step 2-step 4 is repeated, until the ratio shared by water level sequence of obeying each alternative probability density function keeps stable.

(6) collect the data sample of Monte Carlo simulation, comprise water level sequence, probability density function that water level sequence is obeyed and parameter thereof, model parameter corresponding to water level sequence and boundary condition.

(7) stepwise regression analysis and cross-entropy analysis are carried out to the parameter of water level sequence probability density function and mode input variable (model parameter and boundary condition).

the susceptibility results of groundwater level probability distribution and discussion

By the frequency analysis to groundwater level, the water level sequence Normal Distribution of 83.83%, obeys logarithm normal, gamma-2, logarithm gamma-2, P-III, logarithm P-III, to be uniformly distributed and the ratio shared by water level sequence of unknown distribution is respectively: 83.72%, 83.48%, 83.28%, 1.23%, 1.03%, 9.44%, and 13.05%.Therefore, most water level sequence Normal Distribution (or lognormal distribution).Further, when the parameter of gamma-2 distribution α(shaper parameter) is a positive integer ntime, can be regarded as nthe summation of individual exponential distribution.By central limit law, when n(be about 20000 in this example) time larger, gamma-2 distribution will level off to normal distribution.Therefore, the ratio shared by these two distributions is close.

Select the water level sequence of Normal Distribution as analytic target, using the mean and variance of water level sequence as output variable, the input parameter of groundwater model and boundary condition, as input variable (as shown in table 5), carry out stepwise regression analysis and cross-entropy analysis respectively.

The input variable of table 5 groundwater level probability distribution sensitivity analysis and numbering thereof

Variable	Numbering	Variable	Numbering
				Water table aquifer KAverage	1	Intake rate average	11
Aquitard KAverage	2	Inbound traffics variance	12
				Bearing course KAverage	3	Go out flow rate average	13
Water table aquifer specific yield	4	Outflow variance	14
				Water table aquifer coefficient of storage	5	Rate of drawing water average	15
Bearing course coefficient of storage	6	Rate of drawing water variance	16
				Riverbed infiltration coefficient	7	Stream stage average	17
General head boundary hydraulic conductivity	8	Stream stage variance	18
				Precipitation rate average	9	General border head average	19
Precipitation rate variance	10	General border head variance	20

Fig. 2 is the stepwise regression analysis result of groundwater level average and variance.( a), ( b), ( c) represent output variable respectively yfor inspection well 1, the average of the water level sequence of 2,3, ( d), ( e), ( f) represent that output variable y is inspection well 1 respectively, the variance of the water level sequence of 2,3.For the average of water level sequence, front 3 uncertain factors are variable 1,3,19.For the variance of water level sequence, front 3 uncertain factors are variable Isosorbide-5-Nitrae, 7.In regression model, the absolute value of the regression coefficient of variable is general less, and for mean regression model, maximal value is about 0.3, and mean value is about 0.12.For Tobin's mean variance model, maximal value is about 0.4, and mean value is about 0.18.This is mainly due to the average of these water level sequences and not simple linear relationship between variance and input variable, but more complicated nonlinear relationship.

Fig. 3 is the cross-entropy analysis result of groundwater level average and variance.( a), ( b), ( c) represent output variable respectively yfor inspection well 1, the average of the water level sequence of 2,3, ( d), ( e), ( f) represent output variable respectively yfor inspection well 1, the variance of the water level sequence of 2,3.Front 3 uncertain factors of the mean and variance of water level sequence are respectively variable 1,3,19, and variable Isosorbide-5-Nitrae, and 7, coming to the same thing of this and stepwise regression analysis.But the variable that regression coefficient is less has different uncertain importance recognition results from cross-entropy analysis.Therefore, stepwise regression analysis can identify the uncertain factor of key, and can not analyze the common uncertain factor.

conclusion

For the defect of traditional groundwater model Uncertainty Analysis Method in research contents, frequency analysis combines with Sensitivity Analysis by the present invention, sensitivity analysis is carried out to the probability distribution of groundwater model output variable, identifies the crucial uncertain factor.By a desirable groundwater model, using observation station underground water table as model output variable, select distribution common in 7 as alternative probability density function, and carry out test of hypothesis.Stepwise regression analysis and cross-entropy analysis are carried out to the mean and variance of the water level sequence of Normal Distribution.The method can obtain the Probability Characteristics of output variable, effectively identifies the uncertain factor of output variable.Thus provide feedback for the data collection effort of uncertainty analysis, reduce the uncertainty of model.

As mentioned above, although represented with reference to specific preferred embodiment and described the present invention, it shall not be construed as the restriction to the present invention self.Under the spirit and scope of the present invention prerequisite not departing from claims definition, various change can be made in the form and details to it.

Claims (1)

1. groundwater model exports a Uncertainty Analysis Method, it is characterized in that comprising the following steps:

Frequency analysis:

(1) in the frequency analysis process of groundwater model output variable, select normal state, lognormality, gamma-2, logarithm gamma-2, p-III, logarithm p-III and be uniformly distributed as alternative probability density function;

(2) according to principle of maximum entropy, parameter estimation is carried out to normal state, lognormality, gamma-2, logarithm gamma-2, p-III, logarithm p-III and uniformly distributed function;

Wherein, method for parameter estimation is as follows:

(A) be uniformly distributed

Probability density function is:

Parameter estimation is: ${\mathrm{\θ}}_1=\frac{1}{n-1}({\mathrm{nX}}_{\min }-X_{\max }),{\mathrm{\θ}}_2=\frac{1}{n-1}({\mathrm{nX}}_{\max }-X_{\min })---\left(2\right)$

θ ₁, θ ₂represent equally distributed upper and lower border, n is sample size, X _max, X _minbe respectively the maxima and minima of DS;

(B) normal distribution

Probability density function is: $f\left(x\right)=\frac{1}{\sqrt{2\mathrm{\π}}\mathrm{\σ}}\exp [-\frac{{(x-\mathrm{\μ})}^2}{{2\mathrm{\σ}}^2}]---\left(3\right)$

Parameter estimation is: $\mathrm{\μ}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i/n,{\mathrm{\σ}}^2=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(\mathrm{\μ}-x_i)}^2/n,$ μ represents the average of sample, and σ is sample canonical variance, σ ²represent the variance of sample, (x ₁..., x _n) be DS, n is sample size;

(C) gamma-2 distribution

Probability density function is: $f(\mathrm{\α},\mathrm{\β},x)=\frac{x^{\mathrm{\α}-1}{\mathrm{\β}}^{-\mathrm{\α}}{e}^{(-x/\mathrm{\β})}}{\mathrm{\Γ}\left(\mathrm{\α}\right)}---\left(4\right)$

Based on principle of maximum entropy, parameter estimation is:

$\left\{\begin{array}{c}E\left[x\right]=\mathrm{\α}/\mathrm{\β}\\ E\left[\ln x\right]=\ln (1/\mathrm{\β})+\mathrm{\ψ}\left(\mathrm{\α}\right)\\ \mathrm{\ψ}\left(x\right)=d\left[\ln \mathrm{\Γ}\left(x\right)\right]/\mathrm{dx}=\ln x-\frac{1}{2x}-\frac{1}{{12x}^2}+\frac{1}{{120x}^4}-\frac{1}{{252x}^6}\\ \mathrm{\Γ}\left(x\right){\∫}_{}^0{y}^{x-1}\exp (-y)\mathrm{dy}\end{array}\right.---\left(5\right)$

α, β represent form parameter and the scale parameter of gamma-2 distribution function respectively;

(D) P-III distribution

Probability density function is: $f\left(x\right)=\frac{{\mathrm{\β}}^{\mathrm{\α}}}{\mathrm{\Γ}\left(\mathrm{\α}\right)}{(x-c)}^{\mathrm{\α}-1}{e}^{-\mathrm{\β}(x-c)},\mathrm{\α}>0,x\≥c---\left(6\right)$

Parameter estimation based on POME, P-III distribution is:

$\left\{\begin{array}{c}\mathrm{\α}=\frac{4}{{C_s}^2}\\ \mathrm{\β}=\frac{2}{\stackrel{\‾}{x}{C}_v{C}_s}\\ c=\stackrel{\‾}{x}(1-\frac{{2C}_v}{C_s})\\ \frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\ln [x_i-\stackrel{\‾}{x}+{\mathrm{\σ}}_x{\mathrm{\α}}^{1/2}]=\frac{1}{2}\ln \mathrm{\α}+\ln {\mathrm{\σ}}_x-\frac{1}{2\mathrm{\α}}-\frac{1}{{12\mathrm{\α}}^2}+\frac{1}{{120\mathrm{\α}}^4}-\frac{1}{{252\mathrm{\α}}^6}\end{array}\right.---\left(7\right)$

α, β, c represent the form parameter of P-III distribution function, scale parameter and location parameter respectively, σ _xfor sample canonical variance, (x ₁..., x _n) be DS, n is sample size, C _sfor the coefficient of skew, C _vfor coefficient of dispersion;

(E) lognormality, logarithm gamma-2, logarithm p-III distribute

First Logarithm conversion is carried out to DS, more corresponding parameter estimation is carried out to lognormality, logarithm gamma-2 and logarithm P-III distribution;

(3) Chi-squared inspection is selected, for groundwater model output variable selects suitable probability density function;

Sensitivity analysis:

Wherein, the step of Chi-squared inspection is as follows:

(31) by k-1 number, number axis is divided into k interval (-∞, t ₁], (t ₁, t ₂] ..., (t _k-2, t _k-1], (t _k-1,+∞], k ≈ 1.87 (n-1) ^0.4, n is sample size;

(32) sample sequence (x is calculated ₁, x ₂..., x _n) fall into quantity n in each interval _i, i=1,2 ..., k, and calculate for selecting probability density function f ₀x () is in the Probability p in each interval _i:

$\left\{\begin{array}{c}{p}_1=P(X\&le;t_1)=F_0\left(t_1\right)\\ p_2=P(t_1<X\&le;t_2)=F_0\left(t_2\right)-F_0\left(t_1\right)\\ ......\\ p_{k-1}=P(t_{k-2}<X\&le;t_{k-1})=F_0\left(t_{k-1}\right)-F_0\left(t_{k-2}\right)\\ p_k=P(t_{k-1}<X)=1-F_0\left(t_{k-1}\right)\end{array}\right.---\left(8\right)$

(33) Chi-squared statistic χ is calculated ²:

${\mathrm{\&chi;}}^2=\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}\frac{{(n_i-{\mathrm{np}}_i)}^2}{{\mathrm{np}}_i}---\left(9\right)$

(34) confidence level α is specified, if think that sample sequence is obeyed for selecting probability density function, otherwise refusal;

(4) select stepwise regression analysis method, Linear correlative analysis is carried out to the Probability Characteristics of groundwater model output variable and mode input variable;

(5) select cross-entropy analytical approach, the Probability Characteristics of lower water model output variable and the correlationship of mode input variable are analyzed.