Sensitivity analysis method of multi-process model system, storage medium and equipment

Abstract

The invention discloses a sensitivity analysis method of a multi-process model system, a storage medium and equipment, and relates to the technical field of underground water systems, and the method comprises the steps: obtaining an underground water system and parameters of a multi-process model; obtaining a parameter sensitivity index, a model sensitivity index and a process sensitivity index by using a Sobol method based on variance under the conditions of parameter uncertainty of a single model, model and parameter uncertainty of a plurality of process models of the system and uncertainty of a plurality of processes of the system; and obtaining AMA sensitivity indexes of model uncertainty, parameter uncertainty and process uncertainty by using an AMA method. And quantizing contributions of parameters, processes and models to system output according to various sensitivity indexes. The method makes up for the limitation that the result of a single sensitivity analysis method has deviation, and can be applied to complex problems with a large number of processes, models and parameters.

Description

Technical Field

[0001] The present invention relates to the technical field of groundwater systems, and in particular to a sensitivity analysis method, storage medium and equipment for a multi-process model system. Background Art

[0002] Numerical simulation methods represented by reactive solute transport models are powerful tools for simulating groundwater flow, solute transport, and biogeochemical reactions. With the deepening of groundwater-related research, groundwater models are becoming increasingly large and complex, and the running time cost of the models is also increasing. Without the help of statistical analysis methods, it is often very difficult to understand and improve complex groundwater models. At the same time, due to various reasons such as the lack of relevant data and researchers' insufficient understanding of groundwater processes, there are uncertainties in parameters and model structures in groundwater models. These uncertainties will lead to uncertainty in model output and have a negative impact on the prediction accuracy of the model. Sensitivity analysis is an effective tool for identifying important sources of uncertainty and improving models. Sensitivity analysis of complex groundwater models can identify important uncertainty factors in the model, guide researchers to focus on important uncertainty factors in model improvement, data collection, and parameter calibration, and provide technical support for reasonably simplifying model structures, guiding data collection for field work, and improving model performance.

[0003] In actual groundwater modeling, there are usually multiple possible concepts or mathematical models to explain the hydrological system under study. Previous sensitivity analysis studies only evaluated the sensitivity of parameters based on a given model, and the results are not applicable to multi-model systems. In addition, the key input uncertainty in groundwater modeling is ignored: the uncertainty of the groundwater model. At the same time, since each sensitivity analysis method contains different aspects of uncertainty, it is usually difficult to meet all the needs of modelers using only one method, and the evaluation results are prone to bias. Summary of the invention

[0004] The purpose of the present invention is to solve the problem of deviation in the results of a single system sensitivity analysis method and to propose a sensitivity analysis method for a multi-process model system, comprising the following steps:

[0005] S1. Obtain a groundwater system of a multi-process model, and obtain parameters of each process model of the multi-process driven groundwater system;

[0006] S2. Under the condition of parameter uncertainty of a single process model, the variance-based Sobol global sensitivity analysis method is used to obtain the parameter sensitivity index of the single process model, and the AMA method is used to obtain the AMA sensitivity index of the parameters of the single process model;

[0007] Under the condition of model and parameter uncertainty of multiple process models of groundwater system, the variance-based Sobol global sensitivity analysis method is used to obtain the total parameter sensitivity index and the total model sensitivity index, and the AMA method is used to obtain the AMA sensitivity index of the model and parameter uncertainty of the groundwater system;

[0008] Under the conditions of multiple process uncertainties in the groundwater system, the variance-based Sobol global sensitivity analysis method is used to obtain the process sensitivity index, and the AMA method is used to obtain the AMA sensitivity index of the process uncertainty of the groundwater system.

[0009] S3. Based on the various sensitivity indices obtained in S2, quantify the contribution of parameters, processes, and models to the output of the groundwater system.

[0010] Furthermore, under the condition of parameter uncertainty of the single process model, the parameter sensitivity index of the single process model is obtained using the variance-based Sobol global sensitivity analysis method:

[0011] The variance of the output of a single process model is expressed as:

[0012]

[0013] Where V(Y) represents the variance of the output of the single process model; Represents the parameter X i The first-order variance of Indicates that Y is in X i Under the condition of X ～i The expected value of; Y represents the output of the single process model; X i represents the i-th parameter of the single process model; X ～i Indicates the division by X i The set of all parameters except represents the variance contributed by the interaction between the i-th parameter and the j-th parameter; Indicates that Y is in X i and X j Under common conditions, about X ～ij The expected value of V j Represents the parameter X j The first-order variance of X j represents the jth parameter of the single process model; X ～ij Indicates the division by X ij The set of all parameters except V 12...k represents the variance contributed by the interaction between the 1st, 2nd, ..., kth parameters;

[0014] About X i The parameter sensitivity index is expressed as:

[0015]

[0016] Among them, S i Represents parameter X i Parameter sensitivity index of

[0017] The total parameter sensitivity index of the variance-based Sobol global sensitivity analysis method is expressed as:

[0018]

[0019] in, Represents the overall parameter sensitivity index.

[0020] Furthermore, under the model and parameter uncertainties of multiple process models of groundwater systems, the variance-based Sobol global sensitivity analysis method is used to obtain the total parameter sensitivity index and the total model sensitivity index as follows:

[0021] The variance of the output of the groundwater system is expressed as:

[0022] V(Δ)=V(X)+V(M)

[0023] V(X)=E M V X|M E(Δ|X,M)

[0024] V(M)=V M E X|M E(Δ|X,M)

[0025] Where V(Δ) represents the variance of the output of the groundwater system, Δ represents the output of the groundwater system, V(X) represents the variance caused by the uncertainty of the values ​​of all parameters in the system, X represents the set of all system parameters, V(M) represents the variance caused by the uncertainty of the model structure of the process model of the system, M represents the set of process models of the groundwater system, and E M represents the expected term in the model, V X|M represents the variance of the parameter set X corresponding to the process model M, E(Δ|X,M) represents the expectation of the system output Δ under the conditions of X and M, V M represents the variance of the process model M, E X|M Represents the expectation of the parameter set X corresponding to the process model M;

[0026] The sensitivity index of the overall parameter is expressed as:

[0027]

[0028] Among them, S X Indicates the sensitivity index of the total parameter;

[0029] The sensitivity index of the total model (inter-model variance) is expressed as:

[0030]

[0031] Among them, S M Represents the sensitivity index of the overall model.

[0032] Furthermore, under the conditions of multiple process uncertainties in the groundwater system, the process sensitivity index is obtained using the variance-based Sobol global sensitivity analysis method:

[0033] The variance of the output of the groundwater system is expressed as:

[0034]

[0035] Among them, V(Δ) represents, M K represents the set of all process models of process K, M ～K represents the rest of the processes in the system except process K, represents the variance contributed by process K and the interaction between process K and other parameters, About M ～K The conditional expectation of Indicates that the system output Δ is in M ～K Under the condition of K The variance of express About M ～K The variance of Indicates M ～K The variance of Indicates that the system output Δ is in M ～K Under the condition of K expectations;

[0036] The first-order process sensitivity index and the overall process sensitivity index are expressed as:

[0037]

[0038]

[0039] Among them, PS K represents the sensitivity index of process K, Indicates M K The variance of Indicates that the system output Δ is in M K Under the condition of ～K Expectations, P.S. TK Represents the overall process sensitivity index.

[0040] Furthermore, under the condition of parameter uncertainty of the single process model, the AMA sensitivity index of the parameters of the single process model obtained by the AMA method is expressed as:

[0041]

[0042] in, AMA sensitivity index of the parameters of the single process model, X i represents the i-th parameter, SM[Δ] is the unconditional moment of the single process model output Δ; Indicates that the parameter X is considered i For all parameters except , the model outputs the moment of Δ; Represents the quantized fixed parameter X i After that, the model outputs the reduced value of the moment of Δ; is about the uncertain parameter X i conditional expectation.

[0043] Furthermore, under the model and parameter uncertainties of multiple process models of the groundwater system, the AMA sensitivity index of the model and parameter uncertainties of the groundwater system is obtained using the AMA method and is expressed as:

[0044]

[0045] in, AMA sensitivity index representing model and parameter uncertainty of groundwater systems, is the jth model M j The i-th uncertain parameter in ω(M j ) is the model M j The weight of ; SM[Δ] represents the unconditional moment of the groundwater system output Δ; The selected model is M j Finally, consider the model M j All parameters X j , the conditional moment of the system output Δ; Refers to the model M j As a condition, consider the model M j Divide parameters When all parameters except , the system outputs the conditional moment of Δ; Indicates the difference parameter All parameters except It is about the uncertain parameters The conditional expectation of ; || means finding the absolute value.

[0046] Furthermore, under the conditions of multiple process uncertainties of the groundwater system, the AMA sensitivity index of the process uncertainty of the groundwater system is obtained using the AMA method and expressed as:

[0047]

[0048] in, AMA sensitivity index representing process uncertainty in groundwater systems; P k represents the kth process; N M is the number of system models; is the system model M j In the process P k The number of all parameters of M j represents the j-th system model, is the system model M j In the process P k The i-th uncertain parameter of ; Refers to the system model M j In addition to the parameters In addition, regarding the process P k All parameters of ω(M j ) is the model M j The weight of ; SM[Δ] represents the unconditional moment of the groundwater system output Δ; The selected model is M j Finally, consider the model M j All parameters X j , the conditional moment of the system output Δ; Refers to the model M j As a condition, consider the model M j Divide parameters When all parameters except , the system outputs the conditional moment of Δ; About The conditional expectation of ; || means finding the absolute value.

[0049] Furthermore, the mean value of the system output Δ is expressed as:

[0050]

[0051] Where, the expected value E[Δ] of the system output Δ represents the mean value of the system output Δ; N M is the number of system models; ω(M j ) is the model M j The weight of Indicates that the system output Δ is in M j Under the condition, about X j Expected value;

[0052] The variance of the system output Δ is expressed as:

[0053]

[0054] Where V[Δ] represents the variance of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X i The variance of

[0055] The skewness of the system output Δ is expressed as:

[0056]

[0057] Among them, γ[Δ] represents the skewness of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X i The variance of Indicates that the system output Δ is in M j Under the condition, about X i skewness;

[0058] The kurtosis of the system output Δ is expressed as:

[0059]

[0060] Where k[Δ] represents the kurtosis of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X i The kurtosis of .

[0061] The present invention also proposes a computer-readable storage medium, wherein the computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the sensitivity analysis method of the multi-process model system is implemented.

[0062] The present invention also proposes an electronic device, comprising a processor and a memory, wherein the processor and the memory are interconnected, wherein the memory is used to store a computer program, the computer program comprises computer-readable instructions, and the processor is configured to call the computer-readable instructions to execute the above-mentioned sensitivity analysis method of the multi-process model system.

[0063] The beneficial effects brought by the technical solution provided by the present invention are:

[0064] The present invention proposes to jointly use the variance-based Sobol global sensitivity analysis method and the AMA method to conduct a comprehensive global sensitivity analysis of the uncertainties of processes, models and parameters in multiple models, which makes up for the limitation of the bias in the results of the previous use of a single sensitivity analysis method. The method of the present application can be used for the sensitivity analysis of complex groundwater models. By combining the variance-based Sobol global sensitivity analysis method and the AMA method under the conditions of a multi-model system, the relative contribution of the uncertainties of processes, models and parameters in the system to the uncertainty of the target output is comprehensively and accurately evaluated, and detailed sensitivity information is provided, which can be applied to complex problems with a large number of processes, models and parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 is a flow chart of a sensitivity analysis method of a multi-process model system according to an embodiment of the present invention;

[0066] Figure 2 It is a block diagram of an electronic device in an exemplary embodiment of the present invention. DETAILED DESCRIPTION

[0067] To make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0068] The flowchart of the sensitivity analysis method of the multi-process model system of the embodiment of the present invention is as follows: Figure 1 , specifically including the following steps:

[0069] S1. Obtain a groundwater system of a multi-process model and obtain parameters of each process model of the multi-process driven groundwater system.

[0070] Specifically, the groundwater system is driven by multiple processes (such as A, B, ...), and each process in the system has multiple optional process models and related parameters. For example, process A has a process model Related parameters corresponding to each process model

[0071] The parameters of each process model are generated through parameter sampling. According to the actual situation of the model, determine what kind of distribution the parameters follow. The parameters follow normal distribution or uniform distribution, represented by N and U respectively. Enter the mean and variance of the parameter value range under normal distribution, or the lower and upper bounds of the parameter value range under uniform distribution, and place the results of random sampling of the required parameters in a matrix.

[0072] S2. Conduct sensitivity analysis of the groundwater system.

[0073] 1. Analyze using the variance-based Sobol global sensitivity analysis method

[0074] (1) Under the condition of parameter uncertainty of a single process model, Sobol's method implicitly assumes that variance is sufficient to describe uncertainty and uses variance to measure the uncertainty of the model output. The parameter sensitivity index of the single process model is obtained using the variance-based Sobol global sensitivity analysis method.

[0075] The output of the single process model is expressed as:

[0076] y=f(x)=f(x1,x2,...,x k )

[0077] Where y represents the output of a single process model under a set of deterministic parameters, f(x) represents the model function with x as the independent variable, x = x1, x2, ..., x k represents a set of deterministic parameters of a single process model, k represents the number of a set of deterministic parameters of a single process model, each parameter of a single process model has several selectable values, X i represents the i-th parameter of the single process model, x i Represents X i An optional value, X i Represents x i A collection of random input values.

[0078] The Sobol method expands f(x) into increasing formal dimensions:

[0079]

[0080] Where f0 is a constant. represents all functions with one deterministic parameter in a set of deterministic parameters as a variable, represents all functions with two deterministic parameters as variables in a set of deterministic parameters, f 12...k (x1,x2,...,x k ) represents a function with all deterministic parameters in a set of deterministic parameters as variables. Add all functions with each parameter of k deterministic parameters as independent variables, any two parameters as independent variables, any three parameters as independent variables, and so on until all k parameters are used as variables to obtain a model function with a set of deterministic parameters as independent variables.

[0081] In the presence of parameter uncertainty, each parameter becomes a random variable that varies within its uncertainty range. Based on the parameter uncertainty of the single process model, the output of the single process model is expressed as Y, and the variance of the output of the single process model is expressed as:

[0082]

[0083] Where V(Y) represents the variance of the output of the single process model, Represents the parameter X i The first-order variance of X i Importance of the variance of the model output Y, represents the sum of the first-order variances of all parameters, Indicates that Y is in X i Under the condition of X ～i The expected value of Y represents the output of the single process model, X i represents the i-th parameter of the single process model, represents the variance contributed by the interaction between the i-th parameter and the j-th parameter, that is, the second-order variance of the i-th parameter and the j-th parameter, represents the sum of the second-order variances of all parameters; Indicates that Y is in X i and X j Under common conditions, about X ～ij The expected value, V j Represents the parameter X j The first-order variance of X j represents the jth parameter of the single process model, V 12...k It represents the variance contributed by the interaction between the 1st, 2nd, ..., kth parameters, that is, the Kth-order variance of the parameters. The variance of the output of a single process model is the sum of all the variances of the 1st-kth order (k is the number of parameters).

[0084] The first-order parameter sensitivity index is expressed as:

[0085]

[0086] Among them, S i Represents parameter X i The parameter sensitivity index of .

[0087] The total parameter sensitivity index of the variance-based Sobol global sensitivity analysis method is expressed as:

[0088]

[0089] in, represents the overall parameter sensitivity index, V i +V ij +...+V 12...k represents the total order variance of the variance-based Sobol global sensitivity analysis method, which measures the i The main impact and X i The interaction between X and other parameters i The effect on the total variance of Y.

[0090] Based on the total variance law, the total parameter sensitivity index can be estimated with the same computational cost as the first-order parameter sensitivity index, thus avoiding the high amount of calculation and more efficiently calculating the total effect sensitivity index and the interaction between parameters. Through the total variance law, the total variance of the model output can be decomposed into:

[0091]

[0092] in, Represents parameter X ～i Variance of contribution; X ～i means except X i All remaining parameters except ; Indicates confirmation X ～i The average value of the variance of Y when the true value is Represents X ～i Determine X under the condition i The mean value of Y when the true value is Indicates that Y is in X ～i Under the condition of X i The variance of .

[0093] The formula of the total parameter sensitivity index of the Sobol global sensitivity analysis method based on variance is:

[0094]

[0095] (2) When there is model uncertainty, assume that there are K models that can be used to describe the system, and the multi-model system Δ = M(X), where Δ is the output of the system. First, the variance of the model output is decomposed using the total variance law according to the model uncertainty to obtain:

[0096] V(Δ)=V(X)+V(M)=E M V(Δ|M)+V M E(Δ|M)

[0097] Where V(Δ) represents the variance of the output of the groundwater system; Δ represents the output of the groundwater system; V(X) represents the variance within the model, that is, the variance caused by the uncertainty of the values ​​of all parameters in the system; X represents the set of all system parameters; E M V(Δ|M) represents the expected value of V(Δ|M) under the confirmation of model M; M represents the set of process models of the groundwater system; V(Δ|M) represents the variance of the output of the groundwater system under the condition of M; V(M) represents the inter-model variance, that is, the variance caused by the uncertainty of the model structure brought about by the existence of multiple optional models in the system; M represents the set of process models of the groundwater system; V ME(Δ|M) represents the variance of E(Δ|M) under the uncertainty of model M; E(Δ|M) represents the expected value of the output of the groundwater system under the condition of M.

[0098] V(X)=E M V X|M E(Δ|X,M)

[0099] V(M)=V M E X|M E(Δ|X,M)

[0100] Among them, E M represents the expected term in the model, V X|M represents the variance of the parameter set X corresponding to the process model M, E(Δ|X,M) represents the expectation of the system output Δ under the conditions of X and M, V M represents the variance of the process model M, E X|M Represents the expectation about the parameter set X corresponding to the process model M.

[0101] The sensitivity index of the overall parameter is expressed as:

[0102]

[0103] Among them, S X Represents the sensitivity index of the overall parameter.

[0104] The sensitivity index of the overall model is expressed as:

[0105]

[0106] Among them, S M Represents the sensitivity index of the overall model.

[0107] Expected Item E M Calculated using the model averaging method:

[0108]

[0109] Among them, P(M j ) is the jth model M in the system j Corresponding weights, in a system, the weights of all possible system models satisfy ∑P(M j )=1, J means that the system has a total of J models.

[0110]

[0111]

[0112] Among them, N M represents the number of system models, V[Δ|M j] indicates M j The output variance of the system under the condition, E[Δ] represents the expectation of the system output, that is, the mean output of the system, E[Δ|M j ] indicates M j The output expectation of the system under the condition, that is, M j The mean output of the system under these conditions.

[0113] Considering the contribution of uncertainty of a single parameter, the variance within the model is further decomposed:

[0114]

[0115] in, Indicates that all parameters except parameter X are fixed to the mean of their value ranges. The variance of model M at this time, that is, the variance contributed by parameter X, is calculated. The value of this item is obtained for all system models containing parameter X and the model is averaged. This item is the main contribution of parameter X. It means that after fixing parameter X to a certain value, considering the remaining parameters in model M, the variance of model M is calculated. Since the value of parameter X is uncertain, it is necessary to calculate the average value of this item over the range of parameter X. If parameter X is fixed, the more the total variance of the model decreases on average, the greater the impact of parameter X on the total variance, and the more important this parameter is. M represents the expected value of the model M, Represents the parameter X corresponding to the process model M i The variance of Represents the parameter X corresponding to the process model M ～i The expected value of E(Δ|X,M)|X i It represents the expectation of the system output Δ under the conditions of X and M, Represents the parameter X corresponding to the process model M i The expected value of X i |M represents the parameter X corresponding to the process model M i , Represents the parameter X corresponding to the process model M ～i The variance of .

[0116] According to the contribution of uncertainty of a single parameter, the variance within the model is decomposed into the first-order parameter sensitivity index and the total parameter sensitivity index of the variance-based Sobol global sensitivity analysis method, which are expressed as:

[0117]

[0118]

[0119] Among them, E M V X|ME(Δ|X,M) represents the mean of all parameters except parameter X that are fixed within their value ranges, and calculates the variance of model M caused by parameter X and the variance caused by the interaction between parameter X and other parameters. This term represents the variance contributed by parameter X, that is, how much variance of model M is left if all parameters except parameter X are fixed. The larger the value of this term, the greater the variance caused by parameter X and the variance caused by the interaction between parameter X and other parameters, that is, the greater the influence of parameter X on the variance of model output.

[0120] (3) Process modeling may require consideration of multiple possible process models. The combination of these models forms multiple possible system models. When the process is uncertain, the variance-based Sobol global sensitivity analysis method is used to obtain the process sensitivity index.

[0121] The variance of the output of the groundwater system is expressed as:

[0122]

[0123] Where V(Δ) represents the variance of the output of the groundwater system, M K represents the set of all process models of process K, M ～K represents the rest of the processes in the system except process K, represents the variance contributed by process K and the interaction between process K and other parameters, About M ～K The conditional expectation of Indicates that the system output Δ is in M ～K Under the condition of K The variance of express About M ～K The variance of Indicates M ～K The variance of Indicates that the system output Δ is in M ～K Under the condition of K expectations.

[0124] The first-order process sensitivity index and the overall process sensitivity index are expressed as:

[0125]

[0126]

[0127] Among them, PS K represents the sensitivity index of process K, Indicates M K The variance of Indicates that the system output Δ is in M K Under the condition of～K Expectations, P.S. TK Represents the overall process sensitivity index.

[0128] 2. Use the AMA method for analysis (AMA method reference: "Moment-based metrics for global sensitivity analysis of hydrological systems").

[0129] (1) Under the condition of parameter uncertainty of a single process model, the AMA sensitivity index of the parameters of the single process model obtained using the AMA method is expressed as:

[0130]

[0131] in, AMA sensitivity index of the parameters of the single process model, SM[Δ] is the unconditional moment of the output Δ of the single process model; the conditional moments include: mean (first-order moment), variance (second-order moment), skewness (third-order moment) and kurtosis (fourth-order moment); Indicates that the parameter X is considered i For all parameters except for i Indicates the division parameter X i All parameters except Represents the quantized fixed parameter X i After that, the model outputs the reduced value of the moment of Δ; is about the uncertain parameter X i conditional expectation.

[0132] (2) Under the conditions of model and parameter uncertainty of multiple process models of groundwater systems, the AMA sensitivity index of model and parameter uncertainty of groundwater systems is obtained using the AMA method and is expressed as:

[0133]

[0134] in, AMA sensitivity index representing model and parameter uncertainty of groundwater systems, is the jth model M j The i-th uncertain parameter in ω(M j ) is the model M j The weight of ; SM[Δ] represents the unconditional moment of the groundwater system output Δ; The selected model is M j Finally, consider the model M j All parameters X j , the conditional moment of the system output Δ; Refers to the model Mj As a condition, consider the model M j Divide parameters When all parameters except , the system outputs the conditional moment of Δ; Indicates the difference parameter All parameters except It is about the uncertain parameters The conditional expectation of ; || means finding the absolute value.

[0135] If the system substitution model is fixed as M j When the difference of the moment of the total output Δ is larger, the more important the model is. This term is called the contribution term of model selection; the contribution term of parameter selection Quantified the considered model M j Divide parameters The conditional moment of the total output when all parameters except j The conditional moment difference of the total output when all parameters are in the model M j In the fixed parameters After that, model M j The moment difference of the output result, the larger the difference, the better the parameter The more important.

[0136] (3) Under the conditions of multiple process uncertainties in the groundwater system, the AMA sensitivity index of the process uncertainty of the groundwater system is obtained using the AMA method and is expressed as:

[0137]

[0138] The contribution to model selection represents the inclusion process P k The contribution of all system models to the uncertainty of the system output. In this multi-model system, the system alternative model is fixed as M j When the total output Δ is larger, the difference in the moment is larger, which means that when the system model is fixed to M j When the moment of the system output decreases, the process includes the process P k The weighted average of this value of all system models can be obtained to obtain the process P k The contribution of the model choice is common to all system models involved. The contribution to parameter selection is the sum of each included process P k In the system model, about the process P k The sum of the contributions of all parameters to the uncertainty of the model output. Quantified the considered model M j Divide parameters The conditional moment of the total output when all parameters except jHow much the conditional moment of the total output is reduced when all parameters are in the process P k All the processes P in the system model involved k The contribution item of parameter selection can be obtained by summing the items of the relevant parameters and averaging the models. k Sensitivity index The larger the value, the more important the process is.

[0139] in, AMA sensitivity index representing process uncertainty in groundwater systems; P k represents the kth process; N M is the number of system models; is the system model M j In the process P k The number of all parameters of M j represents the j-th system model, is the system model M j In the process P k The i-th uncertain parameter of ; Refers to the system model M j In addition to the parameters In addition, regarding the process P k All parameters of ω(M j ) is the model M j The weight of ; SM[Δ] represents the unconditional moment of the groundwater system output Δ. The selected model is M j Finally, consider the model M j All parameters X j , the conditional moment of the system output Δ; Refers to the model M j As a condition, consider the model M j Divide parameters When all parameters except , the system outputs the conditional moment of Δ; About The conditional expectation of ; || means finding the absolute value.

[0140] The mean (first-order moment) of the system output Δ is expressed as:

[0141]

[0142] Among them, the expected value E[Δ] of the system output Δ represents the mean value of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X j expected value.

[0143] The variance (second-order moment) of the system output Δ is expressed as:

[0144]

[0145] Where V[Δ] represents the variance of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X i The variance of .

[0146] The skewness (third-order moment) of the system output Δ is expressed as:

[0147]

[0148] Among them, γ[Δ] represents the skewness of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X i The variance of Indicates that the system output Δ is in M j Under the condition, about X i The skewness of .

[0149] The kurtosis (fourth-order moment) of the system output Δ is expressed as:

[0150]

[0151] Where k[Δ] represents the kurtosis of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X i The kurtosis of .

[0152] Indicates that when considering model M j In the case of The conditional moments of Δ are calculated by Monte Carlo method for different values ​​of The parameter is discretized by binning operation in the range of values, and the number of bins is N bins (Use the same N for all parameters bins value), calculate the conditional moment inside each box Calculated by the formulas of first-order moment, second-order moment, third-order moment, and fourth-order moment. bins The value of is iteratively calculated until the relative changes of all research quantities are less than a fixed threshold to determine the convergence of the results.

[0153] S3. Based on the various sensitivity indices obtained in S2, quantify the contribution of parameters, processes, and models to the output of the groundwater system.

[0154] The variance-based Sobol method uses the model averaging method to deal with model uncertainty in multi-model systems, ignoring the possible impact of model selection. The AMA sensitivity analysis method explicitly considers the impact of model selection, can evaluate the impact of the selection of each system model in the multi-model system on the uncertainty of the system output, and can quantitatively evaluate the risk of selecting any model. At the same time, the AMA method not only considers the variance of the output, but also considers the mean, skewness and kurtosis of the output, which can enrich the sensitivity results based on the Sobol method and provide comprehensive and detailed sensitivity information for complex multi-model systems. This paper proposes to use the above two comprehensive global sensitivity analyses in combination to make up for the shortcomings of the two methods, better play the advantages of the two global sensitivity analysis methods to provide rich and detailed sensitivity information for groundwater model construction, so as to better understand how the uncertainty of model input is transformed into the uncertainty of model output, and quantify the relative contribution of various input uncertainties to output uncertainty, and screen the most important influencing factors.

[0155] In an embodiment of the present invention, the AMA sensitivity index results of the parameters under the multi-model condition are shown in Tables 1, 2, 3, and 4, wherein Table 1 is the result of the AMA sensitivity index of the parameters under the multi-model condition based on the mean, Table 2 is the result of the AMA sensitivity index of the parameters under the multi-model condition based on the variance, Table 3 is the result of the AMA sensitivity index of the parameters under the multi-model condition based on the skewness, and Table 4 is the result of the AMA sensitivity index of the parameters under the multi-model condition based on the kurtosis.

[0156] Table 1

[0157]

[0158] Table 2

[0159]

[0160] Table 3

[0161]

[0162] Table 4

[0163]

[0164] In an exemplary embodiment, a computer-readable storage medium is included, wherein the computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the sensitivity analysis method of the multi-process model system is implemented.

[0165] See also Figure 2 In an exemplary embodiment, an electronic device is also included, including at least one processor, at least one memory, and at least one communication bus.

[0166] Wherein, a computer program is stored in the memory, and the computer program includes computer-readable instructions. The processor calls the computer-readable instructions stored in the memory through the communication bus to execute the sensitivity analysis method of the multi-process model system.

[0167] The above description of the disclosed embodiments enables one skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to one skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but rather to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A sensitivity analysis method for a multi-process model system, characterized in that: The following steps are involved: S1. Obtain a groundwater system of a multi-process model, and obtain parameters of each process model of the multi-process driven groundwater system; S2. Under the condition of parameter uncertainty of a single process model, the variance-based Sobol global sensitivity analysis method is used to obtain the parameter sensitivity index of the single process model, and the AMA method is used to obtain the AMA sensitivity index of the parameters of the single process model; Under the condition of model and parameter uncertainty of multiple process models of groundwater system, the variance-based Sobol global sensitivity analysis method is used to obtain the total parameter sensitivity index and the total model sensitivity index, and the AMA method is used to obtain the AMA sensitivity index of the model and parameter uncertainty of the groundwater system; Under the conditions of multiple process uncertainties in the groundwater system, the variance-based Sobol global sensitivity analysis method is used to obtain the process sensitivity index, and the AMA method is used to obtain the AMA sensitivity index of the process uncertainty of the groundwater system. S3. Based on the various sensitivity indices obtained in S2, quantify the contribution of parameters, processes, and models to the output of the groundwater system.

2. The sensitivity analysis method of a multi-process model system according to claim 1, characterized in that: Under the condition of parameter uncertainty of a single process model, the parameter sensitivity index of a single process model is obtained using the variance-based Sobol global sensitivity analysis method: The variance of the output of a single process model is expressed as: Where V(Y) represents the variance of the output of the single process model; Represents the parameter X i The first-order variance of Indicates that Y is in X i Under the condition of X ～i The expected value of; Y represents the output of the single process model; X i represents the i-th parameter of the single process model; X ～i Indicates the division by X i The set of all parameters except represents the variance contributed by the interaction between the i-th parameter and the j-th parameter; Indicates that Y is in X i and X j Under common conditions, about X ～ij The expected value of V j Represents the parameter X j The first-order variance of X j represents the jth parameter of the single process model; X ～ij Indicates the division by X ij The set of all parameters except V 12...k represents the variance contributed by the interaction between the 1st, 2nd, ..., kth parameters; About X i The parameter sensitivity index is expressed as: Among them, S i Represents parameter X i Parameter sensitivity index of The total parameter sensitivity index of the variance-based Sobol global sensitivity analysis method is expressed as: in, Represents the overall parameter sensitivity index.

3. The sensitivity analysis method of a multi-process model system according to claim 2, characterized in that: Under the model and parameter uncertainties of multiple process models of groundwater systems, the variance-based Sobol global sensitivity analysis method is used to obtain the total parameter sensitivity index and the total model sensitivity index as follows: The variance of the output of the groundwater system is expressed as: V(Δ)=V(X)+V(M) V(X)=E M V X|M E(Δ|X,M) V(M)=V M E X|M E(Δ|X,M) Where V(Δ) represents the variance of the output of the groundwater system, Δ represents the output of the groundwater system, V(X) represents the variance caused by the uncertainty of the values ​​of all parameters in the system, X represents the set of all system parameters, V(M) represents the variance caused by the uncertainty of the model structure of the process model of the system, M represents the set of process models of the groundwater system, and E M represents the expected term in the model, V X|M represents the variance of the parameter set X corresponding to the process model M, E(Δ|X,M) represents the expectation of the system output Δ under the conditions of X and M, V M represents the variance of the process model M, E X|M Represents the expectation of the parameter set X corresponding to the process model M; The sensitivity index of the overall parameter is expressed as: Among them, S X Indicates the sensitivity index of the total parameter; The sensitivity index of the total model (inter-model variance) is expressed as: Among them, S M Represents the sensitivity index of the overall model.

4. The sensitivity analysis method of a multi-process model system according to claim 3 is characterized in that: Under the conditions of multiple process uncertainties in the groundwater system, the process sensitivity index is obtained using the variance-based Sobol global sensitivity analysis method: The variance of the output of the groundwater system is expressed as: Among them, V(Δ) represents, M K represents the set of all process models of process K, M ～K represents the rest of the processes in the system except process K, represents the variance contributed by process K and the interaction between process K and other parameters, About M ～K The conditional expectation of Indicates that the system output Δ is in M ～K Under the condition of K The variance of express About M ～K The variance of Indicates M ～K The variance of Indicates that the system output Δ is in M ～K Under the condition of K expectations; The first-order process sensitivity index and the overall process sensitivity index are expressed as: Among them, PS K represents the sensitivity index of process K, Indicates M K The variance of Indicates that the system output Δ is in M K Under the condition of ～K Expectations, P.S. TK Represents the overall process sensitivity index.

5. The sensitivity analysis method of a multi-process model system according to claim 1, characterized in that: Under the condition of parameter uncertainty of a single process model, the AMA sensitivity index of the parameters of the single process model obtained using the AMA method is expressed as: in, AMA sensitivity index of the parameters of the single process model, X i represents the i-th parameter, SM[Δ] is the unconditional moment of the single process model output Δ; Indicates that the parameter X is considered i For all parameters except , the model outputs the moment of Δ; Represents the quantized fixed parameter X i After that, the model outputs the reduced value of the moment of Δ; is about the uncertain parameter X i conditional expectation.

6. The sensitivity analysis method of a multi-process model system according to claim 5, characterized in that: Under the condition of model and parameter uncertainty of multiple process models of groundwater system, the AMA sensitivity index of model and parameter uncertainty of groundwater system is obtained by using AMA method, which is expressed as: in, AMA sensitivity index representing model and parameter uncertainty of groundwater systems, is the jth model M j The i-th uncertain parameter in ω(M j ) is the model M j The weight of ; SM[Δ] represents the unconditional moment of the groundwater system output Δ; The selected model is M j Finally, consider the model M j All parameters X j , the conditional moment of the system output Δ; Refers to the model M j As a condition, consider the model M j Divide the parameters When all parameters except , the system outputs the conditional moment of Δ; Indicates the difference parameter All parameters except is about the uncertain parameter X i j The conditional expectation of ; || means finding the absolute value.

7. The sensitivity analysis method of a multi-process model system according to claim 6, characterized in that: Under the conditions of multiple process uncertainties in the groundwater system, the AMA sensitivity index of the process uncertainty of the groundwater system obtained using the AMA method is expressed as: in, AMA sensitivity index representing process uncertainty in groundwater systems; P k represents the kth process; N M is the number of system models; is the system model M j In the process P k The number of all parameters of M j represents the j-th system model, is the system model M j In the process P k The i-th uncertain parameter of ; Refers to the system model M j In addition to the parameters In addition, regarding the process P k All parameters of ω(M j ) is the model M j The weight of ; SM[Δ] represents the unconditional moment of the groundwater system output Δ; The selected model is M j Finally, consider the model M j All parameters X j , the conditional moment of the system output Δ; Refers to the model M j As a condition, consider the model M j Divide the parameters When all parameters except , the system outputs the conditional moment of Δ; About The conditional expectation of ; || means finding the absolute value.

8. The sensitivity analysis method of a multi-process model system according to claim 7, characterized in that: The mean value of the system output Δ is expressed as: Where, the expected value E[Δ] of the system output Δ represents the mean value of the system output Δ; N M is the number of system models; ω(M j ) is the model M j The weight of Indicates that the system output Δ is in M j Under the condition, about X j Expected value; The variance of the system output Δ is expressed as: Where V[Δ] represents the variance of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X i The variance of The skewness of the system output Δ is expressed as: Among them, γ[Δ] represents the skewness of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X i The variance of Indicates that the system output Δ is in M j Under the condition, about X i skewness; The kurtosis of the system output Δ is expressed as: Where k[Δ] represents the kurtosis of the system output Δ, Indicates that the system output Δ is in M j Under the condition, about X i The kurtosis of .

9. A computer-readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, the method according to any one of claims 1 to 8 is implemented.

10. An electronic device, characterized in that: The method comprises a processor and a memory, wherein the processor and the memory are interconnected, wherein the memory is used to store a computer program, the computer program comprises computer-readable instructions, and the processor is configured to call the computer-readable instructions to execute the method according to any one of claims 1 to 8.

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