A transient electromagnetic adaptive lateral constraint inversion method
The Bayesian method is used to construct an adaptive lateral constraint inversion model, dynamically adjust the constraint intensity of the resistivity spatial distribution model, solve the problem of excessive smoothing of lateral constraints in the existing technology, improve the resolution ability of local uneven bodies, and realize the realization of More accurate inversion results.
Patent Information
- Application Number
- CN202310971596.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-08-03
- Publication Date
- 2024-12-20
- Estimated Expiration
- 2043-08-03
AI Technical Summary
When the existing transient electromagnetic inversion method deals with local uneven body, the lateral constraints are prone to excessive smoothing, resulting in the loss of local uneven body information of the inversion result, and the lack of detailed prior information makes it difficult to accurately set the constraint intensity, affecting the inversion effect. .
The Bayesian method is used to construct the adaptive lateral constraint inversion method. By constructing the lateral roughness and prior distribution of the resistivity spatial distribution model, the Bayesian inversion model uses the covariance matrix as a hyperparameter and dynamically adjusts the constraint intensity. Implement transient electromagnetic adaptive lateral constraint inversion.
While maintaining inversion stability, it improves the ability to distinguish local uneven bodies, can determine the lateral constraint intensity by itself, and restore more accurate lateral boundaries, which is suitable for situations where local uneven bodies exist within the detection range.
Smart Images
Figure CN116992668B_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the field of transient electromagnetic exploration in the fields of metal mines, groundwater, oil and gas resources, and in particular relates to a transient electromagnetic adaptive lateral constraint inversion method. Background Art
[0002] The transient electromagnetic method obtains the electrical information of the underground medium by transmitting an electromagnetic source signal underground and then recording the electromagnetic field response generated by the electromagnetic induction of the underground medium. Inversion is the main means of extracting electrical information from transient electromagnetic data. By establishing a mathematical and physical model between the transient electromagnetic response and the underground medium, the electrical parameters of the underground medium are inferred from the actual observed data. Since inferring the model from the data is a typical ill-posed problem with multiple solutions and instability, it is necessary to use regularization or constraints to obtain a stable and unique solution.
[0003] Lateral constraint is the most commonly used constraint in transient electromagnetic data inversion. By constructing the model lateral roughness operator, the roughness of the model parameters of adjacent measurement points in the model space is calculated, and it is introduced into the objective function of the inversion to obtain an overall laterally smooth model. Lateral constraint inversion can reduce the multi-solution nature of transient electromagnetic inversion, so that the inversion can stably obtain a unique solution, and has a good application effect in an environment where the geological structure is layered as a whole. However, when there are local inhomogeneities in the detection area, lateral constraints will over-smooth the local inhomogeneities, making it impossible for the inversion results to restore the information of the local inhomogeneities.
[0004] By setting the lateral constraint strength, the recognition effect of the inversion results for local inhomogeneities can be improved. On the one hand, it is necessary to improve stability through constraints to obtain an overall laterally smooth model; on the other hand, when encountering local inhomogeneities, the constraint strength needs to be weakened to avoid losing information about the local inhomogeneities. Therefore, it is particularly important to set a reasonable constraint strength, and it is often necessary to use detailed prior information to improve the inversion effect. However, detailed and accurate prior information is difficult to obtain, making it impossible to accurately set the constraint strength, resulting in insufficient application of lateral constraint inversion in non-layered models. Summary of the invention
[0005] The present invention proposes a transient electromagnetic adaptive lateral constraint inversion method based on the Bayesian method, which improves the resolution of local inhomogeneities while retaining the lateral constraint inversion stability of transient electromagnetic data, especially when there are local inhomogeneities within the detection range.
[0006] To achieve the above object, the present invention provides a transient electromagnetic adaptive lateral constraint inversion method, comprising the following steps:
[0007] Constructing a lateral roughness of a resistivity spatial distribution model and constructing a prior distribution based on the lateral roughness;
[0008] Constructing a Bayesian inversion model, introducing the prior distribution into the Bayesian inversion model to achieve lateral constraints;
[0009] The covariance matrix controlling the constraint strength in the prior distribution is used as a hyperparameter, and based on the Bayesian inversion model, the parameters of the resistivity spatial distribution model are obtained to realize transient electromagnetic adaptive lateral constraint inversion.
[0010] Optionally, constructing the lateral roughness of the resistivity spatial distribution model includes:
[0011] The roughness operator S of the resistivity spatial distribution model is defined as d :
[0012]
[0013] Based on the roughness operator S d , obtaining the lateral roughness of the resistivity spatial distribution model;
[0014] The lateral roughness of the resistivity spatial distribution model is: R m =S d m
[0015] Among them, M is the number of horizontal nodes in the resistivity spatial distribution model, L is the number of vertical grids in the resistivity spatial distribution model, and R m is the lateral roughness of the resistivity spatial distribution model, and m is the parameter vector of the resistivity spatial distribution model.
[0016] Optionally, constructing a prior distribution based on the lateral roughness includes:
[0017]
[0018] Among them, N is the prior distribution satisfied by the lateral roughness of the constructed model, D is the rank of the roughness operator, C h is a D×D order covariance matrix.
[0019] Optionally, constructing the Bayesian inversion model includes:
[0020] Based on the proposed distribution and the prior distribution, obtaining an alternative model;
[0021] Calculating the candidate model to obtain a likelihood function;
[0022] Obtaining an acceptance scale factor based on the prior distribution, the proposed distribution, and the likelihood function;
[0023] The acceptance scale factor is used to determine whether to update the resistivity spatial distribution model, and the Bayesian inversion model is constructed.
[0024] Optionally, based on the suggested distribution and the prior distribution, obtaining the candidate model includes:
[0025] Determining parameter information of the resistivity spatial distribution model using the prior distribution;
[0026] Based on the parameter information of the resistivity spatial distribution model and the current resistivity spatial distribution model, and using the suggested distribution, the candidate model is obtained.
[0027] Optionally, the suggested distribution is:
[0028]
[0029] Among them, q is the proposed distribution, m is the current resistivity spatial distribution model, m′ is the alternative model, and σ is the variance of the Gaussian distribution, which represents the step size used to generate the alternative model.
[0030] Optionally, the likelihood function is calculated by the following formula:
[0031]
[0032] Where p(d|m) is the likelihood function, d is the data vector, F is the forward operator that maps the resistivity spatial distribution model parameters to data, T represents the transpose of the matrix, and C d is the error matrix of the data.
[0033] Optionally, the acceptance scale factor is calculated by the following formula:
[0034]
[0035] Among them, α(m′|m) is the receiving scale factor, m′ is the alternative model, p(m′) is the prior distribution of the alternative model, p(m) is the prior distribution of the current resistivity spatial distribution model, p(d|m′) is the likelihood function value of the alternative model, q(m|m′) is the proposed distribution of the current model generated by the alternative model, and q(m′|m) is the proposed distribution of the alternative model generated by the current resistivity spatial distribution model.
[0036] Optionally, using the acceptance scale factor to determine whether to update the resistivity spatial distribution model includes:
[0037] using the accepted scaling factor, determining whether the candidate model is accepted;
[0038] If the candidate model accepts the acceptance scale factor, the candidate model enters the sample set of the resistivity spatial distribution model to update the resistivity spatial distribution model and obtain an updated model;
[0039] If the candidate model rejects the acceptance scale factor, the current resistivity spatial distribution model is maintained, and a new candidate model is continuously generated from the proposed distribution and the current resistivity spatial distribution model.
[0040] Optionally, taking the covariance matrix controlling the constraint strength in the prior distribution as a hyperparameter, obtaining the parameters of the resistivity spatial distribution model based on the Bayesian inversion model, and realizing transient electromagnetic adaptive lateral constraint inversion includes:
[0041] The covariance matrix controlling the constraint strength in the prior distribution is used as a hyperparameter, and based on the Bayesian inversion model, the hyperparameter and a plurality of resistivities are sampled and updated, and the transmission and transformation electromagnetic observation data are fitted to obtain the value of the hyperparameter and a plurality of resistivity values;
[0042] The lateral constraint is determined based on the value of the hyperparameter, and the parameters of the resistivity spatial distribution model are obtained based on a number of the resistivity values, so as to realize transient electromagnetic adaptive lateral constraint inversion.
[0043] The present invention has the following beneficial effects:
[0044] The transient electromagnetic adaptive lateral constraint inversion method of the present invention can restore more accurate lateral boundaries, can decide whether to impose strong constraints on the lateral constraints, and can retain the resolution of local target bodies while maintaining the inversion stability; while retaining the lateral constraint inversion stability of transient electromagnetic data, the present invention also improves the resolution of local inhomogeneous bodies, especially when there are local inhomogeneous bodies within the detection range. BRIEF DESCRIPTION OF THE DRAWINGS
[0045] The drawings constituting a part of the present application are used to provide a further understanding of the present application. The illustrative embodiments and descriptions of the present application are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:
[0046] Figure 1 A schematic diagram of a process flow of a transient electromagnetic adaptive lateral constraint inversion method according to an embodiment of the present invention;
[0047] Figure 2 A priori distribution diagram containing lateral constraints of different strengths proposed in an embodiment of the present invention;
[0048] Figure 3 The real model and its lateral constraint strength proposed in the embodiment of the present invention, wherein (a) is a model containing local high resistivity and low resistivity target bodies, and (b) is the constraint strength value at different grids of the model;
[0049] Figure 4The constraint strength distribution at different model units proposed in the embodiment of the present invention, wherein a) is Figure 3 The posterior distribution of the hyperparameters at grid A in the middle, b) is Figure 3 The posterior distribution of the hyperparameters at grid B in the middle, c) is Figure 3 The posterior distribution of the hyperparameters at the grid C in the middle, d) is Figure 3 The posterior distribution of the hyperparameters at the middle grid D;
[0050] Figure 5 The lateral smoothness of the model obtained by different inversion methods proposed in the embodiments of the present invention, where (a) is adaptive lateral constraint inversion; (b) is no constraint added; (c) is fixed-strength lateral constraint inversion; and (d) is conventional lateral constraint inversion based on deterministic methods. DETAILED DESCRIPTION
[0051] It should be noted that, in the absence of conflict, the embodiments and features in the embodiments of the present application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.
[0052] It should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer executable instructions, and that, although a logical order is shown in the flowcharts, in some cases, the steps shown or described can be executed in an order different from that shown here.
[0053] Based on the Bayesian method, this paper designs an adaptive strength lateral constraint inversion method, which can improve the resolution of local inhomogeneities while retaining the stability of lateral constraint inversion of transient electromagnetic data, especially when there are local inhomogeneities within the detection range. The method is mainly divided into two parts: the Bayesian algorithm framework and the construction of adaptive constraints:
[0054] The first part is to establish a Bayesian algorithm framework for extracting resistivity information from transient electromagnetic data. The Bayesian method is a random search method based on probability theory. In transient electromagnetic inversion, alternative models are generated from the model space through the proposed distribution, and the degree of fit between the response of the alternative model and the transient electromagnetic data is used to determine whether to accept the model, thereby constructing a Monte Carlo-Markov chain of model samples, and obtaining the posterior distribution satisfied by the model parameters through the model samples in the chain, obtaining the inversion solution, and at the same time inferring the uncertainty of the solution.
[0055] The second part is to construct lateral constraints with adaptive and variable strength in Bayesian inversion. This is done by introducing model roughness into the prior distribution of Bayesian inversion. The parameters controlling the strength of the constraints in the prior distribution are regarded as hyperparameters. When the model parameters are sampled in the inversion, the hyperparameters controlling the strength of the constraints are sampled, so that the strength of the constraints is determined by the data and the inversion algorithm itself during the inversion process.
[0056] like Figure 1 As shown, this embodiment provides a transient electromagnetic adaptive lateral constraint inversion method, which specifically includes the following steps:
[0057] Construct the lateral roughness of the resistivity spatial distribution model and construct a prior distribution based on the lateral roughness;
[0058] Construct a Bayesian inversion model and introduce prior distribution into the Bayesian inversion model to achieve lateral constraints;
[0059] The covariance matrix controlling the constraint strength in the prior distribution is taken as a hyperparameter. Based on the Bayesian inversion model, the parameters of the resistivity spatial distribution model are obtained to realize transient electromagnetic adaptive lateral constraint inversion.
[0060] The existing lateral constraint inversion is implemented based on a deterministic inversion algorithm, which requires artificial setting of the constraint strength, and the inversion effect for local inhomogeneities is limited. This embodiment proposes an inversion method in which the constraint strength is determined by inversion and data. First, a basic framework of transient electromagnetic Bayesian inversion is constructed to achieve mapping from electromagnetic field response to resistivity. Secondly, a lateral roughness operator of the resistivity spatial distribution model is constructed, and the lateral roughness of the resistivity spatial distribution model is introduced into the prior distribution to achieve lateral constraints, and the parameter controlling the constraint strength in the prior distribution is used as a hyperparameter, and the value of the parameter is determined by Bayesian inversion and data, thereby realizing a lateral constraint inversion method with adaptive constraint strength in each area of the model.
[0061] (1) Bayesian inversion is based on the Bayesian theorem, which updates the inference of model parameters by sampling and verifying the parameters of the resistivity spatial distribution model. The Bayesian theorem is:
[0062]
[0063] Among them, m is the resistivity spatial distribution model parameter vector, d is the data vector, p(m) is the prior distribution of the resistivity spatial distribution model parameters, p(d|m) is the likelihood function, which represents the degree of fit of the model to the data, and p(m|d) is the posterior distribution of the resistivity spatial distribution model parameters, which represents the understanding of the model parameters after integrating the information in the data.
[0064] Bayesian inversion is usually implemented through numerical sampling methods. Its basic process is as follows: first, the value range and probability value of each parameter in the model space are determined by the prior distribution; then, within the sampling range determined by the prior distribution, the current resistivity spatial distribution model is used to generate an alternative model m′ using the proposed distribution, and the likelihood function is calculated based on the alternative model; the acceptance scale factor α composed of the proposed distribution, the prior distribution and the likelihood function determines whether to accept the alternative model.
[0065] The recommended distribution used in this embodiment is:
[0066]
[0067] Among them, q is the proposed distribution, m is the current resistivity spatial distribution model, m′ is the alternative model, and σ is the variance of the Gaussian distribution, which represents the step size used to generate the alternative model.
[0068] The likelihood function is calculated as follows:
[0069]
[0070] Where F is the forward operator that maps the resistivity spatial distribution model parameters to data, T is the transpose of the matrix, and C d is the error matrix of the data.
[0071] The acceptance scale factor is calculated as follows:
[0072]
[0073] Among them, q is the proposed distribution, p(d|m′) is the likelihood function value of the alternative model, q(m|m′) is the proposed distribution of the current resistivity spatial distribution model generated by the alternative model, and q(m′|m) is the proposed distribution of the alternative model generated by the current resistivity spatial distribution model.
[0074] The acceptance ratio factor determines whether the alternative model is accepted. If accepted, the alternative model enters the model sample set and is converted into the current resistivity spatial distribution model. If rejected, the existing resistivity spatial distribution model is maintained and the next round of sampling is carried out.
[0075] Continue to generate new candidate models based on the proposed distribution and the current resistivity spatial distribution model, and carry out the next round of sampling until the convergence condition is met. The convergence condition can be reaching the target number of samples or reaching the maximum number of iterations.
[0076] (2) Define the roughness operator S of the resistivity spatial distribution model parameters as follows: d :
[0077]
[0078] Model roughness R m for:
[0079] R m =S d m (5)
[0080] In order to introduce model roughness into Bayesian inversion, the following prior distribution of model roughness is constructed:
[0081]
[0082] Among them, N is the prior distribution satisfied by the lateral roughness of the constructed model, D is the rank of the roughness operator, C h is a D×D order covariance matrix, which is used to set the constraint strength of each resistivity spatial distribution model parameter. Combining equations (5) and (6), the prior distribution satisfied by the model parameters is obtained as follows:
[0083]
[0084] in, Take C in the inversion h is a diagonal matrix, so that each adjacent measurement point can be given an independent constraint strength. Since in the calculation of the acceptance scale factor in formula (3), there is no need to independently calculate the prior distribution, only the proportion of the prior distribution needs to be calculated, so the introduced prior distribution is:
[0085]
[0086] The above process constructs a multivariate Gaussian prior distribution with a mean of zero imposed on the model parameters through the prior distribution. Figure 2 As shown in the figure, it shows how the prior distribution imposes constraints of different strengths. When different roughnesses are imposed on the resistivity spatial distribution model, the probabilities of different resistivity spatial distribution model parameters in the prior distribution are different; when the constraint strength is fixed, the probability of small roughness (R1) is greater than the probability of large roughness (R2); when stronger constraints are imposed, the probability difference between different roughnesses increases. Since Bayesian sampling is more likely to accept models with large prior probabilities, lateral constraints are introduced in the sampling process.
[0087] The final posterior distribution is:
[0088] After completing Bayesian sampling based on the above prior distribution, the posterior distribution satisfied by the resistivity spatial distribution model parameters under different hyperparameters can be obtained. The complete posterior distribution of the resistivity spatial distribution model parameters can be obtained by integration:
[0089] p(m|d)=∫p(m|d,C)p(C|d)dC (9).
[0090] The covariance matrix that controls the constraint strength in the prior distribution is taken as a hyperparameter. Based on the Bayesian inversion model, the hyperparameters and the resistivity at different depths are sampled and updated, and the transient electromagnetic observation data are fitted to obtain the values of the hyperparameters and the resistivity values at different depths. The values of the hyperparameters can determine the strength of the lateral constraint, and the change of resistivity with depth can obtain the parameters of the resistivity spatial distribution model, thereby realizing transient electromagnetic adaptive lateral constraint inversion.
[0091] The adaptive lateral constraint inversion method proposed in this embodiment can adaptively determine the strength of the lateral constraint. The adaptive lateral constraint inversion is performed on the transient electromagnetic data generated by the resistivity spatial distribution model containing different units, such as Figure 3 As shown, Figure 3 (a) is the designed model, Figure 3 (b) is the hyperparameter distribution of each model unit obtained by Bayesian inversion, where Figure 3 (b) The labeled A, B, C, and D are the hyperparameters of four randomly selected model units. The results show that at the lateral boundaries of the model, the constraint strength is significantly weakened, so that the inversion has the ability to accurately restore the lateral boundaries and will not be smoothed by the lateral constraints. In this process, no restrictions are imposed on the strength of the lateral constraints. The strength of the constraints is completely evaluated by the data and the inversion method in Bayesian sampling to generate the constraint strength of the model at each grid. This shows that the method proposed in the present invention can decide whether to impose strong constraints on the lateral constraints, and can retain the ability to resolve local target bodies while maintaining the stability of the inversion.
[0092] The adaptive lateral constraint inversion proposed in this embodiment regards the lateral constraint strength as a hyperparameter and samples it in the Bayesian inversion. While obtaining the posterior distribution of the model parameters, the posterior distribution of the constraint strength can also be obtained, such as Figure 4 As shown, Figure 4 a), Figure 4 b) Figure 4 c) and Figure 4 d) are respectively from Figure 3 (b) The posterior distribution of the hyperparameters of the four model units A, B, C, and D selected. The results show that different constraint strengths are obtained at different model parameters by the data and the inversion process. The distribution of these constraint strengths can provide additional information for the analysis and interpretation of the inversion results.
[0093] like Figure 5 As shown, Figure 5 (a) is the lateral roughness of the inversion result obtained by the adaptive lateral constraint inversion of the present invention, Figure 5 (b) is the unconstrained inversion result. Figure 5 (c) is the constrained inversion with fixed strength, Figure 5 (d) is a conventional deterministic constrained inversion. The comparison of the inversion results shows that when the lateral constraints are applied, the multi-solution inversion results are significantly reduced, the smoothness at the interface is improved, and it is more in line with the actual model. Compared with the deterministic inversion method or the fixed-strength lateral constraint inversion method, the proposed adaptive lateral constraint inversion method can restore more accurate lateral boundaries.
[0094] The above is only a preferred specific implementation of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed in the present application should be included in the protection scope of the present application. Therefore, the protection scope of the present application should be based on the protection scope of the claims.
Claims
1. A transient electromagnetic adaptive lateral constraint inversion method, characterized in that: The following steps are involved: Constructing a lateral roughness of a resistivity spatial distribution model and constructing a prior distribution based on the lateral roughness; Constructing a Bayesian inversion model, introducing the prior distribution into the Bayesian inversion model to achieve lateral constraints; Constructing the Bayesian inversion model includes: Based on the proposed distribution and the prior distribution, obtaining an alternative model; Calculating the candidate model to obtain a likelihood function; Obtaining an acceptance scale factor based on the prior distribution, the proposed distribution, and the likelihood function; Determining whether to update the resistivity spatial distribution model using the acceptance scale factor, and constructing the Bayesian inversion model; The covariance matrix controlling the constraint strength in the prior distribution is used as a hyperparameter, and based on the Bayesian inversion model, the parameters of the resistivity spatial distribution model are obtained to realize transient electromagnetic adaptive lateral constraint inversion; Taking the covariance matrix controlling the constraint strength in the prior distribution as a hyperparameter, based on the Bayesian inversion model, obtaining the parameters of the resistivity spatial distribution model, and realizing transient electromagnetic adaptive lateral constraint inversion includes: The covariance matrix controlling the constraint strength in the prior distribution is used as a hyperparameter, and based on the Bayesian inversion model, the hyperparameter and a plurality of resistivities are sampled and updated, and the transmission and transformation electromagnetic observation data are fitted to obtain the value of the hyperparameter and a plurality of resistivity values; The lateral constraint is determined based on the value of the hyperparameter, and the parameters of the resistivity spatial distribution model are obtained based on a number of the resistivity values, so as to realize transient electromagnetic adaptive lateral constraint inversion.
2. A transient electromagnetic adaptive lateral constraint inversion method as claimed in claim 1, characterized in that: The lateral roughness for constructing the resistivity spatial distribution model includes: Define the roughness operator of the resistivity spatial distribution model : Based on the roughness operator , obtaining the lateral roughness of the resistivity spatial distribution model; The lateral roughness of the resistivity spatial distribution model is: in, is the number of transverse nodes in the resistivity spatial distribution model, is the number of vertical grids in the resistivity spatial distribution model, is the lateral roughness of the resistivity spatial distribution model, is the parameter vector of the resistivity spatial distribution model.
3. A transient electromagnetic adaptive lateral constraint inversion method as claimed in claim 2, characterized in that: Constructing a prior distribution based on the lateral roughness includes: in, The prior distribution of the lateral roughness of the constructed model satisfies: is the rank of the roughness operator, for The covariance matrix of order .
4. A transient electromagnetic adaptive lateral constraint inversion method as claimed in claim 1, characterized in that: Based on the suggested distribution and the prior distribution, obtaining the candidate model includes: Determining parameter information of the resistivity spatial distribution model using the prior distribution; Based on the parameter information of the resistivity spatial distribution model and the current resistivity spatial distribution model, and using the suggested distribution, the candidate model is obtained.
5. A transient electromagnetic adaptive lateral constraint inversion method as claimed in claim 4, characterized in that: The proposed distribution is: in, For the proposed distribution, is the current resistivity spatial distribution model, As an alternative model, is the variance of the Gaussian distribution, which represents the step size used to generate the candidate model.
6. A transient electromagnetic adaptive lateral constraint inversion method as claimed in claim 5, characterized in that: The likelihood function is calculated by the following formula: in, is the likelihood function, is the data vector, In order to map the resistivity spatial distribution model parameters into the forward operator of the data, represents the transpose of a matrix, is the error matrix of the data.
7. A transient electromagnetic adaptive lateral constraint inversion method as claimed in claim 6, characterized in that: The acceptance scale factor is calculated by the following formula: in, is the receiving scale factor, As an alternative model, is the prior distribution of the candidate model, is the prior distribution of the current resistivity spatial distribution model, is the likelihood function value of the alternative model, The proposed distribution of the current resistivity spatial distribution model generated by the alternative model, Proposed distributions for generating alternative models from the current resistivity spatial distribution model.
8. A transient electromagnetic adaptive lateral constraint inversion method as claimed in claim 1, characterized in that: Determining whether to update the resistivity spatial distribution model using the acceptance scale factor includes: using the accepted scaling factor, determining whether the candidate model is accepted; If the candidate model accepts the acceptance scale factor, the candidate model enters the sample set of the resistivity spatial distribution model to update the resistivity spatial distribution model and obtain an updated model; If the candidate model rejects the acceptance scale factor, the current resistivity spatial distribution model is maintained, and a new candidate model is continuously generated from the proposed distribution and the current resistivity spatial distribution model.