A tunnel surrounding rock parameter inversion method based on displacement pressure double targets

Abstract

The application provides a tunnel surrounding rock parameter inversion method based on displacement pressure double targets, and belongs to the technical field of tunnel engineering, and the method comprises the following steps: determining displacement and pressure optimization targets, determining inversion parameters by using a single factor analysis method, performing Latin hypercube sampling to obtain sample parameter combinations, and constructing learning samples through numerical simulation; selecting a preset proxy model, training and evaluating the learning samples, and obtaining an optimal proxy model; according to the optimal proxy model, constructing an initial population, combining displacement and pressure, optimizing by using a non-dominated sorting genetic algorithm, obtaining an optimal population, obtaining a Pareto optimal solution set, obtaining an engineering preferred solution by using an ideal solution similarity sorting method, and performing forward numerical simulation checking to obtain optimal inversion parameters; and the application solves the problems of single optimization target, lack of parallel displacement and pressure constraints, subjectivity of weights, and large deviation between calculation results and actual measurement in the existing tunnel surrounding rock parameter inversion.

Description

Technical Field

[0001] This invention belongs to the field of tunnel engineering technology, and in particular relates to a method for inverting tunnel surrounding rock parameters based on dual objectives of displacement and pressure. Background Technology

[0002] In underground engineering, the accurate acquisition of rock mechanics parameters is a core prerequisite for design and construction, and directly affects the reliability of numerical analysis and engineering safety. Traditional acquisition methods (indoor tests, field tests, and geophysical surveys, etc.) are often affected by sample disturbance, size effect and site constraints, and are not representative and applicable enough. When they are directly used for numerical simulation, there are often systematic deviations between the calculation results and the field measurements.

[0003] To narrow the gap between simulation and actual measurement, inverse analysis methods based on field monitoring data have received widespread attention. The basic idea is to construct a mapping relationship between parameters and monitoring response on the one hand, and to use optimization algorithms to find the optimal parameter combination that is consistent with the monitoring data within the feasible parameter space on the other hand. It can be seen that the accuracy of the mapping and the effectiveness of the optimization are the two key aspects of inverse analysis.

[0004] In terms of mapping construction, existing technologies mainly include four categories: (1) analytical or semi-analytical solutions: can give closed expressions under idealized boundaries and constitutive assumptions, which are physically clear, but difficult to cover complex geological conditions; (2) physical simulation: can intuitively reproduce key mechanisms, but is limited by scale effects and repeatability; (3) numerical simulation, such as finite element or finite difference: can characterize nonlinearity and heterogeneity, but the cost of a single simulation is high, and the computational burden is heavy in iterative optimization; (4) surrogate model: uses machine learning to learn high-dimensional nonlinear mapping in the "sample-response" space, which significantly reduces the cost of a single prediction while ensuring accuracy, and is suitable for coupling with multi-objective evolutionary optimization to carry out parameter inversion.

[0005] In terms of optimization strategies, early studies mostly focused on displacement monitoring as a single objective, supplemented by intelligent optimization to achieve parameter inversion. For example, BP neural networks were used to replace time-consuming finite element methods and combined with genetic algorithms; support vector machines were combined with improved particle swarm / gray wolf / differential evolution; or hybrid algorithms such as GA-PSO-BP (genetic and particle swarm optimization of BP neural networks) were used for coupled optimization. These works verified the effectiveness of "displacement information + intelligent optimization", but under deep burial and complex geological conditions, the displacement index has strong dispersion, which can easily lead to unstable inversion results.

[0006] To improve robustness, existing technologies are gradually introducing multi-source observation and adopting a multi-objective framework: for example, multi-objectives are transformed into single-objectives through multi-index synthesis, or seepage parameters are inverted by combining seepage information such as head and flow rate; existing technology research shows that the multi-objective framework can make fuller use of monitoring information, but most practices still rely on empirical weighting to integrate objectives, which inevitably introduces subjectivity.

[0007] To avoid subjective weighting, multi-objective evolutionary algorithms provide a natural, weightless paradigm. Among them, the Non-Dominated Sorting Genetic Algorithm (NSGA-II) is widely used in engineering optimization due to its efficient non-dominated sorting and preservation of solution diversity. In geotechnical engineering, existing research has combined NSGA-II with backpropagation neural networks to conduct multi-objective optimization using multi-directional displacement, or proposed improved algorithms to enhance inversion accuracy. Explorations have also emerged of orthogonal design + ANN + GA and improved multi-objective particle swarm optimization for relevant parameter identification. However, current research largely still divides multiple objectives within the single physical domain of "displacement." Work on simultaneously treating displacement and pressure as dual objectives and coupling them with multi-objective algorithms is relatively insufficient. Considering the technical specifications' requirement for simultaneous displacement and pressure monitoring of tunnels with large deformation at level three, joint inversion of multi-source data has a stronger necessity and engineering practicality.

[0008] In summary, existing technologies still have limitations in terms of reliance on a single objective, subjectivity caused by empirical weights, and the trade-off between mapping accuracy and computational efficiency in complex geological conditions. Therefore, there is an urgent need for a technology that can: construct a dual-objective inversion framework by utilizing both displacement and pressure monitoring information in parallel, reduce iteration overhead through surrogate models, and organically couple with unweighted multi-objective evolutionary optimization, thereby improving the robustness, interpretability, and consistency with actual measurements of the inversion. Summary of the Invention

[0009] To address the aforementioned shortcomings in existing technologies, this invention provides a method for inverting tunnel surrounding rock parameters based on dual displacement and pressure objectives. This method solves the problems of existing tunnel surrounding rock parameter inversion methods, such as a single optimization objective, lack of parallel constraints on displacement and pressure, subjective weighting, and significant deviations between calculated and measured results.

[0010] To achieve the above objectives, the technical solution adopted by this invention is: a method for inverting tunnel surrounding rock parameters based on a dual objective of displacement and pressure, comprising the following steps: S1. Construct a three-dimensional numerical model based on geological conditions and excavation methods, determine displacement and pressure targets, use single-factor analysis to determine inversion parameters, and perform Latin hypercube sampling to obtain sample parameter combinations. Call the three-dimensional numerical model to complete the simulation and construct learning samples. S2. Train and evaluate the learning samples in the preset agent model set to obtain the optimal agent model; S3. Based on the optimal surrogate model, an initial population is constructed. Combining the dual objectives of displacement and pressure, a non-dominated sorting genetic algorithm is used for optimization to obtain the optimal population. S4. Obtain the Pareto front solution set from the optimal population, select the Pareto engineering preference solution using the ideal solution similarity ranking method, perform forward numerical simulation verification on the Pareto engineering preference solution, and output the optimal inversion parameters based on the inversion effect evaluation.

[0011] The beneficial effects of this invention are as follows: Addressing the difficulty in obtaining parameters under the complex geological conditions of the Zhaotong tunnel, a multi-objective tunnel surrounding rock parameter inversion method is proposed. This method is based on dual objectives of displacement and pressure, combined with a non-dominated sorting genetic algorithm and an ideal solution similarity ranking method. Sensitivity analysis is used to screen inversion parameters, Latin hypercube sampling and numerical simulation are employed to construct samples, and preset surrogate models are compared to determine the optimal surrogate model. Using displacement and pressure as dual objectives, a Pareto front is generated using a non-dominated sorting genetic algorithm, and the solution is selected using an ideal solution similarity ranking method. This method effectively identifies surrounding rock parameters, and the results are highly consistent with monitoring data, demonstrating good engineering applicability.

[0012] Further, S1 includes the following steps: S101. Construct a three-dimensional numerical model based on geological conditions and excavation methods; S102. Based on the field measurement data, determine the optimization targets for displacement and pressure; S103. Using single-factor analysis, calculate the sensitivity of preset typical mechanical parameters to apparent displacement and surrounding rock pressure, and use the parameter with the highest sensitivity as the inversion parameter based on the average sensitivity. S104. Based on the inversion parameters, Latin hypercube sampling is used to divide each parameter interval into preset equal parts and randomly select a sample point within each equal part to obtain the sample parameter combination. S105. Input the sample parameter combination into the three-dimensional numerical model to calculate the simulated output of apparent displacement and surrounding rock pressure, and integrate it with the corresponding sample parameter combination to form the output dataset, which constitutes the learning sample.

[0013] Furthermore, the expression for calculating the sensitivity of the preset typical mechanical parameters to apparent displacement and surrounding rock pressure is as follows: ; in, Indicates parameters Sensitivity Indicates the first i The baseline value of each input parameter. Indicates the first i The perturbation value of each input parameter. Indicates parameters Below the baseline value of the output parameter. This indicates the amount of change in the output parameters.

[0014] The beneficial effects of the above-mentioned further solutions are as follows: By breaking through the limitations of the past, which only used displacement as a single target, the present invention introduces displacement and pressure monitoring data as dual targets for parameter inversion, which can more comprehensively characterize the stress-deformation characteristics of the tunnel surrounding rock and significantly improve the stability and engineering applicability of the results. Furthermore, through sensitivity analysis, the response parameters most sensitive to displacement and pressure responses were obtained, which are key control factors for tunnel parameter inversion, thus improving the accuracy and efficiency of the inversion.

[0015] Furthermore, S2 includes the following steps: S201. Select a preset number of preset proxy models, and map the sample parameter combination and simulation output in the learning samples to a preset interval to obtain the mapped learning samples. Then train each preset proxy model to obtain the trained preset proxy model. S202. The mapped learning samples are processed using the trained pre-set proxy model to obtain normalized prediction values. S203. Based on five-fold cross-validation, the preset evaluation index of each preset agent model is calculated and compared by normalized predicted values ​​and mapped learning samples to obtain the comprehensive evaluation result of each preset agent model. S204. Based on the comprehensive evaluation results of each preset agent model, the optimal agent model is obtained.

[0016] The beneficial effects of the above-mentioned further solutions are as follows: By selecting a preset number of preset surrogate models and performing optimal surrogate model selection, the present invention improves the accuracy and feasibility of tunnel surrounding rock parameter inversion and reduces prediction costs.

[0017] Furthermore, step S3 includes the following steps: S301. Based on the optimal surrogate model, construct an initial population with the relative root mean square error of displacement and pressure as the fitness function. ; S302. Based on the fast non-dominated sorting, the current population is stratified, and the crowding distance is calculated in each stratum to obtain the comparison criteria of priority of rank and crowding distance order. S303. Use the binary tournament operator to select the parent generation, and perform crossover and mutation on the selected individuals to obtain the offspring. ; S304. Merge parent and child generations. The non-dominated sorting and crowding distance calculations were re-executed, and the next generation population was obtained by truncation using a comparison criterion. ; S305. Determine whether the current iteration has reached the maximum number of generations or meets the preset termination condition. If not, move the next generation population. As a new generation and returned to S302, if so, the non-dominated solution set obtained with relative root mean square error as fitness will be used as the optimal population and output.

[0018] The beneficial effects of the above-mentioned further scheme are as follows: The present invention uses the relative root mean square error as the fitness function and selects the fitness function. It uses a non-dominated sorting genetic algorithm to handle the conflict optimization of displacement and pressure. It does not require preset weights and performs non-dominated sorting on the two objectives, which improves the sorting speed and the accuracy of population optimization. It enables the two heterogeneous physical quantities of displacement and pressure as parallel objectives for inversion, realizing the removal of subjective weights and taking into account deformation and safety.

[0019] Furthermore, S3 also includes a comparative post-evaluation: A unified dual evaluation system of dimensionless assessment and error contribution rate is established. The fitness calibers of root mean square error (RMSE) and relative RMSE are compared, and the relative RMSE is determined as the fitness function accordingly. Specifically: A1. Without affecting evolutionary selection, run a parallel solution with root mean square error as fitness once independently, and map its Pareto front to a unified dimensionless coordinate system according to the mean of the observed dimensions to obtain the dimensionless targets of displacement and pressure. A2. Calculate the contribution rates of displacement and pressure errors to obtain the error contribution rates of root mean square error and relative root mean square error. Combine this with the dimensionless evaluation results to obtain the fitness evaluation results. A3. Based on the fitness evaluation results, the relative root mean square error is used as the fitness function.

[0020] Furthermore, the expression for the dimensionless target after displacement and pressure mapping is as follows: ; ; in, This represents the dimensionless target after displacement mapping. Represents the mean absolute value of the observed displacement. This represents the dimensionless objective after pressure mapping. This represents the average absolute value of the observed pressure.

[0021] Furthermore, the expressions for the contribution rates of displacement and pressure errors are as follows: ; ; in, This represents the contribution rate of displacement error. Indicates the contribution rate of pressure error. Indicates displacement error. This indicates pressure error.

[0022] The beneficial effects of the above-mentioned further scheme are as follows: By introducing a unified dimensionless evaluation and error contribution rate dual evaluation system, and combining displacement and pressure dual objectives, this invention maps the frontier point to dimensionless coordinates and verifies the scale invariance and fairness of the relative root mean square error as a fitness function by calculating the error contribution rate, effectively avoiding the bias caused by dimensional differences.

[0023] Furthermore, step S4 includes the following steps: S401. Perform non-dominated screening and combination on all individuals in the optimal population to obtain the Pareto front solution set; S402. The Pareto front solution set is sorted by the ideal solution similarity ranking method combined with engineering decision preferences, and Pareto engineering preference solutions are selected. S403. Perform forward numerical simulation verification on the Pareto engineering preference solution. If the average relative error of displacement and the average relative error of pressure are both less than the preset threshold, output the optimal inversion parameters; otherwise, return to S3 to continue iterative optimization.

[0024] The beneficial effects of the above-mentioned further scheme are as follows: This invention generates the Pareto front through a non-dominated sorting genetic algorithm, and selects the engineering preference solution by combining the ideal solution similarity sorting method. Through forward numerical simulation, the optimal inversion parameters are obtained, which can flexibly meet different engineering needs, reduce the displacement average relative error and pressure error, and improve the reliability and engineering promotion value of the inversion method. Attached Figure Description

[0025] Figure 1 This is a flowchart of the method of the present invention.

[0026] Figure 2 This is a flowchart of the inverse analysis framework in this embodiment.

[0027] Figure 3 This is a longitudinal section view of the coal seam in the tunnel in this embodiment.

[0028] Figure 4 This is a diagram showing the layout of support and monitoring points in this embodiment.

[0029] Figure 5 This is a graph showing the on-site displacement monitoring data in this embodiment.

[0030] Figure 6 This is a graph showing the on-site pressure monitoring data in this embodiment.

[0031] Figure 7This is a schematic diagram of the three-dimensional numerical model and excavation process in this embodiment.

[0032] Figure 8 This represents the distribution of Latin hypercube sampling in three-dimensional space and the empirical cumulative distribution in this embodiment.

[0033] Figure 9 This is a performance comparison chart of the various proxy models in this embodiment.

[0034] Figure 10 This is a comparison chart of the predicted and actual values ​​of the GPR model, taking D1 and P1 as examples in this embodiment, and a residual distribution chart.

[0035] Figure 11 This is the Pareto front plot in this embodiment.

[0036] Figure 12 This is a comparison chart of fitness curves for different optimization methods in this embodiment.

[0037] Figure 13 This is a comparison chart of the TOPSIS method and the traditional weighted GA in this embodiment.

[0038] Figure 14 This is a graph showing the positive calculation error for different TOPSIS weights in this embodiment. Detailed Implementation

[0039] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0040] Before describing this embodiment, the following terms will be explained: TOPSIS: Ideal Solution Similarity Ranking Method; Pareto solution: Pareto solution; NSGA-II: Non-dominated sorting genetic algorithm-II; RelRMSE: Relative root mean square error; RMSE: Root Mean Square Error; Coefficient of determination; E : Elastic modulus; Poisson's ratio; : Angle of internal friction; coh: cohesive force; tens: tensile strength; ECR: Error Contribution Rate; WGA: Weighted Genetic Algorithm; OAT: Single Factor Analysis; FLAC3D: Three-dimensional finite difference simulation software; The trainbr algorithm is a network training function that optimizes and updates weights and biases based on Levenberg-Marquardt optimization. Squared-exponential kernel: the kernel with squared exponential values; Bagging: Guided aggregation algorithm; RBF kernel: Radial basis function kernel.

[0041] Example In this embodiment, a back-analysis framework oriented towards dual objectives of displacement and pressure is proposed, employing a surrogate model combined with the overall approach of NSGA-II multi-objective optimization: at the objective level, displacement and pressure are treated as parallel optimization objectives; at the mapping level, considering the three-dimensional coupling and iterative overhead of tunnel surrounding rock problems, this invention selects numerical simulation-generated samples and surrogate models to carry the mapping route as the main technical route for mapping construction, reducing iterative computation costs through numerical simulation and surrogate models; at the solution level, a Pareto front with good coverage is obtained through NSGA-II to support engineering trade-offs; at the decision level, TOPSIS is used to rank and select candidate solutions, thereby matching different engineering priorities (such as safety priority, deformation priority, or equilibrium trade-off); to ensure the fairness and verifiability of multi-objective comparisons, error contribution rate (ECR) and unit perturbation test are proposed to demonstrate the scale invariance when using RelRMSE as fitness, and to mitigate the implicit bias introduced by dimensional differences.

[0042] like Figure 1 As shown, this invention provides a method for inverting tunnel surrounding rock parameters based on a dual objective of displacement and pressure, and its implementation method is as follows: S1. Based on geological conditions and excavation methods, construct a three-dimensional numerical model, determine the displacement and pressure targets, use single-factor analysis to determine inversion parameters, and perform Latin hypercube sampling to obtain sample parameter combinations. Call the three-dimensional numerical model to complete the simulation and construct learning samples. The specific steps are as follows: S101. Construct a three-dimensional numerical model based on geological conditions and excavation methods; S102. Based on the field measurement data, determine the optimization targets for displacement and pressure.

[0043] In this embodiment, according to as follows Figure 2The process shown is to study a tunnel with a total length of 16.26 kilometers and a maximum burial depth of about 990 meters. It is a deep-buried extra-long tunnel with steep terrain, large surface undulations, a relative elevation difference of about 1,500 meters, and complex geological conditions. like Figure 3 As shown, in the section from DK377+364 to 485, the tunnel penetrates Carboniferous coal-bearing strata, mainly composed of shale and sandstone interbedded with coal seams. The rock mass has well-developed joints and fissures, making sampling difficult. Figure 2 As shown in the figure. This section also encountered a gas outburst zone, with the gas content of coal seams M1-7 to M1-9 reaching 12.53 m³ / t and a relative pressure of 1.83 MPa, classifying it as a high-gas, complex area. Conventional geophysical exploration and laboratory tests are both limited, making it difficult to directly obtain reliable rock mechanics parameters. Therefore, the key mechanical properties of the surrounding rock and coal seams were identified by relying on field monitoring data and inversion methods. Based on previous hydraulic fracturing tests and in-situ stress inversion analysis, this section is dominated by horizontal tectonic stress, with a maximum horizontal principal stress of 35.9 MPa, a minimum horizontal principal stress of 20.3 MPa, and a vertical principal stress of 28.6 MPa. The direction of the maximum principal stress is basically perpendicular to the tunnel axis. The high in-situ stress pattern poses a severe challenge to the stability of the tunnel.

[0044] In this embodiment, as Figure 4 As shown, to adapt to complex geological conditions, the tunnel cross-section adopts a perfect circle and is equipped with double-layer initial support (thicknesses of 27cm and 23cm respectively, with allowances for deformation of 30cm and 20cm respectively). Given that the excavation cross-sectional area exceeds 200m², the construction adopts the "three-step and invert arch" method: the upper step is 5.5m high, the middle and lower steps are 3.6m each, the invert arch is 3.7m high, and the invert arch lags behind the tunnel face by about 36m. The initial support closely follows the excavation to form a ring, and the secondary lining and invert arch concrete filling are appropriately delayed to ensure the stability of the surrounding rock and reserve space for deformation release. On-site monitoring was conducted, encompassing both displacement and pressure information: displacement monitoring included crown settlement and convergence of the upper, middle, and lower steps; pressure monitoring included stress at the crown, side arch waists, and invert. These two types of monitoring complement each other; the former reflects the overall deformation characteristics of the surrounding rock-structure system, while the latter characterizes the load level borne by the support system. The monitoring points were arranged as follows: Figure 4 As shown; monitoring points were deployed as the tunnel face advanced and continuous monitoring continued until the secondary lining construction was completed, resulting in typical data sequences, such as... Figure 5 and Figure 6 As shown; Strictly adhering to engineering geological conditions and excavation methods, and to minimize boundary effects, a structure with dimensions of [missing information] was constructed. The three-dimensional model contains 173,754 nodes and 1,013,284 elements. Given that the M1-7~9 coal seams and their interbedded gangue layers are relatively thin and exhibit a combined layering characteristic at the arch crown, the three layers of pulverized coal and interbedded gangue are simplified to a single layer. Since the lining is far from the working face (approximately 150m), and the field monitoring data only covers the period before lining construction, the model does not explicitly simulate the lining effect, only considering the stress contribution of the initial and second supports; the anchor bolts are simulated using cable elements; the support effect of the steel arch frame and steel mesh is converted and incorporated into the equivalent parameters of the initial and second supports; the results are as follows: Figure 7 The 3D model diagram and construction sequence diagram shown are provided. The 3D numerical model can be found in [link to model]. Figure 7 Left side; the excavation process strictly followed the actual "three-stage + invert arch" sequence on site, and the construction sequence is shown in [link to construction schedule]. Figure 7 Right side; the material constitutive model adopts the Mohr-Coulomb failure criterion; the normal displacement is constrained by the four boundaries, the bottom is completely fixed, and the top is a free surface.

[0045] S103. Using single-factor analysis, calculate the sensitivity of preset typical mechanical parameters to apparent displacement and surrounding rock pressure, and use the parameter with the highest sensitivity as the inversion parameter based on the average sensitivity.

[0046] In this embodiment, due to the numerous mechanical parameters involved in the surrounding rock and coal seam in this section, considering all parameters simultaneously during the inversion would not only significantly increase computational complexity but also potentially lead to instability in the optimization process. Therefore, it is necessary to conduct sensitivity analysis before the inversion to identify the key parameters that have the most significant impact on displacement and pressure response, in order to ensure the simplicity of the model and the reliability of the inversion results.

[0047] Therefore, this paper employs single-factor analysis (OAT): while keeping other parameters constant, ten preset typical mechanical parameters of the surrounding rock and coal seam are disturbed one by one; by comparing the monitoring responses before and after the disturbance, the sensitivity of each parameter to four apparent displacements and four surrounding rock pressures is calculated, and the average value is taken as the comprehensive evaluation. Considering the dimensional differences of different physical quantities, the sensitivity is defined as the relative rate of change to achieve dimensionless processing, resulting in the following expression for the sensitivity: ; in, Indicates parameters Sensitivity Indicates the baseline value of the input parameter. This represents the perturbation value of the i-th input parameter. Indicates parameters Below the baseline value of the output parameter. Indicates the amount of change in the output parameters; The disturbance range of the parameters is determined based on the railway tunnel design specifications and the hydrogeological conditions of this section, as shown in Table 1. Table 1 is the disturbance range table of the single factor analysis method. Table 1

[0048] Benchmark values ​​for shale and coal seams Input the data into FLAC3D to perform excavation calculations and obtain the baseline response of displacement and pressure. Y Subsequently, each typical mechanical parameter was perturbed according to the amplitude in Table 1, and its average sensitivity was calculated, as shown in Table 2. Table 2 is the average sensitivity table of each typical mechanical parameter. Table 2

[0049] Table 2 shows that the elastic modulus of the surrounding rock is... E Poisson's ratio With internal friction angle It exhibits the highest sensitivity to both displacement and pressure, significantly higher than cohesion and tensile strength; the overall sensitivity of coal seam parameters is relatively low, only... E and This has some impact on the results; therefore, the surrounding rock parameter with the highest sensitivity is selected. E , as well as As inversion parameters, the remaining parameters are fixed at baseline values ​​to reduce computational load and improve optimization efficiency.

[0050] S104. Based on the inversion parameters, Latin hypercube sampling is used to divide each parameter interval into preset equal parts and randomly select a sample point within each equal part to obtain the sample parameter combination.

[0051] In this embodiment, to fully cover the inversion parameter: elastic modulus during model training. E Poisson's ratio and internal friction angle The parameter space is represented by a Latin hypercube sampling method (LHS). LHS divides each parameter interval into several equal parts and randomly selects a sample point within each part, thus achieving uniform coverage of the parameter space. Compared to simple random sampling, LHS achieves higher representativeness with fewer samples, making it particularly suitable for high-dimensional nonlinear problems. This invention generates 100 parameter combinations within a given range of three inversion parameters, serving as input for subsequent numerical simulations. like Figure 8 As shown, the sampling results indicate that the parameter combination is uniformly distributed in three-dimensional space, and its empirical cumulative distribution function (ECDF) is highly consistent with the theoretical uniform distribution, verifying the representativeness and uniformity of the sample; laying the foundation for the next step of numerical simulation forward calculation.

[0052] S105. Input the sample parameter combination into the three-dimensional numerical model to calculate the simulated output of apparent displacement and surrounding rock pressure, and integrate it with the corresponding sample parameter combination to form the output dataset, which constitutes the learning sample.

[0053] In this embodiment, the 100 sets of parameters generated by LHS are combined ( E , as well as The values ​​are substituted into the calculation in sequence. For each calculation, four apparent displacements and four surrounding rock pressures are extracted as outputs, which are denoted as the D series (D1–D4) and the P series (P1–P4) respectively, forming a total of 100×11 input-output dataset (3 input parameters + 8 outputs), which are used for subsequent surrogate model training and multi-objective optimization.

[0054] Numerical simulations yielded 100 sets of displacement and pressure response data, forming a high-quality sample set required for training the surrogate model, thus obtaining the learning samples. The learning samples can effectively characterize the nonlinear mapping between parameters and output. Given the large scale of the 3D mesh model and the high cost of a single computation, Latin hypercube sampling (LHS) was used for sample generation to achieve uniform coverage of the parameter space with a smaller sample size. This approach ensured representativeness while maintaining computational efficiency, resulting in a reasonable distribution of training data and laying a reliable foundation for subsequent surrogate model construction and multi-objective optimization.

[0055] S2. Train and evaluate the learning samples in the preset agent model set to obtain the optimal agent model. The specific steps are as follows: S201. Select a preset number of preset proxy models, and map the sample parameter combination and simulation output in the learning samples to a preset interval to obtain the mapped learning samples. Then train each preset proxy model to obtain the trained preset proxy model. S202. The mapped learning samples are processed using a trained pre-defined proxy model to obtain normalized prediction values.

[0056] In this embodiment, based on the sample data obtained by HLS, four types of surrogate models commonly used in the field of geotechnical engineering are selected for comparison: Backpropagation Neural Network (BPNN), Gaussian Process Regression (GPR), Random Forest (RF), and Support Vector Machine (SVM); predictors are built for each of the eight output variables, resulting in a total of 32 sub-models. In terms of data preprocessing, the normalization factor is first estimated only on the training set, and then the input parameters and output variables are uniformly mapped to a preset value. The interval is used to eliminate dimensional differences and mitigate the impact of inconsistent numerical scales; the specific structure and hyperparameter settings of each surrogate model are shown in Table 3. Except for those listed in the table, the default values ​​are used for all other settings. Table 3

[0057] The mathematical expression for the proxy model is as follows: ; in, This indicates the output predicted by the surrogate model. The functional form representing the surrogate model (specifically: Backpropagation Neural Network (BPNN), Gaussian Process Regression (GPR), Random Forest (RF), and Support Vector Machine (SVM)). Represents the input variable vector. This represents the parameters of the proxy model; each preset proxy model is trained to obtain trained proxy models, and the trained proxy models are used to process the mapped learning samples to obtain normalized prediction values.

[0058] In this embodiment, the proxy models used include Backpropagation Neural Network (BPNN), a multi-layer feedforward neural network that adjusts weights and biases through the backpropagation error algorithm to minimize the difference between the predicted output and the actual target. Its core includes two processes: forward calculation of output and backward error update. Backpropagation Neural Network (BPNN) can approximate complex nonlinearities, but it is sensitive to the sample size and has the risk of local extrema. Gaussian process regression (GPR) is a nonparametric regression method based on a Bayesian framework. It models the output by defining a Gaussian process in the input space (described by the mean function and covariance function / kernel function). It is particularly suitable for small sample data and high-dimensional problems, and can provide prediction uncertainty. Random forest (RF) is a machine learning method based on decision tree ensemble. It achieves regression by constructing multiple decision trees and averaging their predictions. It is insensitive to noise and outliers and is suitable for handling data irregularities in geological surveys. Support Vector Machine (SVM) is a supervised learning algorithm based on kernel methods. In regression problems, it models the problem by maximizing the margin hyperplane and using an insensitive loss function. It relies on the kernel function to handle nonlinear small sample problems, but is sensitive to the kernel and penalty parameters.

[0059] S203. Based on five-fold cross-validation, the preset evaluation index of each preset agent model is calculated and compared by normalized predicted values ​​and mapped learning samples to obtain the comprehensive evaluation result of each preset agent model. S204. Based on the comprehensive evaluation results of each preset agent model, the optimal agent model is obtained.

[0060] In this embodiment, model evaluation employs 5-fold cross-validation repeated 5 times (5×5CV) to improve the stability and robustness of the results. Performance metrics include relative root mean square error (RelRMSE), root mean square error (RMSE), and coefficient of determination (CQD). (where RelRMSE and RMSE should be as small as possible) The larger the better, as shown in the formula below. It should be noted that the surrogate model evaluation only compares the performance of different models. Therefore, normalized data is used to calculate the evaluation index during the cross-validation phase to ensure that different physical quantities can be compared within a unified system. After determining the optimal surrogate model, to ensure the engineering interpretability of the optimization results, the subsequent multi-objective optimization phase recalculates based on the inverse normalized physical scale. Relative root mean square error (RelRMSE), root mean square error (RMSE), and coefficient of determination ( The calculation expression for ) is as follows: ; ; ; in, This represents the relative root mean square error. Indicates the number of samples. Indicates the first i The normalized true value of each sample. Indicates the first i Normalized predicted values ​​for each sample, The mean of the absolute values ​​of the normalized true values. This indicates the prevention of local minima where the denominator is zero. This represents the root mean square error. The coefficient of determination is represented by the coefficient of determination. The coefficient of determination is the mean of the normalized true values. like Figure 9 As shown, the RelRMSE, RMSE, and [other parameters] of the four surrogate models under 5-fold cross-validation are [statistics]. Comparison results; considering the differences in dimensions and scales of different physical quantities, RelRMSE was used as the primary criterion, and RMSE was used in sequence with other criteria when necessary. As an auxiliary criterion, the results in the figure show that GPR is the best overall across all outputs. Its average RelRMSE is approximately 0.070, significantly lower than SVM (0.216), BPNN (0.255), and RF (0.287). In terms of absolute error, GPR's average RMSE ≈ 0.033, also better than SVM (0.090), BPNN (0.112), and RF (0.124). Regarding trend fitting ability, GPR's average... It is higher than SVM (0.827), BPNN (0.708) and RF (0.704); based on RelRMSE as the dominant factor, RMSE and The comprehensive evaluation, aided by this method, determined GPR as the optimal surrogate model for subsequent optimization analysis.

[0061] In this embodiment, the GPR model is used to retrain all samples, resulting in 8 sub-models for subsequent optimization, thus obtaining trained samples. Taking the apparent displacement and surrounding rock pressure parameters D1 and P1 as examples, to intuitively understand the true error magnitude, a comparison chart of predicted and actual values ​​and a residual distribution histogram are plotted based on the inverse normalized data, as shown below. Figure 10 As shown; the predicted value of D1 is highly consistent with the actual value, and the scatter points are closely aligned. Reference line; residuals are nearly zero and symmetrically distributed, with small training / test differences and no significant systematic bias. The RelRMSE obtained in a single evaluation is only 0.002, and the RMSE is 1.05. The value is approximately 1.000, indicating high prediction accuracy and stability. In contrast, the scatter plots of P1 are relatively dispersed, with an error level higher than that of the displacement channel, but the residuals remain approximately symmetrical and concentrated near zero. The RelRMSE of P1 in a single training iteration is 0.017, and the RMSE is 37.11, corresponding to... The value is approximately 0.962, indicating that although the accuracy is not as good as displacement prediction, it accurately reflects the pressure change trend within an acceptable error range. After selecting the optimal surrogate model, GPR is run separately again and saved as the basis for subsequent NSGA-II.

[0062] S3. Based on the optimal surrogate model, construct an initial population. Combining the dual objectives of displacement and pressure, optimize the population using a non-dominated sorting genetic algorithm to obtain the optimal population. The specific steps are as follows: S301. Based on the optimal surrogate model, construct an initial population with the relative root mean square error of displacement and pressure as the fitness function. .

[0063] In this embodiment, based on the optimal surrogate model, a fitness function is constructed using the relative root mean square error of displacement and pressure as the fitness function, and based on a preset scale. Random creation is performed to construct the initial population. .

[0064] In this embodiment, S3 further includes a comparative post-evaluation: Establish a unified dual evaluation system of dimensionless assessment and error contribution rate, compare the fitness caliber of root mean square error and relative root mean square error, and determine the relative root mean square error as the fitness function accordingly. To select a fitness function, we combined displacement and pressure optimization objectives. In conventional optimization, the fitness function is often measured by the root mean square error (RMSE) of the true values ​​to determine prediction bias; the smaller the RMSE, the closer the model is to the observations, and the better its engineering interpretability. However, when optimizing different physical quantities (such as displacement and pressure) simultaneously, relying solely on dimensional absolute errors can introduce comparison bias: objectives with larger magnitudes will dominate the optimization process. For the dual-objective setting (displacement and pressure) in this study, to avoid cross-dimensional unfairness, we compared the use of RMSE and relative root mean square error (RelRMSE). Considering the equal importance of each monitoring point, we weighted the four displacement and four pressure indices equally. like Figure 11 As shown, the Pareto fronts obtained using RMSE and RelRMSE as fitness are presented. Since the two have different dimensions, the original coordinates cannot be directly compared; therefore, a unified dimensionless evaluation framework combining error contribution rate (ECR) is proposed for comparison. Let any front point obtained using RMSE be... Mapping it to dimensionless coordinates, the calculation expression is as follows: ; ; in, This represents the dimensionless target after displacement mapping. Represents the mean absolute value of the observed displacement. This represents the dimensionless objective after pressure mapping. This represents the average absolute value of the observed pressure. After mapping RMSE to RelRMSE space, the comparison of the two Pareto fronts is as follows: Figure 11 As shown in (c), the hypervolume (HV), reverse generation distance (IGD), and [other parameters] are calculated in the same dimensionless space. Indicators; Results showed that the frontiers obtained from the two fitness levels were qualitatively equivalent under a unified scale: the relative difference in HV was <0.2%, and the difference in IGD was similar. Their magnitudes are similar; Furthermore, to summarize the relative contributions of the two types of targets to the overall dimensionless error, the error contribution rate (ECR) is defined, with the denominator being the sum of the eight output errors and the numerator being the sum of the four displacement (or pressure) errors. The expressions for the displacement and pressure error contribution rates are as follows: ; ; in, This represents the contribution rate of displacement error. Indicates the contribution rate of pressure error. Indicates displacement error. The pressure error is represented; since the units of displacement and pressure are different, ECR is only used as a constructive proportion to explain the percentage, and is not summed for physical quantities; and combined with the dimensionless evaluation results, the fitness evaluation results are obtained. In the fitness evaluation results, taking the pressure error contribution rate as an example, the rates were 90.87% and 88.55% under both RMSE and RelRMSE calibers, respectively, indicating that the solution sets obtained by both fitness levels were pressure-dominated. A unit sensitivity test was conducted: after changing the pressure unit from kPa to MPa, the ECR was directly calculated using RMSE, and the pressure proportion plummeted to 0.99%, indicating a low unit sensitivity index. It was 89.88%; while under the RelRMSE caliber... The results verify the scale invariance and comparative fairness of dimensionless error, and support the priority use of RelRMSE as the fitness and interpretation index in multi-objective optimization across physical quantities, and the use of relative root mean square error as the fitness function.

[0065] In this embodiment, since the two physical quantities of displacement and pressure will cause deviation, when selecting the fitness function, NSGA-II is performed once using the relative root mean square error and once using the root mean square error. By comparing the above-mentioned unified dimensionless evaluation and error contribution rate dual evaluation system, the relative root mean square error is finally selected as the fitness function.

[0066] S302. Based on the fast non-dominated sorting, the current population is stratified, and the crowding distance is calculated in each stratum to obtain the comparison criteria of priority of rank and crowding distance order. S303. Use the binary tournament operator to select the parent generation, and perform crossover and mutation on the selected individuals to obtain the offspring. ; S304. Merge parent and child generations. The non-dominated sorting and crowding distance calculations were re-executed, and the next generation population was obtained by truncation using a comparison criterion. ; S305. Determine whether the current iteration has reached the maximum number of generations or meets the preset termination condition. If not, move the next generation population. As a new generation and returned to S302, if so, the non-dominated solution set obtained with relative root mean square error as fitness will be used as the optimal population and output.

[0067] In this embodiment, based on the fitness function, the crossover and mutation operations of the genetic algorithm are used to adjust the initial population. Function: Generates a preset size. descendant population The parent generation and the offspring generation together form a group with a pre-set size of twice the size. new population ; Using the non-dominated sorting method Sort and stratify; calculate crowding distance within the same frontier. And sort them; For population The next generation population is selected using a tournament operator. We prioritize retaining individuals at the front of the non-dominated front and select solutions with better diversity within the front based on the crowding distance to obtain the optimal diversity solution. Repeat the process of generating the next generation population until the next iteration is reached. t Reaching the maximum number of generations n Alternatively, other termination conditions can be met to obtain the optimal population.

[0068] In this embodiment, the performance of the traditional weighted genetic algorithm WGA and NSGA-II is compared. WGA uses five different weighting schemes. Fitness curves for different optimization methods are shown below. Figure 12 As shown, by Figure 12 It can be seen that NSGA-II has the fastest convergence rate for both displacement and stress objective functions, and its convergence value is the second smallest in both directions. In the displacement direction, it is only greater than... The WGA algorithm, in the stress direction, is only greater than The WGA algorithm; it is worth noting that when At that time, the weighted GA basically did not decrease in the displacement direction. This is mainly because the pressure error is larger than the displacement error. After giving the pressure error a larger weight, the displacement error optimization will be "completely" ignored, resulting in the fitness curve in the displacement direction basically not decreasing.

[0069] S4. Obtain the Pareto front solution set from the optimal population, select the Pareto engineering preference solution using the ideal solution similarity ranking method, perform forward numerical simulation verification on the Pareto engineering preference solution, and output the optimal inversion parameters based on the inversion effect evaluation. The specific steps are as follows: S401. Perform non-dominated screening and combination on all individuals in the optimal population to obtain the Pareto front solution set; S402. The Pareto front solution set is sorted by the ideal solution similarity ranking method combined with engineering decision preferences, and Pareto engineering preference solutions are selected. S403. Perform forward numerical simulation verification on the Pareto engineering preference solution. If the average relative error of displacement and the average relative error of pressure are both less than the preset threshold, output the optimal inversion parameters; otherwise, return to S3 to continue iterative optimization.

[0070] In this embodiment, the optimal population is output based on the optimal population. All individuals constitute the Pareto optimal solution set; in parameter inversion, displacement and pressure monitoring data have different physical dimensions, and conflicts often exist during the optimization process: reducing displacement error may increase pressure error; unlike single-objective optimization, the goal of multi-objective optimization is not to find a single solution, but to obtain a set of mutually compromised Pareto optimal solutions, the principle of which is expressed as: ; in, This represents the objective function of the optimization algorithm. This represents the single objective function to be minimized. Indicates the number of multiple objectives.

[0071] In this embodiment, the Ideal Solution Similarity Ranking Method (TOPSIS) is a multi-objective decision analysis method that ranks and selects the optimal solution by calculating the distance between the candidate solution and the ideal solution and the negative ideal solution. For different engineering needs, based on the TOPSIS method and the Pareto front, different combinations of parameters with engineering preferences can be given, such as: (1) Engineering safety is the priority, and the stability of the surrounding rock (usually represented by pressure error) is more critical than the deformation (displacement error), that is, the weight of stress error should be higher than that of displacement error; (2) Deformation control requirements, high-speed railway tunnels have strict limits on deformation, and excessive deformation of the initial support will intrude into the lining design section, that is, the weight of displacement error should be higher than that of stress error; (3) Comprehensive consideration, using the field measured data as a benchmark, comprehensively considering the solution where both displacement and stress errors are stable.

[0072] The TOPSIS method is compared with the WGA weights, and the results are plotted on the Pareto front, as shown below. Figure 13 As shown, under the same weight setting (e.g., 0.5:0.5), there are significant differences in the solutions selected by TOPSIS and WGA: WGA is more inclined to reduce the pressure objective (PressRelRMSE is smaller); the results show that WGA is actually minimizing the weighted sum during the search process. Due to the discreteness of the learning samples, the surrogate model's prediction accuracy for pressure is lower than that for displacement. WGA tends to prioritize reducing pressure error during optimization; while TOPSIS does not change the optimization process, but performs post-processing selection based on the existing Pareto front solution set. Therefore, it is more likely to select a solution where one objective is optimal while the other objective remains acceptable. In summary, NSGA-II combined with TOPSIS is more suitable for engineering applications, as it can comprehensively reflect the overall trade-offs between objectives and provide intuitive and reasonable decision-making references based on Pareto solutions. In contrast, although WGA achieves multi-objective optimization in form, it is more inclined to a weighted sum-based single search and is difficult to fully reflect the balance between objectives. Therefore, in actual engineering inversion and parameter optimization, NSGA-II combined with TOPSIS can better meet the engineering requirements for robustness and interpretability.

[0073] In this embodiment, the optimal solutions with TOPSIS weights of (0.3, 0.7), (0.5, 0.5), and (0.7, 0.3) are selected as representative solutions prioritizing engineering safety, comprehensive consideration, and deformation control. The optimal solutions are shown in Table 4, which is a table of typical TOPSIS weight solutions. These solutions are then substituted into FLAC3D for forward calculation, resulting in a comparison between numerical calculations and on-site measured values ​​under different decision-making conditions. Figure 14 As shown; Table 4

[0074] Depend on Figure 14 It can be seen that the displacement error of the parameters obtained by NSGA-II is significantly smaller than the stress error. This is because the optimal surrogate function GPR of NSGA-II fits the displacement better. When the TOPSIS weight is (0.3, 0.7), the calculated pressure data is basically consistent with the measured pressure data, with an average relative error of only 4.61% and a displacement error of 3.28%. When the TOPSIS weight is (0.5, 0.5), the average relative error of displacement is 2.28% and the relative error of stress is 5.55%. When the TOPSIS weight is (0.7, 0.3), the average relative error of displacement is 1.94% and the relative error of stress is 6.57%. This shows that the NSGA-II displacement-pressure bi-objective optimization method using GPR as a surrogate model is reliable in the back analysis of tunnel surrounding rock parameters and can better balance the errors between displacement and stress. The optimal inversion parameters are obtained, and the method for inverting tunnel surrounding rock parameters based on the dual objectives of displacement and pressure is completed.

Claims

1. A method for inverting tunnel surrounding rock parameters based on a dual-objective displacement-pressure model, characterized in that, Includes the following steps: S1. Construct a three-dimensional numerical model based on geological conditions and excavation methods, determine displacement and pressure targets, use single-factor analysis to determine inversion parameters, and perform Latin hypercube sampling to obtain sample parameter combinations. Call the three-dimensional numerical model to complete the simulation and construct learning samples. S2. Train and evaluate the learning samples in the preset agent model set to obtain the optimal agent model; S3. Based on the optimal surrogate model, an initial population is constructed. Combining the dual objectives of displacement and pressure, a non-dominated sorting genetic algorithm is used for optimization to obtain the optimal population. S4. Obtain the Pareto front solution set from the optimal population, select the Pareto engineering preference solution using the ideal solution similarity ranking method, perform forward numerical simulation verification on the Pareto engineering preference solution, and output the optimal inversion parameters based on the inversion effect evaluation.

2. The tunnel surrounding rock parameter inversion method based on displacement and pressure dual objectives according to claim 1, characterized in that, S1 includes the following steps: S101. Construct a three-dimensional numerical model based on geological conditions and excavation methods; S102. Based on the field measurement data, determine the optimization targets for displacement and pressure; S103. Using single-factor analysis, calculate the sensitivity of preset typical mechanical parameters to apparent displacement and surrounding rock pressure, and use the parameter with the highest sensitivity as the inversion parameter based on the average sensitivity. S104. Based on the inversion parameters, Latin hypercube sampling is used to divide each parameter interval into preset equal parts and randomly select a sample point within each equal part to obtain the sample parameter combination. S105. Input the sample parameter combination into the three-dimensional numerical model to calculate the simulated output of apparent displacement and surrounding rock pressure, and integrate it with the corresponding sample parameter combination to form the output dataset, which constitutes the learning sample.

3. The tunnel surrounding rock parameter inversion method based on displacement and pressure dual objectives according to claim 2, characterized in that, The expression for calculating the sensitivity of the preset typical mechanical parameters to apparent displacement and surrounding rock pressure is as follows: in, Indicates parameters Sensitivity Indicates the first i The baseline value of each input parameter. Indicates the first i The perturbation value of each input parameter. Indicates parameters Below the baseline value of the output parameter. This indicates the amount of change in the output parameters.

4. The tunnel surrounding rock parameter inversion method based on displacement and pressure dual objectives according to claim 1, characterized in that, S2 includes the following steps: S201. Select a preset number of preset proxy models, and map the sample parameter combination and simulation output in the learning samples to a preset interval to obtain the mapped learning samples. Then train each preset proxy model to obtain the trained preset proxy model. S202. The mapped learning samples are processed using the trained pre-set proxy model to obtain normalized prediction values. S203. Based on five-fold cross-validation, the preset evaluation index of each preset agent model is calculated and compared by normalized predicted values ​​and mapped learning samples to obtain the comprehensive evaluation result of each preset agent model. S204. Based on the comprehensive evaluation results of each preset agent model, the optimal agent model is obtained.

5. The tunnel surrounding rock parameter inversion method based on displacement and pressure dual objectives according to claim 1, characterized in that, S3 includes the following steps: S301. Based on the optimal surrogate model, construct an initial population with the relative root mean square error of displacement and pressure as the fitness function. ; S302. Based on the fast non-dominated sorting, the current population is stratified, and the crowding distance is calculated in each stratum to obtain the comparison criteria of priority of rank and crowding distance order. S303. Use the binary tournament operator to select the parent generation, and perform crossover and mutation on the selected individuals to obtain the offspring. ; S304. Merge parent and child generations. The non-dominated sorting and crowding distance calculations were re-executed, and the next generation population was obtained by truncation using a comparison criterion. ; S305. Determine whether the current iteration has reached the maximum number of generations or meets the preset termination condition. If not, move the next generation population. As a new generation and returned to S302, if so, the non-dominated solution set obtained with relative root mean square error as fitness will be used as the optimal population and output.

6. The tunnel surrounding rock parameter inversion method based on displacement and pressure dual objectives according to claim 5, characterized in that, S3 also includes a comparative post-evaluation: A unified dual evaluation system of dimensionless assessment and error contribution rate is established. The fitness calibers of root mean square error (RMSE) and relative RMSE are compared, and the relative RMSE is determined as the fitness function accordingly. Specifically: A1. Without affecting evolutionary selection, run a parallel solution with root mean square error as fitness once independently, and map its Pareto front to a unified dimensionless coordinate system according to the mean of the observed dimensions to obtain the dimensionless targets of displacement and pressure. A2. Calculate the contribution rates of displacement and pressure errors to obtain the error contribution rates of root mean square error and relative root mean square error. Combine this with the dimensionless evaluation results to obtain the fitness evaluation results. A3. Based on the fitness evaluation results, the relative root mean square error is used as the fitness function.

7. The method for inverting tunnel surrounding rock parameters based on dual displacement and pressure objectives according to claim 6, characterized in that, The expression for the dimensionless target after displacement and pressure mapping is as follows: in, This represents the dimensionless target after displacement mapping. Represents the mean absolute value of the observed displacement. This represents the dimensionless objective after pressure mapping. This represents the average absolute value of the observed pressure.

8. The method for inverting tunnel surrounding rock parameters based on dual displacement and pressure objectives according to claim 6, characterized in that, The expressions for the contribution rates of displacement and pressure errors are as follows: in, This represents the contribution rate of displacement error. Indicates the contribution rate of pressure error. Indicates displacement error. The ECR represents the pressure error. Since the units of displacement and pressure are different, the ECR is only used as a constructive proportion to explain the percentage of error and is not used to sum the physical quantities.

9. The method for inverting tunnel surrounding rock parameters based on dual displacement and pressure objectives according to claim 1, characterized in that, S4 includes the following steps: S401. Perform non-dominated screening and combination on all individuals in the optimal population to obtain the Pareto front solution set; S402. The Pareto front solution set is sorted by the ideal solution similarity ranking method combined with engineering decision preferences, and Pareto engineering preference solutions are selected. S403. Perform forward numerical simulation verification on the Pareto engineering preference solution. If the average relative error of displacement and the average relative error of pressure are both less than the preset threshold, output the optimal inversion parameters; otherwise, return to S3 to continue iterative optimization.

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