Frequency domain induced polarization complex resistivity fitting method

Abstract

The invention discloses a frequency domain induced polarization complex resistivity fitting method, and relates to the technical field of geophysical exploration data processing, and the method comprises the specific steps: firstly, collecting and preprocessing frequency domain induced polarization exploration full-band complex resistivity data, and outputting a standardized data set; then, constructing a fully-coupled mathematical model, calculating a theoretical response value, determining parameters and establishing a positive operator; then parameter sensitivity partitioning is carried out, and constraint conditions are set; carrying out inversion fitting by adopting a self-adaptive partition iterative algorithm; and finally, evaluating the quality in two dimensions, and outputting a final complex resistivity fitting result after correction. According to the invention, by constructing the fully-coupled mathematical model, the problem of considering the induced polarization and electromagnetic coupling effect is solved, and the fitting accuracy of the full-band complex resistivity data is improved; a self-adaptive partition iteration inversion fitting algorithm is adopted, the convergence speed is increased, a two-dimensional evaluation system is designed, the geological rationality of fitting parameters is guaranteed, and reliable parameter support is provided for frequency domain induced polarization exploration.

Description

Technical Field

[0001] This invention relates to the field of geophysical exploration data processing technology, specifically a frequency-domain induced electrostatic complex resistivity fitting method. Background Technology

[0002] Frequency-domain induced polarization (IP) exploration is an important technique in geophysical exploration used to detect the electrical characteristics of underground geological bodies. Complex resistivity fitting, as the core component of this technique, directly determines the accuracy of interpreting underground geological structures, mineral resource distribution, and hydrogeological conditions. This technique collects the amplitude and phase data of complex resistivity of underground media at different frequencies, combines them with mathematical models for parameter inversion fitting, and extracts the geological information behind the data. It is now widely used in mineral resource exploration, hydrogeological surveys, engineering geological investigations, and other fields. In actual exploration, the electrical response of underground media includes both induced polarization and electromagnetic coupling effects, and the performance characteristics of these two effects differ significantly at different frequency bands. This places higher demands on the comprehensiveness and accuracy of complex resistivity fitting methods. In order to improve the matching degree between the fitting results and actual geological conditions, the industry urgently needs a complex resistivity fitting method that can simultaneously take into account both effects and adapt to the characteristics of data across the entire frequency band, in order to solve the parameter inversion problem in complex exploration scenarios.

[0003] Traditional frequency-domain induced resistivity fitting methods often construct mathematical models based on a single effect, making it difficult to accurately characterize the frequency-domain response characteristics of both induced polarization and electromagnetic coupling effects simultaneously. This can easily lead to biased fitting results due to incomplete effect characterization. While some methods attempt to consider both effects, they fail to differentiate parameter sensitivity characteristics across different frequency bands, employing a uniform fitting strategy across the entire frequency band. This prevents each parameter from fully utilizing its high-sensitivity frequency range, resulting in insufficient accuracy in parameter inversion. Furthermore, traditional fitting algorithms often lack specificity in initial value selection and fail to incorporate effective constraints based on prior geological information during iteration, leading to deviations from the actual geological range. Fixed iteration strategies also result in slow convergence. In addition, traditional fitting quality evaluation often focuses solely on data matching, without constructing quantitative quality control indicators to identify residual sources or verifying parameter validity with geological information. Consequently, some fitting results, while meeting data fitting requirements, lack practical geological interpretation value. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a frequency-domain induced polarization (IP) complex resistivity fitting method. First, it completes the acquisition and standardization preprocessing of full-band IPI data. Then, it constructs a fully coupled mathematical model integrating IPI and electromagnetic coupling effects and establishes an integrated forward modeling operator. Subsequently, it analyzes parameter sensitivity to divide into sub-bands, sets geological constraints, and constructs a fitting weight matrix. An adaptive partitioning iterative algorithm is used to perform inversion fitting, selectively updating parameters for each sub-band and dynamically adjusting the iteration strategy. Finally, a dual-dimensional quality evaluation is used to identify and correct residual sources. After verifying the geological validity of the parameters, the fitting results are output. This method considers the characterization of both effects, improves fitting accuracy and convergence efficiency, ensures the geological interpretation value of the results, and provides reliable technical support for frequency-domain IPI exploration.

[0005] To solve the above-mentioned technical problems, the present invention provides the following technical solution: a frequency-domain induced polarization complex resistivity fitting method, the specific steps of which are as follows: S1, Full-band induced polarization data acquisition and preprocessing: Acquire full-band complex resistivity data of induced polarization exploration in the frequency domain, obtain the measured values ​​of amplitude and phase at each frequency point, perform standardized preprocessing in combination with construction parameters and geological prior information, and output a standardized complex resistivity dataset. S2, Fully Coupled Forward Response Calculation: Based on the standardized dataset, a fully coupled mathematical model including induced polarization effect and electromagnetic coupling effect is constructed. The theoretical response value of complex resistivity across the entire frequency band is calculated by solving the fully coupled mathematical model, the full set of parameters to be inverted is determined, and an integrated forward modeling operator is established. S3, Parameter sensitivity partitioning and constraint establishment: The sensitivity of each parameter to be inverted in different frequency bands is calculated using the integrated forward modeling operator. Based on the sensitivity distribution characteristics, continuous sub-frequency bands are divided. Combined with geological prior information, the geological constraint intervals of each parameter are set and a fitting weight matrix is ​​constructed. S4, Adaptive partitioned iterative inversion fitting: Based on the standardized dataset, integrated forward modeling operator, fitting weight matrix and geological constraint interval, an adaptive partitioned iterative algorithm is used for inversion fitting. The parameters are iteratively updated in each sub-band. After each iteration, the parameters are checked to see if they meet the geological constraints and the iterative strategy is adaptively adjusted. After convergence, the optimal inversion parameter set and the full-band fitting response are output. S5, Dual-dimensional fitting quality evaluation and result output: The optimal inversion parameter set and the full-band fitting response are evaluated in a dual-dimensional manner. By dividing the sub-bands and constructing quantitative quality control indicators, the source of fitting residuals is identified and targeted corrections are made to the high residual frequency bands. After verifying the geological effectiveness of the parameters, the final complex resistivity fitting result is output.

[0006] Furthermore, the frequency band for acquiring the full-band complex resistivity data of the frequency domain induced polarization exploration covers 0.001Hz to 100kHz, with no less than 4 uniform frequency points per octave, and no less than 4 sets of repeated observations completed at each measuring point; the standardized preprocessing includes invalid frequency point removal, noise suppression, elevation correction, terrain correction, missing data completion, and format unification. Invalid frequency point removal adopts the repeated observation relative error threshold discrimination, noise suppression adopts the 5-point sliding window mid-range smoothing method, elevation correction adopts the uniform half-space geodetic model to complete the resistivity amplitude normalization processing, terrain correction adopts the finite element method to complete the distortion correction processing, and missing data completion adopts the logarithmic linear interpolation method or the Kriging interpolation method.

[0007] Furthermore, the integrated forward response equation of the fully coupled mathematical model is expressed as follows: In the formula, These are the theoretical responses of complex resistivity at different angular frequencies. The set of all parameters to be inverted. For the angular frequencies that correspond one-to-one with the sampling frequency points, The imaginary unit, Zero-frequency resistivity, Polarizability It is a time constant. The frequency correlation coefficient, The equivalent inductance of the ground, It is the equivalent capacitance of the ground.

[0008] Furthermore, the complete set of parameters to be inverted consists of zero-frequency resistivity. Polarizability Time constant Frequency correlation coefficient Four induced polarization characteristic parameters, and the equivalent inductance of the ground. equivalent capacitance of ground It consists of two electromagnetic coupling characteristic parameters; the integrated forward modeling operator is a computational model constructed based on the integrated forward response equation, taking the full set of parameters to be inverted and the angular frequency as input, and the theoretical response value of complex resistivity as output; the integrated forward modeling operator has boundary autonomous conditions: when and When the integrated forward response equation degenerates into the standard Cole-Cole model response equation, the expansion process of the standard Cole-Cole model is as follows: based on the standard Cole-Cole model, the original four parameters characterizing the induced polarization effect—zero-frequency resistivity, polarizability, time constant, and frequency correlation coefficient—and their corresponding polarization response expressions are retained. Two new parameters characterizing the electromagnetic coupling effect—the equivalent inductance of the ground and the equivalent capacitance of the ground—are added. A modulation term characterizing the frequency-varying impedance characteristics of the electromagnetic coupling effect is constructed. The modulation term is coupled with the polarization response expression of the standard Cole-Cole model in a product form to form a fully coupled mathematical model containing both the induced polarization effect and the electromagnetic coupling effect. The expanded model retains the ability of the standard Cole-Cole model to characterize the induced polarization effect and can accurately characterize the frequency domain response of the electromagnetic coupling effect through the added parameters. At the same time, boundary self-consistency conditions are set. When the added equivalent inductance of the ground is 0 and the equivalent capacitance of the ground approaches infinity, the expanded model automatically reverts to the standard Cole-Cole model.

[0009] Furthermore, the sensitivity of each parameter to be inverted in different frequency bands is calculated using the partial derivative method. The partial derivative of the complex resistivity response with respect to each parameter to be inverted is calculated point by point at each frequency, and the magnitude of the partial derivative is used as a quantitative index of parameter sensitivity. The continuous sub-frequency bands are divided into three continuous non-overlapping frequency bands: a low-frequency induced polarization sensitive band, a mid-frequency transition band, and a high-frequency electromagnetic coupling sensitive band. The frequency band division is determined based on the average sensitivity ratio of induced polarization parameters to electromagnetic coupling parameters. The fitting weight matrix is ​​assigned values ​​according to the divided continuous sub-frequency bands. In the low-frequency induced polarization sensitive band, the assigned values ​​for induced polarization parameters range from 0.9 to 1.0, and the assigned values ​​for electromagnetic coupling parameters range from 0 to 0.1. In the high-frequency electromagnetic coupling sensitive band, the assigned values ​​for electromagnetic coupling parameters range from 0.9 to 1.0, and the assigned values ​​for induced polarization parameters range from 0 to 0.1. In the mid-frequency transition band, the parameter values ​​gradually change linearly with the increase of the logarithm of the frequency.

[0010] Furthermore, the specific range of the geologically constrained interval is: zero-frequency resistivity: Polarizability: Time constant: Frequency correlation coefficient: Equivalent inductance of ground: Equivalent capacitance to ground: .

[0011] Furthermore, the frequency-varying decoupling adaptive regularization fitting objective function expression used in the adaptive partitioning iterative algorithm is as follows: In the formula, To fit the objective function value, The set of all parameters to be inverted. For Hadama accumulation, The fitting weight matrix constructed in step S3, This refers to the measured complex resistivity vector corresponding to the standardized complex resistivity dataset output in step S1. The complex resistivity theoretical response vector calculated by the integrated forward modeling operator established in step S2. This is an adaptive damping factor that adjusts in real time according to the fitting residuals. For geologically constrained regularization matrices, The initial values ​​of the parameters to be inverted are: The symbol for L2 norm calculation.

[0012] Furthermore, the initial values ​​of the parameters to be inverted are obtained from the measured data of the high-sensitivity frequency bands of the corresponding parameters. The initial values ​​of the induced polarization parameters are obtained by linear fitting of the measured data of the low-frequency induced polarization sensitive band, and the initial values ​​of the electromagnetic coupling parameters are obtained by analytical solution calculation of the measured data of the high-frequency electromagnetic coupling sensitive band. During the multiple iterations, only the high-value parameters in each sub-frequency band are updated. After each iteration, the parameters that exceed the geological constraint range are corrected by boundary values. The adaptive damping factor is adjusted in real time according to the rate of decrease of the fitting residual. The iteration convergence condition is that the sum of squares of the fitting residuals decreases below the preset accuracy threshold or the number of iterations reaches the preset maximum number of iterations.

[0013] Furthermore, the expression for the two-dimensional residual quality control discriminant operator used in the two-dimensional quality evaluation is as follows: In the formula, The quality control criteria for the k-th sub-band of the full frequency band. The sub-band number is the number assigned to each sub-band within the full frequency band. Let be the quality control weight matrix for the k-th sub-band. For Hadama accumulation, The residual vector for fitting the amplitude of the k-th sub-band. Let be the magnitude of the measured complex resistivity amplitude vector in the k-th sub-frequency band. Let be the phase fitting residual vector of the k-th sub-frequency band. The notation for L1 norm calculation. Noise weighting coefficient, The symbol for the Pearson correlation coefficient is given. The residual vector for fitting the amplitude of the k-th sub-band. The residual vector for the amplitude fitting of the (k-1)th sub-band. This is the symbol for absolute value calculation.

[0014] Furthermore, the full-band sub-bands are divided according to the logarithmic frequency distribution, with no fewer than 5 sub-bands, and the number of frequency points contained in each sub-band does not differ by more than 2. The determination of the source of fitting residuals is based on the Pearson correlation coefficient of the fitting residuals of adjacent frequency points. When the correlation coefficient is less than 0.3, the residuals are determined to be mainly random noise; when the correlation coefficient is greater than or equal to 0.3, the residuals are determined to be mainly system fitting deviation. Targeted corrections include local iterative corrections corresponding to system fitting deviations and sub-band weight adjustments corresponding to random noise. Geological validity verification is completed based on the matching degree between the fitting parameters and geological prior information.

[0015] Compared with existing technologies, this frequency-domain induced complex resistivity fitting method has the following advantages: I. This invention constructs a fully coupled mathematical model that integrates the induced polarization effect and the electromagnetic coupling effect. It reasonably expands the standard model and sets boundary self-consistency conditions, which not only retains the original model's accurate representation of the induced polarization effect, but also achieves an effective characterization of the frequency domain response of the electromagnetic coupling effect. This solves the problem that traditional methods cannot simultaneously take into account both effects. At the same time, it relies on an integrated forward modeling operator to achieve unified forward modeling calculation of all parameters. Combined with parameter sensitivity partitioning to divide characteristic sub-frequency bands, it establishes a fitting weight matrix in a targeted manner, so that the parameter fitting of different frequency bands is more in line with their own response characteristics. This greatly improves the overall integrity and accuracy of fitting the complex resistivity data of the whole frequency band and effectively avoids the parameter deviation problem caused by fitting a single frequency band.

[0016] Second, this invention employs an adaptive partitioned iterative inversion fitting algorithm. Initial values ​​are obtained based on high-sensitivity frequency bands for parameters. During the iteration process, only high-value parameters in each sub-band are updated, and boundary corrections are applied to the parameters in conjunction with geological constraint intervals. Combined with an adaptive damping factor dynamically adjusted with the fitting residuals, the iteration process becomes more targeted and stable, accelerating the fitting convergence speed. Simultaneously, a two-dimensional fitting quality evaluation system is designed, constructing quantitative quality control indicators to accurately identify the sources of fitting residuals. Differentiated correction strategies are adopted for different residual types, and geological validity verification ensures the geological rationality of the fitting parameters. This allows the final output complex resistivity fitting results to possess both data matching and geological interpretability, providing more reliable parameter support for frequency domain induced polarization exploration.

[0017] Other advantages, objectives and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination or study, or may be learned from the practice of the invention. Attached Figure Description

[0018] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort.

[0019] Figure 1 A schematic diagram of the overall process for the frequency domain induced complex resistivity fitting method. Figure 2 Input-output relationship diagram for each step of the frequency domain induced complex resistivity fitting method; Figure 3 The flowchart shows the iterative adjustment strategy for the adaptive damping factor. Detailed Implementation

[0020] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the following detailed description of the specific implementation methods, structures, features, and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided below. Example

[0021] Fitting of frequency-domain induced complex resistivity in metal ore exploration areas.

[0022] This embodiment is applied to an exploration area of ​​magmatic hydrothermal metallic deposits. The terrain in this area is characterized by gently undulating hills and complex stratigraphic lithology. Significant electromagnetic coupling interference exists between underground geological bodies. Therefore, it is necessary to accurately capture the induced polarization and electromagnetic characteristic parameters of underground mineralization bodies through frequency-domain induced polarization complex resistivity fitting. This provides precise geophysical data support for the spatial location of mineralization bodies and the delineation of exploration target areas. The fitting process strictly follows the five steps of this invention, as detailed below. Figure 1 As shown: S1 full-band induced polarization data acquisition and preprocessing: A high-density induced polarization (IP) survey network was deployed across the entire metal exploration area to systematically collect complex resistivity data across the entire frequency band of frequency domain IPI exploration. The collected frequency bands fully covered 0.001Hz to 100kHz, with 6 uniform frequency points set at each octave. Six sets of repeated observations were completed for each measuring point, which greatly improved the richness and reliability of the raw data and accumulated sufficient and high-quality basic data for subsequent fitting analysis. After data acquisition, standardized preprocessing was carried out based on geological prior information such as the construction survey line layout parameters and lithological distribution patterns of the exploration area. This included invalid frequency point removal, noise suppression, elevation correction, topographic correction, missing data completion, and format standardization. Invalid frequencies caused by instrument malfunctions or external electromagnetic interference were eliminated using a repeated observation relative error threshold to ensure data validity. A 5-point sliding window midpoint smoothing method was used to suppress noise, reducing random noise interference from the environment and instruments. Elevation correction of resistivity amplitudes was performed based on a uniform half-space geodetic model to eliminate the influence of topographic elevation differences on resistivity amplitudes. The finite element method was used to correct electrical distortions caused by gentle hilly terrain, restoring the true electrical characteristics of underground geological bodies. A small number of missing data points caused by measurement point obstruction or temporary instrument malfunctions were completed using logarithmic linear interpolation to fill data gaps and ensure data continuity. Finally, a standardized complex resistivity dataset was output after full format standardization, providing a standardized and usable data foundation for subsequent fully coupled forward response calculations.

[0023] S2 fully coupled forward response calculation: Based on the standardized complex resistivity dataset output in step S1, a fully coupled mathematical model encompassing both induced polarization (IP) and electromagnetic coupling effects is constructed. By numerically solving this fully coupled mathematical model, the theoretical response value of complex resistivity across the entire frequency band is accurately calculated, thus achieving a theoretical characterization of the combined effects of IIP and electromagnetic coupling in the underground geological bodies of the exploration area. The set of parameters to be inverted is defined to include four IIP characteristic parameters: zero-frequency resistivity, polarizability, time constant, and frequency correlation coefficient; and two electromagnetic coupling characteristic parameters: equivalent inductance of the earth and equivalent capacitance of the earth. An integrated forward modeling operator is constructed based on the integrated forward modeling response equation. The expression of the integrated forward modeling response equation is: In the formula, These are the theoretical responses of complex resistivity at different angular frequencies. The set of all parameters to be inverted. For the angular frequencies that correspond one-to-one with the sampling frequency points, The imaginary unit, Zero-frequency resistivity, Polarizability It is a time constant. The frequency correlation coefficient, The equivalent inductance of the ground, The operator takes the set of all parameters to be inverted and the angular frequency as input and the theoretical response value of complex resistivity as output, and strictly satisfies the boundary autonomous condition. When the local equivalent inductance is 0 and the earth equivalent capacitance approaches infinity, the integrated forward modeling response equation can automatically degenerate into the standard Cole-Cole model response equation. This achieves a fully coupled and accurate characterization of the induced polarization effect and the electromagnetic coupling effect, while retaining the classical characterization capability of the standard Cole-Cole model for the induced polarization effect. This provides a scientific and efficient computational model support for subsequent parameter sensitivity analysis and inversion fitting.

[0024] S3 parameter sensitivity partitioning and constraint establishment: Using the integrated forward modeling operator established in step S2, the sensitivity of each parameter to be inverted is comprehensively calculated in different frequency bands using the partial derivative method. The partial derivative of the complex resistivity response with respect to each parameter is precisely calculated at each frequency point. The magnitude of the partial derivative is used as a quantitative indicator of parameter sensitivity, achieving precise quantification of the response characteristics of each parameter in different frequency bands. Based on the average sensitivity ratio of induced polarization parameters to electromagnetic coupling parameters, the entire frequency band is precisely divided into three continuous, non-overlapping sub-bands: a low-frequency induced polarization sensitive band, a mid-frequency transition band, and a high-frequency electromagnetic coupling sensitive band. The sensitivity of induced polarization parameters in the low-frequency band is significantly higher than that of electromagnetic coupling parameters, accurately capturing the induced polarization characteristics of underground mineralization. The sensitivity of electromagnetic coupling parameters in the high-frequency band is dominant, effectively characterizing the electromagnetic coupling effect in the exploration area. The mid-frequency band is a smooth transition range between the two types of parameter sensitivity, adapting to the gradual change characteristics of the parameter response. Combined with the prior geological information of the metal exploration area, geological constraint ranges for each parameter are strictly set according to a preset range. The zero-frequency resistivity constraint is 10⁻²Ω・m to 10⁻²Ω・m. 5 Ω・m, polarizability constraint of 0 to 1, time constant constraint of 10⁻³s to 10³s, frequency correlation coefficient constraint of 0.05 to 1.0, and equivalent ground inductance constraint of 10⁻ 7 From H to 10³H, the equivalent earth capacitance constraint is 10⁻¹. 0 From F to 10⁻²F, the parameter value range is limited by geological constraints to avoid parameter values ​​without geological significance during the inversion process. Simultaneously, a fitting weight matrix is ​​constructed. When assigning values ​​to the low-frequency induced polarization (IP) sensitive segment, IPA parameters are assigned 0.95 and electromagnetic coupling parameters are assigned 0.05 to strengthen the fitting weight of IPA parameters in the low-frequency range. When assigning values ​​to the high-frequency electromagnetic coupling sensitive segment, electromagnetic coupling parameters are assigned 0.95 and IPA parameters are assigned 0.05 to highlight the fitting weight of electromagnetic coupling parameters in the high-frequency range. Within the mid-frequency transition segment, parameter values ​​gradually change linearly with the logarithm of the frequency, adapting to the transitional patterns of the two types of parameter sensitivity. This achieves differentiated and precise weight allocation for parameters in different frequency bands, improving the targeting and accuracy of subsequent inversion fitting.

[0025] S4, Adaptive partitioning iterative inversion fitting: Based on the standardized dataset from step S1, the integrated forward modeling operator from step S2, the fitting weight matrix from step S3, and the geological constraint interval, an adaptive partitioning iterative algorithm is used for inversion fitting. During the fitting process, the frequency-varying decoupling adaptive regularized fitting objective function is used as the core calculation basis. The expression for the frequency-varying decoupling adaptive regularized fitting objective function is: In the formula, To fit the objective function value, The set of all parameters to be inverted. For Hadama accumulation, The fitting weight matrix constructed in step S3, This refers to the measured complex resistivity vector corresponding to the standardized complex resistivity dataset output in step S1. The complex resistivity theoretical response vector calculated by the integrated forward modeling operator established in step S2. This is an adaptive damping factor that adjusts in real time according to the fitting residuals. For geologically constrained regularization matrices, The initial values ​​of the parameters to be inverted are: The L2 norm is used to calculate the symbol, ensuring the scientific rigor and accuracy of the fitting process. First, the initial values ​​of the parameters to be inverted are accurately obtained from the measured data of the high-sensitivity frequency bands of each parameter. Then, the initial values ​​of the induced polarization (IP) parameters are obtained through linear fitting of the measured data of the low-frequency IPI sensitive band, ensuring that these initial values ​​match the IPI characteristics of the mineralized bodies in the exploration area. Finally, the initial values ​​of the electromagnetic coupling (EMC) parameters are calculated using analytical solutions from the measured data of the high-frequency EMC sensitive band, ensuring that these initial values ​​match the actual electromagnetic coupling conditions in the exploration area. Reasonable initial values ​​effectively improve the iteration convergence speed. During the iteration process, only high-value parameters within each sub-band are updated. Specifically, the low-frequency induced polarization sensitive band focuses on updating induced polarization parameters, the high-frequency electromagnetic coupling sensitive band focuses on updating electromagnetic coupling parameters, and the mid-frequency transition band adjusts both types of parameters synchronously according to linearly gradual weights. This achieves precise iteration of parameters in different zones. After each iteration, all parameters are comprehensively verified to ensure they meet the geological constraints. Parameters exceeding the constraints are directly corrected by boundary values ​​to ensure that all parameters remain geologically reasonable. Simultaneously, the adaptive damping factor is adjusted in real-time based on the rate of decrease of the fitting residuals, allowing the iteration process to adapt to the changing patterns of the fitting residuals, further improving iteration efficiency and fitting accuracy. In this fitting iteration, the convergence condition is set to the sum of squares of the fitting residuals decreasing below a preset accuracy threshold. After convergence, the optimal inversion parameter set and its corresponding full-band fitting response are output, providing a core object for subsequent fitting quality evaluation. Figure 3 As shown.

[0026] S5, Two-dimensional fitting quality evaluation and output: A two-dimensional quality assessment is performed on the optimal inversion parameter set and the full-band fitted response output in step S4. The two-dimensional residual quality control discriminant operator is used as the quantitative evaluation basis. The expression of the two-dimensional residual quality control discriminant operator is as follows: In the formula, The quality control criteria for the k-th sub-band of the full frequency band. The sub-band number is the number assigned to each sub-band within the full frequency band. Let be the quality control weight matrix for the k-th sub-band. For Hadama accumulation, The residual vector for fitting the amplitude of the k-th sub-band. Let be the magnitude of the measured complex resistivity amplitude vector in the k-th sub-frequency band. Let be the phase fitting residual vector of the k-th sub-frequency band. The notation for L1 norm calculation. Noise weighting coefficient, The symbol for the Pearson correlation coefficient is given. The residual vector for fitting the amplitude of the k-th sub-band. The residual vector for the amplitude fitting of the (k-1)th sub-band. Using absolute value calculations, a comprehensive and accurate quantitative evaluation of the fitting quality is achieved from both amplitude and phase dimensions. First, the entire frequency band is divided into 6 sub-bands according to the logarithmic frequency distribution, ensuring that the number of frequency points within each sub-band differs by no more than 2, making the sub-band division more closely reflect the frequency distribution characteristics. A corresponding quality control weight matrix is ​​matched to each sub-band to achieve differentiated evaluation of the fitting quality of each sub-band. By accurately calculating the amplitude and phase fitting residuals of each sub-band, and combining the Pearson correlation coefficient calculation results, the source of the fitting residuals is accurately identified. If the Pearson correlation coefficient of the fitting residuals of adjacent frequency points is less than 0.3, the residuals are determined to be mainly random noise; if the correlation coefficient is greater than or equal to 0.3, the residuals are determined to be mainly system fitting bias. Accurate identification of the residual source provides a clear direction for subsequent targeted corrections. To address system fitting bias, parameters are optimized using local iterative correction, further reducing system bias and improving fitting accuracy. Random noise is mitigated by adjusting sub-frequency band weights, reducing its interference with the fitting results and making them more closely reflect the true characteristics of underground geological bodies. After residual correction, the geological validity of the optimal inversion parameter set is comprehensively verified by combining prior geological information from the metal exploration area. After confirming a high degree of matching between the parameters and the induced polarization and electromagnetic characteristics of mineralized bodies and stratigraphic lithology within the region, the final complex resistivity fitting results are output. These results include the optimal inversion parameter set, full-band fitting response curves, and fitting evaluation reports for each sub-frequency band. This provides comprehensive, accurate, and reliable geophysical technical support for the spatial positioning of mineralized bodies, delineation of exploration targets, and comprehensive geological interpretation in this metal exploration area.

[0027] In summary, this frequency-domain induced resistivity fitting method has been fully implemented across the entire technical process in magmatic hydrothermal metal exploration scenarios, effectively addressing the issues of strong electromagnetic coupling interference and insufficient accuracy in induced resistivity characteristic parameter inversion in complex geological environments such as gentle hills. The method solidifies the fitting foundation through full-band standardized data processing, achieves simultaneous characterization of induced resistivity and electromagnetic coupling effects using a fully coupled forward model, improves fitting efficiency through partitioned iteration and differentiated weight configuration, and ensures the geological validity of parameters through dual-dimensional quality control and geological constraints. The final fitting results can provide accurate and reliable geophysical support for locating metal mineralization bodies and delineating exploration target areas. Example

[0028] Frequency-domain induced complex resistivity fitting in hydrogeological exploration areas.

[0029] This embodiment is applied to a hydrogeological exploration area of ​​loose rock, where shallow aquifers and clay interlayers are interbedded, resulting in complex and unevenly distributed groundwater. Due to differences in electrical properties, the shallow media exhibit significant electromagnetic coupling effects. Therefore, it is necessary to accurately obtain the induced polarization and electromagnetic parameters of the underground media through frequency-domain induced polarization complex resistivity fitting. This provides a scientific geophysical basis for aquifer stratification, groundwater enrichment evaluation, and borehole layout. The fitting process is strictly implemented according to the method steps of this invention. Specific implementation details are as follows: Figure 2 As shown: S1, Full-band induced polarization data acquisition and preprocessing: In the hydrogeological exploration area, a systematic deployment of induced polarization (IP) exploration points was conducted using the profiling method to collect standardized complex resistivity data across the entire frequency band, covering a range from 0.001 Hz to 100 kHz. To balance exploration efficiency with raw data accuracy, four uniform frequency points were set for each octave band, and four sets of repeated observations were performed for each point, ensuring data reliability while improving overall exploration efficiency. After data collection, standardized preprocessing was carried out based on prior geological information such as the spacing between construction measurement points, the vertical distribution of strata, and the depth of groundwater levels in the exploration area. This included invalid frequency point removal, noise suppression, elevation correction, topographic correction, missing data completion, and format standardization. Invalid frequency points were removed by using the relative error threshold of repeated observations, and abnormal data caused by various interferences were filtered out. A five-point sliding window midpoint smoothing method was used to suppress environmental and instrument noise and reduce noise interference caused by electrical fluctuations in shallow media. Elevation correction of resistivity amplitude was performed based on a uniform half-space geodetic model to eliminate interference from the exploration process. The impact of elevation differences in the gently sloping terrain on resistivity measurement results was investigated. Finite element method was used to correct terrain distortion in the gently sloping terrain of the exploration area, restoring the true electrical response of the shallow aquifer and clay interlayer. Kriging interpolation was used to fill in missing data caused by measurement point obstruction and surface obstacles, leveraging the spatial interpolation advantages of Kriging to ensure spatial continuity of the data. After a complete process of format standardization, a standardized complex resistivity dataset was output, ensuring the data's standardization, continuity, and usability, providing high-quality data support for subsequent fully coupled forward response calculations.

[0030] S2, Fully Coupled Forward Response Calculation: Based on the standardized complex resistivity dataset output in step S1, a fully coupled mathematical model integrating induced polarization (IP) and electromagnetic coupling effects is constructed. This model is solved using specialized numerical methods to accurately calculate the theoretical response values ​​of complex resistivity across the entire frequency band, thus achieving a comprehensive theoretical characterization of the combined effects of IPI and electromagnetic coupling in the shallow subsurface media of the exploration area. The set of parameters to be inverted includes four IPI characteristic parameters: zero-frequency resistivity, polarizability, time constant, and frequency correlation coefficient; and two electromagnetic coupling characteristic parameters: equivalent inductance of the earth and equivalent capacitance of the earth. An integrated forward modeling operator is constructed based on the integrated forward modeling response equation. The expression of the integrated forward modeling response equation is as follows: In the formula, These are the theoretical responses of complex resistivity at different angular frequencies. The set of all parameters to be inverted. For the angular frequencies that correspond one-to-one with the sampling frequency points, The imaginary unit, Zero-frequency resistivity, Polarizability It is a time constant. The frequency correlation coefficient, The equivalent inductance of the ground, The operator takes the set of all parameters to be inverted and the angular frequencies corresponding to the acquisition frequency points as inputs, and outputs the theoretical response value of complex resistivity. It strictly follows the boundary autonomous conditions. When the local equivalent inductance is 0 and the earth equivalent capacitance approaches infinity, the integrated forward modeling response equation can automatically degenerate into the standard Cole-Cole model response equation. This operator not only fully retains the classical characterization ability of the standard Cole-Cole model for induced polarization, but also accurately describes the electromagnetic coupling effect of the shallow medium in the exploration area through the newly added electromagnetic coupling characteristic parameters. It is adapted to the shallow geophysical characteristics of the hydrogeological exploration area and provides an efficient and accurate calculation model for subsequent parameter sensitivity zoning and inversion fitting.

[0031] S3, Parameter sensitivity partitioning and constraint establishment: Using the integrated forward modeling operator constructed in step S2, the sensitivity of each parameter to be inverted is systematically calculated in different frequency bands using the partial derivative method. The partial derivative of the complex resistivity response with respect to each parameter is precisely solved point-by-point, and the magnitude of the partial derivative is used as a quantitative indicator of parameter sensitivity, achieving precise quantification and characterization of the response patterns of each parameter in different frequency bands. Based on the average sensitivity ratio of induced polarization parameters to electromagnetic coupling parameters, the entire frequency band is precisely divided into three continuous, non-overlapping sub-bands: a low-frequency induced polarization sensitive band, a mid-frequency transition band, and a high-frequency electromagnetic coupling sensitive band. This division method is highly adapted to the frequency domain response characteristics of shallow media in the hydrogeological exploration area. The low-frequency band can accurately capture the differences in induced polarization characteristics between shallow aquifers and clay interlayers, the high-frequency band can effectively characterize the electromagnetic coupling effect between shallow media, and the mid-frequency band achieves a smooth transition between the two types of parameter sensitivities. Combined with the prior hydrogeological information of the exploration area, the geological constraint ranges for each parameter are strictly set according to the preset range, where the zero-frequency resistivity is from 10⁻²Ω・m to 10⁻²Ω・m. 5 Ω・m, polarization 0 to 1, time constant 10⁻³s to 10³s, frequency correlation coefficient 0.05 to 1.0, and ground equivalent inductance 10⁻ 7 From H to 10³H, the equivalent capacitance of the ground is 10⁻¹. 0From F to 10⁻²F, the reasonable range of parameter values ​​is effectively limited by the geological constraint interval, avoiding parameter values ​​that do not conform to hydrogeological characteristics during the inversion process and ensuring the geological rationality of the parameters. Simultaneously, a fitting weight matrix is ​​constructed. When assigning values ​​to the low-frequency induced polarization sensitive section, induced polarization parameters are set to 0.9 and electromagnetic coupling parameters to 0.1, strengthening the fitting effect of induced polarization parameters on the induced polarization characteristics of shallow media in the low-frequency range. When assigning values ​​to the high-frequency electromagnetic coupling sensitive section, electromagnetic coupling parameters are set to 0.9 and induced polarization parameters to 0.1, highlighting the fitting effect of electromagnetic coupling parameters on the electromagnetic coupling effect in the high-frequency range. In the mid-frequency transition section, parameter values ​​gradually change linearly with the increase of the logarithm of the frequency, adapting to the transition law of the sensitivity of the two types of parameters, achieving precise weight configuration of parameters in different frequency bands, making the subsequent inversion fitting more consistent with the actual geophysical characteristics of the exploration area.

[0032] S4, Adaptive partitioning iterative inversion fitting: Based on the standardized dataset from step S1, the integrated forward modeling operator from step S2, the fitting weight matrix from step S3, and the geological constraint interval, an adaptive partitioning iterative algorithm is used for inversion fitting. During the fitting process, the frequency-varying decoupling adaptive regularized fitting objective function is used as the core calculation basis. The expression for the frequency-varying decoupling adaptive regularized fitting objective function is: In the formula, To fit the objective function value, The set of all parameters to be inverted. For Hadama accumulation, The fitting weight matrix constructed in step S3, This refers to the measured complex resistivity vector corresponding to the standardized complex resistivity dataset output in step S1. The complex resistivity theoretical response vector calculated by the integrated forward modeling operator established in step S2. This is an adaptive damping factor that adjusts in real time according to the fitting residuals. For geologically constrained regularization matrices, The initial values ​​of the parameters to be inverted are: The L2 norm is used for calculation notation to ensure the scientific rigor and accuracy of the inversion fitting process.

[0033] First, the initial values ​​of the parameters to be inverted are accurately obtained. The initial values ​​of induced polarization (IP) parameters are obtained from measured data in the low-frequency IPI sensitive section through linear fitting, ensuring that the initial values ​​closely match the differences in IPI characteristics between the shallow aquifer and clay interlayer. The initial values ​​of electromagnetic coupling (EMC) parameters are calculated from measured data in the high-frequency EMC sensitive section through analytical solutions, ensuring that the initial values ​​match the actual electromagnetic coupling situation in the shallow layer of the exploration area. Reasonable initial value settings effectively shorten the iteration convergence time and improve the efficiency of the fitting process. During the iteration process, only high-value parameters in each sub-frequency band are updated. The low-frequency band focuses on updating IPI parameters to accurately characterize the IPI characteristics of the shallow medium, while the high-frequency band focuses on updating EMC parameters to effectively characterize the electromagnetic coupling effect in the shallow layer. The mid-frequency band adjusts the two types of parameters according to a linearly gradual weighting to adapt to the transition law of parameter sensitivity, achieving precise iterative updates of parameters in different zones. After each iteration, all parameters are verified to ensure they are within the geological constraints. Parameters outside these constraints are immediately corrected for boundary values ​​to guarantee that they consistently conform to the geological characteristics of the hydrogeological exploration area. Simultaneously, the adaptive damping factor is adjusted in real-time based on the rate of decrease in the fitting residuals, allowing the damping factor to adapt to the residual changes during the fitting process, further improving the accuracy and efficiency of the iterative fitting. The iteration convergence condition is set to either reaching a preset maximum number of iterations or the sum of squared fitting residuals falling below a preset accuracy threshold. Upon meeting either convergence condition, the optimal inversion parameter set and its corresponding full-band fitting response are immediately output, providing the core analytical object for subsequent two-dimensional quality evaluation.

[0034] S5, Two-dimensional fitting quality evaluation and output: A two-dimensional quality evaluation is performed on the optimal inversion parameter set and the full-band fitting response obtained in step S4. The two-dimensional residual quality control discriminant operator is used as the quantitative evaluation standard to achieve a comprehensive and accurate quantitative assessment of the fitting quality from both amplitude and phase dimensions. The expression for the two-dimensional residual quality control discriminant operator is as follows: In the formula, The quality control criteria for the k-th sub-band of the full frequency band. The sub-band number is the number assigned to each sub-band within the full frequency band. Let be the quality control weight matrix for the k-th sub-band. For Hadama accumulation, The residual vector for fitting the amplitude of the k-th sub-band. Let be the magnitude of the measured complex resistivity amplitude vector in the k-th sub-frequency band. Let be the phase fitting residual vector of the k-th sub-frequency band. The notation for L1 norm calculation. Noise weighting coefficient, The symbol for the Pearson correlation coefficient is given. The residual vector for fitting the amplitude of the k-th sub-band. The residual vector for the amplitude fitting of the (k-1)th sub-band. The absolute value calculation sign ensures the reliability of the fitting results. The entire frequency band is divided into 5 sub-bands according to the logarithmic distribution of frequencies, ensuring that the difference in the number of frequency points within each sub-band does not exceed 2. This sub-band division closely reflects the natural distribution characteristics of frequencies. A corresponding quality control weight matrix is ​​assigned to each sub-band to achieve differentiated and accurate evaluation of the fitting quality of different sub-bands. The amplitude and phase fitting residual vectors of each sub-band are accurately calculated. The Pearson correlation coefficient of the fitting residuals between adjacent frequency points is used to identify the source of the residuals. A correlation coefficient less than 0.3 indicates random noise as the dominant factor, while a correlation coefficient greater than or equal to 0.3 indicates system fitting bias as the dominant factor. Accurate identification of the residual source provides a clear direction for subsequent targeted corrections, avoiding blind corrections that could negatively impact the fitting results. Local iterative corrections were implemented to address system fitting biases, and relevant parameters were refined to further reduce system biases and improve the fit between the fitting results and the true characteristics of the shallow subsurface medium. Sub-band weight adjustments were made to address random noise, reducing its interference with the fitting results and ensuring that the fitted parameters more accurately reflect the geophysical characteristics of the shallow aquifer and clay interlayers. After targeted residual correction, the geological validity of the optimal inversion parameter set was comprehensively verified by combining prior hydrogeological information from the exploration area. After confirming a high degree of matching between the parameters and the electrical, induced polarization, and electromagnetic characteristics of the shallow aquifer and clay interlayer, the final complex resistivity fitting results were output. These results include the optimal inversion parameter set, full-band fitting response maps, fitting quality evaluation results for each sub-band, and geological interpretations of the parameters. This provides scientific, accurate, and comprehensive geophysical technical support for aquifer stratification, groundwater richness assessment, precise borehole layout, and groundwater development and utilization planning in this hydrogeological exploration area.

[0035] In summary, this frequency-domain induced resistivity fitting method, implemented throughout the entire process, effectively adapts to the complex geological conditions of shallow aquifers and clay interlayers in hydrogeological exploration scenarios involving loose rocks. It resolves the problem of induced resistivity parameter fitting distortion caused by electromagnetic coupling effects resulting from differences in the electrical properties of shallow media. The method ensures data continuity through standardized preprocessing, accurately characterizes the frequency domain response features of the medium using an integrated forward modeling operator, improves fitting accuracy through adaptive iterative algorithms, and guarantees parameter matching through precise residual correction and geological validity verification. The final results provide comprehensive scientific technical support for aquifer delineation, water-bearing capacity evaluation, and groundwater exploration.

[0036] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A frequency-domain induced complex resistivity fitting method, characterized in that, The specific steps of this method are as follows: S1, Full-band induced polarization data acquisition and preprocessing: Acquire full-band complex resistivity data of induced polarization exploration in the frequency domain, obtain the measured values ​​of amplitude and phase at each frequency point, perform standardized preprocessing in combination with construction parameters and geological prior information, and output a standardized complex resistivity dataset. S2, Fully Coupled Forward Response Calculation: Based on the standardized dataset, a fully coupled mathematical model including induced polarization effect and electromagnetic coupling effect is constructed. The theoretical response value of complex resistivity across the entire frequency band is calculated by solving the fully coupled mathematical model, the full set of parameters to be inverted is determined, and an integrated forward modeling operator is established. S3, Parameter sensitivity partitioning and constraint establishment: The sensitivity of each parameter to be inverted in different frequency bands is calculated using the integrated forward modeling operator. Based on the sensitivity distribution characteristics, continuous sub-frequency bands are divided. Combined with geological prior information, the geological constraint intervals of each parameter are set and a fitting weight matrix is ​​constructed. S4, Adaptive partitioned iterative inversion fitting: Based on the standardized dataset, integrated forward modeling operator, fitting weight matrix and geological constraint interval, an adaptive partitioned iterative algorithm is used for inversion fitting. The parameters are iteratively updated in each sub-band. After each iteration, the parameters are checked to see if they meet the geological constraints and the iterative strategy is adaptively adjusted. After convergence, the optimal inversion parameter set and the full-band fitting response are output. S5, Dual-dimensional fitting quality evaluation and result output: The optimal inversion parameter set and the full-band fitting response are evaluated in a dual-dimensional manner. By dividing the sub-bands and constructing quantitative quality control indicators, the source of fitting residuals is identified and targeted corrections are made to the high residual frequency bands. After verifying the geological effectiveness of the parameters, the final complex resistivity fitting result is output.

2. The frequency-domain induced complex resistivity fitting method according to claim 1, characterized in that, In step S1, the frequency band of the full-band complex resistivity data acquisition for frequency domain induced polarization exploration covers 0.001Hz to 100kHz, with no less than 4 uniform frequency points set for each octave, and no less than 4 sets of repeated observations completed for each measuring point; the standardized preprocessing includes invalid frequency point removal, noise suppression, elevation correction, terrain correction, missing data completion, and format unification processing. Invalid frequency point removal adopts the repeated observation relative error threshold discrimination, noise suppression adopts the 5-point sliding window mid-range smoothing method, elevation correction adopts the uniform half-space geodetic model to complete the resistivity amplitude normalization processing, terrain correction adopts the finite element method to complete the distortion correction processing, and missing data completion adopts the logarithmic linear interpolation method or the Kriging interpolation method.

3. The frequency-domain induced complex resistivity fitting method according to claim 1, characterized in that, In step S2, the integrated forward response equation of the fully coupled mathematical model is expressed as follows: In the formula, These are the theoretical responses of complex resistivity at different angular frequencies. The set of all parameters to be inverted. For the angular frequencies that correspond one-to-one with the sampling frequency points, The imaginary unit, Zero-frequency resistivity, Polarizability It is a time constant. The frequency correlation coefficient, The equivalent inductance of the ground, It is the equivalent capacitance of the ground.

4. The frequency-domain induced complex resistivity fitting method according to claim 3, characterized in that, In step S2, the complete parameter set to be inverted consists of zero-frequency resistivity. Polarizability Time constant Frequency correlation coefficient Four induced polarization characteristic parameters, and the equivalent inductance of the ground. equivalent capacitance of ground It consists of two electromagnetic coupling characteristic parameters; the integrated forward modeling operator is a computational model constructed based on the integrated forward response equation, taking the full set of parameters to be inverted and the angular frequency as input, and the theoretical response value of complex resistivity as output; the integrated forward modeling operator has boundary autonomous conditions: when and At that time, the integrated forward response equation degenerates into the standard Cole-Cole model response equation.

5. The frequency-domain induced complex resistivity fitting method according to claim 1, characterized in that, In step S3, the sensitivity of each parameter to be inverted in different frequency bands is calculated using the partial derivative method. The partial derivative of the complex resistivity response with respect to each parameter to be inverted is calculated point by point at each frequency, and the magnitude of the partial derivative is used as a quantitative index of parameter sensitivity. The continuous sub-frequency bands are divided into three continuous non-overlapping frequency bands: a low-frequency induced polarization sensitive band, a mid-frequency transition band, and a high-frequency electromagnetic coupling sensitive band. The frequency band division is determined based on the average sensitivity ratio of induced polarization parameters to electromagnetic coupling parameters. The fitting weight matrix is ​​assigned values ​​according to the divided continuous sub-frequency bands. In the low-frequency induced polarization sensitive band, the assigned values ​​of induced polarization parameters range from 0.9 to 1.0, and the assigned values ​​of electromagnetic coupling parameters range from 0 to 0.

1. In the high-frequency electromagnetic coupling sensitive band, the assigned values ​​of electromagnetic coupling parameters range from 0.9 to 1.0, and the assigned values ​​of induced polarization parameters range from 0 to 0.

1. In the mid-frequency transition band, the parameter values ​​gradually change linearly with the increase of the logarithm of the frequency.

6. The frequency-domain induced complex resistivity fitting method according to claim 1, characterized in that, In step S3, the specific range of the geological constraint interval is: zero-frequency resistivity: Polarizability: Time constant: Frequency correlation coefficient: Equivalent inductance of ground: Equivalent capacitance to ground: .

7. The frequency-domain induced complex resistivity fitting method according to claim 1, characterized in that, In step S4, the frequency-varying decoupling adaptive regularization fitting objective function expression used in the adaptive partitioning iterative algorithm is: In the formula, To fit the objective function value, The set of all parameters to be inverted. For Hadama accumulation, The fitting weight matrix constructed in step S3, This refers to the measured complex resistivity vector corresponding to the standardized complex resistivity dataset output in step S1. The complex resistivity theoretical response vector calculated by the integrated forward modeling operator established in step S2. This is an adaptive damping factor that adjusts in real time according to the fitting residuals. For geologically constrained regularization matrices, The initial values ​​of the parameters to be inverted are: The symbol for L2 norm calculation.

8. The frequency-domain induced complex resistivity fitting method according to claim 7, characterized in that, In step S4, the initial values ​​of the parameters to be inverted are obtained from the measured data of the high-sensitivity frequency bands of the corresponding parameters. The initial values ​​of the induced polarization parameters are obtained by linear fitting of the measured data of the low-frequency induced polarization sensitive band, and the initial values ​​of the electromagnetic coupling parameters are obtained by analytical solution calculation of the measured data of the high-frequency electromagnetic coupling sensitive band. During the multiple iterations, only the high-value parameters in each sub-frequency band are updated. After each iteration, the parameters that exceed the geological constraint range are corrected by boundary values. The adaptive damping factor is adjusted in real time according to the rate of decrease of the fitting residual. The iteration convergence condition is that the sum of squares of the fitting residuals decreases below the preset accuracy threshold or the number of iterations reaches the preset maximum number of iterations.

9. The frequency-domain induced complex resistivity fitting method according to claim 1, characterized in that, In step S5, the expression for the two-dimensional residual quality control discriminant operator used in the two-dimensional quality evaluation is as follows: In the formula, The quality control criteria for the k-th sub-band of the full frequency band. The sub-band number is the number assigned to each sub-band within the full frequency band. Let be the quality control weight matrix for the k-th sub-band. For Hadama accumulation, The residual vector for fitting the amplitude of the k-th sub-band. Let be the magnitude of the measured complex resistivity amplitude vector in the k-th sub-frequency band. Let be the phase fitting residual vector of the k-th sub-frequency band. The notation for L1 norm calculation. Noise weighting coefficient, The symbol for the Pearson correlation coefficient is given. The residual vector for fitting the amplitude of the k-th sub-band. The residual vector for the amplitude fitting of the (k-1)th sub-band. This is the symbol for absolute value calculation.

10. The frequency-domain induced complex resistivity fitting method according to claim 1, characterized in that, In step S5, the full-band sub-bands are divided according to the logarithmic frequency distribution law, with no less than 5 sub-bands and the number of frequency points contained in each sub-band not exceeding 2. The source of fitting residual is determined based on the Pearson correlation coefficient of the fitting residuals of adjacent frequency points. When the correlation coefficient is less than 0.3, the residual is determined to be mainly random noise. When the correlation coefficient is greater than or equal to 0.3, the residual is determined to be mainly system fitting deviation. Targeted correction includes local iterative correction corresponding to system fitting deviation and sub-band weight adjustment corresponding to random noise. Geological validity verification is based on the degree of matching between the fitted parameters and prior geological information.

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