KR100806207B1 - Multi-frequency ip inversion method - Google Patents

Abstract

A multi-frequency IP(Induced Polarization) inversion method is provided to invert multiple materials at the same time by utilizing a relationship of the materials by extending an inverted algorithm to the maximum. A multi-frequency IP inversion method includes the steps of: defining the distribution of complex electrical resistivity in a space as a model vector, (a) modeling the distribution as complex potential by using the model vector, and defining complex impedance measured by an induced polarization survey as a data vector(S110); and (b) defining an inversion-objected function applying a similarity limitation condition between multiple frequency materials and calculating the distribution of complex electrical resistivity by simultaneously inverting the complex frequency materials with the inversion-objected function(S130). The multi-frequency IP inversion method further includes a step of (c) calculating a frequency characteristic parameter by using the distribution of the complex electrical resistivity after the step (b).

Description

Multi-frequency IP inversion method

1 is a flowchart of a multi-frequency IP inversion method according to an embodiment of the present invention.

FIG. 2 illustrates a numerical model set up to observe characteristics of inversion applied to similarity constraints between multi-frequency data according to an embodiment of the present invention.

Figure 3 (a) is a cross-sectional view showing the electrical resistivity cross-section of the results of the independent inversion of the five IP data of only the noise level, Figure 3 (b) is the amount of model change of all the data according to an embodiment of the present invention The electric resistivity cross section is shown in the inversion result using the objective function that limits the sum of the differences.

Figure 4 (a) shows the phase cross-section of the result of inverting the five IP data that differ only in noise level independently, Figure 5 (b) is the difference in the amount of model change of all the data according to an embodiment of the present invention The phase cross section is shown in the inversion result using the objective function that limits the sum of.

FIG. 5 illustrates a frequency-specific difference between an estimated value and an actual value for an inversion center block according to an embodiment of the present invention.

The present invention relates to an induction polarization (IP) inversion method, and more particularly, to a multi-frequency IP inversion method in a frequency domain using complex electrical resistivity of induction polarization sensing data.

Inductive polarization exploration has been used mainly in the exploration of metal minerals in the past, and the development of probes enables fine IP measurements in the soil layer. The application of is increasing (Vanhala et al., 1992, Weller and Borner, 1996).

Induced polarization is known to occur due to mechanical, electrical and chemical changes inside the medium while current is flowing into the medium, but the mechanism of occurrence thereof is not known. However, through experimental studies on various samples, chemical changes in the medium are considered to be important factors.

Most of the inductive polarization exploration data are interpreted using the chargeability suggested by Seigel (1959) (Oldenburg and Li, 1994), but recently, efforts have been made to define and analyze IP phenomena using complex electrical resistivity. Weller et al., 1996; Kemna, et al, 2000), a study on frequency domain IP inversion analysis using complex electrical resistivity has also been published.

The analysis technique using complex electrical resistivity is an analysis technique in the frequency domain and has the advantage of analyzing the particle size and distribution in the medium and its type using size and phase information. In the IP exploration in the frequency domain, the data for each frequency are independent of each other, so that the inverse analysis is performed separately and then the IP parameters are analyzed through the analysis. In case of inverting a large number of data individually, there is a high possibility that an artificial error of inversion is included, and there is a limitation that it is difficult to use the correlation of each data.

An object of the present invention is to develop a multi-data simultaneous inversion scheme utilizing a plurality of data to maximize the correlation between the numerical modeling and inversion algorithm using complex electrical resistivity in the IP inversion method.

In order to achieve the above object, the multi-frequency IP inversion method of the present invention is an inversion method for calculating a groundwater component gun from induced polarization (IP) sensing data, and (a) models a complex electrical resistivity distribution in space. Defining as a vector, modeling with a complex potential using the model vector, and defining a complex impedance measured by inductive polarization exploration as a data vector; and (b) defining an inversion objective function to which the similarity constraint condition is applied between the multi-frequency data, and calculating a complex electrical resistivity distribution by simultaneously inverting a plurality of frequency data using the inversion objective function.

In addition, after the step (b), (c) may further comprise the step of calculating the frequency characteristic parameter using the complex electrical resistivity distribution.

In the present invention, the inversion is a complex L ₂ between the data measured by inductive polarization exploration and the data modeled by complex potential using the model vector. It is repeated until the size of norm is below the reference value.

The present invention calculates a smoothed complex electrical resistivity distribution by simultaneously inverting a plurality of frequencies by using an inverse objective function applied with a constraint between multiple frequency data, and indirectly calculating frequency characteristic parameters using the same. Use The IP inversion method of the present invention is more stable than the parameter calculation method through the post-processing of the simple inversion result, and has the advantage of flexibility in the selection of a theoretical model for defining the broadband IP characteristics. .

In addition, in the step (a), the complex electrical resistivity distribution in space is defined as a model vector and converted into a complex potential using the model vector to model. Specifically, the electrical resistivity is expressed in the form of a complex number for the spatial frequency of the region to be modeled, the governing equation for the spatial frequency is defined, and the solution of the governing equation for the selected spatial frequency is Fourier transformed into the complex potential in the spatial model. .

Here, the governing equation is the following Equation A calculated through Fourier transform for the direction of assuming that there is no change of the model vector in the direction to be modeled in the region to be modeled. The mixed boundary condition of Equation B is applied as the boundary condition for artificial boundary treatment in numerical modeling.

Equation A

Equation B

는 푸리에 영역에서의 복소전위,

는 공간주파수,

은 경계에 수직인 벡터,

는 복소 전기비저항,

는 경계의 특성을 반영하는 계수임.)(In Equations A and B,

Is the complex potential in the Fourier region,

Is the spatial frequency,

Is a vector perpendicular to the boundary,

Complex electrical resistivity,

Is a coefficient that reflects the nature of the boundary.)

)가 계산된다.The complex potential calculated by Equation A is multiplied by a coefficient according to the geometrical electrode arrangement to obtain a complex impedance (

) Is calculated.

In the step (a), the model vector m * and the data vector d * are defined by the following equation C.

Equation C

는 분할된 역산 블록의 복소 전기비저항,

는 복 소 전위의 복소 임피던스,

은 역산블록의 개수,

는 측정자료의 개수임.)(In Math C above

Is the complex electrical resistivity of the divided inversion block,

Is the complex impedance of the complex potential,

Is the number of inversion blocks,

Is the number of measurement data.)

In addition, in the step (c) of the present invention, the inversion objective function uses a regular equation of Equation D, which is finally obtained by applying a smoothing constraint to a reference model.

[Equation D]

는 자코비안 행렬,

은 평활화 제한 연산자,

는 수치 모델링 연산자,

는 모델 벡터,

는 데이터 벡터,

는 제한의 정도를 조절하는 라그랑지 계수(Lagrangian multiplier)임.)(Equation D above

The Jacobian procession,

Is the smoothing limit operator,

Is a numerical modeling operator,

Model vector,

Is the data vector,

Is the Lagrangian multiplier that controls the degree of restriction.)

를 통해서 모델의 평활화 제한 조건을 가하여 이루어지며, 최종적으로 얻어지는 정규방정식은 수학식 D와 같다.The inversion of the present invention is achieved by minimizing the complex L ₂ norm between the measured data and the data calculated through modeling, and the Lagrangian coefficient.

It is made by applying the smoothing constraint of the model through, and finally the regular equation is obtained as in Equation D.

In addition, in the step (b) of the present invention, it is preferable that the inverse objective function is defined by applying a constraint that minimizes the sum of the differences between the respective model correction vectors Δm * . In this case, the inverse objective function Φ becomes Equation E below applying such a constraint. In Equation E below, the first term and the second term mean a simple sum of the inverse objective functions of each individual frequency data, and the last term is a constraint for minimizing the sum of the difference of model variation amounts of all frequency data.

[Equation E]

는 평활화 제한 연산자,

는 수치 모델링 연산자,

는 모델 벡터,

는 데이터 벡터,

및

는 제한의 정도를 조절하는 라그랑지 곱수임.)(Equation E above

Is the smoothing limit operator,

Is a numerical modeling operator,

Model vector,

Is the data vector,

And

Is the Lagrange product that controls the degree of restriction.)

On the other hand, in addition to applying a condition for limiting the sum of the difference in model variation of all frequency data as shown in Equation E, a constraint for minimizing the sum of the difference between two adjacent model modification vectors ( Δm * ) is applied. It is more preferable to define the inverse objective function. In this case, the inverse objective function is expressed by Equation F below applying such a constraint.

Equation F

는 평활화 제한 연산자,

는 수치 모델링 연산자,

는 모델 벡터,

는 데이터 벡터,

및

는 제한의 정도를 조절하는 라그랑지 곱수임.)(Equation F above

Is the smoothing limit operator,

Is a numerical modeling operator,

Model vector,

Is the data vector,

And

Is the Lagrange product that controls the degree of restriction.)

Hereinafter, a multi-frequency IP inversion method according to an embodiment of the present invention will be described in detail with reference to the accompanying drawings.

1 is a flowchart of a multi-frequency IP inversion method according to an embodiment of the present invention.

As shown in Figure 1, the multi-frequency IP inversion method of the present invention, (a) IP numerical modeling and the definition of the data vector and model vector (S110) and (b) inverse constraints between the multi-frequency data It comprises a simultaneous inversion step (S130) using the objective function.

In addition, after step (b) (S130), (c) may further comprise the step of calculating the frequency characteristic parameter using the complex electrical resistivity distribution calculated through inversion.

First, in step (a), the complex electrical resistivity distribution in space is defined as a model vector, and the complex impedance measured by inductive polarization exploration is defined as a data vector. Express the electrical resistivity (or electrical conductivity) in the form of a complex number for the spatial frequency of the area to be modeled, define the governing equation for the spatial frequency, solve the governing equation for the selected spatial frequency, and then convert it to the complex potential in the spatial To model.

의 시간변화의 주파수 영역에서 수학식 1과 같이 복소수 형태로 표현된다. 실수 성분 및 허수 성분은 모두 주파수에 따라 변한다.If the electrical conductivity of the rock includes polarization,

In the frequency domain of the time change of is expressed in a complex form as shown in equation (1). Real and imaginary components both vary with frequency.

는 각 주파수를

는 복소 상수를 의미한다. 수학식 1에서 실수 성분은 전기전도도와 허수 성분은 분극효과와 연관된다. 수학식 1의 전기전도도 값은 다음의 수학식 2와 같이 크기와 위상값으로 나타낼 수 있다.here,

Each frequency

Denotes a complex constant. In Equation 1, the real component is the electrical conductivity and the imaginary component is associated with the polarization effect. The conductivity value of Equation 1 may be represented by a magnitude and a phase value as shown in Equation 2 below.

In Equation 2, the phase means a phase difference due to polarization between the transmission current and the reception voltage. In order to model IP phenomena, it is necessary to set up an appropriate model that takes into account frequency dependence on conductivity or resistivity. In general, a cole-cole model and a constant phase angle (CPA) model are used.

The governing equation is derived to numerically model the measured data expressed in the form of complex numbers as above. In an embodiment of the present invention, it is assumed that there is no model change in the direction of the direction y in the region to be modeled (two-dimensional model), and through the Fourier transform for the direction of the direction, for the spatial frequency k as shown in Equation 3 below: Obtain a two-dimensional governing equation.

는 푸리에 영역에서의 복소전위,

는 주향 방향에 대한 푸리에 주파수,

는 복소전기비저항을 나타낸다.In Equation 3,

Is the complex potential in the Fourier region,

Is the Fourier frequency,

Denotes the complex electrical resistivity.

In addition, as boundary conditions for artificial boundary surface treatment in numerical modeling, a mixed boundary condition such as Equation 4 below was applied.

은 경계에 수직인 벡터를 의미하며,

는 경계의 형태를 의미하는 상수이다. In Equation 4,

Means a vector perpendicular to the boundary,

Is a constant representing the shape of the boundary.

) 가 계산된다.The differential equation of Equation 3 is solved using the finite element method, and modeling is performed by converting the complex potentials calculated for several selected spatial frequencies k into the complex potentials in space through Fourier transform. The complex potential calculated in this way is multiplied by the distance coefficient according to the geometry of the electrode array.

) Is calculated.

Next, each frequency model vector m * and data vector d * are defined as in Equation 5.

는 분할된 역산 블록의 복소 전기비저항,

는 복소 전위의 복소 임피던스,

은 역산블록의 개수,

는 측정자료의 개수를 나타낸다.In Equation 5,

Is the complex electrical resistivity of the divided inversion block,

Is the complex impedance of the complex potential,

Is the number of inversion blocks,

Indicates the number of measurement data.

As described above, after defining a data vector and a model vector, IP numerical modeling is performed, and then an inversion is performed by defining an inversion objective function.

In the step (b), by applying a similarity constraint condition between the multi-frequency data to define the inversion objective function, using the same to invert a plurality of frequency data at the same time to calculate the complex electrical resistivity distribution (S130).

를 통해서 모델의 평활화 제한 조건을 가하여 이루어진다. 최종적으로 얻어지는 정규방정식은 아래와 같은 수학식 6의 형태를 가지게 된다.Inversion is achieved by minimizing the complex L ₂ norm between measured and modeled data, and the Lagrangian coefficient.

This is done by applying the smoothing constraint of the model. Finally, the regular equation obtained has the form of Equation 6 below.

는 자코비안 행렬,

은 평활화 제한 연산자,

는 수 치 모델링 연산자,

는 모델 벡터,

는 데이터 벡터,

는 제한의 정도를 조절하는 라그랑지 계수(Lagrangian multiplier)이다.In Equation 6,

The Jacobian procession,

Is the smoothing limit operator,

Is a numerical modeling operator,

Model vector,

Is the data vector,

Is the Lagrangian multiplier that controls the degree of restriction.

Inversion is performed repeatedly until the size of the complex L ₂ norm is below a predetermined level. Compared with the conventional electrical resistivity inversion, the process is the same, but there is a difference in that all vectors and matrices are made of complex numbers by using complex electrical resistivity.

In order to simultaneously invert a plurality of frequency data, it is necessary to define a corresponding objective function. However, as shown in the differential equation of Equation 1, each frequency data can be defined independently of the frequency only when a mathematical model such as Cole-cole model is assumed. .

In case of using correlations for multi-frequency data by using a mathematical model, it is easy to apply the constraint between each frequency data, which has the advantage that smoothed variable distribution can be calculated. It is very difficult to change the theoretical model because related variables are directly used for inversion. On the other hand, in the case of using the inverted complex electrical resistivity linear regression method, it is impossible to impose constraints between the multi-frequency data, but the numerical model has the advantage of unlimited.

The present invention attempts an approach that combines the advantages of both techniques. In the inversion process, the complex electrical resistivity is directly used, but the smoothed complex electrical resistivity distribution is calculated by applying a constraint that minimizes the sum of the difference of each model correction vector ( Δm * ) for each other data. A method of indirectly calculating the characteristic parameters was used.

This method has a more stable parameter calculation than a method of calculating a parameter through post-processing of a simple inversed result, and has flexibility in selecting a theoretical model for defining broadband IP characteristics.

In the present invention, a condition for limiting the difference in the amount of change in the inversion iteration was added as a restriction condition for each frequency. This constraint is based on the assumption that the inversion results of each frequency data are similar. In fact, the inversion results for each frequency will not be very different, so it is considered a relatively reasonable constraint.

Equation (7) shows the objective function ( Φ ) of the multi-frequency data inversion applying this constraint. The first and second terms represent the simple sum of the inverse objective function of each individual frequency data, and the last term is a constraint that minimizes the sum of the differences in the model variation of all frequency data.

는 평활화 제한 연산자,

는 수치 모델링 연산자,

는 모델 벡터,

는 데이터 벡터,

및

는 제한의 정도를 조절하는 라그랑 지 계수이다.In Equation 7,

Is the smoothing limit operator,

Is a numerical modeling operator,

Model vector,

Is the data vector,

And

Is the Lagrangian coefficient that controls the degree of restriction.

If Equation 7 is expressed as a determinant, Equation 8 is shown below.

In Equation 8, the main diagonal component is a regular equation of the inversion objective function of each individual frequency data, and the auxiliary diagonal component acts as a constraint for limiting the sum of each individual data.

Equation (7) applies a condition for limiting the sum of the difference in the model variation of all frequency data, but in some cases it may cause a problem that all the frequency data is inverted to the same structure.

In this case, it would be better to impose constraints on two adjacent data rather than all data. If the multi-frequency data are properly listed, the inversion of two adjacent frequency data will not be very large, so it may be a more appropriate choice. In this case, the inversion objective function Φ is defined as in Equation 9 below.

If Equation 9 is expressed as a determinant, Equation 10 is summarized below.

Comparing Equation 10 with Equation 8, it can be seen that the main diagonal components are almost the same but the auxiliary diagonal components exist only in two adjacent frequency data. By applying these constraints, the inversion result of each individual frequency probe can be calculated more stably.

In the present invention, a frequency characteristic parameter is calculated through post-processing using a complex electrical resistivity calculated through multi-frequency inversion where a multi-frequency data is applied with a constraint using frequency-specific correlation. In this case, it is possible to set an arbitrary model since it has flexibility in selecting a theoretical model representing the frequency characteristics of the complex electrical resistivity.

Here, the Cole-Cole model, which is commonly used as an example for extracting frequency characteristic parameters from multi-frequency data, is selected to confirm the applicability of the algorithm.

For the Cole-Cole model, the complex electrical resistivity for each frequency is given by Equation 11 below.

는 DC에서의 전기비저항을 의미하며,

는 충전성(chargeability),

는 시간상수,

는 완화(relaxation)상수로서 분할된 역산블록의 j번째 요소의 값들을 의미한다. 본 발명에서는 다중주파수 역산 알고리듬으로 계산된 각 개별 블록의 값들을 변형된 Levenberg-Marquardt 알고리듬을 이용하여 계산하였다. 이 알고리듬은 역산된 복소 전기비저항과 예측된 전기비저항 값과의 RMS 값을 최소화함으로서 이루어진다. In Equation 11,

Means electrical resistivity at DC,

Chargeability,

Is the time constant,

Denotes the values of the j th element of the inversion block divided as a relaxation constant. In the present invention, the values of each individual block calculated by the multi-frequency inversion algorithm are calculated using the modified Levenberg-Marquardt algorithm. This algorithm is achieved by minimizing the RMS value of the inverted complex resistivity and the predicted resistivity value.

Hereinafter, the numerical experiment results for examining the effectiveness of the IP inversion method of the present invention will be described.

In order to examine the numerical characteristics of the objective function that the variation of each repetition step is limited to the difference, the above objective function was applied with a total of five data sets with different noise levels for the same model.

FIG. 2 illustrates a numerical model set up to observe characteristics of inversion applied to similarity constraints between multi-frequency data according to an embodiment of the present invention.

The model used for the test is a model in which two ideal bands exist. On the one hand, an ideal body having an electrical resistivity equal to a background value of 100 Ohm-m but having a phase of about 30 mrad was assumed. Although it has a phase value of 5 mrad which is a background value, only the electrical resistivity was set to have an outlier value of 200 Ohm-m. In addition, each noise level was set to be a maximum of 5, 30, 10, 2, 2% of the average value of the calculated apparent resistivity and phase. It is assumed that there are 30% and 10% noise-rich data among the data with relatively good noise level.

Figure 3 shows the electrical resistivity cross-section of the results of the inversion by using the objective function of limiting the sum of the difference between the results of the individual inversion and the difference of all the frequency data model variables. Figure 3 (a) is a cross-sectional view showing the electrical resistivity cross-section of the results of the independent inversion of the five IP data of only the noise level, Figure 3 (b) is the amount of model change of all the data according to an embodiment of the present invention The electric resistivity cross section is shown in the inversion result using the objective function that limits the sum of the differences. Noise levels are 5, 30, 10, 2, and 2% from the top, respectively, and the unit is Ohm-m.

In the second image section where the noise level is 30%, the high level noise shows a difference from the model given the shape of the ideal band on the image section, and it is confirmed that the abnormal band which does not exist in the input model appears in the cross section. In the third section with the noise level of 10%, the distortion of the image due to the high noise is confirmed in the inversion result. However, when the objective function limited in the present invention is applied, all of the image distortion is removed and the ideal band appears clearly. have. In particular, it can be seen that the distortion of the image section due to the high noise level is effectively removed.

Figure 4 shows the phase cross section of the inversion experiment results for the same model.

Figure 4 (a) shows the phase cross-section of the result of inverting the five IP data that differ only in noise level independently, Figure 5 (b) is the difference in the amount of model change of all the data according to an embodiment of the present invention The phase cross section is shown in the inversion result using the objective function that limits the sum of. Noise levels are 5, 30, 10, 2, and 2% from the top, respectively, in mrad.

The overall aspect is similar to that of the electrical resistivity image cross section of FIG. 3. In the case of 30% and 10% high-noise data, the distortion in the phase cross-section is shown to be strong.However, the inverse calculation using the objective function limited in this study has effectively removed this distortion in the cross-section. can confirm.

Here, only the results of the constraints using all the data are shown, and the results of the constraints using only the adjacent data are not presented. However, the aspect of the inversion result does not differ much from the case of the limitation on the whole, but there is only a slight difference in the inversion value is closer to the actual value and the boundary is more clearly inverted.

[Table 1] Cole-Cole Variables Calculated in Ideal Center Block

Table 1 shows an example of applying the developed SIP parameter estimation algorithm. For Cole-cole model of Equation 11, DC resistivity is 10 Ohm-m, charging rate is 0.15, time constant is 10, relaxation constant is 0.55. The estimation result of the ideal block among the inversion results for the model included in the 100 Ohm-m, 0 mrad background medium is compared with the actual value. A total of nine frequencies ranging from 0.01 to 100 Hz were used to estimate the variables. Inverse data was performed using the multi-frequency algorithm developed in this study, adding random noise by 5% to all data.

In order to apply the SIP parameter estimation algorithm, it is necessary to determine whether each inverted block has SIP characteristics. The model used in the present invention assumes that only the ideal body has SIP characteristics, and after calculating the phase ratio of the low frequency and high frequency data, the SIP estimation algorithm is applied only when the value is above a certain size. It is inevitable that the resolution of the core is inevitable due to the limitation of the surface exploration, and the estimated values also show a lot of error at the bottom. For the same reason, although the time constant and relaxation constant did not give the SIP characteristics at the bottom, the results were estimated to be outliers. The estimated values are somewhat different from the actual values, but show that they are looking for relatively accurate values. The electrical resistivity was estimated to be higher than actual, and the remaining values were estimated to be lower than actual.

FIG. 5 illustrates a frequency-specific difference between an estimated value and an actual value for an inversion center block according to an embodiment of the present invention. (A) of FIG. 5 compares real components, and (b) compares imaginary components.

The real component shows a little difference from the actual value due to the influence of noise, but it follows the change pattern well. In the imaginary component, the two results agree well despite the influence of noise. Aspect is confirmed. This suggests that the proposed parameter estimation algorithm provides an approximation to the actual value.

In the present invention, in order to invert multiple frequency data simultaneously, we defined a new objective function that limits the total sum of the difference in model variation for each inversion step, and applies the effect to five numerical results with different noise levels in the same model. Confirmed. Assuming that each inverted result is not very different from each other, by limiting the amount of change in the model, the error of interpretation due to noise can be reduced in the inversion analysis of the high noise data contained between the data of good noise level. Could confirm.

A Cole-Cole model parameter estimation algorithm was developed using the modified Levenberg-Marquardt algorithm for multi-frequency parameter estimation. In addition, although only Cole-cole model is assumed to explain SIP characteristics, various numerical models can be set according to the characteristics of the surrounding medium, and thus, more valid results can be obtained in connection with indoor experimental data.

According to the present invention described above, it is possible to provide a reliable IP inversion method capable of inverting multiple data at the same time by maximizing the correlation between a plurality of data by expanding the numerical modeling and inversion algorithm using complex electrical resistivity. have.

Claims (7)

As an inversion method for calculating the groundwater composition of the IP from induced polarization (IP) survey data, (a) defining a complex electrical resistivity distribution in space as a model vector, modeling at a complex potential using the model vector, and defining a complex impedance measured by inductive polarization exploration as a data vector; (b) defining a inversion objective function to which the similarity constraint condition is applied between the multi-frequency data, and calculating a complex electrical resistivity distribution by simultaneously inverting a plurality of frequency data using the inversion objective function; Multi-frequency IP inversion method comprising a. The method according to claim 1, after the step (b), (c) calculating a frequency characteristic parameter using the complex electrical resistivity distribution; The multi-frequency IP inversion method further comprises. The method according to claim 1, wherein the inversion is, Complex L ₂ between the data measured by the induced polarization exploration and the data modeled at the complex potential using the model vector A multi-frequency IP inversion method that is performed repeatedly until the size of norm is below the reference value. The method according to claim 1, wherein in step (b), The inverse objective function is defined by applying a constraint that minimizes the sum of the differences between each model correction vector ( Δm * ). The method of claim 4, wherein the inverse objective function Φ is A multi-frequency IP inversion method characterized by the following Equation applying the above constraints. [Equation E]

(Equation E above

Is the smoothing limit operator,

A numerical modeling operator,

Model vector,

Is the data vector,

And

Is the Lagrange coefficient that controls the degree of restriction.) The method according to claim 1, The inverse objective function is defined by applying a constraint that minimizes the sum of the differences between two adjacent model correction vectors ( Δm * ). The method according to claim 6, wherein the inverse objective function Φ , The multi-frequency IP inversion method characterized in that the following equation to which the constraint is applied. Equation F

(Equation F above

Is the smoothing limit operator,

Is a numerical modeling operator,

Model vector,

Is the data vector,

And

Is the Lagrange product that controls the degree of restriction.)

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