Fractional order polarization equivalent ring device with any frequency dispersion coefficient and design method of fractional order polarization equivalent ring device

Abstract

The invention relates to the field of electromagnetic exploration, in particular to a fractional-order polarization equivalent ring device with any frequency dispersion coefficient and a design method thereof, the fractional-order polarization equivalent ring device comprises a mutual inductance coil and a fractional-order polarization equivalent circuit connected with the mutual inductance coil in parallel, and the fractional-order polarization equivalent circuit comprises an induction branch and a plurality of polarization branches which are connected in parallel; the induction branch comprises at least one induction resistor, and the polarization branch comprises at least one polarization resistor and at least one polarization inductor which are connected in series. And calculating mutual inductance coil parameters and inductive resistance values according to geometrical parameters of the polarization body, performing frequency domain fitting on the fractional order polarization effect of any frequency dispersion coefficient based on a rational function approximation algorithm, and calculating resistance values and inductance values in a polarization branch according to a fitted frequency domain polynomial. The number of the polarization branches is equal to the approximation order of the rational function approximation algorithm. The equivalent circuit quantitative characterization of the fractional order polarization effect of any frequency dispersion coefficient can be realized, and the response rule of the polarization medium under the excitation of the alternating magnetic field is accurately reflected through the equivalent ring device.

Description

Technical Field

[0001] The present invention relates to the field of geophysical exploration, and more specifically, to a fractional-order polarization equivalent loop device with an arbitrary dispersion coefficient and a design method thereof. Background Art

[0002] In the field of geophysical exploration, detecting deep metal ores is the main future development trend. When using electromagnetic detection methods, the external alternating electromagnetic field generates induction effects and polarization effects in underground polymetallic ore bodies. Compared with induction effects and resistivity information, the polarization effect of polymetallic ores is more sensitive.

[0003] CN102096113A discloses a time-domain ground-air electromagnetic detection system and a calibration method, which uses a closed abnormal loop induction signal recording device to realize the test and calibration of the detection system. However, the abnormal loop structure used in this method is simple, equivalent to a series connection of a resistor and an inductor, and can only record induction current and induction voltage and simulate induction effects. In fact, valuable target bodies (such as metal ores, etc.) are also very sensitive to polarization effects. Simply studying induction effects can no longer meet the requirements of geophysical exploration and ignores the complex characteristics of multiphase polarized media.

[0004] CN109885865A discloses a method for calculating equivalent electrical parameters of layered media based on the iterative Debye model, which calculates the equivalent electrical parameters of horizontally layered structure materials based on the Debye model equivalent circuit. The Debye model used in this method is an integer-order model of polarization effect, and the corresponding dispersion coefficient is 1, while the dispersion coefficient range of the actual fractional-order polarization effect is greater than 0 and less than 1. The Debye model and its equivalent circuit cannot characterize the fractional-order characteristics of polarization effects.

[0005] CN113504571B discloses a polarization equivalent loop device and a design method for multiphase conductive media, and designs a polarization equivalent loop device composed of a parallel connection of an RC circuit and a coil. This method uses the GEMTIP model to deduce the RC circuit parameters, only considers three parameters of the conductivity, polarizability, and time constant of the polarization medium, and sets the dispersion coefficient parameter characterizing the fractional-order characteristics of the polarization effect to 1. Therefore, it is still an integer-order model of the polarization effect. The dispersion coefficient range of the actual fractional-order polarization effect is greater than 0 and less than 1.

[0006] It is a consensus in the field of geophysical exploration that the polarization effect has fractional-order characteristics, and the dispersion coefficient represents the fractional-order characteristics of the polarized medium. At present, the research on the equivalent circuit model of the polarization effect only stays at the integer order. How to characterize the fractional-order characteristics of the polarization effect is an urgent problem to be solved. Therefore, it is necessary to propose new devices and methods to achieve quantitative simulation of the fractional-order polarization effect, to reflect the law of the induction-polarization effect of the underground medium under magnetic field excitation, and to combine numerical simulation and experimental verification. Summary of the Invention

[0007] The purpose of the present invention is to provide a fractional-order polarization equivalent loop device and a design method with arbitrary dispersion coefficients in view of the deficiencies and problems of the above-mentioned existing technologies.

[0008] The present invention is implemented as follows.

[0009] The first aspect of the present invention provides a fractional-order polarization equivalent loop device with arbitrary dispersion coefficients, including a mutual inductance coil and a fractional-order polarization equivalent circuit connected in parallel with the mutual inductance coil, wherein the fractional-order polarization equivalent circuit is composed of 1 induction branch and multiple polarization branches connected in parallel; the induction branch includes at least one induction resistor, and the polarization branch includes at least one polarization resistor and at least one polarization inductor connected in series.

[0010] Furthermore, the mutual inductance coil receives the excitation of an external time-varying magnetic field and generates an induced electromotive force, generating currents in the induction branch and the polarization branch.

[0011] The second aspect of the present invention provides a design method for a fractional-order polarization equivalent loop device with arbitrary dispersion coefficients, and the method includes:

[0012] S1 Calculate the inductance L of the mutual inductance coil according to the side length D and thickness r of the underground polarized medium, where the magnetic permeability μ 0 = 4π×10 -7 H / m;

[0013]

[0014] S2 Calculate the induction resistor of the induction branch in the equivalent loop device of the fractional-order polarization effect according to the zero-frequency conductivity σ 0 of the polarized medium and the earth geometric parameter β

[0015] S3 Based on the rational function approximation algorithm, approximate and fit the complex conductivity σ(ω) of the polarized medium in the frequency domain to obtain the frequency-domain fitting polynomial of the fractional-order polarization effect with arbitrary dispersion coefficient c;

[0016] S4 Calculate the polarization resistor R of the i-th polarization branch in the equivalent loop device of the fractional-order polarization effect respectively according to the fitting coefficients of the frequency-domain fitting polynomiall and the polarization inductor L i , the number of polarization branches is equal to the fitting order of the rational function approximation algorithm;

[0017] S5 connects the induction-polarization circuit formed by the parallel connection of the induction branch and the polarization branch with the mutual inductance coil in parallel, which is the equivalent loop device for the fractional-order polarization effect with any dispersion coefficient.

[0018] Furthermore, the approximation range of the rational function approximation algorithm in S3 is 1 Hz - 10 6 Hz, the approximation order is n, calculate the minimum value of the rational approximation error between the rational approximation function and the complex conductivity σ(ω) of the polarization medium, and realize the solution of the coefficients of the rational approximation function;

[0019] Perform partial fraction expansion on the rational approximation function to obtain the frequency-domain fitting polynomial.

[0020] Furthermore, S3 specifically includes:

[0021] The rational approximation function is:

[0022]

[0023] where j is the imaginary unit, ω is the angular frequency, d r (ω) and b w (ω) are coefficients to be solved, r = 1, 2,..., m; w = 1, 2,..., n, m = n or n + 1; γ(ω), θ(ω), and ψ(ω) are real functions of the rational approximation function respectively;

[0024] Decompose the complex conductivity σ(ω) of the polarization medium into the real part characteristic Reσ(ω) and the imaginary part characteristic Imσ(ω), that is, σ(ω) = Reσ(ω) + j·Imσ(ω), then the rational approximation error ξ(ω) after eliminating the denominator is:

[0025]

[0026] Calculate the minimum value of |ξ(ω)| based on the linear programming method to realize the solution of the coefficients of the rational approximation function ;

[0027] Perform partial fraction expansion on the rational approximation function to obtain the frequency-domain fitting polynomial:

[0028]

[0029] where r 1 , r 2 , …, r nand p 1 , p 2 , …, p n are fitting coefficients.

[0030] Further, the polarization resistance R l and the polarization inductance L i in the i-th polarization branch are calculated as follows:

[0031]

[0032] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0033] Based on the rational function approximation algorithm, the present invention establishes a fractional-order polarization equivalent circuit model, constructs the mapping relationship between the fractional-order polarization effect and the equivalent circuit parameters, designs the circuit according to the variation law of the electromagnetic field, and realizes the unity of the electromagnetic field and the circuit. The fractional-order polarization equivalent loop device of the present invention receives the excitation of the external magnetic field through the mutual inductance coil and verifies and simulates the law of the fractional-order polarization effect of the underground medium under the magnetic field excitation, combining numerical simulation and experimental verification. Description of the Drawings

[0034] Figure 1 is the structural diagram of the fractional-order polarization equivalent loop device with any dispersion coefficient provided by the embodiment of the present invention;

[0035] Figure 2 is the circuit model diagram of the fractional-order polarization equivalent loop device with any dispersion coefficient provided by the embodiment of the present invention;

[0036] Figure 3 is the design flow chart of the fractional-order polarization equivalent loop device with any dispersion coefficient provided by the embodiment of the present invention. Detailed Embodiments

[0037] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0038] Due to the unknown nature of the underground environment, the equivalent circuit model provides an idea for simulating the polarization effect. For example, the Debye equivalent circuit is used to represent the Cole-Cole model with a dispersion coefficient of 1. However, the existing integer-order equivalent circuit model cannot characterize the fractional-order characteristics of the polarization effect of actual polymetallic ores. In order to more accurately quantitatively characterize the fractional-order characteristics of the polarization effect, based on the traditional integer-order, the present invention constructs an equivalent circuit for any fractional-order polarization effect based on the multiple zero-pole approximation method and designs a fractional-order polarization equivalent loop device with any dispersion coefficient.

[0039] Example 1, seeFigure 1 Combined with Figure 2 As shown, a fractional-order polarization equivalent loop device with an arbitrary dispersion coefficient according to the present invention has a structure as Figure 1 shown. The device includes:

[0040] A mutual inductance coil and a fractional-order polarization equivalent circuit in a parallel structure. The fractional-order polarization equivalent circuit is composed of 1 induction branch and multiple polarization branches in parallel; the induction branch includes an induction resistor, and the polarization branch is composed of a polarization resistor and a polarization inductor in series; the induction resistor, the polarization resistor, and the polarization inductor all adopt precision resistors and precision inductors, and the mutual inductance coil is wound with a thin wire, so that the resistance of the mutual inductance coil can be ignored relative to the induction resistor and the polarization resistor; the function of the mutual inductance coil is to receive the excitation of an external time-varying magnetic field and generate an induced electromotive force, generating currents in the induction branch and the polarization branch. The current in the induction branch, which is a pure resistance branch, is the induction current, and the total parallel current of the resistor-inductor series branches is the polarization current. The total current in the mutual inductance coil is the sum of the induction current and the polarization current.

[0041] It should be noted that the induction resistor included in the induction branch here can be 1 induction resistor, or it can be understood as a combination of multiple induction resistors. The polarization resistor of the polarization branch can also be understood as a total polarization resistor, or it can be a series-parallel combination of resistors, and the polarization inductor can also be understood as a total polarization resistor, or it can be a series-parallel combination of inductors.

[0042] There are no restrictions on the diameter of the mutual inductance coil and the number of turns of the thin wire. Adjust according to the actual usage situation. Just meet the inductance required by parameters such as the side length D and thickness r of the underground polarization medium.

[0043] Example 2, a design method for a fractional-order polarization equivalent loop device with an arbitrary dispersion coefficient in Example 1, the design process is as Figure 3 shown. The method includes:

[0044] S1 Consult geological data, determine the side length D and thickness r of the underground polarization medium, and calculate the inductance L of the mutual inductance coil, where the magnetic permeability μ 0 = 4π×10 -7 H / m.

[0045]

[0046] S2 According to the zero-frequency conductivity σ 0 of the polarization medium and the earth's geometric parameter β, calculate the induction resistor

[0047] S3 approximates and fits the complex conductivity σ(ω) of the polarizable medium in the frequency domain based on the rational function approximation algorithm, and obtains the frequency domain fitting polynomial of the fractional polarization effect with any dispersion coefficient c.

[0048] Among them, the approximation range of the rational function approximation algorithm is 1 Hz - 10 6 Hz, the approximation order is n, and the rational approximation function is:

[0049]

[0050] Among them, j is the imaginary unit, ω is the angular frequency, d r (ω) and b w (ω) are the coefficients to be solved, r = 1, 2, …, m; w = 1, 2,..., n, m = n or n + 1; γ(ω), θ(ω), and ψ(ω) are the real functions of the rational approximation function respectively.

[0051] Decompose the complex conductivity σ(ω) of the polarizable medium into the real part characteristic Reσ(ω) and the imaginary part characteristic Imσ(ω), that is, σ(ω) = Reσ(ω) + j·Imσ(ω), then the rational approximation error ξ(ω) after eliminating the denominator is:

[0052]

[0053] Calculate the minimum value of |ξ(ω)| based on the linear programming method, and the coefficient solution of the rational approximation function can be realized. Perform partial fraction expansion on the rational approximation function , and we can get:

[0054]

[0055] Among them, r 1 , r 2 , …, r n and p 1 , p 2 , …, p n are the fitting coefficients.

[0056] S4 calculates the polarization resistance R l and the polarization inductance L i of the i-th polarization branch in the equivalent loop device of the fractional polarization effect respectively according to the fitting coefficients of the frequency domain fitting polynomial. The calculation method is as follows:

[0057]

[0058] The number of polarization branches is equal to the fitting order of the rational function approximation algorithm.

[0059] S5 connects the induction-polarization circuit formed by the parallel connection of the induction branch and the polarization branch to the mutual inductance coil in parallel, which is the equivalent loop device for the fractional-order polarization effect with any dispersion coefficient.

[0060] When a time-varying magnetic field is applied to the polarization equivalent loop device, the currents in the mutual inductance coil and each branch can accurately reflect the electromagnetic response law of the polarized medium.

Claims

1. A fractional-order polarization equivalent ring device with arbitrary dispersion coefficient, characterized in that: It includes a mutual inductance coil and a fractional-order polarization equivalent circuit connected in parallel with the mutual inductance coil, wherein the fractional-order polarization equivalent circuit includes an induction branch and multiple polarization branches connected in parallel; the induction branch includes at least one induction resistor, and the polarization branch includes at least one polarization resistor and at least one polarization inductor connected in series.

2. The fractional-order polarization equivalent ring device with arbitrary dispersion coefficient according to claim 1, characterized in that: The mutual inductance coil receives excitation from an external time-varying magnetic field and generates an induced electromotive force, thereby generating a current in the induction branch and the polarization branch.

3. A method for designing a fractional-order polarization equivalent ring device with arbitrary dispersion coefficient according to claim 1 or 2, characterized in that: include: S1 calculates the inductance L of the mutual inductance coil based on the side length D and thickness r of the underground polarized medium, where the magnetic permeability μ0=4π×10 - 7 H / m; S2 calculates the inductive resistance of the inductive branch in the equivalent ring device of the fractional-order polarization effect according to the zero-frequency conductivity σ0 of the polarized medium and the ground geometric parameter β S3 approximates and fits the complex conductivity σ(ω) of the polarized medium in the frequency domain based on the rational function approximation algorithm, and obtains the frequency domain fitting polynomial of the fractional polarization effect with arbitrary dispersion coefficient c; S4 calculates the polarization resistance R of the i-th polarization branch in the equivalent ring device of the fractional-order polarization effect according to the fitting coefficient of the frequency domain fitting polynomial. l and polarization inductance L i , the number of polarization branches is equal to the fitting order of the rational function approximation algorithm; S5 connects the induction-polarization circuit composed of the induction branch and the polarization branch in parallel with the mutual inductance coil, which is an equivalent ring device of the fractional-order polarization effect of arbitrary frequency dispersion coefficient.

4. The design method according to claim 3, characterized in that: The approximation range of the rational function approximation algorithm in S3 is 1Hz-10 6 Hz, the approximation order is n, the minimum value of the rational approximation error between the rational approximation function and the complex conductivity σ(ω) of the polarized medium is calculated, and the coefficients of the rational approximation function are solved; The frequency domain fitting polynomial is obtained by performing partial fraction expansion on the rational approximation function.

5. The design method according to claim 4, characterized in that: S3 specifically includes: Rational approximation function for: Where j is the imaginary unit, ω is the angular frequency, and d r (ω) and b w (ω) is the coefficient to be solved, r=1,2,…,m; w=1,2,…,n, m=n or n+1; γ(ω), θ(ω), and ψ(ω) are rational approximation functions Real function of ; The complex conductivity σ(ω) of the polarized medium is decomposed into the real characteristic Reσ(ω) and the imaginary characteristic Imσ(ω), that is, σ(ω) = Reσ(ω) + j·Imσ(ω). Then, after eliminating the denominator, the rational approximation error ξ(ω) is: Calculate the minimum value of |ξ(ω)| based on the linear programming method to realize the rational approximation function Solve for the coefficients of ; Rational approximation function Perform partial fraction expansion to obtain the frequency domain fitting polynomial: Among them, r1, r2, …, r n and p1,p2,…,p n is the fitting coefficient.

6. The method according to claim 5, characterized in that Polarization resistance R in the i-th polarization branch l and polarization inductance L i The calculation method is as follows:

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