Research Of Complex Resistivity 3D Electromagnetic Forward And Inversion

The resistivity and induced polarization of geologic body generally show thecharacteristics of complex resistivity. Some scholars have expressed the complexresistivity by some mathematical models, for example the Cole-Cole model. It has beenproved that the electrical anomaly of geologic body can be quantitative estimated byCole-Cole model parameters (direct resistivity ρ 0, chargeability m , time constant τ andfrequency dependencec ). It is very important to study the methods to obtain the complexparameters of geologic body. There are some shortcomings in traditional inversionmethods because they base on the hypothesis about apparent resistivity and trueparameters. The hypothesis of electromagnetic coupling effect can bring error for theresult of inversion;and the traditional methods can't inverse the geometry parameters. Inorder to solve the problems above, we studied the methods for complex resistivity 3Delectromagnetic forward and inversion, and developed an algorithm to directly inverse thetrue Cole-Cole model parameters.Firstly, the paper analyzed some complex resistivity mathematical models andequivalent circuit models, then discussed the frequency spectrum characters of Cole-Colemodel. In addition, the paper presented an improved traditional complex resistivityinversion method.Secondly, On the basis of analysis about the complex resistivity mathematical models,the Cole-Cole model is introduced into the 3D electromagnetic modeling. The complexresistivity 3D electromagnetic forward calculation included surface electric dipole andcross-borehole magnetic dipole source. In the forward modeling, the problem of integralwith Bessel function is especially discussed. For the shortcoming of fast Hankeltransforms, the Gaussian quadrature and continued fraction methods have been used tocalculate the integral. The results of calculation indicated that the method is alwaysfeasible and accurate.For the surface electric dipole source, the response of 3D electromagnetic hasfollowing features. In low frequency (<100Hz), the values of secondary electric field E sxvaries with complex resistivity parameters, and the imaginary secondary electric fieldcomponent are more prominent. The absolute values of real secondary electric fieldcomponent Re( E sx) increased with chargeability ( m ), and the imaginary secondaryelectric field component Im( E sx) is in the same phase. The absolute values ofimaginary secondary electric field component Im( E sx) increased with the timeconstant (τ ). In addition, variations of frequency dependence (c ) also bring aboutnotable difference of Im( E sx).For the cross-borehole magnetic dipole source, we found there are some notabledifference between secondary magnetic field H szof complex resistivity 3D body andthe H szof pure resistivity 3D body in the shape of curve. The result indicated that theresponse of vertical magnetic field is also brought about by the induced polarization andelectromagnetic induction. The induced polarization response is more prominent at10~100Hz. The values of real secondary magnetic field component Re( H sz) are varieswith complex resistivity parameter. The negative anomaly of Re( H sz) increased withdecrease of the time constant (τ ). The absolute values of real secondary magnetic fieldcomponent are increased with augment of the chargeability ( m ). The negative anomalyof Re( H sz) is also increased with decrease of the frequency dependence ( c ). Inaddition, the values of imaginary secondary magnetic field component Im( H sz) arealmost changeless in different complex resistivity parameters.Finally, the paper develops the partial derivative matrix that reflects the rate of realmeasurement field to the Cole-Cole model parameters. It is achieved that the complexresistivity 3D electromagnetic inversion base on surface electric dipole andcross-borehole magnetic. In the inversion, the convergence rate of dc resistivity ( ρ 0) istime constant (τ ) is slowest, and the results of τ are strongly affected by the first guess.On the other side, due to strong correlativity between time constant (τ ) and chargeability( m ), the time constant (τ ) can't be accurate inversed when all the complex resistivityparameters take part in the inversion. For this problem, considering the chargeability ( m )can be measured or estimated before the inversion calculation. So we fixed the values ofthe chargeability ( m ) in the inversion, and then inversed other resistivity parameters( ρ 0, τ ,c ). Through the method we can obtain accurate result of inversion. The resultsof theoretical calculation example indicate that the method about inversing resistivityparameters of 3D geologic body from electromagnetic information is feasible.The production of research further developed the theory of forward and inversion forcomplex resistivity method, and offers a new method for the cross-boreholeelectromagnetic probing. It has important theoretical and practical significance.