The Math Problems And Application In The Transient Electromagnetic Field

In recent years, the transient electromagnetic method is widely used in the practicalengineering geological exploration. It is a kind of time-domain electromagnetic explorationmethod. The theory of it is laying a loop on the surface, exiting the loop by step wave or othertype current field source to produce the transition process field in earth. In the moment ofshutting the power, there exits the electromagnetic field. Measuring the attenuation signal andanalyzing the feature of the second induction electromagnetic field, which produced byunderground medium with time change. Then, the conductive capacity and the position of theunderground non-homogeneous substance can be judged. Along with the application of theseismic techniques into the transient electromagnetic method, people pay more and moreattention in wave-field inverse transformation to get more and more accurate undergroundgeological information. Formerly the magnetic source or electrical source is regarded asdipole. Vector and scalar potential function has been analyzed to match the result of constantcurrent field to TEM response.In this thesis, based on the idea that time-varying point charge instead of dipolehypothesis, the time-domain analysis formula of transient electromagnetic field is deduced.Firstly the total space time-domain analytic formula of the homogeneous damped waveequation is deduced by using the method of variable separation. Using orthogonality of thetrigonometric function system, undetermined coefficients in the analytic formula isdetermined. And the total space time-domain Green function solution of the active wave fieldis expressed.Fllowed, based on the total space, the homogeneous half-space time-domain analyticformula of the homogeneous damped wave equation is deduced by using the method ofvariable separation. The vector equation of electromagnetic field is given through theMaxwell equation. The time-domain analytic formula of active wave field and passive wavefield on homogeneous half-space can be got.Then, combining the boundary conditions of electromagnetic field, the time domainresponse formula of transient wave electromagnetic field in the horizontally-layered mediumcan be got from the homogeneous half space.This method of strictly deriving electromagnetic field’s analysis formula in time domainnot only avoid the errors caused by the traditional dipole hypothesis and the transformationbetween the time domain and frequency domain, but also provided the theory basis forsolving the problems of the time-domain response and the people engaged in geologicalexploration.Finally, the problem of wave-field inverse transformation was solved by using themethod of continuous normalization. To obtain the regularization parameter, deviationprinciple and iteration formula with three order convergence are used. By comparing theresults of numerical calculation and the known function of wave field, the feasibility and the effectiveness of the proposed method is proved. Comparing with discrete regularization, inthis thesis, the continuous regularization has the advantages such as structure preserving andfaster convergence speed.