A Theoretical Study Of Wave Field Transformation And Synthetic Aperture Imaging To The Diffusive Transient Electromagnetic Field

Transient electromagnetic(TEM)data has weak resolution and low detection accuracy due to its own diffusion field characteristics.While the wave field can better reflect the interface of underground medium,and the resolution is higher.The so-called wave field transformation of TEM field refers to the transformation of low frequency electromagnetic field in time domain into fictititious wave field satisfying wave equation through integral equation.Then,with the help of some mature techniques developed in the wave field,the transformed "wave field" is analyzed and processed,so as to detect the underground target body and improve the precision of TEM data interpretation.The mathematical relationship between the TEM field and corresponding fictitious wave field is first derived for multiple electromagnetic field components(electric field,magnetic field,and induction electromotive force).Due to wave field transformation equations belong to the first-class Fredholm integration equation,which is a very ill-posed inverse problem.Considering that the conjugate gradient(CG)method is quite efficient for solving system of linear equations whose coefficient matrix has tightly clustered spectrum,in this paper,a new pre-conditioned regularized conjugate gradient(PRCG)method is proposed to realize the transformation from diffusion equation to wave equation.Firstly,the characteristic of the kernel function of the integral equation is analyzed.By comparing several different discretization methods,the discrete mode 1,which has the narrowest time window in q domain and the smallest condition number of coefficient matrix,is chosen to discretize the integral equation.The symmetric positive definite coefficient matrix is constructed,and regularization algorithm is combined with CG method.At each step of the iteration,the regularized linear system itself is solved by CG method.In addition,regularized linear system can be pre-conditioned by SSOR preconditioner.The SSOR pre-conditioned significantly reduces the condition number,and the regularization improves the spectral properties of the coefficient matrix,while the CG iteration implements calculation for wave field.The method is verified by the constant wave field U(q)=1.The result of transformed wave field is in good agreement with that of theoretical.The comparison with the results of RCG method and pre-existing PRCG method shows that the algorithm proposed in this paper converges faster,and more stable.Using the algorithm developed in this paper,we analyze the characteristics of the wave field for layered geoelectric models and summarize its propagation properties.The sensitivity of the wave field to resistivity difference and the effective resolution depth of this algorithm under the condition of specific parameters are quantitatively analyzed based on the two-layer geoelectric model of G and D.Based on Q,A,H,K three-layer geoelectric model,the effects of interlaminar reflection and aliasing interference on layered media are analyzed,and the resolution effect of this algorithm on thin layers is given.The wave field transformation of the noise-containing signal shows that by increasing the regularization parameters accordingly,a very stable wave field transformation solution can still be obtained,which proves that the proposed algorithm is stable,reliable and practical.At present,TEM data interpretation is still dominated by single-point processing.Single-point processing not only reduces the efficiency of interpretation,but also ignores the correlation among the measured data.Due to the similar coherence of multi-aperture TEM,the reflection echo of the same geological body at adjacent survey points has a good correlation.Therefore,synthetic aperture imaging of TEM signal can be performed.Based on the idea of synthetic aperture focusing technique of ultrasound imaging,in this paper,we present a new synthetic aperture focusing technique for TEM imaging,which uses delay related superposition algorithm to realize beamforming and multi-point data synthesis.After transformed wave field is obtained,the delay correlation superposition of the adjacent measurement point signals centered on certain measuring point is carried out,and the traditional single point processing method is developed into a multi-point synthesis method similar to that of the linear array,and synthesizing point by point.Thus,the aperture length of the receiving coil and the signal-to-noise ratio of the data are greatly improved,so as to improve the resolution and interpretation accuracy of the transient electromagnetic method.The synthetic aperture focusing technique of TEM proposed in this paper is verified by different layered models.The imaging results show that this method can suppress the noise,highlight the weak anomaly,and improve the ability of TEM method to distinguish the electrical interface.The analysis of the measured data shows that wave field transformation and synthetic aperture focusing process can realize the imaging of the low-resistance ore body area,and the electrical interface of the low-resistance anomaly area is better reflected,compared with the resistivity inversion section.