Fast Three-Dimensional Integral Equation Algorithm For Airborne Electromagnetic Forward Modeling

With the rapid economic growth of the country,the existing mineral resources can not meet the demand.Electromagnetic(EM)methods are often used for mineral resource exploration.The electromagnetic surveys play an important role in good ground conditions.However,hidden resources are difficult to find in the places that with poor geological conditions,because electromagnetic methods cannot be performed,so airborne electromagnetic(AEM)exploration technology has become an important method.Compared with the differential method,the integral equation(IE)technique occupies a great advantage in three-dimensional numerical calculation.It only needs to discrete the anomalous area,so that the coefficient matrix is small and takes up less memory.But for large anomalous regions,the discretization matrix is dense,which greatly limits the solution efficiency of the IE method.At present,the IE method is divided into direct solution and iterative solution method.In the case of a small number of grids,the speed of direct solution is fast;when the number of meshes increases,the huge matrix inversion is difficult to achieve,so for large complex anomalous regions,the iterative solution method has more advantages.For conventional integral equation methods,a large amount of computation time consumes when solving large linear equations.Therefore,Born(1993)proposed an approximate solution,which avoids solving large-scale equations,the solution speed is very fast but has low calculation accuracy;Zhdanov(1996)proposed Quasi-linear(QL)approximation、Quasi-analytical(QA)approximation methods and so on,those methods improve the accuracy of the approximate solution.However,these approximate methods can only achieve satisfied computational results when the specific conditions are stisfied.Based on the characteristics of AEM using mobile source and large amount of data acquisition,with the advantage of approximation method,this paper proposes fast Fourier transform high order Quasi-analytical(FFT-HOQA)approximation method and Fast Multigrid Quasi-linear approximation(FMGQL)method.In the case of ensuring accuracy,the forward efficiency of AEM is greatly improved.in addition,the computational efficiencies of the conjugate gradient(CG)method,biconjugate gradient(BICG)method,and stable biconjugate gradient(BICGSTAB)method are compared in this paper.The solution to the integral equations of multiple anomaly and irregular models is studied.After numerical simulation and comparative study,this paper draws the following conclusions:(1)The FFT-HOQA approximation method is an effective algorithm for AEM forward modeling,the calculation speed is higher than the traditional integral equation method and Marcoair.The computational advantage is obvious,especially in the case of a large number of grids;(2)Compared with the traditional method,the fast Fourier transform can greatly accelerate the product of the matrix and the vector and improve the calculation efficiency;(3)The use of the contract operator technique can guarantee the convergence of the dyadic Green’s function coefficients in any case,which makes the application of the FFT-HOQA approximation method more extensive,using OpenMP parallel acceleration technology can simultaneously calculate the forward modeling of multiple points,which can save the calculation cost;(4)The combination of FMGQL and fast Fourier transform can realize the matrix-vector multiplication,greatly improve the solution speed.At the same time,FMGQL is more suitable for the solution of large grids;(5)It has a good result that using inhomogeneous background to calculate multiple anomalous bodies models and using the "unity method" to calculate irregular models.