Efficient 3D Forward Modeling Method And L-BFGS Inversion For Transient Electromagnetic Based On Backward Euler And Direct-spliting Scheme

The transient electromagnetic method(TEM)has been widely used in many fields such as engineering survey and geological investigation because of its large detection depth,sensitive to low-resistance bodies,simple device form and high efficiency.3D forward modeling is a tool to learn the transient electromagnetic response laws of different targets and a basis to carry out 3D inversion.However,the current 3D TEM time-domain finite-difference(FDTD)method is computationally inefficient and timeconsuming,which seriously restricts the development and application of 3D inversion.To address the above problem,this thesis proposes an efficient FDTD method based on the Backward Euler(BE)time difference method and the Direct Splitting(DS)strategy.The numerical stability of the new method was verified theoretically.The complex frequency shifted perfectly matched layer(CFS-PML)boundary conditions of the method are established.Based on this new algorithm,3D TEM fast forward modeling is realized.Then,3D TEM inversion is carried out by the L-BFGS optimization algorithm.In forward modeling theory:due to the BE time difference method,the time-step size is no longer restricted by the Courant-Friedrich-Lewy(CFL)stability condition,and the number of time steps is significantly reduced.The Direct Splitting(DS)strategy has the effect of decoupling the electric and magnetic field components and reducing the order of the coefficient matrix as well as reconstructing the matrix.The coefficient matrices of the new equations are tridiagonal matrices with low order.The new algorithm is called BEDS-FDTD.To further speed up 3D forward modeling,this thesis adopts a bilinear transformation(BT)method based on the new equations and re-derive theoretically the CFS-PML absorption boundary.The BEDS-FDTD algorithm code was written based on the OpenACC instruction model and the NVIDIA HPC SDK.After the above techniques,the speed of 3D forward modeling is greatly improved.The full waveform response is calculated in 8s for a model with 50×50×50 grids,and the full waveform response is calculated in less than 240s for a model with 200×200×200 grids.In 3D inversion:an objective function with data fitting terms and constraint terms is constructed.The inverse optimization equation is established based on the L-BFGS algorithm.The BEDS-FDTD method is used for the forward modeling.The inversion algorithm adopts the method of implicitly calculating the sensitivity matrix,namely,calculating the product of transpose of the sensitivity matrix and the residual vector by quasi-forward modeling.Usually,the coefficient matrix of quasi-forward modeling equation is transpose of the coefficient matrix of the forward modeling equation,and solving these large sparse matrices consume a lot of memory and time.To circumvent this problem,this thesis derives the expression of the field from n+1 moments to n moments in inverse time sequence to obtain the governing equations of quasi-forward modeling.Importantly,during the quasi-forward modeling process,the coefficient matrix is still a low order and principal diagonal-dominated tridiagonal matrix.Then,the quasi-forward modeling process is computed on the GPU to speed up the solution and avoid frequent communication between the GPU and the host memory during the calculation of the sensitivity matrix.In inversion validation and engineering applications:to verify the effectiveness of the inversion algorithm,a total of four different complex theoretical models of two device forms,loop source and grounded line source,are used to perform the inversion.The inversion results all show that the inversion algorithm in this thesis can effectively recover the electrical and structural characteristics of the theoretical model.Furthermore,the 3D inversion takes less than 30 min for the model with 55×55×55 grids,which is a huge improvement in computational speed compared with the current 3D inversion algorithm.Finally,3D TEM inversion is carried out for the measured data of a limestone mining in Hepu,Guangxi.The inversion results match well with the real geological situation and collapse distribution locations,which verifies the practicality of 3D forward modeling and inverse software developed in this thesis.