Research On Finite-element Method Based On Unstructured Grids For Airborne EM Modeling

Without needing ground accessment,airborne electromagnetic(AEM)is an effective geophysical exploration tool,especially suitable for explorations in areas with complex topography,such as deserts,mountains,swamps etc.In recent years,as more and more people realized that this method has high sensitivity and large exploration depth,it has been widely used in fields of mineral exploration,groundwater and environmental and engineering investigations.Until now,AEM data interpretation mainly bases on imaging and one-dimensional(1D)inversions.However,for the practical explorations,the underground media usually have complex structures.Approximation of two-or three-dimensional(2D/3D)complex models using 1D model will bring large errors into AEM interpretation results.For these reasons,apart from 1D interpretation,in target areas one needs to use 2D and 3D model to invert AEM data.Forward modeling is the basis of inversion and data interpretation.An effective algorithm can greatly increase AEM modeling speed and improve the modeling accuracy that can further raise the efficiency of 2D and 3D inversions.In this paper,I systematically study frequency-/time-domain AEM modeling for 2D and 3D models.Due to the fact that unstructured grids can greatly fit the physical interface,while the finite-element method can calculate EM responses for arbitrarily complex models,I use the finite-element method based on unstructured grids to model AEM responses.To obtain an effective unstructured grids that increase the forward modeling speed and accuracy,I create a goal-oriented adaptive strategy for the finite-element method based on the unstructured grids for AEM modeling procedure.To avoid the problem that the EM field changes sharply in the area close to the source and to increase the modeling accuracy,I deal in this paper with 2D and 3D forward modeling using the scattered field method with the primary field as the source.For 2D AEM modeling,the Fourier transform is first applied to the Maxwell equations,so that the forward modeling is solved in the wavenumber domain.Considering that the EM filed calculated by 2D modeling has a constant direction along the strike and satisfies the continuity conditions at the interface,the scalar shape functions are used for the finite-element method.For 3D AEM modeling,the vector Helmholtz equation is solved in the space domain directly.As the EM field does not abide the continuity condition at the interfaces and the divergence condition cannot be guaranteed by the scalar shape function in each element,the vector shape functions are used in 3D modeling.The large linear systems are solved by the robust direct decomposition solvers such as MUMPS and PARDISO that have a high accuracy.To obtain 2D frequency-domain EM field on each node,an inverse Fourier transform is applied to the waveform-domain EM field,while the 2D and 3D time-domain EM fields are calculated by transforming the frequency-domain fields via a Hankel transform.For the goal-oriented adaptive technology,I redesign the posterior error estimation algorithm and develop the weighted posterior error formula.As the normal component of the current density for scattered field does not satisfy the continuity condition at the interface,I propose in this paper to estimate the posterior error utilizing the continuity condition of the normal component of the current density of real part of the scattered field that is particularly suitable for the scattered field problem.By summing up the weighted posterior errors for different frequencies,I derive the weighted posterior error formula for multi-frequency AEM adaptive modeling problem.For numerical experiments,I first model and analyze AEM responses in profile for 2D topography using my 2D modeling algorithm.The modeling results indicate that the AEM responses for a topographic earth bear a pattern that has mirror image relationship with the topography.The effect of topography mainly occurs at the high-frequency range or early time-channels.Further,I analyze the AEM responses in a plane for 3D topographic model using my 3D modeling algorithm.From the modeling results,I find that the imaginary part of high-frequency signal has more capability to describe the topography,while the real part of low-frequency signal can better reflect the abnormal body embedded under the earth surface.B and d B/dt field contain different information of the earth media.Interpreting B and d B/dt data together can help reveal more details of the earth structures.By presenting the final adaptive meshes and model results calculated by the goal-oriented adaptive method,I verified the effectiveness of my algorithm,while at the same time summarize the rule of meshes for discretizing models: the lower the frequency or the higher the resistivity,the larger the refined domain must be used,and the coarser the meshes can be used;on the other hand,the higher the frequency or the lower the resistivity,the smaller the refined domain can be used,while the finer meshes must be used.Finally,I calculate 3D AEM responses for arbitrarily anisotropic models and analyze the effect of anisotropy on AEM responses.I draw the conclusions from the modeling results that the anisotropic abnormal body mainly affects the low-frequency and later time-channel signal,while the anisotropic surrounding rock has a large effects on signal of all frequencies and time-channels.Besides,the rotation angle of the surrounding rock's anisotropic conductivity has a relationship with the form of AEM responses,which can help us deduce the rotation angle of the conductivity tensor.The theory and algorithms addressed in this paper lay a foundation for the identification of anomalous AEM signal and 2D / 3D AEM inversions.