I have developed an efficient three-dimensional forward modeling algorithm based on radiation boundary conditions for controlled-source electromagnetic data. The proposed algorithm derives computational efficiency from a stretch-free discretization, air-free computational domain, and a better initial guess for an iterative solver. A technique for estimation of optimum grid stretching for multi-frequency modeling of electromagnetic data is described. This technique is similar to the L-curve method used for the estimation of the trade-off parameter in inversion. Using wavenumber-domain analysis, it is illustrated that as one moves away from the source, the electromagnetic field varies smoothly even in case of a complex model. A two-step modeling algorithm based on radiation boundary conditions is developed by exploiting the smoothness of the electromagnetic field. The first step involves a coarse grid finite-difference modeling and computation of a radiation boundary field vector. In the second step, a relatively fine grid modeling is performed with radiation boundary conditions. The fine grid discretization does not include stretched grid and air medium. An initial solution derived from coarse grid modeling is used for fine grid modeling. Numerical experiments demonstrate that the developed algorithm is one order faster than the finite-difference modeling algorithm in most of the cases presented.
<p>Geoelectric non-destructive imaging and monitoring of the earth's subsurface requires robust and adaptable numerical methods to solve the governing differential equation. Most of the time, the DC data is acquired along a straight line. Hence, we solve the DC problem for the 2D case. But the source for the DC method exhibits a 3D nature. To account for the source's 3D nature, the 2D DC resistivity modeling is often carried out in the wavenumber domain. There have been studies that suggest ways for the selection of optimum wavenumbers and weights. But, this does not guarantee a universal choice of wavenumbers. The chosen wavenumbers and related weights strongly influence the precision of the resulting solution in the space domain. Many forward modeling studies demonstrate that selecting effective wavenumbers is challenging, especially for complicated models with topography, anisotropy, and significant resistivity differences. Moreover, forward modeling requires many wavenumbers as the models get more complex.&#160;</p> <p>This study focuses on developing a method that can completely omit wavenumbers for 2D DC resistivity modeling. The present work finds its motivation in a numerical experiment on a simple half-space model. Since the analytical response for such a model can be easily calculated, we match the analytical solution against the responses obtained from various wavenumbers and weights used in the literature. All the responses deviated from the analytical solution after a certain distance, and none of them were found to be accurate for large offsets. It was discovered after thorough testing of the numerical scheme that the wavenumbers selected for the forward modeling significantly impacted how practical the approach is for large offsets.&#160;</p> <p>To overcome this problem, a new boundary condition is derived and implemented in the existing numerical scheme. The numerical scheme chosen to perform the forward modeling is Mimetic Finite Difference Method (MFDM). We consider that the source is placed on the origin of the coordinate system. This removes the dependency of the source term, expressed in the Fourier domain, on the wavenumber. The solution obtained by solving the resulting equation will be an even function of the wavenumber and be real-valued. This ensures that the potential in the space domain for the 2D model will also be a real-valued even function with a symmetry about a plane perpendicular to the strike direction and passing through the origin. Because the first-order derivative of an even function at the plane of symmetry vanishes, mathematically, it can be expressed as a Neumann boundary condition at the considered plane. Therefore, we propose a scheme to solve the 2D resistivity problem in the space domain using the boundary condition mentioned here.</p> <p>The developed algorithm is tested on isotropic and anisotropic two-layer models with large contrasts. It is found that the numerical solutions obtained using the modified boundary condition described above show considerable accuracy even for large offsets when compared with the analytical solution. On the other hand, the results obtained using available wavenumbers in the literature are also compared and are found to deviate considerably from the analytical solution at large offsets.</p>
I examine the possibility of using a three-dimensional modeling algorithm in studying tunnel detectability using the electromagnetic gradiometer response. A detailed comparison of different source-receiver configurations reveals that the electromagnetic gradiometer response is stronger for a configuration when the transmitter and receiver are orthogonal to each other than the parallel configurations. Orthogonal configurations perform better at a relatively lower frequency than the parallel configuration. The electromagnetic gradiometer response enhances with the broadside offset defined as the distance between transmitter and receiver pair. The response of a tunnel is generally weak; consequently, deviation from the configuration where both receivers are equidistant from the transmitter masks the response of a tunnel. The impact analysis of tunnel-floor conductivity revealed that a one order higher conductive floor does not change the behavior of response compared to a tunnel without a conductive floor. However, the two-order higher-conductive floor changes the shape of the response curves, yet the change in the magnitude is not significant. The presence of a metal conductor substantially enhances the response of an electromagnetic gradiometer system. The parallel configuration is more suitable for the depth estimation of the tunnel than the orthogonal configuration. The distortion in the electromagnetic gradiometer response due to small heterogeneity in the case of orthogonal configuration depends on the broadside offset. For a zero broadside offset, the response is severely distorted by the small-scale inhomogeneity. For broadside offset like 20 m, the impact of small-scale inhomogeneity is almost invisible.
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