ALBERT

Description: 〈span〉〈div〉Summary〈/div〉The investigation of using a novel radial basis function-based meshfree method for forward modelling magnetotelluric data is presented. The meshfree method, which can be termed radial basis function-based finite difference (RBF-FD), uses only a cloud of unconnected points to obtain the numerical solution throughout the computational domain. Unlike mesh-based numerical methods (for example, grid-based finite difference, finite volume and finite element), the meshfree method has the unique feature that the discretization of the conductivity model can be decoupled from the discretization used for numerical computation, thus avoiding traditional expensive mesh generation and allowing complicated geometries of the model be easily represented. To accelerate the computation, unstructured point discretization with local refinements are employed. Maxwell’s equations in the frequency domain are re-formulated using $\mathbf {A}$-ψ potentials in conjuction with the Coulomb gauge condition, and are solved numerically with a direct solver to obtain magnetotelluric responses. A major obstacle in applying common meshfree methods in modelling geophysical electromagnetic data is that they are incapable of reproducing discontinuous fields such as the discontinuous electric field over conductivity jumps, causing spurious solutions. The occurrence of spurious, or non-physical, solutions when applying standard meshfree methods is removed here by proposing a novel mixed scheme of the RBF-FD and a Galerkin-type weak-form treatment in discretizing the equations. The RBF-FD is applied to the points in uniform conductivity regions, whereas the weak-form treatment is introduced to points located on the interfaces separating different homogeneous conductivity regions. The effectiveness of the proposed meshfree method is validated with two numerical examples of modelling the magnetotelluric responses over three-dimensional conductivity models.〈/span〉

Description: 〈span〉〈div〉SUMMARY〈/div〉The investigation of using a novel radial-basis-function-based mesh-free method for forward modelling magnetotelluric data is presented. The mesh-free method, which can be termed as radial-basis-function-based finite difference (RBF-FD), uses only a cloud of unconnected points to obtain the numerical solution throughout the computational domain. Unlike mesh-based numerical methods (e.g. grid-based finite difference, finite volume and finite element), the mesh-free method has the unique feature that the discretization of the conductivity model can be decoupled from the discretization used for numerical computation, thus avoiding traditional expensive mesh generation and allowing complicated geometries of the model be easily represented. To accelerate the computation, unstructured point discretization with local refinements is employed. Maxwell’s equations in the frequency domain are re-formulated using $\mathbf {A}$-ψ potentials in conjunction with the Coulomb gauge condition, and are solved numerically with a direct solver to obtain magnetotelluric responses. A major obstacle in applying common mesh-free methods in modelling geophysical electromagnetic data is that they are incapable of reproducing discontinuous fields such as the discontinuous electric field over conductivity jumps, causing spurious solutions. The occurrence of spurious, or non-physical, solutions when applying standard mesh-free methods is removed here by proposing a novel mixed scheme of the RBF-FD and a Galerkin-type weak-form treatment in discretizing the equations. The RBF-FD is applied to the points in uniform conductivity regions, whereas the weak-form treatment is introduced to points located on the interfaces separating different homogeneous conductivity regions. The effectiveness of the proposed mesh-free method is validated with two numerical examples of modelling the magnetotelluric responses over 3-D conductivity models.〈/span〉

Description: 〈span〉〈div〉SUMMARY〈/div〉A study is presented using a meshfree approach and a radial basis function generated finite difference (RBF-FD) method for numerically modelling three-dimensional (3-D) gravity data. The gravity responses, i.e., vertical gravity and gravity gradients, are obtained by solving the partial differential equation (PDE), that is, the Poisson’s equation for gravitational potential. The meshfree approach discretises PDEs using exclusively a cloud of unconnected nodes, instead of traditional tessellated meshes as used by mesh-based numerical methods such as finite difference, finite element and finite volume. Thus, the potentially computationally expensive and unstable creation and manipulation of 3-D meshes can be entirely avoided. A new type of finite-smoothness radial basis functions (RBFs), namely, the quintic-order polyharmonic spline (PHS) RBF, is proposed here in the RBF-FD frame for solving the gravity problem. Previous geophysical data modelling studies using RBF-FD have employed the infinitely smooth RBFs, such as the popular Gaussian (GA) RBFs. Here, both GA and PHS RBFs were tested with different numbers of nodes per meshfree subdomain and with various shape parameter values (only GA RBFs have a shape parameter). The test results show that the PHS RBF-FD method is more computationally efficient than the GA RBF-FD counterpart. To achieve more efficiency, unstructured node distributions are proposed in discretising the density models. For both quasi-uniform and unstructured node distributions, numerical results from the proposed PHS RBF-FD demonstrate that the computed vertical gravity and gravitational potential values agree well with analytical solutions with a reasonable number of degrees of freedom. A comparison study of modelling a complex density model with the PHS RBF-FD scheme and nodal finite-element method shows that the RBF-FD scheme generates sparse, asymmetric, linear systems of equations, supports unstructured nodal discretisation and local refinement, and can have non-linear 〈span〉h〈/span〉-convergence under refinement. Finally, the proposed RBF-FD method was applied to obtain the vertical gravity and gravity gradients over a real-world density model, where the benefits of meshfree discretisation are clearly illustrated.〈/span〉