Spectral-infinite-element simulation of Gravity and Magnetic data

Ph.D. Project, Queen's University, Canada, 2023

Title: Spectral-infinite-element simulation of Gravity and Magnetic data
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Abstract:

In potential field geophysics, imaging techniques are crucial in estimating source property distributions within two- and three-dimensional spaces. Central to this field are the gravity and magnetic methods, which are governed by the unbounded Poisson/Laplace equations. Traditional approaches for solving these equations are based on direct integral methods, where the computational cost is proportional to the number of observational data points. This dependency poses significant challenges in simulating complex models that require higher data points. This study introduces the Spectral-Infinite-Element Method (SIEM) as a potential solution to address these challenges. SIEM uses nodal quadrature to integrate the spectral element method with the mapped infinite element method. Poisson’s equation is solved within the domain of interest using spectral elements in conjunction with the Gauss-Legendre-Lobatto (GLL) quadrature. For areas outside the domain, the Gauss-Radau quadrature effectively captures the infinite elements in outer space. A distinct advantage of SIEM over traditional integral methods lies in its higher-order discretization capability, which allows for a more accurate representation of complex models. Furthermore, SIEM is independent of the number of observational data points, making it an efficient method for simulating large-scale complex models. The efficacy of SIEM was tested through synthetic simulations of gravity and magnetic methods and compared with analytical solutions. The results demonstrate that SIEM offers a higher accuracy and efficiency over traditional methods. This method provides a robust and scalable solution to complex 3D models with computational ease.

Introduction:

Potential field geophysics is governed by the unbounded Poisson/Laplace equations
The traditional direct integral method used to solve this equation poses challenges in computational cost, singularity, scaling with data points, and limiting complex model simulations
This study introduces the Spectral-Infinite-Element Method (SIEM) as a potential solution to address these challenges
SIEM combines the spectral element method with mapped infinite elements via nodal quadrature
Gauss-Legendre-Lobatto and Gauss-Radau quadratures inside and outside the domain, respectively
SIEM performance is validated against synthetic simulations and analytical solutions

Meshing

A spherical block model was meshed using hexahedral elements, as shown in the figures below.

Meshing of the Spherical Model — Figure 2: Meshing of the spherical block model using hex elements.

Magnetic Modelling using SPECFEM

One block model with magnetic intensity \(M = 10 \, \text{Am}^{-1}\), radius \(2 \, \text{m}\), and depth \(5 \, \text{m}\) from the centre is modeled, as illustrated below.