3D sparse inversion of magnetic amplitude data when strong remanence exists

Abstract

Three-dimensional inversion for susceptibility distributions is a common approach for quantitative interpretation of magnetic data. However, this approach will fail when strong remanence exists because the total magnetization direction is unknown. Magnetic amplitude inversion can reduce remanence effects and thus improve reconstructed results. In this paper, we propose a sparse magnetic amplitude inversion method which minimizes an L₀-like-norm of model parameters subject to bound constraints. By using the iteratively reweighed least squares technique, the bound-constrained L₀-like-norm sparse inversion is transformed to a sequence of bound-constrained nonlinear least squares subproblems. To deal with the bound constraints, we use a logarithm barrier algorithm to solve each subproblem. Compared with the classical L₂-norm inversion method, the proposed sparse method utilizes the known physical property information to produce binary results characterized by sharp boundaries. This method is tested on synthetic data produced by a dipping dyke model and a field data set acquired in Australia.

Appendix

The logarithm barrier method for bound-constrained L₂-norm inversion of magnetic amplitude data

Li and Oldenburg (2003) and Li et al. (2010) have developed practical logarithm barrier methods for solving bound-constrained nonlinear least squares problems in magnetic inversion. We summarized their methods (Li and Oldenburg, 2003; Li et al. 2010) here. First, for simplicity, we omit the superscripts and rewrite Eq. 9 as

$$\begin{aligned} \hbox{min} \quad \varphi_{c} (\varvec{m}) = \varphi_{d} (\varvec{m}) + \mu \varphi_{cm} (\varvec{m}) \hfill \\ {\text{s}} . {\text{t}} .\;\quad \varvec{m}_{\hbox{min} } \le \varvec{m} \le \varvec{m}_{\hbox{max} } . \hfill \\ \end{aligned}$$

(11)

The logarithm barrier method approximates Eq. 11 as a sequence of unconstrained minimizations, making the inequality constraints implicit in the new objective function by adding a barrier term:

$$\hbox{min} \quad \varphi (\varvec{m}) = \varphi_{c} (\varvec{m}) + \lambda \varphi_{\lambda } (\varvec{m}),$$

(12)

where λ is a barrier parameter and will be decreased during minimization. φ_λ is a barrier function and has the form

$$\varphi_{\lambda } (\varvec{m}) = - \sum\limits_{j = 1}^{M} {\left[ {\ln (\frac{{m_{j} - m_{\hbox{min} } }}{{m_{\hbox{max} } - m_{\hbox{min} } }}) + \ln (\frac{{m_{\hbox{max} } - m_{j} }}{{m_{\hbox{max} } - m_{\hbox{min} } }})} \right]} .$$

(13)

Applying one step of Gauss–Newton method for Eq. 12 at the kth iteration, we obtain

$$\begin{aligned} &\left\{ {(\varvec{J}^{(k)} )^{T} \varvec{W}_{d}^{T} \varvec{W}_{d} \varvec{J}^{(k)} \; + \;\mu \varvec{Z}^{T} \varvec{R}^{T} \varvec{RZ}\; + \;\lambda^{(k - 1)} [(\varvec{X}^{(k)} )^{ - 2} + (\varvec{Y}^{(k)} )^{ - 2} ]} \right\}\Delta \varvec{m} \hfill \\ &= (\varvec{J}^{(k)} )^{T} \varvec{W}_{d}^{T} \varvec{W}_{d} [\varvec{d} - \varvec{F}(\varvec{m}^{(k - 1)} )]\; - \;\mu \varvec{Z}^{T} \varvec{R}^{T} \varvec{RZm}^{(k - 1)} \; + \;\lambda^{(k - 1)} [(\varvec{X}^{(k)} )^{ - 1} \; + \;(\varvec{Y}^{(k)} )^{ - 1} ]{\mathbf{1}}, \hfill \\ \end{aligned}$$

(14)

where J^(k) is the Jacobian matrix (Li et al. 2010) at the kth iteration, Δm is the descent direction of objective function, X^(k) and Y^(k) are diagonal matrixes with m^(k−1) − m_min and m^(k−1) − m_max on their main diagonals, and 1 is the column vector with all entries 1. Once the descent direction has been computed, the solution of Eq. 11 can be iteratively solved with appropriate choice of a step length and careful update of the barrier parameter. The algorithm for solving Eq. 11 is summarized as follows:

• 1. Initialize $\varvec{m}^{(0)} = {\mathbf{0}}{\mathbf{.001}}$, $\lambda^{(0)} = {{\varphi_{c} (\varvec{m}^{(0)} )} \mathord{\left/ {\vphantom {{\varphi_{c} (\varvec{m}^{(0)} )} {\varphi_{\lambda } (\varvec{m}^{(0)} )}}} \right. \kern-0pt} {\varphi_{\lambda } (\varvec{m}^{(0)} )}}$, and k = 1.

• 2. Form X^(k) and Y^(k) from m^(k−1).

• 3. Solve Eq. 14 for Δm.

• 4. $\varvec{m}^{(k)} \leftarrow \varvec{m}^{(k - 1)} + \gamma \beta^{(k)} \Delta \varvec{m}$, where $\gamma = 0.925$ and

$$\beta^{(k)} = \left\{ {\begin{array}{*{20}c} {1,} & {{\text{if}}\;\varvec{m}_{\hbox{max} } > \varvec{m}^{(k - 1)} + \Delta \varvec{m} > \varvec{m}_{\hbox{min} } } \\ {\hbox{min} \left( {\begin{array}{*{20}c} {\mathop {\hbox{min} }\limits_{{\Delta m_{i} < 0}} \frac{{\varvec{m}_{i}^{(k - 1)} - m_{\hbox{min} } }}{{\left| {\Delta \varvec{m}_{i} } \right|}},} & {\mathop {\hbox{min} }\limits_{{\Delta m_{i} > 0}} \frac{{m_{\hbox{max} } - \varvec{m}_{i}^{(k - 1)} }}{{\Delta \varvec{m}_{i} }}} \\ \end{array} } \right),} & {{\text{otherwise}} .} \\ \end{array} } \right.$$

• 5. $\lambda^{(k)} \leftarrow [1 - \hbox{min} (\beta^{(k)} ,\gamma )]\lambda^{(k - 1)}$.

• 6. Terminate on convergence or when k attains a specified maximum number of iterations k_max. Otherwise, $n \leftarrow n + 1$ and go to Step 2.