Some aspects of neural network parameter optimization for joint inversion of gravitational and magnetic fields

TL;DR

The article considers the issue of determining in joint field inversion not only the geometric distribution of sources, but also their physical intensities, and examines in detail the possibilities of optimizing some elements of the neural networks and the algorithms used.

Abstract

We consider the optimization of a neural network previously developed by the authors for the joint inversion of 3D gravitational and magnetic fields in the context of mineral exploration. The distinctive feature of this neural network is that it solves ill-posed (ill-conditioned) inverse problems. The neural network implements a special two-level algorithm. The lower level of the algorithm uses two neural networks with equivalent architectures. The first of them computes the gravitational field sources in a given domain from measurements of this field on a remote surface. The second neural network processes magnetic field measured on the same surface to find magnetic sources in the same domain. The found source distributions are used at the upper level of the algorithm to calculate their structural residual, which determines the degree of difference (closeness) of their geometries. As a result, minimizing this residual, when training a neural network at the upper level, implements a computational algorithm that yields geometrically close source distributions of different fields. The article examines in detail the possibilities of optimizing some elements of the neural networks and the algorithms used (datasets, training process, specific form of loss functions, etc.) Test calculations for model problem demonstrate high quality of joint inversion by our optimized neural networks approach. Calculations were also carried out for the joint processing of real-feald data from gravity and magnetic exploration in Jussara region, Goias State, Brazil. The article also considers the issue of determining in joint field inversion not only the geometric distribution of sources, but also their physical intensities.

Figures (16)

• Figure 1: Architecture of the used "U-Net" type network by layers. The input is a tensor of dimensionality $1\times32\times32$. It corresponds to the grid values of size $32\times32$ of the gravitational or magnetic field measured in a rectangle, which is divided into cells by a two-dimensional Cartesian grid of dimensionality 32x32. The output of the network is a tensor of size $16\times32\times32$. It contains the grid values of the source density in a three-dimensional domain, which is divided into cells by a Cartesian grid of dimensionality $16\times32\times32$.
• Figure 2: Training scheme of the proposed two-level neural network, including two "U-Net" type networks and a structural residual.
• Figure 3: Comparison of the behavior of the curves $\operatorname{Loss}_{\text{train}}(\text{epoch})$ and $\operatorname{Loss}_{\text{test}}(\text{epoch})$ for different values of the parameter $\alpha$ (solid lines) and for $\alpha=0$ (dots).
• Figure 4: An example of reconstructing gravimetric and magnetic sources. The upper figures correspond to the distribution of gravitational field sources, the lower ones represents the distribution of magnetic field sources. The left column shows the model exact solutions, the middle column shows the result of separate source reconstruction, and the right column shows the result of joint reconstruction.
• Figure 5: Left: dependence of accuracy $\operatorname{Loss}_{\text{result}}$ on $\alpha$; Right: dependence of the iteration number $\operatorname{iter}_{\text{stop}}$ on $\alpha$.