🔎 Overview

Invert4geom provides a series of tools for conducting a specific style of gravity inversion. Many gravity inversions aim to model the density distribution of the subsurface. These are commonly used to identify bodies of anomalous densities, such as igneous intrusions or ore deposits. The typical way these are performed is to discretize the subsurface into a series of finite volumes, such as cubes or prisms, where the shape of the volumes doesn’t change. The inversion then alters the density values of each of these volumes to match the observed gravity anomaly. In these inversions the density values changes, while the geometry of the volumes remains unchanged. These types of inversions may be referred to as density inversions. Here, instead, we are performing geometric inversions.

Geometric inversions are essentially the opposite. The density values of the volumes in the discretized model remain unchanged, while their geometry is altered. Here we use layers of vertical right-rectangular prisms and alter their tops and bottoms during the inversion. Typically use cases for these style of inversion are modeling the topography of the Moho, the contact between sediment and basement, or the shape of the seafloor in locations where it is not easily mapped.

Currently, this package is only intended to perform inversions using right rectangular prisms. Other types of volumes, such as tesseroids, are currently not implemented.

Much of this software was developed as part of my Ph.D. thesis. For detailed description of the theory and implementation of this inversion, as well as many synthetic tests and a real-world application to modelling bathymetry, see chapter 3 and 4 of my thesis, available here. The code was originally included in this GitHub repository, but much of it has been migrated here.

Modules

Invert4Geom consists of 7 modules:

Inversion

This contains the core tools for performing the inversion. These tools allow for creating the Jacobian matrix, performing the least squares solution, running the inversion, determining when to end the inversion, and updating the misfit and gravity values between each iteration.

Cross Validation

This module contains the code necessary for performing the various cross-validation routines. These routines are split into 2 categories; gravity and constraints cross validation.

Gravity cross validations

The gravity cross validations are those which split the gravity data into testing and training sets, perform the inversion with the training set, and compare the forward gravity of the inverted topography with the un-used testing data. This is a Generalized Cross Validation, specifically a hold-out cross-validation, as described in Uieda & Barbosa (2017). This is used here for estimated the optimal regularization damping parameter during the inversion.

Constraint cross validations

The constraint cross validations are those which use apriori points of known elevation of the surface of in interest to determine optimal inversion parameters. The inversion is performed without including these constraint points, and the inverted surface is compared with the elevations of these constraints points to give a score. This style of validation is used here for estimating the optimal reference level and density contrast of the surface of interest.

Regional

This module contains tools for estimating the regional gravity field. In many styles of inversions, the regional component of the gravity misfit should be removed to help isolate the gravity effects of the layer of interest. This is common in inversions for topography, bathymetry, or sedimentary basins. We provide four methods of estimating the regional component. 1) Low pass filtering the data, 2) fitting a surface to the data with a user defined trend, 3) fitting a set of deep equivalent sources to the data and predicting the gravity effect of those sources, and 4) using apriori constraints to determine the regional field, by assuming the residual component is low at the those points.

Optimization

Functions for performing optimizations. This uses the package Optuna which given a parameter space, an objective function, and a sampling routine, will perform optimization to minimize (or maximize) the object function. This is used in the Regional module for finding the optimal regional separation methods.

Plotting

Various function for plotting maps and graphs used throughout the other modules.

Utils

Utilities used throughout the other modules.

Synthetic

This module has tools for creating synthetic topography and contaminating data with random Gaussian noise. This is mostly used in testing and in the User Guide notebooks.