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Gravity Inversion

Gravity Inversion#

This Inversion Test Problem explores the gravitational acceleration of a three-dimensional (example 1) and pseude-2D (example 2) density model onto specified receiver locations. In this example, only the z-component of the gravityational force is calculated. The underlying code itself is capable of calculating all three gravity components and six gradiometry components and could be modified quickly if there is the need.

The gravitational acceleration is calculated using Newton’s law of universal gravitation:

\[ g (r) =- G \frac{ m} {r^2} \]

With G being the gravitational constant, r is the distance of the mass to the receiver and m is the overall mass of the model, which depends on the density \(\rho\) and the volume V:

\[ m = \int_V {\rho(r) dV} \]

Here, we solve volume integral for the vertical component of \(g\) analytically, using the approach by Plouff et al., 1976:

\[ g_z(M,N)=G \rho \sum_{i=1}^2 \sum_{j=1}^2 \sum_{k=1}^2 (-1)^{i+j+k} [tan^{-1} \frac{a_ib_j}{z_k R_{ijk}} - a_i ln(R_{ijk} + b_j) - b_j ln(R_{ijk} + a_i)] \]

with \(R_{ijk}=\sqrt{a_i^2 + b_j^2 + z_k^2}\) and \(a_i, b_j, z_k\) being the distances from receiver N to the nodes of the current prism M (i.e. grid cell) in x, y, and z directions. It is assumed that \(\rho=const.\) within each grid cell. For more information, please see the original paper:

Plouff, D., 1976. Gravity and magnetic fields of polygonal prisms and application to magnetic terrain corrections. Geophysics, 41(4), pp.727-741

For further reading, see also Nagy et al., 2000:

Nagy, D., Papp, G. and Benedek, J., 2000. The gravitational potential and its derivatives for the prism. Journal of Geodesy, 74(7), pp.552-560

Example details:

  1. Model: Density values on a regularly spaced, rectangular grid. Example-model one is a 3D cube of low density (10 \(kgm^{-3}\)) containing a centrally located high-density cube (1000 \(kgm^{-3}\)). Example-model two repeats Figure 2 of Last and Kubik, 1983, which means a pseudo-2D model containing zero-density background cells and centrally high-density cells in the shape of a cross (1000 \(kgm^{-3}\)).

    Last, B.J. and Kubik, K., 1983. Compact gravity inversion. Geophysics, 48(6), pp.713-721

  2. Returned data: Gravitational acceleration (vertical component).

  3. Forward: The volume integral is solved analytically following the above described approach by Plouff et al., 1976.

Contribution Metadata for Gravity calculation from a density model

This example implements a simple gravityforward problem. The model represents density withinthe earth on a 3D Cartesian grid.

Author:

Hannes Hollmann

Contact:

Hannes Hollmann (hannes.hollmann@anu.edu.au)

Example usage for GravityInversion#

Code block below is generated automatically based on our tests and inspection of the example objects. We’d like to also refer you to the API Reference for more about the Espresso API.

 1import espresso
 2
 3# Create a GravityInversion object
 4my_gravity_inversion = espresso.GravityInversion(example_number=1)
 5
 6# Guaranteed API
 7model_size     = my_gravity_inversion.model_size
 8data_size      = my_gravity_inversion.data_size
 9null_model     = my_gravity_inversion.starting_model
10good_model     = my_gravity_inversion.good_model
11given_data     = my_gravity_inversion.data
12synthetic_data = my_gravity_inversion.forward(good_model)
13
14# Optional API
15cov_matrix         = my_gravity_inversion.covariance_matrix
16jacobian           = my_gravity_inversion.jacobian(good_model)
17model_fig          = my_gravity_inversion.plot_model(good_model)
18data_fig           = my_gravity_inversion.plot_data(given_data)

Additional attributes to explore: [list_capabilities].

LICENCE#

Copyright 2021-2022 InLab All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

  1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

  2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

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