# Gravity Density#

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

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 GravityDensity#

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


Additional attributes to explore: [list_capabilities].