harmonica.prism_gravity

harmonica.prism_gravity(coordinates, prisms, density, field, dtype='float64', disable_checks=False)

Gravitational fields of right-rectangular prisms in Cartesian coordinates

The gravitational fields are computed through the analytical solutions given by [Nagy2000] and [Nagy2002], which are valid on the entire domain. This means that the computation can be done at any point, either outside or inside the prism.

This implementation makes use of the modified arctangent function proposed by [Fukushima2019] (eq. 12) so that the potential field to satisfies Poisson’s equation in the entire domain. Moreover, the logarithm function was also modified in order to solve the singularities that the analytical solution has on some points (see [Nagy2000]).

Warning

The z direction points upwards, i.e. positive and negative values of upward represent points above and below the surface, respectively. But remember that the g_z field returns the downward component of the gravitational acceleration so that positive density contrasts produce positive anomalies.

Parameters

• coordinates (list or 1d-array) – List or array containing easting, northing and upward of the computation points defined on a Cartesian coordinate system. All coordinates should be in meters.

• prisms (list, 1d-array, or 2d-array) – List or array containing the coordinates of the prism(s) in the following order: west, east, south, north, bottom, top in a Cartesian coordinate system. All coordinates should be in meters. Coordinates for more than one prism can be provided. In this case, prisms should be a list of lists or 2d-array (with one prism per line).

• density (list or array) – List or array containing the density of each prism in kg/m^3.

• field (str) –

  Gravitational field that wants to be computed. The available fields are:

  • Gravitational potential: potential

  • Downward acceleration: g_z

• dtype (data-type (optional)) – Data type assigned to the resulting gravitational field. Default to np.float64.

• disable_checks (bool (optional)) – Flag that controls whether to perform a sanity check on the model. Should be set to True only when it is certain that the input model is valid and it does not need to be checked. Default to False.

Returns

result (array) – Gravitational field generated by the prisms on the computation points.

Examples

Compute gravitational effect of a single a prism

>>> # Define prisms boundaries, it must be beneath the surface
>>> prism = [-34, 5, -18, 14, -345, -146]
>>> # Set prism density to 2670 kg/m³
>>> density = 2670
>>> # Define a computation point above its center, at 30 meters above the
>>> # surface
>>> coordinates = (130, 75, 30)
>>> # Define three computation points along the easting axe at 30m above
>>> # the surface
>>> coordinates = ([-40, 0, 40], [0, 0, 0], [30, 30, 30])
>>> # Compute the downward component of the gravitational acceleration that
>>> # the prism generates on the computation points
>>> gz = prism_gravity(coordinates, prism, density, field="g_z")
>>> print("({:.5f}, {:.5f}, {:.5f})".format(*gz))
(0.06551, 0.06628, 0.06173)

Define two prisms with positive and negative density contrasts

>>> prisms = [[-134, -5, -45, 45, -200, -50], [5, 134, -45, 45, -180, -30]]
>>> densities = [-300, 300]
>>> # Compute the g_z that the prisms generate on the computation points
>>> gz = prism_gravity(coordinates, prisms, densities, field="g_z")
>>> print("({:.5f}, {:.5f}, {:.5f})".format(*gz))
(-0.05379, 0.02908, 0.11235)