harmonica.point_gravity
Contents
harmonica.point_gravity¶
- harmonica.point_gravity(coordinates, points, masses, field, coordinate_system='cartesian', parallel=True, dtype='float64')[source]¶
Compute gravitational fields of point masses.
It can compute the gravitational fields of point masses on a set of computation points defined either in Cartesian or geocentric spherical coordinates.
The gravitational potential field generated by a point mass with mass \(m\) located at a point \(Q\) on a computation point \(P\) can be computed as:
\[V(P) = \frac{G m}{l},\]where \(G\) is the gravitational constant and \(l\) is the Euclidean distance between \(P\) and \(Q\) [Blakely1995].
In Cartesian coordinates, the points \(P\) and \(Q\) are given by \(x\), \(y\) and \(z\) coordinates, which can be translated into
northing
,easting
andupward
, respectively. If \(P\) is located at \((x, y, z)\), and \(Q\) at \((x_p, y_p, z_p)\), the distance \(l\) can be computed as:\[l = \sqrt{ (x - x_p)^2 + (y - y_p)^2 + (z - z_p)^2 }.\]The gradient of the potential, also known as the gravitational acceleration vector \(\vec{g}\), is defined as:
\[\vec{g} = \nabla V\]and has components \(g_{northing}(P)\), \(g_{easting}(P)\) and \(g_{upward}(P)\) given by
\[g_{northing}(P) = - \frac{G m}{l^3} (x - x_p),\]\[g_{easting}(P) = - \frac{G m}{l^3} (y - y_p)\]and
\[g_{upward}(P) = - \frac{G m}{l^3} (z - z_p).\]We define the downward component of the gravitational acceleration as the opposite of \(g_{upward}\) (remember that \(z\) points upwards):
\[g_{z}(P) = \frac{G m}{l^3} (z - z_p).\]On a geocentric spherical coordinate system, the points \(P\) and \(Q\) are given by the
longitude
,latitude
andradius
coordinates, i.e. \(\lambda\), \(\varphi\) and \(r\), respectively. On this coordinate system, the Euclidean distance between \(P(r, \varphi, \lambda)\) and \(Q(r_p, \varphi_p, \lambda_p)\) can be calculated as follows [Grombein2013]:\[l = \sqrt{ r^2 + r_p^2 - 2 r r_p \cos \Psi },\]where
\[\cos \Psi = \sin \varphi \sin \varphi_p + \cos \varphi \cos \varphi_p \cos(\lambda - \lambda_p).\]The radial component of the acceleration vector on a local North-oriented system whose origin is located on the point \(P(r, \varphi, \lambda)\) is given by [Grombein2013]:
\[g_r(P) = - \frac{G m}{l^3} (r - r_p \cos \Psi).\]We define the downward component of the gravitational acceleration \(g_z\) as the opposite of the radial component:
\[g_z(P) = \frac{G m}{l^3} (r - r_p \cos \Psi).\]Warning
When working in Cartesian coordinates, the z direction points upwards, i.e. positive and negative values of
upward
represent points above and below the surface, respectively. But remember that theg_z
field returns the downward component of the gravitational acceleration.Warning
When working in geocentric spherical coordinates, remember that the
g_z
field returns the downward component of the gravitational acceleration on the local North oriented coordinate system. It is equivalent to the opposite of the radial component, therefore it’s positive if the acceleration vector points inside the spheroid.- Parameters
coordinates (list or array) – List or array containing the coordinates of computation points in the following order:
easting
,northing
andupward
(if coordinates given in Cartesian coordinates), orlongitude
,latitude
andradius
(if given on a spherical geocentric coordinate system). Alleasting
,northing
andupward
should be in meters. Bothlongitude
andlatitude
should be in degrees andradius
in meters.points (list or array) – List or array containing the coordinates of the point masses in the following order:
easting
,northing
andupward
(if coordinates given in Cartesian coordinates), orlongitude
,latitude
andradius
(if given on a spherical geocentric coordinate system). Alleasting
,northing
andupward
should be in meters. Bothlongitude
andlatitude
should be in degrees andradius
in meters.masses (list or array) – List or array containing the mass of each point mass in kg.
field (str) –
Gravitational field that wants to be computed. The available fields coordinates are:
Gravitational potential:
potential
Downward acceleration:
g_z
Northing acceleration:
g_northing
Easting acceleration:
g_easting
- Tensor components:
g_ee
g_nn
g_zz
g_en
g_ez
g_nz
coordinate_system (str (optional)) – Coordinate system of the coordinates of the computation points and the point masses. Available coordinates systems:
cartesian
,spherical
. Defaultcartesian
.parallel (bool (optional)) – If True the computations will run in parallel using Numba built-in parallelization. If False, the forward model will run on a single core. Might be useful to disable parallelization if the forward model is run by an already parallelized workflow. Default to True.
dtype (data-type (optional)) – Data type assigned to resulting gravitational field. Default to
np.float64
.
- Returns
result (array) – Gravitational field generated by the
point_mass
on the computation points defined incoordinates
. The potential is given in SI units, the accelerations in mGal and the Marussi tensor components in Eotvos.