harmonica.point_gravity#
- harmonica.point_gravity(coordinates, points, masses, field, coordinate_system='cartesian', parallel=True, dtype='float64')[source]#
Compute gravitational fields of point masses.
Compute the gravitational potential, gravitational acceleration and tensor components generated by a collection of point masses on a set of observation points defined either in Cartesian or geocentric spherical coordinates.
Warning
The vertical direction points upwards, i.e. positive and negative values of
upward
represent points above and below the surface, respectively. Butg_z
field returns the downward component of the gravitational acceleration so that positive density contrasts produce positive anomalies. The same applies to the tensor components, i.e. theg_ez
is the non-diagonal easting-downward tensor component.Important
The gravitational potential is returned in \(\text{J}/\text{kg}\).
The gravity acceleration components are returned in mGal (\(\text{m}/\text{s}^2\)).
The tensor components are returned in Eotvos (\(\text{s}^{-2}\)).
- Parameters:
- coordinates
list
of
arrays
List of arrays 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
orarray
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
orarray
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
Easting acceleration:
g_e
Northing acceleration:
g_n
Downward acceleration:
g_z
- 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
.- parallelbool (
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.
- dtypedata-type (
optional
) Data type assigned to resulting gravitational field. Default to
np.float64
.
- coordinates
- 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.
- result
Notes
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 itâ€™s positive if the acceleration vector points inside the spheroid.
Examples using harmonica.point_gravity
#
Point Masses in Cartesian Coordinates