"""
Forward modelling for point masses
"""
import numpy as np
from numba import jit
from ..constants import GRAVITATIONAL_CONST
from .utils import check_coordinate_system, distance_cartesian, distance_spherical_core
[docs]def point_mass_gravity(
coordinates, points, masses, field, coordinate_system="cartesian", dtype="float64"
):
r"""
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
:math:`m` located at a point :math:`Q` on a computation point :math:`P` can
be computed as:
.. math::
V(P) = \frac{G m}{l},
where :math:`G` is the gravitational constant and :math:`l` is the
Euclidean distance between :math:`P` and :math:`Q` [Blakely1995]_.
In Cartesian coordinates, the points :math:`P` and :math:`Q` are given by
:math:`x`, :math:`y` and :math:`z` coordinates, which can be translated
into ``northing``, ``easting`` and ``upward``, respectively. If :math:`P`
is located at :math:`(x, y, z)`, and :math:`Q` at :math:`(x_p, y_p, z_p)`,
the distance :math:`l` can be computed as:
.. math::
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 :math:`\vec{g}`, is defined as:
.. math::
\vec{g} = \nabla V
and has components :math:`g_{northing}(P)`, :math:`g_{easting}(P)` and
:math:`g_{upward}(P)` given by
.. math::
g_{northing}(P) = - \frac{G m}{l^3} (x - x_p),
.. math::
g_{easting}(P) = - \frac{G m}{l^3} (y - y_p)
and
.. math::
g_{upward}(P) = - \frac{G m}{l^3} (z - z_p).
We define the downward component of the gravitational acceleration as the
opposite of :math:`g_{upward}` (remember that :math:`z` points upwards):
.. math::
g_{z}(P) = \frac{G m}{l^3} (z - z_p).
On a geocentric spherical coordinate system, the points :math:`P` and
:math:`Q` are given by the ``longitude``, ``latitude`` and ``radius``
coordinates, i.e. :math:`\lambda`, :math:`\varphi` and :math:`r`,
respectively. On this coordinate system, the Euclidean distance between
:math:`P(r, \varphi, \lambda)` and :math:`Q(r_p, \varphi_p, \lambda_p)` can
be calculated as follows [Grombein2013]_:
.. math::
l = \sqrt{ r^2 + r_p^2 - 2 r r_p \cos \Psi },
where
.. math::
\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 :math:`P(r, \varphi, \lambda)`
is given by [Grombein2013]_:
.. math::
g_r(P) = - \frac{G m}{l^3} (r - r_p \cos \Psi).
We define the downward component of the gravitational acceleration
:math:`g_z` as the opposite of the radial component:
.. math::
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 the
``g_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`` and ``upward`` (if
coordinates given in Cartesian coordiantes), or ``longitude``,
``latitude`` and ``radius`` (if given on a spherical geocentric
coordinate system).
All ``easting``, ``northing`` and ``upward`` should be in meters.
Both ``longitude`` and ``latitude`` should be in degrees and ``radius``
in meters.
points : list or array
List or array containing the coordinates of the point masses in the
following order: ``easting``, ``northing`` and ``upward`` (if
coordinates given in Cartesian coordiantes), or ``longitude``,
``latitude`` and ``radius`` (if given on a spherical geocentric
coordinate system).
All ``easting``, ``northing`` and ``upward`` should be in meters.
Both ``longitude`` and ``latitude`` should be in degrees and ``radius``
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``
coordinate_system : str (optional)
Coordinate system of the coordinates of the computation points and the
point masses.
Available coordinates systems: ``cartesian``, ``spherical``.
Default ``cartesian``.
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 in ``coordinates``.
The potential is given in SI units, the accelerations in mGal and the
Marussi tensor components in Eotvos.
"""
# Organize dispatchers and kernel functions inside dictionaries
dispatchers = {
"cartesian": jit_point_mass_cartesian,
"spherical": jit_point_mass_spherical,
}
kernels = {
"cartesian": {
"potential": kernel_potential_cartesian,
"g_z": kernel_g_z_cartesian,
"g_northing": kernel_g_northing_cartesian,
"g_easting": kernel_g_easting_cartesian,
},
"spherical": {
"potential": kernel_potential_spherical,
"g_z": kernel_g_z_spherical,
},
}
# Sanity checks for coordinate_system and field
check_coordinate_system(coordinate_system)
if field not in kernels[coordinate_system]:
raise ValueError("Gravitational field {} not recognized".format(field))
# Figure out the shape and size of the output array
cast = np.broadcast(*coordinates[:3])
result = np.zeros(cast.size, dtype=dtype)
# Prepare arrays to be passed to the jitted functions
coordinates = tuple(np.atleast_1d(i).ravel() for i in coordinates[:3])
points = tuple(np.atleast_1d(i).ravel() for i in points[:3])
masses = np.atleast_1d(masses).ravel()
# Sanity checks
if masses.size != points[0].size:
raise ValueError(
"Number of elements in masses ({}) ".format(masses.size)
+ "mismatch the number of points ({})".format(points[0].size)
)
# Compute gravitational field
dispatchers[coordinate_system](
*coordinates, *points, masses, result, kernels[coordinate_system][field]
)
result *= GRAVITATIONAL_CONST
# Convert to more convenient units
if field in ("g_easting", "g_northing", "g_z"):
result *= 1e5 # SI to mGal
return result.reshape(cast.shape)
@jit(nopython=True)
def jit_point_mass_cartesian(
easting, northing, upward, easting_p, northing_p, upward_p, masses, out, kernel
): # pylint: disable=invalid-name
"""
Compute gravitational field of point masses in Cartesian coordinates
Parameters
----------
easting, northing, upward : 1d-arrays
Coordinates of computation points in Cartesian coordinate system.
easting_p, northing_p, upward_p : 1d-arrays
Coordinates of point masses in Cartesian coordinate system.
masses : 1d-array
Mass of each point mass in SI units.
out : 1d-array
Array where the gravitational field on each computation point will be
appended.
It must have the same size of ``easting``, ``northing`` and ``upward``.
kernel : func
Kernel function that will be used to compute the gravitational field on
the computation points.
"""
for l in range(easting.size):
for m in range(easting_p.size):
out[l] += masses[m] * kernel(
easting[l],
northing[l],
upward[l],
easting_p[m],
northing_p[m],
upward_p[m],
)
@jit(nopython=True)
def kernel_potential_cartesian(
easting, northing, upward, easting_p, northing_p, upward_p
):
"""
Kernel function for gravitational potential field in Cartesian coordinates
"""
distance = distance_cartesian(
(easting, northing, upward), (easting_p, northing_p, upward_p)
)
return 1 / distance
@jit(nopython=True)
def kernel_g_z_cartesian(easting, northing, upward, easting_p, northing_p, upward_p):
"""
Kernel for downward component of gravitational gradient in Cartesian coords
"""
distance = distance_cartesian(
(easting, northing, upward), (easting_p, northing_p, upward_p)
)
# Remember that the ``g_z`` field returns the downward component of the
# gravitational acceleration. As a consequence, it is multiplied by -1.
# Notice that the ``g_z`` does not have the minus signal observed at the
# compoents ``g_northing`` and ``g_easting``.
return (upward - upward_p) / distance ** 3
@jit(nopython=True)
def kernel_g_northing_cartesian(
easting, northing, upward, easting_p, northing_p, upward_p
):
"""
Kernel function for northing component of gravitational gradient in
Cartesian coordinates
"""
distance = distance_cartesian(
(easting, northing, upward), (easting_p, northing_p, upward_p)
)
return -(northing - northing_p) / distance ** 3
@jit(nopython=True)
def kernel_g_easting_cartesian(
easting, northing, upward, easting_p, northing_p, upward_p
):
"""
Kernel function for easting component of gravitational gradient in
Cartesian coordinates
"""
distance = distance_cartesian(
(easting, northing, upward), (easting_p, northing_p, upward_p)
)
return -(easting - easting_p) / distance ** 3
@jit(nopython=True)
def jit_point_mass_spherical(
longitude, latitude, radius, longitude_p, latitude_p, radius_p, masses, out, kernel
): # pylint: disable=invalid-name
"""
Compute gravitational field of point masses in spherical coordinates
Parameters
----------
longitude, latitude, radius : 1d-arrays
Coordinates of computation points in spherical geocentric coordinate
system.
longitude_p, latitude_p, radius_p : 1d-arrays
Coordinates of point masses in spherical geocentric coordinate system.
masses : 1d-array
Mass of each point mass in SI units.
out : 1d-array
Array where the gravitational field on each computation point will be
appended.
It must have the same size of ``longitude``, ``latitude`` and
``radius``.
kernel : func
Kernel function that will be used to compute the gravitational field on
the computation points.
"""
# Compute quantities related to computation point
longitude = np.radians(longitude)
latitude = np.radians(latitude)
cosphi = np.cos(latitude)
sinphi = np.sin(latitude)
# Compute quantities related to point masses
longitude_p = np.radians(longitude_p)
latitude_p = np.radians(latitude_p)
cosphi_p = np.cos(latitude_p)
sinphi_p = np.sin(latitude_p)
# Compute gravitational field
for l in range(longitude.size):
for m in range(longitude_p.size):
out[l] += masses[m] * kernel(
longitude[l],
cosphi[l],
sinphi[l],
radius[l],
longitude_p[m],
cosphi_p[m],
sinphi_p[m],
radius_p[m],
)
@jit(nopython=True)
def kernel_potential_spherical(
longitude, cosphi, sinphi, radius, longitude_p, cosphi_p, sinphi_p, radius_p
):
"""
Kernel function for potential gravitational field in spherical coordinates
"""
distance, _, _ = distance_spherical_core(
longitude, cosphi, sinphi, radius, longitude_p, cosphi_p, sinphi_p, radius_p
)
return 1 / distance
@jit(nopython=True)
def kernel_g_z_spherical(
longitude, cosphi, sinphi, radius, longitude_p, cosphi_p, sinphi_p, radius_p
):
"""
Kernel for downward component of gravitational gradient in spherical coords
"""
distance, cospsi, _ = distance_spherical_core(
longitude, cosphi, sinphi, radius, longitude_p, cosphi_p, sinphi_p, radius_p
)
delta_z = radius - radius_p * cospsi
return delta_z / distance ** 3