# Copyright (c) 2018 The Harmonica Developers.
# Distributed under the terms of the BSD 3-Clause License.
# SPDX-License-Identifier: BSD-3-Clause
#
# This code is part of the Fatiando a Terra project (https://www.fatiando.org)
#
"""
Forward modelling for point masses
"""
import numpy as np
from choclo.constants import GRAVITATIONAL_CONST
from choclo.point import (
gravity_e,
gravity_ee,
gravity_en,
gravity_eu,
gravity_n,
gravity_nn,
gravity_nu,
gravity_pot,
gravity_u,
gravity_uu,
)
from numba import jit, prange
from .utils import check_coordinate_system, distance_spherical_core
[docs]
def point_gravity(
coordinates,
points,
masses,
field,
coordinate_system="cartesian",
parallel=True,
dtype="float64",
):
r"""
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. But ``g_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. the ``g_ez`` is the non-diagonal **easting-downward** tensor
component.
.. important::
- The gravitational potential is returned in
:math:`\text{J}/\text{kg}`.
- The gravity acceleration components are returned in mGal
(:math:`\text{m}/\text{s}^2`).
- The tensor components are returned in Eotvos (:math:`\text{s}^{-2}`).
Parameters
----------
coordinates : list of arrays
List of arrays containing the coordinates of computation points in the
following order: ``easting``, ``northing`` and ``upward`` (if
coordinates given in Cartesian coordinates), 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 coordinates), 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``
- 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``.
Default ``cartesian``.
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 in ``coordinates``.
The potential is given in SI units, the accelerations in mGal and the
Marussi tensor components in Eotvos.
Notes
-----
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.
"""
# Sanity checks for coordinate_system
check_coordinate_system(
coordinate_system, valid_coord_systems=("cartesian", "spherical")
)
# 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
kernel = get_kernel(coordinate_system, field)
dispatcher(coordinate_system, parallel)(
*coordinates, *points, masses, result, kernel
)
# Invert sign of gravity_u, gravity_eu, gravity_nu
if field in ("g_z", "g_ez", "g_ze", "g_nz", "g_zn"):
result *= -1
# Convert to more convenient units
if field in ("g_e", "g_n", "g_z"):
result *= 1e5 # SI to mGal
tensors = ("g_ee", "g_nn", "g_zz", "g_en", "g_ez", "g_nz", "g_ne", "g_ze", "g_zn")
if field in tensors:
result *= 1e9 # SI to Eotvos
return result.reshape(cast.shape)
def dispatcher(coordinate_system, parallel):
"""
Return the appropriate forward model function
"""
dispatchers = {
"cartesian": {
True: point_mass_cartesian_parallel,
False: point_mass_cartesian_serial,
},
"spherical": {
True: point_mass_spherical_parallel,
False: point_mass_spherical_serial,
},
}
return dispatchers[coordinate_system][parallel]
def get_kernel(coordinate_system, field):
"""
Return the appropriate kernel
"""
kernels = {
"cartesian": {
"potential": gravity_pot,
"g_e": gravity_e,
"g_n": gravity_n,
"g_z": gravity_u,
# diagonal tensor components
"g_ee": gravity_ee,
"g_nn": gravity_nn,
"g_zz": gravity_uu,
# non-diagonal tensor components
"g_en": gravity_en,
"g_ez": gravity_eu,
"g_nz": gravity_nu,
"g_ne": gravity_en,
"g_ze": gravity_eu,
"g_zn": gravity_nu,
},
"spherical": {
"potential": potential_spherical,
"g_z": gravity_u_spherical,
"g_n": None,
"g_e": None,
},
}
if field not in kernels[coordinate_system]:
raise ValueError("Gravitational field '{}' not recognized".format(field))
kernel = kernels[coordinate_system][field]
if kernel is None:
raise NotImplementedError
return kernel
# ------------------------------------------
# Kernel functions for Spherical coordinates
# ------------------------------------------
@jit(nopython=True)
def 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 * GRAVITATIONAL_CONST
# Acceleration components
# -------------------
@jit(nopython=True)
def gravity_u_spherical(
longitude, cosphi, sinphi, radius, longitude_p, cosphi_p, sinphi_p, radius_p
):
"""
Kernel for upward component of gravitational acceleration
Use spherical coordinates
"""
distance, cospsi, _ = distance_spherical_core(
longitude, cosphi, sinphi, radius, longitude_p, cosphi_p, sinphi_p, radius_p
)
delta_z = radius - radius_p * cospsi
return -GRAVITATIONAL_CONST * delta_z / distance**3
def point_mass_cartesian(
easting,
northing,
upward,
easting_p,
northing_p,
upward_p,
masses,
out,
forward_func,
):
"""
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``.
forward_func : func
forward_func function that will be used to compute the gravitational
field on the computation points. It could be one of the forward
modelling functions in :mod:`choclo.point`.
"""
for l in prange(easting.size):
for m in range(easting_p.size):
out[l] += forward_func(
easting[l],
northing[l],
upward[l],
easting_p[m],
northing_p[m],
upward_p[m],
masses[m],
)
def point_mass_spherical(
longitude, latitude, radius, longitude_p, latitude_p, radius_p, masses, out, kernel
):
"""
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 prange(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],
)
# Define jitted versions of the forward modelling functions
point_mass_cartesian_serial = jit(nopython=True)(point_mass_cartesian)
point_mass_cartesian_parallel = jit(nopython=True, parallel=True)(point_mass_cartesian)
point_mass_spherical_serial = jit(nopython=True)(point_mass_spherical)
point_mass_spherical_parallel = jit(nopython=True, parallel=True)(point_mass_spherical)