# Source code for harmonica.forward.point_mass

# Copyright (c) 2018 The Harmonica Developers.
#
# This code is part of the Fatiando a Terra project (https://www.fatiando.org)
#
"""
Forward modelling for point masses
"""
import numpy as np
from numba import jit, prange

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", parallel=True, 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 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 - 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. 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. """ # 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 ) 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)
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": 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, "g_northing": None, "g_easting": 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 @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 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 def point_mass_cartesian( easting, northing, upward, easting_p, northing_p, upward_p, masses, out, kernel ): # pylint: disable=invalid-name,not-an-iterable """ 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 prange(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], ) def point_mass_spherical( longitude, latitude, radius, longitude_p, latitude_p, radius_p, masses, out, kernel ): # pylint: disable=invalid-name,not-an-iterable """ 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 # pylint: disable=invalid-name 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)