# Source code for harmonica.forward.prism

```
# 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 prisms
"""
import numpy as np
from numba import jit, prange
from ..constants import GRAVITATIONAL_CONST
[docs]def prism_gravity(
coordinates,
prisms,
density,
field,
parallel=True,
dtype="float64",
disable_checks=False,
):
"""
Gravitational fields of right-rectangular prisms in Cartesian coordinates
The gravitational fields are computed through the analytical solutions
given by [Nagy2000]_ and [Nagy2002]_, which are valid on the entire domain.
This means that the computation can be done at any point, either outside or
inside the prism.
This implementation makes use of the modified arctangent function proposed
by [Fukushima2019]_ (eq. 12) so that the potential field to satisfies
Poisson's equation in the entire domain. Moreover, the logarithm function
was also modified in order to solve the singularities that the analytical
solution has on some points (see [Nagy2000]_).
.. warning::
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 so that positive density
contrasts produce positive anomalies.
Parameters
----------
coordinates : list or 1d-array
List or array containing ``easting``, ``northing`` and ``upward`` of
the computation points defined on a Cartesian coordinate system.
All coordinates should be in meters.
prisms : list, 1d-array, or 2d-array
List or array containing the coordinates of the prism(s) in the
following order:
west, east, south, north, bottom, top in a Cartesian coordinate system.
All coordinates should be in meters. Coordinates for more than one
prism can be provided. In this case, *prisms* should be a list of lists
or 2d-array (with one prism per line).
density : list or array
List or array containing the density of each prism in kg/m^3.
field : str
Gravitational field that wants to be computed.
The available fields are:
- Gravitational potential: ``potential``
- Downward acceleration: ``g_z``
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 the resulting gravitational field. Default to
``np.float64``.
disable_checks : bool (optional)
Flag that controls whether to perform a sanity check on the model.
Should be set to ``True`` only when it is certain that the input model
is valid and it does not need to be checked.
Default to ``False``.
Returns
-------
result : array
Gravitational field generated by the prisms on the computation points.
Examples
--------
Compute gravitational effect of a single a prism
>>> # Define prisms boundaries, it must be beneath the surface
>>> prism = [-34, 5, -18, 14, -345, -146]
>>> # Set prism density to 2670 kg/m³
>>> density = 2670
>>> # Define a computation point above its center, at 30 meters above the
>>> # surface
>>> coordinates = (130, 75, 30)
>>> # Define three computation points along the easting axe at 30m above
>>> # the surface
>>> coordinates = ([-40, 0, 40], [0, 0, 0], [30, 30, 30])
>>> # Compute the downward component of the gravitational acceleration that
>>> # the prism generates on the computation points
>>> gz = prism_gravity(coordinates, prism, density, field="g_z")
>>> print("({:.5f}, {:.5f}, {:.5f})".format(*gz))
(0.06551, 0.06628, 0.06173)
Define two prisms with positive and negative density contrasts
>>> prisms = [[-134, -5, -45, 45, -200, -50], [5, 134, -45, 45, -180, -30]]
>>> densities = [-300, 300]
>>> # Compute the g_z that the prisms generate on the computation points
>>> gz = prism_gravity(coordinates, prisms, densities, field="g_z")
>>> print("({:.5f}, {:.5f}, {:.5f})".format(*gz))
(-0.05379, 0.02908, 0.11235)
"""
kernels = {"potential": kernel_potential, "g_z": kernel_g_z}
if field not in kernels:
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)
# Convert coordinates, prisms and density to arrays with proper shape
coordinates = tuple(np.atleast_1d(i).ravel() for i in coordinates[:3])
prisms = np.atleast_2d(prisms)
density = np.atleast_1d(density).ravel()
# Sanity checks
if not disable_checks:
if density.size != prisms.shape[0]:
raise ValueError(
"Number of elements in density ({}) ".format(density.size)
+ "mismatch the number of prisms ({})".format(prisms.shape[0])
)
_check_prisms(prisms)
# Compute gravitational field
dispatcher(parallel)(coordinates, prisms, density, kernels[field], result)
result *= GRAVITATIONAL_CONST
# Convert to more convenient units
if field == "g_z":
result *= 1e5 # SI to mGal
return result.reshape(cast.shape)
def dispatcher(parallel):
"""
Return the parallelized or serialized forward modelling function
"""
dispatchers = {
True: jit_prism_gravity_parallel,
False: jit_prism_gravity_serial,
}
return dispatchers[parallel]
def _check_prisms(prisms):
"""
Check if prisms boundaries are well defined
Parameters
----------
prisms : 2d-array
Array containing the boundaries of the prisms in the following order:
``w``, ``e``, ``s``, ``n``, ``bottom``, ``top``.
The array must have the following shape: (``n_prisms``, 6), where
``n_prisms`` is the total number of prisms.
This array of prisms must have valid boundaries.
Run ``_check_prisms`` before.
"""
west, east, south, north, bottom, top = tuple(prisms[:, i] for i in range(6))
err_msg = "Invalid prism or prisms. "
bad_we = west > east
bad_sn = south > north
bad_bt = bottom > top
if bad_we.any():
err_msg += "The west boundary can't be greater than the east one.\n"
for prism in prisms[bad_we]:
err_msg += "\tInvalid prism: {}\n".format(prism)
raise ValueError(err_msg)
if bad_sn.any():
err_msg += "The south boundary can't be greater than the north one.\n"
for prism in prisms[bad_sn]:
err_msg += "\tInvalid prism: {}\n".format(prism)
raise ValueError(err_msg)
if bad_bt.any():
err_msg += "The bottom radius boundary can't be greater than the top one.\n"
for prism in prisms[bad_bt]:
err_msg += "\tInvalid tesseroid: {}\n".format(prism)
raise ValueError(err_msg)
def jit_prism_gravity(
coordinates, prisms, density, kernel, out
): # pylint: disable=invalid-name,not-an-iterable
"""
Compute gravitational field of prisms on computations points
Parameters
----------
coordinates : tuple
Tuple containing ``easting``, ``northing`` and ``upward`` of the
computation points as arrays, all defined on a Cartesian coordinate
system and in meters.
prisms : 2d-array
Two dimensional array containing the coordinates of the prism(s) in the
following order: west, east, south, north, bottom, top in a Cartesian
coordinate system.
All coordinates should be in meters.
density : 1d-array
Array containing the density of each prism in kg/m^3. Must have the
same size as the number of prisms.
kernel : func
Kernel function that will be used to compute the desired field.
out : 1d-array
Array where the resulting field values will be stored.
Must have the same size as the arrays contained on ``coordinates``.
"""
# Iterate over computation points and prisms
for l in prange(coordinates[0].size):
for m in range(prisms.shape[0]):
# Iterate over the prism boundaries to compute the result of the
# integration (see Nagy et al., 2000)
for i in range(2):
for j in range(2):
for k in range(2):
shift_east = prisms[m, 1 - i]
shift_north = prisms[m, 3 - j]
shift_upward = prisms[m, 5 - k]
# If i, j or k is 1, the shift_* will refer to the
# lower boundary, meaning the corresponding term should
# have a minus sign
out[l] += (
density[m]
* (-1) ** (i + j + k)
* kernel(
shift_east - coordinates[0][l],
shift_north - coordinates[1][l],
shift_upward - coordinates[2][l],
)
)
@jit(nopython=True)
def kernel_potential(easting, northing, upward):
"""
Kernel function for potential gravitational field generated by a prism
"""
radius = np.sqrt(easting ** 2 + northing ** 2 + upward ** 2)
kernel = (
easting * northing * safe_log(upward + radius)
+ northing * upward * safe_log(easting + radius)
+ easting * upward * safe_log(northing + radius)
- 0.5 * easting ** 2 * safe_atan2(upward * northing, easting * radius)
- 0.5 * northing ** 2 * safe_atan2(upward * easting, northing * radius)
- 0.5 * upward ** 2 * safe_atan2(easting * northing, upward * radius)
)
return kernel
@jit(nopython=True)
def kernel_g_z(easting, northing, upward):
"""
Kernel for downward component of gravitational acceleration of a prism
"""
radius = np.sqrt(easting ** 2 + northing ** 2 + upward ** 2)
kernel = (
easting * safe_log(northing + radius)
+ northing * safe_log(easting + radius)
- upward * safe_atan2(easting * northing, upward * radius)
)
return kernel
@jit(nopython=True)
def safe_atan2(y, x):
"""
Principal value of the arctangent expressed as a two variable function
This modification has to be made to the arctangent function so the
gravitational field of the prism satisfies the Poisson's equation.
Therefore, it guarantees that the fields satisfies the symmetry properties
of the prism. This modified function has been defined according to
[Fukushima2019]_.
"""
if x != 0:
result = np.arctan(y / x)
else:
if y > 0:
result = np.pi / 2
elif y < 0:
result = -np.pi / 2
else:
result = 0
return result
@jit(nopython=True)
def safe_log(x):
"""
Modified log to return 0 for log(0).
The limits in the formula terms tend to 0 (see [Nagy2000]_).
"""
if np.abs(x) < 1e-10:
result = 0
else:
result = np.log(x)
return result
# Define jitted versions of the forward modelling function
# pylint: disable=invalid-name
jit_prism_gravity_serial = jit(nopython=True)(jit_prism_gravity)
jit_prism_gravity_parallel = jit(nopython=True, parallel=True)(jit_prism_gravity)
```