# Source code for harmonica.equivalent_sources.spherical

```
# 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)
#
"""
Equivalent sources for generic harmonic functions in spherical coordinates
"""
import warnings
import numpy as np
from numba import jit
from sklearn.utils.validation import check_is_fitted
import verde as vd
import verde.base as vdb
from .utils import (
pop_extra_coords,
predict_numba_serial,
predict_numba_parallel,
jacobian_numba_serial,
jacobian_numba_parallel,
)
from ..forward.utils import distance_spherical
[docs]class EquivalentSourcesSph(vdb.BaseGridder):
r"""
Equivalent sources for generic harmonic functions in spherical coordinates
These equivalent sources can be used for:
* Spherical coordinates (geographic coordinates must be converted before
use)
* Regional or global data where Earth's curvature must be taken into
account
* Gravity and magnetic data (including derivatives)
* Single data types
* Interpolation
* Upward continuation
* Finite-difference based derivative calculations
They cannot be used for:
* Joint inversion of multiple data types (e.g., gravity + gravity
gradients)
* Reduction to the pole of magnetic total field anomaly data
* Analytical derivative calculations
Point sources are located beneath the observed potential-field measurement
points by default [Cooper2000]_. Custom source locations can be used by
specifying the *points* argument. Coefficients associated with each point
source are estimated through linear least-squares with damping (Tikhonov
0th order) regularization.
The Green's function for point mass effects used is the inverse Euclidean
distance between the grid coordinates and the point source:
.. math::
\phi(\bar{x}, \bar{x}') = \frac{1}{||\bar{x} - \bar{x}'||}
where :math:`\bar{x}` and :math:`\bar{x}'` are the coordinate vectors of
the observation point and the source, respectively.
Parameters
----------
damping : None or float
The positive damping regularization parameter. Controls how much
smoothness is imposed on the estimated coefficients.
If None, no regularization is used.
points : None or list of arrays (optional)
List containing the coordinates of the equivalent point sources.
Coordinates are assumed to be in the following order:
(``longitude``, ``latitude``, ``radius``). Both ``longitude`` and
``latitude`` must be in degrees and ``radius`` in meters.
If None, will place one point source below each observation point at
a fixed relative depth below the observation point [Cooper2000]_.
Defaults to None.
relative_depth : float
Relative depth at which the point sources are placed beneath the
observation points. Each source point will be set beneath each data
point at a depth calculated as the radius of the data point minus
this constant *relative_depth*. Use positive numbers (negative numbers
would mean point sources are above the data points). Ignored if
*points* is specified.
parallel : bool
If True any predictions and Jacobian building is carried out in
parallel through Numba's ``jit.prange``, reducing the computation time.
If False, these tasks will be run on a single CPU. Default to True.
Attributes
----------
points_ : 2d-array
Coordinates of the equivalent point sources.
coefs_ : array
Estimated coefficients of every point source.
region_ : tuple
The boundaries (``[W, E, S, N]``) of the data used to fit the
interpolator. Used as the default region for the
:meth:`~harmonica.EQLHarmonicSph.grid` method.
"""
# Set the default dimension names for generated outputs
# as xr.Dataset.
dims = ("spherical_latitude", "longitude")
# Overwrite the defalt name for the upward coordinate.
extra_coords_name = "radius"
# Define dispatcher for Numba functions with or without parallelization
_predict_kernel = {False: predict_numba_serial, True: predict_numba_parallel}
_jacobian_kernel = {False: jacobian_numba_serial, True: jacobian_numba_parallel}
def __init__(
self,
damping=None,
points=None,
relative_depth=500,
parallel=True,
):
self.damping = damping
self.points = points
self.relative_depth = relative_depth
self.parallel = parallel
# Define Green's function for spherical coordinates
self.greens_function = greens_func_spherical
[docs] def fit(self, coordinates, data, weights=None):
"""
Fit the coefficients of the equivalent sources.
The data region is captured and used as default for the
:meth:`~harmonica.EQLHarmonicSpherical.grid` method.
All input arrays must have the same shape.
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (``longitude``, ``latitude``, ``radius``, ...).
Only ``longitude``, ``latitude``, and ``radius`` will be used, all
subsequent coordinates will be ignored.
data : array
The data values of each data point.
weights : None or array
If not None, then the weights assigned to each data point.
Typically, this should be 1 over the data uncertainty squared.
Returns
-------
self
Returns this estimator instance for chaining operations.
"""
coordinates, data, weights = vdb.check_fit_input(coordinates, data, weights)
# Capture the data region to use as a default when gridding.
self.region_ = vd.get_region(coordinates[:2])
coordinates = vdb.n_1d_arrays(coordinates, 3)
if self.points is None:
self.points_ = (
coordinates[0],
coordinates[1],
coordinates[2] - self.relative_depth,
)
else:
self.points_ = vdb.n_1d_arrays(self.points, 3)
jacobian = self.jacobian(coordinates, self.points_)
self.coefs_ = vdb.least_squares(jacobian, data, weights, self.damping)
return self
[docs] def predict(self, coordinates):
"""
Evaluate the estimated equivalent sources on the given set of points.
Requires a fitted estimator
(see :meth:`~harmonica.EQLHarmonicSpherical.fit`).
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (``longitude``, ``latitude``, ``radius``, ...).
Only ``longitude``, ``latitude`` and ``radius`` will be used, all
subsequent coordinates will be ignored.
Returns
-------
data : array
The data values evaluated on the given points.
"""
# We know the gridder has been fitted if it has the coefs_
check_is_fitted(self, ["coefs_"])
shape = np.broadcast(*coordinates[:3]).shape
size = np.broadcast(*coordinates[:3]).size
dtype = coordinates[0].dtype
coordinates = tuple(np.atleast_1d(i).ravel() for i in coordinates[:3])
data = np.zeros(size, dtype=dtype)
self._predict_kernel[self.parallel](
coordinates, self.points_, self.coefs_, data, self.greens_function
)
return data.reshape(shape)
[docs] def jacobian(
self, coordinates, points, dtype="float64"
): # pylint: disable=no-self-use
"""
Make the Jacobian matrix for the equivalent sources.
Each column of the Jacobian is the Green's function for a single point
source evaluated on all observation points.
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (``longitude``, ``latitude``, ``radius``, ...).
Only ``longitude``, ``latitude`` and ``radius`` will be used, all
subsequent coordinates will be ignored.
points : tuple of arrays
Tuple of arrays containing the coordinates of the equivalent point
sources in the following order:
(``longitude``, ``latitude``, ``radius``).
dtype : str or numpy dtype
The type of the Jacobian array.
Returns
-------
jacobian : 2D array
The (n_data, n_points) Jacobian matrix.
"""
# Compute Jacobian matrix
n_data = coordinates[0].size
n_points = points[0].size
jac = np.zeros((n_data, n_points), dtype=dtype)
self._jacobian_kernel[self.parallel](
coordinates, points, jac, self.greens_function
)
return jac
[docs] def grid(
self,
upward,
region=None,
shape=None,
spacing=None,
dims=None,
data_names=None,
**kwargs,
): # pylint: disable=arguments-differ
"""
Interpolate the data onto a regular grid.
The grid can be specified by either the number of points in each
dimension (the *shape*) or by the grid node spacing. See
:func:`verde.grid_coordinates` for details. All grid points will be
located at the same `upward` coordinate. Other arguments for
:func:`verde.grid_coordinates` can be passed as extra keyword arguments
(``kwargs``) to this method.
If the interpolator collected the input data region, then it will be
used if ``region=None``. Otherwise, you must specify the grid region.
Use the *dims* and *data_names* arguments to set custom names for the
dimensions and the data field(s) in the output :class:`xarray.Dataset`.
Default names will be provided if none are given.
Parameters
----------
upward : float
Upward coordinate of the grid points.
region : list = [W, E, S, N]
The west, east, south, and north boundaries of a given region.
shape : tuple = (n_north, n_east) or None
The number of points in the South-North and West-East directions,
respectively.
spacing : tuple = (s_north, s_east) or None
The grid spacing in the South-North and West-East directions,
respectively.
dims : list or None
The names of the northing and easting data dimensions,
respectively, in the output grid. Default is determined from the
``dims`` attribute of the class. Must be defined in the following
order: northing dimension, easting dimension.
**NOTE: This is an exception to the "easting" then
"northing" pattern but is required for compatibility with xarray.**
data_names : list of None
The name(s) of the data variables in the output grid. Defaults to
``['scalars']``.
Returns
-------
grid : xarray.Dataset
The interpolated grid. Metadata about the interpolator is written
to the ``attrs`` attribute.
"""
# We override the grid method from BaseGridder so it takes the upward
# coordinate as a positional argument. We disable pylint
# arguments-differ error because we intend to make this method
# different from the inherited one.
# Ignore extra_coords if passed
pop_extra_coords(kwargs)
# Grid data
# We always pass projection=None because that argument it's intended to
# be used only with Cartesian gridders.
grid = super().grid(
region=region,
shape=shape,
spacing=spacing,
dims=dims,
data_names=data_names,
projection=None,
extra_coords=upward,
**kwargs,
)
return grid
[docs] def scatter(
self,
region=None,
size=None,
random_state=None,
dims=None,
data_names=None,
projection=None,
**kwargs,
):
"""
.. warning ::
Not implemented method. The scatter method will be deprecated on
Verde v2.0.0.
"""
raise NotImplementedError
[docs] def profile(
self,
point1,
point2,
size,
dims=None,
data_names=None,
projection=None,
**kwargs,
):
"""
.. warning ::
Not implemented method. The profile on spherical coordinates should
be done using great-circle distances through the Haversine formula.
"""
raise NotImplementedError
class EQLHarmonicSpherical(EquivalentSourcesSph):
"""
DEPRECATED, use ``harmonica.EquivalentSourcesSph`` instead.
This class exists to support backward compatibility until next release.
"""
def __init__(
self,
damping=None,
points=None,
relative_depth=500,
parallel=True,
):
warnings.warn(
"The 'EQLHarmonic' class has been renamed to 'EquivalentSources' "
+ "and will be removed on the next release, "
+ "please use 'EquivalentSources' instead.",
FutureWarning,
)
super().__init__(
damping=damping,
points=points,
relative_depth=relative_depth,
parallel=parallel,
)
@jit(nopython=True)
def greens_func_spherical(
longitude, latitude, radius, point_longitude, point_latitude, point_radius
):
"""
Green's function for the equivalent sources in spherical coordinates
Uses Numba to speed up things.
"""
distance = distance_spherical(
(longitude, latitude, radius), (point_longitude, point_latitude, point_radius)
)
return 1 / distance
```