# 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
import verde as vd
import verde.base as vdb
from numba import jit
from sklearn.utils.validation import check_is_fitted
from ..forward.utils import distance_spherical
from .utils import (
jacobian_numba_parallel,
jacobian_numba_serial,
pop_extra_coords,
predict_numba_parallel,
predict_numba_serial,
)
[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.EquivalentSources.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.EquivalentSourcesSph.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.EquivalentSourcesSph.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"):
"""
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,
):
"""
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.
# 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, # noqa: U100
size=None, # noqa: U100
random_state=None, # noqa: U100
dims=None, # noqa: U100
data_names=None, # noqa: U100
projection=None, # noqa: U100
**kwargs, # noqa: U100
):
"""
.. warning ::
Not implemented method. The scatter method will be deprecated on
Verde v2.0.0.
"""
raise NotImplementedError
[docs] def profile(
self,
point1, # noqa: U100
point2, # noqa: U100
size, # noqa: U100
dims=None, # noqa: U100
data_names=None, # noqa: U100
projection=None, # noqa: U100
**kwargs, # noqa: U100
):
"""
.. 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
