# Source code for harmonica.equivalent_sources.cartesian

```
# 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 Cartesian 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_cartesian
[docs]class EquivalentSources(vdb.BaseGridder):
r"""
Equivalent sources for generic harmonic functions (gravity, magnetics).
These equivalent sources can be used for:
* Cartesian coordinates (geographic coordinates must be project before use)
* Gravity and magnetic data (including derivatives)
* Single data types
* Interpolation
* Upward continuation
* Finite-difference based derivative calculations
They cannot be used for:
* Regional or global data where Earth's curvature must be taken into
account
* 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:
(``easting``, ``northing``, ``upward``).
If None, will place one point source below each observation point at
a fixed relative depth below the observation point [Cooper2000]_.
Defaults to None.
depth : float
Parameter used to control the depth at which the point sources will be
located.
If ``depth_type`` is equal to ``"relative"``, the ``depth`` specifies
the 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 elevation of the data point minus
this *depth*. Use positive numbers (negative numbers would mean point
sources are above the data points).
If ``depth_type`` is equal to ``"constant"``, the ``depth`` specifies
the constant depth at which the point sources are placed beneath the
observation points. Every source point will be located at this *depth*.
Use positive numbers (negative numbers would mean point sources are
located above the zeroth level).
This parameter is ignored if *points* is specified.
Defaults to 500.
depth_type : str
Strategy used for setting the depth of the point sources.
The two available strategies are ``"constant"`` and ``"relative"``.
This parameter is ignored if *points* is specified.
Defaults to ``"relative"``.
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.EQLHarmonic.grid` method.
"""
# Set the default dimension names for generated outputs
# as xr.Dataset.
dims = ("northing", "easting")
# Overwrite the defalt name for the upward coordinate.
extra_coords_name = "upward"
# 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,
depth=500,
depth_type="relative",
parallel=True,
**kwargs,
):
self.damping = damping
self.points = points
self.depth = depth
self.depth_type = depth_type
self.parallel = parallel
# Define Green's function for Cartesian coordinates
self.greens_function = greens_func_cartesian
# Check if depth_type is valid
if depth_type not in ("constant", "relative"):
raise ValueError(
f"Invalid depth type '{depth_type}'. Should be either be 'constant' or 'relative'."
)
# Check if relative_depth has been passed (will be deprecated)
if "relative_depth" in kwargs:
warnings.warn(
"The 'relative_depth' parameter is deprecated, please use "
+ "the 'depth' paramter and set 'depth_type' to 'relative_depth' instead. ",
FutureWarning,
)
# Override depth and depth_type
self.depth, self.depth_type = kwargs["relative_depth"], "relative"
[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.EQLHarmonic.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: (``easting``, ``northing``, ``upward``, ...).
Only ``easting``, ``northing``, and ``upward`` 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_ = self._build_points(coordinates)
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
def _build_points(self, coordinates):
"""
Generate coordinates of point sources based on the data points
Locate the point sources following the chosen ``depth_type`` strategy.
If ``depth_type`` is equal to ``"relative"``, the point sources will be
placed beneath the observation points at a depth calculated as the
elevation of the data point minus the ``depth``.
If ``depth_type`` is equal to ``"constant"``, the point sources will be
placed beneath the observation points at the same height equal to minus
``depth``.
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (``easting``, ``northing``, ``upward``, ...).
Only ``easting``, ``northing``, and ``upward`` will be used, all
subsequent coordinates will be ignored.
Returns
-------
points : tuple of arrays
Tuple containing the coordinates of the equivalent point sources,
in the following order: (``easting``, ``northing``, ``upward``).
"""
if self.depth_type == "relative":
return (
coordinates[0],
coordinates[1],
coordinates[2] - self.depth,
)
if self.depth_type == "constant":
return (
coordinates[0],
coordinates[1],
-self.depth * np.ones_like(coordinates[0]),
)
return None
[docs] def predict(self, coordinates):
"""
Evaluate the estimated equivalent sources on the given set of points.
Requires a fitted estimator (see :meth:`~harmonica.EQLHarmonic.fit`).
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (``easting``, ``northing``, ``upward``, ...). Only
``easting``, ``northing`` and ``upward`` 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: (``easting``, ``northing``, ``upward``, ...).
Only ``easting``, ``northing`` and ``upward`` 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:
(``easting``, ``northing``, ``upward``).
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,
projection=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']``.
projection : callable or None
If not None, then should be a callable object
``projection(easting, northing) -> (proj_easting, proj_northing)``
that takes in easting and northing coordinate arrays and returns
projected northing and easting coordinate arrays. This function
will be used to project the generated grid coordinates before
passing them into ``predict``. For example, you can use this to
generate a geographic grid from a Cartesian gridder.
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
grid = super().grid(
region=region,
shape=shape,
spacing=spacing,
dims=dims,
data_names=data_names,
projection=projection,
extra_coords=upward,
**kwargs,
)
return grid
[docs] def scatter(
self,
region=None,
size=300,
random_state=0,
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,
upward,
size,
dims=None,
data_names=None,
projection=None,
**kwargs,
): # pylint: disable=arguments-differ
"""
Interpolate data along a profile between two points.
Generates the profile along a straight line assuming Cartesian
distances and the same upward coordinate for all points. Point
coordinates are generated by :func:`verde.profile_coordinates`. Other
arguments for this function can be passed as extra keyword arguments
(``kwargs``) to this method.
Use the *dims* and *data_names* arguments to set custom names for the
dimensions and the data field(s) in the output
:class:`pandas.DataFrame`. Default names are provided.
Includes the calculated Cartesian distance from *point1* for each data
point in the profile.
To specify *point1* and *point2* in a coordinate system that would
require projection to Cartesian (geographic longitude and latitude, for
example), use the ``projection`` argument. With this option, the input
points will be projected using the given projection function prior to
computations. The generated Cartesian profile coordinates will be
projected back to the original coordinate system. **Note that the
profile points are evenly spaced in projected coordinates, not the
original system (e.g., geographic)**.
Parameters
----------
point1 : tuple
The easting and northing coordinates, respectively, of the first
point.
point2 : tuple
The easting and northing coordinates, respectively, of the second
point.
upward : float
Upward coordinate of the profile points.
size : int
The number of points to generate.
dims : list or None
The names of the northing and easting data dimensions,
respectively, in the output dataframe. 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 dataframe. Defaults
to ``['scalars']`` for scalar data,
``['east_component', 'north_component']`` for 2D vector data, and
``['east_component', 'north_component', 'vertical_component']`` for
3D vector data.
projection : callable or None
If not None, then should be a callable object ``projection(easting,
northing, inverse=False) -> (proj_easting, proj_northing)`` that
takes in easting and northing coordinate arrays and returns
projected northing and easting coordinate arrays. Should also take
an optional keyword argument ``inverse`` (default to False) that if
True will calculate the inverse transform instead. This function
will be used to project the profile end points before generating
coordinates and passing them into ``predict``. It will also be used
to undo the projection of the coordinates before returning the
results.
Returns
-------
table : pandas.DataFrame
The interpolated values along the profile.
"""
# We override the profile 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)
# Create profile points and predict
table = super().profile(
point1,
point2,
size,
dims=dims,
data_names=data_names,
projection=projection,
extra_coords=upward,
**kwargs,
)
return table
class EQLHarmonic(EquivalentSources):
"""
DEPRECATED, use ``harmonica.EquivalentSources`` instead.
This class exists to support backward compatibility until next release.
"""
def __init__(
self,
damping=None,
points=None,
depth=500,
depth_type="relative",
parallel=True,
**kwargs,
):
warnings.warn(
"The 'EQLHarmonic' class has been renamed to 'EquivalentSources' "
+ "and will be deprecated on the next release, "
+ "please use 'EquivalentSources' instead.",
FutureWarning,
)
super().__init__(
damping=damping,
points=points,
depth=depth,
depth_type=depth_type,
parallel=parallel,
**kwargs,
)
@jit(nopython=True)
def greens_func_cartesian(east, north, upward, point_east, point_north, point_upward):
"""
Green's function for the equivalent sources in Cartesian coordinates
Uses Numba to speed up things.
"""
distance = distance_cartesian(
(east, north, upward), (point_east, point_north, point_upward)
)
return 1 / distance
```