Source code for harmonica.equivalent_layer.harmonic

# 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 layer for generic harmonic functions in Cartesian coordinates
"""
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 EQLHarmonic(vdb.BaseGridder): r""" Equivalent-layer for generic harmonic functions (gravity, magnetics, etc). This equivalent layer 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 It 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 point sources used as the equivalent layer. Coordinates are assumed to be in the following order: (``easting``, ``northing``, ``upward``). If None, will place one point source bellow each observation point at a fixed relative depth bellow 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 elevation 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 point sources used to build the equivalent layer. 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, relative_depth=500, parallel=True, ): self.damping = damping self.points = points self.relative_depth = relative_depth self.parallel = parallel # Define Green's function for Cartesian coordinates self.greens_function = greens_func_cartesian
[docs] def fit(self, coordinates, data, weights=None): """ Fit the coefficients of the equivalent layer. 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_ = ( 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 layer 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 layer. 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 point sources used as equivalent layer 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
@jit(nopython=True) def greens_func_cartesian(east, north, upward, point_east, point_north, point_upward): """ Green's function for the equivalent layer in Cartesian coordinates Uses Numba to speed up things. """ distance = distance_cartesian( (east, north, upward), (point_east, point_north, point_upward) ) return 1 / distance