# 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)
#
"""
Frequency domain filters meant to be applied on regular grids
"""
import numpy as np
[docs]def derivative_upward_kernel(fft_grid, order=1):
r"""
Filter for upward derivative in frequency domain
Return a :class:`xarray.DataArray` with the values of the frequency domain
filter for computing the upward derivative. The filter is built upon the
frequency coordinates of the passed ``fft_grid`` and is defined as follows:
.. math::
g(\mathbf{k}) = |\mathbf{k}| ^ n
where :math:`\mathbf{k}` is the wavenumber vector
(:math:`\mathbf{k} = 2\pi \mathbf{f}` where :math:`\mathbf{f}` is the
frequency vector) and :math:`n` is the order of the derivative.
Parameters
----------
fft_grid : :class:`xarray.DataArray`
Array with the Fourier transform of the original grid.
Its dimensions should be in the following order:
*freq_northing*, *freq_easting*.
Use :func:`xrft.xrft.fft` and :func:`xrft.xrft.ifft` functions to
compute the Fourier Transform and its inverse, respectively.
order : int
The order of the derivative. Default to 1.
Returns
-------
da_filter : :class:`xarray.DataArray`
Array with the kernel for the upward derivative filter in frequency
domain.
References
----------
[Blakely1995]_
See also
--------
harmonica.derivative_upward
"""
# Catch the dims of the Fourier transformed grid
dims = fft_grid.dims
# Grab the coordinates of the Fourier transformed grid
freq_easting = fft_grid.coords[dims[1]]
freq_northing = fft_grid.coords[dims[0]]
# Convert frequencies to wavenumbers
k_easting = 2 * np.pi * freq_easting
k_northing = 2 * np.pi * freq_northing
# Compute the filter for upward derivative in frequency domain
da_filter = np.sqrt(k_easting**2 + k_northing**2) ** order
return da_filter
