harmonica.filters.ifft

harmonica.filters.ifft(fft_grid, *, prefix='freq_')

Compute the inverse Fast Fourier Transform of a 2D regular grid.

If the frequency coordinates have a shift attribute, it will be used to shift the coordinates in the spatial domain to such value.

Important

Assumes that the fft_grid is shifted: it was passed to numpy.fft.fftshift. The outputs of the harmonica.filters.fft function satisfy this condition.

Parameters:

fft_gridxarray.DataArray

Array with a regular grid defined in the frequency domain. Its dimensions should be in the following order: freq_northing, freq_easting.

prefixstr, optional

Prefix used for the name of the frequency coordinates and dimensions.

Returns:

gridxarray.DataArray

Array with the inverse Fourier transform of the passed grid.

Notes

This function implements the discrete inverse Fourier Transform of 2D regular grids. It’s based on the following definition of the inverse Fourier Transform of a \(G:\mathbb{R}^2 \rightarrow \mathbb{R}\) function in the frequency domain:

\[\mathcal{F}^{-1}[G](f_x, f_y) = \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} G(f_x, f_y) e^{2 \pi i f_x x} e^{2 \pi i f_y y} \text{d}f_x \text{d}f_y\]

If we consider two discretized spaces for the \(f_x\) and \(f_y\) frequencies, both evenly spaced with steps equal to \(\Delta f_x\) and \(\Delta f_y\), respectively, then the \((n, m)\) element of the discrete inverse Fourier Transform of \(G\) can be defined as follows:

\[\mathcal{F}^{-1}[G]_{(n, m)} = \sum\limits_{j=0}^{N-1} \sum\limits_{k=0}^{M-1} G(f_x_j, f_y_k) e^{2 \pi i f_x_j x_n} e^{2 \pi i f_y_k y_m} \Delta f_x \Delta f_y\]

This function differs from the plain numpy.fft.ifftn function since it implements the true amplitude and the true phase corrections.