# harmonica.filters.reduction_to_pole_kernel¶

harmonica.filters.reduction_to_pole_kernel(fft_grid, inclination, declination, magnetization_inclination=None, magnetization_declination=None)[source]

Filter for reduction to the pole in frequency domain

Return a xarray.DataArray with the values of the frequency domain filter for applying a reduction to the pole on magnetic data. The filter is built upon the frequency coordinates of the passed fft_grid and is defined as follows:

$g(\mathbf{k}) = \frac{1}{\Theta_m \Theta_f}$

with

$\Theta_m = m_z + i \frac{m_e k_e + m_n k_n}{|\mathbf{k}|}$
$\Theta_f = f_z + i \frac{f_e k_e + f_n k_n}{|\mathbf{k}|}$

where $$\hat{\mathbf{f}} = (f_e, f_n, f_z)$$ is a unit vector parallel to the geomagnetic field and $$\hat{\mathbf{m}} = (m_e, m_n, m_z)$$ is a unit vector parallel to the magnetization vector of the source. The $$f_e$$, $$f_n$$, $$m_e$$, $$m_n$$ are the easting and northing components while the $$f_z$$ and $$m_z$$ are the downward coordinates. Each of these components can be obtained from the inclination and declination angles of the geomagnetic field ($$I$$ and $$D$$, respectively) and for the magnetization vector ($$I_m$$ and $$D_m$$, respectively):

$\begin{split}\begin{cases} f_e = \sin D \cos I \\ f_n = \cos D \cos I \\ f_u = \sin I \end{cases}\end{split}$
$\begin{split}\begin{cases} m_e = \sin D_m \cos I_m \\ m_n = \cos D_m \cos I_m \\ m_u = \sin I_m \end{cases}\end{split}$
Parameters
• fft_grid (xarray.DataArray) – Array with the Fourier transform of the original grid. Its dimensions should be in the following order: freq_northing, freq_easting. Use xrft.xrft.fft and xrft.xrft.ifft functions to compute the Fourier Transform and its inverse, respectively.

• inclination (float in degrees) – The inclination of the inducing Geomagnetic field.

• declination (float in degrees) – The declination of the inducing Geomagnetic field.

• magnetization_inclination (float in degrees or None) – The inclination of the total magnetization of the anomaly source. If None, the magnetization_inclination will be set equal to the inclination, neglecting remanent magnetization and self demagnetization. Default None.

• magnetization_declination (float in degrees) – The declination of the total magnetization of the anomaly source. If None, the magnetization_declination will be set equal to the declination, neglecting remanent magnetization and self demagnetization. Default None.

Returns

da_filter (xarray.DataArray) – Array with the kernel for the reduction to the pole filter in frequency domain.

References

[Blakely1995]