geoana.em.fdem.MagneticDipoleWholeSpace.magnetic_flux_density

MagneticDipoleWholeSpace.magnetic_flux_density(xyz)

Magnetic flux density for the harmonic magnetic dipole at a set of gridded locations.

For a harmonic magnetic dipole oriented in the \(\hat{u}\) direction with moment amplitude \(m\) and harmonic frequency \(f\), this method computes the magnetic flux density at the set of gridded xyz locations provided.

The analytic solution is adapted from Ward and Hohmann (1988). For a harmonic magnetic dipole oriented in the \(\hat{x}\) direction, the solution at vector distance \(\mathbf{r}\) from the dipole is:

\[\begin{split}\mathbf{B}(\mathbf{r}) = \frac{\mu m}{4\pi r^3} e^{-ikr} \Bigg [ \Bigg ( \frac{x^2}{r^2}\hat{x} + \frac{xy}{r^2}\hat{y} + \frac{xz}{r^2}\hat{z} \Bigg ) ... \\ \big ( -k^2r^2 + 3ikr + 3 \big ) \big ( k^2 r^2 - ikr - 1 \big ) \hat{x} \Bigg ]\end{split}\]

where

\[k = \sqrt{\omega^2 \mu \varepsilon - i \omega \mu \sigma}\]

Parameters:

xyz(…, 3) numpy.ndarray

Gridded xyz locations

Returns:

(n_freq, …, 3) numpy.ndarray of complex

Magnetic flux density at all frequencies for the gridded locations provided. Output array is squeezed when n_freq and/or n_loc = 1.

Examples

Here, we define an z-oriented magnetic dipole and plot the magnetic flux density on the xz-plane that intercepts y=0.

>>> from geoana.em.fdem import MagneticDipoleWholeSpace
>>> from geoana.utils import ndgrid
>>> from geoana.plotting_utils import plot2Ddata
>>> import numpy as np
>>> import matplotlib.pyplot as plt

Let us begin by defining the electric current dipole.

>>> frequency = np.logspace(1, 3, 3)
>>> location = np.r_[0., 0., 0.]
>>> orientation = np.r_[0., 0., 1.]
>>> moment = 1.
>>> sigma = 1.0
>>> simulation = MagneticDipoleWholeSpace(
>>>     frequency, location=location, orientation=orientation,
>>>     moment=moment, sigma=sigma
>>> )

Now we create a set of gridded locations and compute the magnetic flux density.

>>> xyz = ndgrid(np.linspace(-1, 1, 20), np.array([0]), np.linspace(-1, 1, 20))
>>> B = simulation.magnetic_field(xyz)

Finally, we plot the real and imaginary components of the magnetic flux density.

>>> f_ind = 1
>>> fig = plt.figure(figsize=(6, 3))
>>> ax1 = fig.add_axes([0.15, 0.15, 0.40, 0.75])
>>> plot2Ddata(
>>>     xyz[:, 0::2], np.real(B[f_ind, :, 0::2]), vec=True, ax=ax1, scale='log', ncontour=25
>>> )
>>> ax1.set_xlabel('X')
>>> ax1.set_ylabel('Z')
>>> ax1.autoscale(tight=True)
>>> ax1.set_title('Real component {} Hz'.format(frequency[f_ind]))
>>> ax2 = fig.add_axes([0.6, 0.15, 0.40, 0.75])
>>> plot2Ddata(
>>>     xyz[:, 0::2], np.imag(B[f_ind, :, 0::2]), vec=True, ax=ax2, scale='log', ncontour=25
>>> )
>>> ax2.set_xlabel('X')
>>> ax2.set_yticks([])
>>> ax2.autoscale(tight=True)
>>> ax2.set_title('Imag component {} Hz'.format(frequency[f_ind]))

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