geoana.em.fdem.ElectricDipoleWholeSpace.current_density#

ElectricDipoleWholeSpace.current_density(xyz)#

Current density for the harmonic current dipole at a set of gridded locations.

For an electric current dipole oriented in the \(\hat{u}\) direction with dipole moment \(I ds\) and harmonic frequency \(f\), this method computes the current density at the set of gridded xyz locations provided.

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

\[\begin{split}\;\;\;\; \mathbf{J}(\mathbf{r}) = \frac{\sigma I ds}{4\pi (\sigma + i \omega \varepsilon) r^3} & e^{-ikr} ... \\ & \Bigg [ \Bigg ( \frac{x^2}{r^2}\hat{x} + \frac{xy}{r^2}\hat{y} + \frac{xz}{r^2}\hat{z} \Bigg ) \big ( -k^2r^2 + 3ikr + 3 \big ) \big ( k^2 r^2 - ikr - 1 \big ) \hat{x} \Bigg ]\end{split}\]

where

\[k = \sqrt{\omega^2 \mu \varepsilon - i \omega \mu \sigma}\]
Parameters
xyz(n, 3) numpy.ndarray

Gridded xyz locations

Returns
(n_freq, n_loc, 3) numpy.array of complex

Current 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 x-oriented electric dipole and plot the current density on the xz-plane that intercepts y=0.

>>> from geoana.em.fdem import ElectricDipoleWholeSpace
>>> 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_[1., 0., 0.]
>>> current = 1.
>>> sigma = 1.0
>>> simulation = ElectricDipoleWholeSpace(
>>>     frequency, location=location, orientation=orientation,
>>>     current=current, sigma=sigma
>>> )

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

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

Finally, we plot the real and imaginary components of the current 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(J[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(J[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]))

(Source code, png, pdf)

../../_images/geoana-em-fdem-ElectricDipoleWholeSpace-current_density-1.png