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 $I d s$ 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 $r$ from the current dipole is:

\begin{aligned} J (r) = \frac{σ I d s}{4 π (σ + i ω ε) r^{3}} & e^{- i k r} . . . \\ [(\frac{x^{2}}{r^{2}} \hat{x} + \frac{x y}{r^{2}} \hat{y} + \frac{x z}{r^{2}} \hat{z}) (- k^{2} r^{2} + 3 i k r + 3) (k^{2} r^{2} - i k r - 1) \hat{x}] \end{aligned}

where

k = \sqrt{ω^{2} μ ε - i ω μ σ}

Parameters:

xyz(…, 3) numpy.ndarray: Gridded xyz locations

Returns:

(n_freq, …, 3) numpy.ndarray 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