geoana.em.tdem.ElectricDipoleWholeSpace.current_density

ElectricDipoleWholeSpace.current_density(xyz)

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

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

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

\[\begin{split}{\bf e_e}(t) = \frac{Ids}{4\pi \sigma r^2} & \Bigg [ \Bigg ( \frac{x^2}{r^2}\mathbf{\hat x} + \frac{xy}{r^2}\mathbf{\hat y} + \frac{xz}{r^2}\mathbf{\hat z} \Bigg ) \Bigg ( 3 \, \textrm{erf}(\theta r) - \bigg ( \frac{4}{\sqrt{\pi}}\theta^3 r^3 \; ... \\ & + \frac{6}{\sqrt{\pi}} \theta r \bigg ) e^{-\theta^2 r^2} \Bigg ) - \Bigg ( \textrm{erf}(\theta r) - \bigg ( \frac{4}{\sqrt{\pi}} \theta^3 r^3 + \frac{2}{\sqrt{\pi}} \theta r \bigg ) e^{-\theta^2 r^2} \Bigg ) \mathbf{\hat x} \Bigg ]\end{split}\]

where

\[\theta = \Bigg ( \frac{\mu\sigma}{4t} \Bigg )^{1/2}\]

Parameters:

xyz(…, 3) numpy.ndarray

Gridded xyz locations

Returns:

(n_time, …, 3) numpy.ndarray of float

Current density at all times for the gridded locations provided.

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.tdem 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.

>>> time = np.logspace(-6, -2, 3)
>>> location = np.r_[0., 0., 0.]
>>> orientation = np.r_[1., 0., 0.]
>>> current = 1.
>>> sigma = 1.0
>>> simulation = ElectricDipoleWholeSpace(
>>>     time, 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(-10, 10, 20), np.array([0]), np.linspace(-10, 10, 20))
>>> J = simulation.current_density(xyz)

Finally, we plot the current density at the desired locations/times.

>>> t_ind = 0
>>> fig = plt.figure(figsize=(4, 4))
>>> ax = fig.add_axes([0.15, 0.15, 0.8, 0.8])
>>> plot2Ddata(xyz[:, 0::2], J[t_ind, :, 0::2], ax=ax, vec=True, scale='log')
>>> ax.set_xlabel('X')
>>> ax.set_ylabel('Z')
>>> ax.set_title('Current density at {} s'.format(time[t_ind]))

(Source code, png, pdf)