geoana.em.tdem.ElectricDipoleWholeSpace.magnetic_flux_density

ElectricDipoleWholeSpace.magnetic_flux_density(xyz)

Magnetic flux 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 magnetic flux 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:

\[\mathbf{b}(t) = \frac{\mu Ids}{4 \pi r^3} \bigg ( \textrm{erf}(\theta r) - \frac{2}{\sqrt{\pi}} \theta r \, e^{-\theta^2 r^2} \bigg ) \big ( - z \, \mathbf{\hat y} + y \, \mathbf{\hat z} \big )\]

where

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

Parameters:

xyz(n, 3) numpy.ndarray

Gridded xyz locations

Returns:

(n_time, n_loc, 3) numpy.array of float

Transient magnetic field at all times for the gridded locations provided. Output array is squeezed when n_time and/or n_loc = 1.

Examples

Here, we define a z-oriented electric dipole and plot the magnetic flux density on the xy-plane that intercepts z=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_[0., 0., 1.]
>>> 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 magnetic flux density.

>>> xyz = ndgrid(np.linspace(-10, 10, 20), np.linspace(-10, 10, 20), np.array([0]))
>>> B = simulation.magnetic_flux_density(xyz)

Finally, we plot the magnetic flux 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], B[t_ind, :, 0:2], ax=ax, vec=True, scale='log')
>>> ax.set_xlabel('X')
>>> ax.set_ylabel('Y')
>>> ax.set_title('Magnetic flux density at {} s'.format(time[t_ind]))

(Source code, png, pdf)