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 $r$ from the dipole is:

\begin{array}{r} B (r) = \frac{μ m}{4 π 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{array}

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: 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]))

(Source code, png, pdf)

../../_images/geoana-em-fdem-MagneticDipoleWholeSpace-magnetic_flux_density-1.png