from geoscilabs.em.HarmonicVMDCylWidget import HarmonicVMDCylWidget
vmd = HarmonicVMDCylWidget()
from matplotlib import rcParams
rcParams['font.size'] = 16
Here, we show fields and fluxes that result from placing a harmonic vertical magnetic dipole (VMD) source over a layered Earth.
There are two commonly used models for describing a harmonic VMD source: 1) as an infinitessimally small bar magnet that experiences a harmonic magnetization in the vertical direction, and 2) as an infinitessimally small horizontal loop of wire carrying a harmonic current. In either case, the harmonic VMD creates a primary harmonic magnetic field in its viscinity; which is vertical at the location of the source.
True dipole sources do not exist in nature however they can be approximated in practice. For geophysical applications, small wire loops carrying harmonic current are used to approximate harmonic VMD sources. These EM sources may be placed on the Earth's surface (ground-based surveys) or flown through the air (airborne surveys). Because the source's primary field is harmonic, it induces anomalous currents in the Earth. The distribution and strength of the induced currents depends on the frequency of the harmonic VMD source and the subsurface conductivity distribution. The induced current ultimately produce secondary magnetic fields which can be measured by one or more receivers.
In this app, we explore the following:
The geological scenario being modeled is shown in the figure below. Here, we assume the Earth is comprised of 3 layers. Each layer can have a different electrical conductivity ($\sigma$). However, a constant magnetic susceptibility ($\chi$) is used for all layers; where $\mu_0$ is the magnetic permeability of free space and $\mu = \mu_0 (1 +\chi)$. The thicknesses of the top two layers are given by $h_1$ and $h_2$, respectively.
In this case, a harmonic VMD source (Tx) is used to excite the Earth, and the Earth's FEM response (secondary magnetic field) is measured by a receiver (Rx). In practice, the transmitter and receiver may be placed near the Earth's surface or in the air. The source may also operate at a variety of frequencies. Because we are operating in the frequency domain, fields and currents within the region have both real and imaginary components.
To understand the fields and currents resulting from a harmonic VMD source over a layered Earth we have two apps:
Follow the exercise in a linear fashion. Some questions may use parameters set in a previous question.
Q1: Set $\sigma_1$, $\sigma_2$ and $\sigma_3$ to arbitrary conductivity values. Based on the geometry of the problem, which components (x, y, z) of each field (E, B, Bsec or J) are zero? Run the Fields app and set AmpDir = None. Next, try different combinations of Field and Comp. Does the simulation match what you initially thought?
Q2: Re-run the Fields app to set parameters back to default. Set Field = E, AmpDir = None, Comp = y and Re/Im = Amp to plot the magnitude of Ey (may also want to use log-scale). Are there significant electric fields in the air as well as in the Earth? What about if you set Field = J? Why is this?
Q3: Re-run the Fields app to set parameters back to default. Set Field = E, AmpDir = None and Comp = y to plot Ey. Change the Re/Im parameter from Re to Im. Is the electric field in the Earth in phase or out of phase with the primary field? Why? Hint: think about Faraday's law.
Q4a: Re-run the Fields app to set parameters back to default. Set Field = E, AmpDir = None, Comp = y and Re/Im = Amp to plot the magnitude of Ey (may also want to use log-scale). Now slowly increase the frequency (f index slider). What happens to the electric field (Ey) in the ground? Why? Hint: think about the penetration depth of the primary signal.
Q4b: Now lower the conductivity of layer 1, layer 2 and layer 3 to 0.0001 S/m while imaging the plot after each layer is changed. What happens to the electric field? Why is this? Hint: think about the penetration depth of the primary signal. What would you expect to happen if you increased the layer conductivities to 0.1 S/m?
Q5: Re-run the Fields app to set parameters back to default. Set Field = B, AmpDir = Direction, Re/Im = Re and f index = 1 for a vector plot of the primary magnetic field; since at low enough frequencies, the secondary field is negligible compared to the primary field. Now switch Field = Bsec and use Re/Im to answer the following:
We use this app to simulate the fields and currents everywhere due to a harmonic VMD source. The fields and induced currents depend on the frequency of the harmonic VMD source and the subsurface conductivity distribution. You will use the app to change various parameters in the model and see how the fields and currents change.
vmd.InteractivePlane_Layer()
Using this app, we show how the fields observed at the receiver location depend on the parameters set in the previous app. Note that if you want to see changes in the data due to changes in the model, you MUST re-run the previous app.
vmd.InteractiveData_Layer()
EM fields will be depenent upon a number of parameters, using a simple half-space model ($\sigma_1=\sigma_2=\sigma_3$) explore how EM fields and data changes upon below four parameters.
E1: Effects of frequency?
E2: Effects of Tx height?
E3: Effects of Conductivity?
E4: Effects of Susceptibility?