EM Fundamentals
Presentation highlight: Basic experiment to the response function
EM fundamentals is the third presentation in the DISC. Following the presentation on DC resistivity (and a break for coffee!), we dive into the fundamental physics of EM.
Building from DC resistivity, some of the motivating factors for looking to an EM survey over DC resistivity include:
- needing to cover large areas or deal with rugged terrain when conducting a ground-based, labour intensive survey may be impractical;
- encountering terrain that is difficult to couple with and inject current into;
- the “shielding problem” (introduced at the end of the DC resistivity presentation). Currents follow the path of least resistance and a thin, highly resistive layer can prevent currents from exciting targets of interest that lie beneath it.
In a basic EM experiment, we have a transmitter, which could be on the ground or towed in a bird. The transmitter is a loop of wire that we put a time-varying current through. A time-varying current produces a time-varying magnetic field that generates currents inside a conductive body. Those currents produce secondary magnetic fields that can be measured by the receiver.
To go into a bit more detail, we will look at one piece at a time, starting with the transmitter. The current driven through the transmitter could be a transient, which promotes looking at the physics in the time-domain, or a harmonic, which promotes looking at the physics in the frequency domain. Often when EM is first introduced (including in Ward and Hohmann, an authoritative resource on EM geophysics), people start by considering the physics in the frequency domain, build up understanding there, and then translate that to the time domain. In the DISC presentations, we go the other way, we instead build a ground-up understanding in the time domain and then move to frequency.
For EM experiments, we will work in the quasi-static regime, which means that we neglect the displacement current (when we move to GPR, which is conducted at much higher frequencies, this term will need to be included).
Ampere’s law tells us that if you have a current, that current gives rise to a magnetic field. The curl here in the equation shows that there is a circulation associated with the magnetic field. For a straight wire, the magnetic fields circulate around it; this can be seen with the right-hand rule… thumb=current, fingers=magnetic field. If you take a straight wire and loop it in on itself, the resulting magnetic field lines look like a dipole, as described. There are two important things to note in the equation that describes the magnetic fields that results from a current loop: (1) the moment depends upon the product of the current and the area, (2) the fields fall off geometrically as 1/r³.
Faraday’s law describes how a time varying flux, b, gives rise to electric fields everywhere (formally, b is called the magnetic flux density, but often, we will casually refer to it as the magnetic field). If we change b with time, that will generate an electric field everywhere. Often, we are interested in currents, so we introduce Ohm’s law, which connects electric fields with currents. In a conductor, a time-varying magnetic flux, we will induce currents.
A circuit model can be a helpful way to explain Faraday’s law. In this circuit, we have a coil hooked up to a voltmeter and we have a magnet. If we start at steady state, with the magnetic stationary, we see that the fields lines look like a dipole, and there is a flux of those lines through the loop, but since it is constant, there is no voltage in the circuit. If we move the magnet towards the loop, the magnetic flux through it increases and a voltage is generated in the circuit (a negative voltage — due to Lenz’ law). When we pull the magnet back, this decreases the flux and again induces a voltage in the circuit (this time it is positive).
In the time domain case, we start with constant current in the transmitter (the primary current, red loop), which generates a constant magnetic flux through the target loop (green). We then turn off the current and there is a large change in flux. This drives a current (the secondary current) through the target loop. The currents will decay as t/τ, where τ is a time-constant that depends on the inductance and resistance of the target. This is captured in the response functions: currents decay as exp(-t/τ).
What happens when we have a harmonic pulse? The main difference here is that in the time-domain example, there was only one change (from currents on to currents off), for a harmonic waveform, the currents are always changing. If we look at the two coil example in the frequency domain, we again start with the primary current (red), which is a harmonic that varies with a given frequency. The secondary current has to oscillate at same frequency, but can be different from the primary in 2 ways: its amplitude and its phase (the difference in where the primary reaches the maximum).
The secondary can be decomposed into 2 parts: a part that is in-phase with the primary, which is varying as cos(ωt), and a part that is out-of-phase with the primary and varies as sin(ωt). The in-phase portion is often also referred to as the real part, and the out-of-phase may be referred to as the quadrature or imaginary component. This terminology is often a point of confusion: “imaginary fields??”. All it is is the portion of the signal that is varying out-of-phase with respect to the primary currents.
How do we think about the phase lag? There are two parts (see the centre slide in the figure above): (1) the π/2 comes from the negative sign in Lenz’ law, (2) the ωL/R term is the induction number, this is what dictates the phase shift. Now we can start to figure out how this response function is working. For the frequency domain, we can plot the induction number and the response function (as shown in the slide on the right in the above figure). When induction number is small, most of the signal is in the quadrature component, whereas when it is large, most of the signal is in the real component.
Whether you have a step off or a harmonic current driven through the transmitter, we induce currents in the target. For a time domain experiment, the currents decay, for a harmonic, we partition the currents some to the in-phase and some to the quadrature, and in either, the behaviour depends on the physical properties of the target.
The currents generated in the target produce secondary magnetic fields (according to ampere’s law). These magnetic fields can be measured by receivers which either sample the magnetic field (b) or the time rate of change of the magnetic field, a voltage, (db/dt). These measured signals contain information about the physical properties of the subsurface.
App highlight: EM induction 2 loop app
To explore concepts of EM induction, there are 2 apps that have been developed, one for a harmonic transmitting current and the other for a transient transmitting current. Devin Cowan has been a significant contributor in the development of both of these. Following the presentation on the EM fundamentals, this app is used in the DISC course to recap concepts from the lecture.
