Signal Basics

2 Signal Basics

There are three closely-related questions central to geophysical measurement:

What is the source signal?
What is the measured signal?
What is the Earth’s response?
1. What physical property does one wish to measure?
2. How does one get a transfer function (response function) from the measured signal?
3. How does one get the desired property from the transfer function?

Of Question 1: ”What is the source signal?”, the answer we have thus far is ”Powergrid noise.” That in itself is not particularly illuminating, so lets look at the receiver. What signal can it measure?

As previously described, the receiver appeared to consist of a single field coil antenna approximately 24” long and 1.5” diameter, closely wrapped with sufficient turns of reasonably sized wire. I could not see if it was ferrite core, or air. This was connected to an amplifier and 24-bit/48-KHz digital recorder. There was also a digital multimeter. In operation the field coil was held lengthwise against the surface of the ground and a data stream recorded for sixty seconds. The antenna’s azimuth was not recorded, it appeared random. The antenna’s inclination was not measured either, but the ground surface was roughly horizontal. The amplifier’s pass-band was said to be 700 Hz - 12 KHz.

This setup can record the magnetic intensity $\vec{B}$ in the direction of the antenna. That is all. What can this tell us? In absence of a more definitive answer to Question 1, not a whole lot. Question 1 may be made more specific:

May we approximate the source by a plane wave?
What do we know of its angle of incidence and polarization?
What is its Power Spectral Density?
What is the noise floor of the amplifier, how does one know when the signal exceeds it, and by how much?

Presumably, the multimeter was to help answer the noise floor question. The question is whether it is sufficient, and what calibrations were made on the instrument before and after data acquisition.

Rather early on one must make the usual assumption about the earth being a passive linear system, that what one gets out of it depends entirely upon what one puts in. In particular, there are no subsurface sources: all transient subsurface currents are induced by the source signal from the surface.

We make the plane wave assumption, and further assume (based thus far upon nothing whatsoever) that the angle of incidence is roughly normal. Note: angles less than 45 or 50 deg. may qualify as ”roughly normal”. e.g. AMT sources. At present I’ve no particular reason to think powergrid radiation to be even that close to normal, particularly near transmission lines and relatively far from towns such as at the September survey site. It is part of Question 1 that should be addressed. Neither would I expect the $\vec{B}$ field from such a source to necessarily be particularly horizontal, but make the assumption regardless. Where does it get us?

Write the magnetic field $\vec{H} = \vec{B} ∕ μ_{0}$ as $\vec{H} = H_{x} \hat{x}$ where $\hat{x}$ is horizontal unit vector in direction of antenna coil. Assume an $exp (- i ω t)$ time dependence ([6, convention]) and decompose the field at earth’s surface $z = z_{0}$ into upward and downward traveling components:

H_{x} (ω) = A (ω) e^{i k (ω) \cdot (z - z_{0})} + B (ω) e^{- i k (ω) \cdot (z - z_{0})}

(1)

where

k^{2} = ω^{2} μ ϵ (1 - i δ) and δ = \frac{σ}{ω ϵ}

(2)

Conductivity $σ = 0$ in the atmosphere where the EM signal is recorded, but otherwise generally dominates $ω ϵ$ by several orders of magnitude. (More in Section 3.) In free space

\begin{array}{rcl} ϵ = ϵ_{0} & = & \frac{1 0^{7}}{4 π c^{2}} = 8.854 \times 1 0^{- 12} F/m, & (3) \\ μ = μ_{0} & = & 1 ∕ (c^{2} ϵ_{0}) = 4 π \times 1 0^{- 7} = 1.257 \times 1 0^{- 6} H/m & (4) \\ k (ω) & = & ω ∕ c & (5) \end{array}

In (eq. 1) $H_{x}$ is the Fourier Transform of the total magnetic field component as received by the antenna. $A (ω)$ is the FT of the incident field. $B (ω)$ is the FT of the earth’s response. If we set $z = z_{0}$ at the antenna location the exponential factors are unity and we may write (1) as

\begin{array}{rcl} H_{x} (ω) & = & A (ω) [1 + R (ω)], where & (6) \\ R (ω) & = & B (ω) ∕ A (ω) & (7) \end{array}

$R$ is the classic reflection coefficient for the $x$ component of the incident magnetic field. One submits $R$ contains all the information that may possibly be obtained from this single antenna at a single location. If we know $A (ω)$ we may invert (7):

\begin{array}{rcl} R (ω) & = & H_{x} (ω) ∕ A (ω) - 1 & (8) \end{array}

This is simple frequency-domain deconvolution. But, one must first know $A (ω)$ . And $A (ω)$ was not measured. Hence the question ”What is the source signal?”

In his verbal description Vendor makes a point about their method being keyed about finding the zeroes – both of magnitude and particularly phase – of the second derivative of some function $F (ω)$ with respect to depth $z$ . In lieu of reviewing his patents or other information, we do not here specify how one obtains $F$ from $H_{x}$ , or $z$ from $ω$ , but for simplicity assume

\begin{array}{rcl} F (H_{x} (ω)) & = & H_{x} (ω) (because that is what’s measured), and & (9) \\ z & = & z (ω) & (10) \end{array}

where $z$ is some reasonably obtainable function, e.g. $z (ω) = V_{p} ∕ (π ω)$ or $z (ω) = 500 \sqrt{ρ ∕ (2 π ω)}$ m as naive examples. We do not pretend these assumptions are correct, only that they can illuminate some of the issues confronting signal interpretation. Neglecting the chain-rules between $ω$ and $z$ , and between $F$ and $H_{x}$ we see terms like

\begin{array}{rcl} H_{x} (ω) & = & A (ω) [1 + R (ω)] & (11) \\ \frac{d H_{x}}{d ω} & = & \frac{d A}{d ω} [1 + R (ω)] + A (ω) \frac{d R}{d ω} & (12) \\ \frac{d^{2} H_{x}}{d ω^{2}} & = & \frac{d^{2} A}{d ω^{2}} [1 + R (ω)] + A (ω) \frac{d^{2} R}{d ω^{2}} + 2 \frac{d A}{d ω} \frac{d R}{d ω} & (13) \end{array}

But if $A (ω)$ is the FT of ”power grid noise”, what does that make its derivatives?

Let’s make an analogy to a familiar mechanical device. Suppose you have a single Vibroseis source and a single geophone which you may place at but one location, and that location is right next to the vibrator plate. For simplicity assume your measurement is made with the plate firmly coupled to exposed bedrock. Adjust vibropower to taste. Now, you might think you might know how to deconvolve the vibrator source from the little squiggly lines you coax from your single solitary geophone. But, you would be wrong. Because unbeknownst to you, while you were back at the office vibrating a desk, the geo-elves have borked your source, and that vibrator engine is now random and chaotic.

Not random. That would be too easy. Random and chaotic. If you knew the source vibrations were truly band-limited random white noise, then you know its Power Spectral Density and after sufficiently long acquisition time some bright seismo-guy might coerce some reflection information from that single lonely geophone. But it isn’t, and he can’t.

Think about ”power grid noise”. Is it band-limited random white noise, or something more realizable and common? For example, could it not be dominated by a random set of more coherent sources? If so, about how many elements are in the set? How does this number change with time? Do individual sources come and go? How might each source change with time? What can you say about their PSD’s? What is the total power grid PSD, and how does it change with time? As a presumably stationary instance, how strong are the 60 Hz harmonics, and how do they affect signal processing?

In eq. (11), the source signal $A (ω)$ and its derivatives are complex quantities, involving both magnitude and phase. As previously noted, if one actually knows what $A (ω)$ is, then one can just do the division at eq. (7) and be done with it. Here we see that uncertainties in $A$ lead to greater uncertainties in its first and second derivatives. As we shall later see in Section (3.2), it is likely the second derivative of the desired reflection coefficient $d^{2} R (ω) ∕ d ω^{2}$ is probably quite small: any attempt to get information about it from the recorded signal is likely to be swamped by noise in the $A (ω)$ source.

What about time-stacking, can it help? Probably. If by time-stacking one means ”sample the data and compute FFT’s over $N$ time sub-intervals, then average the frequency components of each sub-interval”, then this is roughly equivalent to computing the FFT over the entire interval, but without the resulting finer ”resolution” in $Δ f$ . Either way, $A (ω)$ probably depends on time, while $R (ω)$ does not. Eqs (11) become

\begin{array}{rcl} 〈 H_{x} (ω) 〉 & = & 〈 A (ω) 〉 [1 + R (ω)] & (14) \\ \frac{d 〈 H_{x} 〉}{d ω} & = & \frac{d 〈 A 〉}{d ω} [1 + R (ω)] + 〈 A (ω) 〉 \frac{d R}{d ω} & (15) \\ \frac{d^{2} 〈 H_{x} 〉}{d ω^{2}} & = & \frac{d^{2} 〈 A 〉}{d ω^{2}} [1 + R (ω)] + 〈 A (ω) 〉 \frac{d^{2} R}{d ω^{2}} + 2 \frac{d 〈 A 〉}{d ω} \frac{d R}{d ω} & (16) \end{array}

which would reduce some of the fluctuations in $A (ω)$ and its derivatives w.r.t. $ω$ over the total time interval sampled, which in this case was 60 seconds. How much this actually helps, and what additional intrinsic $d A ∕ d ω$ dependencies might be exposed, remain part of Question 1.

Figure 2 illustrates a representative Magneto Telluric Power Spectral Density. Vendor is careful to distinguish Free Mode from MT, but this figure is one I had at hand, and covers the frequency range of interest. Any ”power-grid noise” will be in addition to the MT background. (Or vice versa, depending on one’s point of view.) One suggests variations in $A (ω)$ anything like those illustrated might introduce difficulties in the accurate estimation of $R$ and its derivatives w.r.t. $ω$ in (14). Of particular importance is the realization that very few processes, either natural or due to man, are truly random. ”Band-width limited white noise” is an idealization that rarely occurs.

[Picture]

Figure 1: This example is taken from [5, Fig 1.1]. Note that MT power spectra taken at other locations or other times will differ – frequently significantly – in detail.

Similarly, Figure Fig. (2) shows a representative 5-component Magneto Telluric signal recorded at a sampling rate $Δ t = 2$ s over a 30 min time window. Although this time scale is somewhat longer than the $Δ t \sim 1 0^{- 4}$ s of the audio band, the figure illustrates a potential problem in assuming a priori that a single field component can represent in any way the entire field strength. Importantly, for this particular MT signal, the signal details differ markedly between the x and y components. Again, Vendor stresses that Free Mode is not MT. But if audio band ”powergrid noise” has similar variability, one must question the utility of a single component recorded at random azimuth, as appeared to have been done at the September Survey. One hopes Vendor can share some insight about the polarization dependencies of his source, and how they might or might not effect the results of his method,

[Picture]

Figure 2: This example is taken from [5, Fig 4.1], and represents a 30-minute segment of five-component MT time series recorded at Mt. Doreen in central Australia. MT field components recorded at the same location at other times will differ – perhaps significantly – in detail. Looking at the x and y component of the

B

field, one sees some long-duration similarities, as well as some marked short-term differences where one component is increasing while the other decreases. For this reason MT surveys take great care to acquire at least four, and preferably five, field components. (Practical considerations usually preclude direct acquisition of

E_{z}

Vendor mentions the Singularity Expansion Method (SEM) in analogy with Sonar and Radar target characterization, and that method’s related singular modes (aka ”free modes”, or ”free modes of decay”). These fall in the category of characterization of the earth’s response and its transfer function. We discuss these in context with their relation to the reflection coefficient $R (ω)$ in the Section (3.4). In particular, free mode excitation and eigenfrequencies are completely described within $R (ω)$ . As such they do not provide a pass on deconvolving the source.

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