Quantifying AEM system characteristics using a ground loop
Aaron C. Davis1 and James Macnae1 1 RMIT
University, Applied Physics, GPO Box 2476V, Melbourne, VIC 3001, Australia E-mail:
[email protected];
[email protected] (April 11, 2008) GEO-2007-0266.R2 Running head: Ground loop AEM
ABSTRACT Quantitative interpretation of time-domain airborne electromagnetic (AEM) data is hampered by uncertainty in altimetry, system geometry, transmitter waveform, data averaging and timing. We present a simple calibration method that serves to define these issues by use of a closed multi-turn loop of known electrical and physical properties that is insulated from the ground beneath it. By predicting the secondary response of the AEM receiver and comparing it to the measured data, we have identified and quantified systematic errors mentioned above in several systems. Additionally, we identify an alternative sub-process that uniquely calculates altimeter and geometry errors by measuring the current induced in a ground loop of known properties and comparing it to predictions. The ground-loop method is best used over resistive cover to minimize limitations caused by non-uniform conductive ground and is a calibration tool that makes AEM data consistent with quantitative models. Fluctuating geometrical errors caused by bird swing limit the accuracy of applying the geometry corrections from one fly-over to an entire survey.
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INTRODUCTION Recent attention has been paid to various difficulties in the quantitative interpretation of AEM data. Won and Smits (1987) identified bird swing, altimeter error and calibration errors in amplitude, phase and base level adjustment as possible causes of error in the determination of water depths in coastal and shallow water regions. Their findings were elaborated upon by Deszcz-Pan et al. (1998) who attempted to reduce layered earth inversion errors using ground truth information obtained from boreholes and ground surveys in the Florida Everglades National Park. One systematic source of error identified in the Everglades study was altimetry. The survey was conducted with a helicopter-mounted radar altimeter; it was seen that measured and inverted bird altitudes agreed for lines flown in one direction but disagreed by as much as 2 m in the other (Deszcz-Pan et al., 1998). The authors attributed these differences to changes in ‘bird lift’ which correspondingly caused error in the estimated altitude of the towed bird. Other researchers have also discovered errors and problems caused by altimetry. For example, Brodie and Lane (2003) discussed the variable differences between laser and radar altimetry depending on surface conditions. Laser altimeters housed in towed birds, which estimate ground clearance at the level of the instrument, are subject to slant range error caused by bird swing (Holladay et al., 1997; Davis et al., 2006). This error can be corrected by monitoring the changes in the orientation of the towed bird (Kratzer et al., 2007; Sørensen and Auken, 2004; Vrbancich et al., 2007). A further error source is caused by changes in the geometric coupling of AEM systems to the earth caused by bird swing. This has been mentioned by Won and Smits (1987) and also by Holladay et al. (1997); it has been analyzed by Yin and Fraser (2004) using a superposed dipole and generalized by Fitterman and Yin (2004) for rigid boom frequency-
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domain systems. Bird swing for time-domain systems has also been discussed. Smith (2001) developed an inversion method that tracks the offset of the receiver from the transmitter in a fixed-wing system. Bird separation and attitude are important when trying to detect good conductors during on-time (Hefford et al., 2006); and are of fundamental importance when inverting AEM data for bathymetry (Vrbancich and Smith, 2005; Wolfgram and Vrbancich, 2007). In addition to variable and systematic errors in geometry, AEM data can be affected by other issues that are more closely related to the operation of systems used on survey. In time-domain systems, the transmitter waveform needs to be well known, as does the peak transmitter current and receiver gains. Errors in any of these quantities degrades the quantitative interpretation of data. As an example, when using HoistEM data to estimate seawater depths in Sydney Harbour, Vrbancich and Fullagar (2004) were compelled to use unrealistic values of seawater conductivity in order to achieve reasonable agreement between inverted and measured depths. The receiver gains were found to be incorrect for the survey, and the HoistEM data were corrected by applying a post-survey calibration based on measured depths that subsequently achieved sub-metre accuracy in bathymetry (Vrbancich and Fullagar, 2007). Another aspect discussed in the Everglades National Park study was the identification of possible errors that can occur during the historic calibration process of the particular system used there. Fitterman (1998) analyzed the errors that can arise due to poorly positioned calibration coils or by conducting the calibration over conductive ground. He recommended using a jig to effect the calibration of helicopter EM systems; and this idea has been advanced so that calibration coils are now internal in the RESOLVE system. In this paper, we separate errors into those that are flight-path dependent—such as bird swing, altimetry and transmitter-receiver separation—and those 3
that are independent of flight-path. These errors, such as response averaging, transmitter waveform, receiver windows and gains, can be classified under the title of ‘system errors’, and are best dealt with by a calibration procedure. Calibration of AEM systems can be problematic. For many systems, accurate calibration cannot be achieved on the ground, particularly if it is conductive. The main reason for this is because the system geometry cannot be reproduced while the aircraft used for towing the receiver is in the hangar or on the airstrip. For fixed-wing systems, some transmitters cannot be run at full power on the ground because they rely on the rushing air of flight to provide adequate cooling. For time-domain electromagnetic systems, one of the greatest difficulties is determining transmitter waveform since the sensitive receiver electronics which are optimized for off-time measurement would be over range during the on-time. In addition to this, it is often difficult to precisely determine the relative time between the transmitter turn-off and the receiver turn-on. Various calibration processes have been applied, for example flying over regions of known geological structure (Ley-Cooper and Macnae, 2007), water bodies of known conductivity (Ley-Cooper et al., 2006; Kovacs and Holladay, 1990; Nelson, 1973) and areas of high resistivity or high altitude base-level adjustments (Vrbancich et al., 2000). Until now, no consistent calibration procedure has yet been designed that can be applied to all AEM systems.
METHOD An airborne time-domain system can be calibrated by flying the system over a closed multiturn loop of known resistance and inductance. The transmitter’s magnetic pulse generates an eddy current in the ground loop described by Faraday’s law of induction. The magnitude of the eddy current produced in the ground loop is determined by the current waveform 4
of the transmitter and the mutual inductance between the transmitter and the ground loop. The eddy current generated in the ground loop decays at a constant exponential rate determined by the loop resistance R and self-inductance L, both of which can be measured or estimated. The electromagnetic field generated by the eddy current, called the secondary field, is then detected by the receiving loop of the EM system. The magnitude of the secondary response is determined by the mutual inductance between the receiver and the ground loop, scaled by the amount of current flowing in the ground loop. Since inductance is determined by geometry, the signal measured in the receiver is a function of the positions of all three loops. This is shown schematically in Figure 1. The ground loop position is known by ensuring that it is accurately surveyed when it is laid out. This may be achieved on level ground by obtaining the position of each corner with a GPS receiver and making sure that the loop wire is straight when it is deployed. The position of the transmitter is generally known or approximated: for example, if the system is mounted on a fixed wing aircraft, we can assume that the transmitter loop is fixed with respect to the GPS antenna, and that the separation of the transmitter from the earth is measured with a laser or radar altimeter. The geometry specified by the contractor should be regarded as nominal since it is not usually measured on survey. Further, the towed receiver position with respect to the aircraft is not routinely measured and is often not well known; it is a function of airspeed and changes continuously due to bird swing. Operators also may let out variable lengths of tow cable to reduce wear and tear.
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SECONDARY RESPONSE DUE TO THE PRESENCE OF A GROUND LOOP The electromotive force (emf) generated in one closed loop due to current I(t) flowing in another is proportional to M the mutual inductance between them (e.g., Griffiths, 1989):
E = −M
d I(t). dt
(1)
Using equation 1, and following Grant and West (1965), the emf generated in the ground calibration loop EL (t) due to the AEM transmitter is
EL (t) = −MT L
d IT (t), dt
(2)
where IT (t) is the transmitter current waveform and MT L is the mutual inductance between the transmitter and the ground loop. Since the emf generated in the loop is equal to the potential drops around the loop due to its resistance R and self-inductance L, we obtain
EL (t) = RIL (t) + L
d IL (t), dt
(3)
which can be rearranged (using equation 2) to the following first-order differential equation describing the current in the ground loop due to the transmitter current:
d R −MT L d IL (t) + IL (t) = IT (t). dt L L dt
(4)
The current induced in the ground loop creates an emf ER (t) in the AEM receiver. Using Faraday’s law of induction, equation 1, the AEM receiver emf due to the ground-loop current
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IL (t) is ER (t) = −MRL
d IL (t), dt
(5)
where MRL is the mutual inductance between the ground loop and the receiver. Equations 4 and 5 form a set of differential equations that can be used to predict the secondary response in the receiver. We solve the differential equations 4 and 5 by viewing them as an input/output system using L, the Laplace transform operator as discussed, for example, by Simmons (1991).
Ground loop response
We analyze the ground-loop current and the AEM receiver emf by assuming that the transmitter current IT (t) can be expressed as IT (t) = IT iT (t), where IT is the peak transmitter current and iT (t) is a waveform that begins at time t = 0 and −1 ≤ iT (t) ≤ 1. Applying the Laplace transform to the ground-loop induced current equation 4, using the relationship just defined for the transmitter current, and solving for L[IL (t)] yields
L[IL (t)](s + R/L) = −
MT L IT sL[iT (t)], L
(6)
where s is the Laplace parameter. The general expression of the induced ground-loop current is obtained by applying the inverse Laplace transform L− to the Laplace transform equation 6 above: IL (t) = −
MT L IT − s L L[iT (t)] . L s + R/L
(7)
This shows that the current induced in a ground loop due to a transmitter current waveform of any shape is given by the convolution of the transmitter waveform with the natural 7
response of the ground loop. In the case of a step transmitter current (the Laplace transform of which is 1/s (e.g., Spiegel, 1995)), we see that the ground-loop current reduces to
MT L IT − 1 IL (t) = − L , L s + R/L
(8)
which, upon evaluating the inverse Laplace transform, yields
IL (t) = −
MT L IT −t/τ . e L
(9)
For a step transmitter current, the ground-loop current has the expected exponential decay of time constant τ = L/R, whose magnitude is modified by the peak transmitter current IT , self-inductance L and the purely geometrical inductive coupling of the transmitter loop to the ground loop MT L . If the induced current in the ground loop is measured, the peak of the envelope is used to calibrate the AEM system geometry by altering the altitude and displacement of the transmitter-receiver arrangement. In turn, this adds a second step to the error detection process that determines an altitude correction for the flyover. If the groundloop current is synchronized with a GPS clock, then other geometrical parameters like in-line distance shifts can also be accounted for in this step. Once geometry is determined, the measured and predicted secondary response is compared for the flyover to determine system errors.
AEM receiver response
The AEM receiver response due to the induced current in the ground loop is obtained by placing the general expression of the ground-loop current, equation 7, into the differential
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equation 5 of the receiver emf. Using the Laplace transform, the secondary response at the receiver becomes MRL MT L IT − s2 L L[iT (t)] . ER (t) = L s + R/L
(10)
This is the receiver response due to the current induced in a ground loop from a transmitter current of any waveform. Assuming that the transmitter current has amplitude IT at time t = 0, the emf generated in the airborne receiver becomes
ER (t) =
MRL MT L IT − s L , L s + R/L
(11)
which yields ER (t) =
MRL MT L IT L
δ(t) −
R −t/τ e L
.
(12)
This result is consistent with Liu (1998). The factor MRL MT L /L is the inductive limit of the system response and is analogous to the coupling coefficient defined by Grant and West (1965). In practical time-domain systems, the transmitter waveform is repeated with period T ; the secondary response is sampled in finite time windows and is then stacked over a number of half waveforms of period T /2. Returning to the general receiver response equation 10, and following the derivation of Stolz and Macnae (1998), the response at time t = (tk + tk+1 )/2 (where k is an index) is
MRL MT L IT τ (e−tk /τ − e−tk+1 /τ ) L tk+1 − tk 1 s − ′ L L[i (t)] . s + R/L T (1 + e−T /2τ )
ER (tk , tk+1 , T ) =
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(13)
The first term represents the geometric coupling; the second term accounts for the integration of e−t/τ at window k from tk to tk+1 . The third term is an analytic Taylor series that represents the contribution due to stacking a waveform of period T (Stolz and Macnae, 1998), while the fourth term is the transfer function that convolves the impulse response to the transmitter waveform (or, in this case, i′T (t)).
RESULTS We now present results from a number of the experimental tests conducted using different AEM systems in the period from 2003 to 2007.
HoistEM
Our first example is from the HoistEM helicopter time-domain system operating at 25 Hz base-frequency and flown over a 100×100 m 3-turn loop of wire of calculated self-inductance L = 7.91 mH and measured resistance R = 25.7 Ω (τ = 0.296 ms), placed on a resistive hilltop (0.1 S of cover over a resistive basement of < 1 mS/m) near Paraburdoo in the Pilbara region of Western Australia. Positions of the ground loop were obtained by a GPS receiver. In this experiment, a data acquisition (DAS) unit was attached across a 1 Ω resistor used to close the ground loop. The Roland Edirol UA-5 USB analog to digital converter is designed for audio capture and records 96 000 samples/s at 24-bit precision through 2 channels. It is band limited to -2 dB at 20 Hz and 20 000 Hz. To calibrate the UA-5 DAS (which uses analog volume controls on the input), a reference sine wave of known frequency and VRM S was fed into the second channel. The voltage measured across the 1 Ω resistor closing the ground loop thus translated directly to induced current. While
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descriptions of the HoistEM receiver, transmitter and waveform are given by (Boyd, 2004) and also by (Vrbancich and Fullagar, 2007), we have included our own brief description in Appendix A. Once the ground loop and DAS were connected and operating, the HoistEM system was flown over the ground loop in north-south and south-north directions. The two flight lines most pertinent to this paper, along with the position of the ground loop, are shown in Figure 2. Measuring the current induced in the ground loop allowed a two-part calibration of the HoistEM system. By comparing the measured current to the predicted current, we calculated the HoistEM transmitter altitude above the ground loop by using the expression for the ground-loop induced current, equation 7. Using the flight path recovery (r) and the GPS positions of the ground loop (r′ ), as well as the peak transmitter currents measured by the HoistEM system during survey (IT (peak) (t)), the peak ground-loop current was predicted as a function of positions r and r′ and time t:
IL(peak) (r, r′ , t) = 0.936
MT L (r)IT (peak) (t) , L(r′ )
(14)
where the value 0.936 arises from the convolution of the HoistEM waveform shape and the time constant of the ground loop. For our predicted peak current calculation, we assumed that the center of the transmitter was fixed directly beneath the GPS antenna mounted on the tail of the helicopter. The transmitter was modelled as a hexagon 24 m in diameter. For the vertical distance of the transmitter above the ground loop, we used the altitude measured by the helicopter-mounted radar altimeter (minus a nominal vertical offset of 21 m). Additionally, to account for a constant change in the assumed 21 m offset, we
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allowed a vertical offset ∆h to be present in the altitude (and hence, r) when calculating predicted peak current in the ground loop. Excellent agreement between measured and predicted results for line 1 was achieved with ∆h = −2 m, thereby predicting that the total radar altimeter-transmitter vertical offset was 19 m. However, because the Edirol UA-5 operated on a clock independent of the HoistEM system, we needed to time shift the predicted peak current to match the current measured by the DAS. The result of our altitude correction process is shown in Figure 3. The predicted peak amplitude curves are shown in gray; the actual measured current through successive cycles is shown in black. An interesting feature in this figure is the presence in the measured current of the very clear null values at 3.4 s and 8.8 s. In the presence of resistive ground, these zero current points mark the times when the HoistEM transmitter creates zero net magnetic flux through the ground loop. This ensures us that the response measured in the receiver is due to the ground loop itself and not the earth underneath (which would provide a consistent non-zero background). After we calibrated the altimeter offset for line 1, we predicted the receiver response using equation 13. When we compared our results to the measured data, the curves did not match. The predicted and measured response curves for channels 2, 7 and 12 are shown in Figure 4. Not only are the shapes of the responses different (the predicted responses are less rounded), but the amplitudes are very different and the peaks of the measured responses are shifted about 20 m from the predicted ones. Because we calibrated the altitude of the transmitter for line 1 using our peak current prediction, equation 14, we assumed that the predicted and measured receiver responses differed by a distance shift ∆d. Furthermore, we assumed that the processed data had some undocumented filtering process applied to it. In order to approximate the filtering process, we applied a simple 12
boxcar filter of averaging time corresponding to n samples of our predicted response. We then minimized the difference between the predicted and measured curves for channel 7 of the HoistEM data in a least squares sense. We chose channel 7 because it is a midtime channel that is located approximately one (ground loop) decay constant τ after the transmitter shut-off. By minimizing the error E between measured and predicted curves according to the relation N X (FP (∆d, n)i − Fi )2 , E=
(15)
i=1
where FP (∆d, n)i is the predicted profile based on changes in distance along line and averaging window at every point i; and Fi is the measured response at every point. Figure 5 shows that applying the optimum n=17 (i.e., averaging the data for 1.7 s) and a distance shift of ∆d = 18.5 m does not greatly improve the agreement between real and predicted curves. Both predicted and measured curves have a similar shape, and the peaks match spatially, but it is clear that the predicted curves are still much larger. This problem is most probably associated with the HoistEM receiver amplitude gains: a problem identified by Vrbancich and Fullagar in 2004. They had to use unrealistic values for the conductivity of seawater in Sydney Harbour to model their bathymetry data. In their paper, they applied a constant shift to the measured survey altitude to account for amplitude differences. However, we showed in Figure 3 that the HoistEM system was measuring altitude properly within 2 m or so; we now know that if the Sydney Harbour HoistEM system was identical to the one used our experiment, the correction that Vrbancich and Fullagar applied was probably incorrect. To achieve a better fit between predicted and measured data, we altered the error
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minimization calculation in equation 15 by adding a receiver gain multiplier A:
E=
N X i=1
2 FP (n, ∆d, A)i − Fi ,
(16)
where FP (n, ∆d, A) is the modified predicted response at each fiducial for delay channel 7. We found that E is effectively minimized for n = 17 (averaging for 1.7 s), ∆d = 18.5 m and A = 0.647. The result of the same corrections applied to delay channels 2, 7 and 10, Figure 6a, shows remarkable agreement between measured and predicted values. For each channel, the difference between the predicted and measured curves is less than 5 mV, which implies that the timing windows are most likely correctly specified. An interesting feature of Figure 6b, which shows the residual error between the measured and predicted responses, is its sinusoidal nature. During this flyover, the Hoist was being towed at a groundspeed of about 20 m/s, so the distance range 0–300 m represents about 15 s of flying. The wavelength of the residual (60 m) therefore corresponds to a period of about 3 s. While this period is much too short for an in-line bird swing (Davis et al., 2006), it may be due to the transmitter-receiver assembly pitching or rolling (Davis, 2007). Applying the calibration factors of n = 17, ∆d = 18.5 m and A = 0.649 to the predicted responses for line 2 yielded a slight disagreement between measured and predicted response, Figure 7. Although the width and shape of the predicted peaks matched in position along the line after applying the calibration factors, their amplitude did not agree when compared to the measured data. The disagreement was simply rectified by allowing the vertical offset between helicopter radar altimeter and towed bird transmitter position to be changed from ∆h = 19.3 m to ∆h = 16.9 m; the predicted responses then matched the measured. By maintaining ∆d, n, and A, the corrections obtained from matching delay channel 7 on
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line 1 made the predictions match the measured data on line 2. In order to ensure that the altitude change of 2.4 m between lines was reasonable, we examined the current measured in the ground loop during the line 2 flyover. Figure 7 shows the peak current predicted in the ground loop using ∆h = 19.3 m (gray-dashed) and using ∆h = 16.9 m (gray-solid). Using ∆h = 16.9 m implies that altitude correction may vary by 2–3 m on successive flights. It appears from these experiments that at the time they were conducted, the HoistEM system suffered from a problem in its receiver gains. Only after adjusting them to about 65% of their nominal value could they be made to agree with the measured responses; this finding is corroborated by Vrbancich and Fullagar (2007). Apart from the by now familiar in-line offset (in this case 18.5 m behind the helicopter GPS antenna) and the expected radar altimetry errors of ∼2.5 m, the gain-corrected HoistEM system appears to yield excellent results in terms of predicted to measured response comparisons. The greatest discrepancy between the predicted and the gain-corrected measured responses is what we believe to be stacking and filtering in the measured data. Whilst waveform stacking is a useful and expected process for a contractor to apply to the raw data, there was also some averaging applied to the data that served to spread the measured data out by about 1.7 s. It may be that the stacking process has the effect of filtering and spreading the data, and as we have not been able to find any mention in the literature of filtering in the HoistEM response, we assume that this is the case. In their paper, Vrbancich and Fullagar (2007) stress the need for accurately calibrated data to do meaningful interpretation. The series of tests conducted here shows clearly that in 2003, when these tests were made, the main systematic error was receiver gain.
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VTEM
The next time-domain AEM system that we discuss is the VTEM system of Geotech Airborne, flown over a ground loop placed on resistive ground in Botswana (0.2 S of cover over a resistive basement of 1 mS/m). The VTEM system is a helicopter-borne time-domain system whose transmitter (in 2004) had 5 turns of wire in a 26 m diameter dodecahedron. The receiver, a vertical axis coil of 100 turns, was positioned in the center of the loop and had a diameter of 1.1 m (Witherly et al., 2004). The base frequency for this test was 25 Hz with a trapezoidal transmitter waveform of approximately 45% duty cycle. In 2004, VTEM was powered from the alternator of the helicopter at a peak transmitter current of ∼120 A. The waveform increased linearly from 0 A to 120 A in 1 ms, remained constant for 7 ms, then shut off in 1 ms. The off-time measurements at the receiver started 9.08 ms after the start of the transmitter waveform. Lateral position of the bird was estimated from the GPS antenna position of the helicopter (mounted on the tail), while vertical position was estimated by subtracting a nominal 50 m from the altitude of the helicopter, measured with a radar altimeter mounted on its underside. The ground loop was a 100 × 100 m loop of wire laid out in 4 turns and the system was flown over it 12 times. In this paper, we will discuss only two lines: line 94 and 93. Figure 8 shows the receiver response for the ground-loop flyover of line 94. All channels are shown; the response of the ground loop is very obvious in this plot, starting at 100 m and finishing by 220 m. The system’s response to the ground itself is easily seen between 0–80 m and 230–300 m, and is about 12% of the overall early time signal strength. In this experiment, neither the resistance R nor the self-inductance L of the loop was measured; in order to find the decay constant τ of the ground, we took the mean of the signal measured
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for each delay channel over the peak ground-loop responses from 130–180 m, marked with solid vertical lines in Figure 8. Using delay channels 14–26, we fitted an exponential decay to the measured data. The fit, shown in Figure 9 along with the actual decay curve, has a decay constant of τ = 4.62 ms. Clearly, the late time VTEM data is closely fitted by the exponential decay on channels 14–26. The measured response at earlier time channels 1–13 does not match the exponential decay: This is due to ground conductivity. We calculated the response of the VTEM receiver using the flight recovery data for line 94. By changing distance along line ∆d and the vertical separation between the radar altimeter and transmitter/reciever ∆h and using the error fitting equation,
E=
N X (FP (∆d, ∆h)i − Fi )2 ,
(17)
i=1
where FP (∆d, ∆h)i is the predicted response, we forced the prediction curve to agree with the measured curve along channel 14 (approximately τ = 4.62 ms delay). We found that the error was minimized with ∆d = −7.04 m and ∆h = −50.1 m. Using these parameters, we recalculated the predicted response for channels 1, 20 and 26. The comparison between the predicted and measured curves are shown in Figure 10a. With reasonable parameters for ∆h and ∆d, there is excellent agreement between measured and predicted response for all delay channels 14 through 26 shown in Figure 10; this also shows that the timing windows are correct. With this in mind, and assuming that a good fit for one line would obtain a good fit for another, we attempted to fit the airborne response for line 93. We found that we could not make the predicted responses agree with the measured ones by using the error minimization equation 17. While the response curves in Figure 10 were remarkably symmetric, the responses for line 93 showed an obvious asym-
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metry. We found we had to alter our error minimization equation 17 by adding a term θb which accounted for pitch of the towed transmitter and receiver assembly. The new fitting equation, E=
N X (FP (∆d, ∆h, θb )i − Fi )2 ,
(18)
i=1
was minimized for line 93, channel 14, with the parameters ∆d = 5.5 m, ∆h = −52.0 m and θb = −9.33◦ . The correction was applied to all channels, and is shown in Figure 11a. As before, the prediction for the earliest channel 1 does not fit the measured curve; this is due to the 0.2 S of conductive cover beneath the loop.
AeroTEM
The final experiments that we present are for the AeroTEM II helicopter-borne time-domain system (Balch, 2004; Balch and Boyko, 2003; Balch et al., 2002). The AeroTEM transmitter current waveform was a 1.0981 ms-wide triangular half pulse, operating at 125 Hz with ∼3 ms of off-time and a peak current of ∼270 A. The ground loop used in this experiment was an 80×80 m single-turn solid wire loop with measured resistance R = 1.35 Ω and calculated self-inductance L = 695 µH. The AeroTEM system was slung beneath a helicopter using a (nominal) 55 m cable. The altitude of the transmitter and receiver was estimated from the altitude of the helicopter, as measured with a radar altimeter, minus the cable length. The transmitter loop was an 8-turn octagon of point-to-point diameter of 5 m, and was flown at a survey height of approximately 30 m. For this altitude, it was a reasonable approximation to model the receiver coil as a dipole receiver, concentrically mounted in the plane of the transmitter. Position of the towed bird system was predicted from the position solution of a GPS antenna mounted on the
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helicopter. The bird’s easting and northing was assumed to be that of the GPS, since a lag correction may already have been applied. The AeroTEM system was flown 9 times over the ground loop, which was placed on resistive ground (0.2 S of cover over resistive basement of 1 mS/m). In each case, the flight line of the system was across the corners of the loop, and not perpendicularly to one side. For each line, we calculated the predicted ground-loop response, allowing for changes in distance along line ∆d, altitude error ∆h and averaging n, obtaining a best fit for delay channel 10 (a mid-time channel) and applied the corrections to the other channels. Our results showed that the best fit parameters for ∆d and ∆h changed for each repeat line. The distribution of ∆d for 9 lines was −4.2 ± 3.9 m, while the distribution for ∆h was −1.0 ± 2.6 m. This few metres of variance is similar to the HoistEM altitude variations and is expected, since we know that birds towed behind aircraft act as a pendulum, with swing length of a few metres dependent on the length of the tow cable (Davis et al., 2006). We also found that length n of the averaging window applied to the predicted curves varied in order to optimally match measured responses. We found that the predicted average was between 1.5 s and 2.4 s, with a mean of 1.8 ± 0.3 s. We believe that the variation was due to the geometrical consequences of the orientation of the flight lines over the loop itself. As mentioned, all lines were flown across the corners of the ground loop. In a previous paper, Davis et al. (2006) showed that the oscillation of towed birds consists of both in-line and cross-line swings. We believe that the spread in optimum predicted averaging times is caused by cross-line bird swing during loop flyover. Any sideways deviation of the towed bird causes a change in the width of the ground-loop response; and is exacerbated by the system being flown diagonally over the ground loop.
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RECOMMENDATIONS The ground-loop method proposed in this paper depends on two main factors to enable detection of systematic errors in AEM systems. The first one is that the ground beneath the loop must be sufficiently resistive. Conductive ground will cause tertiary currents to be induced in the ground loop, thereby complicating the signal generated from the transmitter alone. In the two-part calibration discussed in the HoistEM section, the extra signal could also interfere with the altitude corrections. In our derivation of the secondary response at the receiver, we have not accounted for extra signals produced in the ground loop due to conductive cover and, in our opinion, such a derivation would unnecessarily complicate a procedure that has been proven to work so well over resistive terrain. Additionally, the ground itself produces signal at the receiver that, as we have shown with the VTEM example, interferes with the comparison between predicted and measured signals in the early channels. From this consideration, we produced a basic test to determine the maximum allowable ground conductance in order to still achieve a reasonable secondary response to determine errors in the receiver amplitudes and timing windows. Let us consider the VTEM example, Figure 10. For channel 14 (approximately τ after the transmitter shut-off) the difference between the predicted and measured signal was about 2%. One condition is that the conductance of the cover beneath the loop must be such that the signal produced by it must be under this background level of error. We therefore suggest that the conductance of the cover should be under 3 S. With 3 S cover, the time constant τ of the loop must be more than 0.3 ms. However, τ must be short enough to prevent overlap of the ground-loop response into the next transmitter waveform half-cycle. Thus, for a ground loop with a time constant > 0.3 ms, the base frequency of the AEM system must be under 500 Hz for a valid calibration, which was the case for the examples shown. Similarly, for a ground loop with 20
a time constant greater than 3 ms (in our VTEM experiment, τ = 4.62 ms), the system base-frequency must be under 50 Hz, which was met by the 25 Hz base-frequency used. For a fixed-wing time-domain system, where the separation between transmitter and receiver is much greater and the survey altitude is higher, we recommend ground conductance of less than 1.5 S; and that the ground loop should have a time constant τ > 0.3 ms. The limit of conductive cover can be increased up to about 3 S provided that τ > 1 ms. Of course, with such a large ground-loop time constant, the base-frequency of the AEM system should then be less than 150 Hz. The second recommendation concerns the geometry and electrical properties of the ground loop, which in turn determine the decay constant τ of the loop and the uniformity of the signal during flyover. The loop should be accurately surveyed and preferably laid out on flat ground to maximize geometrical accuracy. The time constant of the loop, determined by L and R, should be such that some signal will persist to the mid-time channels of the receiver, yet ideally have disappeared by the next transmitter half-cycle. In order to ascertain the time constant of the loop, its resistance needs to be predicted from wire specifications then measured when it is laid out. The self-inductance can easily be calculated from well-known inductance formulas. If the current induced in the ground loop is measured, the time constant may be predicted from that. The system should be flown over the ground loop at normal incidence, as in the HoistEM example, and not over the corners (AeroTEM example). Furthermore, to limit geometrical errors, we suggest in future using a rectangular ground loop, with the widest side perpendicular to the flight line. Frequency-domain systems can also be calibrated for geometry and altitude by the same procedure, but we do not illustrate that in this paper.
21
CONCLUSION Experiments were performed on 3 different time-domain AEM systems under different conditions; and each experiment illustrated different aspects of the ground-loop error detection method. Measurement of the current induced in the ground loop provides an excellent two-part calibration of an AEM system. Just as receiver response can be predicted from system geometry and timing information, the magnitude of the peak current induced in the ground loop can be predicted from the electrical properties of the loop, the peak transmitter current and geometry alone. This step precedes the timing and receiver response calibration step and allows us to focus on system geometry. This technique was well illustrated with the HoistEM experiments in Western Australia. The prediction of peak induced current allowed us to determine the geometry of the HoistEM transmitter in flight (altitude and GPS antenna-transmitter separation), and use the corrected geometry in the prediction of the receiver response. The result was clear, showing that the measured response exhibited significant averaging as well as a problem with receiver amplitudes. The VTEM system provided another insight to the use of our method: While most lines provided an excellent fit, on one line we found it impossible to fit the predicted responses to the measured responses by changing vertical position and gain alone. It was necessary to add a pitch parameter to the fitting process to account for the pendulum motion of the bird over the ground loop. The ground-loop error detection method in resistive terrain is successful. Wire loops are easy to lay out, very inexpensive and can be left on the calibration line for the duration of the survey. Their electrical properties are very well understood and highly configurable (i.e., it is easy to change the resistance and self-inductance of the loop). Multichannel
22
data loggers are relatively inexpensive and configurable GPS receivers are easily obtained allowing for synchronous recording of loop current. It is an accurate, fast and inexpensive method that can be used almost anywhere to determine the systematic errors in any timedomain AEM system provided the conditions mentioned in the previous section are met. Multiple passes exhibit the geometry changes effected by the pendulum motions of the bird, if they have not otherwise been measured. We believe that the ground-loop method is a viable and valuable way to determine systematic errors in AEM systems. Good results can be achieved despite the problems with geometry, as the method provides a simple input/output approach to match the transmitter current waveform to receiver responses. However, the method cannot be used to provide a systematic correction to geometry that can be applied to the transmitter-GPS antenna separation for the entire survey; until the issue of accurate system geometry can be resolved, the ground-loop method can only provide geometry checks for single lines. In this paper, we hinted at the information that can be gathered if the current in the ground loop is monitored during flyover. Not only does it allow us to determine geometry for the flyover, it also gives us direct information about the transmitter waveform. We have shown that the ground-loop current is the convolution of the loop impulse response with the derivative of the transmitter current waveform: if we know the transfer function of the measurement device used to measure the ground current, it is possible to deconvolve the signal and obtain the transmitter waveform. This is the subject of a future paper.
ACKNOWLEDGEMENTS The authors would like to thank the sponsors of AMIRA project P407B: AngloGold Ashanti; CRC LEME; CVRD; DSTO; Geoscience Australia; Noranda Inc/Falconbridge; Rio Tinto 23
Exploration; Xstrata Copper and Xstrata Nickel. We also acknowledge support from ARC Linkage LP 0348409; and Andrew Boyd, Don Hunter, Nick Ebner and Tim Munday for field work and data collection. Aaron Davis thanks the ASEG, the SEG and the GW Hohmann Trust for scholarship support. We would also like to thank the four anonymous referees for their valuable input.
APPENDIX A DESCRIPTION OF HOISTEM The HoistEM receiver was a 20-bit A/D converter that sampled in window widths of 112.7 µs (Boyd, 2004). Each window was divided into 4 sub-windows of width 25.325 µs and an ‘integration’ space of 11.4 µs. The sampling and stacking scheme was as follows: the first 12 delay channels were one sample window wide (25.325 µs), starting 5.0654 ms after the transmitter current turn-on. Next, 2 channels that were the full 112.7 µs window wide, sampled for 101.3 µs each. The remaining channels stacked an increasing number of 112.7 µs receiver windows, starting at 2 for channel 15, through to 17 receiver windows at channel 27. The HoistEM transmitter consisted of a single turn of stranded aluminium wire. The frame was constructed in a hexagonal shape and had a radius of 12 m. The transmitter current waveform was a 25% duty-cycle wave, powered by a 25 horsepower motor mounted on one skid of the helicopter doing the survey (Boyd, 2004). Vrbancich and Fullagar (2007) describe the HoistEM current pulse as quasi-trapezoidal with an exponential rise-time of approximately 500 µs, followed by a fast 40 µs linear shut-off at 5 ms. Peak currents were between 300 A and 340 A. A representative example of the HoistEM transmitter waveform and the receiver channels is shown in Figure A-1. Because HoistEM is a concentric-loop
24
system, the receiver had to be shut-off during the transmitter turn-on and turn-off to prevent damage to the circuitry; hence, there was no way to monitor the waveform during survey.
25
REFERENCES Boyd, G. W., 2004, HoistEM—a new airborne electromagnetic system, in PACRIM 2004, Adelaide, South Australia, 19–22 September. Brodie, R. and R. Lane, 2003, The importance of accurate altimetry in AEM surveys for land management: Exploration Geophysics, 34, 77–81. Davis, A. C., 2007, Quantitative characterisation of airborne electromagnetic systems: PhD. Thesis, RMIT University. Davis, A. C., J. Macnae, and T. Robb, 2006, Pendulum motion in airborne HEM systems: Exploration Geophysics, 37, 355–362. Deszcz-Pan, M., D. V. Fitterman, and V. F. Labson, 1998, Reduction of inversion errors in helicopter EM data using auxiliary information: Exploration Geophysics, 29, 142–146. Fitterman, D. V., 1998, Sources of calibration errors in HEM data: Exploration Geophysics, 29, 65–70. Fitterman, D. V. and C. Yin, 2004, Effect of bird maneuver on frequency-domain helicopter EM response: Geophysics, 69, 1203–1215. Grant, F. S. and G. F. West, 1965, Interpretation Theory in Applied Geophysics: McGrawHill Book Company. Griffiths, D. J., 1989, Introduction to Electrodynamics: Prentice Hall, 2 edition. Hefford, S. W., R. S. Smith, and C. Samson, 2006, Quantifying the effects that changes in transmitter-receiver geometry have on the capability of an airborne electromagnetic survey system to detect good conductors: Exploration and Mining Geology, 15, 43–52. Holladay, J. S., B. Lo, and S. K. Prinsenberg, 1997, Bird orientation effects in quantitative airborne electromagnetic interpretation of pack ice thickness sounding, in Oceans ’97, MTS (Marine Technology Society)/IEEE, volume 2, 1114–1116. 26
Kovacs, A. and J. S. Holladay, 1990, Sea-ice thickness measurement using a small airborne electromagnetic sounding system: Geophysics, 55, 1327–1337. Kratzer, T., J. Vrbancich, G. Boyd, and K. Matthews, 2007, Real-time kinematic tracking of towed AEM birds: Exploration Geophysics, 38, 132–143. Ley-Cooper, Y. and J. Macnae, 2007, Amplitude and phase correction of helicopter EM data: Geophysics, 72, F119–F126. Ley-Cooper, Y., J. Macnae, T. Robb, and J. Vrbancich, 2006, Identification of calibration errors in helicopter electromagnetic (HEM) data through transform to the altitudecorrected phase-amplitude domain: Geophysics, 71, G27–G34. Liu, G., 1998, Effect of transmitter current waveform on airborne TEM response: Exploration Geophysics, 29, 35–41. Nelson, P. H., 1973, Model results and field checks for a time-domain, airborne EM system: Geophysics, 38, 845–853. Simmons, G. F., 1991, Differential Equations With Applications and Historical Notes: McGraw-Hill, 2 edition. Smith, R. H., 2001, Tracking the transmitting-receiving offset in fixed-wing transient EM systems: methodology and application: Exploration Geophysics, 32, 14–19. Sørensen, K. and E. Auken, 2004, SkyTEM—a new high-resolution helicopter transient electromagnetic system: Exploration Geophysics, 35, 191–199. Spiegel, M. R., 1995, Mathematical handbook of formulas and tables: McGraw-Hill, Inc. Stolz, E. M. and J. C. Macnae, 1998, Evaluating EM waveforms by singular-value decomposition of exponential basis functions: Geophysics, 63, 64–74. Vrbancich, J., G. Boyd, and K. Mathews, 2007, Seatem—a new airborne electromagnetic system for bathymetric mapping and seafloor characterisation, in EGM 2007 International
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Workshop, Capri, Italy, 2007. Vrbancich, J., P. Fullagar, and J. Macnae, 2000, Bathymetry and seafloor mapping via one dimensional inversion and conductivity depth imaging of AEM: Exploration Geophysics, 31, 603–610. Vrbancich, J. and P. K. Fullagar, 2004, Towards seawater depth determination using the helicopter HoistEM system: Exploration Geophysics, 35, 292–296. ——–, 2007, Improved seawater depth determination using corrected helicopter time domain electromagnetic data: Geophysical Prospecting, 55, 407–420. Vrbancich, J. and R. Smith, 2005, Limitations of two approximate methods for determining the AEM bird position in a conductive environment: Exploration Geophysics, 35, 365– 373. Witherly, K., D. Irvine, and E. Morrison, 2004, The Geotech VTEM time domain helicopter EM system, in ASEG 17th Geophysical Conference and Exhibition, Sydney, 2004. Wolfgram, P. and J. Vrbancich, 2007, Layered earth inversions of AEM bathymetry data incorporating aircraft attitude and bird offsset—a case study of torres strait: Exploration Geophysics, 38, 144–149. Won, I. J. and K. Smits, 1987, Airborne electromagnetic bathymetry, in Fitterman, D. V., ed., Developments and applications of modern airborne electromagnetic surveys: U.S. Geological Survey Bulletin 1925, 155–164. Yin, C. and D. C. Fraser, 2004, Attitude corrections of helicopter EM data using a superposed dipole model: Geophysics, 69, 431–439.
28
LIST OF FIGURES 1
Diagram of the concept for calibrating an AEM system using a ground loop. The
height of the transmitter above the ground loop is measured using an altimeter (or GPS positions). The mutual inductances between transmitter, ground loop and receiver are used to calculate the expected response. 2
HoistEM flight lines 1 (dots, flying north-south) and 2 (open diamonds, flying
south-north) are shown in the area around the ground loop laid out on resistive terrain. Corners of the ground loop are marked with open squares. 3
Measured current induced in ground loop (black) as a result of the HoistEM fly-
over (line 1). The gray curves are the predicted values of IL(peak) (r, r′ , t), calculated from measured peak transmitter current, estimated ground-loop self induction and flight path recovery. 4
Measured (dots) and predicted (open circles) HoistEM receiver response for chan-
nels 2, 7 and 12, line 1. Disagreement between the measured and predicted responses can only be corrected by changing translation ∆d and averaging n. 5
Measured (dots) and predicted (open circles) HoistEM response (line 1) after an
average of 1.7 s (n = 17) and a line shift of ∆d = 18.5 m was applied to the predicted response. Predicted curves are of the correct width, and in the right place, but amplitudes do not match. 6
(a) Measured (dots) and predicted (open circles) HoistEM response after an aver-
age of 1.7 s (n = 17), a line shift of ∆d = 18.5 m, and an amplitude scaling of A = 0.694 is applied to the predicted response of line 1. (b) Each predicted delay channel fits the measured response quite well, with the residuals for each delay channel < 5 mV . 7
Current measured in the ground loop as the HoistEM was flown over (line 2, black), 29
as well as peak currents predicted by changing the vertical offset between measured radar altimeter and transmitter from ∆h = 19.3 m (gray dashed) to ∆h = 16.9 m (gray solid). Height error may be speed or pendulum dependent. 8
Voltage measured by the VTEM receiver for line 94. Ground-loop response is easily
seen from 100 m to 220 m. Vertical lines at 130 m and 180 m mark boundaries of the data used to calculate a mean loop response. 9
Mean decay response measured over the loop (solid dots) compared to an exponen-
tial decay of characteristic time τ = 4.62 ms forced to fit VTEM channel 14 (open circles). Misfit at early times is due about approximately 0.2 S of conductive cover. 10
(a) Measured (solid dots) and predicted (open circles) VTEM response for chan-
nels 1, 14, 20 and 26 of line 94 after changes of ∆h = −50.1 m and ∆d = −7.04 m were made to the predicted responses. All late time channels are fit very well by the decay, while measured channel 1 is substantially larger than the predicted values due to the conductive ground. (b) Differences between fitted and measured values for each fiducial of channels 14, 20 and 26. Misfit in channel 1 is not shown. 11
(a) Measured (solid dots) and predicted (open circles) VTEM response for chan-
nels 14, 20 and 26 of line 93 with fitting parameters ∆d = 5.5 m, ∆h = −52.0 m and θb = −9.33◦ . The addition of bird pitch corrects the shape of the prediction: it closely matches the measured curve. The predicted response of channel 1 is not shown. (b) Differences between measured and fitted curves for channels 14, 20 and 26. A-1 Positive half-cycle of the HoistEM transmitter current waveform, normalized to 1, with receiver windows marked by rectangles.
30
Line of flight
IT
Transmitter
Receiver VR
h
MTL
MRL
Trace of flightline on ground
Ground loop VL, IL
Figure 1: Diagram of the concept for calibrating an AEM system using a ground loop. The height of the transmitter above the ground loop is measured using an altimeter (or GPS positions). The mutual inductances between transmitter, ground loop and receiver are used to calculate the expected response. Davis & Macnae – GEO-2007-0266.R2
31
37600
Flight lines for HoistEM test, Western Australia
74
37500
74
37450
74
37400
74
37350
74
37300 73350 5
Northing (m)
37550
Line 1 →
74
Line 1 Line 2
Line 2 →
74
73400
5
73450
5
73500
5
73550
5
73600
5
73650
5
Easting (m)
Figure 2: HoistEM flight lines 1 (dots, flying north-south) and 2 (open diamonds, flying south-north) are shown in the area around the ground loop laid out on resistive terrain. Corners of the ground loop are marked with open squares. Davis & Macnae – GEO-2007-0266.R2
32
Measured and predicted peak ground-loop current HoistEM 400 300
Current (mA)
200
Current null
100 0 −100 −200 −300 −400
0 N
Measured Predicted 2
4
6 Time (s)
8
10 S
Figure 3: Measured current induced in ground loop (black) as a result of the HoistEM flyover (line 1). The gray curves are the predicted values of IL(peak) (r, r′ , t), calculated from measured peak transmitter current, estimated ground-loop self induction and flight path recovery. Davis & Macnae – GEO-2007-0266.R2
33
Measured and predicted HoistEM response, line 1 250
2
Measured Predicted
200
Response (mV)
150
7
100
12
50
2
7
12
0
−50
0 N
50
100
150 200 Distance (m)
250
300 S
Figure 4: Measured (dots) and predicted (open circles) HoistEM receiver response for channels 2, 7 and 12, line 1. Disagreement between the measured and predicted responses can only be corrected by changing translation ∆d and averaging n. Davis & Macnae – GEO-2007-0266.R2
34
Measured and predicted HoistEM response, line 1 n = 17; ∆d = 18.5 m 250 Measured Predicted
2
200
2
Response (mV)
150 7
7
100
12 50
12
0
−50
0 N
50
100
150 200 Distance (m)
250
300 S
Figure 5: Measured (dots) and predicted (open circles) HoistEM response (line 1) after an average of 1.7 s (n = 17) and a line shift of ∆d = 18.5 m was applied to the predicted response. Predicted curves are of the correct width, and in the right place, but amplitudes do not match. Davis & Macnae – GEO-2007-0266.R2
35
a)
Measured and predicted HoistEM response, line 1 n = 17; ∆d = 18.5 m; A = 0.694 150 2
Response (mV)
100 7 50 12 0 Measured Predicted
Residual (mV)
b)
−50
0
50
100
0 N
50
100
10
150
200
250
300
150 200 Distance (m)
250
300 S
5 0 −5
Figure 6: (a) Measured (dots) and predicted (open circles) HoistEM response after an average of 1.7 s (n = 17), a line shift of ∆d = 18.5 m, and an amplitude scaling of A = 0.694 is applied to the predicted response of line 1. (b) Each predicted delay channel fits the measured response quite well, with the residuals for each delay channel < 5 mV . Davis & Macnae – GEO-2007-0266.R2
36
Measured and predicted peak ground-loop current HoistEM, line 2 400 300
Current (mA)
200 100 0 −100 −200 Measured ∆h = 19.3 m ∆h = 16.9 m
−300 −400
0 N
2
4
6 Time (s)
8
10 S
Figure 7: Current measured in the ground loop as the HoistEM was flown over (line 2, black), as well as peak currents predicted by changing the vertical offset between measured radar altimeter and transmitter from ∆h = 19.3 m (gray dashed) to ∆h = 16.9 m (gray solid). Height error may be speed or pendulum dependent. Davis & Macnae – GEO-2007-0266.R2
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Secondary response above ground loop: VTEM Early time
0.3
Response (mV)
0.25
0.2
0.15
0.1
0.05 Late time 0
0 E
50
100
150 200 Distance (m)
250
300 W
Figure 8: Voltage measured by the VTEM receiver for line 94. Ground-loop response is easily seen from 100 m to 220 m. Vertical lines at 130 m and 180 m mark boundaries of the data used to calculate a mean loop response. Davis & Macnae – GEO-2007-0266.R2
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Measured and predicted decay curve: VTEM, line 94 0.3
Response (mV)
0.25
Channel 14
0.2 0.15
Average response τ = 4.62 ms
0.1 0.05
−1
10
0
10 Time after Tx shutoff (ms)
1
10
Figure 9: Mean decay response measured over the loop (solid dots) compared to an exponential decay of characteristic time τ = 4.62 ms forced to fit VTEM channel 14 (open circles). Misfit at early times is due about approximately 0.2 S of conductive cover. Davis & Macnae – GEO-2007-0266.R2
39
Measured and predicted VTEM response, line 94 ∆h = −50.1 m; ∆d = −7.4 m
a)
Measured Predicted
0.3 1 0.25 Response (mV)
14 0.2 20
0.15 0.1
26 0.05
Residual (mV)
b)
0
0
50
100
0
50
100
150
200
250
300
150 200 Distance (m)
250
300
0.01 0 −0.01
E
W
Figure 10: (a) Measured (solid dots) and predicted (open circles) VTEM response for channels 1, 14, 20 and 26 of line 94 after changes of ∆h = −50.1 m and ∆d = −7.04 m were made to the predicted responses. All late time channels are fit very well by the decay, while measured channel 1 is substantially larger than the predicted values due to the conductive ground. (b) Differences between fitted and measured values for each fiducial of channels 14, 20 and 26. Misfit in channel 1 is not shown. Davis & Macnae – GEO-2007-0266.R2
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Measured and predicted VTEM response, line 93 ∆h = −52.0 m; ∆d = 5.5 m; θb = −9.33◦
a) 0.4
Measured Predicted 1
0.35
Response (mV)
0.3
14
0.25 20
0.2 0.15 0.1
26
0.05
Residual (mV)
b)
0
0
50
100
0 W
50
100
0.01
150
200
250
300
150 200 Distance (m)
250
300 E
0 −0.01
Figure 11: (a) Measured (solid dots) and predicted (open circles) VTEM response for channels 14, 20 and 26 of line 93 with fitting parameters ∆d = 5.5 m, ∆h = −52.0 m and θb = −9.33◦ . The addition of bird pitch corrects the shape of the prediction: it closely matches the measured curve. The predicted response of channel 1 is not shown. (b) Differences between measured and fitted curves for channels 14, 20 and 26. Davis & Macnae – GEO-2007-0266.R2
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HoistEM transmitter waveform and receiver windows 1
IT(t)
0.5 0 −0.5 −1 0
5
10 Time (ms)
15
20
Figure A-1: Positive half-cycle of the HoistEM transmitter current waveform, normalized to 1, with receiver windows marked by rectangles. Davis & Macnae – GEO-2007-0266.R2
42
