Effects of full transmitting-current waveforms on transient electromagnetics : Insights from modeling the Albany graphite deposit

In transient electromagnetic (TEM) methods, the full transmitting-current waveform, not just the abrupt turn-off, can have effects on the measured responses. A 3D finite-element timedomain forward-modeling solver was used to investigate these effects. This was motivated by an attempt to match, via forwardmodeling, real data from the Albany graphite deposit in northern Ontario, Canada. Initial modeling results for homogeneous halfspaces illustrate the effects that a full waveform can have on TEM responses, especially the durations of the steady stage and turn-off time. For the Albany data set, a geophysical conductivity model was developed from a geologic model that itself had been constructed predominantly from drillhole information. The conductivities of the various geologic units in the model were first estimated based on typical conductivity values for the respective rock types, then adjusted to fit the measured TEM data as closely as possible. We found that the TEM responses differed significantly from the pure step-off response and that incorporating the effects of the full waveform (particularly the linear ramp turn-off) greatly improved the match between observed and computed responses, especially for the early measurement times. In addition, this Albany example illustrates the presence of sign changes in TEM data caused primarily by localized conductivity targets.


INTRODUCTION
Transient electromagnetic (TEM) methods have been widely used for mineral exploration, hydrogeologic investigations, and other applications (Nabighian and Macnae, 1988;Flores and Peralta-Ortega, 2009;Costabel et al., 2017).The 3D inversion is still not common when interpreting such data.However, as is widely appreciated, 3D forward modeling can play a fundamental role in survey design and interpretation of TEM data.
In the past few decades, several important advances have been made in 3D time-domain forward modeling solvers that enable them to handle the complexities of real-life situations.For example, several finite-element solvers that use unstructured tetrahedral grids and thus have the capacity to deal with complex-shaped topography and geoelectric bodies have been developed (Börner et al., 2008;Um et al., 2010;Li et al., 2018).In addition, TEM data can be affected by anisotropic conductivity (Collins et al., 2006) and induced polarization (IP) effects (Flores and Peralta-Ortega, 2009), and 3D forward modeling solvers have been developed that specifically take such effects into account (Um et al., 2010;Marchant et al., 2014).
Another complicating factor when dealing with real-life TEM data is that the time variation of the current in the transmitter is not trivial.Figure 1 shows a characteristic variation of a full waveform for a "bipolar" cycle.There is an "on-time," during which the transmitter current is on, and currents are being induced and energized in the earth's subsurface, and an "off-time," during which the current in the transmitter is off, the induced current system in the subsurface evolves and decays, and measurements of the resulting magnetic field are made.The on-time typically includes three parts: a turn-on stage, a steady stage, and a turn-off stage (Everett, 2013).The current rises slowly during the turn-on stage, and then it stays constant during the steady stage.This is to allow the currents induced in the subsurface by the turn-on stage to decay to zero.The turn-off stage is typically short to increase the strength of the currents induced in the subsurface, but not so short that the waveform cannot be controlled.To mitigate system noise such as DC bias and coherent noise, the whole on-time-off-time pair for a positive current is repeated with the current in the opposite sense to complete one full bipolar cycle (shown in Figure 1).This whole cycle is then repeated multiple times during the measurement process to perform stacking to enhance the signal-to-noise ratio (S/N).
Other transmitter current waveforms have also been used besides those shown in Figure 1.For example, a half-sine on-time is used by GEOTEM systems (Smith et al., 1996), and a saw-tooth waveform, which has no off-time as such and effectively measures the step response of the subsurface, is used by UTEM systems (West et al., 1984).
The effects of a full waveform on TEM responses have previously been investigated using 1D modeling.For example, Raiche (1984) derives the formulas used to evaluate TEM responses of a layered half-space for a linear ramp turn-off, prior to which the waveform is constant.This method requires that the Laplace transform of the turn-off waveform is known.Fitterman and Anderson (1987) present a more general procedure that is based on the step function response and that can be used with any turn-off waveform.A full transmitter waveform, including a slow turn-on stage, can have nonnegligible effects on TEM responses, especially for a conductive medium in which the induced currents persist longer.Taking the bipolar transmitter waveform used by Geonics EM-37 as an example, Asten (1987) computes full-waveform TEM responses using existing impulse-response forward algorithms combined with a convolution procedure with truncation and residual estimation after an even number of terms.Liu (1998) investigates the effects of different transmitter waveforms on airborne TEM responses in mineral exploration, as do Chen et al. (2012).Christiansen et al. (2011) study the effects of turnoff time and waveform repetition on 1D inverted models for helicopter TEM data.
To date, little work has been done to investigate full-waveform effects when carrying out 3D modeling.Yang and Oldenburg (2012) consider the waveform effects on TEM responses when performing 3D inversions of Mt.Milligan versatile time-domain electromagnetic (VTEM) field data.Sun et al. (2013) analyze the effects of the turn-off time on TEM responses using an FDTD forward solver.Qi et al. (2017) incorporate full waveforms into their finite-element time-domain (FETD) codes for airborne TEM methods, but they did not analyze the effects of these waveforms on TEM responses.
Here, the FETD forward-modeling solver of Li et al. (2018) is used.This FETD approach follows that originally proposed by Um et al. (2010) for the marine TEM scenario, which uses a grounded-line source.This type of finite-element time-stepping method has since been applied to 3D numerical modeling for airborne and on-land loop-source TEM systems (Yin et al., 2016;Li et al., 2018).We use this FETD approach to investigate the effects of the full transmitter current waveform when considering basic models as well as real-life models relevant to mineral exploration.

METHOD
The governing equation of the FETD method of Um et al. (2010) is the curl-curl equation for the total electric field e, which, in the quasistatic regime, is where t is the time, σ is the conductivity, and μ is the magnetic permeability.In this study, we approximate the permeability by that of free space, namely, μ ¼ μ 0 .The term j s e is the electric current density of the source.
The implementation of the FETD solver considered here (Li et al., 2018) can deal with a complex-shaped transmitting loop, which is viewed as a combination of electric dipoles (EDs), each of which may have its own direction.An ED can be further decomposed into two horizontal EDs along the x-and y-directions and one vertical ED along the z-direction.Given that the length of the initial ED is ds, the electric current density of this ED can be represented as follows (Li et al., 2018): where δ is Dirac's delta function and I is the amplitude of the current, which is a time-dependent function.For the three decomposed EDs, c x , c y , and c z are the length coefficients and (x x ; y x ; z x ), (x y ; y y ; z y ), and (x z ; y z ; z z ) are the central points.
Applying the Galerkin method to equation 1, we obtain the finiteelement formulation as where u is a vector of size N, and A and B are N × N matrices, where N is the number of degrees of freedom.For a tetrahedral cell te, the elements of u are ½ e te , where e te j is the approximated electric field corresponding to the jth edge.The elements of A and B are where N j is the first-order vector basis function, which is described in detail in Jin (2014).The term S is nonzero only for elements containing a segment of the transmitting loop, and it is expressed as The first-order backward Euler method is used to discretize equation 3 in time for a full waveform with a linear turn-on stage as shown in Figure 1a.The resulting formula is ðA þ ΔtBÞu kþ1 ðtÞ ¼ Au k ðtÞ − ΔtS kþ1 ; (7) where the superscript k is an integer equal to or greater than 0 that denotes the time-stepping index.For a full waveform with a nonlinear turn-on stage, as shown in Figure 1b, the second-order backward differentiation formula is used (Um et al., 2010).Then, equation 3 can be discretized as ð3A þ 2ΔtBÞu kþ2 ðtÞ ¼ Að4u kþ1 ðtÞ − u k ðtÞÞ − 2ΔtS kþ2 : (8) The electric field is zero everywhere before the current in a transmitter loop source is switched on.Hence, the initial condition is that the electric field is set to zero everywhere.In the study of Li et al. (2018), a step-off waveform is considered, and the electric field is also set to zero within the whole computational domain before the current is turned off.The implementation of Li et al. (2018) can therefore only be used to study off-time TEM responses.However, the algorithm presented here has the capacity of studying the TEM responses for a full transmitter current waveform.In the following study, the time-stepping scheme is that the time step increases five times every 200 iterations, which was demonstrated to be accurate and efficient by Li et al. (2018).
The linear systems of equations 7 and 8 are solved by MUMPS, an open-source package of direct solvers (Amestoy et al., 2006).After obtaining the approximated electric field uðtÞ for the measurement time t, the approximated time derivative of the magnetic induction ∂bðtÞ∕∂t in each tetrahedral cell can be obtained by

EXAMPLES
This section has two parts: half-space examples and the Albany graphite deposit example.The conclusions obtained from the halfspace examples, which comprise a thorough, guided study, will be incorporated when synthesizing the ∂b z ∕∂t response for the Albany example.

Half-space examples
The examples provided here are homogeneous half-spaces of 25 and 0.0004 S∕m, for which the air conductivity is 10 −8 S∕m.A 100 × 100 m loop is laid on the surface of these half-spaces.The observation location is at the loop center.The computational domain is set to 1 and 100 km in all three directions for the 25 and 0.0004 S∕m half-spaces, respectively.The number of edges is 284,087 for the 25 S∕m half-space, and it is 302,354 for the 0.0004 S∕m scenario when using TetGen 1.5.1-beta1(Si, 2015) to generate unstructured tetrahedral meshes.The computing environment is as follows: Linux Ubuntu operation system, dual Intel Xeon E5-2640 v3, 2.6 GHz CPU, and 64 GB of RAM.

Effects of the steady-time duration on ∂b z ∕∂t
We first studied the effect of the duration of the steady time on the ∂b z ∕∂t responses measured at times after the turn-off time.

Full-waveform effects on TEM responses E257
The FETD solver presented above and the convolution method are used to evaluate the ∂b z ∕∂t response for the first half-cycle of the waveform shown in Figure 1a.As shown in equation 10, the convolution method refers (in this study) to calculating the impulse response f s excited by a pure step turn-off source using the FETD solver and then convolving it with the derivative of the transmitting-current waveform to calculate its corresponding ∂b z ∕∂t response f w : Here, a trapezoidal numerical integration is used to solve equation 10.Table 1 and Figure 2 show the five waveforms used here.Waveforms 1 and 2 use a steady-state initial condition followed by a linear ramp off to approximate a step-off response after an infinite on time.For waveforms 1 and 2, the linear-ramp durations are 10 −8 and 10 −7 s, respectively, which are also the unit spacing in the trapezoidal numerical integration.For waveforms 3-5, the turn-on current is a linear ramp, its duration is 2 ms, and the total on-time is set to 8, 40, and 200 ms.The initial time step is 10 −6 s for the turn-on stage.For the turn-off stage, the initial time step is set to 10 −8 s for waveform 1, and 10 −7 s for waveforms 2-5.
Figure 3a shows the ∂b z ∕∂t response evaluated by the analytic method for a pure step turn-off (Li et al., 2016) and by the FETD solver for waveforms 1 and 2. For the half-spaces of 25 and 0.0004 S∕m, these two types of solution coincide well with each other, except the FETD solution for waveform 2 at measurement times earlier than 0.001 ms.The FETD solution for waveform 1 is also reused in the convolution method.Figure 3b-3d shows the ∂b z ∕∂t response for the times after the turn-off time calculated by the FETD solver and the convolution method for waveforms 3-5, respectively.For both the half-spaces, the responses calculated by these two methods present a good agreement and get gradually closer to the analytic turn-off response as the length of the steadytime increases.For example, the analytic solution is obviously larger than the two types of numerical solution for waveform 3 at all measurement times (Figure 3b), and it nearly coincides with the numerical solutions for waveform 5 (Figure 3d) for the 25 S∕m half-space.
The underlying mechanism that is generating the responses described in the preceding paragraph is better explained using vector maps of the electric field (Figure 4) during the on-time and offtime stages.As is well-known, a current system is induced in the subsurface by the current in the transmitting loop turning on.This current system then decays after the turn-on stage is finished.Another induced current system is generated when the current is turned off.In Figure 4a and 4c, the electric fields are observed at 8 and 200 ms, at which the ramp turn-off is initiated in waveforms 3 and 5, respectively.In other words, they are the initial field for calculating the impulse response excited by a ramp turn-off current.
Table 1.The waveforms and their corresponding computation time and the number of iterations for the homogeneous halfspaces when studying the effects of the steady-time duration on the ∂b z ∕∂t response.1) and by the convolution method for waveforms 3-5 (in Table 1), respectively.The origin for the horizontal axis in all panels corresponds to the end of the turn-off time.Full-waveform effects on TEM responses

E259
Because the direction of the second induced current system is opposite to the first (see in Figure 4), the response of the second system is counteracted by the response of the previous one if its decay is not finished.If the steady stage and hence on-time is long enough, the first induced system will have become relatively weak by the time of the turn-off stage and will therefore have little or no effect on the off-time measurements.This is the reason why the electric field in Figure 4d is relatively stronger than the one in Figure 4b.However, as the length of the steady stage increases, so too does the length of one complete cycle of the waveform, and hence the time taken to stack a given number of responses to get a good S/N.The relative errors between the FETD solutions (shown in Figure 3) for the four waveforms and the analytic solution at measurement times after the turn-off time reveal other details.As can be seen from Figure 5, the curves of the relative error of both the half-spaces for waveforms 3-5 are intermixed with each other first, and then gradually separate from 0.001 ms for the 25 S∕m half-space, and from 2 ms for the 0.0004 S∕m one.This demonstrates that the duration of the steady time has stronger effects on the ∂b z ∕∂t response for the 25 S∕m half-space.The duration of the ramp turn-off also has an effect on the accuracy of the convolution method mainly for early measurement times.For example, the relative errors in the responses at the measurement times earlier than 0.01 ms for waveform 3-5 decreases dramatically as the turnoff duration is shortened to from 10 −7 s (waveform 2) to 10 −8 s (waveform 1) for the 0.0004 S∕m half-space (in Figure 5d).It is not necessary to use such a short duration for the ramp turn-off for synthesizing a realistic model because the measurement time is always larger than 0.01 ms.
As shown in Table 1, the total computation time for waveforms 3-5 using the convolution method is approximately 25 min (based on waveform 2), and it increases to 180 min for the FETD solver.As a side note, the computation time always depends on the number of iterations in FETD.Here, a relatively conservative time-stepping scheme is used to obtain more accurate numerical solutions.The number of iterations for every stage is also shown in Table 1.Moreover, the FETD approach requires the recalculation of the whole time-stepping sequence from the start of the turn-on stage for a different waveform for the same earth model.In contrast, the convolution approach can efficiently compute the responses for different waveforms for the same model once the impulse response has been generated for the model.Hence, the convolution method can be more efficient than the FETD solver for the full-waveform modeling.

Effects of the turn-on time on ∂b z ∕∂t
We then investigated the effect of the length of the turn-on time on the ∂b z ∕∂t response for the first half of the bipolar waveform (see Figure 1a).The turn-on current is a linear ramp with durations of 0.001, 0.01, 0.1, and 2 ms.We consider here, the total duration of the turn-on and steady stages to be fixed (at 8 ms), meaning the steady-stage shortens as the turn-on time lengthens.The turn-off time is 0.01 ms.As shown in Figure 6a and 6b, the ∂b z ∕∂t response for these four turn-on times coincides well with each other, but they still exhibit slight differences.Figure 6c and 6d shows the ratio between the off-time ∂b z ∕∂t response for the turn-on time of 0.001 ms and the turn-on times of 0.01, 0.1, and 2 ms.For the half-space of 25 S∕m, the ratios decrease from 1 to approximately 0.88 at all 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Time (s)  (c) The ratio between the off-time responses for the turn-off time of 0.0001 ms and the turn-off times of 0.001, 0.01, and 0.1 ms.The origin for the horizontal axis corresponds to the end of the turn-off time in all panels.Figure 6.Effects of the turn-on time on ∂b z ∕∂t for the turn-on times of 0.001, 0.01, 0.1, and 2 ms.(a and b) The impulse response for half-space of 25 and 0.0004 S∕m, respectively. (c and d) The ratio between the off-time responses for the turn-on time of 0.001 ms and the turn-on times of 0.01, 0.1, and 2 ms for these two half-spaces.The origin for the horizontal axis corresponds to the end of the turn-off time in all panels.measurement times as the duration of the turn-on stage increases.For the half-space of 0.0004 S∕m, the ratios are close to 1 until 1 ms, and then they begin to decrease, especially for the turn-on time of 2 ms.For example, the ratio between the responses for the turn-on times of 0.001 and 2 ms is smaller than 0.9 at the measurement time of 80 ms.These phenomena for the half-spaces indicate that the effects of the length of the turn-on time on the ∂b z ∕∂t response are relatively trivial, but they should not be neglected especially for a long turn-on stage.

Effects of the turn-off time on ∂b z ∕∂t
We next studied the effect of the turn-off time on the ∂b z ∕∂t response for a positive current.The total time of the turn-on and steady stages is again 8 ms.Turn-off times of 0.0001, 0.001, 0.01, and 0.1 ms are considered.As shown in Figure 7a and 7b, the ∂b z ∕∂t responses for the turn-off times of 0.01 and 0.1 ms are significantly weaker in comparison with the ones for the other two turn-off times when the measurement time is earlier than 0.01 ms.This suggests that the turn-off time has a strong effect on ∂b z ∕∂t for the times after the turn-off time, regardless of the half-space conductivity.These results are consistent with those observed when investigating the effect of the length of the turn-off time using 1D codes for homogeneous half-space (and layered earth) models (Asten, 1987).Figure 7c shows the curves of the ratio between the off-time responses for the turn-off time of 0.0001 ms and the turn-off times of 0.001, 0.01, and 0.1 ms.The overall trend is that the ratio gradually increases up to 1 for both the half-spaces.A distinct characteristic is that the ratio for the 0.0004 S∕m half-space is smaller than the one for the 25 S∕m half-space at measurement times earlier than 10 ms.This demonstrates that the effects of the turn-off time on the ∂b z ∕∂t response are more significant in a resistive environment.

Effects of waveform repetition
Finally, for the homogeneous half-spaces, we investigated the effects of waveform repetition on the ∂b z ∕∂t response for the linear and nonlinear turn-on stages (as illustrated by Figure 1).Here, we adopted a full bipolar waveform with five repetitions, for which the first half-cycle uses a positive current, and the second one uses a negative current.For the positive and negative currents, the turn-on, steady, and turn-off times were set to 2, 6, and 0.01 ms, respectively.The nonlinear turn-on current is based on the formula used by the Crone Pulse electromagnetic (EM) system (Dyck and West, 1984): where c is a time constant determined by the inductance and resistance of the loop.Here, it was set to 0.8 ms.
Figure 8 shows the ∂b z ∕∂t response for a full bipolar waveform with five repetitions and for a 0.0004 S∕m half-space.The  responses for the linear and nonlinear turn-on stages agree well with each other at all measurement times.Figure 9a and 9b shows the curves for the relative differences of the response measured at times after the turn-off time between the first and second halves within the same repetition.For the half-spaces of 25 and 0.0004 S∕m, the relative-difference curves for five repetitions are nearly parallel with each other except the parts for the 0.0004 S∕m half-space at early times, at which point the relative differences are extremely small.The relative difference within the same repetition decreases significantly as the repetition number increases.For example, the relative differences at the measurement time of 46.86 ms are 7.48%, 0.68%, 0.19%, 0.08%, and 0.04% for the repetitions from one to five and for the 0.0004 S∕m half-space.In practice, measurements are made after at least several and usually thousands of periods of the transmitted waveform, so there should not be any difference between the positive current response and the negative current response.Figure 9 also shows that the waveform-repetition effects on the ∂b z ∕∂t response have a strong relationship with the half-space conductivity, as do the turn-off effects.
Through the above analyses, the steady-time duration, turn-on time, turn-off time, and waveform repetition have effects on the ∂b z ∕∂t response measured at times after the turn-off time.Furthermore, the effects of the steady-time duration and turn-on time mainly appear at late times, the effects of the turn-off time always appear at early times, and the waveform-repetition effects appear at all measurement times after the turn-off time.In addition, these effects are also related to the half-space conductivity.

The Albany graphite deposit example
The second example we present here is a real-life scenario.The Albany graphite deposit, a rare type of igneous-hosted, fluidderived graphite deposit, is located in the Superior Province of the Canadian Shield at the terrane boundary between the Quetico Subprovince to the south and the Marmion Subprovince to the north (Conly and Moore, 2015;Legault et al., 2015).The basement is composed of Precambrian granitoids and metasedimentary rocks to the south and dominantly tonalite to granodiorite to the north.These rocks are intruded by the younger Nagagami River alkalic intrusive complex; the Albany graphite deposit occurs within a quartz monzonite dominant intrusive host rock that is likely part of the complex (Conly and Moore, 2015).The basement rocks are covered with up to 15 m of relatively thin, flat-lying Paleozoic limestone, sandstones, shales, dolostones, and siltstones and up to 50 m of overburden.
The Albany graphite deposit was discovered by Zenyatta Ventures Ltd. through follow-up of a 2010 airborne TEM and magnetic geophysical survey (Legault et al., 2015).A subsequent ground TEM survey was used to better delineate the two graphite deposits (the "east pipe" and "west pipe") in 2013.The Albany Project ground TEM survey included two transmitting loops.Loop 1 had an irregular shape due to the topography, being an approximately 1200 × 1500 m rectangular loop controlled by 86 points (see Figure 10).The loop 1 survey used the in-loop configuration, with 295 observation locations within the loop (see Figure 10).The survey was performed using a Crone Pulse EM system.For the synthetic modeling study presented here, only data from the loop 1 survey were considered.
The synthetic Albany model that was used for numerical modeling was built mainly based on the drillhole data (Ross and Masun, 2014).The computational domain was 20 × 20 × 20 km in extent and was divided into 2,331,286 tetrahedral elements and 2,703,949 edges by TetGen 1.5.1-beta1(Si, 2015).This model consists of 11 domains, the geometric relationships of which are shown in Figure 11.We built the wireframes for the domains numbered 1-4, which include the overburden and sedimentary-rock layer.These wireframes can be directly used by TetGen.Zenyatta Ventures Ltd. provided the surfaces for the domains numbered 5-11 that had been constructed from drill logs.However, these surfaces intersected with and passed through each other in several places and therefore could not be directly used to generate tetrahedral grids.The surfaces were hence edited and reconstructed in a way that ensured watertight contacts between interfaces and that no surfaces passed through others or themselves.This editing and reconstructing of the wireframe model was performed using FacetModeller (Lelièvre et al., 2018).
The conductivities used for the Albany model are shown in Table 2.These values were chosen based on typical conductivities for the known rock types and then refined by trial and error to make the synthetic TEM data match the field data.
There are two major differences between the geologic model provided by Zenyatta and the synthetic geophysical one that gave the best fit to the real TEM data.One difference is the thicknesses for the overburden and sedimentary rocks.These two thicknesses as obtained from drilling data are 50 and 15 m, respectively (Legault et al., 2015).To better fit the real data, these thicknesses had to be adjusted to 10 and 36 m, respectively, in the geophysical model.One possible explanation is that the shallower overburden has a relatively high conductivity compared with the deeper part, whose conductivity is close to that of the sedimentary rocks below.Another difference is the geometric relationship between the barren sill, east pipe, and west pipe.In the geologic model, the two pipes Full-waveform effects on TEM responses E263   2.
are cut by the barren sill, whereas, due to it being easier to construct, the two pipes intrude through the sill in our synthetic geophysical model.The FETD modeling solver was used to synthesize the ∂b z ∕∂t response for this example.The waveform used approximates that of the Crone Pulse EM system.The total time for the turn-on time, steady stage, and turn-off time was 50 ms (i.e., a quarter of a cycle), and the duration of the turn-off time was 1.5 ms (Khan, 2013).As there is no actual value of the turn-on time used for the Albany graphite example, we designed the turn-on waveform based on equation 11.Three different values of time constant c (0.6, 0.8, and 1.0 ms) and two different turn-on times (5 and 10 ms) were considered.
Figure 12 shows the full-waveform ∂b z ∕∂t response for the observation location 1000Nav-P09.For both turn-on times, the curves for the different values of the time constant have some small differences at the times earlier than 5 ms, and then they coincide well with each other.It further demonstrates that the turn-on waveform has minor effects on the ∂b z ∕∂t response for this real-life example.This is the same as what was observed for the half-space examples in the first part of this section.Figure 13a and 13c shows the response curves for the field data and the synthetic data using the waveforms shown in Table 3.The ∂b z ∕∂t curve for waveform 1 is not able to match the field data for measurement times smaller than 1 ms, whereas the ones for waveforms 2-4 agree with the field data.This suggests that the effects of the Full-waveform effects on TEM responses E265 turn-off time on the responses are very strong and mainly appear at the early measurement times for the Albany graphite deposit, just as for the half-space examples.As shown in Figures 13b  and 13d, the relative differences between the field data and the responses for waveforms 2-4 are mostly larger than 20% for measurement times larger than 1 ms, and the relative differences for waveform 2 seems to be smaller than the ones for the other two waveforms.This result indicates that a more accurate numerical solution is not obtained when incorporating the effects of the steadytime duration, turn-on time, and waveform repetition into synthesizing data for the Albany model.The inaccuracies are caused by the difference between the synthetic Albany model and the true one.In addition, it takes more computation time if the effects of the steady-time duration, turn-on time, and waveform repetition are included in the FETD solution (shown in Table 3).Hence, it is worthwhile to take into account the turn-off effects, but not necessarily the other three parameters, of a full waveform when synthesizing field data.
The sign change for the observation location 1000Nav-P14 (Figure 13c) should be noted.Its very existence in the field data and the synthetic data obtained by the FETD solver that does not consider any IP effect indicates that these sign reversals can be purely caused by the large conductivity contrast between the graphite ore body and the host rocks, as well as the ore body's irregular 3D shape.Therefore, for this data set, Table 3.The waveforms and their corresponding computation times used for synthesizing the ∂b z ∕∂t response for the Albany graphite deposit example.The time constant c is set to 0.8 ms for the nonlinear turn-on stage in waveforms 3 and 4. A full bipolar waveform has been repeated twice before measuring ∂b z ∕∂t for waveform 4. the IP effect is not necessarily the primary cause for the sign reversal phenomenon despite that graphite ore deposits are known to have strong IP effects (Pelton et al., 1978).The response at this observation location, as for all locations within the loop, is positive at all times for a homogeneous half-space.Li et al. (2017) previously illustrate the mechanism for this sign-change phenomenon at the Ovoid Zone massive sulfide ore body located in Labrador, Canada.
The range of the synthesized responses is nearly the same as that for the field data, and there is a good agreement between the patterns of computed and real data (Figures 14 and 15).However, the spatial variation of the real data is relatively complex, especially for the late times (Figure 14).This is most likely because some or all of the various geologic units present in the true subsurface are not particularly uniform, as was assumed in the geophysical model.

CONCLUSION
The work presented here has demonstrated the successful modeling of a real TEM data set for a real-life mineral exploration example.The data set was from a large-loop ground survey with observation locations inside the loop.The survey was over the Albany graphite deposit, Canada, which comprises two mostly vertical electrically conductive graphitic pipes.A critical aspect of being able to model the data set was the incorporation of the full transmitter current waveform, particularly the linear ramp turn-off.This was accomplished via time stepping, in an FETD modeling approach, throughout the turn-on, steady stage, and turn-off of the waveform.As other authors have reported, firstorder backward Euler time stepping is sufficient for a waveform made up of linear segments, whereas a second-order backward differentiation formula is required for more complicated waveforms.The geophysical conductivity model of the Albany deposit was described in terms of an unstructured tetrahedral mesh.This mesh was constructed from a wireframe geologic model that had been derived from drillhole information, albeit after editing the wireframe surfaces to repair infeasible intersections and holes.The conductivities of the various geologic units were adjusted by trial and error, starting from typical values for the various rock types, until the computed data matched the real data.Once the linear ramp turn-off had been incorporated into the modeling, a good match between modeled and real data was obtained.This match included the sign changes observed in the real data, which can be accounted for by the 3D effects of the localized, conductive graphitic pipes.

Figure 4 .
Figure 4. Vector maps of the electric field e at the surface of a homogeneous half-space of 25 S∕m.(a and b) Waveform 3 and (c and d) waveform 5 listed in Table 1.The measurement times are relative to the turn-on of the current in all panels.The black square indicates the transmitting loop of 100 × 100 m.

Figure 5 .Figure 7 .
Figure5.The relative errors between the numerical solutions for waveforms 3-5 listed in Table1and the analytic solutions for a pure step turn-off.(a and b) The half-spaces of 25 S∕m and (c and d) the half-spaces of 0.0004 S∕m.The origin for the horizontal axis in all panels corresponds to the end of the turn-off time.

Figure 8 .
Figure8.The ∂b z ∕∂t response for a full bipolar waveform with five repetitions and for a 0.0004 S∕m half-space.For positive and negative currents, the turn-on, steady, and turn-off times are 2, 6, and 0.01 ms, respectively.

Figure 9 .
Figure 9.The relative differences of the ∂b z ∕∂t response between the first and the second halves within the same repetition.(a and b) The curves for half-spaces of 25 and 0.0004 S∕m, respectively.The origin for the horizontal axis corresponds to the end of the turn-off time.

Figure 10 .
Figure 10.The transmitting loop and observation points for loop 1.The two black stars indicate the observation locations 1000Nav-P09 (the top star) and 1000Nav-P14 (the bottom star).

Figure 11 .
Figure11.The tetrahedral mesh for loop 1 built by FacetModeller(Lelièvre et al., 2018).(a) A perspective view from the air of the two pipes, (b) a perspective view from west to east of the west pipe, (c) a perspective view from south to north of the two pipes and sill, and (d) a section view of northing 5545700 m for the central part of the mesh.The numbers presented in this figure are the domain numbers shown in Table2.

Figure 13 .Figure 12 .
Figure 13.The ∂b z ∕∂t response and its corresponding relative difference for the waveforms in Table 3 in comparison with the field data.(a and b) The observation location 1000Nav-P09 and (c and d) 1000Nav-P14.

Figure 14 .
Figure 14.Contour maps of the ∂b z ∕∂t field data for loop 1.The dashed white lines indicate the boundary between the regions of positive and negative responses.The two black stars indicate the observation locations 1000Nav-P09 and 1000Nav-P14.

Figure 15 .
Figure15.The same as Figure14, but for the synthetic data.

Table 2 .
The domains in the tetrahedral mesh for the Albany graphite deposit example and their assigned conductivity values.