Moment Estimation from Time Domain Electromagnetic Data

Abstract

Moment representations have been proposed to facilitate the interpretation of geophysical time domain electromagnetic responses. We present a new methodology for estimating these moments from field data for different system waveforms when on-time and off-time measurements are available. Quadrature impulse response moments are estimated by a recursive relation involving moments of the input waveform and moments of the observed response. After adapting this method to time domain electromagnetic applications—in particular, MEGATEM and AeroTEM (AirTEM) airborne electromagnetic systems—we present the results from applying this method on synthetic and real data collected over the Reid–Mahaffy test site in northern Ontario, Canada.

1. Introduction

Time domain electromagnetic (TDEM) methods are used in exploration geophysics to search for subsurface mineralized zones that are conductive [1]. Their principle is as follows: an electric current pulse, associated with a primary magnetic field through Ampère’s law, circulates in the transmitter loop, as illustrated in Figure 1. According to Faraday’s law of induction, this primary magnetic field generates secondary currents in conductive materials in the subsurface. These secondary currents are associated with a secondary magnetic field that is detected by the receiver loop. As the pulse is of limited duration, the secondary signal can be measured while the pulse is on (on-time) and after the pulse is turned off (off-time). Typically, only off-time measurements are recorded since on-time measurements are affected by the primary field. In airborne electromagnetics (AEMs), the transmitter (source) and receiver are either located on the aircraft or are towed below the aircraft (Figure 2).

There have been many advancements since the early development of TDEM techniques to localize mineralized zones with EM measurements. Better understanding of the electromagnetic (EM) principles involved [1,2,3] and the development of computer techniques have led to improvements in various domains. Ref. [4] proposed to localize anomalies and interpret them by comparing them with the results of analog model simulations. Following advances in computer technology, Ref. [5] developed an inversion program to invert AEM data or to retrieve the parameters of the model by numerical curve-fitting using the numerical model of a plate in free space developed by [6]. The model based on the inversion of a plate was extended by [7] to consider host rock to be conductive, layered earth. Refs. [8,9] proposed an algorithm for inverting a sphere in layered earth. The thin sheets in the layered earth model of [7] were included by [10] for an initial model for the probabilistic inversion of AEM data for basement conductors.

Some of these elements were incorporated into the three-step process proposed by [11] and refined by [12], which is as follows: (1) compress the data to decay time-constant space, (2) transform the response to a conductivity–depth image, and (3) parameterize local anomalies with plate-like conductors. Similarly, Ref. [13] proposes to invert AEM data in two phases in order to determine background conductivity models with layered-earth inversions and interpret discrete conductors as electric or magnetic dipoles, while the approach in [14] approximates the subsurface current system using a 3D subsurface grid of either 3D magnetic or electric dipoles, for which locations and orientations are solved by inversion. Some of the most elaborate interpretation techniques invert the complete airborne dataset with sophisticated mathematical techniques and solve the forward model with an integral equation [15,16,17], finite volume method [18], or randomly undersampled forward model [19].

Despite these advances, elaborate EM interpretation techniques are not generalized to all AEM datasets as they are complex and costly in terms of time and computer resources and because of the enormous dataset sizes. For these reasons, simple interpretation tools have the potential of assessing data quality quickly, locating discrete conductors, and facilitating the use of more complex interpretation methods. Some of these tools include the use of parameters such as the apparent conductivity of the subsurface [20,21], the decay time constant [22,23], and inductive and resistive limits [24]. At the inductive limit, current flows do not penetrate conductive bodies, and at the resistive limit, the response varies linearly with conductivity. Resistive limit estimation was used by [21,24,25] as the first step of an EM interpretation process. Refs. [26,27] extended the concepts of these limits to the moments of the quadrature impulse response, which are time integrals of the response signal, as the zero-order and first-order moments correspond to the inductive and resistive limit, respectively. Subsequent authors developed solutions for the moments of a sphere in free-space [27], in homogeneous half-space and excited by a source located on the surface of the ground [28,29], thick horizontal layer [30], and in one-dimensional earth with finite conductance [31]. In the year 2000, CGG and Geophysical Software Solutions collaborated to develop an application called EM-Q based on the research done by [32] on the uniform field approximation of the response of a sphere, for which [8] compared the results to the solution attained in [5]. The moments were also used by [25] as an approximation for the development of a 3D EM inversion program.

However, as TDEM measurements are done with systems using different primary current waveforms, the estimation of the moments from the quadrature impulse response is sometimes problematic. Ref. [27] developed a solution by approximating the TDEM waveform as a simple current box and applied this technique to the data integral from a MEGATEM AEM system [33], for which the current waveform is a half-sine. However, this approach is not applicable to more elaborate waveforms such as those of the AeroTEM AEM system [34], for which the current waveform is a symmetrical triangular pulse. Furthermore, Ref. [35] encountered difficulties with the definition of zero time while analyzing MEGATEM data, which resulted in a noisy moment estimation. Ref. [28] only considers step-response measurements, from which it is possible to estimate moments directly. In conclusion, although there are extensive theoretical developments on the use of moments in the field of TDEM interpretation, estimation of the moments from various waveforms on synthetic or field data has been neglected, slowing the application of this method on real data.

For these reasons, we propose a method for directly estimating the quadrature impulse moments from any waveform and measured signals. This method should increase the use of the moment method and better outline its limitations. The proposed algorithm evaluates the moments of the quadrature impulse response with a recursive relation involving the moments of the pulse current derivative and the moments of the quadrature on-time and off-time responses. Consequently, it requires that on-time measurements are available. After a noise analysis, we apply this method to synthetic AirTEM results (the same waveform as AeroTEM) with a sphere model in layered earth, illustrating the conditions for which moments can be useful for outlining a discrete conductor. This is followed by application of the moment method on field data from MEGATEM and AirTEM systems collected over the Reid–Mahaffy test site, Ontario, Canada.

2. Theory

2.1. Moment Estimation

The quadrature impulse response of a TDEM system can be developed as follows [27]. The EM step response of the subsurface can be written as

$$s\left(t\right)=-Bu\left(t\right)h\left(t\right)$$

(1)

where t is time, B is the response at $t=0$, $u\left(t\right)$ is the unit step-on function, and $h\left(t\right)$ is a dimensionless function characterizing the secondary decay associated with currents in the ground. The total impulse response $r\left(t\right)$ is the time derivative of $s\left(t\right)$:

$$r\left(t\right)=-B\left\{\delta \left(t\right)h\left(t\right)+u\left(t\right){\displaystyle \frac{\partial h\left(t\right)}{\partial t}}\right\}$$

(2)

Analogous to frequency-domain systems, Ref. [26] names the first and second terms of this equation the in-phase and quadrature components, respectively, and focuses on the second term. This is represented as

$$i\left(t\right)=-Bu\left(t\right){\displaystyle \frac{\partial h\left(t\right)}{\partial t}}$$

(3)

The moments of the quadrature impulse response $i\left(t\right)$ are defined as

$$I^n={\int }_0^{\infty }{t}^{n}i\left(t\right)dt$$

(4)

In practice, the primary waveform is different from a step response and is convolved with the impulse response. Furthermore, the receiver coil detects the derivative of the magnetic field. This can be written as

$$y\left(t\right)=x\left(t\right)\ast i\left(t\right)$$

(5)

where $x\left(t\right)$ represents the waveform derivative, * is the convolution operator, and $y\left(t\right)$ is the measured secondary field, from which the on-time primary field and the secondary in-phase field have been removed. This equation represents the result $y\left(t\right)$ of a causal convolution of the system waveform $x\left(t\right)$ and the system transfer function $i\left(t\right)$, where $x\left(t\right)=i\left(t\right)=0$ for $t<0$

$$y\left(t\right)={\int }_0^{\infty }i\left(h\right)x(t-h)dh={\int }_0^{\infty }i(t-h)x\left(h\right)dh$$

(6)

Following [36,37], we develop the moments of the convolution response as

$$Y^n={\int }_0^{\infty }y\left(t\right)t^{n}dt={\int }_0^{\infty }{\int }_0^{\infty }i\left(h\right)x(t-h)t^{n}dhdt$$

(7)

If we replace $u=t-h$, then $t=u+h$ and $du=dt$; then, the expression becomes

$$Y^n={\int }_{-h}^{\infty -h}{\int }_0^{\infty }i\left(h\right)x\left(u\right){(u+h)}^{n}dhdu$$

(8)

Since $x\left(u\right)=0$ for $u<0$ and $\infty -h\simeq \infty$, we can re-write the equation as

$$Y^n={\int }_0^{\infty }{\int }_0^{\infty }i\left(h\right)x\left(u\right){(u+h)}^{n}dhdu$$

(9)

Developing the power of a binomial expansion, it is possible to separate the two integrals in the following way:

$$\begin{array}{cc}{Y}^n\hfill & ={\int }_0^{\infty }{\int }_0^{\infty }i\left(h\right)x\left(u\right)\left[{\sum }_{k=0}^n\left(\begin{array}{c}n\hfill \\ k\hfill \end{array}\right)u^{n-k}{h}^k\right]dhdu\hfill \\ Y^n\hfill & ={\sum }_{k=0}^n\left(\begin{array}{c}n\hfill \\ k\hfill \end{array}\right){\int }_0^{\infty }x\left(u\right)u^{n-k}du{\int }_0^{\infty }i\left(h\right)h^{k}dh\hfill \end{array}$$

(10)

where

$$\left(\begin{array}{c}n\hfill \\ k\hfill \end{array}\right)={\displaystyle \frac{n!}{k!(n-k)!}}$$

(11)

If we define the input and impulse response moments as

$$X^n={\int }_0^{\infty }x\left(t\right)t^{n}dt$$

(12)

and

$$I^n={\int }_0^{\infty }i\left(t\right)t^{n}dt,$$

(13)

we can write

$$Y^n=\sum _{k=0}^n\left(\begin{array}{c}n\hfill \\ k\hfill \end{array}\right)X^{n-k}{I}^k.$$

(14)

For $X^0\ne 0$, if we develop

$$Y^n=\sum _{k=0}^{n-1}\left(\begin{array}{c}n\hfill \\ k\hfill \end{array}\right)X^{n-k}{I}^k+X^0{I}^n,$$

(15)

we can evaluate $I^n$ from $X^k$ and $Y^k$, $k\in [0,n]$, as

$$I^n={\displaystyle \frac{Y^n-{\sum }_{k=0}^{n-1}\left(\begin{array}{c}n\hfill \\ k\hfill \end{array}\right)X^{n-k}{I}^k}{X^0}}.$$

(16)

However, in the case of some AEM systems, the current waveform or derivative is symmetric, with positive and negative lobes, such as in MEGATEM and AeroTEM, and $X^0=0$. Then we need to use the following relation:

$$Y^n=\sum _{k=0}^{n-2}\left(\begin{array}{c}n\hfill \\ k\hfill \end{array}\right)X^{n-k}{I}^k+\left(\begin{array}{c}n\hfill \\ n-1\hfill \end{array}\right)X^1{I}^{n-1}$$

(17)

from which we deduce

$$I^{n-1}={\displaystyle \frac{Y^n-{\sum }_{k=0}^{n-2}\left(\begin{array}{c}n\hfill \\ k\hfill \end{array}\right)X^{n-k}{I}^k}{\left(\begin{array}{c}n\hfill \\ n-1\hfill \end{array}\right)X^1}}$$

(18)

or

$$I^n={\displaystyle \frac{Y^{n+1}-{\sum }_{k=0}^{n-1}\left(\begin{array}{c}n+1\hfill \\ k\hfill \end{array}\right)X^{n+1-k}{I}^k}{\left(\begin{array}{c}n+1\hfill \\ n\hfill \end{array}\right)X^1}}$$

(19)

The next subsections demonstrate how this relation can be applied to the MEGATEM and AeroTEM systems.

2.2. MEGATEM

MEGATEM [33] is a fixed-wing system with a transmitter loop wound around the nose, wingtips, and tail of a DeHavilland Dash-7. The transmitter loop covers an area of 406 m², while the receiver comprises three magnetic coils that are towed 50 m below and 131 m behind the aircraft. As stated before, the current pulse is a half-sine. The primary field extraction [38], also called stripping, is done using a reference waveform that is measured at altitude and by minimizing the functional $\Phi$, where

$$\begin{array}{cc}\Phi \hfill & =\sum _n\left[T\left(t_i\right)-\alpha R\left(t_i\right)\right]\hfill \\ \hfill & =\sum _{n}Q\left(t_i\right)\hfill \end{array}$$

(20)

$T\left(t_i\right)$ is the total field response, $R\left(t_i\right)$ is the reference waveform, and $\alpha$ is an unknown factor that is estimated from the minimization. As [38] points out, this procedure estimates the on-time in-phase and quadrature responses. Accordingly, this approach provides the information required for moment estimation using the proposed method.

2.3. AeroTEM

The AeroTEM system [34] is a helicopter-towed EM system with coincident transmitter and receivers characterized by a triangular current pulse (Figure 3) for which the derivative is a constant with alternating polarity. The voltage sensed by the receiver, which is surrounded by a bucking coil to cancel the primary field, is a series of step-responses, i.e., a first step of positive polarity, a second step of double-negative polarity, and a last step of positive polarity. The on-time field is measured as a sum of two voltages separated by half the period.

The response derivative can be seen as the sum of three step-response of alternating polarities and is expressed as

$$z\left(t\right)=-Bu\left(t\right)h\left(t\right)+2Bu(t-P/2)h(t-P/2)-Bu(t-P)h(t-P)$$

(21)

where P is the pulse width, and B, $u\left(t\right)$, and $h\left(t\right)$ are defined as in Equation (1). This can be decomposed according to time:

$$\begin{array}{ccc}z\left(t\right)\hfill & =-Bh\left(t\right)\hfill & 0<t<P/2\hfill \\ \hfill & =-B\left[h\left(t\right)-2h(t-P/2)\right]\hfill & P/2<t<P\hfill \\ \hfill & =-B\left[h\left(t\right)-2h(t-P/2)+h(t-P)\right]\hfill & P<t\hfill \end{array}$$

(22)

The in-phase $x\left(t\right)$ and quadrature $y\left(t\right)$ responses are, respectively,

$$\begin{array}{ccc}x\left(t\right)\hfill & =-Bh\left(0\right)\hfill & 0<t<P/2\hfill \\ \hfill & =Bh\left(0\right)\hfill & P/2<t<P\hfill \\ \hfill & =0\hfill & P<t\hfill \end{array}$$

(23)

and

$$\begin{array}{ccc}y\left(t\right)\hfill & =-B\left[h\left(t\right)-h\left(0\right)\right]\hfill & 0<t<P/2\hfill \\ \hfill & =-B\left[h\left(t\right)-2h(t-P/2)+h\left(0\right)\right]\hfill & P/2<t<P\hfill \\ \hfill & =-B\left[h\left(t\right)-2h(t-P/2)+h(t-P)\right]\hfill & P<t\hfill \end{array}$$

(24)

The on-time sum measurement typical of the AeroTEM system is

$$\begin{array}{cc}d\left(t\right)\hfill & =z\left(t\right)+z(t+P/2)\hfill \\ \hfill & =-Bh\left(t\right)-B\left[h(t+P/2)-2h\left(t\right)\right]\hfill \\ \hfill & =-B\left[h(t+P/2)-h\left(t\right)\right]\hfill \end{array}$$

(25)

If we assume that $h(t+P/2)<<h\left(t\right)$, we can approximate that $d\left(t\right)\simeq Bh\left(t\right)$. Using this approximation, $y\left(t\right)$ can be estimated from Equation (23) as

$$\begin{array}{cc}y\left(t\right)\hfill & \simeq d\left(0\right)-d\left(t\right)\hfill \\ y(t+P/2)\hfill & \simeq 2d\left(t\right)-d\left(0\right)\hfill \end{array}$$

(26)

2.4. Incomplete Moments

In practice, response moments are estimated between the early and late times, as represented by the following equation [28]:

$$Y^n={\int }_0^{t_1}y\left(t\right)t^{n}dt+{\int }_{t_1}^{t_n}y\left(t\right)t^{n}dt+{\int }_{t_n}^{\infty }y\left(t\right)t^{n}dt$$

(27)

where [28] labels the first and last terms the “head” and the “tail”, respectively, and the middle term is the incomplete moment. As demonstrated by [28], if the waveform is a step, the tail and head can be estimated from the early and late-time apparent resistivity. However, for a complex waveform, this method cannot be applied, and the solution we develop in this paper can only estimate the incomplete moments, which are more complex than for a step response.

The error in this estimation can be assessed by comparing the synthetic moment estimates with the analytic moment estimates provided by [30]. For this purpose, we present results for the AirTEM system [39] (shown in Figure 2), which is characterized by a triangular waveform similar to that of the AeroTEM system. This system employs a horizontal loop transmitter with a diameter of 4.25 m and a vertical magnetic coil receiver (dB/dt) located on the side of the transmitter loop. These are towed by a helicopter (Figure 2), and the EM system loop operates approximately 40 m above the ground. The transmitter waveform is a 1.85 ms bipolar triangular pulse with a frequency of 90 Hz. The time derivative of the magnetic field is sampled with 40 off-time channels between 0.02 and 3.3 ms and on-time differences covering the first half of each on-time pulse. The time domain synthetic half-space response is estimated using the program Airbeo [7] with Hankel filters [40] and frequency-to-time domain transformations [41]. The first two moments are estimated from the half-space response using the method proposed in this paper and are compared to the analytic solutions from [30], as seen in Figure 4.

2.5. Normalization

The moments of a wire loop were developed in [27], which illustrates the distribution of the moments for a simple electromagnetic model. For a wire loop with a time constant ${\tau }_w=L/R$, where L and R are the inductance and resistance, respectively, of the loop, the moments are

$$M^n=Bn!{\tau }_w^n$$

(28)

where B is the inductive limit and n is the moment order. This implies that time normalization will have an impact on the variation with the moment order. Accordingly, smaller time constants can be enhanced by scaling the moments. The moments of a scaled integral are developed as follows. If we define the scaled version of $Y^n$ as $Y_{\nu }^n$, where

$$Y_{\nu }^n={\int }_0^{\infty }y(t/\nu )t^{n}dt,$$

(29)

if $t/\nu =\tau$, this becomes

$$\begin{array}{cc}{Y}_{\nu }^n\hfill & ={\int }_0^{\infty }y\left(\tau \right){\left(\nu \tau \right)}^n\nu d\tau \hfill \\ \hfill & ={\nu }^{n+1}{\int }_0^{\infty }y\left(\tau \right){\tau }^{n}d\tau \hfill \\ \hfill & ={\nu }^{n+1}{Y}^n\hfill \end{array}$$

(30)

Scaling corresponds to multiplication of the moments by the power of a time factor. The impulse response moment from Equation (16) becomes

$$I_{\nu }^n={\displaystyle \frac{Y_{\nu }^{n+1}-{\sum }_{k=0}^{n-1}\left(\begin{array}{c}n+1\hfill \\ k\hfill \end{array}\right)X_{\nu }^{n+1-k}{I}_{\nu }^k}{\left(\begin{array}{c}n+1\hfill \\ n\hfill \end{array}\right)X_{\nu }^1}}$$

(31)

Using a recurrent relation, it can be shown that

$$I_{\nu }^n={\nu }^n{I}^n$$

(32)

This allows enhancing the moments of higher orders, if required.

2.6. Moment Estimation from Noisy Data

In practice, moments are estimated from noisy data, and an analysis of the proposed method requires an evaluation of the impact of noisy data on moment estimation. First, we must estimate the impacts of noisy data on the estimations of waveforms and output moments. These are estimated by numerical integration of the waveform and output signals, which can be written as

$$\begin{array}{c}{X}^n={\sum }_0^{l}X\left(t_l\right)t_l^n\Delta t_l\hfill \\ Y^n={\sum }_0^{m}Y\left(t_m\right)t_m^n\Delta t_m\hfill \end{array}$$

(33)

where $X\left(t\right)$ and $Y\left(t\right)$ are the waveform and output signals, respectively, and $t_l,t_l$ are time samples. Assuming independent measurements and neglecting correlations, the variances of these signals are estimated using the variance formula [42]:

$$s_f=\sqrt{{\left({\displaystyle \frac{\partial f}{\partial x}}\right)}^2{s}_x^2+{\left({\displaystyle \frac{\partial f}{\partial y}}\right)}^2{s}_y^2+{\left({\displaystyle \frac{\partial f}{\partial z}}\right)}^2{s}_z^2+...}$$

(34)

where $s_f$ is the variance of the function $f=f(x,y,z,...)$, and $s_x,s_y,s_z$ are the variances of the parameters $x,y,z$. Developing the partial derivatives of Equation (33), we write

$$\begin{array}{c}{\displaystyle \frac{\partial X^n\left(t_l\right)}{\partial t_l}}=n{t}_l^{n-1}X\left(t_l\right)\Delta t_l\hfill \\ {\displaystyle \frac{\partial Y^n\left(t_m\right)}{\partial t_m}}=n{t}_m^{n-1}Y\left(t_m\right)\Delta t_m\hfill \end{array}$$

(35)

Then, Equation (34) can be developed as

$$\begin{array}{c}{s}_{X^n}={\sum }_0^l{\left[n{t}_l^{n-1}X\left(t_l\right)\Delta t_l\right]}^2{s}_l^2\hfill \\ s_{Y^n}={\sum }_0^m{\left[n{t}_m^{n-1}Y\left(t_m\right)\Delta t_m\right]}^2{s}_m^2\hfill \end{array}$$

(36)

where $s_{X^n}$ and $s_{Y^n}$ are the moment waveform and output variances respectively. Equation (36) shows that these moment variances increase with the moment order.

The impulse moment estimate variance can be developed in the following way. If we simplify Equation (19) for $n=1$, we have

$$I^0={\displaystyle \frac{Y^1}{X^1}}$$

(37)

From Equation (34), we write

$$\begin{array}{c}{s}_{I^0}=\sqrt{{\left({\displaystyle \frac{\partial I^0}{\partial Y^1}}\right)}^2{s}_{Y^1}^2+{\left({\displaystyle \frac{\partial I^0}{\partial X^1}}\right)}^2{s}_{X^1}^2}\hfill \\ s_{I^0}=\sqrt{{\displaystyle \frac{s_{Y^1}^2}{{\left(X^1\right)}^2}}+{\displaystyle \frac{{\left(Y^1\right)}^2{s}_{X_1}^2}{{\left(X^1\right)}^4}}}\hfill \end{array}$$

(38)

In general, from Equation (19), the variance becomes

$$\begin{array}{c}{s}_{I^n}=\sqrt{{\left({\displaystyle \frac{\partial I^n}{\partial Y^{n+1}}}\right)}^2{s}_{Y^n}^2+{\sum }_{k=0}^{n-1}\left[{\left({\displaystyle \frac{\partial I^n}{\partial I^k}}\right)}^2{s}_{I^k}^2\right]+{\sum }_{k=1}^n\left[{\left({\displaystyle \frac{\partial I^n}{\partial X^k}}\right)}^2{s}_{X^k}^2\right]}\hfill \end{array}$$

(39)

Without developing this expression in detail, it is easy to again conclude that the moment variances increase with the moment order.

3. Results

3.1. Synthetic Models

We apply this method on synthetic results from numerical models of a localized target in a conductive environment. We use the model of [9] to represent the sphere response to an AirTEM system in layered earth. The sphere has a radius of 50 m and a conductivity of 10 S/m, and its center is located 100 m below the surface of the ground, while the AirTEM system is at an altitude of 40 m. We start with a sphere in a half-space with 0.1 mS/m conductivity, for which the response is shown in Figure 5. The sphere has a distinctive moment response. The peak moments of a similar sphere with varying conductivity are displayed in Figure 6. This figure shows that the sphere’s peak moments are negligible for conductivity below 1 S/m, even if the sphere has a meaningful off-time response, and the moment reaches a maximum value at about 8 S/m and decreases for conductivities above this value.

We now examine the sphere moments below a conductive overburden of 25 m thickness with varying conductivity. Figure 7 shows the peak nomogram for the overburden with varying conductivity (“anomalous” represents the sphere-only response). At low overburden conductivity, the response comes mainly from the sphere. As the overburden conductivity increases, the sphere peak becomes negative, while the overburden response increases. Figure 8 show the corresponding moments. At low overburden conductivity, the sphere moments dominate. However, above 0.1 S/m, the sphere moments decrease as the overburden moments increase strongly. Figure 8 suggests that we can still observe the sphere response if the overburden is 0.1 S/m (100 mS/m). This is confirmed by the subsequent model in Figure 9, wherein the top layer has a conductivity of 100 mS/m and 25 m thickness and is above a half-space with 0.1 mS/m conductivity. Even if the sphere’s off-time responses are barely visible, the moment responses are clearly visible.

3.2. Field Data

Moment estimation was tested on field data collected with the MEGATEM and the AirTEM systems over the Reid–Mahaffy test site in northern Ontario, Canada [43]. This site is characterized by the presence of several conductors located in a resistive basement below a variably conductive overburden.

3.2.1. MEGATEM

MEGATEM data collected over the Reid–Mahaffy test site in 2002 are available from [44]. In particular, the moments of Line 15, shown in Figure 10a, were analyzed by [27] based on B field estimation. For comparison, moments estimated using our proposed method are presented in Figure 10b. The various bedrock conductors can be clearly identified. Another profile of Line 3 in Figure 11 shows three distinct conductors whose moment signatures are remarkably significant.

3.2.2. AirTEM

AirTEM data were collected for testing purposes over the Reid–Mahaffy test site in 2019. The profiles for Line 15 are displayed in Figure 12a. AirTEM is a helicopter TDEM system with a much smaller loop flown closer to the ground. Consequently, the bedrock conductors are less apparent and are partly masked by the overburden response. Despite this fact, some deep conductors to the north of the profile are visible and can be observed in the moment response shown in Figure 12b. Line 3 conductor responses are shown in Figure 13. Clear moment responses are observed over these distinct conductors.

4. Discussion

Although not exhaustive, the results presented in this study allow a discussion of the strengths and limitations of the moment method. The methodology applied in this paper has been used successfully to interpret synthetic and field data. An important requirement of the method is the availability of on-time quadrature measurements. For the MEGATEM system, there is an established procedure for this purpose. For AeroTEM and AirTEM, some approximation is required. For other systems, this could be a major limitation to the implementation of this method.

The synthetic models show that for small background conductivity and a good conductor (in our models, more than a few S/m), the conductor moments are easily detected. However, as shown in Figure 6, a bad conductor (in our models, less than 1 S/m) will not be detected. Furthermore, when the conductivity of the background is significant, even a good conductor response is not detectable. Another important consideration when applying this technique is to consider the noise present in the data. The noise analysis shows that noise increases with the moment order, which may explain why high-order MEGATEM moments are less noisy than equivalent AirTEM moments, as the MEGATEM data, originating from a much more powerful system, are less noisy than the AirTEM data.

The application of this method on datasets from different systems over the same area allowed for a pertinent comparison of the performances of the systems in question. In the case presented, the Reid–Mahaffy area is covered with significant conductive overburden, and the AirTEM system, with smaller coincident loops, has some difficulty penetrating the overburden. Consequently, the discrete conductor moments are scattered. On the other hand, the MEGATEM system—operating at the same frequency but with much higher power, a larger transmitter loop, and greater transmitter–receiver separation—detects distinct conductors that are deeper below the overburden easily, comparatively speaking. These conductors are clearly defined by moments.

As mentioned in Section 2.4, the method presented in this paper provides an estimate of incomplete moments. This must be considered if results are compared with analytic models, and in that case [28], total moment relations must be applied. However, for numerical models wherein system time windows are included, observed moments can be directly related to model moments.

5. Conclusions

The method presented in this paper extends the application of moment estimation to arbitrary EM system waveforms and generalizes the use of moments; it can provide data quality assessments, conductor locations, and preliminary interpretations. A limitation of this approach is the availability of on-time measurements. For systems that can resolve this limitation, such as MEGATEM and AeroTEM (AirTEM), we show that this method successfully estimates moments and allows a comparison of system performance over a given area. We advocate that this method could be extended to other EM systems—either airborne, ground, or borehole—when on-time measurements are available. Our model also shows that for a conductor to be detected, the conductivity range must be appropriate: good conductors are easier to detect as long as there is contrast with the surrounding environment.

This algorithm can be easily implemented for data post-processing and can be used to glean information about the EM data quality quickly. As such, it is seen a supplement to simplified interpretation methods that only provide single parameters like decay time constants or apparent conductivities.

As this method relies on on-time and off-time measurements, improvements in data collection, like reducing noise and improving calibration methods, should improve the moment estimates. This is particularly true for on-time measurements: these are more difficult to collect, which explains why not all systems measure the full waveform.