For a full description of transient electromagnetics, we refer to one of the many excellent expositions in the literature, e.g., . Briefly, a TEM measurement is carried out by applying an input current pulse to a transmitter coil, ground based or airborne. The flow of current generates a primary magnetic field, which penetrates the ground. The transmitter current is abruptly turned off, which induces eddy currents in the ground. The eddy currents give rise to a secondary magnetic field which is measured with a receiver coil. One single measurement is denoted as a transient, and multiple transients are stacked (averaged) into a sounding curve. The decay of the secondary magnetic field is controlled by subsurface resistivity, which can therefore be inferred by modeling of data.

Receiver coils used for TEM measurements are typically designed for wide bandwidth operation (> 100 kHz) to enable data acquisition as soon as possible after the turn-off of the transmitter pulse. Most of the noise found in TEM data is external noise inductively coupled into the receiver coil. This includes electromagnetic interference from powerlines and electrical infrastructure, interference from radio stations in the very low frequency (VLF), 3 to 30 kHz, and low frequency (LF) 30 to 300 kHz bands, and interference from natural sources such as thunderstorms. Further, TEM data also contains thermal noise from the receiver coil and the amplifier electronic circuits.

Different methods are used to suppress individual noise sources. As an example, powerline noise can be efficiently removed by synchronous detection . This consists of transmitting current pulses of alternating polarity at the powerline fundamental frequency, i.e., 50 or 60 Hz, followed by a sign correction and stacking of data.

Many radio stations in the VLF and LF ranges employ encoding with minimum-shift keying. This is a digital format where the bit stream, s(t) is encoded using two different frequencies, f0 and f1, with frequency difference Δf = f1-f0 and center frequency fc = (f0+f1)/2 . Each bit, normally denoted as a symbol, has a duration Tsym with Δf = 0.5/Tsym. In practice, four symbols are used 1s0(t)=cos⁡(2πfc-Δf/2t),s‾0(t)=-s0(t)s1(t)=cos⁡(2πfc+Δf/2t),s‾1(t)=-s1(t) where the negative sign symbols are used to ensure phase continuity of the signal. The enforcement of phase continuity gives a correlation between adjacent bits which enables efficient decoding of bit streams .

The radio signal modeling and subtraction approach contains four steps . 1.

Estimation and correction of signal parameters using the acquired data.

The final step of TEM data processing is gating of data. We use semi-tapered gates with 50 % overlap . The gates are composed of cosine-shaped rising and falling edges and a flat central section. The center time and gate width are exponentially increasing to produce ten gates per decade of time within the limits set by sampling rate and data record length.

After gating, mean values and standard errors of gate values are computed for plotting and analysis . We also compute the covariance between the gate values. When TEM data contains noise from correlated sources, e.g., VLF radio signals, the correlated noise emerges as distinct, visible patterns in a plot of the associated covariance matrices and these matrices are therefore useful for algorithm comparison .

For assessing and comparing the performance of algorithms, data where the signal parameters and bit content of individual radio signals are known is necessary. For this purpose, we generate synthetic MSK-encoded radio signals using Eq. () with carrier frequencies and bit rates mimicking real VLF and LF radio stations . For each radio signal, we draw a symbol timing between -Tsym/2 and Tsym/2 and signal phase between 0 and 2π from uniform distributions. The amplitudes of the radio signals are also randomly selected between 1 % and 10 % of the TEM signal amplitude. The different synthetic VLF and LF radio signals are summed. Further, white Gaussian noise is added to the signal accounting for the broadband thermal noise in real data. Our model does not take powerline noise or impulsive noise into consideration as these noise sources are adequately handled in standard processing steps. For the numerical experiments we use a 2 MHz sample rate.

The field results presented below are based on the noisy-only data set used by . This data set was collected near Aarhus, Denmark using a multi-turn square coil with an effective area of 80.9 m². After amplification, the signal was continuously sampled for 25 s at a 500 kHz sample rate with a digital oscilloscope. To mimic TEM data, we introduce gaps into the data by setting sample values to zero in the data record corresponding to the transmitter pulse blinding.

A synthetic TEM signal decaying as t-5/2 is added to each segment before processing. This smoothly decaying signal, which corresponds to the signal from a homogeneous half-space, captures the essential physics of a real TEM signal yet allows for easy visual inspection of sounding curves. The amplitudes of the TEM signals are chosen, so that the data in the first 5–10 gates have a high signal-to-noise ratio and the early decaying parts are clearly visible in all cases.

The effect of gaps in data streams is investigated using synthetic data composed of known VLF / LF radio bit streams and white Gaussian noise. This allows us to quantify the bit error rate, which is an important factor in the model-and-subtract approach. Small errors in the estimates of, e.g., phase will lead to a slightly poorer result when a reconstructed VLF radio model is subtracted from data, but bit errors will lead to the subtraction of an outright wrong VLF radio model and consequently a very poor result.

The bit error rate is determined as a function of the gap length, i.e., the duration of transmitter pulses, while keeping the size of the subsequent data record containing the TEM transient signal fixed at 2048 µs. The gap length is varied over a range that includes typical low-moment pulses, 200 µs, and high-moment pulses, 450 µs, in common TEM instruments and extends beyond the 5 ms duration of single bits in a 200 bit s⁻¹ stream.

The results are plotted in Fig. . Each data point and the associated error bar (truncated at zero) are computed by repeating the numerical experiments 50 times. In each case a total bit error rate is computed for the four MSK radio stations in the model. A bit error rate of 0.0 corresponds to correct identification of all bits, a bit error rate of 1.0 corresponds to wrong identification of all bits, and a bit error rate of 0.5 corresponds to 50 % of all bits being correctly identified, like random guessing.

The results show that for all gap lengths up to 5 ms, the bit error rate remains very low and nearly flat. For gap lengths longer than 5 ms, the bit error rate increases significantly. These observation match the Viterbi decoder algorithm, i.e., the correlation between adjacent bits in the MSK bit stream ensures an efficient decoding even as the gap length approaches 5 ms. For gaps longer than 5 ms, this correlation is lost, which leads to the threshold seen in Fig. . Additional analysis of data sets with shorter or longer data records give similar results with longer records slightly improving performance. From this analysis we can conclude that the Viterbi algorithm is generally very efficient and if the gaps are less than 5 ms, bit errors are negligible.

The effect of the adaptive filter in reconstruction of the VLF / LF radio signals in data with gaps is investigated through simulations where a synthetic signal composed of TEM data, interference from four MSK encoded radio signals, and white noise is processed in two manners. In the first case, the radio signals are reconstructed using the estimated amplitude, symbol timing, and symbol phase and subtracted from the data record. In the second case, an adaptive filter fine-tunes the VLF and LF radio signals before subtraction. In both cases we compute the mean square error as a function of the gap length, while keeping the size of the subsequent data record containing the TEM transient signal fixed at 2048 µs. The simulations are repeated 50 times for each gap length to compute mean values and standard deviations. With the chosen settings of the adaptive filter, it converges within about 0.5 ms. To improve signal quality, we cull the first segment where data are affected by the filter adaptation.

The results presented in Fig.  demonstrate that the adaptive filter significantly improves the results. The mean square error is reduced by more than an order of magnitude for all investigated gap lengths, and further, the standard deviation of the mean square error is reduced with the adaptive filter, which we ascribe to generally more accurate modeling in this case. Without the adaptive filter we observe a rise in the mean square error for long gap lengths above the 5 ms bit length, which corresponds with the increase in bit error rate, Fig. . Somewhat surprisingly, the adaptive filter performs well even in this limit. We ascribe this to the nonlinear filtering taking place in the adaptive algorithm, partially compensating for bit errors. From this analysis we can conclude that the algorithms for estimation of amplitude, symbol timing, and phase are generally quite accurate, but that in a use-case like this where very high accuracy is needed for efficient noise removal, the performance can be significantly improved upon by automatic fine tuning with an adaptive filter.

We demonstrate the benefits of the proposed approach using a sand-box example with synthetic VLF / LF radio noise, white Gaussian noise, and a simple TEM signal. The synthetic radio noise is composed of 4 radio signals at frequencies between 10 and 80 kHz and operating at 100 or 200 bits s⁻¹. We construct data records composed of 500 transients interleaved with 200 µs gaps introduced into the model by zeroing out the corresponding intervals. The gaps occur at regular intervals matching an 817 Hz transmitter pulse repetition rate. The TEM signal is modeled as a simple half space with an easily identified t-5/2 decay.

The synthetic data are processed in two manners. First, with a standard approach where the data are gated with semi-tapered gates and then stacked. Second, with the improved approach where the radio stations are first modeled and subtracted before the same gating and stacking is applied.

Plots of the stacked data and associated standard errors for the two processing methods are presented in Fig. . The corresponding covariance matrices are shown in Fig. .

In this idealized example, we see a very large improvement if VLF / LF radio interference is applied prior to gating and stacking. If the data are only gated and stacked, we start to see significant differences between the gated signal and the ideal t-5/2 signal after about 3 µs, but by removing VLF / LF radio interference, significant differences do not occur until well after 10 µs. Further, the standard errors are much smaller when radio interference is removed. For a richer comparison, these observations are also quantified in Table  where gate values and standard errors are given for all odd-numbered gates. The table also gives the improvement in the standard errors computed as the ratio of the standard error with and without VLF / LF removal. This improvement ratio is much larger than unity until about gate 21 at around 125 µs. The reason for this behavior is that for early times, the VLF / LF radio interference for the four stations in the 10 to 80 kHz range is located inside the frequency domain main-lobe of the semi-tapered gates and hence cannot be removed by gating. However, this restriction does not apply to the VLF / LF model and subtract approach. For late times, VLF / LF noise is located outside the main-lobe and the semi-tapered gates are efficiently removing the radio noise. Thus, there are no discernible differences between the two methods in this region.

The plot of covariance matrices in Fig.  show only small differences with or without modeling and subtraction of radio noise, even if there are large improvements in the standard errors of data. The reason for this is that the off-diagonal elements of the covariance matrices are scaled with the diagonal values before plotting and the scaling can therefore hide the improvement. This example is included to show that great care is needed in the interpretation of these covariance matrices.

As a final validation of our approach, we utilize a noise-only data set with real VLF radio signals and other noise. As with the full synthetic example above, we embed a simple t-5/2 TEM signal in the data as this allows for a clear visual interpretation and comparison of the results undisturbed by geological signatures in actual TEM data. The data are again processed either with gating and stacking or with modeling and subtraction of VLF and LF radio signals carried out prior to gating and stacking. The number of samples in a data record length is kept fixed, but due to a four times lower sample rate each transient is four times longer. The sounding curve is again obtained by stacking 500 transients.

The results are presented in Figs.  and , and the data for odd gates are summarized in Table . We observe the same trends in the field data example as in the synthetic data example, but the effects of VLF and LF removal are less pronounced in the sounding curve. This is expected as the field data also contains additional noise sources not found in the synthetic model. The best improvement is seen in the gates centered around 20 to 50 µs. The bandwidths of these gates are in the 20 to 50 kHz range matching the main VLF radio signal components in the data set. For gates centered after about 100 µs, tapered gating is effective in suppressing radio noise and the results with or without modeling and subtraction of radio noise are almost identical.

In this field noise case, we see a dramatic improvement in the covariance matrix plot after removal of VLF and LF noise, Fig. . The structure in the off-diagonal elements seen up to about gate 15 is greatly reduced. We interpret this as the result of efficient suppression of the coherent radio noise.

The data presented in Table  completes the analysis. In the early gates, we see about two-fold improvement in the standard errors after removal of interfering radio signals. This improvement is smaller than observed for the synthetic case, which we ascribe to the simplified noise model. In the late gates where semi-tapered gating is effective, there are no differences in standard errors.