One of the most widely used approaches for measuring the earth's subsurface resistivity is the transient electromagnetic (TEM) method. However, noise and interference from different sources, e.g., radio communication, the instrument, the atmosphere, and power lines, severely taint these types of signals. In particular, radio transmission in the very low-frequency (VLF) range between 3 and 30 kHz is one of the most prominent sources of noise. Transient electromagnetic signals are normally gated to increase the signal-to-noise ratio. A precise selection of gate shapes is required to suppress undesired noise while allowing the TEM signal to pass unaltered. We employ the multi-objective particle swarm optimization technique to choose optimal gate shapes and placements by minimizing an objective function composed of standard error bars, the covariance between gates, and the distortion of the gated signal. The proposed method is applied to both fully sampled synthetic TEM data and to boxcar-gated field data. The best output from the search space of gate shapes was found to be a hybrid combination of boxcar and Hamming gates. The effectiveness of hybrid gating over traditional boxcar and semi-tapered gating is confirmed by an analysis of covariance matrices and error bars. The results show that the developed method effectively suppresses VLF noise in the middle gates, which are gates with center times spanning 30 to 200

The transient electromagnetic (TEM) method is a well-known and well-accepted technology utilized in mineral exploration, groundwater mapping, saltwater boundary mapping, and a variety of other applications

Multiple noise sources interfere with the TEM signals. The noise sources include power-line noise (50 or 60 Hz) and its harmonics, internal instrumental noise (thermal noise, aging of electronic components, etc.), environmental noise (sferic noise from thunderstorms, traffic, etc.), interference from radio communication systems in the low-frequency (LF, 30–300 kHz) and very low-frequency (VLF, 3–30 kHz) range, and motion noise in the moving systems

Hence, this work aims to design an optimum gating scheme that is able to suppress the VLF radios in the gated signal and able to adjust itself regularly, e.g., to track daily changes in noise conditions in monitoring instruments. Its significant contributions are the following:

Design and implementation of a fully sampled 4 MHz synthetic model for generating noiseless and noisy TEM signals

Design of gate banks with selected gate shapes to be utilized for gating TEM signals from moving or stationary instruments

Implementation of a new covariance matrix CM tool for studying the presence of VLF noise in the data

Design of a hybrid gate distribution scheme for suppressing the effect of VLF noise in measured TEM signals

Optimization of gates by minimizing a multi-objective cost function composed of standard errors of gated data, signal distortion, and covariance

Validation of the hybrid gating model with synthetic TEM signals and field data acquired in Denmark with an analog boxcar gating instrument

The remainder of the paper is structured as follows: Sect. 2 provides a background of the TEM system, related work, and motivation. Section 3 provides a detailed discussion and implementation of the proposed model. Details about the experimental setup of the synthetic and towed-TEM model are covered in Sect. 4. The results of the proposed model are discussed in Sect. 5. Finally, conclusions are presented in Sect. 6.

Briefly, the principles of TEM measurements are as follows

The high-frequency content in the early times is captured by selecting short early gate widths, typically on the order of a few microseconds. In comparison, the signals' late-time response and low-frequency content are captured by gates of increasingly longer duration. Gating is the integration of the signal over specified adjacent time windows. Essentially, gating provides a decimation of data, and it corresponds to a filtering of the signal with a filter whose frequency response is obtained by taking the Fourier transform of the gate shape. As a result, gating is critical for signal interpretation and data reduction.

Exemplary waveforms of the TEM system:

Many studies have been conducted to investigate the properties of VLF radio signals in various fields, including lightning–ionosphere interactions and uncrewed airborne geophysical surveys. To study the effect of lightning on the ionosphere, a technique based on probing sub-ionospheric VLF has been investigated

The role of VLF noise in the measurement and analysis of TEM signals is significant. Boxcar gating with a rectangular window using analog integrators has been used widely. However, the frequency response of the boxcar has large side lobes that can let the VLF radios in the measured signal pass through

New fully sampled digital receiver systems can employ any desired gate shape; e.g., a B-spline-based gating scheme has been explored to compress the measured TEM data in the SkyTEM receiver system

However, the semi-tapered and B-spline-based gating schemes have only used an optimization metric about reducing standard error bars to suppress the effect of VLF radios

During the late times, from about 100

Hence, suitable gating methods will increase system performance by increasing SNR (i.e., reduced error bars), reduce the correlation between different gates (i.e., suppressing the effect of VLF), and minimize the distortion of the TEM signal by the gating. Normally the same shape is used for all gates, but the paper investigates the potential gains obtained by allowing for an individually selected shape for each gate. To reduce the size of the search space, the shapes are selected from a bank of eight predefined shapes.

The proposed synthetic model and the optimal and hybrid gating schemes are described in detail in this section. The first part presents the fully sampled synthetic model for generating noiseless and noisy TEM signals. The second part contains a gate bank with its mathematical formulations. Finally, using multi-objective particle swarm optimization (MO-PSO), the paper presents the design of optimum gate selection and hybrid gate distribution in the third part.

The developed fully sampled synthetic model for TEM signals is composed of three types: (a) pure TEM signals, (b) TEM signals with random noise (RN), and (c) TEM signals with RN, power-line harmonics, and VLF noise. The proposed model is designed to mimic the characteristics of the towed-TEM instrument

The timescale for measuring TEM signals ranges from microseconds to a few tens of milliseconds, covering a wide dynamic range. The TEM signal decays approximately as

Electrical power lines generate noise from an electric and magnetic field at the fundamental frequency of transmitted power (50 Hz for Denmark) and its harmonics. The frequency, amplitude, and phase of the power-line harmonics vary with time due to load and demand in the power grid but are approximately constant for short data records, usually a few seconds or less. The power-line or harmonic noise with a fundamental frequency of

TEM signals are affected by broadband noise, which stems from thermal noise in the receiver coil and amplifier electronics, along with atmospheric noise. The paper models this broadband noise as a stationary white noise Gaussian process

Signals from VLF radios are generated using antennas spread across the world, causing interference with TEM systems. Here these VLF radio signals are generated as random bit streams and encoded using minimum shift keying (MSK). The frequency content of these VLF radios overlaps with the frequencies of TEM signals, resulting in reduced data quality

We model the TEM signal as an alternating pulse train along with power-line harmonics, RN, and VLF radio noise to create the final noisy TEM system model. The mathematical formulation for the proposed synthetic model is given as

Examples of VLF radio stations

Example of a decaying TEM signal.

In practice, the power-line frequency is constant over a small duration. The correct choice of

The current work selects standard gates based on their frequency, in particular side lobe properties. However, many other gate shapes can also be employed, but their properties are generally quite similar.

The rectangular or boxcar gate is an array of one defined for a desired interval in time. The mathematical formulation of a boxcar gate is defined as

The Hanning gate has a shape like one half-cycle of the cosine wave with a DC shift of 1, so it always remains positive. It is expressed as

The Hamming gate or window is a tapered gate formed by using a raised cosine like the Hann window, but with nonzero start and end points. It is also called the tapering or apodization function formulated as

The Kaiser gate, also known as the Kaiser–Bessel window, approximates the prolate spheroidal window in which the ratio of the main lobe to the side lobe power spectral density is maximized. The attenuation in the side lobes is controlled by a tuning parameter

The Gaussian gate is called a bell curve gate and is the only function that Fourier-transforms itself with a smooth, nonzero function in the closed form. The mathematically Gaussian gate is denoted as

Tukey's gate is a tapered cosine gate whose first and last edges follow a cosine shape, while the central follows the rectangular window. It is also called a fully tapered gate

Here, a semi-tapered gate is defined as a piecewise flat approximation of the cosine part of the fully tapered or Tukey's window gate

The B-spline gate is formulated using the Cox-de-Boor recursive formula such that it is a composite curve of degree

The suppression of VLF noise is accomplished by solving a multi-objective optimization problem such that a cost function composed of gate standard error bars, covariance between gates, and distortion of the gated TEM signal is minimized by varying the gate shape and gate width. Different gate shapes are inputted to the optimization problem to get the optimal solution. The multi-objective cost function used for obtaining the optimal gating scheme is defined as

The diagonal of the covariance matrix contains the variance of each gate. From this we form the

Finally, a signal-normalized measure of the distortion of the TEM signal is computed by forming the

The MO-PSO is an evolutionary meta-heuristic optimization algorithm that has been used in various engineering and other applications

The fully sampled synthetic TEM model is implemented in MATLAB (R2021a) on a 64-bit Windows operating system with an Intel(R) Core(TM) i7-8550U, 1.80 GHz CPU system with 16 GB of RAM. The current work generates the TEM, VLF, power-line interference, and RN at a 4 MHz sampling frequency. A total of 1000 TEM signals with a 1 ms duration are generated, decaying at a rate of

In the VLF model, a bandwidth of 100 Hz (

The field measurements have been recorded using a ground-based monitoring system based on analog boxcar gating

The section presents the results obtained for the proposed model with synthetic and field data. In particular, the presence of VLF noise in the synthetic data is examined by the patterns produced in the CM, and the improvement by the optimized gating scheme is examined. Fully sampled data generated at the 4 MHz sampling frequency are assembled into 84 boxcar gates and used to form the CM in various noise scenarios (Fig.

The covariance matrices obtained for 84 boxcar gates.

Fully sampled data generated at the 4 MHz sampling frequency are assembled into 84 boxcar gates. After obtaining the 84 boxcar gates, they are re-gated to 30 gates, where the MO-PSO is used to search for the best gate shape or combinations of gate shapes such that it minimizes the cost function. The optimization was tested for three different scenarios, i.e., TEM signals with random noise and either one, four, or eight VLF radios. A total of 1000 transients have been used for each scenario to find the optimal gate combination. Most frequently, a combination of boxcar and Hamming gates was obtained as the best solution, followed by semi-tapered boxcar and Hamming gates. Other gates have also been found as the best solution for the optimization problem; however, they occurred fewer times. From an entire solution space obtained at last, a hybrid combination of only boxcar and Hamming gates has been found to be the best solution for the majority of the transients.

The result of the final gate combination is such that for early times (gate numbers 1–10, from 0.3 to 30

Figure

Covariance matrices obtained after re-gating for TEM

Error bars obtained for the re-gated signal composed of TEM with random and VLF noise in the case of one, four, or eight VLF radios for

In contrast, during the mid-times and late times, the hybrid scheme suppresses the VLF radios better than boxcar gates. Similarly, the analysis of standard error bars reveals that the hybrid gating scheme provides small error bars in the early times, mid-times, and late times. The covariance and standard error bar analysis confirm that hybrid gating improves suppression of VLF radios. To further investigate the model's performance, this work also evaluates two improvement factors defined as

Improvement factors and associated standard deviation for gates 15 to 24 with synthetic data.

Mean value of covariance obtained for non-diagonal elements with different VLF radio combinations.

Mean distortion obtained for gate number 22 with different VLF radio combinations.

Table

The proposed hybrid gating model is also tested on the field data acquired from Kompedal Plantage, Denmark. A total of 313 signals have been used to analyze the effect of VLF noise on the gated data. The readings are measured using a buried monitoring TEM system. Each signal is a stacked combination of 252 transients of high-moment data containing 84 gates. All 252 transients are re-gated for each signal using the boxcar, semi-tapered, and the optimized hybrid gate model. After gating all the transients of each signal, they are stacked to form a

Covariance matrices of field data before and after re-gating.

Error bars obtained for the re-gated signal using field data for

Improvement factors and associated standard deviation for gates 15 to 24 with field data.

Figure

Table

Figure

Comparative analysis of boxcar, semi-tapered, and piecewise flat Hamming gates (gate number 27) centered at 510.46

The total decay time of one TEM signal can roughly be divided into three regions: early times between gates 1 and 10 spanning from 0.3 to 30

The proposed hybrid gating model is configured according to the current towed-TEM system. In the future, data from a fully sampled system will be used for the analysis. The data will be gated using fully sampled gates as shown in Fig.

Comparative analysis of a B-spline (gate number 27) gate centered at 472.94

This section compares the analysis of the current work with existing literature.

Traditionally, the system response for SkyTEM is evaluated using model-based interpolation

The proposed model overcomes these issues mentioned in the above literature. The proposed model considered three constraints for optimization, namely minimization of standard errors, covariance, and distortion. The proposed model extends the analysis to synthetic and field data with varying VLF contents. Analysis shows that the proposed model has obtained collective improvement in standard errors with a minimum standard deviation and suppression in covariance of adjacent gates. The average improvement factor obtained for the hybrid gating scheme is 1.719 and 1.717 with synthetic and field data, respectively. The analysis also reveals that simulations based on fully sampled gates offer higher suppression in VLF radios than the existing semi-tapered and B-spline gates. Thus, the proposed model has collectively surpassed the performance of the existing gating scheme for an entire recording duration. The total time required for selecting an optimized gate for one signal was about 17 min. Also, the time required to select the optimum gate increased with an increase in the number of iterations and search agents. It is impractical to use multi-objective particle swarm optimization on a daily basis on, e.g., remote monitoring systems with limited computing resources. Here, an alternative solution is to use the optimization results to generate a set of a few representative and nearly optimum gate banks. Deciding which of these gate banks is optimum for any given measurement is a low-computing-cost operation.

The means of constraints, i.e., the minimum standard error, minimum correlation between off-diagonal terms, and minimum distortion, are used to obtain the optimal combination of gates. The optimal gate is then used to obtain the final gated signal for synthetic and field data. However, in real scenarios, it is not possible to measure the decay of a signal without noise. Therefore, repetition of simulations with only two constraints (minimum standard error, minimum correlation between off-diagonal terms) has also been considered. It has been observed that with two constraints the choice of gate selection varies more frequently from one gate to another (mainly between hybrid and semi-tapered gates). In addition, the optimum hybrid gate design is obtained for a noise model that provides VLF noise of the same amplitude without considering the fluctuations that occurs in real measurements, i.e., daily changes in environmental and atmospheric conditions leading to a varying strength of VLF radio signals. Such variations may affect the optimal gate placement or their types. A different combination of gate shapes may therefore be obtained due to variations in the environmental conditions in field data.

The CM obtained after gating the field data indicates the presence of a strong pattern at the late times. Thus, the field data appear to be contaminated by another noise source. A detailed study of daily changes could potentially help identify the noise source(s) and be used in the development of a more robust or adaptive algorithm with improved noise suppression. This, however, is delegated to future investigations.

The proposed hybrid gating scheme is a new technique designed for fully sampled and existing TEM systems with analog boxcar gates. The hybrid gating scheme is a combination of boxcar and Hamming gates. The developed gating technique has provided minimal distortion in the final gated signal, and it also minimizes covariance at mid-times and late times as well as standard errors for early and late gates. The developed technique is a promising tool to suppress VLF radio noise during the mid-times and late times. The covariance analysis provides an analysis of VLF noise and other coherent sources. The developed technique is not only applicable to the existing analog boxcar integrator TEM systems, but will also be ready to be deployed on fully sampled digital systems. In the future, analysis of more noise sources and solutions to suppress these sources will be examined.

Data used in this work are available upon request to the corresponding author

SKK: data collection, conceptualization, methodology, writing (original draft), and validation. PM: data collection, validation, reviewing, and editing. PKM: data collection and reviewing. JJL: conceptualization, validation, reviewing, and editing.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research has been supported by Innovation Fund Denmark (grant no. 0177-00085A).

This paper was edited by Lev Eppelbaum and reviewed by two anonymous referees.