Channel modeling for Through-the-Earth (TTE) communication has mainly been based on deterministic and empirical formulations. Deterministic models are limited to homogeneous rock structures while empirical models are only valid for the same configuration as the measurements that were taken. Given the difficulties in implementing underground measurements taking years to achieve enough database for a variety of mines and depths, we propose, as a more versatile solution, realistic simulations of a variety of underground structures to statistically characterize magnetic field attenuation. An algorithm based on the Monte Carlo method is presented and uses the Finite Element Method (FEM) to compute field propagation through each random structure. An empirical model is produced through simulations, and some metrics are evaluated, such as the median of magnetic field intensity, and the propagation loss, specially at the optimum operation frequency, in which channel variability is less spread when compared to all possible frequencies. This model can be used to estimate link budget for TTE communication in coal mines.

Through-the-Earth (TTE) communication is a practical solution to link underground environments to the surface. Such systems are more resistant to accidents in the interior of mines that could affect primary wired networks [

Between 1970 and 1980, the American Bureau of Mines measured the intensity of magnetic fields for Through-the-Earth communication for frequencies between 630 and 3030 Hz in 94 coal mines all over the United States [

All models mentioned above tried to investigate the channel behavior of a complex environment composed of multiple layers of different rocks using models based on 2 or 3 homogeneous layers. By varying the conductivity of the upper layer, these simple models try to fit the channel’s behavior. Yan also proposed a more complex and laborious analytic model that characterizes the propagation medium as a stratified soil with different electrical conductivities for each horizontal layer in [

A feasible way to understand the propagation of electromagnetic waves in complex environments, where analytical formulas become very difficult, is through simulations of the propagation environment in order to generate a database for parameter estimation and modeling of the propagation channel. Often, the simulated propagation environment depends on statistical distributions computed from measurement data of the channel, as in [

In the remainder of this article, we present the best-known models for TTE propagation in Section

A uniform plane wave decays exponentially with distance traveled through the material medium, and such decay can be expressed as a function of skin depth

The simplest approximation for a magnetic field created by an electrically small loop antenna is made by considering vacuum as a homogeneous medium, neglecting any boundary condition [

A more sophisticated model proposed by Wait and Spies in [

Geometry used for the calculation of the magnetic field at the point

All models cited above are in the frequency domain, since

The models previously presented are valid approximations for a few homogeneous layers of soil and air. Still, they may be valid for estimating the propagation loss at a mine site with definite depth and soil conductivity at a specific frequency. In this study, a more simplified model of the multilayered channel is proposed. Using several random scenarios, based on characteristics of coal mines, it was possible to develop an empirical model for the calculation of the

Models in [

In this section, we present the normalized magnetic field for coaxial configuration and the characterization of electric conductivity distribution based on the measured data for setting the simulator, followed by other parameters that model the earth profile.

Considering only the vertical component of the

The propagation loss in coaxial transmission can be found using

The channel behavior tends to be similar to a bandpass filter, which consequently establishes an optimal transmission frequency where propagation loss is minimal.

The data presented in [

Their selection criteria were based on two considerations: the group of mines had to be sufficiently large to represent their physical uniqueness and operation characteristics, and the number of workers inside, due to the correlation between each mine’s size and the quantity of miners. To do this, they used the technique known as optimum allocation, which states that for strata with great variability, it is necessary to perform a larger number of samples.

After a thorough screening, they selected 94 sites, whose depth ranged from 20 to 400 meters, located in ten American states: Pennsylvania, Ohio, West Virginia, Virginia, Tennessee, Kentucky, Illinois, Alabama, Utah, and Colorado.

Due to the complexity and extensiveness of this study, there is no reason to believe that if similar measurements were carried out nowadays, their statistic behavior would be different.

The apparent conductivity estimated through the

The raw data of field intensity is normalized by the magnetic moment for each measurement as in (

There are very few reports or papers on electromagnetic simulations in TTE communications [

Simulation process algorithm. The red boxes represent MATLAB processes, the orange boxes represent VBA processes, and the green boxes represent CST processes.

With the parameters estimated in Section

Multilayer TTE scenario for simulations using FEM.

The transmitting antenna is a one-turn square loop with sides of

Horizontal layers are modeled in such a way so as to represent the sedimentary sequences of rocks found in coal mines [

Regarding the electromagnetic characteristics of rocks, the magnetic permeability was set as a vacuum condition

Using a FEM solver with tetrahedral meshing and absorptive boundary walls, 150 independent and random trials were carried out resulting in more than 200 thousand measurements of field intensity. Such quantity was selected to surpass the number of scenarios considered in [

Since the optimum frequency in TTE transmission is usually below 10 kHz for moderate and great depths (>100 meters), the frequencies were chosen in logarithmic steps from 0.1 to 10 kHz, and the

A useful way to describe the average behavior of _{,} and

The coefficients found are

In order to compare the model with measured data, Figure

Therefore, it is possible to suggest that the simulated model is valid, including the layer choice and conductivity distributions and their parameters.

To supplement the results presented here, a comparison was made between the deterministic models

Comparison of different models and measures of normalized vertical magnetic field for depths of 60, 150, and 210 meters.

It is noticeable in Figure

Observing the shades of trial density, a larger field variability for higher frequencies and depths is observed. This is because the variability of sedimentary layer dimensions is more sensitive to smaller wavelengths, and also the larger number of independent layers for greater depths contributes to the field variation. To better understand channel variability, a 90% prediction interval was computed for data around the median of the

CDF of propagation loss at optimum frequencies for depths of 150, 220, and 300 meters.

This figure shows the evaluation of channel variability observing exclusively simulated field values of minimum attenuation for each trial. The discrete cumulative probability of the propagation loss at the optimal frequency is computed for depths of 150, 220, and 300 meters. The

The proposed stochastic generation of a variety of underground structures for electromagnetic simulations provides more adequate conditions for TTE channel modeling than simplified deterministic models and the empirical models in [

With regard to the CDF of propagation loss, it is observed that for a 300 m depth, which is the worst-case condition, there is an additional loss of 16 dB for the scenario with a cumulative probability of 95% compared to the scenario with 5% of cumulative probability. The variance, which may be seen from data collected at the optimal frequency for each trial, is smaller than that seen for all possible field values at frequencies higher than the optimum. Such smaller variation augments the predictability of the link budget in a TTE communication even though, in praxis, it also depends on atmospheric and manmade noise conditions at each site. The study of electromagnetic noise levels at the surface and underground as in [

The model, presented in this paper, used 150 independent data points to achieve a RMSE of 3.7 dB (while

The magnetic field data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This research was funded by the Vale Institute of Technology (ITV) and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).