At the ultrahigh frequencies common to portable radios, tunnels such as mine entries are often modeled by hollow dielectric waveguides. The roughness condition of the tunnel walls has an influence on radio propagation, and therefore should be taken into account when an accurate power prediction is needed. This paper investigates how wall roughness affects radio propagation in tunnels, and presents a unified ray tracing and modal method for modeling radio propagation in tunnels with rough walls. First, general analytical formulas for modeling the influence of the wall roughness are derived, based on the modal method and the ray tracing method, respectively. Second, the equivalence of the ray tracing and modal methods in the presence of wall roughnesses is mathematically proved, by showing that the ray tracing-based analytical formula can converge to the modal-based formula through the Poisson summation formula. The derivation and findings are verified by simulation results based on ray tracing and modal methods.

Radio propagation in tunnels has been investigated for decades [

The ray tracing method treats radio waves as ray tubes, and the electrical field at any location within a tunnel is represented by a summation of the rays reaching the location [

So far the ray-mode equivalence discussion has been limited to tunnels with smooth walls. It is known that underground mine tunnels often have rough walls, which influence the radio signal propagation in the tunnel. In this paper, we will investigate how surface roughness affects radio propagation in tunnels and discuss the ray-mode equivalence in the presence of wall roughness.

Despite the long history of tunnel propagation research, very few investigations have analyzed the influence of roughness on tunnel propagation. Mahmoud and Wait [

The roughness analyses above are based on the modal method, where the roughness effect is modeled by an additional attenuation constant applied to the dominant mode. In addition, there are some other methods used for analyzing the roughness effect. For example, Martelly and Janaswamy [

In this paper, we proposed a unified ray-mode method for modeling radio propagation in tunnels with rough walls. The contribution of this paper include the following. First, we derived general analytical formulas for modeling the influence of the wall roughness based on the modal and ray tracing methods, respectively. For the modal method, we derived a general roughness attenuation constant for both the dominant mode and higher order modes. In addition to the capability of modeling roughness attenuation for higher modes, the model developed in this paper is more general than Emslie’s model in the sense that it can model radio propagation in tunnels that have different roughnesses on different surfaces. Such flexibility is particularly useful for mines and caves where the roughness condition for each wall could be significantly different. Second, we mathematically proved the equivalence of the ray-mode solution in the presence of wall roughness.

It should be noted that a ray method has been discussed by Emslie

In summary, “rays” in [

The rest of this paper is organized as follows. A roughness-modified reflection coefficient is introduced in Section II. Ray tracing and modal methods for modeling radio propagation in tunnels with rough walls are presented in Section III. The equivalence of the ray tracing and modal methods is also discussed in Section III. Some numerical results and analysis are given in Section IV. Finally, the conclusions are given in Section V.

When a radio wave is incident on a tunnel wall, part of the wave transmits into the wall and the other part is reflected back to the tunnel. It is found that power loss associated with reflections on tunnel walls constitutes the major propagation loss in a tunnel environment [

For a plane wave incident on a smooth surface, it is known that the wave is reflected in the specular direction, given by Snell’s law of reflection. The reflected field can be calculated by multiplying the incident field with the corresponding Fresnel reflection coefficient _{⊥,//}, given by [

Here, the subscripts ⊥ and // denote the perpendicular and parallel polarizations, respectively, and _{⊥,//} is the angle of incidence defined by the angle between the direction of the incident wave and the normal to the surface. Δ_{⊥,//} is a quantity related to surface impedance, and is given by

_{//,⊥} = _{//,⊥}/_{0} are the complex relative permittivity. For grazing incidences where the angle of incidence approaches 90°, we can approximate the Fresnel reflection coefficients _{⊥,//} as [

Radio reflection from a rough surface is usually handled by a stochastic method, since the surface roughness can only be measured statistically. Although other distributions are possible [_{h}_{h}

It is apparent from (_{s}_{s}_{h}

It should be noted that the surface becomes perfectly reflecting at extreme grazing incidence (_{⊥,//} → _{s}

We consider a straight hollow dielectric waveguide with rectangular cross-sectional dimensions, as shown in _{0} is the permittivity of air, and _{a}_{,}_{b}_{0}. A transmitter is located at _{0}, _{0}, 0) and a receiver at

The electric field within a rectangular dielectric waveguide can either be represented by a ray summation based on the ray tracing method, or by a mode summation based on the modal method. In the following, we discuss the two methods for modeling the influence of wall roughness on tunnel propagation in detail.

Based on the ray tracing theory, the electric field at an arbitrary point _{0}, _{0}, 0)] as [

_{t}_{m}_{,}_{n}_{0,0} becomes the point source itself, and the ray path connecting the image _{0,0} and the receiver becomes the line-of-sight path. Some examples of the rays and images in a 2-D rectangular tunnel are shown in _{m}_{n}_{m}_{,}_{n}

In (_{//,⊥} for a tunnel with smooth walls is defined in (

_{a}_{,}_{b}_{a,b}_{0} are the complex relative permitivities for the horizontal and vertical walls, normalized by the vacuum permitivity _{0}. _{a,b}

_{a,b}_{a,b}

Now, we look at the two reflection coefficients (i.e., _{⊥} and _{//}_{s}_{h,i}

Therefore, the electric field inside a tunnel with rough walls can be calculated by

Similarly, for a horizontally polarized source, the electrical field can be calculated as

A comparison between the two electrical field expressions [in (

The modal method views the electrical field in a tunnel as a set of hybrid modes denoted by EH_{p,q}

_{x}_{y}_{z}_{p,q}_{p,q}_{t}_{p,q}_{p,q}_{x,y}

For electrically large tunnels where

It is apparent that (_{1}_{,}_{1} mode.

As shown in [_{p,q}_{x}, θ_{y}_{x}_{y}_{x}, θ_{y}_{x},_{y}_{x,y}_{x,y}

The axial distance _{z,x}

For tunnels with four smooth walls, the attenuation of the

_{x,y}

Now, we evaluate the radio attenuation in tunnels with rough walls. Again, we introduce the scattering factor _{s,i}

Substituting (

_{p,q}

Note that the modal attenuation constant _{p,q}_{p,q}

_{h,}_{2} = _{h,}_{4} and _{h,}_{1} = _{h,}_{3}), after some mathematical manipulation, it can be shown that (

For the dominant mode EH_{1}_{,}_{1}, and with the assumption of equal roughness for all the four walls (_{h,i}_{h}

_{h,i}_{p,q}

It is shown in (_{h,i}_{h,}_{1} and _{h,}_{3}, does not change the overall power attenuation. In other words, for a tunnel with only one rough surface, the propagation behavior is the same no matter whether the rough surface is on the top (ceiling) or the bottom (floor).

A comparison between (_{p,q}

By following a similar procedure given in Section III-B.1, the modal attenuation constant for the vertical polarization case can be derived as:

The roughness modal attenuation constant

A comparison between (

For the ray tracing method, the roughness effect is taken into account by applying a modified Fresnel reflection coefficient to each ray as the ray is reflected by different tunnel walls. For the modal method, the modified Fresnel reflection coefficient is applied to each mode, which is viewed as a mixture of four plane waves. The two methods are two different views of the same problem, and thus should be mathematically equivalent. _{p,q}_{p,q}

The detailed mathematical proof of (

It should be noted that the influence of both transmit and receive antennas’ positions on tunnel propagation has been modeled in _{p,q}_{p,q}_{1}_{,}_{1} is minimized when either the transmitter or the receiver is close to any of the four tunnel walls, and is maximized when antennas are located in the center (_{0} = 0_{0} = 0). This finding has been recently confirmed by measurement results in a train tunnel in [

A close examination of (

In _{p,q}

Ideally, the proposed model should be validated with measurement results in a tunnel with rough surfaces. However, adding controlled surface roughness to a physical tunnel is not very feasible in practical, as we are trying to introduce a stochastic method (based on roughness definition) into a deterministic problem. As a compromise, we will add simulated roughness to a physical tunnel with smooth surfaces and compare numerical results generated based on different modeling methods.

Recently, extensive RF measurements have been performed in a concrete tunnel to support ultrahigh frequency propagation model development research at the National Institute for Occupational Safety and Health [_{h}_{h}_{h}

It can be found from _{h}

To investigate the ray-mode equivalence at short distances, the results for the first 60 m in

This paper investigates the influence of wall roughness on radio propagation in tunnels and mines. Analytical solutions based on the ray tracing and modal methods are derived, respectively, and shown to be equivalent when the frequencies of interest are high and the separation distance between the transmitter and the receiver is sufficiently far. It is found that surface roughness in tunnels introduces additional attenuation to RF signals. The additional attention caused by surface roughness decreases with tunnel dimensions rapidly and linearly increases with wavelength. The developed models are useful for understanding and analyzing radio propagation in mines, where surface roughnesses are generally significant.

The author would like to thank R. Jacksha and T. Plass for their help on taking the measurement data. He would also like to thank Dr. J. Waynert, L. Ko, J. Schall, and A. Mayton for reviewing the manuscript.

Color versions of one or more of the figures in this paper are available online at

Disclaimer

The findings and conclusions in this paper are those of the author(s) and do not necessarily represent the views of the National Institute for Occupational Safety and Health. Mention of any company or product does not constitute endorsement by the NIOSH.

Substituting (

Applying the small angle approximation sin _{x,y}_{x,y}

In (

Substituting [

Substituting (

Following the procedures introduced in [

To separate the even and odd instances of the image order _{1} and _{2} to reformulate _{1} and _{2}, respectively

The values of the aiding variables, such as _{l}_{l}

With the help of the Poisson summation formula, we can convert _{l}_{1}_{2}) into its 2-D Fourier transform _{l}

Substituting (

The integral in (_{1}) = 0 and (_{2}) = 0:

As a result, the integral in (

We assume that the separation distance z is sufficiently large, such that the following approximations hold:

Substituting the stationary point expression in (

For electrically large tunnels where

Substituting (

Substituting (

Based on (_{1}_{,}_{2} can be represented by _{1}_{,}_{2} as

Substituting (

Substituting (

Note that in (_{l}_{l}

_{p,q}_{p,q}

It has been shown in the appendix of [

Substituting (

Cross section of a hollow dielectric waveguide.

Ray tracing method for modeling the influence of wall roughness on radio propagation in tunnels.

Modal method for modeling the influence of wall roughness on radio propagation in tunnels.

Radio reflection from a rough surface.

Influence of surface roughness on tunnel propagation: vertical polarization.

Influence of surface roughness on tunnel propagation: horizontal polarization.

Power attenuation in a tunnel with different surface roughnesses on different walls.

Closer view of the first 60-m data in

Parameters Table

Dimension | Axis | Size | Permitivity | Image | Mode |
---|---|---|---|---|---|

Horizontal | _{a} | ||||

Vertical | _{b} |

Summary of Parameters Used in the Simulation

Parameter | Value | Parameter | Value |
---|---|---|---|

Tunnel width (2a) | 1.83 m | Re {_{a},_{b} | 8.9 |

Tunnel height (2b) | 2.35 m | _{a,b} | 0.15 S/m |

Transmitter height | 1.22 m | f | 0.45, 0.915 GHz |

Receiver height | 1.22 m | 2.45 GHz |

List of Variables

l | m | n | x̄_{l} | x̃_{l} | ȳ_{l} | ỹ_{l} |
---|---|---|---|---|---|---|

1 | 2_{1} | 2_{2} | _{0} − | _{0} − | _{0} − | _{0} − |

2 | 2_{1} | 2_{2} + 1 | _{0} − | _{0} − | 2_{0} − | −_{0} − |

3 | 2_{1} + 1 | 2_{2} + 1 | 2_{0} − | −_{0} − | 2_{0} − | −_{0} − |

4 | 2_{1} + 1 | 2_{2} | 2_{0} − | −_{0} − | _{0} − | _{0} − |