Finite-difference time-domain modelling of through-the-Earth radio signal propagation

Through-the-Earth (TTE) radio has been proposed for emergency communications in locations inaccessible by conventional means, such as underground mines. While the technology is viable, it is unclear how the signal propagates in inhomogeneous media; neither modelling or obtaining a conductivity distribution in the context of TTE radio has been previously attempted. With a robust model, many practical questions can be answered, such as what is the maximum range or the optimal frequency to use, or where the transmitter and receiver should be ideally placed. To this end, a finite-difference time-domain (FDTD) code was developed and optimized for the forward modelling of TTE radio transmissions. This method is computationally intensive, and to improve performance, it was run on a graphics processing unit (GPU). The code was validated against analytical solutions for simple geometries.
The use of a TTE radio signal to estimate the conductivity structure of an operating environment was demonstrated at an experimental coal mine. A transmitter was set up on the surface, and the signal strength in the underground mine was measured at several locations and frequencies. A four-layer conductivity model, with topography, was fitted to this data, using the FDTD code.
TTE radio signal range has been experimentally observed to increase in the presence of elongated conductors. A thin-wire approximation was added to the FDTD code, and conductivity models were fitted to match observed data for propagation along elevator shafts in multistory buildings and along railway tracks. Multistory buildings present a separate challenge due to their conductive reinforced concrete structure. By representing the floors and walls as thin sheets of a given conductance, observed and modelled data were reconciled for a variety of buildings, including above-ground reinforced concrete constructions, a building with steel floors, and a building partially below ground, and for placements of the transmitter and receiver inside or outside the building.
Further work remains in modelling a wider variety of settings where TTE radio could be used, such as buildings with different materials. Another avenue of research is to improve the performance of the modelling code, by refining the numerical method used.

Modelling and optimizing through-the-Earth radio transmissions

In this paper, we propose a uniformly convergent numerical scheme for a class of time-fractional singularly perturbed delay partial differential equations exhibiting a right regular boundary layer. The time-fractional derivative is considered in the Caputo sense with order β∈(0,1). The domain is discretized with a uniform mesh in both time and space directions. The numerical scheme comprises the discretization technique given by the Crank–Nicolson method in the temporal direction and the non-standard finite difference method in the spatial direction. To show the parameter uniform convergence of the proposed method, the boundedness of truncation error and stability analysis are performed. It is shown that the proposed scheme is second-order convergent in the temporal direction and first-order convergent in the spatial direction. Numerical examples are presented, and their numerical simulations confirm the theoretical findings. 

Crank–Nicolson method for solving time-fractional singularly perturbed delay partial differential equations

In this paper, the numerical method for solving a class of generalized fractional advection-diffusion equation (GFADE) is considered. The fractional derivative involving scale and weight factors is imposed for the temporal derivative and is analogous to the Caputo fractional derivative following an integration-after-differentiation composition. It covers many popular fractional derivatives by fixing different weights w(t) and scale functions z(t) inside. The numerical solution of such GFADE is derived via a collocation method, where conventional Legendre polynomials are implemented. Convergence and error analysis of polynomial expansions are studied theoretically. Numerical examples are considered with different boundary conditions to confirm the theoretical findings. By comparing the above examples with those from existing literature, we find that our proposed numerical method is simple, stable and easy to implement. 

Legendre collocation method for new generalized fractional advection-diffusion equation

(Fractional) differential equations have seen increasing use in physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and many other fields over the last two decades, providing a new and more realistic way to capture memory-dependent phenomena and irregularities inside systems using more sophisticated mathematical analysis (see, for example, [...]

/pdf/fractional-order-system-control-theory-and-applications-24meq3se.pdf

Fractional-Order System: Control Theory and Applications

In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis.

/pdf/finite-difference-collocation-method-for-the-generalized-15hfcr03.pdf

Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation

We show how a finite-difference time-domain modelling algorithm can be utilised for efficient 3D inversion of marine magnetotelluric (MT) data, and demonstrate the capability of our inversion on a synthetic salt dome. In comparison to frequency-domain modelling methods which are typically applied, two advantages of the finite-difference time-domain modelling method are (1) that it is an explicit, low-memory method; and (2) that it allows forward and adjoint computation for the electromagnetic field unknowns at all frequencies of interest in a single modelling run.

Efficient 3D MT Inversion Using Finite-difference Time-domain Modelling

A Fully Finite Difference Scheme for Time-Fractional Telegraph Equation Involving Atangana Baleanu Caputo Fractional Derivative

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Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation

Efficient 3D MT Inversion Using Finite-difference Time-domain Modelling

A Fully Finite Difference Scheme for Time-Fractional Telegraph Equation Involving Atangana Baleanu Caputo Fractional Derivative