Fractal-Constrained Crosshole/Borehole-to-Surface Full-Waveform Inversion for Hydrogeological Applications Using Ground-Penetrating Radar

Abstract

Full waveform inversion (FWI) is considered one of the most promising interpretation tools for hydro-geological applications using ground-penetrating radar. However, FWI has had limited practical uptake for several reasons: large computational requirements; an inability to reconstruct loss mechanisms of soil; and the need for a good initial starting model. We aim to address these issues via a novel FWI subject to a fractallycorrelated distribution of water. Initially, the dispersive properties of the soil are expressed as a function of the water fraction using a semi-empirical model. This approach means the permittivity, conductivity, and relaxation mechanisms are all correlated, and therefore sensitivity problems between the permittivity and loss mechanisms no longer affect the performance of FWI. Subsequently, the distribution of the water fraction is constrained to follow a fractal geometry. Fractal-correlated noise is then compressed using principal component analysis (PCA) in order to further reduce the number of the system's unknowns and accelerate FWI. PCA reduces the volume and dimensions of the optimization space, and thus, initialisation is no longer necessary. Lastly, a novel measurement configuration is suggested that uses superposition with all the individual measurements in order to reduce the number of forward models that need to be executed for every iteration of FWI. These enhancements substantially reduce the computational requirements of FWI and therefore eliminate the need for high-performance computers and time-consuming algorithms. The proposed scheme has been successfully tested with several numerical case-studies, which indicate the potential of this approach to become a commerciallyappealing interpretation tool for hydro-geology.

Fractal-Based Orthonormal Basis for Compressing and Reducing the Dimensionality of Full-Waveform Inversion for Hydrogeological Applications Using Ground-Penetrating Radar

“…Full-waveform inversion (FWI) using ground-penetrating radar (GPR) is gaining momentum as a powerful hydrogeological tool for inferring the hydraulic properties of soils between boreholes [1]. Nonetheless, the large computational requirements of FWI make it often unattainable with limited practical uptake [2]. In addition, the inability to accurate reconstruct the loss mechanisms and the need for a good initial model, further reduce the applicability of FWI [1], [2].…”

“…Nonetheless, the large computational requirements of FWI make it often unattainable with limited practical uptake [2]. In addition, the inability to accurate reconstruct the loss mechanisms and the need for a good initial model, further reduce the applicability of FWI [1], [2].…”

“…In order to overcome the aforementioned limitations, we suggest a novel framework that substantially reduces the optimization space of FWI which consequently reduces the overall computational requirements [2]. This methodology assumes that the water fraction of the investigated medium follows a fractal distribution [3].…”

“…Based on that, using a principal components analysis on 3000 randomly generated fractals, we build an orthonormal basis that is fine-tuned for fractal correlated noise. Furthermore, it is proven [2], that fractal correlated noise is highly compressible and can be sufficiently represented with just 30-40 principal components. This reduces the optimization space since now FWI needs to fine-tune just these 30-40 parameters instead of every cell of the investigated medium [2].…”

“…Furthermore, it is proven [2], that fractal correlated noise is highly compressible and can be sufficiently represented with just 30-40 principal components. This reduces the optimization space since now FWI needs to fine-tune just these 30-40 parameters instead of every cell of the investigated medium [2].…”