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].

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].

The involved fractals describe the distribution of the water fraction that is subsequently transformed to dielectric properties via a semi-empirical formula that relates readily available soil properties to the frequency depended complex electric permittivity [4], [5]. Via this approach, we overcome the need for a simultaneous FWI for both permittivity and conductivity [6]. This further reduces the optimization space and overcomes pitfalls associated with reconstructing loss mechanisms [2].

References

[1] Klotzsche, A., Vereecken, H., & Kruk van der J., (2019), Review of Crosshole Ground-Penetrating Radar Full-Waveform Inversion of Experimental Data: Recent Developments, challenges, and Pitfalls, Geophysics, vol. 84, pp. H13-H28.

[2] Giannakis, I, Giannopoulos, A., Warren, C. & Sofroniou, A., (2021), Fractal-Constrained Crosshole/Borehole-to-Surface Full Waveform Inversion for Hydrogeological Applications Using Ground-Penetrating Radar, IEEE Transactions on Geoscience and Remote Sensing, Early Access.

[3] Turcotte, L. (1992), Fractal and Chaos in Geology and Geophysics, Cambrige, UK: The Press Syndicate of the University of Cambridge.

[4] Peplinski, N. R., Ulaby, F. T., & Dobson, M. C., (1995), Dielectric Properties of Soils in the 0.3-1.3 GHz Range, IEEE Transactions on Geoscience and Remote Sensing, vol. 33, no. 3, pp. 803-807.

[5] Giannakis, I., Realistic Numerical Modelling of Ground Penetrating Radar for Landmine Detection, (2016), PhD Thesis Submitted at The University of Edinburgh.

[6] Meles, G. A., Kruk, van der J., Grennhalgh, S. A., Ernst, J. R., Maurer, H & Green, A. G., (2010), A New Vector Waveform Inversion Algorithm for Simultaneous Updating of Conductivity and Permittivity Parameters from Combination Cross/Borehole-to-Surface GPR Data, IEEE Transactions on Geoscience and Remote Sensing, vol. 48, no. 9, pp. 3391-3407.