Fractal-Based Orthonormal Basis for Compressing and Reducing the&#160;Dimensionality of Full-Waveform Inversion for Hydrogeological&#160;Applications Using Ground-Penetrating Radar
Authors: Giannakis Iraklis , Warren Craig , Giannopoulos Antonios , Sofroniou Anastasia 
Published: March 2021

&lt;p&gt;Full-waveform inversion (FWI) using ground-penetrating radar (GPR) is gaining momentum as a&amp;#160;powerful hydrogeological tool for inferring the hydraulic properties of soils between boreholes [1].&amp;#160;Nonetheless, the large computational requirements of FWI make it often unattainable with limited&amp;#160;practical uptake [2]. In addition, the inability to accurate reconstruct the loss mechanisms and the need&amp;#160;for a good initial model, further reduce the applicability of FWI [1], [2].&lt;/p&gt;&lt;p&gt;In order to overcome the aforementioned limitations, we suggest a novel framework that substantially&amp;#160;reduces the optimization space of FWI which consequently reduces the overall computational&amp;#160;requirements [2]. This methodology assumes that the water fraction of the investigated medium follows&amp;#160;a fractal distribution [3]. Based on that, using a principal components analysis on 3000 randomly&amp;#160;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&amp;#160;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].&lt;/p&gt;&lt;p&gt;The involved fractals describe the distribution of the water fraction that is subsequently transformed to&amp;#160;dielectric properties via a semi-empirical formula that relates readily available soil properties to the&amp;#160;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&amp;#160;space and overcomes pitfalls associated with reconstructing loss mechanisms [2].&lt;/p&gt;&lt;p&gt;References&lt;/p&gt;&lt;p&gt;[1] Klotzsche, A., Vereecken, H., &amp; Kruk van der J., (2019), Review of Crosshole Ground-Penetrating Radar Full-Waveform Inversion of Experimental Data: Recent Developments, challenges, and Pitfalls,&amp;#160;Geophysics, vol. 84, pp. H13-H28.&lt;/p&gt;&lt;p&gt;[2] Giannakis, I, Giannopoulos, A., Warren, C. &amp; Sofroniou, A., (2021), Fractal-Constrained&amp;#160;Crosshole/Borehole-to-Surface Full Waveform Inversion for Hydrogeological Applications Using&amp;#160;Ground-Penetrating Radar, IEEE Transactions on Geoscience and Remote Sensing, Early Access.&lt;/p&gt;&lt;p&gt;[3] Turcotte, L. (1992), Fractal and Chaos in Geology and Geophysics, Cambrige, UK: The Press&amp;#160;Syndicate of the University of Cambridge.&lt;/p&gt;&lt;p&gt;[4] Peplinski, N. R., Ulaby, F. T., &amp; 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.&lt;/p&gt;&lt;p&gt;[5] Giannakis, I., Realistic Numerical Modelling of Ground Penetrating Radar for Landmine&amp;#160;Detection, (2016), PhD Thesis Submitted at The University of Edinburgh.&lt;/p&gt;&lt;p&gt;[6] Meles, G. A., Kruk, van der J., Grennhalgh, S. A., Ernst, J. R., Maurer, H &amp; Green, A. G., (2010),&amp;#160;A New Vector Waveform Inversion Algorithm for Simultaneous Updating of Conductivity and Permittivity Parameters from Combination Cross/Borehole-to-Surface GPR Data, IEEE Transactions&amp;#160;on Geoscience and Remote Sensing, vol. 48, no. 9, pp. 3391-3407.&lt;/p&gt;