Focused inversion of vertical radar profile (VRP) traveltime data Giulio Vignoli 1 , Rita Deiana 2 , and Giorgio Cassiani 2 ABSTRACT The reconstruction of the GPR velocity vertical profile from vertical radar profile (VRP) traveltime data is a problem with a finite number of measurements and imprecise data, analogous to similar seismic techniques, such as the shallow down-hole test used for S-wave velocity profiling or the vertical seismic profil- ing (VSP) commonly used in deeper exploration. The uncer- tainty in data accuracy and the error amplification inherent in deriving velocity estimates from gradients of arrival times make this an example of an ill-posed inverse problem. In the frame- work of Tikhonov regularization theory, ill-posedness can be tackled by introducing a regularizing functional (stabilizer). The role of this functional is to stabilize the numerical solution by incorporating the appropriate a priori assumptions about the geometrical and/or physical properties of the solution. One of these assumptions could be the existence of sharp boundaries separating rocks with different physical properties. We apply a method based on the minimum support stabilizer to the VRP traveltime inverse problem. This stabilizer makes it pos- sible to produce more accurate profiles of geological targets with compact structure. We compare more traditional inversion results with our proposed compact reconstructions. Using syn- thetic examples, we demonstrate that the minimum support sta- bilizer allows an improved recovery of the profile shape and velocity values of blocky targets. We also study the stabilizer behavior with respect to different noise levels and different choices of the reference model. The proposed approach is then applied to real cases where VPRs have been used to derive moisture content profiles as a function of depth. In these real cases, the derived sharper profiles are consistent with other evi- dence, such as GPR zero-offset profiles, GPR reflections and known locations of the water table. INTRODUCTION The use of near-surface geophysics as a contribution to the solu- tion of hydrological and environmental problems has become of general acceptance. The corresponding applications are generally referred to under the general name of “hydrogeophysics” (Rubin and Hubbard, 2005; Vereecken et al., 2006). Several techniques have been extensively used for the hydrogeophysical characteriza- tion of the soil and subsoil. Among them, the most popular are elec- trical resistivity tomography (ERT — e.g., Binley and Kemna, 2005) and ground-penetrating radar (GPR — e.g., Davis and Annan, 1989; Annan, 2005). GPR is particularly suitable for vadose zone characterization and monitoring (Cassiani et al., 2006) as elec- tromagnetic wave propagation velocity is directly related to the soil volumetric moisture content (e.g., Greaves et al., 1996; Strobbia and Cassiani, 2007; Wijewardana and Galagedara, 2010) and therefore porosity below the water table (e.g., Turesson, 2006), provided that suitable constitutive relationships are established (e.g., Topp et al., 1980; Roth et al., 1990; Brovelli and Cassiani, 2010). ERT and GPR have great advantages when used in borehole configuration because of the dramatic increase in resolution that can be achieved at depth. In addition, borehole GPR is superior to surface-to-surface GPR because (1) penetration is enhanced by avoiding the shallow, usual- ly electrically conductive, soil layers, and (2) measurement of ve- locity is simplified by using direct waves between transmitter and receiver. As a consequence, a number of hydrogeophysical studies have adopted crosshole GPR (e.g., Hubbard et al., 1997; Alumbaugh et al., 2002; Binley et al., 2002; Binley and Beven, 2003; Rucker and Ferré, 2004; Tronicke et al., 2004; Cassiani and Binley, 2005; Deiana et al., 2007, 2008; Looms et al., 2008a, b). However, these applications require two boreholes at a distance not larger than a few meters, a situation rarely available at sites of Manuscript received by the Editor 18 April 2011; revised manuscript received 9 September 2011; published online 3 February 2012. 1 King Fahd University of Petroleum & Minerals, Earth Sciences Department, Dhahran, Saudi Arabia. E-mail: giulio.vignoli@gmail.com. 2 Università di Padova, Dipartimento di Geoscienze, Padova, Italy. E-mail: rita.deiana@unipd.it; giorgio.cassiani@unipd.it. © 2012 Society of Exploration Geophysicists. All rights reserved. H9 GEOPHYSICS. VOL. 77, NO. 1 (JANUARY-FEBRUARY 2012); P. H9–H18, 8 FIGS. 10.1190/GEO2011-0147.1 Downloaded 05 Feb 2012 to 147.162.110.99. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/