Full-waveform inversion of off-ground, borehole, and surface GPR
Conventional tomograms provided by standard ray-based methods have limited resolution that scales with the diameter of the first Fresnel-zone, primarily because only a fraction of the information contained in the radar data is included in the inversion. Higher resolution radar tomograms with a potential resolution limit as small as half the wavelength can be derived by using full-waveform inversion (FWI) schemes that use the full information content of the recorded trace. The full-waveform inversion can be implemented for a large number of unknowns using gradient-based optimizations, whereas for a limited number of model parameters a combined global and local optimization approach can be used.
Gradient-based full-waveform inversion
The FWI inversion for a large number of unknowns uses gradient-based optimization methods where a good start model is needed to enhance the chance of finding the global minimum instead of a local one. A simultaneous updating of the permittivity and conductivity at each iteration is implemented . A good ray-based start model is needed and if necessary adapted such that the starting model has at least half a wavelength overlap between the measured and synthetic data. In the case of large contrast, a frequency filtering can be included to reduce the non-linearity .
Important steps to apply this method to experimental data include a 3D to 2D conversion and the estimation of the effective wavelet. This new full-waveform inversion was used to obtain high resolution images of a saturated gravel aquifer using an optimized source-receiver setup . Moreover, the full-waveform inversion was able to image a sub-wavelength thickness low-velocity waveguiding layer that acted as a preferential flow path . The excellent fit of amplitudes and phases between the measured and modeled data also for late time arrivals confirms the presence of the low-velocity decimeter scale waveguide. An optimized source receiver setup was investigated by a checkerboard analysis  and an improved formulation was introduced and remaining gradients were investigated . Analysing many crosshole measurements and comparing the porosity estimates from the permittivity results show a good correspondence with Neutron–Neutron logging data. Moreover, estimates of hydraulic permeability based on flowmeter logs confirm the presence of zones of preferential flow in depth interval with high permittivity and high porosity . In addition, a novel amplitude analysis was introduced that explores the maxima and minima of the trace energy that are caused by the presence of these low-velocity waveguides .
 A. Klotzsche, J. van der Kruk, N. Linde, J. Doetsch, H. Vereecken, 2013, 3D characterization of high-permeability zones in a gravel aquifer using 2D crosshole GPR full-waveform inversion and waveguide detection, Geophysical Journal International, 195, 932-944, doi:10.1093/gji/ggt275
 X. Yang, A. Klotzsche, G. Meles, H. Vereecken, J. van der Kruk, 2013, Improvements in crosshole GPR full-waveform inversion and application on data measured at the Boise Hydrogeophysics Research Site, Journal of Applied Geophysics, 99, 114-124, doi:10.1016/j.jappgeo.2013.08.007.
 M. Oberröhrmann, A. Klotzsche, H. Vereecken, J. van der Kruk, 2013, Optimization of acquisition setup for cross-hole GPR fullwaveform inversion using checkerboard analysis, Near Surface Geophysics, 11, 197-209, doi:10.3997/1873-0604.2012045.
 A. Klotzsche, J. van der Kruk, G. A. Meles, H. Vereecken, 2012, Crosshole GPR full-waveform inversion of waveguides acting as preferential flow paths within aquifer systems, Geophysics,77, H57-H62, doi:10.1190/GEO2011-0458.1
 A. Klotzsche, J. van der Kruk, G. A. Meles, J. A. Doetsch, H. Maurer, N. Linde, 2010, Full-Waveform Inversion of Crosshole Ground Penetrating Radar data to Characterize a Gravel Aquifer Close to the River Thur, Near Surface Geophysics, 8, 635-649, doi: 10.3997/1873-0604.2010054
 G. A. Meles, S. Greenhalgh, J. van der Kruk, A.G. Green and H. Maurer, 2011, Taming the non-linearity problem in GPR full-waveform inversion for high contrast media, Journal of Applied Geophysics, 73, 174-186, doi:10.1016/j.jappgeo.2011.01.001
 G. A. Meles, J. van der Kruk, S. Greenhalgh, J. Ernst, H. Maurer, A.G. Green, 2010, A New Vector Waveform Inversion Algorithm for Simultaneous Updating of Conductivity and Permittivity Parameters from Combination Crosshole-/Borehole-to-Surface GPR Data, IEEE Geoscience and Remote Sensing, 48, 3391-3407, doi: 10.1109/TGRS.2010.2046670
Combined global and local full-waveform optimization
Simple geometries such as horizontal layers can be describe with a limited number of model parameters, which enables a gradient-free optimization to invert off-ground and on-ground GPR data. A combined global and local search is performed using the Simplex search algorithm where the measured data is fitted with data calculated for a horizontally layered model using the full-waveform for off-ground and on-ground GPR data. A 3D frequency domain forward model of Maxwell’s equation is used where the integral representation of the electric field is numerically evaluated. An important aspect for a successful inversion is the estimation of the unknown source wavelet, which is straight forward for off-ground GPR  and for horizontally layered media the water content and chloride can be estimated  including chloride gradients . For on-ground GPR, the effective source wavelet is influenced by the subsurface, such that a simultaneous optimization of model parameters and source wavelet is needed . Combined geophysical measurements were carried out over a silty loam with significant variability in the soil texture. The medium properties obtained from the on-ground GPR full-waveform inversion are consistent with Theta probe, electromagnetic resistivity tomography, and electromagnetic induction results and enable the formulation of an empirical relationship between soil texture and soil properties . With the layered setup, several scenarios can be considered and the obtained models can be used as start model for more complicated lateral varying media.
 S. Busch, J. van der Kruk, H. Vereecken, 2014, Improved characterization of fine texture soils using on-ground GPR full-waveform inversion, IEEE, Transactions in Geoscience and Remote Sensing, 52, 3947-3958, doi:10.1109/TGRS.2013.2278297.
 S. Busch, J. van der Kruk, J. Bikowski, H. Vereecken, 2012 Quantitative conductivity and permittivity estimation using full-waveform inversion of on-ground GPR data, Geophysics, 77, H79-H91, doi:10.1190/GEO2012-0045.1
 A. Kalogeropoulos, J. van der Kruk, J. Hugenschmidt, J. Bikowski, E. Bruhwiler, 2013, Full-waveform GPR inversion to assess chloride gradients in concrete, NDT&E International, 57, 74-84, doi:10.1016/j.ndteint.2013.03.003
 A. Kalogeopoulos, J. van der Kruk, H. Hugenschmidt, K. Merz, 2011, Chlorides and Moisture Assessment in Concrete by GPR Full-Waveform Inversion, Near Surface Geophysics, 9, 277-285, doi:10.3997/1873-0604.2010064