Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

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An overview of full-waveform inversion in exploration geophysics

Crosshole radar tomography is a useful tool in diverse investigations in geology, hydrogeology, and engineering. Conventional tomograms provided by standard ray-based techniques have limited resolution, primarily because only a fraction of the information contained in the radar data i.e., thefirst-arrivaltimesandmaximumfirst-cycleamplitudesis included in the inversion. To increase the resolution of radar tomograms,wehavedevelopedaversatilefull-waveforminversion scheme that is based on a finite-difference time-domain solution of Maxwell’s equations. This scheme largely accountsforthe3Dnatureofradar-wavepropagationandincludes an efficient method for extracting the source wavelet from the radar data.After demonstrating the potential of the newschemeontworealisticsyntheticdatasets,weapplyitto two crosshole field data sets acquired in very different geologic/hydrogeologic environments. These are the first applications of full-waveform tomography to observed crosshole radar data.The resolution of all full-waveform tomograms is showntobemarkedlysuperiortothatoftheassociatedraytomograms. Small subsurface features a fraction of the dominant radar wavelength and boundaries between distinct geological/hydrological units are sharply imaged in the fullwaveformtomograms.

Application of a new 2D time-domain full-waveform inversion scheme to crosshole radar data

We have developed a new full-waveform groundpenetrating radar (GPR) multicomponent inversion scheme for imaging the shallow subsurface using arbitrary recording configurations. It yields significantly higher resolution images than conventional tomographic techniques based on first-arrival times and pulse amplitudes. The inversion is formulated as a nonlinear least squares problem in which the misfit between observed and modeled data is minimized. The full-waveform modeling is implemented by means of a finite-difference time-domain solution of Maxwell's equations. We derive here an iterative gradient method in which the steepest descent direction, used to update iteratively the permittivity and conductivity distributions in an optimal way, is found by cross-correlating the forward vector wavefield and the backward-propagated vectorial residual wavefield. The formulation of the solution is given in a very general, albeit compact and elegant, fashion. Each iteration step of our inversion scheme requires several calculations of propagating wavefields. Novel features of the scheme compared to previous full-waveform GPR inversions are as follows: 1) The permittivity and conductivity distributions are updated simultaneously (rather than consecutively) at each iterative step using improved gradient and step length formulations; 2) the scheme is able to exploit the full vector wavefield; and 3) various data sets/survey types (e.g., crosshole and borehole-to-surface) can be individually or jointly inverted. Several synthetic examples involving both homogeneous and layered stochastic background models with embedded anomalous inclusions demonstrate the superiority of the new scheme over previous approaches.

A New Vector Waveform Inversion Algorithm for Simultaneous Updating of Conductivity and Permittivity Parameters From Combination Crosshole/Borehole-to-Surface GPR Data

Crosshole radar techniques are important tools for a wide range of geoscientific and engineering investigations. Unfortunately, the resolution of crosshole radar images may be limited by inadequacies of the ray tomographic methods that are commonly used in inverting the data. Since ray methods are based on high-frequency approximations and only account for a small fraction of the information contained in the radar traces, they are restricted to resolving relatively large-scale features. As a consequence, the true potential of crosshole radar techniques has yet to be realized. To address this issue, we introduce a full-waveform inversion scheme that is based on a finite-difference time-domain solution of Maxwell's equations. We benchmark our new scheme on synthetic crosshole data generated from suites of increasingly complex models. The full-waveform tomographic images accurately reconstruct the following: (1) the locations, sizes, and electrical properties of isolated subwavelength objects embedded in homogeneous media; (2) the locations and sizes of adjacent subwavelength objects embedded in homogeneous media; (3) abrupt media boundaries and average and stochastic electrical property variations of heterogeneous layered models; and (4) the locations, sizes, and electrical conductivities of water-filled tunnels and closely spaced subwavelength pipes embedded in heterogeneous layered models. The new scheme is shown to be remarkably robust to the presence of uncorrelated noise in the radar data. Several limitations of the full-waveform tomographic inversion are also identified. For typical crosshole acquisition geometries and parameters, small resistive bodies and small closely spaced dielectric objects may be difficult to resolve. Furthermore, electrical property contrasts may be underestimated. Nevertheless, the full-waveform inversions usually provide substantially better results than those supplied by traditional ray methods.

Full-Waveform Inversion of Crosshole Radar Data Based on 2-D Finite-Difference Time-Domain Solutions of Maxwell's Equations

We propose a new method for the quantitative resolution analysis in full seismic waveform inversion that overcomes the limitations of classical synthetic inversions while being computationally more efficient and applicable to any misfit measure. The method rests on (1) the local quadratic approximation of the misfit functional in the vicinity of an optimal earth model, (2) the parametrization of the Hessian in terms of a parent function and its successive derivatives and (3) the computation of the space-dependent parameters via Fourier transforms of the Hessian, calculated with the help of adjoint techniques. In the simplest case of a Gaussian approximation, we can infer rigorously defined 3-D distributions of direction-dependent resolution lengths and the image distortion introduced by the tomographic method. We illustrate these concepts with a realistic full waveform inversion for upper-mantle structure beneath Europe. As a corollary to the method for resolution analysis, we propose several improvements to full waveform inversion techniques. These include a pre-conditioner for optimization schemes of the conjugate-gradient type, a new family of Newton-like methods, an approach to adaptive parametrization independent from ray theory and a strategy for objective functional design that aims at maximizing resolution. The computational requirements of our approach are less than for a typical synthetic inversion, but yield a much more complete picture of resolution and trade-offs. While the examples presented in this paper are rather specific, the underlying idea is very general. It allows for problem-dependent variations of the theme and for adaptations to exploration scenarios and other wave-equation-based tomography techniques that employ, for instance, georadar or microwave data

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1365-246X.2011.05218.x

Resolution analysis in full waveform inversion

High‐frequency electromagnetic‐wave propagation phenomena associated with borehole georadar experiments are complex. To improve our understanding of the governing physical processes and radiative properties of borehole georadar antenna systems, we have developed a modelling tool based on a finite‐difference time‐domain (FDTD) solution of Maxwell's equations in cylindrical coordinates. The computational domain is bounded by cylindrical symmetry conditions along the left edge of the model and uniaxial perfectly matched layer (UPML) absorbing boundary conditions along the top, bottom and right model edges. An accurate and efficient grid‐refinement technique allows us to account for detailed aspects of borehole georadar antenna systems, slim boreholes and materials with very high dielectric permittivities, such as water. Numerical experiments reveal that the radiation patterns of finite‐size Wu–King‐type antennae and infinitesimal electric dipoles in dry boreholes differ only slightly from the analytic solution of an infinitesimal electric dipole in a homogeneous full‐space. In contrast, there are substantial differences between the radiation patterns of antennae placed in water‐filled boreholes and their analytic full‐space equivalents without boreholes. The effects of placing the antennae in air‐ and water‐filled boreholes are explored using data acquired in crystalline rock and alluvial sediments, respectively. In both cases, simulations based on realistic transmitter antennae located in boreholes and spatially corrected receiver radiation patterns provide better agreement between the observed and modelled data than simulations based on infinitesimal transmitter and receiver dipoles.

Realistic FDTD modelling of borehole georadar antenna radiation: methodolgy and application

Waveform-based tomographic imaging of crosshole georadar data is a powerful method to investigate the shallow subsurface because of its ability to provide images of electrical properties in near-surface environments with unprecedented spatial resolution. A critical issue with waveform inversion is the a priori unknown source signal. Indeed, the estimation of the source pulse is notoriously difficult but essential for the effective application of this method. Here, we explore the viability and robustness of a recently proposed deconvolution-based procedure to estimate the source pulse during waveform inversion of crosshole georadar data, where changes in wavelet shape with location as a result of varying near-field conditions and differences in antenna coupling may be significant. Specifically, we examine whether a single, average estimated source current function can adequately represent the pulses radiated at all transmitter locations during a crosshole georadar survey, or whether a separate source wavelet estimation should be performed for each transmitter gather. Tests with synthetic and field data indicate that remarkably good tomographic reconstructions can be obtained using a single estimated source pulse when moderate to strong variability exists in the true source signal with antenna location. Only in the case of very strong variability in the true source pulse are tomographic reconstructions clearly improved by estimating a different source wavelet for each transmitter location.

Waveform Inversion of Crosshole Georadar Data: Influence of Source Wavelet Variability and the Suitability of a Single Wavelet Assumption

High-frequency electromagnetic wave propagation phenomena associated with borehole georadar experiments are complex. To improve our understanding of the governing physical processes, we present a suitable finitedifference time-domain (FDTD) solution of Maxwell’s equations in cylindrical coordinates. An important feature of this algorithm is the use of a powerful grid refinement technique that enables us to account efficiently for detailed design aspects of georadar antennas as well as materials with very high dielectric permittivities. This type of modeling provides the basis for improving the ray-based inversion of the first-cycle amplitudes and/or for performing the full-waveform inversion of crosshole georadar data. We first validate the accuracy of the algorithm with respect to the solutions for an infinitesimal electric dipole source as well as a wire-type dipole antenna and then apply it to explore the radiative properties of realistic antenna designs used in borehole georadar.

Realistic Modeling of Borehole Georadar Antenna Radiation

A New Vector Waveform Inversion Algorithm for Simultaneous Updating of Conductivity and Permittivity Parameters From Combination Crosshole/Borehole-to-Surface GPR Data (vol 48, pg 3391, 2010)

J. R. Ernst

Papers

Application of a new 2D time-domain full-waveform inversion scheme to crosshole radar data

Realistic FDTD modelling of borehole georadar antenna radiation: methodolgy and application

Waveform Inversion of Crosshole Georadar Data: Influence of Source Wavelet Variability and the Suitability of a Single Wavelet Assumption

Realistic Modeling of Borehole Georadar Antenna Radiation

A New Vector Waveform Inversion Algorithm for Simultaneous Updating of Conductivity and Permittivity Parameters From Combination Crosshole/Borehole-to-Surface GPR Data (vol 48, pg 3391, 2010)