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

Quantitative imaging of the elastic properties of the subsurface at depth is essential for civil engineering applications and oil- and gas-reservoir characterization. A realistic synthetic example provides for an assessment of the potential and limits of 2D elastic full-waveform inversionFWIof wide-aperture seismic data for recovering high-resolution P- and S-wave velocity models of complex onshore structures. FWI of land data is challengingbecauseoftheincreasednonlinearityintroducedbyfreesurface effects such as the propagation of surface waves in the heterogeneous near-surface. Moreover, the short wavelengths of the shear wavefield require an accurate S-wave velocity starting modeliflowfrequenciesareunavailableinthedata.Weevaluated different multiscale strategies with the aim of mitigating the nonlinearities. Massively parallel full-waveform inversion was implemented in the frequency domain. The numerical optimization relies on a limited-memory quasi-Newton algorithm that outperforms the more classic preconditioned conjugate-gradient algorithm. The forward problem is based upon a discontinuous Galerkin DG method on triangular mesh, which allows accuratemodelingoffree-surfaceeffects.Sequentialinversionsofincreasingfrequenciesdefinethemostnaturallevelofhierarchyin multiscale imaging. In the case of land data involving surface waves, the regularization introduced by hierarchical frequency inversions is not enough for adequate convergence of the inversion. A second level of hierarchy implemented with complexvalued frequencies is necessary and provides convergence of the inversion toward acceptable P- and S-wave velocity models. Amongthepossiblestrategiesforsamplingfrequenciesintheinversion, successive inversions of slightly overlapping frequency groups is the most reliable when compared to the more standard sequential inversion of single frequencies. This suggests that simultaneous inversion of multiple frequencies is critical when consideringcomplexwavephenomena.

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Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion

SUMMARY For the last 30 yr, since Tarantola’s pioneering theoretical study of waveform inversion, geophysicists and applied mathematicians have utilized a waveform inversion to delineate the earth’s structures. However, successful applications of waveform inversion to real data are nominal. The failures are mainly caused by the high non-linearity of the waveform inversion and the real data containing insufficient low-frequency components. We propose a waveform inversion algorithm that is robust and is not sensitive to the initial model by exploiting the wavefield in the Laplace domain and the adjoint property of the wave equation. The wavefield in the Laplace domain is equivalent to the zero frequency component of the damped wavefield. Therefore, the inversion of Poisson’s equation in electrical prospecting can be viewed as a waveform inversion problem, exploiting the zero frequency component of an undamped wavefield. Since our inversion algorithm in the Laplace domain is strongly associated with the nature of DC inversion for Poisson’s equation in electrical prospecting, our algorithm can generate a velocity model that is equivalent to a long-wavelength velocity model (a smooth velocity model). Through numerical tests, we found that the Laplace-transformed wavefields for optimally low Laplace damping constants are critically needed in our algorithm as the low-frequency components are necessary in the waveform inversion of the frequency domain. We note that the objective function by l2 norm of the logarithmic wavefield in the Laplace domain behaves as if it has no local minimum points in low and high Laplace damping constants. Moreover, we note that the forward modelling in the Laplace domain could be accurately calculated by using a coarser grid of several hundreds of metres than that of the frequency domain. Numerical tests of the salt dome model with high-velocity contrast demonstrate the robustness of our algorithm to both synthetic and real data. To apply our algorithm to real data more successfully, we need to improve the accuracy of the Laplace transformation for the wavefields in the time domain. However, our waveform inversion in the Laplace domain can be successfully applied to real data since our algorithm has more advantages in forward modelling and inversion than those in the frequency domain.

Waveform inversion in the Laplace domain

SUMMARY

An application of full-waveform tomography to dense onshore wide-aperture seismic data recorded in a complex geological setting (thrust belt) is presented.



The waveform modelling and tomography are implemented in the frequency domain. The modelling part is solved with a finite-difference method applied to the visco-acoustic wave equation. The inversion is based on a local gradient method. Only the P-wave velocity is involved in the inversion. The inversion is applied iteratively to discrete frequency components by proceeding from low to high frequencies. This defines a multiscale imaging in the sense that high wavenumbers are progressively incorporated in images. The linearized waveform tomography requires an accurate starting velocity model that has been developed by first-arrival traveltime tomography.



After specific pre-processing of the data, 16 frequency components ranging between 5.4 and 20 Hz were inverted. Ten iterations were computed per frequency component leading to 160 tomographic models. The waveform tomography has successfully imaged southwest-dipping structures previously identified from other geophysical data as being associated with high-resistivity bodies. The relevance of the tomographic images is locally demonstrated by comparison of a velocity–depth function extracted from the waveform tomography models with a coincident vertical seismic profiling (VSP) log available on the profile. Moreover, comparison between observed and synthetic seismograms computed in the (starting) traveltime and waveform tomography models demonstrates unambiguously that the waveform tomography successfully predicts for wide-angle reflections from southwest-dipping geological structures.



This study demonstrates that the combination of first-arrival traveltime and frequency-domain full-waveform tomographies applied to dense wide-aperture seismic data is a promising approach to quantitative imaging of complex geological structures. Indeed, wide-aperture acquisition geometries offer the opportunity to develop an accurate background velocity model for the subsequent waveform tomography. This is critical, because the building of the macromodel remains an open question when only near-vertical reflection data are considered.

/pdf/multiscale-imaging-of-complex-structures-from-multifold-wide-4qt74jyimb.pdf

Multiscale imaging of complex structures from multifold wide-aperture seismic data by frequency-domain full-waveform tomography: application to a thrust belt

Although waveform inversion has been studied extensively since its beginning 20 years ago, applications to seismic field data have been limited, and most of those applications have been for global-seismology- or engineering-seismology-scale problems, not for exploration-scale data. As an alternative to classical waveform inversion, we propose the use of a new, objective function constructed by taking the logarithm of wavefields, allowing consideration of three types of objective function, namely, amplitude only, phase only, or both. In our waveform inversion, we estimate the source signature as well as the velocity structure by including functions of amplitudes and phases of the source signature in the objective function. We compute the steepest-descent directions by using a matrix formalism derived from a frequency-domain, finite-element/finite-difference modeling technique. Our numerical algorithms are similar to those of reverse-time migration and waveform inversion based on the adjoint state of the wave equation. In order to demonstrate the practical applicability of our algorithm, we use a synthetic data set from the Marmousi model and seismic data collected from the Korean continental shelf. For noisefree synthetic data, the velocity structure produced by our inversion algorithm is closer to the true velocity structure than that obtained with conventional waveform inversion. When random noise is added, the inverted velocity model is also close to the true Marmousi model, but when frequencies below 5 Hz are removed from the data, the velocity structure is not as good as those for the noise-free and noisy data. For field data, we compare the time-domain synthetic seismograms generated for the velocity model inverted by our algorithm with real seismograms and find that the results show that our inversion algorithm reveals short-period features of the subsurface. Although we use wrapped phases in our examples, we still obtain reasonable results. We expect that if we were to use correctly unwrapped phases in the inversion algorithm, we would obtain better results.

/pdf/waveform-inversion-using-a-logarithmic-wavefield-2coiak7dam.pdf

Waveform inversion using a logarithmic wavefield

Linearized inversion of surface seismic data for a model of the earth’s subsurface requires estimating the sensitivity of the seismic response to perturbations in the earth’s subsurface. This sensitivity, or Jacobian, matrix is usually quite expensive to estimate for all but the simplest model parameterizations. We exploit the numerical structure of the finite‐element method, modern sparse matrix technology, and source‐receiver reciprocity to develop an algorithm that explicitly calculates the Jacobian matrix at only the cost of a forward model solution. Furthermore, we show that we can achieve improved subsurface images using only one inversion iteration through proper scaling of the image by a diagonal approximation of the Hessian matrix, as predicted by the classical Gauss‐Newton method. Our method is applicable to the full suite of wave scattering problems amenable to finite‐element forward modeling. We demonstrate our method through some simple 2‐D synthetic examples.

/pdf/efficient-calculation-of-a-partial-derivative-wavefield-17eprwv474.pdf

Efficient calculation of a partial-derivative wavefield using reciprocity for seismic imaging and inversion

Because of its computational efficiency, prestack Kirchhoff depth migration remains the method of choice for all but the most complicated geological depth structures. Further improvement in computational speed and amplitude estimation will allow us to use such technology more routinely and generate better images. To this end, we developed a new, accurate, and economical algorithm to calculate first-arrival traveltimes and amplitudes for an arbitrarily complex earth model. Our method is based on numerical solutions of the wave equation obtained by using well-established finite-difference or finite-element modeling algorithms in the Laplace domain, where a damping term is naturally incorporated in the wave equation. We show that solving the strongly damped wave equation is equivalent to solving the eikonal and transport equations simultaneously at a fixed reference frequency, which properly accounts for caustics and other problems encountered in ray theory. Using our algorithm, we can easily calculate first-arrival traveltimes for given models. We present numerical examples for 2-D acoustic models having irregular topography and complex geological structure using a finite-element modeling code.

/pdf/traveltime-and-amplitude-calculations-using-the-damped-wave-2c8j7ahjbm.pdf

Traveltime and amplitude calculations using the damped wave solution

This work was financially supported by the National Laboratory Project
of the Ministry of Science and Technology, Brain Korea 21 Project of the
Ministry of Education, grant No. R05-2000-00003 from the Basic Research
Program of the Korea Science & Engineering Foundation, and grant No.
PM10300 from the Korea Ocean Research & Development Institute.

/pdf/prestack-depth-migration-using-straight-ray-technique-srt-m2rsw60xdq.pdf

Harry Y. Lim

Papers

Efficient calculation of a partial-derivative wavefield using reciprocity for seismic imaging and inversion

Traveltime and amplitude calculations using the damped wave solution

Prestack depth migration using straight ray technique(SRT)