Model For Full Waveform Inversion Research Articles

Efficient snapshot-free reverse time migration and computation of multiparameter gradients in full-waveform inversion

Computing images in reverse time migration and model parameter gradients from adjoint wavefields in full-waveform inversion (FWI) requires the correlation of a forward-propagated wavefield with another reverse-propagated wavefield. Although in theory only two wavefield propagations are required, one forward propagation and one reverse propagation, it requires storing the forward-propagated wavefield as a function of time to carry out the correlations, which is associated with significant input/output (I/O) cost. Alternatively, three wavefield propagations can be carried out to reverse propagate the forward-propagated wavefield in tandem with the reverse-propagated wavefield. We have determined how highly accurate reverse time migrated images and FWI model parameter gradients for anisotropic elastic FWI can be efficiently computed without significant disk I/O using two wavefield propagations by means of the principle of superposition.

SPiRaL: a multiresolution global tomography model of seismic wave speeds and radial anisotropy variations in the crust and mantle

SUMMARYSPiRaL is a joint global-scale model of wave speeds (P and S) and anisotropy (vertical transverse isotropy, VTI) variations in the crust and mantle. The model is comprised of >2.1 million nodes with five parameters at each node that capture velocity variations for P- and S-waves travelling at arbitrary directions in transversely isotropic media with a vertical symmetry axis (VTI). The crust (including ice, water, sediments and crystalline layers) is directly incorporated into the model. The default node spacing is approximately 2° in the lower mantle and 1° in the crust and upper mantle. The grid is refined with ∼0.25° minimum node spacing in highly sampled regions of the crust and upper mantle throughout North America and Eurasia. The data considered in the construction of SPiRaL includes millions of body wave traveltimes (crustal, regional and teleseismic phases with multiples) and surface wave (Rayleigh and Love) dispersion. A multiresolution inversion approach is employed to capture long-wavelength heterogeneities commonly depicted in global-scale tomography images as well as more localized details that are typically resolved in more focused regional-scale studies. Our previous work has demonstrated that such global-scale models with regional-scale detail can accurately predict both teleseismic and regional body wave traveltimes, which is necessary for more accurate location of small seismic events that may have limited signal at teleseismic distances. SPiRaL was constructed to predict traveltimes for event location and long-period waveform dispersion for seismic source inversion applications in regions without sufficiently tuned models. SPiRaL may also serve as a starting model for full-waveform inversion (FWI) with the goal of fitting waves with periods 10–50 s over multiple broad regions (thousands of kilometres) and potentially the globe. To gain insight to this possibility, we simulated waveforms for a small set of events using SPiRaL and independent waveform-based models for comparison. For the events tested, the performance of the traveltime-based SPiRaL model is shown to be generally on par with regional 3-D waveform-based models in three regions (western United States, Middle East, Korean Peninsula) suggesting SPiRaL may serve as a starting model for FWI over broad regions.

Joint wave-equation traveltime inversion of diving/direct and reflected waves for P- and S-wave velocity macromodel building

Full-waveform inversion (FWI) suffers from the local minima problem and requires a sufficiently accurate starting model to converge to the correct solution. Wave-equation traveltime inversion (WETI) is an effective tool to retrieve the long-wavelength components of the velocity model. We have developed a joint diving/direct and reflected wave WETI (JDRWETI) method to build P- and S-wave velocity macromodels. We estimate the traveltime shifts of seismic events (diving/direct waves and PP- and PS-reflections) through the dynamic warping scheme and construct a misfit function using the time shifts of diving/direct and reflected waves. We derive the adjoint wave equations and the gradients with respect to the background models based on the joint misfit function. We apply the kernel decomposition scheme to extract the kernel of the diving/direct wave and the tomography kernels of PP- and PS-reflections. For an explosive source, the kernels of the diving/direct wave and PP-reflections and the kernel of the PS-reflections are used to compute the P- and S-wave gradients of the background models, respectively. We implement JDRWETI by a two-stage inversion workflow: First, we invert the P- and S-wave velocity models using the P-wave gradients, and then we improve the S-wave velocity model using the S-wave gradients. Numerical tests on synthetic and field data sets reveal that the JDRWETI method successfully recovers the long-wavelength components of P- and S-wave velocity models, which can be used for an initial model for the subsequent elastic FWI. Moreover, the JDRWETI method prevails over the existing reflection WETI method and the cascaded diving/direct and reflected wave WETI method, especially when large velocity errors are present in the shallow part of the starting models. The JDRWETI method with the two-stage inversion workflow can give rise to reasonable inversion results even for the model with different P- and S-wave velocity structures.

The impact of acquisition geometry on full-waveform inversion updates

Full-waveform inversion (FWI) is a high-resolution model-building technique that uses the entire recorded seismic data content to build the earth model. Conventional FWI usually utilizes diving and refracted waves to update the low-wavenumber components of the velocity model. However, updates are often depth limited due to the limited offset range of the acquisition design. To extend conventional FWI beyond the limits imposed by using only transmitted energy, we must utilize the full acquired wavefield. Analyzing FWI kernels for a given geology and acquisition geometry can provide information on how to optimize the acquisition so that FWI is able to update the velocity model for targets as deep as basement level. Recent long-offset ocean-bottom node acquisition helped FWI succeed, but we would also like to be able to utilize the shorter-offset data from wide-azimuth data acquisitions to improve imaging of these data sets by developing the velocity field with FWI. FWI models are heading toward higher and higher wavenumbers, which allows us to extract pseudoreflectivity directly from the developed velocity model built with the acoustic full wavefield. This is an extremely early start to obtaining a depth image that one would usually produce in much later processing stages.

Full-Waveform Inversion for Imaging Faulted Structures: A Case Study from the Japan Trench Forearc Slope

Full-waveform inversion (FWI) of limited-offset marine seismic data is a challenging task due to the lack of refracted energy and diving waves from the shallow sediments, which are fundamentally required to update the long-wavelength background velocity model in a tomographic fashion. When these events are absent, a reliable initial velocity model is necessary to ensure that the observed and simulated waveforms kinematically fit within an error of less than half a wavelength to protect the FWI iterative local optimization scheme from cycle skipping. We use a migration-based velocity analysis (MVA) method, including a combination of the layer-stripping approach and iterations of Kirchhoff prestack depth migration (KPSDM), to build an accurate initial velocity model for the FWI application on 2D seismic data with a maximum offset of 5.8 km. The data are acquired in the Japan Trench subduction zone, and we focus on the area where the shallow sediments overlying a highly reflective basement on top of the Cretaceous erosional unconformity are severely faulted and deformed. Despite the limited offsets available in the seismic data, our carefully designed workflow for data preconditioning, initial model building, and waveform inversion provides a velocity model that could improve the depth images down to almost 3.5 km. We present several quality control measures to assess the reliability of the resulting FWI model, including ray path illuminations, sensitivity kernels, reverse time migration (RTM) images, and KPSDM common image gathers. A direct comparison between the FWI and MVA velocity profiles reveals a sharp boundary at the Cretaceous basement interface, a feature that could not be observed in the MVA velocity model. The normal faults caused by the basal erosion of the upper plate in the study area reach the seafloor with evident subsidence of the shallow strata, implying that the faults are active.

Reflection Full Waveform Inversion with Decoupled Elastic-wave Equations in Inhomogeneous Medium

Reflection full-waveform inversion (RFWI) can reduce the nonlinearity of inversion providing an accurate initial velocity model for full-waveform inversion (FWI) through the tomographic components (low-wavenumber). However, elastic-wave reflection full-waveform inversion (ERFWI) is more vulnerable to the problem of local minimum due to the complicated multi-component wavefield. Our algorithm first divides kernels of ERFWI into four subkernels based on the theory of decoupled elastic-wave equations. Then we try to construct the tomographic components of ERFWI with only single-component wavefields, similarly to acoustic inversions. However, the S-wave velocity is still vulnerable to the coupling effects because of P-wave components contained in the S-wavefield in an inhomogeneous medium. Therefore we develop a method for decoupling elastic- wave equations in an inhomogeneous medium, which is applied to the decomposition of kernels in ERFWI. The new decoupled system can improve the accuracy of S-wavefield and hence further reduces the high-wavenumber crosstalk in the subkernel of S-wave velocity after kernels are decomposed. The numerical examples of Sigsbee2A model demonstrate that our ERFWI method with decoupled elastic-wave equations can efficiently recover the low-wavenumber components of the model and improve the precision of the S-wave velocity.

Correlative Full-Intensity Waveform Inversion

Full-waveform inversion (FWI) is considered an effective technique for building high-resolution velocity models by fitting observed seismology waveforms. When the observed waveforms lack low frequencies or when the starting model is dissimilar to the true model, FWI usually suffers from cycle-skipping problems. To mitigate this difficulty, we propose a new correlative objective function that matches the phase differences between the seismic-waveform intensities to provide a good starting model. The waveform intensity separates the frequency band of the original data into an ultralow-frequency part and a higher frequency part, even when the original data lacking in low-frequency information. As the low-frequency part of the intensity is less prone to the cycle-skipping problem, it can be used to build an initial model for FWI. Furthermore, the source wavelets, in practice, are estimated from the observed data, which may be inaccurate in amplitude, phase shift, and time delay. To mitigate the inaccurate-source problem, a Wiener filter is constructed to build a source-independent objective function. Applications to the Marmousi model and Chevron blind-test model demonstrate that the proposed method converges to an acceptable initial model for conventional FWI. It is highly efficient and insensitive to inaccurate source wavelets, including inaccurate amplitude, phase shift, and time delays.

Wave-equation migration velocity analysis in the common-offset domain to avoid cycle-skipping for FWI

Full waveform inversion (FWI) has been used successfully to build high-resolution acoustic velocity models for imaging but can fail when the starting model is far from accurate because the absence of low frequency information can lead to conversion to local minima. Wave-equation based migration velocity analysis (WEMVA), is an iterative method that has successfully been used to provide low-frequency starting models for full waveform inversion (FWI) and avoid cycle-skipping. In this work, we extend WEMVA to the common-offset domain by minimizing an image residual in the common-offset domain, which can be formed after either reverse-time migration (RTM) or least-squares migration (LSM). Although very expensive in comparison to RTM, LSM has an increased amplitude fidelity that helps to ensure that the image residual used in WEMVA is largely a function of moveout instead of errors in amplitude, thus increasing the likelihood of converging to a global minimum. We demonstrate the method, theory and results on three synthetic models of increasing complexity.

Seismic inversion by Newtonian machine learning

We present a wave-equation inversion method that inverts skeletonized seismic data for the subsurface velocity model. The skeletonized representation of the seismic traces consists of the low-rank latent-space variables predicted by a well-trained autoencoder neural network. The input to the autoencoder consists of seismic traces, and the implicit function theorem is used to determine the Fréchet derivative, i.e., the perturbation of the skeletonized data with respect to the velocity perturbation. The gradient is computed by migrating the shifted observed traces weighted by the skeletonized data residual, and the final velocity model is the one that best predicts the observed latent-space parameters. We denote this as inversion by Newtonian machine learning (NML) because it inverts for the model parameters by combining the forward and backward modeling of Newtonian wave propagation with the dimensional reduction capability of machine learning. Empirical results suggest that inversion by NML can sometimes mitigate the cycle-skipping problem of conventional full-waveform inversion (FWI). Numerical tests with synthetic and field data demonstrate the success of NML inversion in recovering a low-wavenumber approximation to the subsurface velocity model. The advantage of this method over other skeletonized data methods is that no manual picking of important features is required because the skeletal data are automatically selected by the autoencoder. The disadvantage is that the inverted velocity model has less resolution compared with the FWI result, but it can serve as a good initial model for FWI. Our most significant contribution is that we provide a general framework for using wave-equation inversion to invert skeletal data generated by any type of neural network. In other words, we have combined the deterministic modeling of Newtonian physics and the pattern matching capabilities of machine learning to invert seismic data by NML.

The Kefalonia Transform Fault: A STEP fault in the making

Vertical edges along subducted slabs have been recognized in the majority of subduction zones. Surprisingly, slab edges evolved into Subduction-Transform-Edge-Propagator (STEP) faults in only a few regions; the conditions under which STEPs form are special. It is relevant to constrain the conditions that facilitate STEP fault initiation because they leave a clear geological footprint in the overriding plate, whereas vertical tears generally do not. We therefore study a candidate region for STEP fault initiation in the western Hellenic Subduction Zone.We investigate the structure and seismicity of the shallow western Hellenic Subduction Zone using a recent full-waveform inversion model which both captures details of crustal and upper-mantle structure, yielding constraints in the depth interval from 10 to 200 km where lithosphere-mantle interactions have tectonic expressions. The western end of the Hellenic Subduction Zone is fragmented near the Kefalonia Transform Fault. We identify a separate Epirus lithospheric fragment that is roughly vertical below the southern Albanides. We also identify a new and major contrast within the lithospheric mantle of the Ionian ocean basin, which aligns with a gradient in free-air anomalies.In the overriding plate, the Kefalonia Transform Fault zone accommodates right lateral strike-slip deformation. We interpret this fault zone as a proto-STEP fault that formed simultaneously with Pliocene fragmentation of the Epirus fragment.Comparing the recent evolution of the NW Hellenic slab edge with currently active STEPs indicates that along-trench variations in convergence velocity are a prerequisite for STEP fault initiation. Such velocity variations may result from subduction of continental crust along part of the trench. Resistance to sea ward tear propagation by a mechanically strong subducting plate may prevent variations in convergence velocity to occur, and thus STEP fault initiation. The amount of time over which a velocity contrast persists will also be relevant for STEP fault initiation. Mechanical coupling between upper- and lower-plate, and the deformability of the upper-plate appear to also play a role in the initiation of the STEP fault once a slab is fragmented.

Bayesian trans-dimensional full waveform inversion: synthetic and field data application

SUMMARYSeismic full waveform inversion (FWI) is a state-of-the-art technique for estimating subsurface physical models from recorded seismic waveform, but its application requires care because of high non-linearity and non-uniqueness. The final outcome of global convergence from conventional FWI using local gradient information relies on an informative starting model. Bayesian inference using Markov chain Monte Carlo (MCMC) sampling is able to remove such dependence, by a direct extensive search of the model space. We use a Bayesian trans-dimensional MCMC seismic FWI method with a parsimonious dipping layer parametrization, to invert for subsurface velocity models from pre-stack seismic shot gathers that contain mainly reflections. For the synthetic study, we use a simple four-layer model and a modified Marmousi model. A recently collected multichannel off-shore seismic reflection data set, from the Lord Howe Rise (LHR) in the east of Australia, is used for the field data test. The trans-dimensional FWI method is able to provide model ensembles for describing posterior distribution, when the dipping-layer model assumption satisfies the observed data. The model assumption requires narrow models, thus only near-offset data to be used. We use model stitching with lateral and depth constraints to create larger 2-D models from many adjacent overlapping submodel inversions. The inverted 2-D velocity model from the Bayesian inference can then be used as a starting model for the gradient-based FWI, from which we are able to obtain high-resolution subsurface velocity models, as demonstrated using the synthetic data. However, lacking far-offset data limits the constraints for the low-wavenumber part of the velocity model, making the inversion highly non-unique. We found it challenging to apply the dipping-layer based Bayesian FWI to the field data. The approximations in the source wavelet and forward modelling physics increase the multimodality of the posterior distribution; the sampled velocity models clearly show the trade-off between interface depth and velocity. Numerical examples using the synthetic and field data indicate that trans-dimensional FWI has the potential for inverting earth models from reflection waveform. However, a sparse model parametrization and far offset constraints are required, especially for field application.

Convolution-based multi-scale envelope inversion

Envelope inversion (EI) is an efficient tool to mitigate the nonlinearity of conventional full waveform inversion (FWI) by utilizing the ultralow-frequency component in the seismic data. However, the performance of envelope inversion depends on the frequency component and initial model to some extent. To improve the convergence ability and avoid the local minima issue, we propose a convolution-based envelope inversion method to update the low-wavenumber component of the velocity model. Besides, the multi-scale inversion strategy (MCEI) is also incorporated to improve the inversion accuracy while guaranteeing the global convergence. The success of this method relies on modifying the original envelope data to expand the overlap region between observed and modeled envelope data, which in turn expands the global minimum basin of misfit function. The accurate low-wavenumber component of the velocity model provided by MCEI can be used as the migration model or an initial model for conventional FWI. The numerical tests on simple layer model and complex BP 2004 model verify that the proposed method is more robust than EI even when the initial model is coarse and the frequency component of data is high.

Velocity-attenuation model from check-shot drift trends in North Sea well data

Seismic attenuation distorts phase and narrows bandwidth in seismic surveys. It is also an exploration attribute, as, for example, gas or overpressure, may create attenuation anomalies. Compensating attenuation in imaging requires accurate models. Detailed attenuation models may be obtained using full-waveform inversion (FWI) or attenuation tomography, but their accuracy benefits from reliable starting models and/or constraints. Seismic attenuation and velocity dispersion are necessarily linked for causal linear wave propagation such that higher frequencies travel faster than lower frequencies in an attenuative medium. In publicly released well data from the Norwegian North Sea, we have observed systematic positive linear trends in check-shot drift when comparing (lower frequency) time-depth curves with (higher frequency) integrated sonic transit times. We observe velocity dispersion consistent with layers having constant seismic attenuation. Adapting a previously published method, and assuming an attenuation-dispersion relationship, we use drift gradients, measured over thick stratigraphic units, to estimate interval P-wave attenuation and tentatively interpret its variation in terms of porosity and fluid mobility. Reflectivity modeling predicts a very low attenuation contribution from peg-leg multiples. We use the attenuation values to develop a simple regional relationship between P-wave velocity and attenuation. Observed low drift gradients in some shallower units lead to an arch-shaped model that predicts low attenuation at both low and high velocities. The attenuation estimates were broadly comparable with published effective attenuation values obtained independently nearby. This general methodology for quickly deriving a regional velocity-attenuation relationship could be used anywhere that coincident velocity models are available at seismic and sonic frequencies. Such relationships can be used for fast derivation (from velocities) of starting attenuation models for FWI or tomography, constraining or linking velocity and attenuation in inversion, deriving models for attenuation compensation in time processing, or deriving background trends in screening for attenuation anomalies in exploration.

The effectiveness of a pseudo‐inverse extended Born operator to handle lateral heterogeneity for imaging and velocity analysis applications

ABSTRACTWave equation–based migration velocity analysis techniques aim to construct a kinematically accurate velocity model for imaging or as an initial model for full waveform inversion applications. The most popular wave equation–based migration velocity analysis method is differential semblance optimization, where the velocity model is iteratively updated by minimizing the unfocused energy in an extended image volume. However, differential semblance optimization suffers from artefacts, courtesy of the adjoint operator used in imaging, leading to poor convergence. Recent findings show that true amplitude imaging plays a significant role in enhancing the differential semblance optimization's gradient and reducing the artefacts. Here, we focus on a pseudo‐inverse operator to the horizontally extended Born as a true amplitude imaging operator. For laterally inhomogeneous models, the operator required a derivative with respect to a vertical shift. Extending the image vertically to evaluate such a derivative is costly and impractical. The inverse operator can be simplified in laterally homogeneous models. We derive an extension of the approach to apply the full inverse formula and evaluate the derivative efficiently. We simplified the implementation by applying the derivative to the imaging condition and utilize the relationship between the source and receiver wavefields and the vertical shift. Specifically, we verify the effectiveness of the approach using the Marmousi model and show that the term required for the lateral inhomogeneity treatment has a relatively small impact on the results for many cases. We then apply the operator in differential semblance optimization and invert for an accurate macro‐velocity model, which can serve as an initial velocity model for full waveform inversion.

Frequency-dependent reflection wave-equation traveltime inversion from walkaway vertical seismic profile data

To accurately image the geologic structures from walkaway vertical seismic profile (VSP) data, it is necessary to estimate the subsurface velocity field with high resolution and enhanced illumination of deep reservoirs. Because full-waveform inversion (FWI) suffers from cycle skipping when the initial model is far from the true model, we have adopted frequency-dependent reflection wave-equation traveltime inversion (FRWT) to generate the background velocity model for VSP migration. The upgoing reflection data are separated from the original shot gathers, and dynamic warping is used to evaluate the traveltime differences between the observed data and the calculated data. Different frequency bands of the data are inverted in sequence to reconstruct the velocity model with higher resolution. We also implement wavefield decomposition on the gradient field to extract the contributions of reflection components and improve the updated model. The inverted results obtained from FRWT can be used as the initial model for conventional FWI or the velocity model for reverse time migration. The experiments on synthetic data and field data demonstrate that our approach can effectively recover the background velocity model from walkaway VSP data.

Parametric convolutional neural network-domain full-waveform inversion

Most current full-waveform inversion (FWI) algorithms minimize the data residuals to estimate a velocity model based on the assumption that the updated model is the sum of a background model and an estimated model perturbation. We have performed reparameterization of the initial velocity model, by the weights in a convolutional neural network (CNN), to automatically capture the salient features in the initial model, as a priori information. The prior information in CNN weights is iteratively updated as regularization to constrain the CNN-domain inversion to refine the features captured in CNN pretraining by reducing the data misfit. Synthetic examples using a 1D increasing velocity function v(z) and a 2D smoothed version of the correct Marmousi2 model as initial models indicate that the performance of the CNN-domain FWI depends on the existence and accuracy of the prior information in the initial velocity model (i.e., whether features whose positions, shapes, and values are present in the correct model are approximately included in the initial model). Forty different sets of randomly initialized CNN weights are used to parameterize and test CNN-domain FWI, using a 2D smoothed Sigsbee model as the initial velocity model. All 40 sets invert for the Sigsbee salt body more accurately (with a smaller standard deviation of the final rms model errors), by CNN-domain FWI, than FWI does. Features that are not represented within the CNN hidden layers in the initial velocity model, and so cannot be recovered by CNN-domain FWI, can be recovered using the final CNN-domain FWI velocity model as the starting model in a subsequent conventional FWI.

Bayesian transdimensional seismic full-waveform inversion with a dipping layer parameterization

Seismic full-waveform inversion (FWI) has become a popular tool for estimating subsurface models using the amplitude and phase of seismograms. Unlike the conventional gradient-based approach, Bayesian inference using Markov chain Monte Carlo (MCMC) sampling can remove dependence on starting models and can quantify uncertainty. We have developed a Bayesian transdimensional (trans-d) MCMC seismic FWI method for estimating dipping-layer velocity models, in which the number of layers is unknown. A time-domain staggered-grid finite-difference wave equation solver is used for forward modeling. The FWI and MCMC methods are known to be computationally expensive. Two strategies are used to get practical computational performance. A layer-stripping strategy is used to accelerate sampler convergence, and a parsimonious dipping layer parameterization is used so that the MCMC algorithm can search broadly with fewer iterations. The parameters for each layer are velocity, thickness, and lower interface dip angle. We find that this parameterization has sufficient flexibility to invert for narrow 2D velocity models using small offset data. Model stitching is then used to bring several such inversions together to create larger 2D models. In turn, these can be used as starting models for gradient-based adjoint FWI to image complicated geologic settings. Two synthetic 2D numerical examples, including the Marmousi model, are considered, using seismic data dominated by reflections. Creation of good starting models traditionally requires significant human effort. We determine how much of that effort can be substituted with computation.

Full-waveform inversion of salt models using shape optimization and simulated annealing

A good starting model is imperative in full-waveform inversion (FWI) because it solves a least-squares inversion problem using a local gradient-based optimization method. A suboptimal starting model can result in cycle skipping leading to poor convergence and incorrect estimation of subsurface properties. This problem is especially crucial for salt models because the strong velocity contrasts create substantial time shifts in the modeled seismogram. Incorrect estimation of salt bodies leads to velocity inaccuracies in the sediments because the least-squares gradient aims to reduce traveltime differences without considering the sharp velocity jump between sediments and salt. We have developed a technique to estimate velocity models containing salt bodies using a combination of global and local optimization techniques. To stabilize the global optimization algorithm and keep it computationally tractable, we reduce the number of model parameters by using sparse parameterization formulations. The sparse formulation represents sediments using a set of interfaces and velocities across them, whereas a set of ellipses represents the salt body. We use very fast simulated annealing (VFSA) to minimize the misfit between the observed and synthetic data and estimate an optimal model in the sparsely parameterized space. The VFSA inverted model is then used as a starting model in FWI in which the sediments and salt body are updated in the least-squares sense. We partition model updates into sediment and salt updates in which the sediments are updated like conventional FWI, whereas the shape of the salt is updated by taking the zero crossing of an evolving level set surface. Our algorithm is tested on two 2D synthetic salt models, namely, the Sigsbee 2A model and a modified SEG Advanced Modeling Program (SEAM) Phase I model while fixing the top of the salt. We determine the efficiency of the VFSA inversion and imaging improvements from the level set FWI approach and evaluate a few sources of uncertainty in the estimation of salt shapes.

Two-grid full-waveform Rayleigh-wave inversion via a genetic algorithm — Part 2: Application to two actual data sets

We have applied our two-grid genetic-algorithm Rayleigh-wave full-waveform inversion (FWI) to two actual data sets acquired in Luni (Italy) and Grenoble (France), respectively. Because our technique used 2D elastic finite-difference modeling for solving the forward problem, the observed data were 3D to 2D corrected prior to the inversion. To limit the computing time, both inversions focused on predicting low-resolution, smooth models by using quite coarse inversion grids. The wavelets for FWI were estimated directly from the observed data by using the Wiener method. In the Luni case, due to the strong dispersion effects on the data, to strengthen the inversion, envelopes and waveforms were considered in the objective function and an offset-marching strategy was applied. Though no a priori information was exploited, the outcomes of the Luni and Grenoble data inversion were fair. The predicted Luni [Formula: see text] model indicates a strong velocity increase from approximately 3 to 6 m, and velocity inversions have been detected at approximately 2 and 9 m depths. Analyzing the dispersion spectra, it results that the predicted Luni data reasonably reproduced the waveforms related to the fundamental mode and, likely, a small part of those related to the first higher mode. Concerning the Grenoble example, the predicted [Formula: see text] model coincides reasonably well with the long-wavelength structures presented in the [Formula: see text] profiles obtained from nearby boreholes. The data reconstruction is generally satisfactory, and when mismatches occur between the predicted and observed traces, the phase differences are always within half-periods. The fair inversion outcomes suggest that the predicted Luni and Grenoble models would likely be adequate initial models for local FWI, which could further increase the resolution and the details of the estimated [Formula: see text] models.

Application of Laplace Domain Waveform Inversion to Cross-Hole Radar Data

Full waveform inversion (FWI) can yield high resolution images and has been applied in Ground Penetrating Radar (GPR) for around 20 years. However, appropriate selection of the initial models is important in FWI because such an inversion is highly nonlinear. The conventional way to obtain the initial models for GPR FWI is ray-based tomogram inversion which suffers from several inherent shortcomings. In this paper, we develop a Laplace domain waveform inversion to obtain initial models for the time domain FWI. The gradient expression of the Laplace domain waveform inversion is deduced via the derivation of a logarithmic object function. Permittivity and conductivity are updated by using the conjugate gradient method. Using synthetic examples, we found that the value of the damping constant in the inversion cannot be too large or too small compared to the dominant frequency of the radar data. The synthetic examples demonstrate that the Laplace domain waveform inversion provide slightly better initial models for the time domain FWI than the ray-based inversion. Finally, we successfully applied the algorithm to one field data set, and the inverted results of the Laplace-based FWI show more details than that of the ray-based FWI.