Gradient-based Full-waveform Inversion Research Articles

Subsoil density field reconstruction through 3-D FWI: a systematic comparison between vertical- and horizontal-force seismic sources

SUMMARY Bulk-density (ρ) of soil is an important indicator of soil compaction and type. A knowledge of the spatial variability of in situ soil density is important in geotechnical engineering, hydrology and agriculture. Surface geophysical methods have so far shown limited success in providing an accurate and high-resolution image of 3-D soil-density distribution. In this pursuit, 3-D seismic full-waveform inversion (FWI) is promising, provided the robustness and accuracy of density inversion via this approach can be established in the near-surface scale. However, simultaneous reconstruction of ρ and seismic wave velocities through multiparameter FWI remains a challenging task. Near-surface seismic data are commonly dominated by dispersive surface waves whose velocities are controlled by the value and distribution of shear-wave velocity (VS). One major difficulty in estimating reliably ρ from near-surface seismic data is due to the relatively low sensitivity of the seismic wavefield to ρ compared to seismic velocities. Additionally, the accuracy of the estimated ρ decreases due to error in VS—an issue known as parameter coupling. Parameter coupling makes it difficult to estimate accurately ρ within the framework of conventional gradient-based FWI. More sophisticated optimization approaches (e.g. truncated Newton) can reduce the effect of parameter coupling, but these approaches are commonly not affordable in near-surface applications due to heavy computational burden. In this research, we have investigated how choosing correctly the force direction of the seismic source can contribute to a higher accuracy of ρ estimates through 3-D FWI. Using scattered wavefields, the Hessian, and inversion tests, an in-depth and systematic investigation of data sets corresponding to different force directions has been carried out. A comparison of the scattered wavefields due to a point-localized ρ perturbation for different force directions shows the robustness of the horizontal-force data set to noise compared to the vertical-force data set. Furthermore, for a point-scatterer model, an analysis of the gradients of the misfit function using the Hessian shows that utilizing a horizontal-force source enables one to reconstruct the high-resolution gradient with relatively small parameter coupling. Finally, inversion tests for two different subsoil models demonstrate that 3-D FWI on a horizontal-force-source seismic data set is capable of providing a more accurate 3-D ρ distribution in soil compared to a vertical-force-source data set. Our results show that the use of a horizontal-force source might allow avoiding computationally demanding, costly optimization approaches in 3-D FWI.

A Rayleigh Wave Globally Optimal Full Waveform Inversion Framework Based on GPU Parallel Computing

Conventional gradient-based full waveform inversion (FWI) is a local optimization, which is highly dependent on the initial model and prone to trapping in local minima. Globally optimal FWI that can overcome this limitation is particularly attractive, but is currently limited by the huge amount of calculation. In this paper, we propose a globally optimal FWI framework based on GPU parallel computing, which greatly improves the efficiency, and is expected to make globally optimal FWI more widely used. In this framework, we simplify and recombine the model parameters, and optimize the model iteratively. Each iteration contains hundreds of individuals, each individual is independent of the other, and each individual contains forward modeling and cost function calculation. The framework is suitable for a variety of globally optimal algorithms, and we test the framework with particle swarm optimization algorithm for example. Both the synthetic and field examples achieve good results, indicating the effectiveness of the framework.

Love wave full-waveform inversion for archaeogeophysics: From synthesis tests to a field case

High precision detection of the shallow-surface structure is of great significance to geotechnical engineering and archaeological geophysical research. Surface wave Full waveform inversion (FWI) is attracting more and more attention in shallow seismic exploration. The FWI of Love waves is more straightforward and has a higher precision and resolution compared with Rayleigh waves. In the classic gradient-based FWI algorithm, the gradient amplitude does not scale with increasing depth due to the coupling effect and double scattering; thus, the inversion image cannot represent the significant small-scale characteristic structures in archaeogeophysics. To overcome the above limitations, a pre-conditioned FWI approach was presented, and a complete elastic Love waves FWI workflow was developed for archaeogeophysics. The superior performance of the developed method was verified through synthesis tests and a field experiment. The synthetic tests on a tomb model show that the pre-conditioned approach has a better parameter reconstruction capability, can characterize small-scale anomalies, improves the convergence rate, and has a strong anti-noise ability. In the field experiment, the pre-conditioned approach-based Love waves FWI workflow was adopted to invert the recorded Love waves datasets gathered in Xianyang City, China. The target of the survey was to determine the location and approximate size of a buried ancient tomb to provide reliable guidance for further excavation. The obtained FWI image revealed an underground profile with a depth of 9.5 m, and a low-velocity anomaly with a height of about 1.6 m and a length of about 4.7 m was delineated, which we identified as the tomb. The subsequent excavation confirmed this conclusion. The above results show that the complete Love wave FWI workflow developed has the ability to perform accurate imaging under the condition of few shots.

Seismic full-waveform inversion using minimization of virtual scattering sources

Full-waveform inversion (FWI) is a powerful tool for imaging underground structures with high resolution; however, this approach commonly suffers from the cycle-skipping issue. Recently, various FWI methods have been suggested to address this problem. Such methods are mainly classified into either data-space manipulation or model-space extension. We developed an alternative FWI method that belongs to the latter class. First, we define the virtual scattering source based on perturbation theory. The virtual scattering source is estimated by minimizing the differences between observed and simulated data with a regularization term penalizing the weighted virtual scattering source. The inverse problem for obtaining the virtual scattering source can be solved by the linear conjugate gradient method. The inverted virtual scattering source is used to update the wavefields; thus, it helps FWI to better approximate the nonlinearity of the inverse scattering problem. As the second step, the virtual scattering source is minimized to invert the velocity model. By assuming that the variation of the reconstructed wavefield is negligible, we can apply an approximated full Newton method to the velocity inversion with reasonable cost comparable to the Gauss-Newton method. From the numerical examples using synthetic data, we confirm that the proposed method performs better and more robust than the simple gradient-based FWI method. In addition, we show that our objective function has fewer local minima, which helps to mitigate the cycle-skipping problem.

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.

Elastic SH- and Love-wave Full-Waveform Inversion for shallow shear wave velocity with a preconditioned technique

In contrast to the Rayleigh waves, the system of SH- and Love-wave Full-Waveform Inversion (FWI) is simpler (Only S-wave velocity and density is inverted). However, the gradient is not scaled with the increasing depth in gradient-based FWI. The main reasons for the phenomenon are the limited frequency band of the wavelet, non-uniform coverage of the excitation-receiving points and the double scattering. The illumination can be enhanced by adding a preconditioned operator to the inversion. To compensate the illumination, we presented a preconditioned technique abiding by the theory of inverse scattering in SH- and Love-wave FWI. The synthetic tests demonstrate that the preconditioned technique can expedite the convergence rate and enhance the illumination intensity to the deeper medium compared with the conventional gradient-based FWI. The results also indicate that the preconditioned conjugate gradient (PCG) method has the best parameter reconstruction performance among the local optimization algorithms. Subsequently, we adopt the PCG method to inverse the field data acquired in the village of Čachtice, northwestern Slovakia. In addition, a multiscale strategy is employed and the Wiechert-Herglotz method is followed to estimate the initial S-wave velocity model in inversion. Eventually, this paper illustrates the credibility of the inverted result from the perspectives of waveform fitting, common-offset profile, signal similarity, relative error of channel energy and normalized misfit function values. This study primarily seeks to delineate the foundations of historical buildings under the survey line. According to the range of S-wave velocity, we assert that there are no foundation remains in the subsurface.

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.

Two-grid full-waveform Rayleigh-wave inversion via a genetic algorithm — Part 1: Method and synthetic examples

When reliable a priori information is not available, it is difficult to correctly predict near-surface S-wave velocity models from Rayleigh waves through existing techniques, especially in the case of complex geology. To tackle this issue, we have developed a new method: two-grid genetic-algorithm Rayleigh-wave full-waveform inversion (FWI). Adopting a two-grid parameterization of the model, the genetic algorithm inverts for unknown velocities and densities at the nodes of a coarse grid, whereas the forward modeling is performed on a fine grid to avoid numerical dispersion. A bilinear interpolation brings the coarse-grid results into the fine-grid models. The coarse inversion grid allows for a significant reduction in the computing time required by the genetic algorithm to converge. With a coarser grid, there are fewer unknowns and less required computing time, at the expense of the model resolution. To further increase efficiency, our inversion code can perform the optimization using an offset-marching strategy and/or a frequency-marching strategy that can make use of different kinds of objective functions and allows for parallel computing. We illustrate the effect of our inversion method using three synthetic examples with rather complex near-surface models. Although no a priori information was used in all three tests, the long-wavelength structures of the reference models were fairly predicted, and satisfactory matches between “observed” and predicted data were achieved. The fair predictions of the reference models suggest that the final models estimated by our genetic-algorithm FWI, which we call macromodels, would be suitable inputs to gradient-based Rayleigh-wave FWI for further refinement. We also explored other issues related to the practical use of the method in different work and explored applications of the method to field data.

Enhancing low-wavenumber components of full-waveform inversion using an improved wavefield decomposition method in the time-space domain

Full-waveform inversion (FWI) is a successful technique that attempts to build high-resolution velocity models by minimizing the residuals between observed and calculated data after a series of iterations. However, its successful application still relies on the quality of the estimated initial velocity model, as well as on the design of a suitable hierarchical workflow. If the initial model is far away from the true model, the widely used gradient-based FWI will inevitably get trapped into meaningless local minima. Given that the properties of the low-wavenumber components can help construct the initial model and overcome cycle skipping, we propose a low-wavenumber update FWI method based on the Hilbert transform called the hybrid method. This method provides a new form of FWI in the time-space domain that appears to be capable of enhancing the low-wavenumber updates. In this method, the updates of low wavenumbers and high wavenumbers are divided into two parts. In the first step, a gradient formula based on the Hilbert transform plays a key role in updating the low-wavenumber components. The gradient is formed by cross-correlation of upgoing and downgoing waves of source and residual wavefields. Instead of using the wavefield decomposition method in the frequency-wavenumber domain with high computational complexity, we apply a novel method to extract the upgoing and downgoing wavefields in the time-space domain. In addition to the acceptable computational burden, the memory is reduced greatly. The second step is the standard FWI, and its task is to update the high-wavenumber components. Numerical tests on an anomaly model and the Marmousi model demonstrate the effectiveness of the hybrid method for extracting low-wavenumber components even for noisy data or when low-frequency data are missing.

A two-step elastic full-waveform inversion applied to reflection seismic data for shallow hazard identification

We apply a two-step elastic full-waveform inversion (FWI) to well-site survey (WSS) marine seismic data to estimate high-resolution P-wave (Vp) and S-wave (Vs) velocity models. Our approach combines a first global, genetic-algorithm optimization and a subsequent gradient-based inversion. The broad-band frequency content of the available seismic data makes it possible to extend the frequency range considered in the inversion up to 70 Hz and thus to derive a high-resolution elastic characterization of the shallowest part of the subsurface. The lack of low frequencies and the limited maximum source-to-receiver offset of the WSS acquisition, make the GA inversion particularly crucial as it provides a starting model for the gradient-based FWI that contains the large-medium wavelengths of the seismic velocity field. The following gradient-based FWI yields Vp and Vs models characterized by an improved resolution with respect to the outcomes of GA-FWI. The match between the observed and the predicted seismic data proves the reliability of our predictions.

The reconstruction of the Marmousi-2 geological model using the gradient-based time domain acoustic and elastic Full-Waveform Inversion

Marmousi-2 is an elastic model defined by the P-wave, the Swave velocities (Vp and Vs respectively) and by the densities of the geological bodies that reproduce a geological profile of North Quenguelain the Quanza Basin in Angola. This study is focused on the evaluation of the accuracy and quality of the Vp and Vs estimated models and of their Vp to Vs ratio, using the gradient-based acoustic and elastic Full-Waveform Inversion (FWI). Two different initial models have been tested: the first one obtained with Genetic Algorithms (GA) and the second one obtained by applying a smoothing function to the Marmousi-2 true model. These initial models are characterized by variable Vp/Vs ratios, both in depth and in the horizontal direction, which nicely approximate the true geological setting. The analysis of the different tomographic images obtained by the FWI and the model misfit computation proved the good reconstruction of the Vp and Vs model only when the initial model has Vp/Vs ratios that reproduce the depth and the horizontal variations of the true model; this happens because the background trend of Vp/Vs ratio is preserved from the initial to the reconstructed models and is not much modified by FWI.Furthermore, it was noticed that also the use of an initial model obtained with Genetic Algorithms allows for a good Vp and Vs model reconstruction. However, the final reconstructed model has in this case a lower degree of matching with the true model. Nevertheless, the improvements brought by gradient-based FWI, that is, the decrease of the model misfit between the true model and the reconstructed model, are greater for the GA case compared to the smooth case. Then, starting from a less accurate initial model, the FWI is able to deliver a final model close to the true model.

From classical reflectivity-to-velocity inversion to full-waveform inversion using phase-modified and deconvolved reverse time migration images

To obtain a physical understanding of gradient-based descent methods in full-waveform inversion (FWI), we find a connection between the FWI gradient and the image provided by reverse time migration (RTM). The gradient uses the residual data as a virtual source, and RTM uses the observed data directly as the boundary condition, so the FWI gradient is similar to a time integration of the RTM image using the residual data, which physically converts the phase of the reflectivity to the phase of the velocity. Therefore, gradient-based FWI can be connected to the classical reflectivity-to-velocity/impedance inversion (RVI). We have developed a new FWI scheme that provides a self-contained and physically intuitive derivation, which naturally establishes a connection among the amplitude-preserved RTM, the Zoeppritz equations (amplitude variation with angle inversion), and RVI, and combines them into a single framework to produce a preconditioned inversion formula. In this scheme, the relative velocity update is a phase-modified and deconvolved RTM image obtained with the residual data. Consistent with the deconvolution, the multiscale approach applies a gradually widening low-pass frequency filter to the deconvolved wavelet at early iterations, and then it uses the unfiltered deconvolved wavelet for the final iterations. Our numerical testing determined that the new method makes a significant improvement to the quality of the inversion result.

FWI for model updates in large-contrast media

We describe a new solution for recovering the long-wavelength features of a velocity model in gradient-based full-waveform inversion (FWI). The method uses reflected and transmitted wave modes to recover high-resolution velocity models. The new FWI gradient enables reliable velocity updates deeper than the maximum penetration depth of diving waves and reduces the FWI dependency on recording ultralong offsets. We also discuss a new FWI regularization scheme that overcomes the limitations of the inversion in the presence of high-contrast geobodies and cycle skipping. The solution utilizes a priori information about the earth model in the regularization as an extra term in the objective function. The implementation makes use of the split Bregman method, making it efficient and accurate. Results from applying the new FWI gradient to field data show that we can combine both transmitted and reflected energy in a global FWI scheme to obtain high-resolution velocity models without imprint of the reflectivity on the velocity updates. We illustrate the new regularization method's potential on the BP 2004 velocity benchmark model where our regularized FWI solution is capable of using a simple starting velocity model to deliver a high-quality reconstruction of salt bodies.

A robust gradient for long wavelength FWI updates

We introduce a robust method to produce long wavelength updates in gradient-based Full Waveform Inversion (FWI). The solution introduces dynamic weights in the velocity sensitivity kernel derived from impedance and velocity parameterization of the classical objective function. The new kernel implementation effectively eliminates the migration isochrones produced by the specular reflections, enhances the low wavenumber components in the gradient in heterogeneous media, and is able to deliver velocity updates beyond the penetration depth of diving waves. We use synthetic examples to illustrate how this dynamic weighted FWI gradient successfully recovers the velocity from pre-critical reflections. We also show with dual-sensor streamer data from deep-water Gulf of Mexico how the dynamic weighted FWI gradient can combine both transmitted and reflected energy in a global FWI scheme.

Wavefield compression for adjoint methods in full-waveform inversion

Adjoint methods are a key ingredient of gradient-based full-waveform inversion schemes. While being conceptually elegant, they face the challenge of massive memory requirements caused by the opposite time directions of forward and adjoint simulations and the necessity to access both wavefields simultaneously for the computation of the sensitivity kernel. To overcome this bottleneck, we have developed lossy compression techniques that significantly reduce the memory requirements with only a small computational overhead. Our approach is tailored to adjoint methods and uses the fact that the computation of a sufficiently accurate sensitivity kernel does not require the fully resolved forward wavefield. The collection of methods comprises reinterpolation with a coarse temporal grid as well as adaptively chosen polynomial degree and floating-point precision to represent spatial snapshots of the forward wavefield on hierarchical grids. Furthermore, the first arrivals of adjoint waves are used to identify “shadow zones” that do not contribute to the sensitivity kernel. Numerical experiments show the high potential of this approach achieving an effective compression factor of three orders of magnitude with only a minor reduction in the rate of convergence. Moreover, it is computationally cheap and straightforward to integrate in finite-element wave propagation codes with possible extensions to finite-difference methods.

Estimating a starting model for full-waveform inversion using a global optimization method

Full-waveform inversion (FWI) has become a popular method to estimate elastic earth properties from seismograms. It is formulated as a data-fitting least-squares minimization problem that iteratively updates an initial velocity model with the scaled gradient of the misfit until a satisfactory match between the real and synthetic data is obtained. However, such a local optimization approach can converge to a local minimum if the starting model used is not close enough to an optimal model. We have developed a two-step process in which we first estimate a starting model using a global optimization method. Unlike local optimization methods, a global optimization method starts with a random starting model and is not generally susceptible to be trapped in a local minimum. The starting model for FWI that we aim to estimate is sparsely parameterized and contains a set of interfaces and velocities that are used to represent the entire velocity model. We have obtained the depth of the interfaces and the velocities by minimizing the data misfit in the least-squares sense using a global optimization method called very fast simulated annealing (VFSA). Once the sparse velocity model was obtained from VFSA, we used that as a starting model in a conventional gradient-based FWI to obtain the final model. We have applied the proposed method to one synthetic data set and two field data sets from offshore India. The proposed method was able to estimate a velocity model that was not cycle skipped for realistic frequency bands. We have demonstrated that with the proper choice of model parameterization and optimization parameters, the global and gradient optimization algorithms converge in a finite number of iterations. We have determined that the resulting algorithm is computationally feasible in two dimensions and accurate for practical implementation of FWI.