Full-waveform Inversion Approach Research Articles

Full-waveform inversion for imaging and geologic interpretation: A deepwater Gulf of Mexico case study

Full-waveform inversion (FWI) is commonly used in model-building workflows to improve the resolution of the shallow velocity model and thus has a potentially positive impact on the imaging of deeper targets. This type of inversion commonly makes use of first arrivals from the longest offsets. However, signal from smaller offsets and later times can extend the depth range of the FWI-derived velocity model. Waveform inversion methods that use reflections have been shown to provide greater details and accuracy when deriving velocity models for deepwater exploration and production. The derived velocity sometimes provides an improved migrated image useful for interpretation in complex geology and enhances geologic features such as subsalt sediments, faults, and channels. We have used combination of FWI and a wavefield inversion approach known as reconstructed wavefield inversion (RWI) that makes use of diving waves and reflections to derive a velocity model for a deepwater survey off the coast of Veracruz in the Gulf of Mexico. The velocity model we derived from this approach produces an improved image of the target reservoir, and furthermore contains enough geologic details for direct interpretation. We enhanced the resolution of the velocity model further by performing a poststack amplitude inversion with the FWI + RWI derived velocity used as the input low-frequency model. The resulting high-resolution velocity provides an excellent product for detecting shallow gas anomalies, delineating a gas reservoir in an anticline structure as well as a system of deep, sand-filled channels. The inverted velocity also indicates a better correlation with sonic velocity measured from two blind wells than the initial tomography velocity, indicating the benefits of FWI approaches for quantitative reservoir characterization in deepwater environments.

Bayesian full-waveform tomography with application to crosshole ground penetrating radar data

We present a probabilistic full-waveform inversion (FWI) approach that infers a geostatistical model along with the subsurface structure. Probabilistic FWI with Markov chain Monte Carlo (MCMC) allows for uncertainty quantification and removes the requirement of having a starting model in the cone of attraction of the assumed correct global minimum. We demonstrate our approach on a synthetic and a field data set. For the latter, we compare the results with deterministic FWI and a cone penetration test. Our results compare equally well with the cone penetration test as the deterministic results do. This is a positive result as for the deterministic inversion almost seven times more data were used than for the probabilistic inversion. Furthermore, the probabilistic FWI was able to converge to the posterior distribution starting from randomly drawn models. However, our uncertainty estimates are too narrow, because the necessarily short Markov chains implied by a computationally costly forward problem and the global nature of the model proposal scheme prevented a full exploration of the posterior probability density function. Without prior information such as borehole logs, the algorithm is only able to infer relative electric conductivity values, because the unknown amplitude of the wavelet and the mean of the conductivity are strongly correlated. This study clearly demonstrates the feasibility of probabilistic FWI and highlights the advantages and disadvantages of the approach.

Tomographic full waveform inversion using normalized integral wavefield

Thanks to smooth intrinsic nature of normalized integral wavefield (NIW), full waveform inversion (FWI) using NIW has a potential to retrieve the long wavelength model parameters from seismic data even in the absence of low frequencies. However, if reflection events are dominant in the data, this strategy might fail in rendering a convergent result. To steer the focus of inversion more onto the matching of first arrivals' kinematics, hence the tomographic result, we construct objective functions using the logarithmic NIW and the NIW raised to the power of small positive number for FWI. We derive the gradient formulae of the new objective functions using the adjoint-state method, thus the computational cost of our approaches remains compatible to that of the conventional FWI. Synthetic data example on 2D SEG/EAGE salt-dome model, in which reflections are prevalent, indicates that our FWI approaches generate tomographic convergent results even without low frequency information.

High-Resolution Characterization of Near-Surface Structures by Surface-Wave Inversions: From Dispersion Curve to Full Waveform

Surface waves are widely used in near-surface geophysics and provide a noninvasive way to determine near-surface structures. By extracting and inverting dispersion curves to obtain local 1D S-wave velocity profiles, multichannel analysis of surface waves (MASW) has been proven as an efficient way to analyze shallow-seismic surface waves. By directly inverting the observed waveforms, full-waveform inversion (FWI) provides another feasible way to use surface waves in reconstructing near-surface structures. This paper provides a state of the art review of MASW and shallow-seismic FWI and a comparison of both methods. A two-parameter numerical test is performed to analyze the nonlinearity of MASW and FWI, including the classical, the multiscale, the envelope-based, and the amplitude-spectrum-based FWI approaches. A checkerboard model is used to compare the resolution of MASW and FWI. These numerical examples show that classical FWI has the highest nonlinearity and resolution among these methods, while MASW has the lowest nonlinearity and resolution. The modified FWI approaches have an intermediate nonlinearity and resolution between classical FWI and MASW. These features suggest that a sequential application of MASW and FWI could provide an efficient hierarchical way to delineate near-surface structures. We apply the sequential-inversion strategy to two field data sets acquired in Olathe, Kansas, USA, and Rheinstetten, Germany, respectively. We build a 1D initial model by using MASW and then apply the multiscale FWI to the data. High-resolution 2D S-wave velocity images are obtained in both cases, whose reliabilities are proven by borehole data and a GPR profile, respectively. It demonstrates the effectiveness of combining MASW and FWI for high-resolution imaging of near-surface structures.

On the full-waveform inversion of Lamé parameters in semi-infinite solids in plane strain

We are concerned with imaging the spatially varying Lamé parameters of semi-infinite, arbitrarily heterogeneous solids, when probed by elastic waves in the time domain. We use a full-waveform inversion approach to tackle the inverse medium problem, and seek the Lamé distributions that minimize the misfit between measured and computed responses, subject to the governing PDEs. As is commonly the case, the resulting inverted profiles of the second Lamé parameter (μ) are of better quality than those of the first (λ). To improve the resolution of both Lamé parameters, we discuss the use of three robustifying schemes, namely, source-frequency continuation, regularization factor continuation, and a search direction-biasing scheme. We demonstrate with numerical experiments the effect the schemes have on the inversion process and conclude with an application of the robustified full-waveform method to a challenging adaptation of the Marmousi2 model.

Normalized nonzero-lag crosscorrelation elastic full-waveform inversion

Full-waveform inversion (FWI) is an attractive technique due to its ability to build high-resolution velocity models. Conventional amplitude-matching FWI approaches remain challenging because the simplified computational physics used does not fully represent all wave phenomena in the earth. Because the earth is attenuating, a sample-by-sample fitting of the amplitude may not be feasible in practice. We have developed a normalized nonzero-lag crosscorrelataion-based elastic FWI algorithm to maximize the similarity of the calculated and observed data. We use the first-order elastic-wave equation to simulate the propagation of seismic waves in the earth. Our proposed objective function emphasizes the matching of the phases of the events in the calculated and observed data, and thus, it is more immune to inaccuracies in the initial model and the difference between the true and modeled physics. The normalization term can compensate the energy loss in the far offsets because of geometric spreading and avoid a bias in estimation toward extreme values in the observed data. We develop a polynomial-type weighting function and evaluate an approach to determine the optimal time lag. We use a synthetic elastic Marmousi model and the BigSky field data set to verify the effectiveness of the proposed method. To suppress the short-wavelength artifacts in the estimated S-wave velocity and noise in the field data, we apply a Laplacian regularization and a total variation constraint on the synthetic and field data examples, respectively.

Radius estimation of subsurface cylindrical objects from ground-penetrating-radar data using full-waveform inversion

Ray-based radius estimations of subsurface cylindrical objects such as rebars and pipes from ground-penetrating-radar (GPR) measurements are not accurate because of their approximations. We have developed a novel full-waveform inversion (FWI) approach that uses a full-waveform 3D finite-difference time-domain (FDTD) forward-modeling program to estimate the radius including other object parameters. By using the full waveform of the common-offset GPR data, the shuffled complex evolution (SCE) approach is able to reliably extract the radius of the subsurface cylindrical objects. A combined optimization of radius, medium properties, and the effective source wavelet is necessary. Synthetic and experimental data inversion returns an accurate reconstruction of the cylinder properties, medium properties, and the effective source wavelet. Combining FWI of GPR data using SCE and a 3D FDTD forward model makes the approach easily adaptable for a wide range of other GPR FWI approaches.

Realistic FDTD GPR Antenna Models Optimized Using a Novel Linear/Nonlinear Full-Waveform Inversion

Finite-difference time-domain forward modeling of ground-penetrating radar (GPR) is becoming regularly used in model-based interpretation methods, such as full-waveform inversion (FWI) and machine learning schemes. Oversimplifications in such forward models can compromise the accuracy and realism with which real GPR responses can be simulated, which degrades the overall performance of interpretation techniques. A forward model must be able to accurately simulate every part of the GPR problem that affects the resulting scattered field. A key element, especially for near-field applications, is the antenna system. Therefore, the model must contain a complete description of the antenna, including the excitation source and waveform, the geometry, and the dielectric properties of any materials in the antenna. The challenge is that some of these parameters are not known or cannot be easily measured, especially for commercial GPR antennas that are used in practice. We present a novel hybrid linear/nonlinear FWI approach that can be used, with only knowledge of the basic antenna geometry, to simultaneously optimize the dielectric properties and excitation waveform of the antenna and minimize the error between real and synthetic data. The accuracy and stability of our proposed methodology are demonstrated by successfully modeling a 1.5-GHz commercial antenna from Geophysical Survey Systems, Inc. Our framework allows accurate models of GPR antennas to be developed without requiring detailed knowledge of every component of the antenna. This is significant because it allows commercial GPR antennas, regularly used in GPR surveys, to be more readily simulated.

A self-adjoint velocity-stress full-waveform inversion approach to microseismic source estimation

Accurate estimation of microseismic events in time and space enables important characterization of hydraulic fracture networks. Determining the orientation of the fracture can help differentiate between reactivation of an in situ fracture and a hydraulically induced fracture, which is important in characterizing the effectiveness of the stimulation process. We consider the source as separable in time and space and invert for a complete description of the source in one of these two dimensions assuming the other is known. We recover the wavelet, which includes the source amplitude and time evolution. In space, we recover a description of the source that is distributed and includes an estimate of the moment tensor at every point in the domain. A change of variables applied to the velocity-stress form of the elastic-wave equation ensures that the system is self-adjoint. Thus, full-waveform inversion can be tailored to estimate microseismic events with limited modifications to the forward wave solver. The inversion does not use any a priori assumptions about the form of the source, does not require a good starting guess for accurate source recovery, and is robust in the presence of noise. Applying this technique to wavelet inversion correctly recovers a 30 Hz Ricker wavelet from a zero initial guess with and without noise. Furthermore, for a realistic microseismic event generated from coupled flow and deformation modeling, the algorithm recovers the peak time and approximate shape of the wavelet. In fact, our algorithm recovers as much of the true wavelet as possible given the energy in the observed data. Furthermore, experiments involving a distributed source in the shape of an ellipse illustrate that the inversion scheme can not only estimate the focal mechanism of failure events, but also the geometry of the failure plane.

Seismicity in the block mountains between Halle and Leipzig, Central Germany: centroid moment tensors, ground motion simulation, and felt intensities of two M ≈ 3 earthquakes in 2015 and 2017

On April 29, 2017 at 0:56 UTC (2:56 local time), an MW = 2.8 earthquake struck the metropolitan area between Leipzig and Halle, Germany, near the small town of Markranstadt. The earthquake was felt within 50 km from the epicenter and reached a local intensity of I0 = IV. Already in 2015 and only 15 km northwest of the epicenter, a MW = 3.2 earthquake struck the area with a similar large felt radius and I0 = IV. More than 1.1 million people live in the region, and the unusual occurrence of the two earthquakes led to public attention, because the tectonic activity is unclear and induced earthquakes have occurred in neighboring regions. Historical earthquakes south of Leipzig had estimated magnitudes up to MW ≈ 5 and coincide with NW-SE striking crustal basement faults. We use different seismological methods to analyze the two recent earthquakes and discuss them in the context of the known tectonic structures and historical seismicity. Novel stochastic full waveform simulation and inversion approaches are adapted for the application to weak, local earthquakes, to analyze mechanisms and ground motions and their relation to observed intensities. We find NW-SE striking normal faulting mechanisms for both earthquakes and centroid depths of 26 and 29 km. The earthquakes are located where faults with large vertical offsets of several hundred meters and Hercynian strike have developed since the Mesozoic. We use a stochastic full waveform simulation to explain the local peak ground velocities and calibrate the method to simulate intensities. Since the area is densely populated and has sensitive infrastructure, we simulate scenarios assuming that a 12-km long fault segment between the two recent earthquakes is ruptured and study the impact of rupture parameters on ground motions and expected damage.

The Linearized Bregman Method for Frugal Full-waveform Inversion with Compressive Sensing and Sparsity-promoting

Full-waveform inversion (FWI) reconstructs the subsurface properties from acquired seismic data via minimization of the misfit between observed and simulated data. However, FWI suffers from considerable computational costs resulting from the numerical solution of the wave equation for each source at each iteration. To reduce the computational burden, constructing supershots by combining several sources (aka source encoding) allows mitigation of the number of simulations at each iteration, but it gives rise to crosstalk artifacts because of interference between the individual sources of the supershot. A modified Gauss–Newton FWI (MGNFWI) approach showed that as long as the difference between the initial and true models permits a sparse representation, the $$\ell _1$$ -norm constrained model updates suppress subsampling-related artifacts. However, the spectral-projected gradient $$\ell _1$$ (SPG $$\ell _1$$ ) algorithm employed by MGNFWI is rather complicated that makes its implementation difficult. To facilitate realistic applications, we adapt a linearized Bregman (LB) method to sparsity-promoting FWI (SPFWI) because of the efficiency and simplicity of LB in the framework of $$\ell _1$$ -norm constrained optimization problem and compressive sensing. Numerical experiments performed with the BP Salt model, the Marmousi model and the BG Compass model verify the following points. The FWI result with LB solving $$\ell _1$$ -norm sparsity-promoting problem for the model update outperforms that generated by solving $$\ell _2$$ -norm problem in terms of crosstalk elimination and high-fidelity results. The simpler LB method performs comparably and even superiorly to the complicated SPG $$\ell _1$$ method in terms of computational efficiency and model quality, making the LB method a viable alternative for realistic implementations of SPFWI.

Full waveform inversion of SH‐ and Love‐wave data in near‐surface prospecting

ABSTRACTWe develop a two‐dimensional full waveform inversion approach for the simultaneous determination of S‐wave velocity and density models from SH ‐ and Love‐wave data. We illustrate the advantages of the SH/Love full waveform inversion with a simple synthetic example and demonstrate the method's applicability to a near‐surface dataset, recorded in the village Čachtice in Northwestern Slovakia. Goal of the survey was to map remains of historical building foundations in a highly heterogeneous subsurface. The seismic survey comprises two parallel SH‐profiles with maximum offsets of 24 m and covers a frequency range from 5 Hz to 80 Hz with high signal‐to‐noise ratio well suited for full waveform inversion. Using the Wiechert–Herglotz method, we determined a one‐dimensional gradient velocity model as a starting model for full waveform inversion. The two‐dimensional waveform inversion approach uses the global correlation norm as objective function in combination with a sequential inversion of low‐pass filtered field data. This mitigates the non‐linearity of the multi‐parameter inverse problem. Test computations show that the influence of visco‐elastic effects on the waveform inversion result is rather small. Further tests using a mono‐parameter shear modulus inversion reveal that the inversion of the density model has no significant impact on the final data fit. The final full waveform inversion S‐wave velocity and density models show a prominent low‐velocity weathering layer. Below this layer, the subsurface is highly heterogeneous. Minimum anomaly sizes correspond to approximately half of the dominant Love‐wavelength. The results demonstrate the ability of two‐dimensional SH waveform inversion to image shallow small‐scale soil structure. However, they do not show any evidence of foundation walls.

Joint acoustic full-waveform inversion of crosshole seismic and ground-penetrating radar data in the frequency domain

Integrating crosshole ground-penetrating radar (GPR) with seismic methods is an efficient way to reduce the uncertainty and ambiguity of data interpretation in shallow geophysical investigations. We have developed a new approach for joint full-waveform inversion (FWI) of crosshole seismic and GPR data in the frequency domain to improve the inversion results of both FWI methods. In a joint objective function, three geophysical parameters (P-wave velocity, permittivity, and conductivity) are effectively connected by three weighted cross-gradient terms that enforce the structural similarity between parameter models. Simulation of acoustic seismic and scalar electromagnetic problems is implemented using 2D finite-difference frequency-domain methods, and the inverse problems of seismic FWI and GPR FWI are solved using a matrix-free truncated Newton algorithm. The joint inversion procedure is performed in several hierarchical frequencies, and the three parameter models are sequentially inverted at each frequency. The joint FWI approach is illustrated using three numerical examples. The results indicate that the joint FWI approach can effectively enhance the structural similarity among the models, modify the structure of each model, and improve the accuracy of inversion results compared with those of individual FWI approaches. Moreover, joint inversion can reduce the trade-off between permittivity and conductivity in GPR FWI, leading to an improved conductivity model in which artifacts are significantly decreased.

Quantitative imaging for civil engineering by joint full waveform inversion of surface-based GPR and shallow seismic reflection data

Full waveform inversion (FWI) of ground penetrating radar (GPR) and seismic data is a promising imaging tool for the detailed characterization of latent cavities and tunnels. In this study, surface-based GPR FWI and seismic FWI are combined to quantitatively image the geometry and geophysical attributes of the subsurface. These two FWI methods are integrated into a joint procedure and linked by cross-gradient functions which can provide structural constraints between the parameters. The elastic and electromagnetic properties of the medium (P-wave velocity, permittivity, and conductivity) are linked and simultaneously inverted in the joint FWI approach. This joint FWI approach is tested using three numerical examples. In the first two examples, target objects with various shapes are contained in both homogeneous and inhomogeneous media. In the third example, two sediment-filled cavities and a fractured area are analyzed in the calcarenite layers. The results show that the joint FWI approach can integrate the common structure of models and correct the model structures that are inaccurately reconstructed in the individual FWI methods, providing a comprehensive structural interpretation of the study areas.

Full-waveform inversion of Crosshole GPR data: Implications for porosity estimation in chalk

The Maastrichtian-Danian chalk is a widely distributed hydrocarbon and groundwater reservoir rock in north-western Europe. Knowledge of lateral and vertical heterogeneity and porosity variation in this type of rock is essential, since they critically determine the reservoir properties. We have collected a densely sampled crosshole ground-penetrating radar (GPR) dataset from a highly heterogeneous section of the chalk and inverted it with a full-waveform inversion (FWI) approach. To date, successful crosshole FWI has only been reported for a handful of GPR field data, none of which include strongly heterogeneous environments like the one considered in this study. Testing different starting models shows that all FWI results converge to very similar subsurface structures indicating that the results are robust with regard to local variations in the permittivity starting models and are not very sensitive to the conductivity starting models. Compared to their ray-based counterparts, the obtained FWI models show significantly higher resolution and improved localization of fine-scale heterogeneity. The final FWI permittivity tomogram was converted to a bulk porosity model using the Complex Refractive Index Model (CRIM) and comparisons with plug sample porosities and televiewer image logs verify that variations in the obtained permittivity are related to facies and lithology changes. The inferred porosity varies from 30 to 54%, which is consistent with values in the chalk cores from the investigated boreholes and in agreement with other studies conducted in similar rocks onshore. Moreover, porosities vary significantly over scales of less than a meter both laterally and vertically. The FWI constrains porosity variation with decimeter scale resolution in our 5m (horizontally) by 10m (vertically) model section bridging the gap between what is measured on the core sample scale and the scale typical of hydrogeophysical field experiments conducted to characterize fluid flow in the subsurface. The results provide complementary knowledge to traditional chalk reservoir characterization.

Orthorhombic full-waveform inversion for imaging the Luda field using wide-azimuth ocean-bottom-cable data

Ocean-bottom-cable (OBC) acquisition has become the new trend in the Bohai Bay area as a result of the operational flexibility, better illumination, better multiple elimination, and better signal-to-noise (S/N) ratio it brings for targets at middle-to-deep depths. However, the presence of azimuthal anisotropy poses severe challenges to imaging with wide-azimuth (WAZ) OBC data, in particular when imaging fault planes, which are very sensitive to velocity errors. The fault images can be smeared and fault shadows observed within complex strike-slip fault systems if the azimuthal dependency of wave propagation is not properly modeled and velocity variations across faults are not properly resolved. To address these kinds of imaging challenges in the Luda field WAZ OBC data, we have developed a practical orthorhombic full-waveform inversion (FWI) approach to reconstruct high-resolution models in the presence of azimuthal anisotropy. We demonstrate that our orthorhombic FWI approach can produce a high-resolution velocity model that reconciles the structural discrepancies between seismic images from different azimuths and significantly improves the focusing of the fault planes and the quality of the image beneath the faulting system. The combined effect of these improvements gives a clear uplift in the final seismic image.

Improving the greater K2 area subsalt imaging with advanced seismic acquisition, model building, and imaging technologies — A Gulf of Mexico case study

Subsalt imaging has proven challenging in part because of the complex geometries of the salt canopies, which can yield highly variable illumination on subsalt reflectors. The resulting complexity of the seismic wavefields has also challenged traditional velocity-model-building tools and migration algorithms. A case study for the K2 field illustrates how improved seismic data in the form of long-offset, full-azimuth acquisition and more advanced model building and imaging tools have substantially enhanced the imaging quality of this major subsalt field. The higher fold and more complete azimuthal coverage afforded by the full-azimuth coil data improved the effectiveness of the combined tomographic and full-waveform inversion approach for velocity-model building. Utilization of TTI RTM migration algorithms provided further benefits not only for better delineation of the salt geometry during model building but also in delivering superior final image volumes relative to that which could be provided by less robust algorithms, such as Kirchhoff, beam, and one-way wave equation.

An improved scattering-integral approach for frequency-domain full waveform inversion

An improved scattering-integral approach for frequency-domain full waveform inversion

A Gauss–Newton full-waveform inversion for material profile reconstruction in viscoelastic semi-infinite solid media

An inversion framework employing a Gauss–Newton method is developed to reconstruct material profiles in heterogeneous, viscoelastic, semi-infinite domains. In particular, a full-waveform inversion approach is investigated to image the elastic and attenuating parameters of a layered media. To account for the viscoelasticity of the medium, a Generalized Maxwell Body with one spring and two Maxwell elements in parallel (GMB2) is adopted in the forward and inverse wave propagation problems. Perfectly-matched-layers were introduced as wave absorbing buffers to simulate the semi-infinite extent of the domain. Using transient wave equations endowed with the GMB2 constitutive relation and the PML, a partial-differential-equations-constrained optimization scheme was implemented that lead to classic KKT (Karush–Kuhn–Tucker) conditions including time-dependent state, adjoint, and time-invariant control problems. An optimal solution of the viscoelastic parameters was obtained using a reduced-space approach based on a line search algorithm where the search direction was computed by the Gauss–Newton method. Considerable improvements on the accuracy and convergence rate of solutions were made by the developed Gauss–Newton inversion procedure compared to previous research using the Fletcher–Reeves method.

A computational method for full waveform inversion of crosswell seismic data using automatic differentiation

Full waveform inversion (FWI) is a model-based data-fitting technique that has been widely used to estimate model parameters in Geophysics. In this work, we propose an efficient computational approach to solve the FWI of crosswell seismic data. The FWI problem is mathematically formulated as a partial differential equation (PDE)-constrained optimization problem, which is numerically solved using a gradient-based optimization method. The efficiency and accuracy of FWI are mainly determined by the three main components: forward modeling, gradient calculation and model update which usually involves the gradient-based optimization algorithm. Given the large number of iterations needed by FWI, an accurate gradient is critical for the success of FWI, as it will not only speed up the convergence but also increase the accuracy of the solution. However computing the gradient still remains a challenging task even after the adjoint PDE has been derived. Automatic differentiation (AD) tools have been proved very effective in a variety of application areas including Geoscience. In this work we investigated the feasibility of integrating TAPENADE, a powerful AD tool into FWI, so that the FWI workflow is simplified to allow us to focus on the forward modeling and the model updating. In this paper we choose the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method due to its robustness and fast convergence. Numerical experiments have been conducted to demonstrate the effectiveness, efficiency and robustness of the new computational approach for FWI.