Nonlinear Full Waveform Inversion Research Articles

A Numerical Experiment of Full Waveform Inversion of Complex Structures Concealed Around a Horizontal Hydraulic Fracturing Well Using Perforation Seismic Data

It is difficult to image the details of complex structures concealed around a horizontal hydraulic fracturing well using seismic data from the ground surface. In this paper, an approach is proposed to solve this problem by non-linear full waveform inversion (FWI) using perforation seismic data. The feasibility of the approach was investigated using numerical modeling based on an experimental model built from the well-known SEG/EAGE overthrust model, which contains complex geological structures with faults. First, seismic modeling was performed to produce experimental synthetic data, including three sets of perforation seismic data recorded by different acquisition systems deployed in observation wells and on the ground surface, and another set of conventional seismic reflection data with both sources and receivers deployed on the ground surface. Then, FWI was performed separately on each data set using an initial velocity model which was heavily smoothed to remove the target structures. The inversion results show that the concealed complex structures around the well were successfully recovered by FWI using perforation data, while the benchmark image from the FWI using conventional seismic data was poor. Particularly, the experiments also demonstrated that FWI using perforation seismic data can image the faults around a horizontal hydraulic fracturing well, while this is unable to achieve using conventional ground surface seismic data. This conclusion was also proved to be valid for noisy data deteriorated either by synthetic Gaussian or field noises. Further experiments demonstrated that FWI using perforation data recorded from wells outperformed that of surface data in terms of structure imaging accuracy characterized by quantitative errors.

Self-Supervised Deep Learning for Nonlinear Seismic Full Waveform Inversion

Seismic full waveform inversion (FWI) is able to build high-resolution velocity model based on the full information carried by seismic wave. However, FWI requires an accurate enough initial model to ensure convergence. In this paper, we propose a new nonlinear FWI method to mitigate the initial model dependence problem. Specifically, we firstly propose a nonlinear operator within the hybrid model- and data-driven framework based on the frequency controllable envelope operator (FCEO) and a deep learning architecture U-Net. FCEO is used to obtain the envelope of a band-limited data and U-Net realizes the mapping from this envelope to that corresponding to a lower frequency band. The U-Net is trained in a self-supervised manner that avoids the reliance on labeled data and benefits the generalization ability. Based on the nonlinear operator, a nonlinear FWI method is proposed by defining a new misfit function. In addition, the calculation of gradient is derived using the adjoint-state method. Using numerical examples, we investigate the performance of the proposed nonlinear operator and the new nonlinear FWI method. The results clearly demonstrate that the proposed nonlinear operator is effective in obtaining low-frequency envelope data, and the new nonlinear FWI method has advantages over common method in mitigating cycle-skipping and building an initial model for conventional FWI.

Ultrasound Computed Tomography.

This chapter presents theoretical, numerical, and experimental frameworks for the use of Ultrasound Computed Tomography (USCT) for cortical bone tissue imaging. Most of the research conducted on this topic concerns adult bone, although some work presented in this chapter is specific to the study of child bone. USCT is recognized as a powerful method for soft tissue imaging. In bone imaging, the difficulties arise from the very high impedance contrast between tissues which alters the propagation of the ultrasonic waves and limits the linear inversion algorithms used. Solutions consist in optimally assessing non-linear effects in an iterative approach aiming at local linearization. When the problem can be reduced to the study of a fluid-like cavity buried in an elastic cylinder surrounded by water, the signal processing and/or compound algorithms can be added as an extension to the linear algorithms. The main limitation of these methods is the heavy experimental costs involved. We have then suggested the introduction of purely numerical non-linear full-waveform inversion algorithms. The performances and the limitations of these linear and non-linear methods applied to cortical bone tissue imaging problems are overviewed and discussed.

Nonlinear full waveform inversion of wide-aperture OBS data for Moho structure using a trans-dimensional Bayesian method

SUMMARY Seismic full waveform inversion (FWI) is a powerful method for estimating quantitative subsurface physical parameters from seismic data. As the FWI is a nonlinear problem, the linearized approach updates model iteratively from an initial model, which can get trapped in local minima. In the presence of a high-velocity contrast, such as at Moho, the reflection coefficient and recorded waveforms from wide-aperture seismic acquisition are extremely nonlinear around critical angles. The problem at the Moho is further complicated by the interference of lower crustal (Pg) and upper mantle (Pn) turning ray arrivals with the critically reflected Moho arrivals (PmP). In order to determine velocity structure near Moho, a nonlinear method should be used. We propose to solve this strong nonlinear FWI problem at Moho using a trans-dimensional Markov chain Monte Carlo (MCMC) method, where the earth model between lower crust and upper mantle is ideally parametrized with a 1-D assumption using a variable number of velocity interfaces. Different from common MCMC methods that require determining the number of unknown as a fixed prior before inversion, trans-dimensional MCMC allows the flexibility for an automatic estimation of both the model complexity (e.g. the number of velocity interfaces) and the velocity–depth structure from the data. We first test the algorithm on synthetic data using four representative Moho models and then apply to an ocean bottom seismometer (OBS) data from the Mid-Atlantic Ocean. A 2-D finite-difference solution of an acoustic wave equation is used for data simulation at each iteration of MCMC search, for taking into account the lateral heterogeneities in the upper crust, which is constrained from traveltime tomography and is kept unchanged during inversion; the 1-D model parametrization near Moho enables an efficient search of the trans-dimensional model space. Inversion results indicate that, with very little prior and the wide-aperture seismograms, the trans-dimensional FWI method is able to infer the posterior distribution of both the number of velocity interfaces and the velocity–depth model for a strong nonlinear problem, making the inversion a complete data-driven process. The distribution of interface matches the velocity discontinuities. We find that the Moho in the study area is a transition zone of 0.7 km, or a sharp boundary with velocities from around 7 km s−1 in the lower crust to 8 km s−1 of the upper mantle; both provide nearly identical waveform match for the field data. The ambiguity comes from the resolution limit of the band-limited seismic data and limited offset range for PmP arrivals.

Reducing seismic reflector distortion beneath gas clouds in the Malay Basin using full-wavefield imaging approaches

A seismic example from the Malay Basin is presented, demonstrating improved seismic imaging beneath gas clouds using full-wavefield imaging approaches. Overall imaging concepts, synthetic examples, and field implementation strategies are discussed, and results that tie with well information are presented. Seismic imaging beneath gas clouds using the full-wavefield redatuming technique improves the image by estimating the waveform transmission operators via equivalent-medium representation of the overburden from the gas cloud reflection response for use in a form of multidimensional deconvolution of the wavefield. The other example shown uses the full-wavefield migration seismic imaging technique, which utilizes primaries and higher-order multiples as signals to improve the reflectivity estimation in imaging. The demonstrated full-wavefield imaging approach uses information carried by the gas cloud reflection response to correct seismic image distortion. Removing the internal multiple using conventional demultiple processing in the gas cloud area will also remove the valuable information of the subsurface that it carries. Such multiples must be preserved for this method to be successful. The information is translated into transmission operators that are estimated by simulating the reflection response through an effective medium of the gas cloud overburden. The effective medium is obtained via nonlinear full-waveform inversion techniques from the reflections of the gas cloud overburden area. Finally, a deconvolutional process removes the transmission operators from the gas cloud reflections and recovers the underlying reflectors. Full-wavefield imaging can reconstruct the amplitudes of the reflection response below a gas cloud overburden zone so that the complex transmission imprint on the area underneath is removed properly. The Malay Basin field case study shows that implementation of this approach can provide a reliable amplitude image of the subsurface affected by gas clouds, calibrated and verified by the well information.

Multi-parameter wavefield reconstruction inversion for wavespeed and attenuation with bound constraints and total variation regularization

Wavefield reconstruction inversion (WRI) extends the search space of Full Waveform Inversion (FWI) by allowing for wave equation errors during wavefield reconstruction to match the data from the first iteration. Then, the wavespeeds are updated from the wavefields by minimizing the source residuals. Performing these two tasks in alternating mode breaks down the nonlinear FWI as a sequence of two linear subproblems, relaying on the bilinearity of the wave equation. We solve this biconvex optimization with the alternating-direction method of multipliers (ADMM) to cancel out efficiently the data and source residuals in iterations and stabilize the parameter estimation with appropriate regularizations.Here, we extend WRI to viscoacoustic media for attenuation imaging. Attenuation reconstruction is challenging because of the small imprint of attenuation in the data and the crosstalks with velocities. To address these issues, we recast the multivariate viscoacoustic WRI as a triconvex optimization and update wavefields, squared slowness, and attenuation factor in alternating mode at each WRI iteration. This requires to linearize the attenuation-estimation subproblem via an approximated trilinear viscoacoustic wave equation. The iterative defect correction embedded in ADMM corrects the errors generated by this linearization, while the operator splitting allows us to tailor ℓ1 regularization to each parameter class. A toy numerical example shows that these strategies mitigate crosstalk artifacts and noise from the attenuation reconstruction. A more realistic example representative of the North sea further shows the ability of the method to jointly reconstruct the wavespeed and the attenuation starting from a very crude attenuation-free initial model, the moderate strength of the crosstalk artifacts and the sensitivity of the multi-parameter reconstruction to noise.

Improving full-waveform inversion by wavefield reconstruction with the alternating direction method of multipliers

Full-waveform inversion (FWI) is an iterative nonlinear waveform matching procedure subject to wave-equation constraint. FWI is highly nonlinear when the wave-equation constraint is enforced at each iteration. To mitigate nonlinearity, wavefield-reconstruction inversion (WRI) expands the search space by relaxing the wave-equation constraint with a penalty method. The pitfall of this approach resides in the tuning of the penalty parameter because increasing values should be used to foster data fitting during early iterations while progressively enforcing the wave-equation constraint during late iterations. However, large values of the penalty parameter lead to ill-conditioned problems. Here, this tuning issue is solved by replacing the penalty method by an augmented Lagrangian method equipped with operator splitting (iteratively refined WRI [IR-WRI]). It is shown that IR-WRI is similar to a penalty method in which data and sources are updated at each iteration by the running sum of the data and source residuals of previous iterations. Moreover, the alternating direction strategy exploits the bilinearity of the wave-equation constraint to linearize the subsurface model estimation around the reconstructed wavefield. Accordingly, the original nonlinear FWI is decomposed into a sequence of two linear subproblems, the optimization variable of one subproblem being passed as a passive variable for the next subproblem. The convergence of WRI and IR-WRI is first compared with a simple transmission experiment, which lies in the linear regime of FWI. Under the same conditions, IR-WRI converges to a more accurate minimizer with a smaller number of iterations than WRI. More realistic case studies performed with the Marmousi II and the BP salt models indicate the resilience of IR-WRI to cycle skipping and noise, as well as its ability to reconstruct with high-fidelity, large-contrast salt bodies and subsalt structures starting the inversion from crude initial models and a 3 Hz starting frequency.

Velocity and Density Reconstruction Based on Scattering Angle Separation

The density of the medium under investigation is important for seismic interpretation and rock property analysis. However accurate estimation of density is difficult, because seismic responses to density and velocity are coupled, and there exits a trade-off problem. By analyzing the properties of their radiation patterns, we propose a scattering angle-based inversion strategy for simultaneous estimation of density and velocity. Both the source-side and receiver-side wavefields are decomposed into local angle domain, and then local scattering angles are calculated. Small scattering angles correspond to forward scattering, which are caused by velocity perturbation, so small scattering angles are used only for velocity estimation. Density perturbation is only responsible for backward scattering, which are related to large scattering angles. Therefore by filtering out small scattering angles during density inversion, the crosstalk between velocity and density is greatly reduced. Numerical examples prove that the scattering angle-based inversion method provides much improved density inversion results. Furthermore, the multi-stage strategy also helps velocity to overcome the cycle skipping problem in highly nonlinear full waveform inversion.

Characterization of a viscoelastic heterogeneous object with an effective model by nonlinear full waveform inversion

The determination of equivalent viscoelastic properties of heterogeneous objects remains challenging in various scientific fields such as (geo)mechanics, geophysics or biomechanics. The present investigation addresses the issue of the identification of effective constitutive properties of a binary object by using a nonlinear and full waveform inversion scheme. The inversion process, without any regularization technique or a priori information, aims at minimizing directly the discrepancy between the full waveform responses of a bi-material viscoelastic cylindrical object and its corresponding effective homogeneous object. It involves the retrieval of five constitutive equivalent parameters. Numerical simulations are performed in a laboratory-scale two-dimensional configuration: a transient acoustic plane wave impacts the object and the diffracted fluid pressure, solid stress or velocity component fields are determined using a semi-analytical approach. Results show that the retrieval of the density and of the real parts of both the compressional and the shear wave velocities have been carried out successfully regarding the number and location of sensors, the type of sensors, the size of the searching space, the frequency range of the incident plane pressure wave, and the change in the geometric or mechanical constitution of the bi-material object. The retrieval of the imaginary parts of the wave velocities can reveal in some cases the limitations of the proposed approach.

Retrieval of the equivalent acoustic constitutive parameters of an inhomogeneous fluid-like object by nonlinear full waveform inversion

This study addresses the problem of the acoustic characterization of an inhomogeneous object such as a soft-tissue organ containing a cyst or tumor whose size and/or composition evolve either negatively due to increased disease or positively due to increased response to treatment. The so-called ‘corrupted’ binary object, probed by a transient, acoustic plane wave, is a tube composed of a homogenous fluid-like (or assumed as such) mantle (medium 1: three acoustic constitutive parameters, one geometric parameter) surrounding a homogeneous fluid-like (or assumed as such) core (medium 2: three acoustic constitutive parameters, one geometric parameter), immersed in a spatially-infinite, homogeneous fluid (host medium 0: two acoustic parameters). The complete inversion of the diffracted acoustic field response of this object involves the retrieval of seven (six acoustic and one geometric) parameters, assuming we know beforehand the outer radius of the tube and acoustic parameters of the host. An alternative to this time-consuming, hazardous (due to the ill-posed nature of the) procedure, is to minimize the discrepancy, between the full waveform response of the binary object to a transient plane wave and the response of a homogeneous cylinder (medium characterized by three acoustic parameters, one geometric parameter) to the same transient plane wave, so as to retrieve the (three so-called equivalent) acoustic parameters of the homogeneous object. Thus, the first inverse problem is replaced by a second one (same assumptions concerning the outer radius of the objects, the host medium, the probe radiation and the sensing configuration as the first one) involving the retrieval of only three (instead of six) acoustic parameters. This procedure is potentially useful if the variation of at least one of the three equivalent parameters is sensitive to the variation of a key parameter of the inhomogeneous body (usually the characteristic dimension or the wavespeed of the core) and this variation can be expressed in a simple algebraic form (such as by a mixing formula). It is shown that this situation can arise if the average frequency of the acoustic probe radiation is sufficiently low. A sidelight of this investigation is the discovery that the equivalent constitutive parameters of the homogeneous cylinder are dispersive even when the component materials of the tube are not dispersive.

Acoustic nonlinear full-waveform inversion on an outcrop-based detailed geological and petrophysical model (Book Cliffs, Utah)

Analog outcrops are commonly used to develop predictive reservoir models and provide quantitative parameters that describe the architecture and facies distribution of sedimentary deposits at a subseismic scale, all of which aids exploration and production strategies. The focus of this study is to create a detailed geological model that contains realistic reservoir parameters and to apply nonlinear acoustic full-waveform prestack seismic inversion to this model to investigate whether this information can be recovered and to examine which geological features can be resolved by this process.Outcrop data from the fluviodeltaic sequence of the Book Cliffs (Utah) are used for the geological and petrophysical two-dimensional model. Eight depositional environments are populated with average petrophysical reservoir properties adopted from a North Sea field. These units are termed lithotypes here. Synthetic acoustic prestack seismic data are then generated with the help of an algorithm that includes all internal multiples and transmission effects. A nonlinear acoustic full-waveform inversion is then applied to the synthetic data, and two media parameters, compressibility (inversely related to the square of the compressional wave velocity vP) and bulk density, ρ, are recovered at a resolution higher than the shortest wavelength in the data. This is possible because the inversion exploits the nonlinear nature of the relationship between the recorded data and the medium contrast properties. In conventional linear inversion, these details remain masked by the noise caused by the nonlinear effects in the data. Random noise added to the data is rejected by the nonlinear inversion, contributing to improved spatial resolution. The results show that the eight lithotypes can be successfully recovered at a subseismic scale and with a low degree of processing artifacts. This technique can provide a useful basis for more accurate reservoir modeling and field development planning, allowing targeting of smaller reservoir units such as distributary channels and lower shoreface sands.

Comparison of source-independent methods of elastic waveform inversion

In this paper, we investigate several source-independent methods of nonlinear full-waveform inversion of multicomponent elastic-wave data. This includes iterative estimation of source signature (IES), standard trace normalization (STN), and average trace normalization (ATN) inversion methods. All are based on the finite-element method in the frequency domain. One synthetic elastic crosshole model is used to compare the recovered images with all these methods as well as the known source signature (KSS) inversion method. The numerical experiments show that the IES method is superior to both STN and ATN methods in two-component, elastic-wave inversion in the frequency domain when the source signature is unknown. The STN and ATN methods have limitations associated with near-zero amplitudes (or polarity reversals) in traces from one of the components, which destroy the energy balance in the normalized traces and cause a loss of frequency information. But the ATN method is somewhat superior to the STN method in suppressing random noise and improving stability, as the developed formulas and the numerical experiments show. We suggest the IES method as a practical procedure for multicomponent seismic inversion.

Evidence for a thick free gas layer beneath the bottom simulating reflector in the Makran accretionary prism

Seismic reflection data from the Makran continental margin indicate the presence of a strong and widespread bottom simulating reflector (BSR). We apply a nonlinear full waveform inversion technique to multichannel reflection data from this area, to investigate the detailed velocity structure and hence the origin of the BSR. Our result shows an abrupt decrease in the compressional wave velocity from 2.2 to 1.3 km/s at a depth of 500 m below the sea-bed. The low velocity zone is unusually thick (∼200–350 m), and may contain large quantities of free gas, similar to some of the recently drilled Blake Ridge sites of ODP Leg 164. The voluminous free gas may have been generated by the dissociation of gas hydrates as a consequence of the upward movement of the base of gas-hydrate stability field, relative to the sediment column, due to uplift and sedimentation in the accretionary wedge.

Velocity structure of a bottom simulating reflector offshore Peru: Results from full waveform inversion

Much of our knowledge of the worldwide distribution of submarine gas hydrates comes from seismic observations of Bottom Simulating Reflectors (BSRs). Full waveform inversion has proven to be a reliable technique for studying the fine structure of BSRs using the compressional wave velocity. We applied a non-linear full waveform inversion technique to a BSR at a location offshore Peru. We first determined the large-scale features of seismic velocity variations using a statistical inversion technique to maximise coherent energy along travel-time curves. These velocities were used for a starting velocity model for the full waveform inversion, which yielded a detailed velocity/depth model in the vicinity of the BSR. We found that the data are best fit by a model in which the BSR consists of a thin, low-velocity layer. The compressional wave velocity drops from 2.15 km/s down to an average of 1.70 km/s in an 18 m thick interval, with a minimum velocity of 1.62 km/s in a 6 m interval. The resulting compressional wave velocity was used to estimate gas content in the sediments. Our results suggest that the low velocity layer is a 6–18 m thick zone containing a few percent of free gas in the pore space. The presence of the BSR coincides with a region of vertical uplift. Therefore, we suggest that gas at this BSR is formed by a dissociation of hydrates at the base of the hydrate stability zone due to uplift and subsequently a decrease in pressure.