We recognized that the envelope fluctuation and decay of seismic records carries ultra low-frequency (ULF, i.e., the frequency below the lowest frequency in the source spectrum) signals that can be used to estimate the long-wavelength velocity structure. We then developed envelope inversion for the recovery of low-wavenumber components of media (smooth background), so that the initial model dependence of waveform inversion can be reduced. We derived the misfit function and the corresponding gradient operator for envelope inversion. To understand the long-wavelength recovery by the envelope inversion, we developed a nonlinear seismic signal model, the modulation signal model, as the basis for retrieving the ULF data and studied the nonlinear scale separation by the envelope operator. To separate the envelope data from the wavefield data (envelope extraction), a demodulation operator (envelope operator) was applied to the waveform data. Numerical tests using synthetic data for the Marmousi model pro...

/pdf/seismic-envelope-inversion-and-modulation-signal-model-21kx2j28no.pdf

Seismic envelope inversion and modulation signal model

S U M M A R Y Full waveform inversion (FWI) requires a hierarchical approach to maneuver the complex non-linearity associated with the problem of velocity update. In anisotropic media, the nonlinearity becomes far more complex with the potential trade-off between the multiparameter description of the model. A gradient filter helps us in accessing the parts of the gradient that are suitable to combat the potential non-linearity and parameter trade-off. The filter is based on representing the gradient in the time-lag normalized domain, in which the low scattering angle of the gradient update is initially muted out in the FWI implementation, in what we may refer to as a scattering angle continuation process. The result is a low wavelength update dominated by the transmission part of the update gradient. In this case, even 10 Hz data can produce vertically near-zero wavenumber updates suitable for a background correction of the model. Relaxing the filtering at a later stage in the FWI implementation allows for smaller scattering angles to contribute higher-resolution information to the model. The benefits of the extended domain based filtering of the gradient is not only it’s ability in providing low wavenumber gradients guided by the scattering angle, but also in its potential to provide gradients free of unphysical energy that may correspond to unrealistic scattering angles.

/pdf/scattering-angle-based-filtering-of-the-waveform-inversion-4gwooupktb.pdf

Scattering-angle based filtering of the waveform inversion gradients

The gradient of standard full-waveform inversion (FWI) attempts to map the residuals in the data to perturbations in the model. Such perturbations may include smooth background updates from the transmission components and high wavenumber updates from the reflection components. However, if we fix the reflection components using imaging, the gradient of what is referred to as reflected-waveform inversion (RWI) admits mainly transmission background-type updates. The drawback of existing RWI methods is that they lack an optimal image capable of producing reflections within the convex region of the optimization. Because the influence of velocity on the data was given mainly by its background (propagator) and perturbed (reflectivity) components, we have optimized both components simultaneously using a modified objective function. Specifically, we used an objective function that combined the data generated from a source using the background velocity, and that by the perturbed velocity through Born modeli...

/pdf/simultaneous-inversion-of-the-background-velocity-and-the-4dsi17guis.pdf

Simultaneous inversion of the background velocity and the perturbation in full-waveform inversion

Because modeling for full-waveform inversion (FWI) cannot produce reflections unless the velocity model has the scattering potential (high wavenumbers), using a migration/demigration process to generate modeling data, which is a key step in what is now known as reflection FWI (RFWI), is a credible alternative to tackle the reflection nonlinearity associated with FWI. However, because RFWI depends on a conventional data residual or zero-lag correlation objective function, high nonlinearity can still exist when the true amplitude migration is not used, as well as at far offsets due to cycle skipping. To avoid the cycle skipping and the need for a true amplitude migration, we have developed a correlation-based reflection full-waveform inversion method to update the low-wavenumber components of the velocity model. The success of this method relies on a sensitivity kernel decomposition and a correlation-based objective function. The sensitivity kernel decomposition makes it possible to separate out the contributions of different subkernels and to smear the reflected wave residuals along the “rabbit-ear” wavepath to obtain middle and deep background model estimates. The correlation-based objective function measures differences in kinematic information and behaves in a more linear way than the traditional waveform residual misfit. Moreover, our approach is less sensitive to the frequency content and amplitude information of the seismic data, enabling reliable background velocity estimates to be obtained without the need for low frequencies and full-physics modeling. Because the kinematic features of reflected waves are described correctly, the inversion result of the proposed method can be used as a migration model or an initial model for conventional FWI to achieve a correct high-wavenumber model update.

Correlation-based reflection full-waveform inversion

The availability of low-frequency data is an important factor in the success of full-waveform inversion (FWI) in the acoustic regime. The low frequencies help determine the kinematically relevant, low-wavenumber components of the velocity model, which are in turn needed to avoid convergence of FWI to spurious local minima. However, acquiring data less than 2 or 3 Hz from the field is a challenging and expensive task. We have explored the possibility of synthesizing the low frequencies computationally from high-frequency data and used the resulting prediction of the missing data to seed the frequency sweep of FWI. As a signal-processing problem, bandwidth extension is a very nonlinear and delicate operation. In all but the simplest of scenarios, it can only be expected to lead to plausible recovery of the low frequencies, rather than their accurate reconstruction. Even so, it still requires a high-level interpretation of band-limited seismic records into individual events, each of which can be extr...

Full-waveform inversion with extrapolated low-frequency data

Method for improving convergence in gradient-based iterative inversion of seismic data ( 101 ), especially advantageous for full wavefield inversion The method comprises decomposing the gradient into two (or more) components ( 103 ), typically the migration component and the tomographic component, then weighting the components to compensate for unequal frequency content in the data ( 104 ), then recombining the weighted components ( 105 ), and using the recombined gradient to update ( 106 ) the physical properties model ( 102 )

Tomographically Enhanced Full Wavefield Inversion

Multi-Parameter Full Wavefield Inversion Using Non-Stationary Point-Spread Functions

s, F032, http://dx.doi.org/10.3997/2214-4609.20149730. Shuey, R. T., 1985, A simplification of the Zoeppritz equations: Geophysics, 50, 609–614, http://dx.doi.org/10.1190/1.1441936. Van Leeuwen, T., and W. A. Mulder, 2010, A correlation-based misfit criterion for wave-equation traveltime tomography: Geophysical Journal International, 182, 1383–1394, http://dx.doi.org/10.1111/j.1365-246X.2010.04681.x. © 2017 SEG SEG International Exposition and 87th Annual Meeting Page 1301 D ow nl oa de d 08 /1 7/ 17 to 9 8. 20 0. 21 4. 25 1. R ed is tr ib ut io n su bj ec t t o SE G li ce ns e or c op yr ig ht ; s ee T er m s of U se a t h ttp :// lib ra ry .s eg .o rg /

Time-domain broadband phase-only full-waveform inversion with implicit shaping

Method for reducing computational time in inversion of geophysical data to infer a physical property model ( 91 ), especially advantageous in full wavefield inversion of seismic data. An approximate Hessian is pre-calculated by computing the product of the exact Hessian and a sampling vector composed of isolated point diffractors ( 82 ), and the approximate Hessian is stored in computer hard disk or memory ( 83 ). The approximate Hessian is then retrieved when needed ( 99 ) for computing its product with the gradient ( 93 ) of an objective function or other vector. Since the approximate Hessian is very sparse (diagonally dominant), its product with a vector may therefore be approximated very efficiently with good accuracy. Once the approximate Hessian is computed and stored, computing its product with a vector requires no simulator calls (wavefield propagations) at all. The pre-calculated approximate Hessian can also be reused in the subsequent steps whenever necessary.

Accelerating Full Wavefield Inversion with Nonstationary Point-Spread Functions

We propose a new method to perform wave-equation migration velocity analysis by maximizing the flatness of the angledomain common image gathers. Instead of maximizing the image-stack-power objective function directly with respect to the slowness, we link the objective function to the slowness indirectly through an intermediate moveout parameter. This approach is immune to the cycle-skipping problem, and it produces high-quality gradients. In addition, the proposed method does not require explicit picking of the moveout parameters. Our numerical examples demonstrate the great potential of this method: in the first example where there is a Gaussian-shaped anomaly slowness error, our method produces well-behaved gradient; from our test on the Marmousi models, the proposed method converges to a high-quality model that uniformly flattens the angle-domain common image gathers.

Yaxun Tang

Papers

Tomographically Enhanced Full Wavefield Inversion

Multi-Parameter Full Wavefield Inversion Using Non-Stationary Point-Spread Functions

Time-domain broadband phase-only full-waveform inversion with implicit shaping

Accelerating Full Wavefield Inversion with Nonstationary Point-Spread Functions

Residual moveout-based wave-equation migration velocity analysis