Waveform inversion based on wavefield decomposition

Full-waveform inversion (FWI) is a technique for determining the optimal model parameters by minimizing the seismic data misfit between observed and modeled data. The objective function may be highly nonlinear if the model is complex and low-frequency data are missing. If a data set mainly contains reflections, this problem particularly prevents the gradient-based methods from recovering the long wavelengths of the velocity model. Several authors observed that nonlinearity could be reduced by progressively introducing higher wavenumbers to the model. We have developed a new inversion workflow to solve this problem by breaking down the FWI gradient formula into four terms after wavefield decomposition and then selecting proper terms to invert for the short- and long-wavelength components of the velocity model alternately. Numerical tests applied on a 2D synthetic model indicate that this method is efficient at recovering the long wavelengths of the velocity model using mainly offset-limited reflection events. The source does not need to contain low frequencies. The initial velocity model may have large errors that would otherwise prevent convergence for conventional FWI.

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Full-waveform inversion with extrapolated low-frequency data

Geophysics ◽

10.1190/geo2016-0038.1 ◽

2016 ◽

Vol 81 (6) ◽

pp. R339-R348 ◽

Cited By ~ 51

Author(s):

Yunyue Elita Li ◽

Laurent Demanet

Keyword(s):

Waveform Inversion ◽

Low Frequency ◽

Velocity Model ◽

Propagation Model ◽

Full Waveform Inversion ◽

Frequency Data ◽

Local Minima ◽

Bandwidth Extension ◽

Low Frequencies ◽

Full Waveform

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 extrapolated to a lower (or higher) frequency band from the nondispersive nature of the wave-propagation model. We have used the phase-tracking method for the event separation task. The fidelity of the resulting extrapolation method is typically higher in phase than in amplitude. To demonstrate the reliability of bandwidth extension in the context of FWI, we first used the low frequencies in the extrapolated band as data substitute, to create the low-wavenumber background velocity model, and then we switched to recorded data in the available band for the rest of the iterations. The resulting method, extrapolated FWI, demonstrated surprising robustness to the inaccuracies in the extrapolated low-frequency data. With two synthetic examples calibrated so that regular FWI needs to be initialized at 1 Hz to avoid local minima, we have determined that FWI based on an extrapolated [1, 5] Hz band, itself generated from data available in the [5, 15] Hz band, can produce reasonable estimations of the low-wavenumber velocity models.

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Full-intensity waveform inversion

Geophysics ◽

10.1190/geo2017-0682.1 ◽

2018 ◽

Vol 83 (6) ◽

pp. R649-R658 ◽

Cited By ~ 17

Author(s):

Yike Liu ◽

Bin He ◽

Huiyi Lu ◽

Zhendong Zhang ◽

Xiao-Bi Xie ◽

...

Keyword(s):

Objective Function ◽

Waveform Inversion ◽

Low Frequency ◽

Velocity Model ◽

Low Frequencies ◽

Full Waveform ◽

Seismic Waveform ◽

Using Data ◽

Full Intensity ◽

Starting Model

Many full-waveform inversion schemes are based on the iterative perturbation theory to fit the observed waveforms. When the observed waveforms lack low frequencies, those schemes may encounter convergence problems due to cycle skipping when the initial velocity model is far from the true model. To mitigate this difficulty, we have developed a new objective function that fits the seismic-waveform intensity, so the dependence of the starting model can be reduced. The waveform intensity is proportional to the square of its amplitude. Forming the intensity using the waveform is a nonlinear operation, which separates the original waveform spectrum into an ultra-low-frequency part and a higher frequency part, even for data that originally do not have low-frequency contents. Therefore, conducting multiscale inversions starting from ultra-low-frequency intensity data can largely avoid the cycle-skipping problem. We formulate the intensity objective function, the minimization process, and the gradient. Using numerical examples, we determine that the proposed method was very promising and could invert for the model using data lacking low-frequency information.

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Regularized full-waveform inversion with automated salt flooding

Geophysics ◽

10.1190/geo2018-0146.1 ◽

2019 ◽

Vol 84 (4) ◽

pp. R569-R582 ◽

Cited By ~ 13

Author(s):

Mahesh Kalita ◽

Vladimir Kazei ◽

Yunseok Choi ◽

Tariq Alkhalifah

Keyword(s):

Optimization Problem ◽

Waveform Inversion ◽

Dominant Frequency ◽

Model Parameters ◽

Constant Density ◽

Data Sets ◽

Full Waveform Inversion ◽

Data Set ◽

Low Frequencies ◽

Full Waveform

Full-waveform inversion (FWI) attempts to resolve an ill-posed nonlinear optimization problem to retrieve the unknown subsurface model parameters from seismic data. In general, FWI fails to obtain an adequate representation of models with large high-velocity structures over a wide region, such as salt bodies and the sediments beneath them, in the absence of low frequencies in the recorded seismic signal, due to nonlinearity and nonuniqueness. We alleviate the ill posedness of FWI associated with data sets affected by salt bodies using model regularization. We have split the optimization problem into two parts: First, we minimize the data misfit and the total variation in the model, seeking to achieve an inverted model with sharp interfaces; and second, we minimize sharp velocity drops with depth in the model. Unlike conventional industrial salt flooding, our technique requires minimal human intervention and no information about the top of the salt. Those features are demonstrated on data sets of the BP 2004 and Sigsbee2A models, synthesized from a Ricker wavelet of dominant frequency 5.5 Hz and minimum frequency 3 Hz. We initiate the inversion process with a simple model in which the velocity increases linearly with depth. The model is well-retrieved when the same constant density acoustic code is used to simulate the observed data, which is still one of the most common FWI tests. Moreover, our technique allows us to reconstruct a reasonable depiction of the salt structure from the data synthesized independently with the BP 2004 model with variable density. In the Sigsbee2A model, we manage to even capture some of the fine layering beneath the salt. In addition, we evaluate the versatility of our method on a field data set from the Gulf of Mexico.

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A new parameter set for anisotropic multiparameter full-waveform inversion and application to a North Sea data set

Geophysics ◽

10.1190/geo2015-0349.1 ◽

2016 ◽

Vol 81 (4) ◽

pp. U25-U38 ◽

Cited By ~ 32

Author(s):

Nuno V. da Silva ◽

Andrew Ratcliffe ◽

Vetle Vinje ◽

Graham Conroy

Keyword(s):

North Sea ◽

Waveform Inversion ◽

Real Data ◽

Model Parameters ◽

Full Waveform Inversion ◽

Radiation Patterns ◽

Data Set ◽

Full Waveform ◽

Seismic Image ◽

Central North Sea

Parameterization lies at the center of anisotropic full-waveform inversion (FWI) with multiparameter updates. This is because FWI aims to update the long and short wavelengths of the perturbations. Thus, it is important that the parameterization accommodates this. Recently, there has been an intensive effort to determine the optimal parameterization, centering the fundamental discussion mainly on the analysis of radiation patterns for each one of these parameterizations, and aiming to determine which is best suited for multiparameter inversion. We have developed a new parameterization in the scope of FWI, based on the concept of kinematically equivalent media, as originally proposed in other areas of seismic data analysis. Our analysis is also based on radiation patterns, as well as the relation between the perturbation of this set of parameters and perturbation in traveltime. The radiation pattern reveals that this parameterization combines some of the characteristics of parameterizations with one velocity and two Thomsen’s parameters and parameterizations using two velocities and one Thomsen’s parameter. The study of perturbation of traveltime with perturbation of model parameters shows that the new parameterization is less ambiguous when relating these quantities in comparison with other more commonly used parameterizations. We have concluded that our new parameterization is well-suited for inverting diving waves, which are of paramount importance to carry out practical FWI successfully. We have demonstrated that the new parameterization produces good inversion results with synthetic and real data examples. In the latter case of the real data example from the Central North Sea, the inverted models show good agreement with the geologic structures, leading to an improvement of the seismic image and flatness of the common image gathers.

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

Remote Sensing ◽

10.3390/rs11161839 ◽

2019 ◽

Vol 11 (16) ◽

pp. 1839

Author(s):

Xu Meng ◽

Sixin Liu ◽

Yi Xu ◽

Lei Fu

Keyword(s):

Time Domain ◽

Waveform Inversion ◽

Radar Data ◽

Dominant Frequency ◽

Data Set ◽

Laplace Domain ◽

Full Waveform ◽

Highly Nonlinear ◽

The Time Domain ◽

Ground Penetrating

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

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Assisting salt model building with reflection full-waveform inversion

Interpretation ◽

10.1190/int-2018-0155.1 ◽

2019 ◽

Vol 7 (2) ◽

pp. SB43-SB52 ◽

Cited By ~ 1

Author(s):

Adriano Gomes ◽

Joe Peterson ◽

Serife Bitlis ◽

Chengliang Fan ◽

Robert Buehring

Keyword(s):

Model Building ◽

Waveform Inversion ◽

Low Frequency ◽

Velocity Model ◽

Full Waveform Inversion ◽

Reflection Data ◽

Full Waveform ◽

Velocity Model Building ◽

Wave Limit ◽

Rabbit Ears

Inverting for salt geometry using full-waveform inversion (FWI) is a challenging task, mostly due to the lack of extremely low-frequency signal in the seismic data, the limited penetration depth of diving waves using typical acquisition offsets, and the difficulty in correctly modeling the amplitude (and kinematics) of reflection events associated with the salt boundary. However, recent advances in reflection FWI (RFWI) have allowed it to use deep reflection data, beyond the diving-wave limit, by extracting the tomographic term of the FWI reflection update, the so-called rabbit ears. Though lacking the resolution to fully resolve salt geometry, we can use RFWI updates as a guide for refinements in the salt interpretation, adding a partially data-driven element to salt velocity model building. In addition, we can use RFWI to update sediment velocities in complex regions surrounding salt, where ray-based approaches typically struggle. In reality, separating the effects of sediment velocity errors from salt geometry errors is not straightforward in many locations. Therefore, iterations of RFWI plus salt scenario tests may be necessary. Although it is still not the fully automatic method that has been envisioned for FWI, this combined approach can bring significant improvement to the subsalt image, as we examine on field data examples from the Gulf of Mexico.

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Tackling cycle skipping in full-waveform inversion with intermediate data

Geophysics ◽

10.1190/geo2018-0096.1 ◽

2019 ◽

Vol 84 (3) ◽

pp. R411-R427 ◽

Cited By ~ 8

Author(s):

Gang Yao ◽

Nuno V. da Silva ◽

Michael Warner ◽

Di Wu ◽

Chenhao Yang

Keyword(s):

Objective Function ◽

Waveform Inversion ◽

Synthetic Data ◽

Velocity Model ◽

Full Waveform Inversion ◽

Data Set ◽

Intermediate Data ◽

Full Waveform ◽

L2 Norm ◽

Recorded Data

Full-waveform inversion (FWI) is a promising technique for recovering the earth models for exploration geophysics and global seismology. FWI is generally formulated as the minimization of an objective function, defined as the L2-norm of the data residuals. The nonconvex nature of this objective function is one of the main obstacles for the successful application of FWI. A key manifestation of this nonconvexity is cycle skipping, which happens if the predicted data are more than half a cycle away from the recorded data. We have developed the concept of intermediate data for tackling cycle skipping. This intermediate data set is created to sit between predicted and recorded data, and it is less than half a cycle away from the predicted data. Inverting the intermediate data rather than the cycle-skipped recorded data can then circumvent cycle skipping. We applied this concept to invert cycle-skipped first arrivals. First, we picked up the first breaks of the predicted data and the recorded data. Second, we linearly scaled down the time difference between the two first breaks of each shot into a series of time shifts, the maximum of which was less than half a cycle, for each trace in this shot. Third, we moved the predicted data with the corresponding time shifts to create the intermediate data. Finally, we inverted the intermediate data rather than the recorded data. Because the intermediate data are not cycle-skipped and contain the traveltime information of the recorded data, FWI with intermediate data updates the background velocity model in the correct direction. Thus, it produces a background velocity model accurate enough for carrying out conventional FWI to rebuild the intermediate- and short-wavelength components of the velocity model. Our numerical examples using synthetic data validate the intermediate-data concept for tackling cycle skipping and demonstrate its effectiveness for the application to first arrivals.

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Deep learning for low-frequency extrapolation from multioffset seismic data

Geophysics ◽

10.1190/geo2018-0884.1 ◽

2019 ◽

Vol 84 (6) ◽

pp. R989-R1001 ◽

Cited By ~ 19

Author(s):

Oleg Ovcharenko ◽

Vladimir Kazei ◽

Mahesh Kalita ◽

Daniel Peter ◽

Tariq Alkhalifah

Keyword(s):

Deep Learning ◽

Seismic Data ◽

Signal To Noise Ratio ◽

Waveform Inversion ◽

Low Frequency ◽

Low Frequencies ◽

Seismic Surveys ◽

Full Waveform ◽

Frequency Components ◽

Marine Seismic

Low-frequency seismic data are crucial for convergence of full-waveform inversion (FWI) to reliable subsurface properties. However, it is challenging to acquire field data with an appropriate signal-to-noise ratio in the low-frequency part of the spectrum. We have extrapolated low-frequency data from the respective higher frequency components of the seismic wavefield by using deep learning. Through wavenumber analysis, we find that extrapolation per shot gather has broader applicability than per-trace extrapolation. We numerically simulate marine seismic surveys for random subsurface models and train a deep convolutional neural network to derive a mapping between high and low frequencies. The trained network is then tested on sections from the BP and SEAM Phase I benchmark models. Our results indicate that we are able to recover 0.25 Hz data from the 2 to 4.5 Hz frequencies. We also determine that the extrapolated data are accurate enough for FWI application.

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Image-guided sparse-model full waveform inversion

Geophysics ◽

10.1190/geo2011-0395.1 ◽

2012 ◽

Vol 77 (4) ◽

pp. R189-R198 ◽

Cited By ~ 48

Author(s):

Yong Ma ◽

Dave Hale ◽

Bin Gong ◽

Zhaobo (Joe) Meng

Keyword(s):

Waveform Inversion ◽

Model Space ◽

Model Parameters ◽

Full Waveform Inversion ◽

Widespread Application ◽

Low Frequencies ◽

Sparse Model ◽

Image Guided ◽

Full Waveform ◽

Recorded Data

Multiple problems, including high computational cost, spurious local minima, and solutions with no geologic sense, have prevented widespread application of full waveform inversion (FWI), especially FWI of seismic reflections. These problems are fundamentally related to a large number of model parameters and to the absence of low frequencies in recorded seismograms. Instead of inverting for all the parameters in a dense model, image-guided full waveform inversion inverts for a sparse model space that contains far fewer parameters. We represent a model with a sparse set of values, and from these values, we use image-guided interpolation (IGI) and its adjoint operator to compute finely and uniformly sampled models that can fit recorded data in FWI. Because of this sparse representation, image-guided FWI updates more blocky models, and this blockiness in the model space mitigates the absence of low frequencies in recorded data. Moreover, IGI honors imaged structures, so image-guided FWI built in this way yields models that are geologically sensible.

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Time-domain full-waveform inversion of exponentially damped wavefield using the deconvolution-based objective function

Geophysics ◽

10.1190/geo2017-0057.1 ◽

2018 ◽

Vol 83 (2) ◽

pp. R77-R88 ◽

Cited By ~ 15

Author(s):

Yunseok Choi ◽

Tariq Alkhalifah

Keyword(s):

Objective Function ◽

Frequency Band ◽

Waveform Inversion ◽

Low Frequency ◽

Full Waveform Inversion ◽

Low Frequencies ◽

Full Waveform ◽

Adjoint State ◽

Recorded Data ◽

Adjoint State Method

Full-waveform inversion (FWI) suffers from the cycle-skipping problem when the available frequency-band of data is not low enough. We have applied an exponential damping to the data to generate artificial low frequencies, which helps FWI to avoid cycle skipping. In this case, the least-squares misfit function does not properly deal with the exponentially damped wavefield in FWI because the amplitude of traces decays almost exponentially with increasing offset in a damped wavefield. Thus, we use a deconvolution-based objective function for FWI of the exponentially damped wavefield. The deconvolution filter includes inherently a normalization between the modeled and observed data; thus, it can address the unbalanced amplitude of a damped wavefield. We specifically normalize the modeled data with the observed data in the frequency-domain to estimate the deconvolution filter and selectively choose a frequency-band for normalization that mainly includes the artificial low frequencies. We calculate the gradient of the objective function using the adjoint-state method. The synthetic and benchmark data examples indicate that our FWI algorithm generates a convergent long-wavelength structure without low-frequency information in the recorded data.

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