Localized adaptive waveform inversion: regularizations for Gabor deconvolution and 3-D field data application

Achieving high-resolution intracranial imaging in a safe and portable manner is critical for the diagnosis of intracranial diseases, preoperative planning of craniotomies and intraoperative management during craniotomy procedures. Adaptive waveform inversion (AWI), a variant of full waveform inversion (FWI), has shown potential in intracranial ultrasound imaging. However, the robustness of AWI is affected by the parameterization of the Gaussian penalty matrix and the challenges posed by transcranial scenarios. Conventional AWI struggles to produce accurate images in these cases, limiting its application in critical medical settings. To address these issues, we propose a stabilized adaptive waveform inversion (SAWI) method, which introduces a user-defined zero-lag position for the Wiener filter. Numerical experiments demonstrate that SAWI can achieve accurate imaging under Gaussian penalty matrix parameter settings where AWI fails, perform successful transcranial imaging in configurations where AWI cannot, and maintain the same imaging accuracy as AWI. The advantage of this method is that it achieves these advancements without modifying the AWI framework or increasing computational costs, which helps to promote the application of AWI in medical fields, particularly in transcranial scenarios.

SUMMARYCorrectly interpreting phase events thanks to data processing techniques based on correlation or deconvolution has been the focus of numerous studies in the field of high-resolution seismic imaging using full-waveform inversion. To mitigate the non-convexity of the misfit function and the risk to converge towards non-informative local minima, correlation and deconvolution techniques make it possible to focus on phase information instead of amplitude information and to design more convex misfit function, alleviating the dependency of the full-waveform inversion process on the accuracy of initial models. Such techniques however rely on the assumption that phase events can be compared one by one, or that all the phase events are shifted in time in a similar way. This assumption is not satisfied in practice, which limits the effectiveness of these correlation/deconvolution-based methods. To overcome this issue, we propose to account for the non-stationary relation between observed and predicted data through a local in-time deconvolution technique, based on time–frequency analysis of the signal using a Gabor transform. This makes it possible to estimate instantaneous time-shift between locally coherent phase events. This strategy generalizes the conventional normalized deconvolution technique, which has been popularized under the name of adaptive waveform inversion. To support the introduction of our novel method, we compare it with four misfit functions based respectively on classical cross-correlation, penalized cross-correlation, penalized deconvolution, and adaptive waveform inversion. We analyse the behaviour of these methods on specific scenarios, and then propose a comparison on 2-D synthetic benchmarks. We show how our ‘localized’ adaptive waveform inversion applies in these realistic tests and overcomes some of the limitations of the aforementioned techniques.

Summary Considering the complex nonlinearity between seismic data and perturbations in the model, a waveform optimization problem based on the least square sample-to-sample comparison has proven to be inadequate in dealing with such nonlinearity. A slew of more global comparison based optimizations have proven their values in circumventing such nonlinearity. Among these methods in particular, adaptive waveform inversion (AWI), which is based on the deconvolution operator, offers the right balance between phase matching and amplitude normalization. The key element in the AWI success is the normalization factor, which is inherently offered by the instantaneous traveltime measure used to unwrap the phase. Thus, we recast the AWI problem using a misfit function based on the original generalized instantaneous travel time, which highlights some of the features of AWI and traces their roots. We demonstrate that the misfit function of AWI is actually the least square averaged instantaneous travel time over multiple frequencies. We use the Marmousi model and Chevron 2014 FWI benchmark dataset to verify the effectiveness of the proposed method.

Full waveform inversion (FWI) iteratively improves the accuracy of the model by minimizing the discrepancies between the predicted and the observed data. However, FWI commonly suffers from cycle skipping when the initial model is poor, leading to an erroneous result. To mitigate this problem, we propose deep-learning-backed adaptive waveform inversion (DL-AWI), which introduces a deep twin neural network to precondition the waveforms and compare the ratio of two signals with a zero-lag spike, thereby enhancing the stability of the inversion process. DL-AWI can project the synthetic and observed signals into an extended latent space via several convolutional neural networks (CNNs) with shared weights, which can accelerate the data matching. Compared with classic FWI methods, the proposed DL-AWI provides a wider space for model updates, significantly decreasing the risk of being trapped in local minima. We use synthetic and field examples to validate its efficiency in subsurface model inversion, and the results show that DL-AWI is robust even when a poor initial model is provided.

Adaptive waveform inversion (AWI) provides a means of performing full-waveform inversion (FWI) that appears to be immune to the effects of cycle skipping. However, the form of the AWI algorithm suggests that it could have increased sensitivity to errors in the assumed source wavelet, to noise in the field data, and to inadequacies in the physics used to simulate wave propagation. We examine each of these for a synthetic model. We show that AWI is in fact less sensitive than FWI to errors in the source wavelet, and is no more sensitive to errors in the data and in the modelling than is FWI. It appears likely that the immunity that AWI displays to cycle skipping also contributes to its reduced sensitivity to errors in the assumed source wavelet.

Application of the least-squares inversion method: Fourier series versus waveform inversion

We explore the use of stochastic optimization methods for seismic waveform inversion. The basic principle of such methods is to randomly draw a batch of realizations of a given misfit function and goes back to the 1950s. The ultimate goal of such an approach is to dramatically reduce the computational cost involved in evaluating the misfit. Following earlier work, we introduce the stochasticity in waveform inversion problem in a rigorous way via a technique calledrandomized trace estimation. We then review theoretical results that underlie recent developments in the use of stochastic methods for waveform inversion. We present numerical experiments to illustrate the behavior of different types of stochastic optimization methods and investigate the sensitivity to the batch size and the noise level in the data. We find that it is possible to reproduce results that are qualitatively similar to the solution of the full problem with modest batch sizes, even on noisy data. Each iteration of the corresponding stochastic methods requires an order of magnitude fewer PDE solves than a comparable deterministic method applied to the full problem, which may lead to an order of magnitude speedup for waveform inversion in practice.

SUMMARY Seismic refraction data collected in land or marine surveys are often complicated by small, near-random time shifts of the individual seismograms relative to one another. For the case of fixed-receiver geometry, these shifts arise primarily from basement topography, often masked by sediments, beneath each source. Although a topographic timing correction is normally applied to the seismograms, the effects of poorly-known topography produce residual shifts in the seismograms. The application of additional time corrections (statics) can bias solutions computed using iterative waveform inversion, because a suitable set of statics can make a poor starting velocity model appear to match the data. Without any correction, however, seismograms computed from even the best-fitting laterally homogeneous model may not agree with the data to within one-half period. In this paper I present a formalism for iterative waveform inversion of refraction data containing residual time shifts. In this method the statics are directly modelled in the synthetic seismograms, and new values estimated in the inversion together with new velocity model parameters. The method is similar to studies of statics in seismic reflection data, but in this case values for the statics are estimated simultaneously with the velocity model; with the use of suitable norms the new velocity model is required to be smooth and the statics required to be small. Convergence is achieved when the velocity profile's roughness, the RMS size of the statics, and the waveform misfit are in equilibrium. Because the starting synthetics need not match the data to within one-half period, a much larger set of velocity models used for an initial guess will converge on the family of solutions than for waveform inversions in which time shifts are not used. The original, arbitrary starting model loses its importance after the first iteration, and solutions meeting the convergence criterion appear to be independent of the particular choice of starting model. Numerical tests show the method to be robust: an inaccurate starting model that fits the data only through a biased set of statics is accurately corrected. The method is illustrated on a set of refraction data collected in the vicinity of the East Pacific Rise. In this application two quite different starting velocity models converge to a common family of solutions.

Gas hydrates are frozen methane gas that forms at appropriate pressure and temperature conditions. They are found in the marine sediments along continental margins worldwide. They have the economic potential of being tapped as a fuel source and also have the potential as a “greenhouse” agent after being freed into the atmosphere. In seismic sections, the occurrence of the base of gas hydrates, in some areas is often marked by a bright amplitude reflection. Such reflections follow the sea floor topography and are called bottom‐simulating reflectors (BSR). The BSRs have negative polarity with respect to the sea‐floor reflection and, in a common shot or a CDP gather, the amplitude increases with offset. The negative impedance contrast causing BSRs may be due to negative velocity contrast between hydrated sediments and normal sediment below or due to the presence of free gas at the base of the hydrates. In this paper, we carry out a prestack seismic waveform inversion of multichannel seismic data collected in the offshore of South Carolina to investigate the origin of the BSRs. We apply a multistage seismic waveform inversion for this purpose. A nonlinear optimization method is applied to estimate the low‐frequency component of the velocity, whereas an amplitude‐variation‐with‐offset inversion is applied to determine high‐frequency components of the velocity field. Our detailed seismic waveform inversion along the seismic line results in at least three low‐velocity zones where the velocity is well below the velocity of the normal sediments. Such low‐velocity zones correlate very well with negative fluid factors indicating the presence of free gas. Thus we conclude that the BSR is caused by free gas at the base of the hydrates in this region. We identify at least two other layers of free gas beneath the hydrates. The thickness of each of these layers is below the resolution of the source wavelet. Our results confirm similar findings reported from Ocean Drilling Program drilling and vertical‐seismic‐profiling analysis in the same general area.

A shot-encoding technique can be used in seismic waveform inversion to significantly reduce the computational cost by reducing the number of seismic simulations in the inversion procedure. Here we developed two alternative shot-encoding schemes to perform simultaneous-sources waveform inversion. The first scheme (I) encodes shot gathers with random-phase rotations applied to seismic traces. The second scheme (II) encodes shot gathers with random static time shifts. The well-known polarity encoding scheme (III) is just a special case of the random-phase rotation scheme. The second scheme is a variation of the conventional static shift encoding (IV), but the static time shifts in the second scheme are limited to one period of the dominant frequency. All encoded shot gathers are added up into a single super-shot gather for seismic waveform inversion. We perform the time-domain waveform inversion, using these shot-encoding schemes in conjunction with a restarted L-BFGS algorithm in the iterative inversion. The effectiveness and efficiency analyses demonstrate that the two shot-encoding schemes (I and II) proposed in this paper may improve the convergence of the iterative inversion, reduce the crosstalk effect among shots and consequently produce a subsurface velocity model with a high resolution.

A method is presented for the inversion of waveform data for the three‐dimensional distribution of seismic wave velocities. The method is applied to data from the global digital networks (International Deployment of Accelerometers, Global Digital Seismograph Network); the selected data set consists of some 2000 seismograms corresponding to 53 events and 870 paths. The moment tensors of the earthquakes are determined through an iterative procedure which minimizes the corrupting influence of lateral heterogeneity. A global model is constructed for shear wave velocity, expanded up to degree and order 8 in spherical harmonics, and described by a cubic polynomial in depth for the upper 670 km of the earth's mantle. Although no a priori information is incorporated, the model predictions reproduce much of what is known about the dispersion of mantle waves, for example, high phase velocities for shields, low velocities at ridges, and a strong degree 2 pattern for Rayleigh waves. Since the method makes use of complete waveforms, overtone data are also included. It is shown that the model is reproducible in that substantially the same model can be constructed from each half of the total data set considered independently. The model shows that shields and ridges are major features in the depth interval 25–250 km. The ridges of the southern Pacific and the larger shields persist to 350 km, but the SouthEast Indian Rise is underlain by a high‐velocity anomaly at this depth, as is much of the Mid‐Atlantic Ridge. At 450–650 km the major features are a broad region of high velocities incorporating South America, much of the South Atlantic and parts of West Africa, a broad region of low velocities in the central and eastern Pacific, high velocities in the western Pacific, and a low‐velocity anomaly beneath the Red Sea and the Gulf of Aden. In the absence of a crustal correction, degrees 2 and 3 show a high positive correlation with the geoid; paradoxically, this is largely destroyed when the distribution in crustal thickness is taken into account. Spherical harmonic degrees 4–7 show a significant negative correlation.

Early arrival waveform inversion (EWI) is an efficient waveform inversion method to resolve the near-surface velocity structures with high resolution. However, EWI suffers from cycle-skipping issues, which is similar to the full waveform inversion (FWI), resulting in artifacts produced in the reconstructed near-surface velocity model. These artifacts denote that the inversion process drops to a local minimum and are challenging to remove because it typically couples with the velocity layers or objects. We suggest that the artifacts produced by EWI typically correspond to velocities with large variations, and such velocities could cause local variations in common offset gather (COG) traveltimes after applying static corrections. We develop an annealing static-constrained early arrival waveform inversion (ASEWI) by minimizing the COG first-arrival variances after applying both long- and short-wavelength statics with an annealed process. The static-constrained term can guide the objective function to converge well and avoids dropping into local minima in the early stage of the iterations. As the inversion progresses, the weight of the static-constrained term is gradually weakened for the better fittings of early arrivals. ASEWI does not introduce additional undetermined coefficients due to an annealed scaling factor that is only associated with the number of iterations. Thus, the velocity artifacts produced by the local minimum in EWI can be suppressed by removing the corresponding variations in COG traveltimes. The effectiveness of the ASEWI method is demonstrated by the applications of both synthetic and field data examples. We also introduce the conventional first-arrival traveltime tomography (FTT) and EWI for comparisons. Both synthetic and field data results show that SEWI can improve the imaging resolution, suppress the artifacts, and produce near-surface velocity and static solutions with higher accuracy compared with two other methods.

Ultrasound computed tomography (USCT) holds great promise for improving the detection and management of breast cancer. Because they are based on the acoustic wave equation, waveform inversion-based reconstruction methods can produce images that possess improved spatial resolution properties over those produced by ray-based methods. However, waveform inversion methods are computationally demanding and have not been applied widely in USCT breast imaging. In this work, source encoding concepts are employed to develop an accelerated USCT reconstruction method that circumvents the large computational burden of conventional waveform inversion methods. This method, referred to as the waveform inversion with source encoding (WISE) method, encodes the measurement data using a random encoding vector and determines an estimate of the sound speed distribution by solving a stochastic optimization problem by use of a stochastic gradient descent algorithm. Both computer simulation and experimental phantom studies are conducted to demonstrate the use of the WISE method. The results suggest that the WISE method maintains the high spatial resolution of waveform inversion methods while significantly reducing the computational burden.

Near-surface seismic imaging often plays a significant role in producing quality data processing results for the deep subsurface in land and shallow marine environments. First-arrival traveltime tomography is a common approach for near-surface imaging due to its high efficiency and simplicity. However, the method faces issues of missing hidden layers and resolving the structures with low resolution. On the other hand, waveform inversion should offer better solutions for dealing with these issues, but it may suffer from the cycle-skipping problem. We intend to use the advantages and reduce the disadvantages of the two methods by developing a new strategy of alternately applying traveltime tomography and waveform inversion through iterations. First-arrival traveltime tomography applies a wavefront raytracer and a nonlinear inversion approach. Waveform inversion is a multiscale approach in which a wavelet transform is applied in the data domain to better handle the cycle-skipping problem. By alternating the two inversions rather than performing a joint inversion, we reduce the memory requirements and avoid nonphysical scaling problems between the two approaches. Using one synthetic and two real data examples, we determine that alternating inversions minimize two separate objective functions at the same time and constrain the near-surface structures fairly well compared with the waveform inversion method alone. For the field examples, the new method avoids generating the obvious artifacts and provides results consistent with the geology analysis of those areas.

SUMMARYInversion for seismological structures of the oceanic lithosphere–asthenosphere system is important to understand the mechanisms of plate tectonics. Previous models of the oceanic upper mantle have been primarily obtained via global tomography using surface waveforms. However, besides scarcity of waveform data in the oceanic regions, difficulties in fitting phases for shorter-period components in the previous global tomography have yielded resultant models that possess poor resolutions above $\sim \, 50$ km depth. Recent developments of broad-band ocean-bottom seismometer (BBOBS) arrays provide larger amount of seismic data with epicentral distances of <20°. In this study, we develop an appropriate method to fully utilize the information contained in the shorter-period components of BBOBS arrays. We first fit the envelopes without phase information to analyse the shorter-period components (8–60 s) which are generally unavailable in the conventional phase fitting. We then use the resultant model as our initial model for waveform inversion of the longer periods (12.5–200 s) to fit the phase, which allows us to infer a continuous structure model from the crust to the asthenosphere. We demonstrate the validity of this combined envelope-fitting and waveform inversion method by analysing the waveform data from a BBOBS array that was deployed in the Northwestern Pacific and has recorded events in the vicinity of the Japan Trench to obtain the average velocity structure between the event and station arrays. We independently resolve the crustal compressional and shear wave velocities, and thickness by analysing the envelopes, which minimizes biases in the subsequent waveform inversion. We also find that the waveform inversion improves the resolution in the asthenosphere. Our results suggest that further extension of this method should improve our knowledge of the oceanic lithosphere–asthenosphere system.