Tomographic full waveform inversion (TFWI) by combining full waveform inversion with wave-equation migration velocity analysis
Biondo Biondi and Ali Almomin
By extending the velocity-model domain to subsurface offsets we solve the local-minima problem of data-fitting waveform inversion. We then regularize the extended-model data-fitting inversion with the addition of an image-focusing term to the objective function, therefore achieving robust global convergence of the waveform inversion problem. The method shares with full waveform inversion the advantage of simultaneously solving for all the wavelengths of the model, but it also has the global convergence characteristics of wave-equation migration velocity analysis. Numerical implementation of the proposed inversion method requires the solution of an extended wave-equation where velocity is a convolutional, instead of scalar, operator. The resulting method is therefore computationally intensive, but it can be easily tested in 2D. A simple example with a Gaussian velocity anomaly illustrates how the reflections from the anomaly recorded in the low-frequency components of the data increase the spatial resolution of the final inversion results. Numerical tests performed on synthetic data from a modified Marmousi model demonstrate the global convergence as well the high-resolution potential of the method.
Tomographic full waveform inversion: Practical and computationally feasible approach
Ali Almomin and Biondo Biondi
We provide an alternative formulation to tomographic full-waveform inversion which computes the backscattering and the forward scattering components of the model separately. To maintain high resolution results of tomographic full-waveform inversion, the two components of the gradient are mixed based on a Fourier domain scale separation. This formulation is based on the Born approximation where the medium parameters are broken into a long wavelength and short wavelength components. This approximation has an underlying assumption that the data contain primaries only without multiples. The results of the Marmousi model show that convergence is possible even with large errors in the initial model that would have prevented convergence to conventional full-waveform inversion.
Residual moveout-based wave-equation migration velocity analysis in 3-D
Yang Zhang and Biondo Biondi
In the previous report (SEP-143), we proposed a novel approach to perform Wave-Equation Migration Velocity Analysis (WEMVA) using Residual Moveout (RMO) parameters. The 2-D examples we tested showed that this approach is immune to the cycle-skipping problem, and it produces high-quality gradients. In this paper, we try to address the theoretical and practical challenges of extending this method to three-dimensions. Specifically, we propose a new parameterization for the 3-D angle-domain residual moveout and a generalized 3-D RMO-based WEMVA theory. We also address the practical issues in 3-D transforms between subsurface offset and angle-domain image gathers.
Fast velocity model evaluation with synthesized wavefields
Sophisticated interpretation tools such as automatic image segmentation allow for near-instant creation of multiple, geologically-plausible velocity models for a given imaging target. However, testing each of these models via remigration of the full dataset is not computationally feasible. By using an initial image to synthesize both an areal source function and a Born-modeled dataset, a large number of models can be tested and evaluated in a fraction of the time required for even a single migration of the full dataset. Tests using synthetic 2D data suggest that this method can demonstrate the impact of imaging using different models, even with small differences among them. Furthermore, a measure of image focusing can help quantify the relative merits of the different models.
Early-arrival waveform inversion: Application to cross-well field data
Xukai Shen, Tieyuan Zhu, and Jerry M. Harris
We apply early-arrival waveform inversion to a cross-well field data in order to find the extent of a potential reservoir that terminate somewhere between the two wells. Ray-based tomography result is too smooth to resolve the boundary. On the other hand, waveform inversion is able to better utilize the advantage of short wavelength cross-well data, resulting in a velocity model that defines the potential reservoir boundary more sharply. Details from the waveform inversion result are verified by well-log data.
Imaging with anisotropy and attenuation
VTI migration velocity analysis using RTM
Yunyue (Elita) Li, Peng Shen, and Colin Perkins
We use a vertical transverse isotropic (VTI) two-way propagation engine to perform wave-equation migration velocity analysis. Using reverse time migration (RTM), we have a chance to characterize the wavefields propagating at large angles and to image steeply dipping reflectors. By including VTI Thomsen parameters, we can better describe the properties of the subsurface. We first derive the migration velocity analysis gradient when using a first-order VTI two-way wave-equation. Then, we test our method on a synthetic VTI Marmousi model. The inversion results show that our method can resolve a better velocity model and a better-focused subsurface image.
Early-arrival waveform inversion for near-surface velocity and anisotropic parameters: modeling and sensitivity kernel analysis
Understanding the relationships between vertical velocity, anisotropic parameters ε, δ and traveltime is important both for parametrization of waveform inversion and proper interpretation of waveform inversion results. Forward modeling indicates that traveltime is more sensitive to vertical velocity and ε than to δ. This suggests that δ may be fixed during an inversion for vertical velocity and ε. When vertical velocity and ε are directly parametrized, ε changes little during the inversion. A more balanced sensitivity kernel can be obtained by using different parametrizations, such as the vertical and horizontal velocities, or the logarithms of slowness and ε.
Wave-equation migration velocity analysis for anisotropic models on 2-D ExxonMobil field data
Yunyue (Elita) Li
I apply an anisotropic wave-equation migration velocity analysis (WEMVA) method on a 2-D ExxonMobil dataset. I start from the best anisotropic model available from seismic, walkaway VSPs and check shots. The initial gathers in the subsurface offset domain are already fairly well focused; however, after inversion using our WEMVA method, I obtain an image with better continuity and clearer faults. By looking at the angle gathers, it is easy to distinguish the contribution of the improved η model from the contribution of the velocity model.
Wave-equation migration velocity analysis for VTI media using optimized implicit finite difference
Yunyue (Elita) Li
Anisotropic wave-equation migration velocity analysis (WEMVA) requires fast and accurate wave modeling at all angles. I use an optimized implicit finite difference one-way propagation engine to improve both the efficiency and accuracy of this process. In this implicit finite difference scheme, anisotropic parameters η and δ are mapped into two finite difference coefficients, α and β. When computing the perturbed wavefields from model perturbation, I apply a chain rule to link the wave equation with the actual anisotropic parameters via the finite difference coefficients. I test the implementation by impulse responses in both 2D and 3D. The sensitivity kernels for wave-equation reflection tomography confirm the theoretical understanding that waves have a higher sensitivity for η at large angles and a higher sensitivity for vertical velocity at small angles.
Early-arrival waveform inversion for near-surface velocity and anisotropic parameters: inversion of synthetic data
I test different inversion parametrizations of vertical velocity and anisotropic parameter ε. A model space parametrized by the squares of vertical and horizontal velocity results in vertical velocity and ε updates with opposite signs. On the other hand, a model space parametrized by the logarithm of the vertical velocity squared and ε has more reasonable updates, as well as better data matching. However, ambiguity does exist in the inversion results between vertical velocity and ε. I clearly demonstrate these findings using a synthetic example.
Estimation of Q from surface-seismic reflection data in data space and image space
Estimating seismic attenuation, or Q, is crucial for imaging applications, reservoir characterization and seismic acquisition design. A new method for Q estimation uses central frequency shift to analyze Q versus offset or angle in both the data space and image space. Q migration and Q compensation are also presented here for the image-based Q estimation. Applying this new approach to a 2D synthetic model demonstrate that both data-based and image-based methods are able to accurately estimate Q for a model with flat reflectors and homogeneous medium.
Reducing the cost of waveform imaging
Image gather reconstruction using StOMP
Robert G. Clapp
Constructing 3-D angle gathers through cross-correlation poses a computational problem primarily due to the accompanying increase in volume size which forces the gathers to be stored in a computationally more expensive memory level. Compressive sensing can be used to mitigate this challenge. The correlation volume size can be reduced by both phase encoding and random subsampling. The full correlation gathers can then by reconstructed using an l1 inversion scheme known as Stagewise Orthogonal Matching Pursuit. Preliminary results indicate that almost all angle gather information can be recovered.
Linearised inversion with GPUs
Chris Leader and Robert Clapp
Graphical Processing Units (GPUs) can provide considerable computational advantages over multi-core CPU nodes or distributed networks by locally accelerating certain types of floating point operations. However, when processing and inverting exploration scale seismic datasets we encounter two key problems - compounded disk IO (explicit routing through the host is necessary) and the relatively small memory provided by the GPU (≤ 6 Gbytes, restricting model sizes that can be allocated). As shown in an earlier discussion the IO bottleneck on the adjoint side can be somewhat circumvented by using random domain boundaries. Herein will be discussed how the forward modelling routine must be adapted to create an adjoint pair such that least-squares iterative inversion can be performed. We will then analyse how domain decomposition and P2P communication can be used to propagate over larger model sizes in such a way that communication can be effectively hidden and subsequently we can observe linear scaling.
How incoherent can we be? Phase-encoded linearised inversion with random boundaries
Chris Leader and Ali Almomin
To perform linearised inversion on seismic exploration scale datasets we are continually looking for methods to accelerate computation and reduce data handling overhead. One option to accelerate reverse time imaging is to use random domain boundaries for the source wavefield computation, alleviating much of the required IO in favour of some additional computation. Additionally, data handling problems can be addressed by phase-encoding data (weighting, shifting, summing) and then inverting for a common model between realisations. Both random boundary and phase-encoding methods rely on wavefield incoherency during correlation and stacking to build a clean image. Here we investigate if these can be effectively used together, or if these techniques combined create wavefields that are too incoherent, slowing convergence as a function of cost when compared to linearised inversion without phase-encoding. We show that by using multiple realisations per iteration we can improve convergence and create cleaner reflectivity images.
FWI with different boundary conditions
Xukai Shen and Robert G. Clapp
We compare time-domain Full Waveform Inversion using different boundary schemes: absorbing boundary condition, random boundary condition and continuation of velocity. The absorbing boundary condition requires saving the wavefield, the other two do not, but require extra wavefield modeling. The random boundary condition results in a gradient calculation that is almost as good as the absorbing boundary condition, whereas the continuation of velocity result in a gradient that has strong artifacts. However, the final inversion results using different boundaries are similar where there is a lot of data to contrain the model. When there is not so much data to constrain the inversion, results from using the random boundary condition are similar to those resulting from the absorbing boundary condition, but the continuation of velocity boundary condition does not work as well. We demonstrate this with synthetic examples.
Novel models and data types
Continuous monitoring by ambient-seismic noise tomography
Sjoerd de Ridder
Seismic arrays permanently installed over a hydrocarbon reservoir can record data continuously, even in the absence of active seismic shooting. Here we study ambient-seismic noise tomography as a tool for continuous monitoring. This is of interest for both monitoring production-related changes over a long time scale (years), but it may also be a source of data for monitoring hazards over short time scales (days to weeks). We compare cross-correlations of partial recordings to cross-correlations of the full recording as a function of absolute recording time, inter-station distance and frequency. We use straight-ray tomography to image the virtual-seismic sources for travel times picked after band passes for various frequencies. The correlations converge faster for nearby receiver pairs and lower frequencies than for further receiver pairs and higher frequencies. The convergence rate also depends on the strength of the microseism energy in the ambient seismic field. Features visible in ambient-seismic noise tomography images of Scholte wave group-velocity for various frequency bands are compared to slices of a P wave velocity cube obtained from full waveform inversion of active seismic data. Clear similarities indicate that the Scholte wave group-velocity between 0.15-0.75 Hz forms an image up to a depth of 240 meters. Clear similarities indicate that the Scholte wave group-velocity between 0.15-0.75 Hz forms an image up to a depth of 240 meters. This study shows the feasibility of ambient-seismic noise tomography monitoring of the near surface.
Single frequency 2D acoustic full waveform inversion
Sjoerd de Ridder, Ali Almomin, and Musa Maharramov
Single frequency 2D acoustic full waveform inversion shows promise as an approach to inverting the acoustic, dispersive 2D acoustic wave equation for the underlying velocity cube as a function of frequency. A GPU friendly finite difference time domain kernel and an associated frequency domain optimization scheme are shown to retrieve a Gaussian anomaly via full waveform inversion for several acquisition geometries. Although the wave field is single frequency, the spatial distance of sources and receivers allows their sensitivity kernels to interfere and form a gradient that can recover the anomaly.
Identifying reservoir depletion patterns from production-induced deformations with applications to seismic imaging
The ultimate objective of our research is to study the feasibility of using subsidence measurements for the regularization of linearized waveform inversion of time-lapse data sets, and for differential travel-time tomography. In this paper, we focus on developing a robust framework for inverting pore pressure change in a production reservoir from partial displacement or subsidence data, and on estimating subsurface displacements from the inverted pore pressure change. We discuss potential applications of the resulting displacement fields to the estimation of production-induced change of seismic velocities and impedance. Sensitivity of the proposed framework to the uncertainty and spatial heterogeneity of the poroelastic subsurface parameters is discussed as well.
Correlation energy between surface and borehole stations at the Valhall field
Jason P. Chang and Sjoerd de Ridder
Ocean-bottom cables at the Valhall field have provided an abundance of passive seismic data for testing the potential of seismic interferometry. We cross-correlate recordings from surface stations and 2-km deep borehole stations. Results show correlated energy at frequencies between 0.175 Hz and 1.75 Hz. However, the signal we retrieved is not time-symmetric, as there are multiple arrivals at acausal correlation time lags compared to the one arrival visible at causal time lags. The apexes of the causal events are found at acausal time lags rather than at zero time lag. The virtual source is centered northwest of the borehole stations at the offshore platform. We conclude that these observations are due to the borehole acting as a wave-guide. Because this mechanism does not satisfy the conditions of seismic interferometry, we cannot interpret these cross-correlation results as inter-station Green's functions.
Hunting for microseismic reflections using multiplets
Noha S. Farghal and Stewart A. Levin
Microseismic monitoring of hydraulic fracturing is often used to locate reactivated faults and newly created fractures by locating the microseismic sources that occur during the process. These microseisms generate reflections as well as direct arrivals, but they tend to be fairly weak and quite difficult to locate and image. In this work we identify multiplets, i.e. repeated microseisms originating from about the same subsurface location, and thereby identify their consistent reflections.
Imaging through inversion
Joint imaging with streamer and ocean bottom data
Mandy Wong, Shuki Ronen, and Biondo Biondi
In the past we have shown that up-going and down-going (mirror) imaging can be combined in a joint inversion. We extend the method to joint-inversion of nodes and streamers data. Compared to conventional post-imaging merging, the joint inversion enhances resolution, suppresses migration artifacts, and more importantly, brings up the relative amplitude of true reflectors in the subsurface. We present a linearized inversion scheme for imaging narrow-azimuth and ocean-bottom data. We demonstrate the concept and methodology in 2D with a synthetic Marmousi model.
Imaging with multiples using linearized full-wave inversion
Mandy Wong, Biondo Biondi, and Shuki Ronen
We presents a technique for imaging both primaries and multiples using linearized inversion. When used with a suitable migration velocity model, linearized full-wave inversion (LFWI) makes use of the multiple energy as signal while removing the crosstalk in the image. By using the two-way propagator in both modeling and migration, we can image a class of multiply scattered events. Such events can scatter off sharp-interfaces in the migration velocity many times but only interact with the (reflectivity) model once. We demonstrate the concept and methodology in 2D with a synthetic Sigsbee2B model.
P/S separation of OBS data by inversion in a homogeneous medium
A separation of pressure waves from shear waves in OBS data prior to processing can improve the resulting image. I present a nearly medium-independent P/S separation method, which operates by reconstructing the observed data using a homogeneous medium. The closer the homogeneous medium parameters are to the actual parameters of the medium in which the receivers are planted, the better the separation results. Synthetic tests indicate that the proposed method works reliably for land data even when using wrong medium parameters, but cannot work for OBS data in its current configuration.
Edge-preserving smoothing for segmentation of seismic images
In many disciplines, pre-processing images by smoothing is common prior to automatic image segmentation; however, traditional smoothing blurs boundaries and can be counterproductive, especially for seismic images. Here, an edge-preserving smoothing technique based on directional maximum homogeneity is introduced for 3D seismic images, and tested on both synthetic and field data. Edge-preserving smoothing is shown to both decrease the level of noise in an image, and improve the accuracy of automatic segmentation results. In addition, a ``hybrid'' smoothing technique blending traditional and edge-preserving smoothing combines the advantages of each and produces encouraging results.
Enhanced interpreter-aided salt-boundary extraction using shape deformation
Yang Zhang and Adam D. Halpert
In many marine seismic exploration projects, precise interpretation of the salt-body geometry (which is also called “salt-body segmentation”) is a key component of building the subsurface velocity model. However, segmentation of salt is very human-intensive, even with the help of currently available semi-automatic computer software. This paper addresses the problem of automatically and accurately tracking the salt boundary in a series of neighboring seismic image slices, given an accurate salt segmentation for only one single reference slice. (The reference segmentation can be done manually). We achieve this using a landmark-based shape deformation technique plus SVM (Support Vector Machine) style regression. An example on a 3-D Gulf of Mexico data set demonstrates the effectiveness of our approach.
Decon and Interpolation
Polarity preserving decon in ``N log N'' time
A slight modification to Fourier spectral factorization enables deconvolution to preserve and enhance seismogram polarities. It spikes the center lobe of the Ricker wavelet. It works by tapering at small lags the antisymmetric part of the time-domain representation of the log spectrum.
Decon in the log domain with variable gain
Jon Claerbout, Antoine Guitton, and Qiang Fu
We base deconvolution on the concept of output model sparsity. We improve our method of log spectral parameterization by including time-variable gain. Since filtering does not commute with time variable gain, gain is now done after decon (not before). Results at two survey locations confirm the utility. We resolve a stability issue with a long-needed regularization. An intriguing theoretical aspect shows that log spectral parameterization links penalty functions to crosscorrelation (not autocorrelation) statistics of outputs.
Integral operator quality from low order interpolation
Stewart A. Levin
In most discussions of interpolation methods, it is the worst-case behavior that dominates the analysis. From a systems point of view, one really should analyze how that interpolation is used in producing an end product in order to determine the interpolation's suitability. In this report I look at the summation operators slant stack, NMO and Kirchhoff migration as the ``systems'' and determine that their output quality can be significantly better than the traditional take on interpolation would suggest. In one scenario, I even found nearest neighbor interpolation did the job even better than linear interpolation.
Recent progress regarding logarithmic Fourier-domain bidirectional deconvolution
Bidirectional deconvolution in the Fourier domain is a new method of removing the mixed phase wavelet from seismic data. I demonstrate that this is self-preconditioned, therefore a scheme that has a preconditioner in the logarithmic Fourier-domain deconvolution is not necessary. I show a simple synthetic test case which incorporates a gain function into the deconvolution method.