Implementation aspects of eigendecomposition‐based high‐resolution velocity spectra

In this paper, we discuss high‐resolution coherence functions for the estimation of the stacking parameters in seismic signal processing. We focus on the Multiple Signal Classification which uses the eigendecomposition of the seismic data to measure the coherence along stacking curves. This algorithm can outperform the traditional semblance in cases of close or interfering reflections, generating a sharper velocity spectrum. Our main contribution is to propose complexity‐reducing strategies for its implementation to make it a feasible alternative to semblance. First, we show how to compute the multiple signal classification spectrum based on the eigendecomposition of the temporal correlation matrix of the seismic data. This matrix has a lower order than the spatial correlation used by other methods, so computing its eigendecomposition is simpler. Then we show how to compute its coherence measure in terms of the signal subspace of seismic data. This further reduces the computational cost as we now have to compute fewer eigenvectors than those required by the noise subspace currently used in the literature. Furthermore, we show how these eigenvectors can be computed with the low‐complexity power method. As a result of these simplifications, we show that the complexity of computing the multiple signal classification velocity spectrum is only about three times greater than semblance. Also, we propose a new normalization function to deal with the high dynamic range of the velocity spectrum. Numerical examples with synthetic and real seismic data indicate that the proposed approach provides stacking parameters with better resolution than conventional semblance, at an affordable computational cost.     

2D multiscale non-linear velocity inversion

An efficient and robust non-linear inversion method for velocity optimization combining a global random search followed by a simplex technique is presented. The background velocity field is estimated at different spatial scales by analysing image gathers after iterative prestack depth migrations. First, the global random search is used to determine the main features/trends of the velocity model (large-scale component). Then, the simplex technique improves the resolution of the velocity field by estimating smaller-scale features. A measure of the quality of the velocity model (objective function) is based on flattening offset events in depth-migrated image gathers. To help constrain the solution, the algorithm can incorporate a priori information about the model and a smoothness condition. This 2D velocity estimation offers the benefit of being semi-automatic (requiring minimal human intervention) as well as providing a global and objective solution (which is a useful approach to an interpretation-derived velocity-estimation technique). The method is applied to a real data set where AVO analysis is carried out after prestack depth migration, as structural effects are non-negligible. It is demonstrated that the method can successfully estimate a laterally inhomogeneous velocity model at a computational cost modest compared with an interpretation-based iterative prestack depth velocity-analysis technique.     

Depth and morphology of reflectors from the non-linear inversion of arrival-time and waveform semblance data. Part I: method and applications to synthetic data

We propose a two-dimensional, non-linear method for the inversion of reflected/converted traveltimes and waveform semblance designed to obtain the location and morphology of seismic reflectors in a lateral heterogeneous medium and in any source-to-receiver acquisition lay-out. This method uses a scheme of non-linear optimization for the determination of the interface parameters where the calculation of the traveltimes is carried out using a finite-difference solver of the Eikonal equation, assuming an a priori known background velocity model. For the search for the optimal interface model, we used a multiscale approach and the genetic algorithm global optimization technique. During the initial stages of inversion, we used the arrival times of the reflection phase to retrieve the interface model that is defined by a small number of parameters. In the successive steps, the inversion is based on the optimization of the semblance value determined along the calculated traveltime curves. Errors in the final model parameters and the criteria for the choice of the best-fit model are also estimated from the shape of the semblance function in the model parameter space. The method is tested and validated on a synthetic dataset that simulates the acquisition of reflection data in a complex volcanic structure. This study shows that the proposed inversion approach is a valid tool for geophysical investigations in complex geological environments, in order to obtain the morphology and positions of embedded discontinuities.     

A fast and robust two-point ray tracing method in layered media with constant or linearly varying layer velocity

A fast and robust method for two-point ray tracing in one-dimensional layered media is presented. This method is applicable to layered models with constant or linearly varying isotropic layer velocity. For given model properties and source and receiver positions, a ray path can be uniquely determined once its ray parameter (i.e. horizontal slowness) is known. The ray parameter can be obtained by numerically solving the nonlinear offset (i.e. source–receiver horizontal distance) equation using Newton's method, which generally works well at near and mid offsets. However, Newton's method becomes hard to converge at large offsets due to the oversensitivity of offset to ray parameter. Based on the analysis of the characteristic of the offset equation, a modified ray parameter is proposed and used to replace the generic ray parameter in numerical calculation. Numerical experiments show that the iteration process becomes stable and converges rapidly with the modified ray parameter. Moreover, a rational function that asymptotically approximates the shape of the offset equation is introduced for obtaining good initial estimates of the modified ray parameter. Numerical tests show that this method is robust in any situation, and an accurate ray parameter can be obtained within two or three iterations for a wide range of model velocity structure and source–receiver distance. Furthermore, the proposed two-point ray tracing method is easy to implement.     

Inversion of reflection seismograms by differential semblance analysis: algorithm structure and synthetic examples1

Seismograms predicted from acoustic or elastic earth models depend very non-linearly on the long wavelength components of velocity. This sensitive dependence demands the use of special variational principles in waveform-based inversion algorithms. The differential semblance variational principle is well-suited to velocity inversion by gradient methods, since its objective function is smooth and convex over a large range of velocity models. An extension of the adjoint state technique yields an accurate estimate of the differential semblance gradient. Non-linear conjugate gradient iteration is quite successful in locating the global differential semblance minimum, which is near the ordinary least-squares global minimum when coherent data noise is small. Several examples, based on the 2D primaries-only acoustic model, illustrate features of the method and its performance.     

Rapid 3-D Earthquake Location using a Hybrid Global–Local Inversion Approach

A novel hybrid approach to earthquake location is proposed which uses a combined coarse global search and fine local inversion with a minimum search routine. The method exploits the advantages of network ray tracing and robust formulation of the Fréchet derivatives to simultaneously update all sampled initial source parameters in the solution space to determine the best solution. Synthetic examples, involving a three-dimensional (3-D) complex velocity model and a challenging source–receiver layout, are used to demonstrate the advantages over direct grid search algorithms in terms of solution accuracy, computational efficiency, and sensitivity to noise. Therefore, this is a promising scheme for earthquake early warning, tsunami early warning, rapid hazard assessment, and emergency response after strong earthquake occurrence.     

The effectiveness of a pseudo-inverse extended Born operator to handle lateral heterogeneity for imaging and velocity analysis applications

Wave equation–based migration velocity analysis techniques aim to construct a kinematically accurate velocity model for imaging or as an initial model for full waveform inversion applications. The most popular wave equation–based migration velocity analysis method is differential semblance optimization, where the velocity model is iteratively updated by minimizing the unfocused energy in an extended image volume. However, differential semblance optimization suffers from artefacts, courtesy of the adjoint operator used in imaging, leading to poor convergence. Recent findings show that true amplitude imaging plays a significant role in enhancing the differential semblance optimization's gradient and reducing the artefacts. Here, we focus on a pseudo-inverse operator to the horizontally extended Born as a true amplitude imaging operator. For laterally inhomogeneous models, the operator required a derivative with respect to a vertical shift. Extending the image vertically to evaluate such a derivative is costly and impractical. The inverse operator can be simplified in laterally homogeneous models. We derive an extension of the approach to apply the full inverse formula and evaluate the derivative efficiently. We simplified the implementation by applying the derivative to the imaging condition and utilize the relationship between the source and receiver wavefields and the vertical shift. Specifically, we verify the effectiveness of the approach using the Marmousi model and show that the term required for the lateral inhomogeneity treatment has a relatively small impact on the results for many cases. We then apply the operator in differential semblance optimization and invert for an accurate macro-velocity model, which can serve as an initial velocity model for full waveform inversion.     

Automatic cross-well tomography by semblance and differential semblance optimization: theory and gradient computation

A technique for automatic cross-well tomography based on semblance and differential semblance optimization is presented. Given a background velocity, the recorded seismic data traces are back-propagated towards the source, i.e. shifted towards time zero using the modelled traveltime between the source and the receiver and corrected for the geometrical spreading. Therefore each back-propagated trace should be a pulse, close to time zero. The mismatches between the back-propagated traces indicate an error in the velocity model. This error can be measured by stacking the back-propagated traces (semblance optimization) or by computing the norm of the difference between adjacent traces (differential semblance optimization).
It is known from surface seismic reflection tomography that both the semblance and differential semblance functional have good convexity properties, although the differential semblance functional is believed to have a larger basin of attraction (region of convergence) around the true velocity model. In the case of the cross-well transmission tomography described in this paper, similar properties are found for these functionals.
The implementation of this automatic method for cross-well tomography is based on the high-frequency approximation to wave propagation. The wavefronts are constructed using a ray-tracing algorithm. The gradient of the cost function is computed by the adjoint-state technique, which has the same complexity as the computation of the functional. This provides an efficient algorithm to invert cross-well data. The method is applied to a synthetic data set to demonstrate its efficacy.     

Fast and accurate earthquake location within complex medium using a hybrid globallocal inversion approach

A novel hybrid approach for earthquake location is proposed which uses a combined coarse global search and fine local inversion with a minimum search routine, plus an examination of the root mean squares (RMS) error distribution. The method exploits the advantages of network ray tracing and robust formulation of the Fréchet derivatives to simultaneously update all possible initial source parameters around most local minima (including the global minimum) in the solution space, and finally to determine the likely global solution. Several synthetic examples involving a 3-D complex velocity model and a challenging source-receiver layout are used to demonstrate the capability of the newly-developed method. This new global-local hybrid solution technique not only incorporates the significant benefits of our recently published hypocenter determination procedure for multiple earthquake parameters, but also offers the attractive features of global optimal searching in the RMS travel time error distribution. Unlike the traditional global search method, for example, the Monte Carlo approach, where millions of tests have to be done to find the final global solution, the new method only conducts a matrix inversion type local search but does it multiple times simultaneously throughout the model volume to seek a global solution. The search is aided by inspection of the RMS error distribution. Benchmark tests against two popular approaches, the direct grid search method and the oct-tree important sampling method, indicate that the hybrid global-local inversion yields comparable location accuracy and is not sensitive to modest level of noise data, but more importantly it offers two-order of magnitude speed-up in computational effort. Such an improvement, combined with high accuracy, make it a promising hypocenter determination scheme in earthquake early warning, tsunami early warning, rapid hazard assessment and emergency response after strong earthquake occurrence.     

An Accurate Method of Computing the Gradient of Seismic Wave Reflection Coefficients (SWRCs) for the Inversion of Stratum Parameters

The optimization inversion method based on derivatives is an important inversion technique in seismic data processing, where the key problem is how to compute the Jacobian matrix. The computational precision of the Jacobian matrix directly influences the success of the optimization inversion method. Currently, most of the AVO (amplitude versus offset) inversions are based on approximate expressions for the Zoeppritz equations to obtain the derivatives of the seismic wave reflection coefficients (SWRCs) with respect to the stratum parameters. As a result, the computational precision and range of applications of these AVO inversions are restricted undesirably. In order to improve the computational precision and to extend the range of applications of AVO inversions, the partial derivative equations of the Zoeppritz equations are established, with respect to the ratios of wave velocities and medium densities. By solving the partial derivative equations of the Zoeppritz equations accurately, we obtained the partial derivative of SWRCs with respect to the ratios of seismic wave velocities and medium densities. With the help of the chain rule for derivatives, the gradient of the SWRCs can be accurately computed. To better understand the behavior of the gradient of the SWRCs, we plotted the partial derivative curves of the SWRCs, analyzed the characteristics of these curves, and gained some new insight into the derivatives. Because only a linear system of equations is solved in our method without adding any new restrictions, the new computational method has both high precision and a quick running speed; it is not only suitable for small incident angles and weak reflection seismic waves but also for large incident angles and strong reflection seismic waves. With the theoretical foundations established in the article, we can further study inversion problems for layered stratum structures and we can further improve the computational speed and precision of the inversions.     

Eliminating 3D pre‐stack migration artefacts by 6D filtering

State‐of‐the‐art 3D seismic acquisition geometries have poor sampling along at least one dimension. This results in coherent migration noise that always contaminates pre‐stack migrated data, including high‐fold surveys, if prior‐to‐migration interpolation was not applied. We present a method for effective noise suppression in migrated gathers, competing with data interpolation before pre‐stack migration. The proposed technique is based on a dip decomposition of common‐offset volumes and a semblance‐type measure computation via offset for all constant‐dip gathers. Thus the processing engages six dimensions: offset, inline, crossline, depth, inline dip, and crossline dip. To reduce computational costs, we apply a two‐pass (4D in each pass) noise suppression: inline processing and then crossline processing (or vice versa). Synthetic and real‐data examples verify that the technique preserves signal amplitudes, including amplitude‐versus‐offset dependence, and that faults are not smeared.     

Dense 3D residual moveout analysis as a tool for HTI parameter estimation

Three‐dimensional residual moveout analysis is the basic step in velocity model refinement. The analysis is generally carried out using horizontal and/or vertical semblances defined on a sparse set of in‐lines or cross‐lines with densely sampled source–receiver offsets. An alternative approach, which we call dense residual moveout analysis (DRMA), is to use all the bins of a three‐dimensional survey but sparsely sampled offsets. The proposed technique is very fast and provides unbiased and statistically efficient estimates of the residual moveout. Indeed, for the sparsest possible offset distribution, when only near‐ and far‐angle stacks are used, the variance of the residual moveout estimate is only 1.4 times larger than the variance of the least‐squares estimate obtained using all offsets. The high performance of DRMA makes it a useful tool for many applications, of which azimuthal velocity analysis is considered here. For a horizontal transverse isotropy (HTI) model, a deterministic procedure is proposed to define, at every point of residual moveout estimation, the azimuthal angle of the HTI axis of symmetry, the Thomsen anisotropy coefficients, and the interval (or root‐mean‐square) velocities in both the HTI isotropy and symmetry planes. The procedure is not restricted by DRMA assumptions; for example, it is also applicable to semblance‐based residual moveout estimates. The high resolution of the technique is illustrated by azimuthal velocity analysis over an oilfield in West Siberia.     

Interval anisotropic parameter estimation from walkaway vertical seismic profiling data

This paper presents a new explicit method for the estimation of layered vertical transverse isotropic (VTI) anisotropic parameters from walkaway VSP data. This method is based on Dix‐type normal moveout (NMO) inversion. To estimate interval anisotropic parameters above a receiver array, the method uses time arrivals of surface‐related double‐reflected downgoing waves. A three‐term NMO approximation function is used to estimate NMO velocity and a non‐hyperbolic parameter. Assuming the vertical velocity is known from zero‐offset VSP data, Dix‐type inversion is applied to estimate the layered Thomsen anisotropic parameters ?, δ above the receivers array. Model results show reasonable accuracy for estimates through Dix‐type inversion. Results also show that in many cases we can neglect the influence of the velocity gradient on anisotropy estimates. First breaks are used to estimate anisotropic parameters within the walkaway receiver interval. Analytical uncertainty analysis is performed to NMO parameter estimates. Its conclusions are confirmed by modelling.     

Improving the gradient of the image‐domain objective function using quantitative migration for a more robust migration velocity analysis

Migration velocity analysis aims at determining the background velocity model. Classical artefacts, such as migration smiles, are observed on subsurface offset common image gathers, due to spatial and frequency data limitations. We analyse their impact on the differential semblance functional and on its gradient with respect to the model. In particular, the differential semblance functional is not necessarily minimum at the expected value. Tapers are classically applied on common image gathers to partly reduce these artefacts. Here, we first observe that the migrated image can be defined as the first gradient of an objective function formulated in the data‐domain. For an automatic and more robust formulation, we introduce a weight in the original data‐domain objective function. The weight is determined such that the Hessian resembles a Dirac function. In that way, we extend quantitative migration to the subsurface‐offset domain. This is an automatic way to compensate for illumination. We analyse the modified scheme on a very simple 2D case and on a more complex velocity model to show how migration velocity analysis becomes more robust.     

基于非因果滤波器的多次波匹配相减方法(英文)

在常规多道匹配滤波方法中的滤波器是物理可实现的因果滤波器,只能实现地震信号序列延迟的滤波。本文提出了最小二乘意义下的非因果多道输入多道输出维纳滤波方法,通过比较多道匹配相减中因果和非因果滤波方法之间的差别,验证了方法的有效性,解决了模型数据滞后于实际数据的情况。而且,通过定义长度随偏移距和层速度变化的滑动时窗,解决了匹配时窗内同相轴数量随偏移距增大而增加的问题。并将上述方法应用到改进的扩展多道匹配相减去除多次波的方法中,利用Pluto1.5理论模拟数据,对非因果滤波器和变长度时窗的匹配相减方法进行测试,取得了很好的去除多次波后的地震数据。     

Automatic cross-well tomography: an application of the differential semblance optimization to two real examples

An automatic tomography algorithm, based on differential semblance optimization (DSO), has been used to invert real cross-well seismic data for the background velocity. The method relies on the first-arrival transmitted waves. Given a background velocity model, the traveltimes between the sources and the receivers are computed, then semblance panels are created by back-propagating the data traces. If the velocity model is correct all the first-arrival transmitted waves will be aligned in the semblance panels. The DSO method consists of finding the background velocity by minimizing the L ₂-norm of the difference between adjacent back-propagated traces. Thanks to the good behaviour of this DSO cost function about the solution, a local (gradient) optimization can be performed. This provides a relatively fast algorithm when ray tracing and analytic computation of the gradient are used.
Unfortunately the method fails in the presence of caustics in the data. However, this difficulty can be circumvented by applying suitable masks to the data. This approach is first applied to a synthetic example then to two real data sets: the McElroy data set recorded in West Texas and the NIMR data set recorded in Oman. The results are quite encouraging and similar to those obtained with classical tomography.     

Source-independent wave-equation based microseismic source location using traveltime inversion

A new source location method using wave-equation based traveltime inversion is proposed to locate microseismic events accurately. With a sourceindependent strategy, microseismic events can be located independently regardless of the accuracy of the source signature and the origin time. The traveltime-residuals-based misfit function has robust performance when the velocity model is inaccurate. The new Fréchet derivatives of the misfit function with respect to source location are derived directly based on the acoustic wave equation, accounting for the influence of geometrical perturbation and spatial velocity variation. Unlike the mostly used traveltime inversion methods, no traveltime picking or ray tracing is needed.Additionally, the improved scattering-integral method is applied to reduce the computational cost. Numerical tests show the validity of the proposed method.     

Depth and morphology of reflectors from the non-linear inversion of arrival times and waveform semblance data. Part II: modelling and interpretation of real data acquired in Southern Apennines, Italy

In order to retrieve a 2D background velocity model and to retrieve the geometry and depth of shallow crustal reflectors in the Southern Apennines thrust belt a separate inversion of first arrival traveltimes and reflected waveforms was performed. Data were collected during an active seismic experiment in 1999 by Enterprise Oil Italiana and Eni-Agip using a global offset acquisition geometry. A total of 284 on-land shots were recorded by 201 receivers deployed on an 18 km line oriented SW–NE in the Val D'Agri region (Southern Apennines, Italy).
The two-step procedure allows for the retrieval of a reliable velocity model by using a non-linear tomographic inversion and reflected waveform semblance data inversion. The tomographic model shows that the P wave velocity field varies vertically from approximately 3 km/s to 6 km/s within 4 km from the Earth's surface. Moreover, at a distance of approximately 11 km along the profile, there is an abrupt increase in the velocity field. In this zone indeed, an ascent from 2 km depth to 0 km above sea level of the 5.2 km/s iso-velocity contour can be noted. The retrieved velocity can be associated with Plio-Pleistocene clastic deposits outcropping in the basin zone and with Mesozoic limestone deposits. The inversion of waveform semblance data shows that a P-to-P reflector is retrieved at a depth of approximately 2 km. This interface is deeper in the north-eastern part of the profile, where it reaches 3 km depth and can be associated with a limestone horizon.     

Migration velocity analysis and waveform inversion

Least‐squares inversion of seismic reflection waveform data can reconstruct remarkably detailed models of subsurface structure and take into account essentially any physics of seismic wave propagation that can be modelled. However, the waveform inversion objective has many spurious local minima, hence convergence of descent methods (mandatory because of problem size) to useful Earth models requires accurate initial estimates of long‐scale velocity structure. Migration velocity analysis, on the other hand, is capable of correcting substantially erroneous initial estimates of velocity at long scales. Migration velocity analysis is based on prestack depth migration, which is in turn based on linearized acoustic modelling (Born or single‐scattering approximation). Two major variants of prestack depth migration, using binning of surface data and Claerbout's survey‐sinking concept respectively, are in widespread use. Each type of prestack migration produces an image volume depending on redundant parameters and supplies a condition on the image volume, which expresses consistency between data and velocity model and is hence a basis for velocity analysis. The survey‐sinking (depth‐oriented) approach to prestack migration is less subject to kinematic artefacts than is the binning‐based (surface‐oriented) approach. Because kinematic artefacts strongly violate the consistency or semblance conditions, this observation suggests that velocity analysis based on depth‐oriented prestack migration may be more appropriate in kinematically complex areas. Appropriate choice of objective (differential semblance) turns either form of migration velocity analysis into an optimization problem, for which Newton‐like methods exhibit little tendency to stagnate at nonglobal minima. The extended modelling concept links migration velocity analysis to the apparently unrelated waveform inversion approach to estimation of Earth structure: from this point of view, migration velocity analysis is a solution method for the linearized waveform inversion problem. Extended modelling also provides a basis for a nonlinear generalization of migration velocity analysis. Preliminary numerical evidence suggests a new approach to nonlinear waveform inversion, which may combine the global convergence of velocity analysis with the physical fidelity of model‐based data fitting.