Tongning Yang

Moveout analysis of wave-equation extended images

Conventionalvelocityanalysisappliedtoimagesproducedby wave-equationmigrationwithacrosscorrelationimagingconditionusesmoveoutinformationfromspacelagsorfocusinginformationfromtimelag.However,morerobustvelocity-estimation methodscanbedesignedtosimultaneouslytakeadvantageofthe semblanceandfocusinginformationprovidedbymigratedimages. Such a velocity estimation requires characterization of the moveout surfaces defined jointly for space- and time-lags extended images. The analytic solutions to the moveout surfaces can be derived by solving the system of equations representing the shifted source and receiver wavefields. The superposition of the surfaces from many experiments shots is equivalent to the envelope for the family of the individual surface. The envelope forms a shape that can be characterized as a cone in the extended space of depth, space lag, and time lag. When imaged with the correct velocity, the apex of the cone is located at the correct reflectiondepthandatzerospaceandtimelags.Whenimagedwith the incorrect velocity, the apex of the cone shifts in the depth direction and along the time-lag axis. The characteristics of the cones are directly related to the quality of the velocity model. Thus, their analysis provides a rich source of information for velocity model-building. Synthetic examples verify the derived formulas characterizing the moveout surfaces. The analytic formulas match the numeric experiments well, demonstrating the accuracy of the formulas. Based on information provided by the extended imaging condition, future application for velocity updates can benefit from the robustness of the depth-focusing analysisandofthehighresolutionofthesemblanceanalysis.

Image‐domain wavefield tomography with extended common‐image‐point gathers

Waveform inversion is a velocity-model-building technique based on full waveforms as the input and seismic wavefields as the information carrier. Conventional waveform inversion is implemented in the data domain. However, similar techniques referred to as image-domain wavefield tomography can be formulated in the image domain and use a seismic image as the input and seismic wavefields as the information carrier. The objective function for the image-domain approach is designed to optimize the coherency of reflections in extended common-image gathers. The function applies a penalty operator to the gathers, thus highlighting image inaccuracies arising from the velocity model error. Minimizing the objective function optimizes the model and improves the image quality. The gradient of the objective function is computed using the adjoint state method in a way similar to that in the analogous data-domain implementation. We propose an image-domain velocity-model building method using extended common-image-point space- and time-lag gathers constructed sparsely at reflections in the image. The gathers are effective in reconstructing the velocity model in complex geologic environments and can be used as an economical replacement for conventional common-image gathers in wave-equation tomography. A test on the Marmousi model illustrates successful updating of the velocity model using commonimage-point gathers and resulting improved image quality.

Image‐domain waveform tomography with two‐way wave‐equation

We propose an image-domain velocity model building method using the two-way wave equation and extended seismic images. We show that common-image-point gathers can effectively extract velocity information from steep reflections imaged with the two-way wave propagator. Such gathers have the advantages over conventional common-image gathers that they are capable of characterizing reflections with arbitrary dip and that they are computationally cheap especially for wide-azimuth imaging. We develop a waveform tomography procedure based on the adjoint-state method and commonimage-point gathers. Synthetic examples show that the information from steep reflections improve the resolution of velocity estimation, thus potentially leading to more accurate and faster converging inversion.

Wave‐equation migration velocity analysis using extended images

Wave-equation migration velocity analysis (WEMVA) is a velocity estimation technique designed to invert for velocity information using migrated images. Its capacity for handling multi-pathing makes it appropriate in complex subsurface regions characterized by strong velocity variation. WEMVA operates by establishing a linear relation between a velocity model perturbation and a corresponding migrated image perturbation. The linear relationship is derived from conventional extrapolation operators and it inherits the main properties of frequency-domain wavefield extrapolation. A key step in implementing WEMVA is to design an appropriate procedure for constructing image perturbations. Using time-lag extended images, one can characterize the error in migrated images by defining the focusing error as the shift of the focused reflection along the time-lag axis. Under the linear approximation, the focusing error can be transformed into an image perturbation by multiplying it with an image derivative taken relative to the time-lag parameter. The resulting image perturbation is thus a mapping of the velocity error in image space. This approach is computationally efficient and simple to implement, and no further assumptions about smoothness and homogeneity of the velocity model and reflector geometry are needed. Synthetic examples demonstrate the successful application of our method to a complex velocity model.

Illumination Compensation for Image-domain Wavefield Tomography

Wavefield tomography and waveform inversion are related techniques that share the need to simulate accurate wavefields in the subsurface. In both cases, models are updated iteratively using gradients computed by, for example, the adjoint state method. The major difference between these techniques is the domain in which one formulates the objective function that compares the observed and simulated wavefields. For image-domain wavefield tomography, the objective function is defined based on a residual evaluated using migrated images. The role of the penalty function is to highlight image defocusing caused by imperfect velocity. This is accomplished by defining an operator which annihilates an image corresponding to the correct velocity. Conventional techniques, like differential semblance optimization, assume that correctly migrated images are completely focused, which is not the case when illumination is poor. In this paper, we address this problem by defining an alternative penalty operator which takes illumination into consideration and leads to more robust and accurate inversion results. This technique is particularly relevant for imaging in complex areas, e.g. sub-salt.

Wave‐equation migration velocity analysis with extended common‐image‐point gathers

Wave-equation migration velocity analysis (WEMVA) is an image-domain velocity model building technique based on band-limited wave propagation and designed especially for complex subsurface environments. It exploits the coherency of reflection events measured in extended images produced by a cross-correlation imaging condition with non-zero lags. Conventional approaches use either space-lags or time-lag common image gathers, in which only partial information of the extended images is used for velocity updates. We propose an WEMVA approach using the complete information from both space-lags and time-lags of extended images. With this approach, the velocity model building benefits both from the robustness of using the time-lag information and from the high resolution of using the space-lags information. Such an implementation is facilitated by using extended common-image-point gathers (CIPs) constructed sparsely along reflections and defined jointly for spaceand time-lags. These CIPs avoid the bias towards nearly-horizontal reflectors so that steeply dipping events are well preserved in the gathers and the corresponding information related to velocity can be used. Also, the computation of the extended images can be avoided in areas where the velocity is known, e.g., inside salt bodies, or areas where the signal-to-noise ratio is too low, e.g., in shadow zones. Using CIPs for WEMVA can reduce the cost of constructing extended images and offer flexibility for the velocity model building.

Waveform Inversion in the Image Domain

Waveform inversion is a costly but accurate technique for model building. This method places huge demands on the data by requiring extremely low frequencies to reduce cycle skipping. One way of addressing this problem is to use an objective function based on correlations instead of differences between the observed and simulated wavefields. This procedure reduces cycle skipping, although it disregards information carried by the wavefield amplitudes. Correlation-based waveform inversion can be implemented either in the data or image domains. The key elements of the image-domain implementation are the objective function, defined using time-lag extended images obtained by wave-equation migration, and its gradient, computed using the adjoint state method. We derive the gradient calculation and compare its elements one-to-one with their data-domain counterparts. The two implementations are similar since they use the same wavefields constructed using the same wave-equation, boundary and initial conditions. The difference is that the objective functions are formulated before or after summation over experiments. The image-domain implementation evaluates the objective function throughout the model space, while the data-domain implementation evaluates the objective function only on the surface. The image-domain method constrains better the velocity model, while preserving all other important characteristics of its data-domain counterpart.

Wave‐equation extended images for semblance and depth focusing velocity analysis

Conventional velocity analysis applied to images produced by wave-equation migration makes use either of moveout information from space-lags, or of focusing information from timelags. However, more robust velocity estimation methods can be designed to take advantage at the same time of the moveout and focusing information provided by the migrated images. Such joint velocity estimation requires characterization of the analytic moveout surface defined for space-lag and time-lag common-image gathers. The analytic expressions for such surfaces are nonlinear, but reduce naturally to the conventional space-lag (linear) and time-lag (nonlinear) moveout functions. A joint migration velocity analysis technique exploiting the entire extended imaging condition information has the potential to benefit from the robustness of depth focusing analysis and of the high resolution of conventional semblance analysis.