Anisotropic model building with uncertainty analysis Andrey Bakulin, Dave Nichols, Konstantin Osypov*, Marta Woodward, Olga Zdraveva, WesternGeco/Schlumberger Summary Velocity estimation is usually an ill-posed problem even for isotropic media. Widespread use of anisotropic imaging has been shown to aid better focusing and positioning. However, it greatly escalates the complexity of the model building and makes the velocity estimation much more ill- posed. Conventional techniques continue to rely on gradient-based methods that deliver a single solution (or realization) of the model to the user. Here we demonstrate an alternative approach that acknowledges the non- uniqueness of the problem. It delivers an entire suite of models that fit the data equally well, allowing the user to select the most geologically plausible solution. Introduction In the past the goal of seismic imaging was to focus the data and provide a high quality subsurface image. In the last decade more emphasis has been placed on delivering a proper depth image that is as close as possible to the actual subsurface structure. To achieve this goal it is no longer enough to simply focus the data, but one has to use a realistic anisotropic depth model to perform such imaging. It is well known that surface seismic data alone cannot uniquely resolve all the parameters of an anisotropic subsurface. In time imaging this is reflected in our ability to resolve NMO velocity and anellipticity but not the vertical velocity, and thus depth (Tsvankin, 2001). In depth imaging there is a more complicated relationship between the resolved parameters. We have more confidence in the imaging velocity and less confidence in the vertical velocity which prevents us from accurately predicting true vertical depths. What is less well appreciated is that we often cannot resolve all the parameters of the model even if we have well data to help constrain the vertical velocity. In a companion presentation Bakulin et al. (2009) showed a significant ambiguity between epsilon and delta in TTI and VTI for a case of joint tomography of common image gathers and vertical VSP traveltimes. In this presentation we will use the methods described by Osypov et al. (2008) to characterize which combinations of parameters are resolved by a joint tomography experiment and which combinations of parameters are unresolved. Linear vs. Non-linear The most general way to describe of the information in a particular combination of data is to define the likelihood function for that data. This is the probability of collecting the data we observe given a particular model. The likelihood is a potentially complicated and non-linear function. If we assume Gaussian noise in the data we can characterize the likelihood using the misfit between the modeled and observed data (Taranatola, 2005). Figure 1 shows a hypothetical data misfit for a two parameter model. We can see that the misfit has multiple minima. We can also see a “trough” of good parameter combinations that all fit the data reasonably well. This trough does not lie along a straight line but is a banana shape. v2 v1 Figure 1: A non-linear misfit function is shown in grey. The linearized approximation is shown as colored contours. The well resolved direction is V1, the poorly resolved direction is V2. If we wish to analyze the multi-dimensional misfit in the neighborhood of the minimum we often use a linearized approximation to the true non-linear problem. This gives a quadratic approximation to the misfit function which is indicated by the elliptical error contours shown in Figure 1. The linearized problem is much more mathematically tractable but we can see that it misses some of the features in the true model. The misfit function only has a single minimum and the trough of good solutions now lies along a straight line which is the long axis of the ellipse. Despite these limitations we will use the linearized approximation to explore the uncertainty in our inversion but we must always keep in mind that this approximation is less valid the further away we sample the model from our local minimum. Methodology We can characterize the information in a joint surface and borehole tomography problem using a linearized approximation to the true problem. In this case the linearization involves the assumption that the rays do not change significantly when model is altered. The basic tomography problem is described by Woodward et al. (2008) and the analysis of uncertainty is described by Osypov et al. (2008). 3720 SEG Houston 2009 International Exposition and Annual Meeting