A STRATEGY FOR TIME-TO-DEPTH CONVERSION AND VELOCITY ESTIMATION

We present a strategy to time-to-depth conversion and velocity estimation based only on the imagewavefront propagation. It has two main features: (1) it computes the velocity field and the traveltime directly, avoiding the ray-tracing step; and (2) it requires only the knowledge of the image-wavefront at the previous time step. As a consequence, our method tends to be faster than usual techniques and does not carry the constraints and limitations inherent to common ray-tracing strategies. We have tested the feasibility of the method on a synthetic Gaussian velocity anomaly as well as on the original Marmousi velocity model and two smoothed versions of it. Moreover, we migrated the Marmousi data set using the estimated depth velocity models. Our results indicate that the present strategy can be used to construct starting models for velocity-model building in depth migration and/or tomographic methods. INTRODUCTION The need to investigate regions with complex geology has encouraged the development of imaging methods that act in the depth domain. Important examples are prestack depth migration (PSDM) and full-waveform tomography (FWT) techniques. However, the application of these methods faces at least two problems: they require (1) an accurate velocity macromodel and (2) large computation power. In contrast, time migration has proved to be a very fast and robust process, making it routinely employed for seismic imaging. Moreover, time velocity-model building is a very well understood process, leading to high-quality migration velocity models in time. Therefore, it is highly desirable to construct starting models for depth techniques from these time-domain velocity models by means of time-to-depth conversion. While vertical conversion from time to depth has been routinely employed for a long time, Hubral (1977) was the first to recognize the need of taking lateral displacements into account. He demonstrated that timeand depth-domain coordinates are interconnected by the so-called image ray. More recently, Cameron et al. (2007, 2008) derived the theoretical relation between the time-migration velocity and seismic velocity using image-ray theory and paraxial ray-tracing theory (Popov et al., 1978; Červeny, 2001; Popov, 2002). Their algorithm consists of image-ray tracing to convert time Dix velocities into ray coordinates velocities and then time-to-depth convert them based on Dijkstra-like fast marching methods (Sethian, 1999a,b). Iversen and Tygel (2008) proposed a similar but more efficient technique that even in 3D requires only a single-azimuth time-migration velocity field as an input to construct the depth velocity field. Despite being a very attractive method, time-to-depth conversion is an ill-posed problem (Cameron et al., 2007). It aggregates the limitations of all involved steps, that is, the constraints involved in time migration, ray-tracing, and Dix-based velocity conversion (Iversen and Tygel, 2008). Thus, regularization is essential for adequate time-to-depth conversion. Such a regularization can be added in two phases of the process: (1) during the estimation of the Dix velocity field from an estimated time-migration velocity field, and (2) during the image-ray tracing. Valente et al. (2009) compared the three conversion techniques of Cameron et al. (2007), Cameron et al. (2008), and Iversen and Tygel (2008). They demonstrated that the 138 Annual WIT report 2014 different procedures react differently to different kinds of regularization. However, although the image-ray trajectories and the resulting depth velocity models depended on the regularization employed, the corresponding final depth images were very similar. The objective of the present work is to present an alternative algorithm for the time-to-depth conversion, which does not require image-ray tracing. Instead, it simulates the propagation of an image-wavefront in the subsurface. Our approach has the advantage of directly computing the velocity field and the traveltime. We test the approach on a synthetic Gaussian example and on the Marmousi data set. For the latter, we use Fourier Finite-difference (FFD) migration based on complex Padé approximations (Amazonas et al., 2007) to evaluate the quality of the depth-migrated images. TIME-TO-DEPTH CONVERSION ALGORITHMS We start with the formulation of the time-to-depth-conversion inverse problem based on image-ray theory and a brief description of some of the proposed newer algorithms. Two-dimensional inverse problem Consider an image point at coordinates ( , ⌧ ) in a time migrated section, with min   max and 0  ⌧ ⌧max. This image point can be associated with an image-ray that has reached the acquisition surface at the position in time ⌧ . The problem of time-to-depth conversion consists of tracing this imageray back into the medium together with a family of rays, associated with a plane wave that tangent to the acquisition surface at , until the time ⌧ is consumed and the ray has reached its position x = (x, z) in the subsurface. In other words, we want to simultaneously solve the set of equations for kinematic and dynamic ray tracing with plane-wave initial conditions. This set can be represented in two dimensions as dx d⌧ = v2(x)p , dp d⌧ = 1 v(x) r x v(x) , (1) @Q @⌧ = v2(x)P , @P @⌧ = 1 v(x) @2v @⌘ Q , where ⌘ denotes the coordinate direction perpendicular to the ray. The initial conditions for image-ray tracing read x(⌧ = 0) = ( , 0) , p(⌧ = 0) = ẑ vDix(x(⌧ = 0)) , Q(⌧ = 0) = 1 , P (⌧ = 0) = 0 (2) where ẑ is the unit vector in the vertical direction. Inversion algorithm As an alternative to simplify the computation of the derivatives along the image-rays, Cameron et al. (2007) proposed an image-ray tracing algorithm which fits the image-wavefront by polynomial curves. Based on this algorithm, Valente (2013) proposed an additional regularization to the fitting, which increases the choice of the degree of the polynomial. Only a little later, Cameron et al. (2008) presented a new ray tracing algorithm, in which the last two equations of system (2) are integrated based on the Lax-Friedrichs method (Lax, 1954), resulting in the scheme P j = Pn j+1 + P n j 1 2 ⌧ 4 2vn j vn j+2 vn j Qj+1 vn j vn j 2 Qj 1 ! , 1 Q j = 1 Qj ⌧ 2 ⇥ (v j ) P j + (v n+1 j ) P j ⇤