Time-to-depth conversion and seismic velocity estimation using time-migration velocity

The objective of this work is to build an efficient algorithm (a) to estimate seismic velocity from time-migration velocity, and (b) to convert time-migrated images to depth. We establish theoretical relations between the time-migration velocity and the seismic velocity in 1 Page 1 of 23 GEOPHYSICS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 or Peer Rview 2-D and 3-D using paraxial ray tracing theory. The relation in 2-D implies that the conventional Dix velocity is the ratio of the interval seismic velocity and the geometrical spreading of the image rays. We formulate an inverse problem of finding seismic velocity from the Dix velocity and develop a numerical procedure for solving it. This procedure consists of two steps: (1) computation of the geometrical spreading of the image rays and the true seismic velocity in the time-domain coordinates from the Dix velocity; (2) conversion of the true seismic velocity from the time domain to the depth domain and computation of the transition matrices from time-domain coordinates to depth. For step 1, we derive a partial differential equation (PDE) in 2-D and 3-D relating the Dix velocity and the geometrical spreading of the image rays to be found. This is a nonlinear elliptic PDE. The physical setting allows us to pose a Cauchy problem for it. This problem is ill-posed. However we are able to solve it numerically in two ways on the required interval of time. One way is a finite difference scheme inspired by the Lax-Friedrichs method. The second way is a spectral Chebyshev method. For step 2, we develop an efficient Dijkstra-like solver motivated by Sethian’s Fast Marching Method. We test our numerical procedures on a synthetic data example and apply them to a field data example. We demonstrate that our algorithms give significantly more accurate estimate of the seismic velocity than the conventional Dix inversion. Our velocity estimate can be used as a reasonable first guess in building velocity models for depth imaging. 2 Page 2 of 23 GEOPHYSICS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 or Peer Rview INTRODUCTION Time-domain seismic imaging is a robust and efficient process routinely applied to seismic data (Yilmaz, 2001; Robein, 2003). Rapid scanning and determination of time-migration velocity can be accomplished either by repeated migrations (Yilmaz et al., 2001) or by velocity continuation (Fomel, 2003). Time migration is considered adequate for seismic imaging in areas with mild lateral velocity variations. However, even mild variations can cause structural distortions of time-migrated images and render them inadequate for accurate geological interpretation of subsurface structures. To remove structural errors inherent in time migration, it is necessary to convert timemigrated images into the depth domain either by migrating the original data with a prestack depth migration algorithm or by depth migrating post-stack data after time demigration (Kim et al., 1997). Each of these options requires converting the time migration velocity to a velocity model in depth. The connection between the timeand depth-domain coordinates is provided by the concept of image ray, introduced by Hubral (1977). Image rays are seismic rays that arrive normal to the Earth’s surface. Hubral’s theory explains how a depth velocity model can be converted to the time coordinates. However, it does not explain how a depth velocity model can be converted to the time-migration velocity. Moreover, image-ray tracing is a numerically inconvenient procedure for achieving uniform coverage of the subsurface. This may explain why simplified image-ray-tracing algorithms (Larner et al., 1981; Hatton et al., 1981) did not find widespread application in practice. Other limitations of image rays are related to the inability of time migration to handle large lateral variations in velocity (Bevc et al., 1995; Robein, 2003). 3 Page 3 of 23 GEOPHYSICS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 or Peer Rview The objective of the present work is to find an efficient method for building a velocity model from time-migration velocity. We establish new ray-theoretic connections between time-migration velocity and seismic velocity in 2-D and 3-D. These results are based on the image ray theory and the paraxial ray tracing theory (Popov and Pšenčik, 1978; Červený, 2001; Popov, 2002). Our results can be viewed as an extension of the Dix formula (Dix, 1955) to laterally inhomogeneous media. We show that the Dix velocity is seismic velocity divided by the geometrical spreading of the image rays. Hence, we use the Dix velocity instead of time migration velocity as a more convenient input. We develop a numerical approach to find (a) seismic velocity from the Dix velocity, and (b) transition matrices from the time-domain coordinates to the depth-domain coordinates. We test our approach on synthetic and field data examples. Our approach is complementary to more traditional velocity estimation methods and can be used as the first step in a velocity model building process. TIME MIGRATION VELOCITY Kirchhoff prestack time migration is commonly based on the following travel time approximation (Yilmaz, 2001). Let s be a source, r be a receiver, and x be the reflection subsurface point. Then the total travel time from s to x and from x to r is approximated as T (s, x) + T (x, r) ≈ T̂ (x0, t0, s, r) (1) where x0 and t0 are effective parameters of the subsurface point x. The approximation T̂ usually takes the form of the double-square-root equation T̂ (x0, t0, s, r) = √ t0 + |x0 − s|2 v2 m(x0, t0) + √ t0 + |x0 − r|2 v2 m(x0, t0) , (2) 4 Page 4 of 23 GEOPHYSICS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 or Peer Rview where x0 and t0 are the escape location and the travel time of the image ray (Hubral, 1977) from the subsurface point x. Regarding this approximation, let us list four cases depending on the seismic velocity v and the dimension of the problem: 2-D and 3-D, velocity v is constant. Equation 2 is exact, and vm = v. 2-D and 3-D, velocity v depends only on the depth z. Equation 2 is a consequence of the truncated Taylor expansion for the travel time around the surface point x0. Velocity vm depends only on t0 and is the root-mean-square velocity: vm(t0) = √ 1 t0 ∫ t0