A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations

Introduction

Time-domain seismic processing has been a popular and effective tool in areas with mild lateral velocity variations (Yilmaz, 2001; Robein, 2003; Bartel et al., 2006). Time migration produces images in the time coordinate as opposed to the usual depth coordinate. The time coordinate, along with time-migration velocities, is determined during time migration by optimizing image qualities. However, it is highly sensitive to lateral velocity changes (Black and Brzostowski, 1994; Bevc et al., 1995). Therefore, the time-migrated image is usually distorted (Cameron et al., 2008,2009; Iversen and Tygel, 2008; Hubral, 1977; Cameron et al., 2007). For this reason, in many cases time migrations are inadequate for accurate geological interpretation of subsurface structures. On the other hand, time-migration velocities can be conveniently and efficiently estimated either by repeated migrations (Yilmaz et al., 2001) or by velocity continuation (Fomel, 2003). Depth migration can handle general media and output images in regular Cartesian depth coordinates. But it requires an accurate interval velocity model construction, which is in practice both challenging and time-consuming. An iterative process of tomographic updates is often employed (Stork and Clayton, 1992; Bishop et al., 1985), where a good initial interval velocity model is crucial for achieving the global minimum. It is thus of great interest to convert the time-migration velocity to the depth domain in order to unravel inherent distortions in time-domain images and to provide a reasonable starting model for building depth-imaging velocities.

The relationship between time and depth coordinates was first explained by Hubral (1977) through the concept of image rays. An interval velocity model can be converted to one in the time domain by tracing image rays that dive into earth with slowness vector normal to the surface. The time coordinates are then defined by the traveltime along image rays and its surface location (Robein, 2003; Larner et al., 1981). However this process is not trivially revertible and it does not reveal directly the connection between time-migration velocity and interval velocity. According to Dix (1955), in a layered medium where $v = v(z)$, the image rays run straightly downward and the time-migration velocities are the root-mean-square (RMS) velocities appearing in a truncated Taylor approximation for traveltimes. The Dix inversion formula is exact in a $v(z)$ medium, where the conversion between time- and depth-domain attributes is theoretically straightforward.

Even a mild lateral velocity variation can cause image rays to bend and invalidate Dix inversion. Cameron et al. (2008,2007) studied and established theoretical relations between the time-migration velocity and the seismic velocity in depth for general media using the paraxial ray-tracing theory. They showed that the conventional Dix velocity is equal to the ratio of the interval velocity and the geometrical spreading of image rays. This is consistent with Dix formula because when $v = v(z)$ the geometrical spreading equals to one everywhere. In order to carry out the time-to-depth conversion in the presence of lateral velocity variations, one can solve a nonlinear partial differential equation (PDE) of elliptic type with boundary conditions on the surface (Cameron et al., 2009). The problem is mathematically ill-posed. Cameron et al. (2009) adopt a two-step numerical procedure for the time-to-depth conversion. The first step is a Lax-Friedrichs-like finite-difference method or a spectral Chebyshev method to solve for geometrical spreading in the time coordinate. The next step is a Dijkstra-like solver motivated by the fast marching method (Sethian, 1999) for velocity conversion and coordinate mapping. In this approach, it is crucial to yield the development of low harmonics and damp the high harmonics during the first stage. Iversen and Tygel (2008) discussed an extension of the time-to-depth conversion problem along 2-D profiles in 3-D. An essential part of the algorithm is the time-stepping (integration) along image-rays. The robustness of these methods may not be satisfactory in practice because of stability issues that arise from the ill-posed nature of the problem.

In this paper, we start with the theoretical relations derived by Cameron et al. (2008,2007) but cast the original problem in a nonlinear iterative optimization framework. This idea is motivated by the observation that, for arbitrary depth velocity model, two of the PDEs can be always satisfied, while the remaining one associated with image-ray geometrical spreading can be rewritten in the form of a cost functional that indicates errors in interval velocity. A key benefit in our formulation is that each iteration will give a global update of the whole velocity model. In contrast, the previous approaches only consider information locally at each time step. Another advantage is that we are able to include regularization in the optimization framework in order to deal with the ill-posedness issue.

The paper is organized as follows. We first show theoretical derivations of all necessary components involved in the proposed approach. Next, we develop a numerical implementation for 2-D time-to-depth conversion. Finally, we test the algorithm on synthetic and field data examples. We conclude the paper by giving some discussions on the proposed method.

A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations