Accurate imaging of surface seismic data without a velocity model
Authors: Brandsberg-Dahl Sverre , Hornby Brian E. , Xiao Xiang 
Published: February 2016

Abstract
We present a method for imaging surface seismic data that does not depend on a background velocity model. Instead of computing Green's functions in a background model, we use direct measurements from VSP data as Green's functions for the migration. These Green's functions will have all propagation effects intrinsically in the measurement: including all arrivals and all kinematic and dynamic effects caused by anisotropy and absorption. The VSP Green's functions can be used in a standard migration scheme for surface seismic that enables us to create a robust image of the sub-surface.
Although the method is strictly only valid for v(z) media, we show that mild lateral variations can be handled. When the subsurface is laterally invariant or smoothly varying, the measured Green's functions can be used to image the surface seismic data accurately in locations away from the well-bore, however, lateral velocity variations will introduce some distortion in the image. Regardless of the velocity distribution, the VSP can always be used to image the surface seismic in the location of the VSP.

Introduction
Seismic migration is the process where wavefields recorded on or near the surface are mapped back into the sub-surface to form an image of the sub-surface structure [1]. Common to all current migration methods is that they rely on a model of the subsurface for computing Green's functions. These computed Green's functions are approximations of the true Green's function in the subsurface [2]. The more accurate the method used for computing the Green's functions, and to some extent also the choice of imaging condition, the better image we will get. Most state-of-the-art migration methods will use Green's functions that include multiple arrivals and some finite-frequency effects, enabling us to produce accurate images of fairly complex geological structures. Such advanced wavefield extrapolation methods are very compute intensive and the migration process can take weeks to complete even on large supercomputers. As expensive as today's state-of-the art methods are, they are based on a simplified computational model. In most standard applications the model is represented by a scalar velocity that is smooth on a wavelength scale, thus ignoring scattering effects from small-scale structure, anisotropy and other intrinsic rock effects. Further, whatever the model is, it has to be derived from the data itself or some other form of remote measurement of velocity in the subsurface. The problem of estimating the velocity can often be more difficult than the imaging process itself and at the very best the solution will be a good, but non-unique solution to an inverse problem based on the seismic data itself. Commonly, much more crude approximations are used, for example only trying to match the predicted first-arrivals to the observed first arrivals in the data [3].
On the other hand, the measured data from a receiver in a vertical seismic profile (VSP) provides the true Green's function between these two locations as long as we assume the source to be impulsive and the receiver to be a point receiver. Such a Green's function will have all propagation effects intrinsically in the measurement, an observation that was made first by Schuster [4] in the context of reverse time migration. The measured signal from the VSP will includes all arrivals within the measured time window and the waveforms will include all kinematic and dynamic effects, like anisotropy and absorption in the subsurface as well as any other mechanism that we do not describe in a typical computational model. These VSP Green's functions can be used in a standard migration scheme for surface seismic that enables us to create a robust image of the sub-surface without the need for a velocity model. The concept of using the VSP to provide Green's functions for migration of surface seismic data is also closely related to the work of Berkhout and co-workers [1], on common focal point (CFP) operators. In their work, they have introduced these operators for use in the migration/imaging of surface seismic data. In our application the VSPs are used in an equivalent way to the CFP operators so we benefit from a rich literature on how to manipulate and apply these operators for imaging of seismic data.