Fresnel volume migration of single-component seismic data

Standard implementations of Kirchhoff prestack depth migration (PSDM) distribute the recorded wavefield along two-way-traveltime isochrons and an image is generated by constructive interference of these isochrons along the actual reflector elements. Beside the recent developments of wave-equation-based approaches, Kirchhoff PSDM is still considered widely as a state-of-the-art technique in obtaining high-quality images of the subsurface, particularly for highly irregular survey layouts and target-oriented imaging tasks. However, for sparse sampling or limited aperture, the resulting image is affected by significant migration noise as a result of limited constructive interference of the back-propagated wavefield. Some modifications have been proposed to reduce these artifacts. These modifications include constructing a specular path of wave propagation, derived from estimates of the emergent angle of coherent phases in the seismogram section, and the mainly heuristic restriction of the imaging operator to the neighborhood of that wavepath. Our approach uses Fresnel volumes to restrict the migration operator in a physically frequency-dependent way. Using the emergent angle at the receiver, determined by a local slowness analysis, a ray is propagated into the subsurface; the back-propagation of the wavefield is restricted to the vicinity of this ray according to its approximated Fresnel volume. This so-called Fresnel volume migration approach enhances image quality significantly compared with standard Kirchhoff PSDM because of the inherent focusing and the restriction of the back-propagation to the region around the actual reflection point.

Beyond ray tomography: Wavepaths and Fresnel volumes

Two techniques that account for the band‐limited nature of seismic data are incorporated into tomographic traveltime inversion schemes. The first technique, the wavepath algorithm, is based upon the wave equation, the Born approximation, and an adjoint method for computing Frechet derivatives. Computation of a single wavepath requires the forward propagation of the seismic wavefield, as well as the reverse propagation of a residual wavefield. The second technique, the Fresnel volume approach, is based upon the paraxial ray approximation. The Fresnel volume algorithm requires little more computation than does conventional ray tracing and an order of magnitude less computer time than our calculation of wavepaths. When the Fresnel volume sensitivity functions are normalized by the area of the Fresnel ellipse perpendicular to the ray, the sensitivity estimates are very similar to the wavepaths. In particular, there is heightened sensitivity to velocity structure near the source and receiver locations. The normalization by the Fresnel ellipse area is necessary to ensure ray theoretical results in the limit of infinite frequency. Tomographic inversion based upon wavepaths or Fresnel volumes is more appropriate when considering the arrival time of the peak of the initial pulse rather than the first‐arrival time. Furthermore, using the traveltime of the peak instead of the first‐arrival time reduces the bias of tomograms to high velocity anomalies. The raypath, wavepath, and Fresnel volume techniques were applied to a set of cross‐borehole traveltime observations gathered at the Grimsel Rock Laboratory. All methods imaged a low velocity fracture zone in the granitic site, in agreement with independent well information. Estimates of model parameter resolution are similar for the wavepath and Fresnel volume schemes. The source‐receiver regions are the most well resolved areas. However, the model parameter resolution computed using a conventional ray‐based formalism is more evenly distributed over the cross‐borehole area.

Fresnel volume ray tracing

The concept of “Fresnel volume ray tracing” consists of standard ray tracing, supplemented by a computation of parameters defining the first Fresnel zones at each point of the ray. The Fresnel volume represents a 3-D spatial equivalent of the Fresnel zone that can also be called a physical ray. The shape of the Fresnel volume depends on the position of the source and the receiver, the structure between them, and the type of body wave under consideration. In addition, the shape also depends on frequency: it is narrow for a high frequency and thick for a low frequency. An efficient algorithm for Fresnel volume ray tracing, based on the paraxial ray method, is proposed. The evaluation of the parameters defining the first Fresnel zone merely consists of a simple algebraic manipulation of the elements of the ray propagator matrix. The proposed algorithm may be applied to any high‐frequency seismic body wave propagating in a laterally varying 2-D or 3-D layered structure (P, S, converted, multiply reflected, etc.). Numerical examples of Fresnel volume ray tracing in 2-D inhomogeneous layered structures are presented. Certain interesting properties of Fresnel volumes are discussed (e.g., the double caustic effect). Fresnel volume ray tracing offers numerous applications in seismology and seismic prospecting. Among others, it can be used to study the resolution of the seismic method and the validity conditions of the ray method.