Geometric Theory Of Seismic Imaging

Abstract

In this paper I provide an overview of main concepts and results by S.V.Goldin in the field of geometric theory of seismic imaging. Then I present some recent results on velocity continuation of seismic images developing his ideas. Introduction Main areas of seismology that S.V. Goldin have contributed to include: statistic methods of signal detection in seismic traces; inverse kinematic problem for layered media; geometric theory to seismic imaging, physics of the earthquake source. In this paper we will discuss geometric theory of seismic imaging following Goldin (1998,2003). Theory of contact mappings in seismic imaging was first discussed in details in (Goldin, 1994). Similar ideas were developed in (Hubral et al., 1996; Tygel et al., 1996). Concept of velocity continuation was first introduced by (Fomel, 1994a). Operators. Let us consider an operator F transforming some input function to another one: : ( ) ( ) F u w → x y , (1) and its adjoint F∗ . Here we will consider the case when x and y are of the same dimension and F is invertible. We usually assume that ( ) u x and ( ) w y contain singularities supported on a piece-wise smooth surfaces Φ and Ψ correspondingly. Popular examples are reflectors present in an image and traveltime surfaces in data. Then we use terms ‘migration’ for F∗ and ‘demigration’ for F in general sense of displacing singularities in ( ) u x into those in ( ) w y . Note that most of seismic processing procedures fall into category of operators (1): modeling, migration, offset data transformation, remigration etc. There are few ways to implement these operators: 1. Boundary-value problems for hyperbolic partialdifferential equations (PDEs). 2. Generalized Radon Transform (GRT) integral operators. 3. Fourier Integral Operators (FIOs). GRT or Kirchhoff type integral operators can be defined as follows: ( ) ( ) ( , ) ( , ) ( ) w a u d δ φ = ∫ y y x y x x x , (2) where ( , ) 0 φ = y x defines summation hypersurfaces. Inverse scattering theory developed in the framework of FIOs is described in (De Hoop, 2003). FIOs represent a class of operators even more general than GRT: ( , , ) ( ) ( , , ) ( ) i w a e u d d φ = ∫∫ y xθ y y x θ x x θ , (3) with some properties on ( , , ) a y xθ and ( , , ) φ y xθ amplitude and phase function correspondingly. Below we will consider a relation between these two types of operators.