Z-99 SEISMIC INTERFEROMETRY, TIME-REVERSAL AND RECIPROCITY KEES WAPENAAR AND JACOB FOKKEMA Section of Applied Geophysics, Department of Geotechnology, Delft University of Technology, The Netherlands Introduction Seismic interferometry is the process of generating new seismic responses by cross-correlating seismic observations at different receiver locations. A first version of this principle was derived by Claerbout [2], who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response. Later he conjectured a similar principle for cross-correlations of 3-D wave fields. In a similar fashion, Schuster [5] introduced the principle of interferometric imaging, i.e. forming an image of the subsurface from cross-correlated seismic traces. Wapenaar et al. [6, 8] used a one-way reciprocity theorem of the correlation type to derive relations between reflection and transmission responses of arbitrary 3-D inhomogeneous media. These relations form a basis for seismic interferometry and they prove Claerbout’s conjecture. Moreover, despite the differences in assumptions, these relations show a strong resemblance with those of Weaver and Lobkis [9] and Derode et al. [3, 4] for the retrieval of the Green’s function from cross-correlations of wave fields in closed and open systems, respectively. Derode et al. [3, 4] derive their expressions for the Green’s function retrieval using physical arguments, exploiting the principle of time reversal invariance of the acoustic wave equation and source-receiver reciprocity. Their approach can be seen as the ‘physical counterpart’ of our derivation based on a reciprocity theorem of the correlation type, which implicitly also makes use of time reversal invariance. The relation between these two approaches is discussed in this paper. Physical derivation of Green’s function retrieval In this section we summarize the ‘physical approach’ of Derode et al. [3, 4] for deriving expressions for Green’s function retrieval. We freely modify their notation, so that it matches our notation in the next section. Consider the configuration in Figure 1a. The shaded area represents a lossless arbitrary inhomogeneous acoustic medium, which is embedded by a homogeneous medium. In the inhomogeneous medium we have denoted two points x A and x B . Our aim is to show that the response at x B due to an impulsive source at x A [i.e., the Green’s function G(x B , x A ,t)] can be obtained by cross-correlating passive measurements of the wave fields at x A and x B due to sources in the embedding homogeneous medium. The derivation starts by considering another physical experiment, namely an impulsive source at x A and receivers at x on a closed surface S in the embedding homogeneous medium. The response at one particular point x on S is illustrated in Figure 1a and is denoted by G(x, x A ,t). Suppose that we record this response for all x on S , revert the time axis, and feed these time-reverted functions G(x, x A , −t) to sources at x on S , see Figure 1b. According to Huygens’ principle the wave field at any point x B in S due to these sources on S is then given by p(x B ,t) ∝  S G(x B , x,t) ∗ G(x, x A , −t)d 2 x, (1) where ∗ denotes convolution, ∝ denotes ‘proportional to’ and x B is considered as a variable. According to this equation, G(x B , x,t) propagates the source function G(x, x A , −t) from x to x B and the result is integrated over all sources on S . Due to the invariance of the acoustic wave equation for time-reversal (assuming the medium is lossless), the wave field p(x B ,t) focusses for x B = x A at t =0. Hence, the wave field p(x B ,t) for arbitrary x B and t can be seen as the response of a virtual source at x A and t =0, i.e., G(x B , x A ,t). Since the source at x A is only virtual, causality is not obeyed and p(x B ,t) is symmetric in EAGE 67th Conference & Exhibition — Madrid, Spain, 13 - 16 June 2005