One‐way acoustic reciprocity and its applications in multiple elimination and time‐lapse seismics

Abstract

In this paper we review our reciprocity theorems for one-way wave fields, modified for lossy media, and we discuss applications in multiple elimination and time-lapse seismics. Introduction One-way wave equations have played a prominent role in seismic processing since the pioneering work of Claerbout [4], Berkhout [2] and others. The reason for this is that a seismic experiment can be explained in terms of ‘downgoing’ waves traveling from the source at the Earth’s surface to a target in the subsurface and ‘upgoing’ waves traveling from the target to the receivers at the surface. One-way wave equations naturally honor this distinction between downgoing and upgoing waves. This paper starts with a review of reciprocity theorems for oneway wave fields. These reciprocity theorems formulate general relations between the one-way wave fields in two different ‘states’. One of these states is an actual seismic experiment, while the other state can either be a computational state (e.g. a wave field propagator), a desired state (e.g. multiple-free data) or another seismic measurement (characterizing time-lapse differences in the target). Fokkema and van den Berg [8] derived seismic processing techniques from Rayleigh’s reciprocity theorem for total acoustic wave fields. In the current paper the oneway reciprocity theorems form the starting point. These theorems provide a theoretical frame-work for current seismic processing techniques based on the one-way wave equations. Some applications will be indicated. One-way reciprocity theorems in media with losses The one-way wave equation and its symmetry properties. We review the acoustic one-way wave equation for downgoing and upgoing waves in an inhomogeneous medium with losses. We introduce a one-way wave vector and a one-way source vector , according to and (1)