One-way acoustic reciprocity and its application in time-lapse seismic By Kees Wapenaar Jacob Fokkema Menno Dillen Pieter Scherpenhuijsen Delft University of Technology Applied Earth Sciences

Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

Problem statement

-= Reference Monitor Difference

Problem statement Difference

Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

Acoustic reciprocity State AState B xAxA xAxA xBxB xBxB

Acoustic Reciprocity (continued) V State AState B P A, V k,A Q A,F k,A  A,  A P B, V k,B Q B,F k,B  B,  B Wave fields Sources Medium n

with Acoustic Reciprocity (continued)

with Acoustic Reciprocity (continued)

Applications: Multiple elimination: van Borselen, Fokkema and van den Berg, Geophysics 1996 Time-lapse seismic: Fokkema and van den Berg, 1996 Acoustic Reciprocity (continued)

Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

or : is the square root operator One-way wave fields

One-way acoustic Reciprocity (horizontally layered medium) hence c1,c2,c1,c2, c1,c2,c1,c2,

One-way acoustic Reciprocity (arbitrary inhomogeneous medium) State AState B xAxA xAxA xBxB xBxB

Flux-normalized one-way wave fields Power flux: Note:

One-way acoustic Reciprocity (horizontally layered medium) hence c1,c2,c1,c2, c1,c2,c1,c2,

One-way acoustic Reciprocity (arbitrary inhomogeneous medium) State AState B xAxA xAxA xBxB xBxB

One-way Acoustic Reciprocity (continued) V State AState B P A, V k,A S A,S k,A  A,  A P B, V k,B Q B,F k,B  B,  B Wave fields Sources Medium

and One-way Acoustic Reciprocity (continued)

Applications Representations of scattered data in inhomogeneous dissipative media: –Generalised Bremmer series expansion –Generalised primary representation Inverse wave field extrapolation in inhomogeneous dissipative media Multiple elimination Time-lapse seismic One-way Acoustic Reciprocity (cont’d)

Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

Time-lapse seismic  State A (reference wave field): State B (monitoring wave field):

Time-lapse seismic

Evaluation of boundary integral AB AB Fokkema and van den Berg (1996) Dillen (2000, Ph.d thesis) Wapenaar et al. (2000, SEG) B A  

Evaluation of boundary integral AB AB Fokkema and van den Berg (1996) Dillen (2000, Ph.d thesis) Wapenaar et al. (2000, SEG) B A  

Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

Numerical example Scherpenhuijsen (2000, M.Sc. Thesis) 

Upgoing wave fields at surface -=

Boundary integrals below reservoir -=

Numerical example (continued) Difference data

Numerical example (continued) Boundary integral

Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

Conclusions Reciprocity theorems formulate a relation between wave fields, sources and medium parameters in two states. Usual (i.e. pressure normalized) one-way wave fields don’t obey reciprocity. Flux-normalized one-way wave fields do obey reciprocity.

Conclusions Applications: Representations of scattered data in inhomogeneous dissipative media Inverse wave field extrapolation in inhomogeneous dissipative media Multiple elimination Time-lapse seismic