### Green's theorem framework for data reconstruction

A. Ramirez
2009 71st EAGE Conference and Exhibition - Workshops and Fieldtrips   unpublished
SUMMARY____________________________________________________________ There is a tremendous and pressing need to improve our ability to effectively extrapolate, interpolateandregularizeseimicdata.Thatdrivestheinterestinmethodsofdatareconstructionin general. In the last decade, there has been an ever increasing attention to methods dealing with interferometryto reconstruct wavefield in new locations(limited to locations where sources or
more » ... utarenewedinterestinGreen'stheorem becausealldifferentapproachestointerferometrycanbederivedfromit.Interferometrictechniques are approximations to Green's theorem. Interferometryin general uses two measure pressure wavefields and two high-frequency and one-way wave approximations to Green's theorem. The approximationsaremadebecausetheexactformuladerivedfromGreen'stheoremrequiresasecond measurement--thenormalderivativeofthepressurefieldattherecordingsurface,hence,withthe approximations onlythe pressure field is needed. If interferometry is used to reconstruct data in environments where strong two-way waves exist (such as in marine surface seismic experiments, where the free-surface reflects all the upgoing energy as strong downgoing energy) errors and artifacts are produced in the synthesized wavefield due to the approximations. There are different waystodealwiththeseartifacts(alsoknownasspuriousmultiples),forexample,DongandSchuster (2008) use shaping filters anditerations to reduce the effect of the artifacts in the reconstructed data(seealsotheUTAM2008annualreport). In this presentation, a different approach is used: The approach starts with Green's theorem, and insteadofusingdirectlytheapproximationsmadebyinterferometry,weusedifferentauxiliary functions to obtain methods for data reconstruction with different degrees of approximations and effectiveness, including methods with no approximations. For example, direct wave interferometry uses the measure pressure data, a reference (constant velocity) Green's function and a single approximation,andshowsaddedvalueinthereconstructedwavefield(seeattachedpaper).Wealso provide two algorithms for data reconstruction purely based on Green's theorem with no approximations.Thefirstonerequiresdualmeasurements(pressurefieldanditsnormalderivative atthemeasurementsurface)andusesareferenceGreen'sfunction.Thesecondonerequiresonlya single measurement (pressure field) and uses a double-Dirichlet Green's function (vanishing at the free-surfaceandatthemeasurement surface). These three techniques allow reconstructing the wavefield anywhere between the measurementsurfaceandthefreesurface.Thethreetechniquesdealandusethefactthattwo-way waves exist when dealing with measurements close to the free-surface. Numerical examples and comparisonswillbeshown. For this presentation the theory, examples, comparisons and conclusions are limited to measurementswithinthewatercolumninamarineexperimentwithcontrolledsources.