Green's theorem framework for data reconstruction
71st EAGE Conference and Exhibition - Workshops and Fieldtrips
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
... 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.