Gaussian‐Beam‐Kirchhoff prestack depth migration

Abstract

SUMMARYDuetoimagingproblemsofaseismicwavefieldincomplexgeologicalregionsbytheuseofzeroorderraytheoryasGreenfunction,wereporton a Gaussian-Beam (GB) type operator in the kernel of a finite-offsettrue-amplitude prestack depth migration Kirchhoff operator. This op-erator is a weighted GB superposistion integral constrained to the pro-jected Fresnel zone of a seismic experiment. The current migrationoperator is applied to a simple synthetic example and is compared tothe conventional Kirchhoff-PSDM in a complex synthetic geologicalmodel of the Solim˜oes Basin (Brazil). The depth images in all thecases showed an increase in horizontal resolution and a reduction ofmigration artifacts.INTRODUCTIONKirchhoff-type time or depth migration has been used as workhorseby oil industry since the pioneering work of Hagedoorn (1954), inwhich the surfaces of maximum convexity were later related to thescalar wave equation and became familiar in the geophysics litera-ture as “Kirchhoff migration” (Schneider, 1978). In the last threedecades, however, Kirchhoff migration in the meaning stated aboveevolved from a single imaging only operator to a inversion operator.This has led to the development of a miscellaneous of other imagingtechniques (Tygel et al., 1993), where the resulting operators are ableto handle AVO/AVA data, those ones important in the characterizationof oil-bearing reserves.The basic algorithm that supports Kirchhoff migration is based on thekinematicanddynamicraytracing(Cervenˇ ´y,2001),i.e.,zeroorderraytheory. However, ray theory may not correctly simulate the seismicwavefield in some determined regions of the velocity model. In orderto bypass these drawbacks, along the years several methods have beenused, including the paraxial ray theory (Bortfeld, 1989). The paraxialray theory is essentially a real theory (Cervenˇ ´y, 2001); its extension toa complex character lead us to the Gaussian beam concept, which areparaxial rays with an amplitude factor that exponentially decays withrespect to the distance from the so called central ray, in a Gaussianmanner. Unlike ray theory, which is a mathematical representation ofthe energy path of the wavefield, the GB’s are physical quantities thatcan be measured, and in being so, they exist in regions of the velocitymodel where common rays (i.e., central rays) cannot be numericallytraced. Besides that, in some situations where the wavefield is notregular (i.e., the amplitudes tend to infinity, in caustic regions), theGB’s are regular, granting the numerical stability of the method.Toourknowledge, thefirstworkstodealwithGB’sasanimagingtoolwereduetoRaz(1987)andCosta(1989). Later,Hill(1990)developeda GB migration for poststack data by using the decomposition of thewavefieldinplanewaves. InHill(2001),theGBmigrationisextendedto 3D prestack data, where using the Kirchhoff downward extrapo-lation principle (Schneider, 1978) and the Claerbout (1971) imagingprinciple, it is shown that in the midpoint coordinates this process canbeseenasalocalslantstackingofeachwavefieldelementdecomposedin beams (Hale, 1992).In this work, we present the theoretical and numerical results of a 3DKirchhoff-type prestack depth migration operator where we make useof a weighted GB superposition integral as a Green function of theimaging problem. In order to adequately control the parameters thatdefine the half-width of a GB, we use as a physical constraint forthe determination of the “beam parameters” the Fresnel volume, es-pecifically speaking the Fresnel zone of a seismic experiment and itscounterpart, projected towards the acquisition surface. By using thediffraction stacking principle (Schleicher et al., 1993), we calculatethe Fresnel volume elements by dynamic raytracing. Then, we de-termine a radius of influence, inside of which we stack the seismicreflections corresponding to reflectors elements in the neighbour of areflection point and that influences on the resolution of the final im-age. Our process is compromised with the true amplitude of reflec-tion events, by correcting seismic data from the effects of geometricalspreading losses. The interaction between the weight-function of theGB superposition integral and the weight-function of the true ampli-tude migration operator has led to a weight-function of the GB super-position integral that is proportional to the value of the Fresnel zonematrix. In our opinion, this leads to a new physical interpretation ofthereferredweight-function,whencomparedtotheanalysisofKlimˇes(1984), wherenoconstrainttothenumberofparaxialraystobesuper-posed on a observation point (i.e., on the geophone) is stated (Ferreiraand Cruz, 2005). We have applied our algorithm to a complex syn-thetic model of the Solimoes Basin, Brazil. The migrated results have˜shown a sensible reduction in the number of migration artifacts and agood horizontal resolution.THEORYThemigrationoperatortobeconsideredisgivenby(FerreiraandCruz,2005)I(M,t)=−12π