Introduction (ps.gz 180K) , (src 421K)

intro1 Efficient and accurate imaging In complex geologic environments, imaging of seismic reflection data requires depth migration driven by an interval velocity model. If the true velocity model of the earth is known, depth migration can back-propagate the seismic events recorded at the earth's surface to the subsurface positions where they were reflected. In recent years, industrial practices in oil exploration have proved that depth migration is a powerful tool for imaging complex subsurface structures in areas of strong lateral velocity variations (Hatton et al., 1981; Claerbout, 1985; Deregowski, 1985). However, studies have also shown that the result of such an imaging method is sensitive to errors in the velocity model (Parkes and Hatton, 1987). If the velocity model used in the migration process does not correctly model the propagation of reflection events, depth migration produces an erroneous image of the subsurface. Therefore, many efforts have been made to develop methods ...

Wavefront propagation by the finite-difference method
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findif2 The traveltimes of seismic wave propagation can be calculated on a regular grid through the process of propagating wavefronts, as first introduced by Vidale (1988). In Vidale's method, local planar or circular wavefronts are propagated by using a finite-difference method. Since then, several different finite-difference schemes have been developed to improve the accuracy and efficiency of the traveltime calculation (Van Trier and Symes, 1991; Podvin and Lecomte, 1991). In this chapter I present another improved finite-difference method for propagating wavefronts. The method can calculate the traveltime field in a model of large velocity contrasts and the geometrical amplitude field in regions where the wave field is regular. The implementation of the method is fully vectorizable. The results of the computation are sufficiently accurate for imaging seismic reflection data. I begin this chapter by reviewing some well-known results of zero-order asymptotic ray theory that are relevant to the calculation of ...

Wavefront propagation by local ray tracing
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wftprg3 Propagating wavefronts by the finite-difference method described in Chapter findif2 has its limitations. Although the method gives accurate traveltimes, it produces large errors in calculating traveltime gradients when velocity varies rapidly. Because geometrical amplitudes honor the zeroth-order transport equation, which involves the traveltime gradients, these errors are reflected in the amplitude calculation. The large errors in traveltime gradients are caused by the discontinuous representation of a velocity field that the finite-difference method assumes, as well as by the first-order finite-difference approximations that the method makes. The traveltime field in a complex velocity medium is generally a multivalued function of subsurface positions because of possible multiple arrivals. But all existing finite-difference methods compute only the traveltime of the first arrival at each subsurface point. One reason is that the wavefront of a first arrival is always continuous; hence finite-difference approximations to the eikonal equation are valid. Another reason is that among different ...

Residual depth migration
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resmig4 The preceding two chapters describe two wavefront propagation methods for calculating the traveltimes and geometrical amplitudes of seismic waves. These methods can also calculate the gradient fields of traveltime and take-off angle. This chapter extends these methods and results to a new application --- residual depth migration. Residual depth migration is a process of converting an image migrated with one interval velocity model to an image migrated with another. As noted in Chapter intro1 , each iteration of depth-migration velocity analysis involves the processes of estimating the perturbations of the interval velocities and then remigrating the data with the updated velocity model. However, complete remigration in each iteration is computationally expensive. When the perturbations of the interval velocities are small, residual depth migration can be considerably more efficient than full depth migration. The high efficiency results from the partial focusing effect of the previous migration. If the depth migration is implemented by the Kirchhoff integral, then the ...

Field data examples
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fiedat5 Efficient and accurate prestack depth migration plays an important role in both the imaging and velocity analysis of seismic reflection data collected in areas of lateral velocity variation. If such an imaging method is implemented using a Kirchhoff integral approach, the largest part of the effort in migrating data is the computation of the traveltimes and geometrical amplitudes of seismic waves in a given interval velocity model. Chapters findif2 and wftprg3 present two wavefront-propagation methods for rapid calculation of the traveltimes and geometrical amplitudes in a general 2-D velocity model. With these two methods, the computational cost of Kirchhoff depth migration is considerably reduced. Chapter resmig4 presents a residual depth migration algorithm that converts an image migrated with one velocity model to an image migrated with another velocity model. This algorithm further reduces the computational cost of depth migration when data are to be migrated with successive velocity models, as in ...

The derivation of two partial differential equations
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pdeapda In this appendix, I derive two partial differential equations that are related to the calculation of the geometrical amplitudes of seismic waves. The first partial differential equation is for the calculation of the take-off angle and its gradient fields. Let be the take-off angle of the ray connecting a surface point and a subsurface point . Because the ray has a unique take-off angle, is constant for any point along the ray. Therefore, the derivative of with respect to the arc length of the ray is zero, which gives The ray equations (equation findif2 oderay of Chapter findif2 ) give

The integral formulation of residual depth migration
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rmgapdb In this appendix, I derive an integral formula for residual depth migration using a kinematic approximation. Residual depth migration is a linear transformation from an image migrated with one velocity model to an image migrated with another velocity model. Suppose I migrate surface data with velocity . Equation resmig4 kmig of Chapter resmig4 gives the integral formula as follows: where stands for the cross-correlation in time, is the impulse response of the half-derivative operator, is the source wavelet, is the directivity function defined in equation resmig4 rsg , and and for or are the geometrical amplitude and traveltime fields in the velocity field, , respectively, with the source at the surface position . Two variables and are determined by the data geometry. For common-shot migration, and ; for constant-offset migration, and . ...

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