On Methods For Fast Implementation And Optimization Of Gaussian Beam Migration

As one of the ray-based migration imaging methods,Gaussian beam migration overcomes the limitation of the standard ray-based migrations on dealing with the imaging problem in the caustic regions,while has the features of the standard ray-based migrations such as high efficiency and steep dip imaging,and can produce high quality images competitive with the reverse time migration.With its flexibility,this method can adapt to various acquisition configurations and complex surface condition.Up to now,the theory of Gaussian beam migration has been developed for acoustic,elastic,anisotropic and attenuated media.Since the Green function in Gaussian beam migration is expressed by the Gaussian beam summation method,the integrals over the ray parameters have to be calculated.The integrals over the ray parameters are two-fold for the two-dimensional condition while four-fold for the three-dimensional condition.Moreover,the integral over the frequencies also has to be calculated at each imaging point to obtain the imaging result.Previously,in order to increase the computational efficiency,the steepest descent approximation was adopted in the literatures to reduce the dimension of the integrals over the ray parameters at the cost of a precision loss.However,the fast implementation of the integral over the frequencies was rarely reported.To solve this problem,we propose a fast algorithm by changing the order of the integrals in the migration formula and then treat the two innermost integrals over the frequencies as a couple of two-dimensional continuous functions with respect to the real and imaginary parts of the total complex traveltime.After determining the sampling ranges and intervals of the real and imaginary parts of the total complex traveltime,a couple of lookup tables corresponding to the two innermost integrals are constructed at the sampling points.Finally,the results of the two innermost integrals are obtained through interpolation in the constructed lookup tables based on the actual values of the real and imaginary parts of the total complex traveltime at the imaging point.Therefore,the calculation of the integral over the frequencies at each imaging point is avoided.Meanwhile,since merely dealing with the integral over the frequencies,the fast algorithm is applicable to both of the migration formulae with and without the steepest descent approximation.The numerical examples under the parallel computation environment show that,when adopting the fast algorithm for the two-dimensional condition,the migration formula with the steepest descent approximation cannot achieve significant improvement in the computational efficiency but will result in a precision loss compared with that without the steepest descent approximation.For the three-dimensional condition,since the dimension of the integrals over the ray parameters is reduced from four to two by the steepest descent approximation,the migration formula with the steepest descent approximation shows significant improvement in the computation efficiency compared with that without the steepest descent approximation.But for the regions of complicated structures,in order to meet the requirement of imaging accuracy,the migration formula without the steepest descent approximation is still needed.Since the above fast algorithm is independent of the property of the media,it is applicable to Gaussian beam migrations in elastic and anisotropic media.Depth migration by the Gaussian beam summation method dealing with single trace can also adopt the similar strategy of the fast algorithm,and the main difference is that the lookup tables corresponding to the two innermost integrals over the frequencies are constructed for each trace.For Gaussian beam migration in attenuated media,due to the introduction of the attenuation factor for compensating the attenuation effect of the media,the two innermost integrals over the frequencies are treated as a couple of three-dimensional continuous functions with respect to the real and imaginary parts of the total traveltime and the total attenuation factor.Furthermore,the related issues about controlled beam migration are also discussed.Controlled beam migration suppresses the effect of the incoherent noises in the real data on the imaging results by picking out the most coherent events during the procedure of local slant stack.As the core technique of CGG geophysical company,the technical details of this method cannot be found out in the open literatures.Here,we employ the method of analytical trace to improve the stability of the calculation of coherence and optimize the definition method of the weight function to protect the coherent events better.Then,we combine the controlled beam migration with the proposed fast algorithm to increase the computational efficiency of the whole migration.At the end,by employing the property that in the constant-dip partial image the constructive parts of the local events have slopes consistent with that of the constant dip while the deconstructive parts of the local events have slopes distinctly different with that of the constant dip,we optimize the migration aperture in the local dip angle domain to improve the imaging results.Likewise,the method of analytical trace is employed to guarantee the stability when computing the coherences of the local events in the constant-dip partial image.