Wavefield representation, propagation and imaging using localized waves: Beamlet, curvelet and dreamlet

This chapter is a review on the application of wavelet transform to wave propagation and imaging. The authors review the development of phase-space localized propagators in heterogeneous media and the application to geophysical, especially seismic imaging. In the first part, phase-space localization, mainly along the line of time-frequency localization is reviewed. Then phase-space localization using generalized wavelet transform applied to wave field and one-way propagator decompositions are reviewed and analyzed. Physically the phasespace localized propagators are beamlet or wavepacket propagators which are propagator matrices for short-range iterative propagation. When asymptotic solutions are applied to the beamlet for long-range propagation, beamlets evolve into global beams. Various asymptotic beam propagation methods have been developed in the past, such as the Gaussian beam, complex ray, Gaussian packet, coherent state, and more recently the curvelet and dreamlet (tensor product of drumbeat and beamlet) methods. Local perturbation method for propagation in strongly heterogeneous media is also briefly described. Curvelet transform and its application to propagation and imaging are reviewed in comparison with the beamlet approach. Finally physical wavelet is introduced and dreamlet as a type of physical wavelet on the observation surface is discussed with the recent applications to seismic data decomposition, compression, extrapolation and imaging. Based on the review and analysis, some conclusions are reached as follows. For wavefield decomposition, beamlet, dreamlet and curvelet transforms have elementary functions of directional wavelets. Beamlet and dreamlet belong to a type of physical wavelet, representing an elementary wave (satisfying wave equation) in various wavefield decomposition schemes using localized building elements (wavelet atoms), such as coherent state, Gabor atom, Gabor-Daubechies frame vector, local trigonometric basis function. Curvelet transform is a specifically defined mathematical transform, characterized by the parabolic scaling. The parabolic scaling law, width.≈length² or its generalization width.wavelength2 is similar to the beam-aperture requirement for asymptotic beam solution: the beamwidth must be smaller than the scale of heterogeneity and much greater than the wavelength, i.e., α>β≫λ where λ is the wavelength, β is the beamwidth, and a is scale of heterogeneity. Optimal beamwidth is reached by balancing the beam geometric spreading which is controlled by the ratio β/λ, and the beam-front distortion which depends on α/β. Using optimal beamwidth, beamlet/dreamlet or curvelet propagator will be sparse in smooth media for short range propagation. For strong and rough heterogeneities, beamlet/dreamlet or curvelet scattering will occur and asymptotic propagator may not work well. In this case, the local perturbation method can be applied, in which the propagator is decomposed into a background propagator and a perturbation operator for each forward marching step. Numerical examples demonstrated the validity of the approach.