True-amplitude one-way wave equation migration in the mixed domain

Abstract

One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium,the extrapolation of the wavefields with the true-amplitude one-way wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain true-amplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.

Compensation for transmission losses based on one-way propagators in the mixed domain

“…From equation 20, the proposed one-way propagator is composed of two terms: transmission-coefficient-compensating amplitudes, and conventional FFD propagator governing amplitudes and phases. Consequently, the algorithm structure might allow the compensation term (equation 21) to be incorporated into other phaseshift operators in the mixed domain, e.g., split-step Fourier method (Stoffa et al, 1990) for media with a weak lateral velocity contrast, a generalized screen propagator (de Hoop et al, 2000), or even amplitude-preserving one-way propagators (Zhang et al, 2005;Vivas and Pestana, 2010). However, this needs to be verified in the future, for it is not the focus of this paper.…”

“…Most current true-amplitude migration algorithms include only geometrical spreading (Zhang et al, 2003(Zhang et al, , 2005Vivas and Pestana, 2010), and Q-compensation based on one-way wave-equation migration has been discussed by many authors (Dai and West, 1994;Mittet et al, 1995;Valenciano et al, 2011;Wang, 2008;Yu et al, 2002), but transmission losses during migration have been paid little attention. Deng and McMechan (2007) presented a twopass recursive algorithm to compensate for transmission losses using the framework of full-wave prestack reverse time migration (RTM) (Chang and McMechan, 1986).…”

Amplitude-Preserved Wave Equation: An Example to Image the Gas Hydrate System

“…Some researchers have done a great deal of work in solving one-way true amplitude equations to produce satisfactory imaging results [23,24], for example, the beamlet propagator is combined with it [25]. As mentioned above, based on the achievements of one-way wave equation migration, the theories of approximation, such as Taylor expansion, are used to solve one-way true amplitude equations [26][27][28][29]. In order to solve the issue of the limited imaging angle, ref.…”

Deconvolution-type imaging condition effects on shot-profile migration amplitudes

“…The OWWE do not propagate the wave field amplitudes properly (Wapenaar, 1990;Godin, 1999) and must therefore, be modified by introducing in the equations a new operator which includes lateral and vertical gradients of the velocity field. The new resulting equations are known as One-Way Wave Equation with True Amplitude (OWWE -TA) (Zhang, 1993;Zhang, Zhang & Bleistein, 2003;Vivas & Pestana, 2010).…”