An Amplitude and Traveltime Calculation Using a Higher-order Parabolic Equation

A higher-order parabolic equation is used to compute the traveltime (phase) and the amplitude in constant density acoustic media. This approach is in the frequency domain, thereby avoiding the high frequency approximation inherent in the Eikonal equation. Intrinsic attenuation can be naturally incorporated into the calculation. The error at large angles of propagation caused by the expansion of the square root operator can be virtually eliminated by adding more terms to the expansion. An efficient algorithm is obtained by applying the alternate direction method. Our results are in excellent agreement with the finite element approach for the range-dependent wedge-shaped benchmark problem. The amplitude and the phase are calculated for a syncline and the Marmousi models. 11-1 Cheng et al. INTRODUCTION Numerical traveltime calculations are of great interest in exploration seismology. Traveltimes are crucial for the prestack seismic migration process. Finite difference schemes are the most computationally efficient traveltime methods (Vidale, 1988; Podvin and Lecomte, 1991; van Trier and Symes, 1991). These schemes are based upon the Eikonal equation, which is a high frequency approximation. However, finite difference schemes provide no amplitude information. The problems caused by using these first arrival times for imaging in complex velocity structures are discussed by Geoltrain and Brac (1993) and Nichols (1994). The parabolic approximation to wave equations is widely used in seismic migration processing (Claerbout, 1985); an example is the 45-degree migration. It is based on the continued fraction expansion of a square root operator. Difficulty with the higherorder differential equation resulting from the higher-order expansion limits its practical application to second order terms. The parabolic approximation is seldom used for forward modeling in exploration seismology. On the_ oJher hand, in the field ofocean acoustics the parabolic equation is the main tool for forward modeling (Jensen et al., 1994). The technique is well developed and widely used. The purpose of this paper is to demonstrate the use of a higher-order parabolic equation to compute the phase (traveltimes) and amplitudes in a complex velocity structure. With a higher-order Pade series expansion, the angle limitation of one-way propagation is virtually eliminated. The traveltimes and amplitudes are computed in the frequency domain and the high frequency approximation is removed. Since amplitudes are important for reliable imaging, intrinsic attenuation becomes an important factor. Attenuation can be naturally and easily incorporated into our frequency domain calculation. The calculated phase (traveltime) can be used directly in the common shot prestack migration. The phase term from the source to the imaging point can be used as the imaging condition to form the migrated section. The amplitude information can be added as a weight to this imaging condition. In the following section, the higher-order parabolic equation is derived and the numerical implementation is described. Finally, the phases and amplitudes of a simple syncline and the Marmousi model (Bourgeois et al., 1991) are calculated as numerical examples. HIGHER-ORDER PARABOLIC EQUATION In this section, we briefly derive the higher-order acoustic parabolic equation and describe its numerical solution method. In cylindrical coordinates (r, ¢, z) for a media with constant density and azimuthal symmetry, the Helmholtz equation at frequency w is: