Scattering attenuation: 2-D and 3-D finite difference simulations vs. theory

Abstract

Scattering of seismic waves is studied by producing synthetic vertical seismic profiling (VSP) seismograms with 2-D and 3-D finite difference modelling in random media. The random models used are Gaussian and band-limited self-similar, or fractal random media. The modelling is performed acoustically, but we believe that, considering the geometry of this study, the results obtained will hold for the elastic case as well. Properties of the random media are discussed, in particular the difference between discrete and continuous media, and the importance of this difference. We show that when using the band-limited Von Karman correlation function when generating the random medium, the size of the model should be greater than 2 πa, where a is the correlation distance, and the grid spacing should be less then a. If not, the medium will not have the proper characteristics. Analytical expressions for scattering attenuation, derived from single scattering theory, can be used to estimate scattering Q from borehole velocity logs, if it is known what minimum scattering angle, θ min, to use. θ min, the minimum angle energy, must be scattered to be regarded as not contributing to the propagating wave. We estimate θ min by comparing Q values estimated from our synthetic VSP seismograms with the analytical expressions. The comparison also shows when the assumption of single scattering is valid. Previous studies in 2-D give a θ min of ∼30°. In this paper, we make a comparison for both the 2-D and 3-D cases, and show that the Q estimate is highly sensitive to how the analysis is done. We show that single scattering theory agrees well with finite difference simulations for self-similar media with low Hurst numbers, but with a somewhat lower θ min of 10–20°. This holds for a range of correlation lengths, a, including the case of infinite, or absence of, a. For Gaussian and exponential media, simulations and theory agree as well with θ min of 10–20°, but only for ka<5, where k is the wave number of the source. For ka>5, simulations and theory diverge, and single scattering theory cannot explain the amplitude attenuation observed in the scattering simulations for these types of media, indicating that it may be difficult to estimate the fractal properties of a medium from seismic data alone. With the difficulties of characterizing the scattering medium, and to estimate the scattering attenuation in the simple case of synthetic data with pure scattering, we conclude that it may be difficult to separate scattering and intrinsic attenuation from real data.