Scale and Frequency Dependence of Reflection and Transmission Coefficients

Well-logs show that heterogeneities occur at many different spatial scales. In this paper, we want to characterize how waves are affected by these heterogeneities, and we study how reflection and transmission coefficients depend on temporal frequency and spatial scale. We use wavelet transformations to filter certain spatial scales from the velocity logs. The scale-filtered logs serve as input for a numerical layer-stack model to calculate reflection and transmission coefficients as functions of frequency and scale. We find that transmission coefficients are largely independent of frequency or scale. They depend mostly on average slowness. Contrarily, reflection coefficients are extremely sensitive to the perturbations of the slownesses, even at low frequencies. INTRODUCTION Seismic velocities obtained from well-logs commonly exhibit strong fluctuations over very short distances. Typically, the sampling interval is well below one meter. On the other hand, seismic properties are also estimated from surface seismic data which have a resolving power of tens of meters. Many researchers are studying how to relate the two different datasets obtained at different spatial wavelengths. We want to pose a slightly different question instead: which spatial scales of heterogeneity affect the propagation of seismic waves of a particular temporal frequency? We will answer this question by calculating reflection and transmission coefficients as functions of temporal frequency of the wave and spatial scale contents of the medium. As we will show, the results for transmission experiments differ dramatically from reflection experiments. We find that transmission is oblivious to the different spatial scales. Contrarily, reflections are extremely sensitive to change of scale content. Even perturbations at scales well below the seismic wavelength affect the resulting reflection parameters dramatically. Specifically, we will concentrate on reflection and transmission magnitudes and phases which relate to phaseand group-slownesses of the layer-stack.