Transmission Effects in the Continuous One-dimensional Seismic Model

Abstract

It is common in petroleum exploration to loher a variety of instruments down a well to record rock properties as a function of depth. These recordings are called well logs and are of many different types. One which was developed in the early fifties is called the sonic log or continuous velocity log (CVL). This log is a recording of the compressional velocity in the vertical direction as a function of depth.With the availability of this log the question arose of its connection with data obtained from seismic surveys. This subject was addressed in a very influential paper by Peterson, Fillipone & Coker. The treatment was highly simplified mathematically. It was based on a layered earth approximation, and only treated the leading term in a decomposition of the seismogram by order of reflection. This leading term has come to be known as the primary reflection synthetic seismogram. Subsequent to this paper, higher order terms were calculated, so-called multiple reflection terms. All these calculations were based upon a layered approximation to the CVL. These layered approximations were physically realizable in the sense that they could actually be constructed from suitable materials and real physical experiments conducted with them. It is, of course, true in such models that there is both a reflection coefficient and transmission coefficient associated with each interface.In the mathematical approximations there has always been a question of whether and how to treat the transmission effects. In order to shed light on this problem we here start with the continuous formulation, rather than with a layered model. We then consider a physically realizable layered approximation. A related layered model without transmission losses is then formulated and by a limiting process converted to a continuous model. This latter model is then compared to the original continuous model to reveal explicitly the effects of transmission losses.It should be emphasized that the interest in mathematical approximations derives from their connection with the inverse problem: that of obtaining the rock properties from seismic data. In these days of fast digital computers there is no longer any problem in computing synthetic seismograms based upon one-dimensional wave propagation theory. For the inverse problem, which is basically a problem in statistical estimation theory, a decomposition of the exact solution by order of reflection remains important, and the effect of transmission losses is a key issue. 519

Nonlinear multichannel impedance inversion by total-variation regularization

“…There are two different nonlinear formulas that, in the case of normal incidence, relate reflection coefficients to AI parameters: One is derived from a continuous earth model, in which the elastic parameters vary continuously with depth (Foster, 1975). The discretization of this model gives the jth component of the reflectivity function as…”

“…Zhang and Yin (2004) and Oliveira et al (2009) use Newton-type schemes to linearize the problem around a reference impedance model, and then they iteratively update it until convergence to a stationary point. In the continuous form of seismic data, it is also possible to linearize the relation between the reflection coefficients and the logarithm of the normalized impedance by assuming reflection coefficients with a small absolute value (Foster, 1975). In this case, the impedance section, in the logarithm domain, can be regularized and recovered in multichannel form via the linear inverse theory.…”

An Effective Acoustic Impedance Imaging Based on a Broadband Gaussian Beam Migration

“…where ρ i+1 , v i+1 and Z i+1 = ρ i+1 v i+1 are density, velocity and AI in the (i + 1) th layer, respectively, and r i , ρ i , v i and Z i = ρ i v i are reflectivity, density, velocity and AI in the i th layer, respectively. If the absolute value of the reflection coefficients is small, this nonlinear relationship between the reflectivity and the AI can be reformulated according to [51]:…”