Peter Hubral’s research while affiliated with Bundesanstalt für Geowissenschaften und Rohstoffe and other places

A derivation and discussion of geometrical spreading factors are given for two- and three-dimensional earth models with curved reflecting boundaries. The spreading factors can be used easily to transform primary reflections in a zero-offset seismic section into true amplitude reflections. These permit an estimation of interface reflection coefficients, either directly or in connection with a true amplitude migration. A seismic section with true amplitude reflections can be described by one physical experiment: the tuned reflector model. 18 refs.

We thank Dr. Szaraniec for his interesting comments, which shed new light on the properties of the normal incidence synthetic seismogram. By the way, his equations (D‐4.1) and (D‐4.2) were also given by Nestler and Rösler [1977, equation (4)], although with an error; they write z k - 1 rather than z k . We are reminded of a saying by our colleague Prof. Dan Loewenthal of Tel Aviv University: the normal incidence discretely layered model is a rich lode from which a seemingly endless series of interesting formulas can be mined. Dr. Szaraniec has unearthed more nuggets, and we hope that his success will inspire others to continue searching. The relation between our results and the theory of continued fractions should, as Dr. Szaraniec points out, provide further insights.

The normal incidence unit impulse reflection response of a perfectly stratified medium is expressible as an autoregressive-moving average (ARMA) model. In this representation, the autoregressive (AR) component describes the multiple patterns generated within the medium. The moving average (MA) component, on the other hand, bears a simple relation to the sequence of reflection coefficients (i.e., primaries only) of the layered structure. An alternate representation of the reflection response can be formulated in terms of a superposition of purely AR time-varying minimum-delay wavellts. Each successive addition of a deeper interface to the layered system gives rise to an AR wavelet whose leading term is equal to the magnitude of the primary reflection originating at this interface. We accordingly call these wavelets 'generalized primaries'. The AR component of every generalized primary contains only those multiple reflections that arise from the addition of its particular interface to the layered medium. Therefore, the impulsive reflection seismogram can be decomposed into a progressively delayed summation of as many generalized primaries as there are refllction coefficients, referred to here as a 'sum AR' representation. -Authors

Among the various dynamic ray tracing systems described by Červený and Hron [1] is one particular linear system of second order that readily provides identical parameters (in the ray centred coordinate system) to those that fall out of the system by Popov and Pšenčík [3, 4]. Hence the initial conditions of the latter system for sources and interfaces can easily be used to provide those for the linear system of second order. © 1980 ACADEMIA Publishing House of the Czechoslovak Academy of Sciences.

The curvatures of a wavefront along a ray in laterally inhomogenous three‐dimensional velocity media separated by curved first‐order interfaces can be computed by a system of equations that includes a Riccati‐type differential matrix equation. This has a simple analytic solution along circular raypaths in media of constant velocity gradients.

A system of three ordinary non-linear first order differential equations is proposed for the computation of the geometrical spreading of the wave front of a seismic body wave in a three-dimensional medium. The variables of the system are the parameters which provide a second order approximation of the wave front.

. The ray series solution of the elastodynamic equation of motion for shear waves propagating through a laterally inhomogeneous three-dimensional medium can be simplified by the use of a particular coordinate system that accompanies the wave front along the ray of investigation. The system is entirely determined by parameters that are obtainable from the ray. The transport equations for the principal shear wave components are then no longer coupled, but reduce to the same type of equation which determines the principal compressional wave component.