Numeric tensor framework: Exploiting and extending Einstein notation

ARNTSEN, B. 1988. Numerical ray generation in the computation of synthetic vertical seismic profiles. Geophysical Prospecting 36,478-503. Ray theories are a class of methods often chosen to compute synthetic seismograms due to their efficiency and ability to deal with complex, three-dimensional inhomogeneous media. To deal with the large number of rays needed to compute synthetic seismograms, a ray generation algorithm is given which is capable of generating a numerical code describing each ray. The code describes a subset of all possible rays by considering only pre-critical reflections. In a horizontally plane-layered medium the generation of rays and computation of amplitudes and traveltimes can be efficiently accomplished by grouping the rays into reflection order and dynamic analogue groups. Expressions summing all unconverted rays and rays with a single mode conversion are given for source and receiver located at arbitrary positions within the medium. Examples of zero-offset synthetic VSPs obtained by this method are given.

Numerical ray generation in the computation of synthetic vertical seismic profiles1

Extraction of Seismic Profiles for One-Dimensional Nonhomogeneous Media Using the Impulse Response Model

This paper had been prepared as a tutorial complement to its companion paper, THE ARRAY MATRIX FOR SIGNAL PROCESSING [1], but could not be presented in the ICASSP-86 framework. Its objective was, and it is so now, to exemplify some of the adaptations and generalizations of matrix methods stemming from Tait's ARRAY MATRIX theory [2]. That theory has since been extended and consolidated as a mathematics of polyxa [3], which can abbet, we believe, the smooth transition of vector-matrix methods into the array processing era. Through these papers we wish to encourage individual and collective adoption of what is now called 'extended' matrix methods and notation, for communicating work on signal processing and in system theory. We exemplify this here on rederivation of the remarkable MARPLE algorithm.

Extended matrix formulation for the Marple algorithm

The diversity of notation permeating the publications on signal processing suggests the need for some consolidation and standardization. We advocate here the array theory and notation of Tait as a standardizing framework for a smooth transition from the vector-matrix era to the array domain and array processing era. A duality between array domain and matrix domain operations blends the flexibility and power of the array domain with the compactness and overview attributes of matrix expressions. The notation facilitates tractability through the super- and subscript patterns for the indices and the range set of the indices. We introduce fundamentals from Tait's work with some consolidation and exemplify notation and manipulations. The relevance for signal processing is demonstrated more fully in a companion paper [8].

The array matrix generalization for signal processing

The impulse response h(z,t) for the wave equation which models the propagation in a one-dimensional inhomogeneous lossless medium is detailed, Pulse spreading and pulse hight reduction versus travel distance are then accommodated as a superposition integral with the lossless response acting as an excitation function on a dispersion model g(z,t).

Impulse response for the one-dimensional inhomogeneous medium with an approximation for attenuation and dispersion

The fundamental tool for simulation and interpretive work on acoustical test data towards extraction of spatial and physical parameters of multilayered media, such as marine sediments or geological strata, is the ideal layered media reflection model. We present here a formulation for the ideal reflection process, when the source and receiver are in different layers. Layer transition times are not restricted to being equal. The wavelet index notation used aids in giving very tractable and compact results. The model gives the response as a sum of all replica pulse wavelets due to a source pulse signal, or equivalently as the sum of impulse wavelets in the reflection sequence model. The new features of this model format are (1) complete details of reflected wavelets to arbitrary reflection multiplicity, (2) systematic count and aggregation of all multipath wavelets with identical arrival times, (3) compact expressions for wavelet amplitudes and delay times. The model is pertinent, for example, for simulation and analysis of seismic responses to source pulses triggered in exploration boreholes.

W. Vetter

Papers

The array matrix generalization for signal processing

Impulse response for the one-dimensional inhomogeneous medium with an approximation for attenuation and dispersion

A raypath reflection model for layered media with source and receiver in different layers

Extended matrix formulation for the Marple algorithm