LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Allan Abrams; manufacturing supervised by Jacqui Ashri.
A recently developed formulation of the inverse source problem as a Fredholm integral equation of the first kind provides motivation for the development of analytical characterizations of the nonuniqueness in the inverse source problem. Nonradiating sources, i. e., sources for which the field is identically zero outside a finite region, are introduced. It is then shown that the null space of the Fredholm integral equation is exactly the class of nonradiating sources.
Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common-midpoint (CMP) spreads is usually well described by the normal-moveout (NMO) velocity defined in the zero-spread limit. In their recent work, Grechka and Tsvankin showed that the azimuthal dependence of NMO velocity generally has an elliptical shape and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for normal-moveout velocity in anisotropic media of arbitrary symmetry.For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The influence of anisotropy on normal-moveout velocity is absorbed by the slowness components of the zero-offset ray (along with the derivatives of the vertical slowness with respect to the horizontal slownesses) -quantities that can be found in a straightforward way from the Christoffel equation. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix-type average of the matrices responsible for the "interval" NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrary anisotropic media. For models with a throughgoing vertical symmetry plane (i.e., if the dip plane of the reflector coincides with a symmetry plane of the overburden), the semi-axes of the NMO ellipse are found by the more conventional rms averaging of the interval NMO velocities in the dip and strike directions.Modeling of normal moveout in the most general heterogeneous anisotropic media requires dynamic ray tracing of only one (zero-offset) ray. Remarkably, the expressions for geometrical spreading along the zero-offset ray contain all the components necessary to build the NMO ellipse. This method is orders of magnitude faster than multi-azimuth, multi-offset ray tracing and, therefore, can be efficiently used in traveltime inversion and in devising fast dip-moveout (DMO) processing algorithms for anisotropic media. This algorithm becomes especially efficient if the model consists of homogeneous layers or blocks separated by smooth interfaces.The high accuracy of our NMO expressions is illustrated by comparison with ray-traced reflection traveltimes in piecewise-homogeneous, azimuthally anisotropic models. We also apply the generalized Dix equation to field data collected over a fractured reservoir and show that P -wave moveout can be used to find the depth-dependent fracture orientation and evaluate the magnitude of azimuthal anisotropy.
An approximate solution is presented to the seismic inverse problem for two-dimensional veloci ty variations . The solution is given as a multiple integral over the data observed at the uppersurface. An acoustic model is used and the reflections are assumed to be sufficiently weak to allow a "l i nearization" procedure in the otherwise non-linear inverse problem . Synthetic examples are presented demonstrating accuracy of the method wi th dipping planes at angles up to 45° and wi th veloci ty variations up to 20%. The method was also tested under automatic gain control , in which case velocity estimates were lost but the method nonetheless sucessfufly migrated the data.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.