AD-A247 553 PL-TR-91-2113 WAVE PROPAGATION IN LATERALLY VARYING MEDIA: A MODEL EXPANSION METHOD" DTIC" Charles B. Archambeau S ELECTE D SD University of Colorado/CIRES Campus Box 449 Boulder, CO 80309 I May 1991 Final Report I August 1988.1 February 1991 Approved for Public Release; distribution unlimited PHILLIPS LABORATORY ~J AIR FORCE SYSTKKS COMIAM HANSCO W AIR FORCE WASE, MASSACHUSM S 01731-5000 I L92-03185
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AD-A247 553
PL-TR-91-2113
WAVE PROPAGATION IN LATERALLY VARYINGMEDIA: A MODEL EXPANSION METHOD"
DTIC"Charles B. Archambeau S ELECTE D
SDUniversity of Colorado/CIRESCampus Box 449Boulder, CO 80309
I May 1991
Final ReportI August 1988.1 February 1991
Approved for Public Release; distribution unlimited
PHILLIPS LABORATORY~J AIR FORCE SYSTKKS COMIAM
HANSCO W AIR FORCE WASE, MASSACHUSM S 01731-5000
I L92-03185
SPONSORED BYDefense Advanced Research Projects Agency
Nuclear Monitoring Research OfficeARPA ORDER NO. 5299
MONITORED BYPhillips Laboratory
Contract F19628-88-K-0033
The views and conclusions contained in this document are those ofthe authors and should not be interpreted as representing theofficial policies, either expressed or implied, of the DefenseAdvanced Research Projects Agency or the U.S. Government.
This technical report has been reviewed and is approved forpublication.
T~tAVAALLZJDONALD H. ECKHARDT, DirectorEarth Sciences Division
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7 7 May 1, 1991 I Final 1 Aug. 1988 - I Feb. 19914. TITLE AND SUBTITLE S. FUNDING NUMBERS
Wave Propagation in Laterally Varying Media: A Model F19628-88-K-0033Expansion Method PR 8A10
______________________________________ TA DA*6. AUTHC R(S) WV AK
Chat les B. Archambeau
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The University of Colorado/CIRES REPORT 'dUMBER
Campus Box 449Boulder, CO. 80309
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Phillips Laboratory AGENCY REPORT NUMBER
Hanscom AFB, MA. 01731-5000 PL-TR-91-2113Contract Manger: James Lewkowicz/LWH
Approved for public release; distribution unlimited
13. ABSTRACT (Maximum 200 words)A general approach, using modes defined on subregions of the medium, has been
deveiop~d to model seismic wave propagation in media with vertically and horizontallyvariable elastic and anelastic properties. The approach is also applicable toacoustic waves in fluid media and electromagnetic wave propagation in laterallyvarying media. The restriction on the medium variability is that it can be repre-sented by step function variations in its properties in both the vertical and hori-zontal directions.
The basic method makes use of normal node expansion of the wave field in eachpartitioned sub-region of the medium within which the medium is uniform in the late-ral directions. Thus the medium is partitioned into laterally uniform zones andcomplete normal mode solutions are obtained for each horizontally layered zone. Inthe analytical devolopment the "zonal eigenvalues and eigenfunctions" are generatedby treating each zone as a layered half space or radially layered sphere, as is ap-propriate for medium geometry. The re;ulting set of modes are then used as a basesfor expansions of the wave fields in ti~e layered subregions. The modes are thenused as bases for expansions of the wa- e fields in each zone at the common boundaries
17. SECURITY CLASSIFICATION 18 SECURITY LASSIFICATION 19 SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS P, GE OF ABSTRACT
Unclassified L Iassi: ied Uncla~isified UL-
NSN 7S40-0I.280-5500 Stamdard Form 298 (Rev 2-89)
295- 102
13. cont.
between the zones where continuity of displacement and traction is required. This results
in the definition of a "lateral propagator" of the wave field 4hen applied to all the zonesmaking up the entire medium and is, in application, very similar to the classical "verticalpropagator" method. The method is .exact wilen the lateral variations are actually discontin-ous step changes in properties. When the actual changes can be approximated as a sequenceof steps the method should be superior in computational accuracy and speed to numericalmethods.
Table of Contents
Zonal Partitioning and Green's Function Representations I...
"Forward" and "Backward Propagating" Mode Expansions...... 5
OrthogonvIity and Normalization Relations for Zonal Eigenfunctions 8
Zonal Boundary Conditions, Projections and Lateral Propagators ................. 12
Summary and Conclusions ................................... ................................... 20
where (a)Gl and (a)gji are the zonal displacement and traction Greens' functions appropriate for
the zone or region V,.* The vertical boundary surfaces of Va are Ea and Et. 1, as indicated in
Figure 1. Here we assume no sources ir, 'd Va and that the Green's functions satisfy all inter-
nal boundary conditions on all horizontal layers in Va. (In this case there are no surface
integrals over intei ial boundaries in (1)). Green's functions in Va can be written in erms of the
eigenvalues ka and eigenfunctions (a)4 for this zone as
Summation over repeated coordinate indices is used throughout. Coordinate ind,ces will appear as lower case latin sub-scnpts and superscripts. The summation convention does not apply to any indices appeanng in parenthesis.
Throughout thia development the "sum" over the eigenvalues ka will be written as a discrete summation b.t it shouldbe understood that in an unbounded medium, such as a layered half space. pat of the wave number spectrum will be con-tinuous In this case the "sum" over ka must be interpreted as a generalized summation involving a regular sum over thedisci. :e part of the spectrum plus an integration c er the continuous part of the wave number spectrum
a- a a+iII I
_ _ _ _ I IS Va Va iI I I
I_ I
m J I_Va-il Va + prI I I
I I II I II I I
Figure I Zonal partitioning of a vertically and laterallyvarying medium into subregions of uniformhorizontal layering.
are introduced, where the inner product is defined over the surface E'a (or Ea-1) as:
<Wj(k ), Xj(ka)> a Wj(ro, k )7y(ro, ka)dao
with summation over the repeated coordinate index (j) implied. An exactly analogous result
holds for LU1; with the suffix "R" replaced by "L" in (24) and (25).
Comparing (25) with the equivalent expressions in (15) - (16), it is clear that the tuner pro-
ducts appeanng in (25) must reduce to delta functions over the angular index m and the mode
eigenvalues kca. In particular, the following orthogonality conditions apply*:
*Where it is obvious from context, the R and L identifying subscripts on the wave numbers Rkc and Lka will besuppressed in order to reduce clutter in the equations.
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Dr. Michel Bouchon Prc f. Ari Ben-MenahemI.R.I.G.M.-B.P. 68 Deartment of Applied Mathematics38402 St. Martin DEeres Weizman Institute of ScienceCedex, FRANCE Rehovot, ISRAEL 951729
Dr. Hilmar Bupgum Dr. Robert NorthNTNF/NORSAR Geophysics DivisionP.O. Box 51 Geological Survey of CanadaN-2007 Kjeller, NORWAY 1 Observatory Crescent
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Dr. Michel Campillo Dr. Frode RingdalObservatoire de Grenoble NTNFINORSAR.I.R.I.G.M.-B.P. 53 P.O. Box 5138041 Grenoble, FRANCE N-2007 Kjeller, NORWAY
Dr. Kin Yip Chun Dr. Jorg SchlittenhardtGeophysics Division Federal Institute for Geosciences & Nat'l Res.Physics Deparunent Postfach 510153University of Toronto D-3000 Hannover 51, FEDERAL REPUBLiC OFOntario, CANADA M5S IA7 GERNMANY
Dr. Alan DouglasMinistry of Defense Universita Degli Studi Di TriesteBlacknest, Brimpton Facolta Di IngegneriaReading RG7-4RS, UNITED KINGDOM Istituto Di Miniere E. Geofisica Applicata, Trieste,
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Dr. Manfred HengerFederal Institute for Geosciences & Nat'l Res.Postfach 510153 Dr. John Woodhot seD-3000 Hanover 5 1, FRG Oxford University
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Ms. Eva Johannisson Oxford OX I3PR, ENGLANDSenior Research OfficerNational Defense Research Inst.P.O. Box 27322S-102 54 Stockholm, SWEDEN
Dr. Fekadu KcbedeGeophysical Observatory, Science FacultyAddis Ababa UniversityP. O. Box 1176Addis Ababa, ETHIOPIA 14