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
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I L92-03185
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MONITORED BYPhillips Laboratory
Contract F19628-88-K-0033
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kolid Earth Geophysics Branch 61id Earth Geophysics B anchEarth Sciences Division Earth Sciences Division
T~tAVAALLZJDONALD H. ECKHARDT, DirectorEarth Sciences Division
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REPOT DCUM NTAIONPAG 0MS No 0704-0188P................. C*~ -.01 ,*o2., - ~; n.e t.-- orl w. ot t ac. 4 t'' .2e:9 at# a~~w~p~h e*,~jA~tgth q atAff-o 860 Cr..e e*.ne :'Ie' ,not d ornmmen rgavd ACqtC-.buratfl "t.ate or., a tq ~~
(01w,"oj~foffltjo O~jingj~q I,Cr. tot .0.CIP9r t.1j :6fdA .*% atT.CoA Neora,-~ se" .crs aoeorste for n-a.vatOo oceatonsand itzont jis .etfeeon08..9.qaw Sqte 204 A.neqors.4 J2202.430i *' oe 'e, l"q.~ n ue ~Oro~Adco~~etOO. iSS t~at'hngtofl DC 201
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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
7. PERFORMING ORGANIZATION NAME(S) AND AOORESSES) 8. PERFORMING ORGANIZATION
The University of Colorado/CIRES REPORT 'dUMBER
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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
14. SUBJECT TERMS 15. NUMBER OF PAGES
Wave Propagationi, Modes, Inh nogelecus media, Propagators 4016. PRICE CODE
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
References ..... .. ...... . ..... 21
NTIS CRA&M
DTIC TAB r-Una'nnounced LJustification.. .........
B y .... .......... ................. ................
Dist;'ibution I
Availabhiity C":,,"Avw n!,d i.1 or
Dist Specal
A-1
I1|
Wave Propagation in Laterally Varying Media: A Modal Expansion Method
by
Charles B. Archambeau
Zonal Partitioning and Green's Function Representations
Consider a two dimensionally varying elastic-anelastic medium, as indicated in Figure 1.
In each zone Vc, a = 1,2, - • M. the medium varies in the vertical direction (z), but is uniform in
the horizontal direction (y or p). The supposition is that the laterally varying medium can be
approximated by a seri s of step variations in material properties in the same way as is done in
the vertical direction.
In V. A e have for he frequency domain displacement field (a)u at any point r within Vc:
(a)ul (r, 0)) = I [tj(r)(a)Gj(r, ro, (o) - u,(ro)(a)g](r, ro; co)]dao (1)
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.
2
(a)i~jrr.;co = 4n (a)jj., ka) (ahft(, ka) (2)mc~ Na(k ,' (o)
where (a)11 is the complex conjugate of (aw and Na, is a normalization constant which may be a
function of frequency oi and the wave number ka. Since the WT)4J, are eigenfunctions for the
region Va, this Green's function satisfies all boundary conditions along the horizontal boun-
daries in Va. (For details see Harvey, 1983.)
Further, since:
= n~o) a(a)Gnir, r; co)1
where niO) is the surface normal to Ea and E,,-, and x/o) are source coordinate variables, then
(a~~I~; r; o) 4i ~(a)qi (r0, ka)(a)WI(r, ka)mJgJr;r;(o 1 Na~k a)c) (f3)
Here m is the angular index for cylindrical coordinates, ka the horizontal wave number
corresponding to the modes in Va and where:
(a)Y.g(r 0, ka) = n~o) C~jyn axp [(a)Wn(ro, ka)] (4)
Because of the horizontal layering in Va, the eigenfunctions (a)Wj and (a)Yj are defined sec-
fionally, that is:
( a ) , J = a ) W J ' ) ( z I z ~ i ! zJ i
with (s) the horizontal layer index in Va-
For the horizontally layered region Va we have that:
=aG (aPG' + "t)Gil(5(a)g~ = (ctRgjl + k9aiJ
3
Here ( 0; amid ( )G/ are the Rayleigh and Love type Green's displacement functions (with Simi-
lar names for the associated Green's tractions) and whe i e
("2Gjl(r, ro; o) = 4n E (0hp1(r0, A) (*)xp1(r, <)
(cL)Gjj(r. ro; co) = 4nIC L &)ko
with Rk. and Lk,, representing the Rayleigh and Love type mode eig nvalues. Likewis
(fgj(r, ro; c) = 4nt 1 (a)T(r 0, Rtka) ~(F , RA.)mpg~cG N R)(ka, w)(7
(aL)gj(r, ro; w) = 4nt E ~~( L k)(h(rJ)(7M,Lkca N&-)(ka, o
In cylindrical coordinates (p, 0, z), the eigenfunctions are (see, for example, Harvey, 1981;:
(c ,(r, Rk,) = (a Dm(Z; Rk.) P(Rk. )+ (aI)Em(z; Rk.) B(Rk~p. ,
(aR)P(r, Rka) = (c)Rm(z, Rkc,) PM(Rkcp,4O)(8+ (a)Sm(z; Rka) Bm(Rka p, 0
f1WrLk)= (a')Fm(z; Lka) Cm(LkaP, {9(c)'P(r, Lk) = (a)T(z; .ka)Cm(LkapO)J
Here P, B and C are the vector cylindrical harmonics defined as:
Pm(kp, ) zJm(kp)em
Bm(kp,o) ap d P + P*[~ Jm(kp)enT1 1 (10)
Cm(kp,4) Wkp__
where
4
Jm(kp) = H ,(kp) + H/i) (kp)
with Jm the cylindrical Bessel function and HA) and HA) the cylindrical Hankel functions.
These vector functions are clearly such that Pmo'Bin = Pm'Cm = BnCm = 0 and also have the
usual functional orthogonality. (e.g. Stratton 1941, Morse and Feshbach. 1953). here 6:, e. and
6# are the unit vectors in cylindrical coordinates. The various "stress-displacement" functions
(a)Dm, O)Em. (a)Rm'.' (a)Tm in (8) are the same as those usually appearing in the ordinary
developments for a laterally homogeneous layered half space -- such as described in Harkrider
(1964): Ben Menahem and Singh (1972), or Harvey (1981).
Similar representations for the eigenfunctions can be given in cartesian and spherical coor-
dinates. (In the latter case the eigenfunctions ,W and LW are usually termed spheroidal and tor-
sional; and P, B and C become vector spherical harmonics). The choice of cylindrical coordi-
nates implies rotational symmetry, that is that the medium is partitioned into zones V. which are
cylindrical shells, with 7igure 1 depicting a cross section at fixed 0. If cartesian coordinates are
used, then Figure 1 represents a cross section at constant y, with properties constant in the ±y
directions. In the development that immediately follows cylindrical coordinates will be used;
however the cartesian and spherical representations are also appropriate and the development
and results are analogous to those for the cylindrical choice.
"Forward" and "Backward Propagating" Mode Expansions
In addition to the eigenfunction expansions of the Green's functions in V,, we can also
expand the displacements and tractions, appearing in (1) in terms of eigenfunctions in V,. In
particular, (a), (ro) and (a)tj(r o) may be expanded in terms of "forward" and "backward" pro-
pagating modes as:
5
()u(r, o)) -(@)u i)(ro. Co) + ()U1 2)(ro0 a)1
()tj(ro, (0) (a)t"(r. (0) + ()tJM(ro, (0)Jwhere the sUp r cripts (1) and (2) denote modes propagating in the positive and negative radial
(p) directions. Specifically,
(0 011:[(*)a)( (qire k) +* (a)a 2)(k) ()w/j 2)(r0, (2(U)t( , to) - £kf(aa,1? (k )(. FV~1)(r0, k ) + (') )(k, ) (a)ijFZ)(ro k )] 1?
m(!2)
when
*'0)W a k0. g) Pp) + ()E,'(zg; k)BA + ()Fm,(zo, ka) CW)elm,#
(13)
(,I,(P)(r , k). ()R m(zo, k ) PW + O)Sm.,(zo; k ) D) + T,(z0; k ) Co1 eim'#
with
PAP) = ei, H)(kp) ; p = 1, 2
BP) = ~ Q~+64 J Hp~)(k4~p) (14)
The coefficients (a)a() (ks) are to be determined from boundary conditions at E. and E,.
these conditions bring the continuity of displacement and traction on these surfaces. On the
other hand, of course, all the functions (W)Dg), :Ea' ), (€)FM), (G)R-),, (a)Sg), and (a)Tg ) are
known functions of the coordinate variables and the intrinsic material properties of ti , internal
horizontal layers, since they are provided by the usuai one-dimensional propagator approach in a
layered half space (e.g.. Harvey, 1981). The explicit forms of the functions are included in the
6
Appendix 1.
Given that 01 and g in (1) can be split into Rayleigh and Love type Green's functions, as
defined in (5)-(9). then it follows that (O)uj can also be split into modal sums involving only (,j
and (OL"p. Therefore:
W au ,P Wu [ ()P) + (CPtulP)] (S
where
£('?pul(P) r., [o) ~ (1) (vk ) (fQ~ji) + ('a,,a 2) (,k;) (TyI)]
(16)(Cr)(p)(ro.)- (a() am aLa() ) Lk; aL)W
potl m'. (L) ()(L;,Tw(1 O
with
('(,,) = [(u)D, tko. ,m.), + (=a.)m,o: .ka),BM)] et'R,{ 17(TA) ,(p)(ro, Lk.) = (Q)Fm°(Z. ; ,kga) CO ) eunw#17
A similar decompo sition applies to the ti action Mt.
It is importart to note that the eigenfunctions used to expand the Green's functions in equa-
tions (2) - (7) are appropriate for the horizontally layered zone in Va and are themselves normal-
ized such that:
< (aWj(P)(ka), (a wj(P)(k )> n (IWP)(kr) (P)(k r)dV = 6(k, -k;) 6Im'
(19)
<N(Lhgj(P)(k.), (ct)y,(P)> . OT (p)(kar) (L) (P)(k'r)dV = 6(ka -k )6m'M
where (a)i denotes the complex conjugate of (a)W, and the right hand side involves the usual
delta functions. Therefore the normalization factors appearing in the Green's function expan-
sions are free parameters that may be chosen so as to appropnately normalize the zonal Green's
functions in V., a = 1,2, ... M.
To obtain the appropriate normalization factors for (G? and (aL)G and, in addition, to
express these Green's functions in forms that are convenient for use with the expanded form for
(a)uj in (15)-(16), it is useful to adopt an expansion form for the Green's functions that is similar
to that for (a)uj in (15). That is, using both (apI ) and (a)Wp,( 2) in the expansion for (a)G1, we
express the Green's functions as:
(a Gj (r, re; o)) = (cGMj'0 + ()Gi2)l( a) G (r, r; co) = (af)G 1) + (PG I12)f (20)
where:
(INGSP) = 4n E (a)ijfP)(ro, Rk ) ( )W,(P)(r, Rka)m, Rka RNip)(ka, ) (21)
(cL0Ge) = 47t E (chf(P)(r 0 , Lka) a)qi(P)(r, Lka)m, k LN~a)(ka , (o)
and similarly for (Ogfj and (L)gjJ, the Green's tractions.
Orthogonality and Normalization Relations for Zonal Eigenfunctions
We can use (15)-(16) in (1) and also substitute (20)-(21) into this representation integral.
Since the representation given by (1) should be of the form of the expansion in (15), we should
obtain by proper choice of the normalization factors, RNa) and .Na), exactly the expansion
given in (15) in terms of forward and backward propagating modes. In particular, from (1) we
have:
(M)u, (r, o) = (a)uj (r, co) + (aL)uj (r, co) (22)
8
with
(0)1ui(r, (o) = :Rt~ (i G j- RUj 1 R da0, ; r c Va'
(a)u,(r, (o) = L1 t1 L G I U (23)LL~l[Ltj (jGi d1a0 r a
Introducing the exphcit eigenfunction expansions from (15)-(16) and (20)-(2 1), we get:
auj(r, co)= (au,(P)(r, c)) (24)pal
'Vu.'P)(r, o)= ( a~ 9 (Rk; ) m-a. RNFa } [{) <aP(P)(Rk ), (a),# a(R)>a.,-i
M',Rk mRk,
- < (a)W(P)(Rk ), (a)T (1)(Rka)>a a-l}(a)Mli l(r , Rka) +{(aYJPP)(Rka), (a~1I(2)(ka)>a a-I
- < j()M1 (ik'), (aff1 (2) (Rk)> a..i} (M12) (r, k.~ (25)
Here terms of the form:
<xpi(k ), Xj(ka)>a, a-i a <Wj(k ), j(ka)>a + <j(k ), Xj(kt)>a.-I
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.
9
< ()Y10%(k) (Q)Wj~P)(k)>p - < (a)(P)(k'), (a)'j(P)(k,)>,
= [ [¢YP)(khro). (a)i(P)(karo) - (a)W(P)(k~ro) • (a TAP)(karo)dao (26)
= [P(k pp) -f)(kaPp) + Bp)(k~pp) 4?)(ktpp) 6m;
with a = , a - I and p = 1,2 and where np = 2tpp. ( !ere pp is the constant value of the radial
coordinate on the surface Ep.) In addition,
< ( MIP) (k'), (a)W(q)(k,) >p - < (ayjP)(k ), (a)fq)(ka)>P =
(27)
[('Pj(P)(k~ro). (a); q)(ktro) - (aP(P)(k~r0). (aqTq))karo)j da: 0
for a3 = a, a - 1 and p * q. Formally identical relations hold for the eigenfunctions (a)VP) and
are obtained by replacing the suffix "R" by "L" in (26) and (27). Here we observe th it the for-
ward and backward propagating modes are completely orthogonal sets. These conditions are
equivalent to those obtained by Herrera (1964) and McGarr and Alsop (1967) and were used by
Kennett (1983) in his development of a formalism for wave propagation in laterally varying
media. In more explicit form, equations (26) and (27) are equivalent to:
ka) (a)Unzo ; ka) - (a)Dm(zo k) (a)Rm(Zo ka) dzo =
11(a)m(zO k, (aL1nz 0 ka m a o
(a)Sm(zo ; k ) (a)-( ; ka) - (a)Em(z o k) (aF n, 70 ka) dz 0 = 6=
where the subscript "R" on the P-SV wave number has also bcen suppressed n these expres-
sions. For the SH modes the analogous orthogonahty relation is easily seen to b
1 [)Tm(zO ; kc)(W)Fm(zo ; ka) - ()Fm(zo; k')(a)Tm(zo ; ka) dzo = , '
10
where the wave numbers and k,, and kc now refer to the SH wave number set Lka. The "vertical
eigenfunctions" in V, are those defined in (13) and are simple expontials in zo. (See Harvey,
1981.) Here also we consider the k, to be discrete infinite sets, so that orthogonality is expressed
by the Kronecker delta 6.
Using these orthogonality relations in (25) gives:
r
"uuj(')(r, o)= Ya ( Ea mt (kP) ( iI')(r, k +)mk ,a
provided we take:
RNI() = [n PW l)(kaPa)' Pi)(kapa) + B) (kap) -iW)(kaPci)}
(28)
+ na-i{P P!(kapaj.) -P,' O(kaPa-I) + B')(kaPa-it)B(kapa- i)
Similarly,
Ruj(2)(r, C'))= (a ) (k.) ( )2)(r,k.)in,ka
provided
= ..Na) n, P&?(~,-k)(kapa) + 13 1)(kapa+ BAk2(kcppci
(29)
11
+ n.. )(kapai) -t P1)(kap._ 1) + B( )(kapa.-) • -2)(kp_0,iP
The results for (a)u are analogous and the normalization factors are:
i,.NI (u = [naCM)(kaPca)'CW(kaPa) + nla-iCA()(kaPa-i)c')(kaPa-1)] (30)
LN2(a ) [naC)(kopa) - 2)(kap) + na-1(31(kaPa))2)(kaa(
Thus, the form of the displacement field in any one of the zones Va is given by
(a)u3(r, o))= E (aml(Rka)(ay(i)(r, Rk,) +(a (2 (Rka) (qy42)(r, ,kc)1m,Rka
+ 1 (Lkc,) (aL9)i( 1)(r, Lko) + (qaa 1 (Lka)"aLi, 2 (r, l.ka) " r V= (32)m,Lka
which is (merely) a sum of P-SV modes propagating in the forward and backward horizontal
directions, plus a similar sum of SH modes. Further, the displacement field in V, is connected to
its values on the boundary surfaces Ec and E,-i by the representations in (23), with the Greens
functions given by the eigenfunction expansions of (20) - (21) and with the normalizations
specified by (28) - (31). Use of these latter representations provide the means of determining the
coefficients ('?a p) and wa (P) in (32), and thereby an explicit expression of the displacement
field in V. in terms of the modes of this horizontall layered region. As will be shown. the
coefficients between all the zones Va, a = 1,2, • • M, are linked by a propagator formalism.
Zonal Boundary Conditions, Projections and Lateral Propagators
Continuity conditions expressing conservation of momentum, mass and energy apply
throughout the medium, however complex the intrinsic matenal properties. In parucula such
conditions apply along the control surfaces E,, separating the zones of uniform lateral ploperties
12
in Figure 1. In the case of a solid medium, with welded contacts at all layer boundaries, the con-
tinuity conditions along the surface E. are:
jat a=a ~)a 1, 2,3 (33)
where the subscript a on the matrix brackets is used to indicate evaluation on the vertical boun-
dary E,, between the zones Va and Va~l
The displacements and tractions in (33) can be expressed in terms of the eigenfunction
expansion of (32). However, since the P-SV and SH modes are decoupled in Va and V+ 1, then
(33) can also be expressed by the decoupled set of relations:
(~A~'~k ) (?~j(P)(Rka)1 u)A(k) [(a+i)R4I1(P)(Rk+i)]] ,m Rka pat I (0)()RJ M Rku-, Put 'iRfP(k~ a]
(34a)
E Z~j~= E)(Lka)= E (] 1 j )tAg)(Lka+,) (a+,)(+) :j- 3M Lka Pu (a3'1FJP)(Lkc1 . a(aM kaikalP(P)(a+ )j Jal
(34b)
where the expansions in P-SV and SH i loves have been substituted for uj and t, on both sides of
(33). A similar set of boundary equations apply to the other vertical boundary of Va, on the sur-
face Ea - 1, in Figure 1. (In this case the matrices are evaluated on E. - I so the matrix indices in
(34) change to (a - 1) throughout, while on the right side of (34) all the eigenvalue and eigen-
function indices change from a + 1) to (a - 1).)
We can extramt expressions for individual mode coefficients (O'Ag?) and ( )Ag), appropri-
ate to the zone V,. in terms of the mode coefficients in the zone Va, I by taking integral inner
products ("project ons") between the displacement and traction eigenfunctions on both sides of
13
(34). Then we can use the (P-SV) orthogonality relations in (26) - (27), along with comparable
orthogonal relations for SH modes. Specifically, using inner product bracket notation as before
in equation (25) and taking the inner products between displacement and traction eigenfunctions
on both sides of (34), we have:
m, ()a pal <(a)qj(P)(kc), (a)X(s)(kn))> = k.d (a+i)A <)(ka+1 ) <(a+1Jj(P)(ka+1 ) , (a)(k(n))>c
(35)
where indices R or L have been suppressed but are implied, with appropnate use depending on
whether j = 1, 2 or j = 3, as indicated by (34a) and (341)). (That is, this equation applies to either
(34a) or (34b)). For specificity, one uses P-SV eigenfunctions and eigenvalues and a subscript
"R" when considering component equations with j = 1 2 and uses SH eigenfunctions and eigen-
values with subscript "L" when considering the j = 3 component equation.) Here k~nl denotes the
specific nth eigenvalue of one particular mode with angular index in'.
Now we can subtract the upper matnx equation in (35) from the lower one and then make
use of the orthogonality relations for P-SV modes in (26) - (27), and the obvious similar pair for
the SH modes, to obtain:
(O)A1 s)(k&1)) = F. j (a +AMg)(k+) <(a+ )WjiP)(k,+ 1) (u)P(s)(kn))> -
<(a+I)P()(k) (WW(kn))>U s = 1, 2 (36)
where we have equated the sums over m, on each side of (35), term by term. ['his equation
again applies to either P-SV or SH modes; however, for P-SV modes j = 1, 2 and or SH modes.
then j = 3. Therefore in (36) the implied summation over the coordinate index is wer j = 1 and
2, for the P-SV case, and for SH modes only the one term, for wh ch j = 3., oc urs. The fiee
14
index (s) denotes the forward and backward horizontally propagating modes, so that (36)
expresses a relationship for both mode types. The factor Na) is the normalization "constant"
appropriate for the different mode types. These factors are given in (28) - (29), for the forward
and backward propagating P-SV modes, and in (30) - (31) for the SH modes.
It can be seen from (36) that a particular mode in Va, at a particular eigenvalue (or wave
number), will be "excited" by all the forward and backward propagating modes in V,+, in the
manner descri ed by the expression on the right side in (36). Thus, all the modes in Va,,, at all
wave number', will contribute to the excitation of any one mode in V. (at a particular wave
number) in pr portion to the sum of the mode coefficients, (a4t)AM)(ka+t), weighted by the inner
product factors given by the bracket term on the right side of (36). Thus the weight factors in
(36) will be called coupling coefficients.
Considering the k.+, eigenvalues as a discrete (infinite) set {kg 1), as was implied for k.
by the use o k n), then we can define the discrete coupling coefficients as
in' ' (a + I ; )(i P (a)kP 3)(k.n)> - <a+YP)(kg1 (Gh#p()(k&>]
(37)
and (36) becomes:
(k~n)) PC' ) (a+l ;a) (a+)a ')(k 1 ) , s=l,2 (38)I p-i
The coupling coefficients can be expressed in more detail when the specific functional
forms of the eigenfunctions appearing in the inner products are used in (37). In this case we can
use the orthogonality of the vector cylinderical harmonics to reduce the coupling factors to sim-
ple integrals over the vertical (z) coordinate on the boundaries of each zone Va. Specifically,
from (37) for the P-SV case, using the eigenfunction expressions given earlier in (13) - (14), one
15
has:
RCi 1 'S) (a+l I a) n [{<(+)D, ()R> - <(a+)R I, (a)D> ''(k pa). P)
+ {<(a+i)EIt (a)Sn> - <(a+')S ,, (a)E> B9)(kQ! .iPa)-)(kn)Pa)l (39)
where n. = 2nPa, with p, denoting the value of the radial coordinate on the surface E,. Further
the various inner products involve the "vertical eigenfunctions" defined in (13) ,nd (17); where
these inner products have explicit forms of the type:
<(alI)DI , (a)Rn> a i (a+')Dm(zo; j%)(a)R~n(zo; k ,))dzo (39a)
with similar expressions for the other products in (39). If these products are compared to those
in (26) and (27) - or more directly to the orthogonality relations involving the vertical eigenfunc.
tions given by the equations following equation (27) it can be seen that the inner products in
(39) reduce to delta functions if the eigenfunctions in the zones V. and V., are the same; that
is. if (a+i)Dm = (G)Dm , (a+i)Rm = (a)R, etc. This, of course, is as it must be, since only when the
physical properties in the two zones are identical will the eigenfunctions be the same and it then
follows that the coupling matrix must be diagonal -that is that the boundary between the two
zones produces no ,ioss mode excitation and is transparent. We see, therefore, that t'ie analyn.
cal expression in (39) for the coupling does indeed have this required property.
The normalization factor for Ca-s) is the ratio HNs(CO / nt which can be redefined as [.\,, a),
where from the previous expressions for RNs(a), in (28) and (29), ttus co istant has the form:
RN,(a) = PQ)(k ,n)p,'-ft)(k(n)pj) + Bs)(k P)' knf(ka p.)
(40)
+Ps-. P)() l)p) p 1 )a-, -m ,.- -,I
16
In an exactly analogous fashion the coupling coefficients for the SH modes are found to be:
LC gn)(a+ Ia)= na) [<( ,)F1, (a)Tn>-<(a+,)T1 ,(a)Fn>]C )(kalpa). 1)(k(n)pa)
(41)
where the inner products are again of the simple form:
<(a .I)F , (a)Tn> X I(a+')Fm(zo ; k&1c1)(a)Tm(Zo k&")) dzo (41 a)
Further, we can again define a new normalization factor LNS(O) W LN,(W) / nt which has the foi m:
L&'(0) = C~)knp)r n)knp.+ fe.11L] (k )C,) !m)(kln)pa..I] (42)L .
The con- rutations involved in determining these coefficients are straightforward, since the
cylinderical t rmonics are tabulated and tihe integrals over the vertical coordinate z4 involve
simple integrn Is of exponentials that can be evaluated analytically. in closed form, for the gen.
eral case.
Since (3.:) constitutes a set of two equations for s = 1 and s = 2, corresponding to forward
and backward propagatng modes and since the sums on the right can clearly be expressed as a
product of ma.rnces, it is natural to write the results in matrix form. Therefore we define:
Oa ) (&))
t (aaS[a u .for s = I and2 (43a)
and a similar column matrix of length (L) denoted [(*+')a (P 1. where the angular index m has
been -uppressed ,n wnnng the mode excitation matrices. Further, we can define coupling
17
m-trices by:
C 11 (p,'C21(PS) CLI(Ps)
C2(Ps ) C 22(P's) CL2(p's)[C ( s)] - - (43b)
C IN(P's) ... C (ps)
for each s and p value, where s - 1,2 and p a 1.2. With these definitions one c, n write the sys-
tem of equations implied by (38) in the form:
[(~aZ) =[C gj) ] [C (2.2)1 [,) a2 (44)
where the forward and backward propagating mode excitation coefficients are, isplayed expli-
citly. In defining the [C6M) I matrices, and in writing the matrix result in (44). the "a indices"
have been suppressed. However, when confusion can arise they should be written as
[Ch,(PA) (ct + I; a)]. etc., since the a indices change when the matrix refers to a boundary other
than -,. (eg. Between the zones Va-. and V., on the surface E,1, the cot ping matrix is
expressed as [Ci(P') (a; a-)]).
Obviously the coupling matrices are square only if L = N. that is if we use as many modes
in V. as in V., to represent the propagating waves. This choice will be adhe: Ad to, from this
point forward, although it is not a necessary condition.
It is evident that the partitioned matrices can t e written in unpartitionc I form as well,
where, with L - N, the mode coefficient matrices are nf d nension (2N x 1) and the coupling
matrix is square and of dimension (2N x 2N). Thus. we can also define mode coefficient
matrices consisting of the (ordered) mode coefficients for the forward and backward propagating
modes in the zones V. and V.+1 as (say):
18
=) [(oa 2[(a(ama]
(45a)
(+ ((a+ia(]a ,
and, similarly, we can define what can appropriately be called a horizontal propagator matnix:
[H ~ ~ ~ c~z [ca+I a. (5b
Now the equation (44) can be written in the more compact form:
[(G)m] = [Hn(a+1, a)] [(a+,m] (46)
and expresses the required conditions between the mode coefficients in neighboring zones.
If we take successive values of a, with c ranging from I to M. I say, then we get
[MM a,] = [Ht(2 ; l)1[(2)rM1]
[(2)mnl a [Hi(3 ; 2)l[(3)mJ
[€"-')m n] = [H 1n(N ; M- 1)lI('4)M 1
Clearly, by notin, in these equations that the indices I and n are just dummy indices providing a
numbering syster i for the eigenvalues, then
I (Imn] = [Hlj (2;, 1)1 [H ,(3 ; 2)] ... IX(M * Ni-I1)] [(Mt)M 1]
by successive su~stitunons. Consequently, we can write, for any 1 > a + 1:
19
This is a propagator equation that connects the mode coefficients in any zone Vp with those in
any other zone Va. In case 1 = a + 1 the equation (47) reduces to equation (46), which connects
the coefficients in any two neighboring zones. Since the coupling coefficients composing :Hin'L J
can be computed from the simple eigenfunction inner products at the zone interfaces, this equa-
tion provides the means of computing mode coefficients that produce displacements and trac-
tions satisfying all the boundary conditions along the vertical beundaries of the medium. Since
the eigenfunctions used already satisfy the boundary conditions along the horizontal boundanes
in each zone, then by use of the horizontal propagator relation all the boundary conditions in the
laterally and vertically "layered" medium being considered can be satisfied.
Summary and Conclusions
The basic method described here makes use of normal mode expansions of the wave field
in each partitioned sub-region of the medium within which the medium is uniform in the lateral
directions. Thus the medium is partitioned into laterally uniform zones and complete normal
mode solutions are obtained fro each horizontally layered zone. In the analytical development
the "zonal eigenvalues and eigenfunctions" are generated by treating each zone as a layered half
space or radially layered sphere, as ia appropriate for th medium geometry. The resulting set of'
modes are then used as bases for expansions of the wave fields in the laycied subiegions. 1i1C
mode expansions defined on the zones are then "connected" by matching (equating) the exact
Green's function representations of the wave fields in each zone at the common boundaries
between the zones where continuity of displacement and traction is required. This results in the
definition of a "lateral propagator" of the wave field when applied to all the zones mak-ng up the
entire medium and is, in application, very similar to the classical "vertical propagator method.
20
The method is exact when the lateral variations are actually discontinuous step changes in pro-
perties. When the actual changes can be approximated as a sequence of step the method should
be supenor in computational accuracy and speed to numencal methods.
In implementations of this method it is only necessary to compute the "zonal" normal
modes once, and subsequently these zonal mode solutions can be combined in a variety of ways,
using the lateral propagator equation, to produce theoretically predicted wave fields in many dif-
ferent laterally varying structures wintout the necessity of a complete recomputation of wave
fields in each new structure. Further, the propagators are analytically defined so that manipula-
tions related to inversion and perturbation calculations can be considered. For these reasons, and
because of its inherent high accuracy, this method should prove useful in modeling seismic wave'
fields in complex media and in inversion studies. In the present study the method is developed
in detail for two dimensionally variable media, using cylindrical coordinates and wave functions.
However, analogous results in rectangular and spherical coordinates may be obtained using the
same procudure and are appropriate for media with variability in all three spatial dimensions.
References
Ben Menahem, A. and S.J. Singh, Computation of Models of Elastic Dislocations in the Earth,
Me hods in Computational Physics, Vol. 12, Academic Press, 1972.
Harkrider, D.G., Surface Waves in Multilayered Media, I. Rayleigh and Love Waves from
Sources in a Multilayered Half Space, Bull. Seism. Soc. of Am., 54. 1964.
Harvey, D., Seismogram Synthesis using normal mode Superpo,,tion: the Locked Mode
Approximation, Geophys. J. Roy. w tron. Soc., 66. 1981
21
Hames, DJ., Contribution to dhe linearized theory of surface wave transmission, J. Geophys
Res, 69,1964.
Kenneth, B.L.N., Guided wave Propagation in laterally varying media - 1. Theore'cal develop-
ment, Geophys. Res., 69.1964.
Mc Garr, A. and L.E. Alsop, Transmission and reflection of Rayleigh waves at vertical boun-
daries, J. Geophys. Res., 72. 1967.
Morse, P.M. and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, 1953.
Stratton, J.A., Electromagnetic Theory, McGraw-Hill, 1941.
22
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Mr. Alfred Lieberman Dr. Frank F. PilotteACDAiVI.OA'State Department Bldg HQ AFTAC(ITRoom 5726 Patrick AFB, FL 32925-6001320 - 21st Street, NWWashington, DC 20451
Stehen Mangino Kade PoleyPhillips LabortorylWH CA.ACIS/TMCHanscom A 9, MA 01731-5000 Room 4X16NHB
Washington, DC 20505
Dr. Robert Masse Mr. lock RachlinBox 25046, Mail Stop 967 U.S. Geological SurveyDenver Federal Center Gology, Rm 3 C136Denver, CO 80225 Mail Stop 928 Natdonal CenterReston. VA 22092
Art McGarr Dr. Robert ReinkeU.S. Geological Survey, MS-977 WINTESG345 Middlefield Road Kirtland AFB, NM 87117-6008Menlo Park, CA 94025
Richard Morrow Dr. Byron RistvetACDAIVI, Room 5741 HQ DNA, Nevada Operations Office320 21st Street N.W An: NVCGWashington, DC 20451 P.O. Box 98539
Las Vegas, NV 89193
Dr. Carl Newton Dr. George RotheLos Alsmos National Laboratory HQ AFTAC/FTRP.O. 'Jox 1663 Patrick AFB, FL 32925-6001Mail Stop C335, Group ESS-3Los Alamos, NM 87545
Dr. iVao Nguyen Dr. Alan S. Ryall, Jr.AFr CflTR DARPA/NMROPatrick AFB, FL 32925 1400 Wilson Boulevard
Arlington. VA 22109-2308
Dr. K..":h H. Olsen Dr. Michael ShoreLos A!unos Sc.endtfic Laboratory Defense Nuc!ear AgencyiSPSSP. 0. Box 1663 6801 Telegaph Road.%I,;, ,: D--c6 Alexadr.- VA 22310Los Alaios, NM 87545
Phillips LaboratoryAtn: XOHanscom AFB, MA 01731-5000
Dr. Lar y Turnbull Phillips LaboratoryCIA-OS WR/ Atm: LWWashihgton, DC 20505 Hanscom AFB, MA 01731-5000
Dr. Thomu Weaver DARPAIPMLos Alamos National labortoy 1400 Wilson BoulevardP.O. Box 1663, Mail Stop C335 Arligton, VA 22209Los Alamos, N 87545
Phillips Laboratory Defense Technical Informaion CenterRsamh Library Cameron StationATFNl: SLL Alexadria, VA 22314 (2 copies)Hanscom APB ,MA 01731-5000
Phillips IAbo -r Defense Intelligence AgencyATM: SUL Directomw for Scientific& Technical IntelligenceKinladAPB, NM 87117-6008 (2 copios) At1: DT1B
Washington, DC 20340-6158
Secretaryo the Air Force AFrACJCA(SAFRD) (STINFO)Washington. DC 20330 Patrick AFB, FL 32925-6001
Office of the Secrmu Defense TACIECDDR&E Battelle Memorial InstituteWashington, DC 20330 505 King Avenue
Columbus, OH 43201 (Final Report Only)
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CONTRACTORS (FOREIGN)
Dr. Ramon Cabre, S.J.Observatorio San CalixtoCasilla 5939La Paz, Bolivia
Prot~ Hans-Peter HazqesInstitute for GeophysicRurUniversity/BochurP.O. Box 1021484630 Bochum 1, FRG
Prof. Eystein HusebyeNTNF/NORSARP.O. Box 51N-2007 Kieller. NORWAY
Prof. Brian .$. KennettResearch School of Earth SciencesInstitute of Advanced StudiesG.P.O. Box 4Canberra 2601, AUSTRALIA
Dr. Bernard MassinonSociete Ra4iorana27 rue Claude Bernard75005 Paris. FRANCE (2 Copies)
Dr. ?,.wre MechelerSociete Radlomana27 rue Claude Bernard75005 Paris, FRANCE
Dr. Svein MykkeltveitNTNF/NORSARP.O. Box 51N-2007 Kjeller, NORWAY (3 copies)
13
FOXEtGU (OTHER)
Dr. Peter Basham Dr. Tormod KvaemaEarth Physics Branch NTNF/NORSARGeological Survey of Canada P.O. Box 51I Observatory Crescent N-2007 Kjeller, NORWAYOttawa, Ontario, CANADA KIA 0Y3
Dr. Eduard Berg Dr. Peter MarshallInstitute of Geophysics Procurement ExecutiveUniversity of Hawaii Ministry of DefenseHonolulu, HI 96822 Blacknest, Brimpton
Reading FG7-4RS, UNITED KINGDOM
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
Ottawa, Ontario, CANADA KIA 0Y3
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,
ITALY
Dr. Manfred HengerFederal Institute for Geosciences & Nat'l Res.Postfach 510153 Dr. John Woodhot seD-3000 Hanover 5 1, FRG Oxford University
Dept of Earth SciercesParks Road
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
