IN
LATERALLY VARYING MEDIA
by
Michael Gerhard Bostock
A thesis submitted for the degree of
Doctor of Philosophy
of
The Australian National University
A B S T R A C T
It is generally acknow ledged th a t th e e a rth exhib its a d o m in an tly rad ial v aria
tion in m a teria l p ro p erties on which a less pronounced, la teral v ariatio n is su p er
im posed. T his la te ra l v ariation reflects a com plex rheology and co n tains im p o rta n t
in fo rm atio n regardin g th e n a tu re of m a jo r dynam ical processes w ithin th e earth .
Seism ic waves are, w ith o u t d o u b t, th e m ost sensitive m eans cu rren tly a t o ur dis
posal for d elin eatin g b o th large and sm all scale e a rth stru c tu re , an d consequently
afford th e b est p ro sp ects for en han cing our know ledge of th e forces w hich sh ap e th e
p la n et. However, a com prehensive u n d erstan d in g of th e effects of e a rth s tru c tu re
on th e passage of seism ic waves is a p rereq uisite to this objective. T his thesis is
concerned w ith th e th eo re tic al d escription of surface wave p ro p ag atio n in several
classes of la terally varying, stratified m edia which m odel th e o u te r layers of th e
e a rth .
T h e first stu d y deals w ith th e p ro p ag atio n of th e regional p h ase L g, usually in te rp re te d as a sum of higher m ode surface waves, in a cru stal w aveguide of variable
thickness. A ray-based m ethod is in tro d u ced which can be used to m easure, in a
se m iq u a n tita tiv e m an n er, various asp ects of sc atterin g from changes in to p o g rap h y
of th e e a r th ’s surface and th e M ohorovicic discontinuity. T h e m eth od is applied to
m odels of central A sia and is used to explain th e observed behav io u r of Lg in this s tru c tu ra lly com plex region.
T h e rem a in d er of th e thesis represents a unified tre a tm e n t of surface wave p ro p
ag atio n in different form s of lateral heterogeneity based on rep resen tatio n theorem s
of classical ela.st0d3ma.mics and wavefield expansions in series of surface wave basis
functions. In th e first of th ree p a rts , th e T -m a trix form ulation of sc a tte rin g from a
single o b stacle is ex ten d ed to surface waves using previously unrecognized o rth o g o
n ality relations which involve integral p ro p erties of th e surface wave eigenfunctions.
T h e T -m a trix depen d s only on th e size and sh ap e of th e o b stacle, is valid in b o th
th e near- an d far-fields, and provides a com plete account of th e sc a tte rin g response
to a given h arm onic wave. N um erical exam ples are presented to d e m o n stra te th e
th e second p a rt to derive a com posite surface wave T -m a trix for an a rb itra ry m ul
tip le o b stacle configuration an d which accounts for th e en tire hierarchy of m u ltip le
sc a tte rin g in teractio n s. T h e im p o rtan ce of m ultiple sc a tte rin g in th e co n tex t of
regional phases such as Lg is exam ined by way of several num erical exam ples. In th e final p a rt th e T -m a trix description is recast in a refle ctio n /tran sm issio n
form alism which is developed to describe sc atterin g from obstacles ex h ib itin g a
degree of q uasi-concentric m ultilayering a b o u t a vertical axis. T his d escrip tio n , al
th o u g h highly idealized from a geophysical perspective, leads to a th eo ry for surface
wave p ro p a g a tio n in m edia ex h ib itin g a continuous v ariatio n in physical p ro perties.
T h e th e o ry relies on th e expansion of th e G reen ’s function for a stratified reference
m edium in term s of th e surface wave basis functions an d is shown to be sim ply th e
3-D ex tensio n of coupled m ode techniques used to m odel surface wave p ro p ag atio n
in w aveguides ex h ib itin g stric tly 2-D lateral heterogeneity. R esults from num erical
ex p erim en ts are p resented to exam ine th e effect of n ear-source h etero gen eity on
P rio r to s u b m ittin g form al ap p licatio n to u n d erta k e P hD stu d y a t th e R esearch
School of E a rth Sciences I expressed to my eventual supervisor, Dr B. L. N. Ken-
n e tt, my wish to work on a th e o retical problem in whole e a rth seismology. My
aim s w ere to acq u a in t myself w ith an a rea of seismology in which I was in terested
b u t largely ig n o ran t, an d to develop som e experience and p erh ap s proficiency w ith
m e th o d s of m a th e m a tic a l an d c o m p u ta tio n a l physics which, I believe, are indis
p en sab le tools for a good seism ologist, w hether th eo retician or o b serv atio n alist. I
owe a g re a t deal to B rian for creatin g an environm ent in which I was able to fully
realize my objectiv es. T h e problem s he suggested I investigate were challenging,
in terestin g an d to pical. He provided me w ith sound guidance and en co u rag em en t,
an d gave u n sp arin g ly of his tim e w henever I wished to discuss difficulties rela ted
or u n re la te d to research. I am also extrem ely g ratefu l to B rian for his efforts in
estab lish in g a research assistan tsh ip which allowed m e to look after ce rta in fiscal
responsibilities I was obliged to assum e as an overseas stu d en t.
P hil C um m ins has been a m uch-valued friend and advisor. He w atched p a
tien tly as I g esticu lated my ideas, discussed various aspects of his own varied re
search in terests and provided me w ith m any avenues of d istra ctio n from seismology,
not th e least of which included old copies of th e G u ard ian Weekly. I have enjoyed
an d b en efitte d from conversations on various aspects of seism ology w ith Roger
B ow m an, D oug C h ristie, A nton H ales, K azuki K oketsu and Ron List. I am g r a te
ful to Colin T hom so n a t Q u een ’s U niversity for first in tro d u cin g me to th e field of
th e o re tic a l seismology, an d for encouraging and su p p o rtin g my ap p licatio n for P hD
c a n d itu re a t th e A .N .U . Staff and stu d e n ts too num erous to m ention by n am e, on
th e seism ology floor an d th ro u g h o u t R .S.E .S, have had a definite influence on th is
thesis by c o n trib u tin g to an atm o sp h ere conducive to ded icated research.
A n u m b er of individuals have helped to m ake my tim e in C a n b e rra a m ore
enjoyable experience, and have th ereb y c o n trib u ted in a significant, th o u g h in d irect,
way to th e com pletion of this thesis. I wish in p a rtic u la r to th a n k L in d a Ayliffe,
they played in this experience which included forays in to re m o te r p a rts of New
S outh W ales, my acq u a in ta n ce w ith w indsurfing, m o rn in g and afte rn o o n ten nis,
and various exercises in food g en eratio n , p reservation an d co n su m p tio n .
I acknow ledge financial assistan ce from th e A u stra lia n N a tio n a l U niversity in
th e form of an A.N.U P o s tg ra d u a te Scholarship. I am g rate fu l to th e d irec to r
of R .S .E .S , K u rt Lam beck, for seeing fit to finance my tr ip to th e 1990 A m erican
G eophysical U nion m eeting in San Fransisco w here I h ad th e o p p o rtu n ity to present
p o rtio n s of this research and discuss my ideas w ith research ers in re la te d fields.
Finally, I th a n k my p aren ts, H ew itt an d Sigrid, a n d m y g ra n d fa th e r, H ugh, for
T h is thesis is based on work condu cted in th e R esearch School of E a r th Sciences
a t T h e A u stralian N atio n al U niversity betw een A ugust 1989 and J u n e 1991 while I
was a full tim e research stu d e n t. T h e work presented here is believed to be original
except w here explicit m ention is m ade in th e te x t to th e work of o th e rs, an d has not
been su b m itte d for any degree or q ualification a t any o th e r in s titu tio n . M uch of th e
m a teria l contained in this thesis has, however, been published or is s u b m itte d for
p u b licatio n . T h e co rrespondence of published p ap ers or m a n u scrip ts w ith ch ap ters
and ap pend ices is no ted below.
C h a p t e r 2:
Bostock, M. G. & B. L. N. K e n n e tt, 1990. T h e effect of 3-D s tr u c tu re on Lg p ro p ag atio n p a tte rn s , Geophys. J. Int., 1 0 1, 355-365.
C h a p t e r 3:
B ostock, M. G ., 1991. Surface wave sc a tte rin g from 3-D o b stacles, Geophys. J. Int., 1 0 4 , 351-370.
C h a p t e r 4, A p p e n d i c e s B C:
B ostock, M. G. & B. L. N. K en n ett, 1991. M ultiple sc a tte rin g of su rface waves
from discrete o bstacles, A ccepted for p u b licatio n in Geophys. J. Int.
C h a p t e r 5, A p p e n d i x D:
B ostock, M. G ., 1991. Reflection an d tran sm ission of surface waves in la terally
varying m edia, S u b m itte d to Geophys. J. Int.
A p p e n d i x A:
Bostock, M. G ., 1990. On th e o rth o g o n ality of surface wave eigenfunctions in
C O NTEN TS
A B ST R A C T
i
ACKNOW LED GEM ENTS
iii
PREFACE
v
CO N TEN TS
vi
1
GENERAL IN T R O D U C T IO N
1
1.1
LATERAL HETEROGENEITY WITHIN THE EARTH
3
1.2
PREVIOUS STUDIES ON SURFACE WAVES IN
HETEROGENEOUS MEDIA
7
1.3
THESIS SCOPE AND ORGANIZATION
9
2
THE EFFECT OF T H R EE-DIM EN SIO N AL ST R U C T U R E
ON Lg
PROPAGATION PATTERNS
13
2.1
INTRODUCTION
14
2.2
THE RAY DIAGRAM METHOD
15
2.3
SIMPLE CRUSTAL STRUCTURES
18
2.4
CENTRAL ASIA
22
2.5
DISCUSSION
38
3
SURFACE WAVE SC ATTERING FROM 3-D OBSTACLES
39
3.1
INTRODUCTION
40
3.2
A FORMALISM FOR SURFACE WAVE BASIS FUNCTIONS
42
3.3
ORTHOGONALITY RELATIONS
47
3.4
THE SURFACE WAVE T-MATRIX
48
3.5
STRUCTURE OF THE T-MATRIX
51
3.6
NUMERICAL EXAMPLES
54
3.7
CIRCULAR CYLINDER
61
3.8
ELLIPTICAL CYLINDER
69
3.9
TAPERED CYLINDER
73
3.10
DISCUSSION
82
4
MULTIPLE SCATTERING OF SURFACE WAVES FROM
DISCRETE OBSTACLES
86
4.1
INTRODUCTION
87
4.2
SURFACE WAVE SCATTERING BY TWO OBSTACLES
89
4.3
SURFACE WAVE SCATTERING BY
N
OBSTACLES
100
4.5
DISCUSSION
122
5
R E FL E C T IO N /T R A N SM ISSIO N OF SURFACE WAVES
IN H ETEROGENEOUS M EDIA
125
5.1
INTRODUCTION
126
5.2
A REFORMULATION
127
5.3
REFLECTION AND TRANSMISSION I
131
5.4
REFLECTION AND TRANSMISSION II
139
5.5
PROPAGATION IN A LATERALLY VARYING MEDIUM
147
5.6
THE RELATIONSHIP WITH KENNETT’S COUPLED MODES 152
5.7
NUMERICAL EXAMPLES
154
5.8
DISCUSSION
171
R EFER EN CES
175
A P P E N D IX A:
ON THE ORTHOGONALITY OF SURFACE
WAVE EIGENFUNCTIONS IN CYLINDRICAL COORDINATES
A1
A P P E N D IX B:
NUMERICAL IMPLEMENTATION OF THE
TRANSLATION OPERATOR
B1
A P P E N D IX
C: BASIS FUNCTION COMPLETENESS AND
NUMERICAL CONSIDERATIONS
Cl
A P P E N D IX D:
A REPRESENTATION FOR THE SURFACE
C H A P T E R 1
O ur u n d e rsta n d in g of th e n a tu re of e a rth s tru c tu re has relied heavily u p o n
th e developm ent of seismology over th e last 100 years. M uch of th e pioneering
work in seism ology was devoted to th e delineation of m a jo r d isco n tin u ities w ithin
th e e a rth , n o ta b ly th e b o u n d aries of th e cru st, m an tle, inner an d o u te r cores; and
subsequently finer an d less-pronounced discontinuities an d g rad ie n ts w ith in th ese
principal zones. T h is work led n a tu ra lly to th e developm ent of seism ological t a
bles (e.g. Jeffreys & B ullen, 1940; H errin, 1968; Dziewonski &; A n d erso n , 1981;
K e n n e tt & E n g d h al, 1991) which define radially sy m m etric e a rth m odels an d in
c o rp o ra te trav e ltim e an d , in recent m odels, also th e perio d s of free o scillatio n of
th e norm al m odes of th e e a rth . T h e success of these m odels in p re d ic tin g travel
tim es and free oscillation periods is testim on y to a d o m in an tly rad ial v aria tio n in
physical p ro p erties w ithin th e e a rth . In recent years, however, nu m ero u s stu dies
have in d icate d th a t a less-pronounced, b u t nonetheless significant la te ra l co m p o
n en t of h eterogeneity also exists a t m any scales th ro u g h o u t th e e a rth com prising
deviatio ns of a few p ercen t. T h e delineation of lateral h etero g en eity a t all d e p th s
and scales has becom e a m a jo r ob jective of earth science since it can prov id e an
ind ication of th e processes which have controlled th e p la n e t’s ev o lution an d which
co ntinue to sh ap e it. Since seismological observations offer w h at are u n d en iab ly
th e b est o p p o rtu n itie s for resolving detailed earth stru c tu re , a sound th e o re tic a l
u n d e rsta n d in g of th e effects of lateral heterogeneity on seism ic wave p ro p a g a tio n
is fu n d am e n tal to this objective. T his thesis concerns th e p ro p ag atio n of seism ic
surface waves in th e o u te rm o st layers of th e e a rth . In o rd er to set th e stag e for
this work we will p resent a brief overview of th e n a tu re of h etero g en eity w ith in th e
e a rth as revealed by seism ological studies w ith p a rtic u la r reference to th e c ru st and
u p p e r m an tle. T his is followed by a review of th e techniques em ployed in recent
studies to d escribe surface wave p ro p ag atio n in a laterally hetero gen eo u s c ru st and
u p p e r m a n tle which provides an indication of those areas w here our u n d e rs ta n d in g
is still incom plete. In th e final section of this ch ap ter we o u tlin e th e o rg an iz atio n
of th e thesis an d th e approach we have ad o p ted to fu rth er th e u n d e rs ta n d in g of
1.1 LATERAL H ETERO G ENEITY W ITH IN THE EARTH
L ateral heterogeneity ap p ea rs to exist th ro u g h o u t th e e a rth an d is m anifest
in a variety of forms. W ith in th e e a r th ’s core, for exam ple, stu d ies of P phases (P o u p in e t et al., 1983) and th e sp littin g of th e e a r th ’s free oscillations (G iard in i et al., 1987) in d icate a stro ng com po nent of asph ericity which has led to th e suggestion th a t th e inner core m ay be an iso tro p ic w ith an axis of sy m m etry coin cid ent w ith
e a r t h ’s axis of ro ta tio n . H eterogeneity in th e e a r th ’s deep interior also exists as
p e rtu rb a tio n s in th e to p o g rap h y of th e core m a n tle b o u n d ary ; in 1972 C leary and
H addon in te rp re te d precursors to th e P K P ph ase as sc a tte re d waves from th e u n d u la tio n s in th e d ep th of this discontinuity. M ore recently, variatio n s in relief of
up to 20 km have been m app ed on th e core-m antle b o u n d ary by inversions of P phase trav e ltim e residuals (C reager &; Jo rd a n , 1986; Morelli & D ziewonski, 1987)
an d b ear im p o rta n t im plications regarding chem ical p ro p erties an d dy nam ics in
th e o u te r core an d lower m antle.
B ody wave traveltim es have been used a t shallow er levels to in vestig ate velocity
h eterogeneity a t a range of scales in th e lower m an tle and overlying tra n s itio n
zone (420 - 670 km ). Long-w avelength s tru c tu re revealed th ro u g h P — an d 5 - w a v e velocity p e rtu rb a tio n s of up to two p ercen t bears little ap p a re n t relatio n to te cto n ic
s tru c tu re a t th e surface (D ziew onski, 1984; H ager et al., 1985), and is difficult to explain th ro u g h sim ple ste a d y -s ta te m odels of m a n tle convection. S m aller-scale
featu res are also evident th ro u g h th is zone, th e m ost n otab le, p erh ap s, b eing th e
p e n e tra tio n of su b d u cted oceanic slabs exten d s to d ep th s of a t least 1000 km as
suggested by C reager & Jo rd a n (1984).
T h e research described in this thesis is u ltim a te ly directed tow ards an im proved
u n d e rsta n d in g of th e effects of la teral heterogen eity in th e cru st an d u p p e r m a n tle
on th e p ro p ag atio n of surface waves. C onsequently, our concern will lie p rim arily
w ith th e ch ara cter of heterogeneity th ro u g h this region. T h e ind icatio n of stud ies
a t a full range of scales is th a t deviations from radially sy m m etric m odels of e a rth
s tru c tu re are, in fact, m ost pronounced w ithin th e u p p er 400 km of th e e a rth . To
available th ro u g h long-period digital seism ic netw orks such as th e G lobal D igital
S eism ograph N etw ork (G D SN ) and th e In te rn a tio n a l D eploym ent of A ccelerom e
ters (ID A ) to invert for global velocity s tru c tu re . N akanishi Sz A nderson (1984); N a ta f et a l., 1984; T an im o to Sz A nderson (1984, 1985) have used p h ase velocity m easurem ents of fu n d am en tal m ode surface waves from a d a ta set co m p risin g 13
earth q u ak es to exp an d velocity d istrib u tio n s in term s of spherical h arm on ics to
o rder 6. W oodhouse and Dziewonski (1984) em ployed a larger d a ta base (53 e a r th
quakes for a to ta l of 870 p ath s) and ad o p ted a wave fittin g pro ced ure to m odel
m a n tle waves an d iterativ ely invert for velocity using a spherical harm o n ic ex p an
sion to order 8. T h e results from th e individu al stu d ies are rem arkably co n sisten t
and include several im p o rta n t o bservations, m ost n o ta b ly th e expression of surface
te cto n ic s tru c tu re well into th e m antle. O lder, m ore sta b le cratonic regions (e.g.
A u stralian , B altic, C an ad ian Shields) are u n derlain by high shear velocity roots
while areas of young cru st, in p a rtic u la r th e m id-ocean ridge system s, are ch ara c
terized by negative u p p er m a n tle velocity anom alies. T hese features are ev id en t to
d ep th s of ap proxim ately 350 km , alth o u g h th e ab so lu te m a g n itu d e of th e anom alies
ap p ea rs to decrease from ±8% a t 50 km d ep th to levels of ±1% below 650 km . T h e
resolution of global studies is restric ted to longer-w avelength features (~2000 km)
owing to lim ited azim uth al p a th coverage an d th e use of long-period waves. At
sh o rte r periods, th e ap p ro x im atio n th a t waves travel g reat circle p ath s is no longer
valid an d th e effects of m u lti-p a th in g m ust be considered.
A tte m p ts to m ap m ore detailed s tru c tu re b en ea th co n tin ents have also involved
observations of long-period b od y waves and fu n d am en tal-m o d e surface waves, al
tho u g h a t higher frequencies th a n global studies. C o m p a rativ e analyses of observed
an d sy n th etic seism ogram s (B urdick, 1981; G ran d Sz H elm berger, 1984) in d icate th a t hetero geneity is stro n g est in th e u p p erm o st m an tle and th a t u p p e r m an tle
s tru c tu re s in crato n ic and tecto n ic regim es differ considerably (differences in ve
locity of up to 10% above 170 km d ep th ). T h e correlation of large-scale velocity
s tru c tu re well into th e m a n tle w ith m a jo r s tru c tu ra l features a t th e surface is also a
G rand, 1987) and surface wave (Snieder, 1988b,c) data. Tom ographic studies have placed tighter constraints on the extent and location of upper m antle heterogeneity. Romanowicz inverted teleseismic P -wave arrival times from western and northern Europe and m apped large scale variations in velocity which corresponded closely to m ajor tectonic regimes such as the Alps and the Baltic shield. However th e models failed to account for a significant fraction of the standard deviations in the d ata, a result a ttrib u te d to substantial scattering from com plicated short wavelength stru ctu re. Snieder employed linear scattering theory to invert Rayleigh waveforms over central and eastern Europe. The reconstructed velocity models implied hori zontal length scales of heterogeneity com parable to or less than the wavelengths of the surface waves (200 km). T he distribution of velocity anomalies correlated well with much of th e known regional stru ctu re although the actual m agnitude of the postulated variations is probably too large for the application of linear scattering theory. In regions with denser and more uniform d a ta distributions, more detailed and confident delineation of stru ctu re is possible (H um phreys et al., 1984; Spak- man et a/., 1988). Spakm an et al. employed a d a ta set of some 500,000 P -wave delay times from earthquakes in the eastern M editerranean and were able to image, quite conclusively, slab penetration at the Hellenic subduction zone to depths of 200+ km beneath the Aegean Sea. U nfortunately, the wider application of detailed inversion studies to other regions of interest is restricted by the non-uniform and frequently sparse distributions of earthquake sources which severely limit horizontal resolution.
th a t w hile u p p er m an tle v elocity-depth profiles from long-period b ody wave stu d ies
generally showed close agreem ent, sim ilar m odels from sh o rt-p erio d stu d ie s were far
m ore com plex an d ex hibited g reater variatio n due to th e effects of d e ta ile d s tru c
tu re. In ord er to describe b o th sh ort- an d long-period ob servations, th ey developed
a class of m odels describing lateral heterog en eity as d e p a rtu re s from a stratified
reference. T h e m odels were devised to acco m m odate th e range of velocity p ro
files derived from sh o rt-p erio d studies and observations of high-frequency, regional
surface wave p ro p ag atio n . R easonable concordance was achieved by h o rizo n tal ve
locity p e rtu rb a tio n s up to ±2% on a scale of 300 - 500 km w ith a v ertical scale which
increased from 100 km a t 200 km d ep th to 200 km a t 900 km. K e n n e tt & N olet
(1990) and K e n n e tt (1991) have em ployed sim ilar concepts to inv estig ate la teral
velocity g rad ien ts in th e u p p er m antle. A rep resen tativ e m odel was co n stru c te d
using th e long w avelength s tru c tu re b en ea th w estern E u ro p e as inferred by N olet
(1990) w ith sup erim p o sed sm all scale p e rtu rb a tio n s (±1% w ith scale len g th s of 300
km ) to satisfy sh o rt-p erio d observations. At d ep th s shallower th a n 210 km this
class of m odel in dicates th a t lateral g rad ien ts m ay app ro ach 0.001 km s " 1 k m -1
and exceed th e corresponding v ertical g radients.
We may conclude th a t p rom in en t s tru c tu ra l features evident a t th e e a r t h ’s
surface are characterized by velocity anom alies which m ay ex ten d significant dis
tances (500+ km ) into th e m antle. S uperim posed on these large-scale variatio n s
are short-w aveleng th p e rtu rb a tio n s in th e la teral veolcity d istrib u tio n . T h e effects
of this com po site lateral heterogeneity on seism ic wave p ro p ag atio n are m ost pro
nounced a t regional scales w ithin th e cru st and u p p er m antle. T h u s in th e case
of surface waves conditions exist for b o th m u lti-p a th in g (L evshin & B erteu ssen,
1979) in sm oothly varying heterog eneity and com plex sc a tte rin g from a b r u p t an d
discontinuous variations in e a rth s tru c tu re (Snieder, 1988c). T h e c h a ra c te r of la t
eral h eterogeneity w ithin th e e a rth an d , indirectly, th e m an n er in w hich it affects
surface wave pro p ag atio n m ust to large degree reflect th e processes w hich govern
1.2 P R E V IO U S S T U D IE S O N S U R F A C E W A V E S IN
H E T E R O G E N E O U S M E D IA
G iven th a t th e ch ara cter of lateral h eterogeneity w ithin th e e a rth in general,
a n d th e cru st and u p p er m an tle in p a rtic u la r, is highly d ep en d e n t u pon th e scale
of investig atio n it is not su rp rizin g th a t various classes of techn iq u e have em erged
to describe surface wave p ro p ag atio n in specific en viro nm en ts. In m ed ia w here th e
v ariatio n s in s tru c tu re are slight and occur over large distances w ith resp ect to th e
d o m in a n t w avelengths of th e surface waves (e.g. > 50 s p ro p ag atio n a t c o n tin e n ta l
scales) a ray based tre a tm e n t is m ost useful. T h e developm ent of a ray th e o ry for
su rface waves analogous to th e geom etric ray th eo ry used in b o d y wave stu d ies has
been considered by several a u th o rs (B re th e rto n , 1967; Ju lia n , 1970; W oodhouse,
1974). W oodhouse em ployed th e elastic wave L agrangian an d th e p ro p a g a to r m a
trix form alism for a stratified m edium to derive eq u atio ns governing slow v ariatio n s
in a m p litu d e, frequency, and w avenum ber of th e surface wave tra in in m edia w ith
g ra d u a l la teral variatio ns. T h e solution of these eq u atio n s by th e m e th o d of c h a r
acteristics yields ray tracin g eq uatio ns which have been em ployed in nu m ero us
su b seq u en t studies. T h e ap p licatio n of ray th eo ry is useful in slowly varying m edia
w here processes such as m ode coupling and reflection are insignificant b u t w here
sim plifications such as th e geom etrical optics (‘g reat circle’) ap p ro x im atio n are no
longer w arran ted . Indeed, it has been shown th a t surface wave ra y tra c in g is, in
fact, essential in th e stu d y of sh o rt-p erio d waves a t global scales (Sobel & von Seg-
gern, 1978) and in th e analysis of free-oscillations using higher o rb it, lo n g-period
waves (S chw artz &; Lay, 1987). W oodhouse & W ong (1984) were able to ex p lain
a m p litu d e anom alies in m an tle waves travelling m ultiple o rb its as d ue to th e fo
cussing and defocussing of ray bundles along p ro p ag atio n s p ath s. Y om ogida & Aki
(1985) derived a m eth o d for synthesizing surface wave w aveform s in sm o o th ly v ary
ing la te ra l heterogeneity using G aussian beam s. T h e m e th o d was ap p lied (1987)
in co n ju n ctio n w ith th e Born ap proxim ation to invert for Rayleigh wave p h ase ve
locities a t periods of 30-80 s in th e Pacific O cean, using b o th a m p litu d e a n d phase
significant im provem ent on m ore conventional g re a t circle d a ta inversions which
em ploy p hase d a ta alone.
In m edia w here la teral heterogeneity exists a t scales co m p arab le to or less th a n
th e w avelengths of th e surface waves som e account m ust be m ade for th e effects
of m ode coupling and sc a tte rin g from a b ru p t or discontinuous stru c tu re . For mild
h etero g en eity this is conveniently accom plished using th e B orn a p p ro x im atio n or,
equivalently, first o rder p e rtu rb a tio n s schem es. Snieder (1988d) has shown th a t
ray geom etrical effects such as rayb end in g an d focussing can be identified w ithin
th e first o rd er Born ap p ro x im atio n for th e surface wave displacem ent field. T his
descrip tio n also allows for th e in teractio n of th e surface wavefield (including m odal
coupling) w ith hetero geneity which exhib its a b ru p t velocity co n tra sts provided th e
a c tu a l m a g n itu d es are sm all. Hence th e m eth o d is very useful in p ractical term s
an d p erm its th e inversion of surface wave d a ta w here m ild la teral v ariatio n s in
e a rth p ro p erties occur a t all scales (see Snieder, 1988b,c). A disad v an tag e how
ever is th a t th e form ulation is only valid a t large d istances and fails to account for
near-field effects. T anim oto (1990) has also applied em ployed linear p e rtu rb a tio n
th eo ry and derived a solution for surface waves in mild la teral heterogeneity which
assum es a p o te n tial rep resen tatio n for Love and Rayleigh waves th a t satisfies th e
scalar H elm holtz equ ation. By em ploying th e G reen ’s fun ction rep resen tatio n th e
orem he derives integral form ulae describing th e displacem ent field p e rtu rb a tio n
in term s of m a te ria l p e rtu rb a tio n s an d kernels involving th e surface wave p o te n
tials. T h e ad v an tag e of this approach is th a t th e kernels m ay be used to illu s tra te
g raphically how th e waves sam ple th e heterogeneous m edium differently a t various
frequencies. A lthough effects such as m ode coupling can in p rinciple be accom o
d a te d , c o m p u ta tio n a l con siderations have lim ited ap p licatio n of th e m eth o d to
perio ds g re a te r th a n 20 s w here coupling can be ignored.
If la teral variations exceed a few percent in ab so lu te m a g n itu d e descriptions
of th e surface wavefield based on linear scatterin g theory, alth o u g h p erh ap s q u ali
ta tiv ely indicative, are no longer q u an tita tiv e ly reliable. T h e tre a tm e n t of surface
m ed ia, as for ex am p le a con tinental m argin, has been extensively stu d ie d by several
a u th o rs (see M alischewsky, 1987). M ost of these stu dies em ploy an ex p ansion of
th e stress an d d isplacem ent fields in term s of m odal eigenfunctions to m inim ize th e
stress or energy flux m ism atch across th e co n tact. R efinem ents to th e ap p ro ach of
th is p roblem include th e work of Kazi (1976) which in co rp o rates b o d y wave con
trib u tio n s th ro u g h th e use of th e Schw inger-Levine v ariatio n al principle, an d th a t
of Its & Y anovskaya (1985) who stu d ied 3-D reflection and tran sm issio n of surface
waves in cid en t a t oblique angles to weakly curved b o u n d aries betw een two differ
ing vertical s tru c tu re s. In m edia w here lateral hetero g eneity exists as p e rtu rb a tio n s
on a reference s tru c tu re surface wave p ro p ag atio n is effectively tre a te d using th e
coupled m ode ap p ro ach of K e n n e tt (1984a). As in th e v ertical co n tact problem
th e stress and displacem ent are ex p anded as a series of w eighted eigenfunctions (in
th is case for th e reference s tru c tu re ), however th e m odal coefficients now evolve as
th e wave proceeds th ro u g h th e heterogeneity. T h e problem m ay be cast in term s
of tran sm issio n an d reflection m atrices which facilitates th e num erical solution by
tra n sfo rm in g a tw o-point b o u n d ary value problem into an initial value problem .
U n fo rtu n ate ly , th e form ulation is stric tly valid for 2-D s tru c tu re s only an d fails to
in c o rp o ra te 3-D effects such as coupling betw een Love an d Rayleigh waves.
T h e m a jo r gaps rem aining in o u r u n d erstan d in g of how surface waves p ro p ag ate
in la te ra lly heterogeneous m edia arise in situ a tio n s w here co n tra sts are large and
linear s c a tte rin g th eo ry is no longer valid. As m en tio n n ed , theories do exist to de
scribe p ro p ag atio n in p articu la r env iro n m ents, such as sub v ertical co n tacts betw een
q u a rte rsp a c e s and m edia ex hib iting stric tly 2-D variatio n s in m a teria l p ro p erties,
how ever consideration of m ore generally varying 3-D s tru c tu re is a problem which
rem ains to large degree unsolved. It is of course possible to tr e a t this p ro blem using
b ru te force num erical techniques such as finite elem ent or finite difference m e th
ods, how ever this is co m p u tatio n ally infeasible for m any p ractical ap p licatio n s. In
a d d itio n we seek to gain physical insight into o p erativ e m echanism s such as m ode
coupling, w avetype conversion and near-field sc atterin g , w hich is not easily afforded
1.3 T H E S I S S C O P E A N D O R G A N I Z A T I O N
It is a p p a re n t th a t th e m a g n itu d e of th e e a r th ’s la teral heterog en eity an d th e
v ariety of form s w hich it m ay assum e are g re a te st in th e cru st an d u p p e r m an tle.
T his h etero g en eity will ex ert a significant influence on th e c h a ra c te r of seism ic wave
p ro p a g a tio n in th e o u te rm o st layers of th e e a rth especially a t higher frequencies,
an d by identifying this influence we o b ta in evidence th a t is fu n d am en tal to a d
dressing one of th e m a jo r objectives of global geophysics: th e delin eatio n of e a rth
s tru c tu re . However, th e fulfillm ent of th is o b jective also requires a com prehensive
u n d e rsta n d in g of elastic wave in teractio n w ith lateral heterogeneity. In m any re
gions, th e cru st an d u p p er m an tle behave as a w aveguide an d a rep resen tatio n of
th e to ta l wavefield in term s of its surface wave com p o n en ts will in c o rp o ra te m ost
of th e m a jo r features observed on regional seism ogram s. T h is rep resen tatio n has
th e ad v an tag e th a t it essentially reduces th e dim ensionality of th e p roblem by one
o rd er an d allows us to co n ce n trate on wave p ro p ag atio n in two dim ensions p arallel
to th e e a r t h ’s surface. A lthough surface waves in a wide variety of laterally h e te ro
geneous en v iro n m en ts are effectively described using various m eth o d s developed in
recent years, m any gaps rem ain in our knowledge of p ro p ag atio n th ro u g h m edia
ex h ib itin g general v ariations in 3-D. T h e o bjectiv e of this thesis is to narrow som e
of th ese gaps.
In c h a p te r 2 we in tro d u ce a sim ple ray-based schem e for describing th e p ro p a
g atio n of th e regional phase Lg w hich relies on an in te rp re ta tio n of cru stally guided waves in term s of m ultiply reflected, constru ctiv ely interfering sh ear waves. O ne
of th e m erits of th e m eth o d is th a t it provides an in d icatio n of how surface waves
in te ra c t w ith variation s in th e to p o g rap h y of m a jo r d iscontinuities (in this case
th e free surface and th e M oho), a process which is in h erently difficult to m odel
using d escrip tion s based on m odal eigenfunction expansions. A lthough it is only
se m i-q u a n tita tiv e th e approach has th e adv an tag e over m ore tra d itio n a l ray trac in g
app ro ach es (cf. W oodhouse, 1974) th a t it provides a m easure of such com plicated processes as m odal coupling and w avetype conversion. S im ulations of c ru sta l s tru c
surface wave p ro p ag atio n from ray tra cin g w ith w ell-docum ented ob servations of
th e Lg phase in th ese stru c tu ra lly com plex regions.
T h e d evelopm ent of an exact th eo ry of surface wave sc a tte rin g from a discrete
o b stacle is th e focus of ch ap ter 3. T h e d erivatio n relies on B e tti’s th eo rem of elas-
to d y n a m ics an d requires th e expansion of th e incident and sc a tte re d fields in term s
of tw o different sets of surface wave basis functions. T h e basis fu nctions incorpo
ra te th e Love an d Rayleigh wave eigenfunctions an d vector cylindrical harm onics
to d escribe th e h orizo n tal dependence. T h e coefficients of th e in cident and s c a t
te re d wavefield expansions are th en rela ted th ro u g h a surface wave T -m a trix ( c /.
W a te rm a n , 1969) w hich com pletely describes th e sc a tte rin g response of an obstacle
a t a given frequency an d depends only on th e o b sta c le ’s geom etrical configuration.
In c o n tra st to previous descriptions based on th e Born ap p ro x im atio n (Snieder,
1986a), th e T -m a trix form ulation rem ains valid for sc a tte re rs ex h ib itin g large ve
locity c o n tra sts an d ex ten d in g over broad regions; an d can be used to exam ine
th e b eh av io u r of th e scattered wave in th e near-field. T h e m eth o d is applied to
in v estig ate th e effects of sc a tte re r dim ension and g eo m etry on th e n a tu re of m odal
coupling an d w avetype (Love-R ayleigh) conversion.
In c h a p te r 4 th e T -m a trix form ulatio n is ex ten d ed to acco m o d ate scatterin g
from two or m ore sc a tte re rs including th e effects of m u ltip le scatterin g . T h e devel
o p m e n t relies on tra n sla tio n o p e ra to rs for th e vector cylindrical harm onics which
effectively describe p ro p ag atio n of sc a tte re d fields betw een sc atterers. T h e resu lt
ing co m p o site T -m a trix can be w ritte n in a form which involves th e individual
T -m a tric e s of all sc atterers and which clearly indicates th e physical processes con
trib u tin g to th e to ta l sc a tte re d field. T h e form u latio n is applied to several sim ple
d u a l-s c a tte re r configurations to ex am ine th e im plications of m ultip le scatterin g in
te ra c tio n s to th e p ro p ag atio n of regional phases such as Lg in th e e a r t h ’s cru st and u p p e r m antle.
We exploit certain properties of th e surface wave basis functions in ch ap ter 5
to in v estig ate p ro p ag atio n and sc a tte rin g in laterally heterogeneous m edia which
m edia co nsisting of discrete enveloping cylindrical layers and find th a t wave p ro p
ag atio n is conveniently described using a refle ctio n /tran sm issio n m a trix form alism
analogous to th a t in th e th eo ry of elastic waves in h o rizon tally stratified m edia.
T his is som ew hat of an academ ic exercise since few te rre stria l en v iron m en ts are
acc u ra te ly m odelled in this way. However it leads us to consider th e form of solu
tion w hen la te ra l hetero geneity exists as 3-D p e rtu rb a tio n s on a stratified reference
m edium , a m ore realistic m odel in a geophysical co n text. We em ploy p e rtu rb a tio n
th e o ry an d in varian t em bedding to illu s tra te how surface wave p ro p ag atio n in such
m edia can be described in term s of reflection an d tran sm issio n pro p erties. T his
is, as we d e m o n stra te , sim ply th e extension of K e n n e tt’s (1984a) coupled m ode
tech n iq u e to th re e dim ensions. T his th eo ry is applied to a su ite of sim ple models
to in v estig ate th e effects of near-source h etero geneity on th e ch a ra c te r of surface
C H A PT E R 2
THE EFFECT OF T H R EE-D IM EN SIO N A L ST R U C T U R E
2.1 IN T R O D U C T IO N
T h e Lg phase p ro p ag ates in th e e a r t h ’s cru st an d d o m in ates seism ogram s a t regional distances. It is ch aracterized by frequencies betw een 1 and 5 Hz and is
often of ex ten d ed d u ra tio n . A lthough th e w avetrain does n o t ex h ib it a clear o nset,
it builds to an a am p litu d e m axim u m a t a g roup velocity close to 3.5 km s -1 and
may, in regions of sed im en tary cover, carry significant energy to group velocities of
2.8 km s _1. R ecently considerable a tte n tio n has focussed on th e sig n atu re of th e Lg w avetrain in deducing c ru sta l m orphology an d as an aid to nuclear d iscrim in atio n ;
in b o th instances an u n d e rsta n d in g of th e m an n er in which Lg in teracts w ith zones of cru stal heterog en eity is a p rim ary o bjective.
T h e sen sitivity of Lg to m a jo r v ariatio n in cru stal s tru c tu re has been widely noted; early studies revealed th a t as little as 100 km of intervening oceanic cru st
is sufficient to elim inate Lg p ro p ag atio n across ocean basins (Ew ing et al., 1957). T h e c h a ra c te r of Lg p ro p ag atio n across certain co n tin en tal areas w here th e cru st is known to be com plex, m ost n o ta b ly cen tral Asia, has been used to co n strain
s tru c tu re and provide insight into th e o p erativ e tecto n ic processes (R uzaikan et al., 1977; K adin sky -C ad e et al., 1981; Ni & B arazangi, 1983). T hese stu d ies have involved th e in te rp re ta tio n of sh o rt-p erio d seism ogram s to q u alitativ ely ch arac
terize broad geographical zones in term s of th e ir am en ab ility to Lg pro p ag atio n . M ore q u a n tita tiv e analysis is possible w ith increased ra y p a th coverage; K en n ett et
al (1985) inverted a com prehensive set of Lg a tte n u a tio n d a ta for p ath s in NW E urope to delineate cru stal heterogen eity in th e N o rth Sea region.
Tw o different description s, which are equivalent in th e case of a stratified
m edium , can be m ade for th e physical n a tu re of Lg in a heterogeneous crust: th e phase can be considered as a sum of higher m ode surface waves whose e n
ergy is m ostly confined to th e c ru st, or altern ativ ely , as th e result of co n stru ctiv e
interference betw een .S-waves m u ltiply reflected betw een th e free surface an d th e
c ru st-m a n tle boundary. For heterogeneous zones, two m ethods have evolved to
T h e coupled m ode technique ( K e n n e tt, 1984a) is based on th e modal in te r
p reta tio n and solves th e b o u n d a ry value problem exactly; however in zones of
exag g erated heterogeneity and variable layer thickness it can becom e c o m p u ta ti o n
ally intractab le. T h e m eth o d of ray d iagram s is a simpler m eans of assessing th e
p ro p ag atio n of Lg in a s em i-q u an titatitiv e m a n n er and is n o t limited by th e sam e circum stances. It involves tracin g a set of rays at fixed slowness th ro u g h a cru stal
layer of variable thickness, and th e d isru p tio n in th e p a t te r n of th e ray sy stem
(or equivalently the reflection points) is used as a m easu re of m odal coupling an d
scattering. K en n ett (1986) com pared th e visually descriptive ray d iagram s with
coupled m ode solutions for simple 2-D e a r th models and established th e validity of
th e ray based technique.
O u r objective in this s tu d y is to ex ten d K e n n e t t ’s (1986) ap p ro ach to 3-D
heterogeneity. After describing a basis for th e in te rp re ta tio n of ray d iagram s we
exam ine two simple e a rth models and investigate th e effects of oblique incidence
on Lg propagation. We then shift our focus to a central Asia to co m p are our predictions of Lg p ropagation with observations from previous studies.
2.2 T H E R A Y D I A G R A M M E T H O D
K e n n e tt (1986) provided a basis for th e in te rp re ta tio n of ray diagram s by c o m
paring th e m with coupled m ode solutions for a suite of 2-D ea rth models an d n o tin g
some simple and consistent relationships. T h e coupled m o d e technique describes
Lg as a sum of modal eigenfunctions, associated with a laterally homogeneous ref erence model, and weighted by m odal coefficients which vary with position. In 2-D
these coefficients satisfy a system of non-linear differential equations which can be
solved exactly in principle. In practice, however, th e m e th o d becomes c o m p u ta
tionally in tractab le in zones of ex ag g erated heterogeneity (g reater th a n 5% velocity
p e r tu r b a tio n or 2 km in b o u n d ary p e r tu r b a tio n for 1 Hz waves) as consideration
of in term o d e coupling am ong an increasing n u m b er of modes becomes necessary.
M aupin (1988) has m ade use of coupled local modes to give a more direct t r e a tm e n t
reco m p u tatio n of modal eigenfunctions. Ray d iagram s, in co n tra st, are most use
ful where the shape of the crustal waveguide varies and becom e more difficult to
in terp ret if a non-uniform velocity s tr u c tu re is im posed. T h e analysis of K en n ett
(1986) included models am en ab le to b o th m ethods, an d d e m o n stra te d the close
correspondence between th e coupled m ode an d ray d ia g ra m results.
T h e correspondence between th e two m e th o d s is m ost easily explained by n o t
ing t h a t for a given frequency th e Lg wave train will comprise a finite num ber of
modes. Each mode is characterized by a specific phase velocity which can be associ
ated with an 5 - wave ray system a t a p articu la r angle to th e vertical. This suggests
th a t th e geometrical regularity of a system of rays launched in some sy stem atic
fashion may be used as an indication of th e coherence and sustained a m p litu d e
of a given m ode in addition to a m easure of th e S'-wave constructive interference
condition.
At fixed frequency, the angle of S'-wave p ro p ag atio n to th e vertical increases as
th e phase velocity decreases. For a given angle to th e vertical, d eterm in ed by the
choice of incident mode, the behaviour after interaction with heterogeneity can be
conveniently characterized by th e spread in prop ag atio n angles. Increased angles to
the vertical correspond to conversion to lower order modes with th e change in angle
p roportional to the spread in in term o d e coupling. Similarly steeper propagation
corresponds to conversion to higher order modes. In general waves incident with
higher phase velocities tend to show a larger variation in ray angle after tra n s m is
sion th ro u g h th e heterogeneity d ue to increased m ultiple reflections in the complex
zone. In those cases where caustics develop in th e ray d iagram s th e equivalent
coupled m ode solutions show considerable variation in th e am p litu d es of th e modal
coefficients with position. T h e crustal models presented in the following analysis
exhibit 3-D heterogeneity. It is no longer possible to c o n stru ct ray p a tte rn s from a
system of plane waves in cross section as in th e 2-D case. R ath er, we have resorted
to plan views representing s u rface/M o h o to p o g rap h y via contours and consider a
single point source em ittin g rays a t a given angle to th e vertical (m odal slowness)
and th e horizontal projection of ra y p ath s provide a m easu re of th e constructive
interference condition in a m an n er analogous to th e p a t te r n of 2-D ray systems
considered by K en n ett (1986). An added feature of th e 3-D analysis is t h a t vve are
able to exam in e th e way in which th e wavefronts tilt as they en co u n ter features
oblique to th e line of propagation. This ‘ray t i l t ’ will lead to a ro tatio n of the
polarization of the 5-waves between th e vertical an d horizontal planes. As a result
energy having an explosive source lying purely in a vertical plane (i.e. represented
solely by higher m ode Rayleigh waves), may after en co u n ter with th e heterogene
ity have some com ponent in th e horizontal plane, which would be in terp rete d as
conversion to higher mode Love waves. Zones of th e model where th e ray p attern s
exhibit a consistent change in tilt can be ex pected to p ro d u ce significant conversion.
We have investigated Lg propagation in two simple models of cru stal stru c tu re an d in reconstructions of th e crustal s tr u c tu re in central Asia based on world to
po g rap h y d a ta , using ray diagrams. All of these models are characterized by simple
variation in surface and basem ent (Moho) topography; physical properties remain
c o n stan t th ro u g h o u t the crust.
A lthough a more sophisticated analysis which inco rp o rates crustal stratification
m ight be im plem ented with slight modification, this has n ot been done for several
reasons. Ray diagram s are a t best s e m iq u an titativ e a n d are m ean t to provide a
simple m eans of assessing Lg p ropagation th ro u g h crustal s tru ctu re . Moreover, although th e shape of the crustal waveguide is th e d o m in a n t factor influencing
Lg, o th e r factors such as small scale heterogeneity in cru stal velocities cannot be entirely ignored. In addition, when we co n stru ct a cru stal model from ju s t the
surface to p o g rap h y we are forced to make a n u m b e r of assu m p tio n s which may not
be well founded, and so a very complex model is not w arran ted .
S hear velocities of 3.5 km s -1 and 4.6 km s -1 were chosen to characterize the
crust and u p p er mantle. For a given phase velocity rays were traced away from a
point source with multiple reflections at the surface an d th e Moho. T h e character
of th e p ro p ag atio n p attern s are indicated by p lo ttin g a plan section of th e rays and
are illustrated on s e p a ra te ray diagram s by ‘tick s’ located p erp en d icu lar to th e ray
a t surface and Moho reflection points. This display enables th e regular ch aracter
of th e Lg wavefront in the homogeneous regions to be com pared with th e d isrupted p a tte r n after passage th ro u g h th e heterogeneity.
2.3 S I M P L E C R U S T A L S T R U C T U R E S
In this section we consider two s tru ctu re s for which th e surface an d Moho to
po g rap h y are represented as piecewise sm o o th surfaces and th e zones of variation
are of limited spatial extent. T h e ray p a tte r n s resulting from these simple config
uratio n s are intended to d e m o n stra te th e effects of simple changes in to p o g rap h y
and aid in th e in te rp reta tio n of more com plicated ray diagram s for central Asia
presented in th e following section.
2.3.1 C ru sta l T h ic k e n in g
T h e first case is of a linear m o u n tain chain and a section across the model
along th e profile A-B is shown. Ray diagram s were exam ined for the full range
of phase velocities associated with Lg propagation (3.5 to 4.6 km s - 1 ) and several general features were observed to characterize th e wavefield em ergent from the
s tru ctu re . In figure 2.1 we show th e ray diagram s for phase velocities of 3.7 km s " 1
and 4.3 km s _l which illustrate th e m a jo r features of the wavefield. P erp en d icu lar
to th e axis of the range we note a central corridor m arked by a continuous and
regular arran g em en t of b o tto m reflection points and little variation in ray density
with azim u th (even th o u g h th e angle to the vertical is increased). T his corridor is
flanked by zones of a lte rn a tin g high and low ray density as th e angle between the
axis and th e wavefront increases.
For p articu lar com binations of source location and p hase velocity, some zones
may exhibit regularity in b oth ray density an d reflection p a tte r n , an d , therefore
represent windows th ro u g h which a significant fraction of th e initial Lg wave energy can pass. As ray incidence becomes increasingly oblique to th e m o u n tain range axis,
rays becom e tra p p e d within the range and few rays p e n e tr a te th e s tr u c tu re to th e
lateral waveguides, especially for e a rth q u ak es in close proximity.
Rays which impinge on th e surface a t angles to th e surface norm al less th a n th e
critical angle defined by Snell’s law will undergo transm ission into th e u p p er m antle
an d are distinguished by filled diam o n d s m arking th e Moho reflection points. Ken-
n e tt (1986) notes t h a t this situ atio n represents a coupling to modes co n stitu tin g
the S n phase. As observed in th e equivalent 2-D model ( K e n n e tt, 1986), refraction after emergence from the m o u n tain range occurs most p rom inently near the m ax i
m um phase velocity for Lg (above 4.4 km s - 1 ). T h e d istrib u tio n of these em ergent refracted rays is dep en d en t on source location and displays little azim u th al pref
erence. In figure 2.2 we show th e polarization of th e S rays for th e sam e phase velocities as figure 2.1 by means of ‘ticks’ which show th e projection onto a h o r
izontal plane of waves which left th e source with energy confined to the vertical
plane. Figure 2.2 reveals t h a t changes in ray tilt develop in th e zones flanking the
central corridor where to pographic gradients becom e increasingly transverse to the
direction of propagation. T hus we expect some tran sfer of energy between vertical
and horizontal planes which would be in terp rete d as a conversion between Love
and Rayleigh waves. This feature introduces f u rth e r complexity to w h at is already
seen to be a very com plicated wavefield.
2.3.2 C r us t a l t h i n n i n g
T h e second case represents a region of crustal th in n in g sim ulating a zone of
localized crustal extension and is designed to illu strate th e effect of local transverse
gradients in topography. In cross section it is characterized by flanks rising from
30 km to 20 km d ep th to a plateau at th e Moho; its configuration in th e horizontal
plane is elliptical. K en n ett (1986) n oted in an analogous 2-D model th a t constric
tion of the crustal waveguide had a m ore severe effect on transm ission of Lg th a n increase in thickness. This is also a p p a r e n t in th e 3-D model (see figure 2.3) where
we again display ray p a tte rn s for phase velocities of 3.7 a n d 4.3 km s - 1 . T h e effect
of th e cru stal thin n in g is to steer rays away from the a n g u lar window defined by
Y
k
m
Y
k
m
D
ep
th
km
8
00
.0
.
.0
800.0
30
.0
.0
Cross section A-B
400.0 800.0 1200.0 1600.0 X km
90 degree tilt
3.70 km/s
400.0 800.0 1200.0 1600.0 X km
[image:29.547.37.491.86.603.2]th e presence of strong gradients in to p o g rap h y oblique to th e ray p a th , gives rise to
caustics in th e ray p a tte rn , and is p articularly noticeable a t higher phase velocities
as in figure 2.3c. T h e constriction of th e waveguide can also lead to waves trav e l
ling across th e top of the s tr u c tu re being refracted into th e m antle, and hence th e
low density of em ergent rays is com plem ented by an ad d itio n al energy loss th ro u g h
conversion to Sn.
In figure 2.4, we show th e ray tilt representation of th e 5-w ave polarisation
for this crustal th in n in g model. T h e th ree dim ensional c h ara cter of th e cru stal
heterogeneity leads to polarization changes for a larger p ro p o rtio n of th e rays th a n
for th e linear m ountain chain of figure 2.2. At th e higher phase velocities th e
polarization ro tatio n is particu larly strong a t th e lower m argin of th e heterogeneity
where th e gradients in Moho to p o g rap h y are nearly tran sv erse to th e ray p ath .
2.4 C E N T R A L A S I A
T h e propagation of Lg is of considerable interest in d eterm in in g th e n a t u r e of th e e a r t h ’s crust in central Asia. E x trem e variations in to p o g rap h ic relief over th e
region shown in figure 2.5 testify to a complex and heterogeneous waveguide. To
th e n orth of the Indian shield (average elevation 200 m), th e Himalayas rise above
6000 m and form th e southern p erim eter of th e T ib e ta n P lateau , a rem ark ab le
physiographic feature covering some 2.5 x 106k m 2 and characterized by an average
elevation of a b o u t 5000 m. T h e Tarim Basin separates th e T ib e ta n P la te a u to th e
northw est from a second m a jo r m o u n tain range, th e Tien Shan.
T h e p a tte r n of Lg propagation varies significantly over th e region (see Figure 2.5; Ruzaikan et al., 1977; Ni & B arazangi, 1983). Specifically, Lg p ro p ag ates very efficiently across th e more tectonically stable areas: th e Indian Shield, Tarim Basin
and E urasian Platform . T h e phase is also present for p a th s along th e H imalayas
and Tien Shan (although less clear); however it is generally weak or ab sen t for p a th s
crossing th e T ib e ta n P lateau . These observations bear im p o r ta n t implications for
the in terp reta tio n of the tecto n ic evolution of th e region. A dditional interest in
Y
k
m
Y
k
m
D
e
p
th
k
m
800
.0
.
.0
___
___
___
___
___
___
_
80
0
.0
30.0
.C
c
Cross section A -B
400.0 800.0 1200.0 1600.0 X km
370 km/s
400.0 800.0 1200.0 1600.0 Xkm
F ig u r e 2 .3 . R eflection p a tte rn s for rays im p in g in g on an e llip tic a l zone o f c ru s ta l
X km
X km
•00 1-25 250 375 5.00
Height km above msl
o
Lg clear
*
Lg weak
►
Lg absent
»% 5.
I. 50.
45.
| . 40.
I-
35-. 3035-.
I _ 25.
20
.b o th Soviet and Chinese nuclear test sites. T h e prom inence of Lg on regional
seismogram s makes th e phase valuable for the detection of small events, b u t the
sensitivity of Lg to th e variation in crustal s tr u c tu re across th e region reduces its
utility for discrim ination and a b e tte r knowledge of th e effects of heterogeneity on
p ro p ag atio n is necessary.
Surface to p o g rap h y for th e following models of th e cru stal s tr u c tu re was con
s tru c te d by sm oothing (25 point average) digital elevation d a t a supplied a t 5' in
tervals from a world to p o g rap h y d a t a base ( E T 0 P 0 5 compiled by th e N ational
Geophysical D a ta Center, Boulder, Colorado). T h e d e p th to th e Moho was de
term in ed by fu rth er sm oothing (81 point average) of th e surface to p o g rap h y and
assum ing 90% isostatic com pensation occurs within th e crust. T his assu m p tio n ,
alth o u g h simplistic, is th o u g h t to be reasonable within th e lim itations of th e ray
m e th o d for a uniform crust.
T h e choice of a suitable phase velocity in representing Lg p ro p ag atio n in the
real e a rth by ray diagram s deserves some discussion. In typical cru stal situ atio n s
th e higher modes co n stitu tin g Lg p ro p ag ate at a range of phase velocities. W ith in
this range certain modes within loosely defined phase velocity windows will d o m
in ate a t different group velocities. T h e onset of Lg on sh o rt-p erio d seismograms
is d eterm in ed by modes a t phase velocities of 3.4 to 3.6 km s -1 (i.e. cru stal shear
velocities). T h e am p litu d e m ax im u m which follows is usually characterized by
slightly greater phase velocities between 3.7 and 4.0 km s -1 whereas typical values
for th e late-arriving Airy phases reach 4.3 to 4.5 km s ~ l . T hese la tte r phases play
an im p o r ta n t role in determ ining th e character of regional seismogram s especially
in zones of heterogeneity where conversion to S n is significant. An additional fac
to r to consider is th e effect of velocity variation w ithin th e crust. A lthough the
sh ap e of th e crustal wave guide is a. m a jo r influence on Lg prop ag atio n , a positive
cru stal velocity gradient in d epth will have some effect on th e configuration of the
ray system s, flattening th em and extending the distances between reflection points.
In o u r simple single layer crustal model this is equivalent to a decrease in phase