this direction and is ra th e r m ore su b d u ed over th e rem a in d er of th e h eterogeneous band. Second, th e a m p litu d e m axim um for Rayleigh waves occurs over a b ro ad lobe cen tered on th e x-axis, w hereas for Love waves th e m ax im u m is d is trib u te d sym m etrically accross two lobes a t angles of less th a n ±30° to th e x-axis. T h e angle a t which th e a ctu al m axim um in th e tran sv erse g rad ie n t of th e p e rtu rb a tio n s occurs is som ew hat g reater, ap p ro x im ately ±35°, and this d iscrep an cy is in fact due to th e effects of m ultip le scatterin g . T h e ch ara cteristics of th e tra n s m itte d m ode (incident plus unconverted sc a tte re d m odes) over th e window are m uch as expected and include increased am p litu d es (> 10%) and increased p hase delays over th e window encom passed by th e low velocity hetero g en eity (see figure 5.12,
R l ^ R l , R4-+R4).
5.8 D IS C U S S IO N
In this c h a p te r we have seen th a t th e surface wave T -m a trix fo rm u latio n in tro d u ce d and developed in ch ap ters 3 and 4 can be viewed in a m ore general re flec tio n /tran sm issio n co n tex t akin to th a t which has been successfully em ployed in describing plan e elastic wave p ro p ag atio n in stratified m edia. T h e essential in g re dients in o u r surface wave description are integral eq u atio n s rela ted to th e classical re p resen tatio n theo rem s of elasto d y n am ics, and a set of o rth o g o n al surface wave basis functions which m ay be used for th e expansion of any field, including th e G re e n ’s fu nction, in a horizontally stratified m edium . T h e general n a tu re of these requ irem ents im plies th a t th e concepts applied here are not re stric te d to th e surface wave problem . T his is obvious when we consider th a t th e T -m a trix ap pro ach to
sc a tte rin g from one or m ore discrete obstacles was first fo rm u lated for aco ustic, an d la ter, electro m ag n etic an d elastic cases in b o th 2-D and 3-D geom etries. In a like m a n n er, th e p resent tre a tm e n t of surface wave pro p ag atio n th ro u g h m ed ia e x h ib it ing a co ntinuous variatio n in m ateria l pro p erties can be ex ten d ed to o th e r classes of wave p ro p ag atio n described by som e variant of th e H elm holtz eq u atio n . All th a t is
required is a com plete basis function expansion of th e G re e n ’s function in th e form derived in A ppendix D. Such expansions are, for exam ple, well know n for b o th
scalar and vector waves in hom ogeneous m edia in a spherical co o rd in a te g eom etry (see M orse & Feshbach, 1953) and were in fact used in th e original deriv atio ns of th e T -m a trix (c f. W aterm an 1969) ra th e r th a n th e som ew hat sim pler arg u m en ts m ade in ch a p te r 3.
T h e theo ry has been applied to in v estigate th e effect of n ear-source, co n tin u ously varying heterogeneity on th e tran sm issio n of surface waves into th e far-held by way of several sim ple m odels. By considering an isotropic (m = 0) incident w aveheld em an atin g from an explosive source we have been able to exam ine th e sc a tte rin g of an incident Rayleigh m ode into th e tra n s m itte d m ode, Love m odes and o th e r Rayleigh m odes. For sm all velocity p e rtu rb a tio n s < 1% we observe th e effects of focussing and defocussing on th e incident m ode as would b e pred icted from geom etrical optics. T h a t is, p o rtio n s of th e w aveheld passing th ro u g h low velocities regions, for exam ple, exh ibit larger am p litu d es an d positive p h ase delays relative to th e incident held. T h is b eh av iour is exem plihed by th e fu n d am en tal R ayleigh m ode in our m odel selection since it in teracts only w ith w eaker p a rts of th e heterogeneity. For certain configurations of m ore stro n gly dehned hetero gen eity (±4% ) signih cant reflection of energy m ay occur resulting in a tra p p in g of energy w ithin th e near source zone and a red uction in th e am p litu d e of th e tra n s m itte d wave over certain azim uths. T his effect m ay be related to variatio n in signal o b served from d is ta n t events a t nearby sta tio n s sited in areas of varying geologic s tru c tu re (B arker et al., 1981).
S c atterin g involving wa.vetype conversion (Rayleigh into Love ) is a m a tte r of som e in terest since regional seism ogram s from explosive sources freq uently reveal large tran sv erse com ponents of d isplacem ent. As an explosive source im p arts no energy to th e Love co m p on en t of th e w aveheld, Love waves m ust be produced th ro u g h th e in teractio n of th e w aveheld w ith h eterogeneity som ew here along th e
p ro p ag atio n p a th (assum ing th a t we are dealing w ith isotropic m edia). A possible c a n d id a te for producing tran sv erse energy given an explosive source is th e in te r action of th e w aveheld w ith hetero geneity in th e vicinity of th e source because Love and Rayleigh waves are, in this regim e, no longer se p a ra te d by co o rd in a te
(e.g. Love waves have b o th radial and tran sv erse co m p o nen ts of disp lacem en t) and th e outgoing wavefields ex h ib it a sin gular ra th e r th a n sinusoidal beh av io u r. In a d d itio n th e in teractio n w ith h eterogeneity obviously affects a larger p o rtio n of th e to ta l wavefield in this situ a tio n th a n for hetero geneity a t g re a te r d istan ces from th e source. T h e conclusion draw n from o u r analysis, how ever, is th a t reasonably continuous la teral variatio n in m aterial p ro p erties does n o t c o n trib u te significantly to R ayleigh-Love coupling for th e m a g n itu d e of p e rtu rb a tio n s involved (±4% in velocity and density). T his does not elim in ate th e p ossibility th a t n ear-so u rce sc a tte rin g may play an im p o rta n t role, b u t ra th e r it is m ore likely th a t th e form of hetero geneity exam ined here is less effective th a n o th e rs in th is reg ard . As we have seen in ch ap ter 4, surface wave sc a tte rin g from discrete obstacles m ay be an effective way of converting Rayleigh energy to Love energy especially if th e s c a tte r e d are sm all 0.01 > ka > 1 .0 and relatively densely d is trib u te d such th a t m u ltip le sc a tte rin g in teractio n s becom e significant. T h e two problem s are rela ted in th a t we can view th e first-order m ultip le sc a tte re d field off one ob stacle as analogous to th e in teractio n of th e source field w ith local heterogeneity. From th is p o in t of view it seems likely th a t energy tran sfer betw een Rayleigh waves and Love waves is m ost efficient from fairly a b ru p t changes in m aterial p ro p erties, i.e. discontinuous s tru c tu re , oriented p erp en d icu lar to th e directio n of p ro p a g a tio n of th e incident wavefield. We could in th eo ry sim ulate this class of h etero g en eity in o u r p e r tu r b atio n schem e by specifying th e heterog en eity in term s of a very large n um b er of a zim u th a l co n trib u tio n s p, b u t this would in tu rn require an even larger nu m b er of azim u th a l term s m in o ur tru n c a te d basis function exp an sio n s, an d is im p ractical
from a num erical sta n d p o in t. T h e logical direction in w hich to proceed would be to em ploy th e T -m a trix form ulation an d specify th e incident field to be th a t of th e p rescrib ed , local explosive source. A ltern ativ ely we m ight place th e source w ithin a locally, laterally hom ogeneous cylindrical volum e em b ed d ed w ithin an otherw ise la terally hom ogeneous half-space an d use th e form ulation of section 5.3.
F inally, we have been restric ted to m odelling relatively sim ple form s of lateral h eterogeneity in this stu d y owing to lim itatio n s in co m p u ter hardw are. As com pu-
ta tio n a l power increases, th e m eth o d m ay be app lied to m ore co m p licated m odels which accu rately describe situ a tio n s in th e real e a rth .