Surface wave propagation in laterally varying media

It is generally acknowledged th a t the earth exhibits a dom inantly radial varia­ tion in m aterial properties on which a less pronounced, lateral variation is super­ im posed. This la teral variation reflects a complex rheology and contains im portan t inform ation regarding the nature of m ajor dynam ical processes w ithin the earth . Seismic waves are, w ithout doubt, the most sensitive means currently at our dis­ posal for delineating both large and small scale earth structu re , and consequently afford th e best prospects for enhancing our knowledge of the forces which shape the planet. However, a comprehensive understanding of the effects of earth structu re on the passage of seismic waves is a prerequisite to this objective. This thesis is concerned with the theoretical description of surface wave propagation in several classes of laterally varying, stratified media which model the outer layers of the earth . T he first study deals with the propagation of the regional phase Lg , usually in terpreted as a sum of higher mode surface waves, in a crustal waveguide of variable thickness. A ray-based m ethod is introduced which can be used to measure, in a sem iquantitative m anner, various aspects of scattering from changes in topography of the e a r th ’s surface and the Mohorovicic discontinuity. The m ethod is applied to models of central Asia and is used to explain the observed behaviour of Lg in this s tructu ra lly complex region. T he rem ainder of the thesis represents a unified trea tm en t of surface wave prop­ agation in different forms of lateral heterogeneity based on representation theorem s of classical ela.st0d3ma. mics and wavefield expansions in series of surface wave basis functions. In the first of three parts , the T -m atrix form ulation of scattering from a single obstacle is extended to surface waves using previously unrecognized orthogo­ nality relations which involve integral properties of the surface wave eigenfunctions. T he T -m atrix depends only on the size and shape of the obstacle, is valid in both the nearand far-fields, and provides a com plete account of the scattering response to a given harm onic wave. Numerical examples are presented to dem onstrate the influence of obstacle size and geom etry on the scattered wavefield. T he transla tion properties of the surface wave basis functions are exploited in the second p art to derive a com posite surface wave T -m atrix for an a rb itra ry mul­ tip le obstacle configuration and which accounts for the entire hierarchy of m ultiple scattering interactions. T he im portance of m ultiple scattering in the context of regional phases such as Lg is exam ined by way of several numerical exam ples. In the final part the T -m atrix description is recast in a reflection/transm ission form alism which is developed to describe scattering from obstacles exhibiting a degree of quasi-concentric m ultilayering about a vertical axis. This description, al­ though highly idealized from a geophysical perspective, leads to a theory for surface wave propagation in m edia exhibiting a continuous variation in physical properties. T he theory relies on the expansion of the G reen’s function for a stratified reference m edium in term s of the surface wave basis functions and is shown to be simply the 3-D extension of coupled mode techniques used to model surface wave propagation in waveguides exhibiting strictly 2-D lateral heterogeneity. Results from num erical experim ents are presented to exam ine the effect of near-source heterogeneity on the wavefield observed at large distances. A C K N O W L E D G E M E N T S Prior to subm itting formal application to undertake PhD study a t the Research School of E arth Sciences I expressed to my eventual supervisor, Dr B. L. N. Kenn e tt, my wish to work on a theoretical problem in whole earth seismology. My aims were to acquaint myself w ith an area of seismology in which I was interested bu t largely ignorant, and to develop some experience and perhaps proficiency with m ethods of m athem atical and com putational physics which, I believe, are indis­ pensable tools for a good seismologist, w hether theoretician or observationalist. I owe a great deal to Brian for creating an environm ent in which I was able to fully realize my objectives. The problem s he suggested I investigate were challenging, in teresting and topical. He provided me with sound guidance and encouragem ent, and gave unsparingly of his tim e whenever I wished to discuss difficulties related or unrelated to research. I am also extrem ely grateful to Brian for his efforts in establishing a research assistantship which allowed me to look after certain fiscal responsibilities I was obliged to assume as an overseas student. Phil Cum m ins has been a much-valued friend and advisor. He w atched p a­ tiently as I gesticulated my ideas, discussed various aspects of his own varied re­ search interests and provided me with many avenues of d istraction from seismology, not the least of which included old copies of the G uardian Weekly. I have enjoyed and benefitted from conversations on various aspects of seismology w ith Roger Bowman, Doug C hristie, Anton Hales, Kazuki Koketsu and Ron List. I am g ra te ­ ful to Colin Thom son at Q ueen’s University for first introducing me to the field of theoretical seismology, and for encouraging and supporting my application for PhD canditu re at the A.N. U. Staff and students too numerous to m ention by nam e, on the seismology floor and throughout R.S.E.S, have had a definite influence on this thesis by contributing to an atm osphere conducive to dedicated research. A num ber of individuals have helped to make my tim e in C anberra a m ore enjoyable experience, and have thereby contributed in a significant, though indirect, way to the com pletion of this thesis. I wish in particu lar to thank Linda Ayliffe, Candice Bowman, David Dwyer, Peter M cCarthy, Andrew McLean, Loani M cRae, Frank Nielson, John Stefanoff, L isette Van Vliet, and Jon W oodhead for the parts they played in this experience which included forays into rem oter parts of New South Wales, my acquaintance with windsurfing, m orning and afternoon tennis, and various exercises in food generation, preservation and consum ption. I acknowledge financial assistance from the A ustralian N ational U niversity in the form of an A.N. U P ostgraduate Scholarship. I am grateful to th e director of R . S.E.S, K urt Lambeck, for seeing fit to finance my trip to the 1990 American Geophysical Union m eeting in San Fransisco where I had the oppo rtun ity to present portions of this research and discuss my ideas w ith researchers in related fields. Finally, I thank my parents, H ew itt and Sigrid, and my grandfather, Hugh, for their em otional support and encouragem ent over the past th ree years. P R E F A C E This thesis is based on work conducted in the Research School of E a rth Sciences a t The A ustralian N ational University between August 1989 and June 1991 while I was a full tim e research student. The work presented here is believed to be original except where explicit m ention is m ade in the text to the work of o thers, and has not been subm itted for any degree or qualification a t any o ther institu tion . Much of the m aterial contained in this thesis has, however, been published or is subm itted for publication. T he correspondence of published papers or m anuscrip ts w ith chapters and appendices is noted below. C h a p te r 2: Bostock, M. G. & B. L. N. K ennett, 1990. The effect of 3-D s tru c tu re on Lg propagation pa tte rn s, Geophys. J. Int., 1 0 1 , 355-365. C h a p ter 3: Bostock, M. G. , 1991. Surface wave scattering from 3-D obstacles, Geophys. J. Int., 104, 351-370. C h a p ter 4, A p p e n d ic e s B C: Bostock, M. G. & B. L. N. K ennett, 1991. M ultiple scattering of surface waves from discrete obstacles, Accepted for publication in Geophys. J. Int. C h a p ter 5, A p p e n d ix D: Bostock, M. G. , 1991. Reflection and transm ission of surface waves in laterally varying media, Subm itted to Geophys. J. Int.