MATHEMATICAL GEOPHYSICS

MATHEMATICAL GEOPHYSICS

MODERN APPROACHES IN GEOPHYSICS formerly Seismology and Exploration Geophysics Managing Editor: G. NOLET, Department of Theoretical Geophysics, University of Utrecht, The Netherlands Editorial Advisory Board: B. L. N. KENNETT, Research School of Earth Sciences, The Australian National University, Canberra, Australia R. MADAR IAGA, Institut Physique du Globe, Universite Paris VI, France R. MARSCHALL, Prakla-Seismos AG, Hannover, F.R.G. R. WO RTE L, Department of Theoretical Geophysics, University of Utrecht, The Netherlands

Mathematical Geophysics A Survey of Recent Developments in Seismology and Geodynamics edited by N. J. VLAAR G. NOLET M. J. R. WORTEL and s. A. P. L. CLOETINGH University of Utrecht, The Netherlands D. Reidel Publishing Company ~. A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP " Dordrecht I Boston I Lancaster I Tokyo

Library of Congress Cataloging in Publication Data Mathematical geophysics. (Modern approaches in geophysics) Includes index. 1. Seismology-Mathematics. 2. Geodynamics-Mathematics. I. Vlaar, N. J., 1933- II. Series. QE539.M33 1987 551.2'2'0151 87-23462 ISBN-13: 978-94-010-7785-9 e-isbn-13: 978-94-009-2857-2 001: 10.1007/978-94-009-2857-2 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland. All Rights Reserved 1988 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 1 st edition 1988 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

CONTENTS Preface I. SEISMOLOGY AND THREE DIMENSIONAL STRUcrURE OF THE EARTH Waves in a 3-D Earth Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 G. Masters and M. Ritzwoller Low frequency seismology and three-dimensional structure - observational aspects. J.Park Free-oscillation coupling theory. K. Yomogida Surface waves in weakly heterogeneous media. R. Snieder On the connection between ray theory and scattering theory for surface waves. 77 A. van den Berg A hybrid solution for wave propagation problems in inhomogeneous media. 85 1 31 53 Large-scale inyersion Chapter 6 Chapter 7 Chapter 8 P. Mora Elastic wave field inversion and the adjoint operator for the elastic wave equation. B.L.N. Kennett and P.R. Williamson Subspace methods for large-scale nonlinear inversion. 139 W. Spakman and G. Nolet Imaging algorithms, accuracy and resolution in delay time tomography. 155 117

vi CONTENTS II. CONVECTION AND LITHOSPHERIC PROCESSES Geomagnetism Chapter 9 J. Bloxham The determination of fluid flow at the core surface from geomagnetic observations. 189 Mantle convection Chapter 10 G.T. Jarvis and W.R. Peltier Long wavelengthfeatures of mantle convection. 209 Chapter 11 F. Quareni and D.A. Yuen Mean-field methods in mantle convection. 227 Chapter 12 P. Machetel and D.A. Yuen Infinite Prandtl number spherical-shell convection. 265 Chapter 13 A.M. Forte and W.R. Peltier Lateral heterogeneity and the geoid: the importance of the surface kinematic constraints. 291 Postglacial rebound Chapter 14 W.R. Peltier Lithospheric thickness. Antarctic deglaciation history. and ocean basin discretization effects in a global model of postglacial sea level change. 325 Chapter 15 M. Nakada and K. Lambeck Non-uniqueness of lithospheric thickness estimates based on glacial rebound data along the east coast of North America. 347 Thermomechanical processes in the lithosphere Chapter 16 S.A.P.L. Cloetingh and MJ.R. Wortel On the mechanics of plate boundary formation. 363 Chapter 17 C.L. Angevine, S.R. Linneman and P.L. Heller Supercontinent breakup: effect on eustatic sea level and the oceanic heat flux. 389 Index 401

PREFACE The contributions to this book follow a topical trend. In several geophysical fields evidence is accumulating concerning the deviation of the earth's structure from radial symmetry. Seismology provides the most adequate resolution for revealing the earth's lateral inhomogeneity on a global to local scale. Lateral structure in the density distribution is also manifest in the earth's gravity field and in the geoid. Asphericity in physical parameters, generally supposed only to vary with the vertical coordinate, has a profound influence on geodynamics. The effects of these deviations from spherical symmetry concern in particular convection theory, post-glacial rebound and the dynamics of the lithosphere and upper mantle in general. At the 16th International Conference on Mathematical Geophysics which was held in Oosterbeek, the Netherlands, in 1986, the need was felt to present the state of the art. Several prospective authors were found interested to contribute to the present book. This Oosterbeek conference was one in a long series of topical conferences starting with the Upper Mantle Project Symposia on Geophysical Theory and Computers in the 1960s, and thence their successors, the conferences on Mathematical Geophysics, until the present. Without exaggeration it can be said that these conferences were at the heart of the main developments in solid earth geophysics. They were by their very character dominated by the application of mathematical and computational physics to the problems connected with studying the earth's interior structure and dynamics. The main developments in global seismology have, since the beginnings, formed the hard core of the symposia. Emphasis has been placed on developments in the theory of propagation and excitation of seismic waves, free oscillations of the earth, data processing, and inverse problems. Whereas much of this work was connected with a radially symmetric, rotating or non-rotating earth model,- the last years have seen a growing interest in lateral deviations from a standard earth. This is witnessed by the increase of contributions in the fields of seismic tomography, the effect of lateral structure on free oscillations, and on the propagation of surface waves. As the resolution of tomographic imaging of the earth increases rapidly, the time is approaching when geological structures on lithospheric and crustal scales can be mapped with great precision, thus providing an important data base for unravelling the history of the earth's crust. Therefore it was felt to be justified to lay extra emphasis on this topic, also for the purpose of searching for connections between the fundaments of seismology and those of seismic exploration methods. Another topic of growing interest is the evolution of both oceanic and continental lithosphere. This is particularly the case given the greater resolution mentioned above, increasing knowledge of the earth's gravity field, and deeper insight into rheological properties of the earth's materials. Greater understanding of plate tectonic mechanisms, vii

viii Preface particularly their connection with deeper seated mantle dynamics, is required to grasp hold of geological reality. Traditionally, the conferences on mathematical geophysics were firmly based within a rather closed system of the mathematical description of mostly observable geophysical fields. Present day geodynamics is becoming more and more multi-disciplinary. Mathematical geophysics may offer the geophysical framework for eliminating speculation on the earth's structure, dynamics, and history. All chapters in this book, deal, one way or another, with geophysical expressions and effects of lateral heterogeneity in the earth. The chapters on seismology deal with different bands in the seismic spectrum and different approaches for resolving lateral heterogeneity from the inversion of imperfect data. Low frequency seismology develops towards a high sophistication and appears to be at the threshold of unravelling the large scale 3-D structure of the earth. Intermediate frequency seismology, as represented by the study of surface waves, employs horizontal asymptotic ray theory for resolving intermediate scale structure, and even incorporates off great circle path effects by means of a scattering Born approximation approach. High frequency body waves are used for tomographic studies which reveal strong lateral heterogeneity, particularly in the upper mantle. Attention is given to the problem of large scale inversion and the resolving power of seismological data sets. Seismology is the most important source of information on the earth's structure and the relevant chapters are indicative of the high standards required in frontline research in this field. Geomagnetic observations may be inverted into the fluid velocity field near the core surface. This type of study is relevant in the light of emerging seismological evidence for lateral core structure. The interpretation of seismological findings in terms of the solid earth's internal and surface dynamics which should find its expression in the gravity field, surface heat flux, and vertical and horizonl:1l motion of the earth's surface appears to be a promising field of research. In this context is knowledge of the earth's internal rheology of primary importance. The earth's rheological behavi,our is manifest in the refined studies of post-glacial rebound. The rheological structure, still considered to be only vertically varying is fundamental to the problem of convective motion in the earth's mantle. The question arises in how far laterally varying rheology, as is clearly revealed by lithosphere and upper mantle structure may invalidate some of the present approach to the earth's internal dynamics. An interesting development 'is the attempt to connect the plate tectonic velocity field, poloidal and toroidal, to lateral heterogeneity and its geoidal expression. The toroidal velocity field of the lithosphere indicates that spherically symmetric viscosity structure may not be warranted. The most obvious laterally varying structure is the earth's strong lithosphere. Understanding the processes of its creation, evolution and destruction would impose very strong constraints on internal dynamics. Whereas long standing traditions treat the lithosphere as circumstantial to mantle flow, thorough knowledge of lithospheric processes may give way to better understanding deeper seated dynamics. Moreover, geology and geophysics present many observables which still await explanation. Here a quantitative approach to Ethospheric evolution and its thermo-mechanical properties might provide the setting for the unravelling of the earth's evolution and dynamics.

Preface ix The editors wish to express their great appreciation to the authors for the high level of their achievements as laid down in the chapters of this volume, and also for their constructive cooperation. Thanks are due to Maarten Remkes and Hong-Kie Thio for their enthusiastic assistence to the editors. Utrecht, July 1987 N.J. Vlaar