Lecture notes in earth sciences springer

Lecture Notes in Earth Sciences Editors: S Bhattacharji, Brooklyn G M Friedman, Brooklyn and Troy H J Neugebauer, Bonn A Seilacher, Tuebingen and Yale Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Athanasios Dermanis Armin Griin Fernando Sansb (Eds.) Geornatic Methods for the Analysis of Data in the Earth Sciences With 64 Figures Springer Editors Professor Dr Athanasios Dermanis The Aristotle University of Thessaloniki Department of Geodesy and Surveying University Box, 503 54006 Thessaloniki, Greece E-mail: dermanis @ topo.auth.gr Professor Fernando Sansb Politecnico di Milano Dipartimento di Ingegneria Idraulica, Ambientale e del Rilevamento Piazza Leonardo da Vinci, 32 20133 Milano, Italy E-mail: fsanso @ ipmtf4.topo.polimi it Professor Dr Armin Griin ETH Honggerberg Institute of Geodesy and Photogrammetry Hi1 D 47.2 8093 Zurich, Switzerland E-mail: agruen @geod.ethz.ch "For all Lecture Notes in Earth Sciences published till now please see final pages of the book" Library of Congress Cataloging-in-Publication Data Die Deutsche Bibliothek - CIP-Einheitsaufnahme Geomatic methods for the analysis of data in the earth sciences Athanasios Dermanis (ed.) - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Lecture notes in earth sciences; 95) ISBN 3-540-67476-4 ISSN 0930-03 17 ISBN 3-540-67476-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law Springer-Verlag is a company in the BertelsmannSpringer publishing group O Springer-Verlag Berlin Heidelberg 2000 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: Camera ready by editors Printed on acid-free paper SPIN: 10768074 3213130-5432 10 PREFACE There has been a time when statistical modeling of observation equations was clear in disciplines like geodesy, geophysics, photogrammetry and practically always based on the conceptual arsenal of least square theory despite the different physical realities and laws involved in their respective observations A little number of (very precise) observations and an even smaller number of parameters to model physical and geometrical laws behind experimental reality, have allowed the development of a neat line of thought where "errors" were the only stochastic variables in the model, while parameters were deterministic quantities related only to the averages of the observables The only difficulty there was to make a global description of the manifold of mean values which could, as a whole, be a very complicated object on which finding the absolute minimum of the quadratic functional1 could be a difficult task, for a general vector of observations This point however was the strong belief in the demostly theoretical, since the accuracy of observations terministic model were such that only a very small part of the manifold of the means was really interested in the minimization process and typically the non-linearity played a minor part in that The enormous increase of available data with the electronic and automatic instrumentation, the possibility of expanding our computations in the number of data and velocity of calculations (a revolution which hasn't yet seen a moment of rest) the need of fully including unknown fields (i.e objects with infinitely many degrees of freedom) among the "parameters" to be estimated have reversed the previous point of view First of all any practical problem with an infinite number of degree of freedom is underdetermined; second, the discrepancy between observations and average model is not a simple noise but it is the model itself that becomes random; third, the model is refined to a point that also factors weakly influencing the observables are included, with the result that the inverse mapping is unstable All these factors have urged scientists in these disciplines to overcome the bounds of least squares theory (namely the idea of "minimizing" the discrepancies between observations and one specific model with a smaller number of parameters) adopting (relatively) new techniques like Tikhonov regularization, Bayesian theory, stochastic optimization and random fields theory to treat their data and analyze their models Of course the various approaches have been guided by the nature of the fields analyzed and the physical laws underlying the measurements in different disciplines (e.g the field of elastic waves in relation to the elastic parameters and their discontinuities in the earth, the gravity field in relation to the earth mass density and the field of gray densities and its discontinuities within digital images of the earth in relation to the earth's surface and its natural or man-made coverage) So, for instance, in seismology, where 1% or even 10% of relative accuracy is acceptable, the idea of random models/parameters is widely accepted and conjugated with other methods for highly non-linear phenomena, as the physics of elastic wave propaNote that in least squares theory the target function is quadratic in the mean vector, not in the parameter vector gation in complex objects like the earth dictates In geodesy deterministic and stochastic regularization of the gravity field is used since long time while non-linearity is typically dealt with in a very simple way, due to the substantial smoothness of this field; in image analysis, on the contrary, the discontinuities of the field are even more important than the continuous "blobs", however these can be detected with nonconvex optimization techniques, some of which are stochastic and lead naturally to a Bayesian interpolation of the field of gray densities as a Markov random field The origin of the lecture notes presented here, is the IAG International Summer School on "Data Analysis and the Statistical Foundations of Geomatics", which took place in Chania, Greece, 25-30 May 1998 and was jointly sponsored by the International Association of Geodesy and the International Society of Photogrammetry and Remote Sensing According to the responses of the attendees (who were asked to fill a questionnaire) the School has been a great success from both the academic and organizational point of view In addition to the above mentioned scientific organizations we would also like to thank those who contributed in various ways: The Department of Geodesy and Surveying of The Aristotle University of Thessaloniki, the Department of Mineral Resources Engineering of Technical University of Crete, the Mediterranean Agronomic Institute of Chania, in the premises of which the school took place, the excellent teachers, the organizing committee and especially Prof Stelios Mertikas who took care of the local organization This school represents a first attempt to put problems and methods developed in different areas one in front of the other, so that people working in various disciplines could get acquainted with all these subjects The scope is to attempt tracking a common logical structure in data analysis, which could serve as a reference theoretical body driving the research in different areas This work has not yet been done but before we can come so far we must find people eager to look into other disciplines; so this school is a starting point for this purposes and hopefully others will follow In any case we believe that whatever will be the future of this attempt the first stone has been put into the ground and a number of young scientists have already had the opportunity and the interest to receive this widespread information The seed has been planted and we hope to see the tree sometime in the future The editors CONTENTS An overview of data analysis methods in geomatics A Dermanis F Sansb A Griin Data analysis methods in geodesy 17 A Dermanis and R Rumrnel Introduction 17 The art of modeling 19 Parameter estimation as an inverse problem 24 3.1 The general case: Overdetermined and underdetermined system without full rank (r and 9m [k,] > The modulus z is needed to have outgoing waves With similar consideration the line-source plane-wave decomposition is: A more intuitive version of the plane-wave decomposition (60) can be obtained with the variable transformation kx + a where a is the angle between the plane-wave direction and the x axis, ic, = cos a and ky = sin a : Appendix B: implement at ion details The phase back-projection is performed directly in the domain (Ic,,, kyo), avoiding the interpolation process The angles (a,, at)are computed by inverting the relations (24) The ambiguity due to the fact that two pairs K, - K t give a single K O (see Figure 13) is resolved with the causal path condition (K, K t ) (r, - r t ) The Fourier transform property of real object function, G(-K,) = 6' (K,), is applied during the phase back-projection; hence, for a given K O , i.e a given K, - Kt, the estimated phase is back-projected also in -KO with the opposite sign + Appendix C : DT with source/receiver directivity function All the real sources (receivers) are not isotropic Their directivity function can be roughly included in the inversion process by excluding the plane-waves (and thus the corresponding object wavenumbers) that are not present in the source (receiver) spectrum Thus the filter F is reformulated as follows: where (a,,a)are the view angles associated with the object wavenumber by means of equation (25), f ( a ) is the directivity function, and { f, (a,) ft (at) > 0) Figure 13: Scheme for the phase back-projection is a boolean expression returning a 011 value A more advanced method consists of applying also a weighting factor that really takes into account the source and receiver directivity functions This can be obtained multiplying each data by f,, (a,,) ft, (at,) and updating the average factor in equation (28) with N1 where N?+Nzin Anyway, the optimal estimate is actually obtained with a LSQR approach that will be attempted in the next future References Aki K Richards PG (1980): Quantitative seismology- Theory and meth0ds.W.H Freeman & Co Carcione JM (1996) Ground-penetrating radar: Wave theory and numerical simulation in lossy anisotropic media Geophysics 61: 1664-1677 Chew WC (1994): Waves and fields in inhomogeneous media IEEE Press, New York Gelius LJ (1995) Generalized acoustic diffraction tomography Geophysical Prospecting 43: 3-30 Devaney AJ (1982) A filtered back propagation algorithm for diffraction tomography Ultrasonic Imaging 4: 336-350 Dickens TA (1994) Diffraction tomography for crosswell imaging of nearly layered media Geophysics 59: 694-706 Harris JM, Wang GY (1996) Diffraction tomography for inhomogeneities in layered background medium Geophysics 61: 570-583 Kak AC, Slaney M (1988): Principles of Computerized Tomographic Imaging IEEE Press, New York Miller D, Oristaglio M, Beylkin G (1987) A new slant on seismic imaging: Migration and integral geometry Geophysics 52: 943-964 Pratt GR, Worthington MH (1988) The application of diffraction tomography to cross-hole seismic data Geophysics 53: 1284-1294 Pratt GR, Worthington MH (1990) Inverse theory applied to multi-source cross-hole tomography Part 1and part Geophysical Prospecting 38: 287-329 Woodward MJ, Rocca F (1988a) Wave equation tomography-11 Stanford Exploration Project 57: 25-47 Woodward MJ, Rocca F (198813) Wave-equation tomography 58th Ann Internat Mtg., Soc Explor Geophys., Expanded Abstract, pp 1232-1235 Woodward MJ (1992) Wave equation tomography Geophysics 57: 15-26 Wu RS, Toksoz MN (1987) Diffraction tomography and multisource holography applied to seismic imaging Geophysics 52: 11-25 [...]... 2000 c Springer- Verlag Berlin Heidelberg 2000 generalized inverse of a matrix has been independently (re)discovered in geodesy, preceding its revival and study in applied mathematics This brings us to the fact that overdetermined problems are, in modern methodology "inverse problems" The study of unknown spatial functions, such as the density of the earth in geophysics, or its gravity potential in geodesy,... identifying lines of separation in the image, which correspond to the "outlines" of the depicted objects Line segments, converted from raster to vector form, may be combined into sets which may be examined for resemblance to prototype line sets corresponding to known object outlines Thus one may identify objects as land parcels, roads, buildings, etc Again the problem is not foreign to a probabilistic point... field, but geodynamics will remain an interdisciplinary field involving different discipline~of geophysics 2 The art of modeling A model is an image of reality, expressed in mathematical terms, in a way, which involves a certain degree of abstraction and simplification The characteristics of the model, i.e., that part of reality included in it and the degree of simplification involved, depends on a particular... description of the dynamics of the interaction of the solid earth with atmosphere, oceans and ice This may imply either discrete models involving a finite number of unknowns (geodetic networks) or continuous models, which in principle involve an infinite number of parameters (gravity field) These basic ideas about modeling were realized, within the geodetic discipline at least, long ago Bruns (1876)... Irregularities in polar motion, earth rotation and nutation occur in response to time variable masses and motion, inside the earth, as well as, between earth system components c The analysis of potential differences between individual points has changed to the detailed determination of the gravity field of the earth as a whole including its temporal variations In short one could say that in addition to... methods in geodesy Athanasios Dermanis Reiner Rummel Departmentof Geodesy and Surveying The Aristotle University of Thessqloniki Institute of Astronomical and Physical Geodesy Technical University of Munich 1 Introduction "Geodesy" is a term coined by the Greeks in order to replace the original term "geometry", which had meanwhile lost its original meaning of "earth or land measuring" (surveying) and... to distinguish (in the sense of determining which is more probable) between alternative models in relation to the same data Usually the original model f :X +Y is compwed with an alternative model f ': X'+Y , where f ' is the restriction of f to a subset X'CX , defined by a means of constrains h(x)=O on the unknown x In practice, within the framework of the linear (or linearized) approach, a linear... having input and output to the others The node is essentially a flexible algorithm, which to a given input produces a specific output The flexibility of the algorithm lies in its ability of self-modification under inputs of known class so that a correct output is produced This modification under known inputs has the character of learning similar to the learning involved in the classification training,... prediction is sought in the class of strictly linear predictors of the form ? = d T b instead of the class ?=dTb + K used in collocation It is interesting to note that the duality, which appears in the Gauss-Markov theorem, between the deterministic least squares principle and the probabilistic principle of minimum mean square estimation error, finds an analogue in the problem of the estimation of an unknown... mean of the available data, or at most a linear trend such as &=ao +alx+a,y in the planar case The problem is that an increasing number s of parameters absorbs increasing amount of information from the data leaving little to be predicted, in fact 6x=0 when s=n , n being the number of observations Again this statistical justification of the norm choice presupposes linearity of the model y= f (x) This means ... data in the earth sciences Athanasios Dermanis (ed.) - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Lecture notes in earth sciences; ... @geod.ethz.ch "For all Lecture Notes in Earth Sciences published till now please see final pages of the book" Library of Congress Cataloging -in- Publication Data Die Deutsche Bibliothek - CIP-Einheitsaufnahme.. .Lecture Notes in Earth Sciences Editors: S Bhattacharji, Brooklyn G M Friedman, Brooklyn and Troy H J Neugebauer, Bonn A Seilacher, Tuebingen and Yale Springer Berlin Heidelberg