ALBERT

Beschreibung / Inhaltsverzeichnis: PREFACE There are problems, when applying statistical inference to the analysis of data, which are not readily solved by the inferential methods of the standard statistical techniques. One example is the computation of confidence intervals for variance components or for functions of variance components. Another example is the statistical inference on the random parameters of the mixed model of the standard statistical techniques or the inference on parameters of nonlinear models. Bayesian analysis gives answers to these problems. The advantage of the Bayesian approach is its conceptual simplicity. It is based on Bayes' theorem only. In general, the posterior distribution for the unknown parameters following from Bayes' theorem can be readily written down. The statistical inference is then solved by this distribution. Often the posterior distribution cannot be integrated analytically. However, this is not a serious drawback, since efficient methods exist for the numerical integration. The results of the standard statistical techniques concerning the linear models can also be derived by the Bayesian inference. These techniques may therefore be considered as special cases of the Bayesian analysis. Thus, the Bayesian inference is more general. Linear models and models closely related to linear models will be assumed for the analysis of the observations which contain the information on the unknown parameters of the models. The models, which are presented, are well suited for a variety of tasks connected with the evaluation of data. When applications are considered, data will be analyzed which have been taken to solve problems of surveying engineering. This does not mean, of course, that the applications are restricted to geodesy. Bayesian statistics may be applied wherever data need to be evaluated, for instance in geophysics. After an introduction the basic concepts of Bayesian inference are presented in Chapter 2. Bayes' theorem is derived and the introduction of prior information for the unknown parameters is discussed. Estimates of the unknown parameters, of confidence regions and the testing of hypotheses are derived and the predictive analysis is treated. Finally techniques for the numerical integration of the integrals are presented which have to be solved for the statistical inference. Chapter 3 introduces models to analyze data for the statistical inference on the unknown parameters and deals with special applications. First the linear model is presented with noninformative and informative priors for the unknown parameters. The agreement with the results of the standard statistical techniques is pointed out. Furthermore, the prediction of data and the linear model not of full rank are discussed. A method for identifying a model is presented and a less sensitive hypothesis test for the standard statistical techniques is derived. The Kalman-Bucy filter for estimating unknown parameters of linear dynamic systems is also given. Nonlinear models are introduced and as an example the fit of a straight line is treated. The resulting posterior distribution for the unknown parameters is analytically not tractable, so that numerical methods have to be applied for the statistical inference. In contrast to the standard statistical techniques, the Bayesian analysis for mixed models does not discriminate between fixed and random parameters, it distinguishes the parameters according to their prior information. The Bayesian inference on the parameters, which correspond to the random parameters of the mixed model of the standard statistical techniques, is therefore readily accomplished. Noninformafive priors of the variance and covariance components are derived for the linear model with unknown variance and covariance components. In addition, informative priors are given. Again, the resulting posterior distributions are analytically not tractable, so that numerical methods have to be applied for the Bayesian inference. The problem of classification is solved by applying the Bayes rule, i.e. the posterior expected loss computed by the predictive density function of the observations is minimized. Robust estimates of the standard statistical techniques, which are maximum likelihood type estimates, the so-called M-estimates, may also be derived by Bayesian inference. But this approach not only leads to the M-estimates, but also any inferential problem for the parameters may be solved. Finally, the reconstruction of digital images is discussed. Numerous methods exist for the analysis of digital images. The Bayesian approach unites some of them and gives them a common theoretical foundation. This is due to the flexibility by which prior information for the unknown parameters can be introduced. It is assumed that the reader has a basic knowledge of the standard statistical techniques. Whenever these results are needed, for easy reference the appropriate page of the book "Parameter Estimation and Hypothesis Testing in Linear Models" by the author (Koch 1988a) is cited. Of course, any other textbook on statistical techniques can serve this purpose. To easily recognize the end of an example or a proof, it is marked by a A or a t~, respectively. I want to thank all colleagues and students who contributed to this book. In particular, I thank Mr. Andreas Busch, Dipl.-Ing., for his suggestions. I also convey my thanks to Mrs. Karin Bauer, who prepared the copy of the book. The assistance of the Springer- Verlag in checking the English text is gratefully acknowledged. The responsibility of errors, of course, remains with the author.

Beschreibung / Inhaltsverzeichnis: PREFACE This monograph is a compendium of revised papers which were originally presented at the "Ron Mather Symposium on Four-Dimensional Geodesy", 28-31 March, 1989, held at the University of New South Wales, Sydney, Australia. The symposium had the enthusiastic support of the International Association of Geodesy and the Australian Academy of Sciences. The symposium served two purposes: to honour the achievements of the late Professor Ron S. Mather, the distinguished Australian geodesist who died in 1978, and to review and report on the latest developments in four-dimensional geodesy. Four-dimensional geodesy is a convenient term for those geodetic principles and techniques which yield position, gravity and their time variations. In the past geodesists have tended to think of the earth as a static body, save from occasional savage earthquakes or volcanic eruptions. So, why the need to coin the term "four-dimensional geodesy") Because it explicitly recognises that time is an integral part of understanding geodetic measurements. But let's first identify the scope of modern geodesy. Geodesy has traditionally been concerned with two separate, though closely related, topics: accurate positioning of objects on the earth's surface, and mapping the earth's external gravity field. These are still the fundamental tasks of geodesy, although the spheres of application have now extended into space. However, present and emerging geodetic measurement technologies for gravity field mapping and positioning are sensitive to defolTnations of the earth's surface and gravity field. Within the geodetic community, this new emphasis on accounting for the time-varying characteristics of position and gravity has fundamental principles; in particular the establishment and maintenance of appropriate global reference systems for geodesy. At the same time, there has been a growing recognition by the earth sciences in general of the important role of geodesy in studying earth deformations, as well as atmosphere and ocean dynamic phenomena. The geodetic measurements, for example, are taken over time scales of hours to decades, and occasionally to a century or longer. Though this is only a small part of the whole deformation spectrum, it is a very important one. Geodesy bridges the low frequency part of the spectrum available from geological observations, with the high frequency end observed from, for example, seismic instrumentation. It's role in atmospheric and oceanographic studies is as a unique, high precision remote sensing tool. The revolution in geodesy is not, however, restricted to the measurement technology only. It is true that without the advances of space geodesy and terrestrial metrology, the notion of four-dimensional geodesy is a rather academic one. These advances, which now reveal time-variable signals above the measurement noise level, have important implications for all geodetic activities. The geodetic activities we refer to can be identified as: experiment design and measurement processes; definition and maintenance of highly stabie geodetic reference systems; data analysis; and interpretation of position and gravity results. Ultra high precision measurements are of little use without sophisticated analysis tools to extract the small signals in the data. The interpretation of geodetic results will be in error if insufficient attention is paid to ensuring that the reference systems to which the results relate are themselves stable. Clearly four-dimensional geodesy is as much about concepts and principles, as about computers and geodetic equipment. This diversity is reflected in the papers selected for this book. They range over topics related to the modem measurement tools, the reduction and analysis techniques, to the interpretation of geodetic results within the context of problems currently being investigated in the earth sciences. We would like to thank the International Association of Geodesy and the Australian Academy of Sciences for sponsorship of the Symposium. Unisearch Ltd., the commercial arm of the University of New South Wales, was the managing agent, and staff members of the School of Surveying and of Unisearch Ltd. were involved in the organisation of the Symposium. We would like to gratefully acknowledge these excellent contributions. Let us express also our gratitude for the useful guidance which we received from Prof. K. Lambeck, A. Prof. A. Stolz and Dr. R. Coleman of the Scientific Advisory Committee and the continuous support given by Prof. E.W. Grafarend. Sincere thanks are due to the authors of the selected papers for agreeing to contribute to this Monograph, and for their positive cooperation during the production of this volume.