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: OPENING ADDRESS On behalf of the Local Organizing Committee, I welcome you all to the first International Workshop on GPS-techniques in surveying and geodesy held at this university. This workshop is designed to bring together experts from various countries and also scientists who carry out, analyze and interpret such measurements with those who work on instrumental and theoretical problems. The workshop focuses hereby on high-precision applications with emphasis on monitoring time-dependent phenomena such as those relevant to geodynamics as well as men-made constructions as those in civil engineering and similar fields. It is astonishing to see how, in spite of all earlier satellite work over the last two decades, GPS-methods became so fast a relevant new technology, in its proper sense, in modern geodesy and surveying besides VLBI and Satellite Laser Ranging (SLR). With the recent development of new dual-frequency receivers the role of GPS-procedures in monitoring large-scale phenomena over big distances will still expand; and the application of kinematical GPS-approaches is of utmost interest in solving high-precision problems. It is indeed fascinating to realize how GPS-methods have become in such a short time a surprisingly efficient and effective, this means : fast, precise and easy to apply, tool which is able to replace already now, after a few years of existence and with an incomplete set of a few out of the 18 satellites (of the final stage), at least partially some expensive, slow and cumbersome classical surveying methods. On the other hand, it cannot be overemphasized that GPS-procedures are still at their beginning and the full spectrum of their capabilities still has to be explored. In Europe, for example, where excellent classical surveying systems do exist the situation is quite different from the situation in other countries such as Canada or the USA. Even within Europe the application types of GPS-methods will vary; for example, in Norway the situation is quite different from central European countries. It is often forgotten, that together with GPS we will have to introduce new concepts and a new thinking in combination with other modern satellite procedures. GPS itself can resolve only a small part of the problems to be solved by modern geodesy but it will open the way to a great variety of new applications and capabilities. Modern global tectonics is just one of the new disciplines of high interest and great practical impact. I could continue in citing other similarly important new fields. GPS is, however, of special importance because it replaces old technologies and fills gaps where modern and efficient tools are most needed. Consequently, also the optimal combination of GPS-methods with new auxiliary and also classical high-precision techniques is of great importance, mainly under the european conditions outlined above. Moreover, the real-time or almost-real-time use of GPS in combination with photogrammetry, inertial geodesy, gravity gradiometry or even classical surveying is of substantial interest. It is indeed important to realize the new concepts in modern satellite and space methods and I, therefore, spoke above of a new "technology" which should be optimally developed as there is a worldwide need of such capabilities and tools. In view of the few active NAVSTAR-satellites in sky in 1988 this is perhaps not the best year for GPS-applications but the right time for a review of the experience gained until now and using it as a base for the planning of the future...