ALBERT

Description / Table of Contents: 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.

Description / Table of Contents: Biolaminated deposits, produced by microbial communities, were studied in modern peritidal environments and in the rock record. The term microbial, mat refers to modern, the term stromatolite to ancient analogs. The term biolaminated deposits was used to encompass both microbial mats and stromatolites. Microbial mat environments studied are the Gavish Sabkha, the Solar Lake, both hypersaline back-barrier systems at the Gulf of Aqaba, Sinai Peninsula, and the "Farbstreifen-Sandwatt" (versicolored sandy tidal flats) on Mellum, an island in the estuary embayment of the southern North Sea coast. Three facies-relevant categories were distinguished: (i) the mat-forming microbiota, (2) environmental conditions controlling mat types and lithology, (3) bioturbation and grazing. Cyanobacteria account for biogenic sediment accretion in all cases studied. Three major groups occur: filamentous cyanobacteria, coccoid unicells with binary fission and those with multiple fission. In the presence of these groups the following mat types evolve: (i) continuously flat (stratiform) L~-laminae (occur in all environments studied); (2) translucent, vertically extended Lv-laminae (only Gavish Sabkha and Solar Lake); (3) nodular granules (only Gavish Sabkha). Basically, the development of mats is controlled by moisture. Thus high-lying parts where the groundwater table runs more than 40 cm below surface are bare of mats. These are: The circular slope and elevated center of the Gavish Sabkha, the shorelines of the Solar Lake and the episodically flooded upper supratidal zone of Mellum Island. The following situations of water supply were found to stimulate mat growth: (i) Capillary movement of groundwater to exposed surfaces, (2) shallowest calm water, both realized in the Gavish Sabkha and the Solar Lake. On Mellum Island, mats form in the lower supratidal zone, which is flooded in the spring tide cycle and wetted during low tide by capillary groundwater. Salinity is almost that of normal seawater, whereas in the Solar Lake, it ranges from 45 °/oo to 180 °/oo and in the Gavish Sabkha, it reaches more than 300 °/oo. Salinity increase is correlated with rising concentrations of magnesium and sulfate ions. In the Gavish Sabkha, episodic sheetfloods cause high-rate sedimentation which is accidental to the living mats. Episodic low-rate sedimentation stimulates the mats to grow through the freshly deposited sediment layer. This occurs predominantly on Mellum Island due to eolian transport. Within the Gavish Sabkha, mineralogy of sediments, community structures, standing crops, redox potentials and pH are highly correlative to the increasing evenness in moisture supply which is realized by the inclination of the system below mean sea level. These conditions bring about a lateral sequence of facies types which include (I) siliciclastic biolaminites at the coastal bar base, (2) nodular to biolaminoid carbonates at saline mud flats, (3) regularly stratified stromatolitic carbonates with ooids and oncoids within the hypersaline lagoon, (4) biolaminated sulfate towardthe elevated center. High-magnesium calcite in facies type 3 precipitates around decaying organic matter and forms also the ooids and oncoids. These occur predominantly within hydroplastic Lv-laminae which provide numerous nucleation centers. Within the Solar Lake, facies type 3 (stromatolitic carbonates with ooids and oncoids) is most important, and grows to extraordinary thickness at the lake's shelf. The regular alternation of dark and light laminae results from seasonally oscillating water depths. These conditions couple back over changing light and salinity intensities to changing dominance structures of mat-building communities. Increasing salinity correlates with decreasing water depth and accounts for the relative abundance of coccoid unicells and diatoms, both active producers of extracellular slimes (Lv-laminae). Water depths locally or temporarily increased favor surface colonization by Mic~ocoleu8 chthonoplastes (Lh-laminae). The biolaminated deposits of the versicolored tidal flats on Mellum Island are similar to facies type 1 of the Gavish Sabkha (siliciclastic biolaminites). Differences exist in the lithology: Sediments upon or through which the mats on Mellum Island grow are made up of clean sand. The grains originate predominantly from re-worked glacial sediments and are rounded to well rounded. By contrast, the strong angularity of siliciclastic grains in the Gavish Sabkha clearly shows their status as primary weathering products. In all environments studied, insects play a significant role. Mainly salt beetles contribute to the lebensspuren spectrum. There is no indication that burrowing and grazing beetles and dipterans are detrimental to the growing mat systems. According to the marine fauna, two distributional barriers exist: (i) physical and (2) biogeochemical factors. Physical barriers are (a) hypersalinity and barrier-closing, which restrict the marine fauna in the Gavish Sabkha and the Solar Lake to a few species, mainly meiofaunal elements such as ostracods and copepods. Only in the Gavish Sabkha, one marine gastropod species occurs which colonizes mud flats of lower salinity. A salinity barrier of about 70 °/oo separates the gastropod habitats from the zones of growing mats. Under reduced salinity, the snails are able to destroy the microbial mats completely. (b) Decreasing regularity of flooding in the microbial mat environment of Mellum Island excludes intertidal deformative burrowers such as cockles and lugworms. However, locally the mats are pierced by numerous dwelling traces. These stem from small polychaetes and amphipod crustaceans which are able to spread over the intertidal-supratidal boundary and settle up to the MHWS-Ievel. Biogeochemical barriers are oxygen depletion within the sediments, high ammonia and sulfide contents, which generate through bacterial break-down of organic matter. Within the highly productive mats of Mic~ocoleu8 chthonoplastes on Mellum Island, dwelling traces of marine polychaetes and amphipod crustaceans disappear due to these conditions. The name of the mat-forming species, Microcoleus chthonoplastes, indicates its capacity to form "soils" (Greek chthonos). While lithology is not altered, the presence of Mic~ocoleu8 mats leads to a habitat change which excludes trace-making "arenophile" invertebrate species and favors "chthonophile" species which do not leave traces. Stromatolitic microstructures studied in rock specimens were interpreted using modern analogs: Microcolumnar buildups in Precambrian stromatolites, ooids and oncoids were compared with those of modern microbial mats. The nodular to biolaminoid facies type found in the Gavish Sabkha was suggested to be an analog to the Plattendolomite facies of Permian Zechstein, North Poland. Studies of the Lower Jurassic ironstone of Lorraine clearly indicate that fungi have been involved in the formation of stromatolites, ooids and oncoids. In conclusion, the comparative study of microstructures in microbial mats and stromatolites reveals a better understanding in both fields. In many cases, it was geology which first revealed the similarity of recent forms to those ancient ones and consequently encouraged research into them.