Search

The Bayesian approach allows an intuitive way to derive the methods of statistics. Probability is defined as a measure of the plausibility of statements or propositions. Three rules are sufficient to obtain the laws of probability. If the statements refer to the numerical values of variables, the so-called random variables, univariate and multivariate distributions follow. They lead to the point estimation by which unknown quantities, i.e. unknown parameters, are computed from measurements. The unknown parameters are random variables, they are fixed quantities in traditional statistics which is not founded on Bayes’ theorem. Bayesian statistics therefore recommends itself for Monte Carlo methods, which generate random variates from given distributions. Monte Carlo methods, of course, can also be applied in traditional statistics. The unknown parameters, are introduced as functions of the measurements, and the Monte Carlo methods give the covariance matrix and the expectation of these functions. A confidence region is derived where the unknown parameters are situated with a given probability. Following a method of traditional statistics, hypotheses are tested by determining whether a value for an unknown parameter lies inside or outside the confidence region. The error propagation of a random vector by the Monte Carlo methods is presented as an application. If the random vector results from a nonlinearly transformed vector, its covariance matrix and its expectation follow from the Monte Carlo estimate. This saves a considerable amount of derivatives to be computed, and errors of the linearization are avoided. The Monte Carlo method is therefore efficient. If the functions of the measurements are given by a sum of two or more random vectors with different multivariate distributions, the resulting distribution is generally not known. TheMonte Carlo methods are then needed to obtain the covariance matrix and the expectation of the sum.

Minimal Detectable Biases (MDBs) or Minimal Detectable Outliers for the Expectation Maximization (EM) algorithm based on the variance-inflation and the mean-shift model are determined for an example. A Monte Carlo method is applied with no outlier and with one, two and three randomly chosen outliers. The outliers introduced are recovered and the corresponding MDBs are almost independent from the number of outliers. The results are compared to the MDB derived earlier by the author. This MDB approximately agrees with the MDB for one outlier of the EM algorithm. The MDBs for two and three outliers are considerably larger than MDBs of the EM algorithm.

Artificial intelligence is interpreted by a machine learning algorithm. Its realization is applied for a two-dimensional grid of points and depends on six parameters which determine the limits of loops. The outer loop defines the width of the grid, the most inner loop the number of scans, which result from the three-dimensional coordinate system for the $$x_i$$-, $$y_i$$-, $$z_i$$-coordinates of a laser scanner. The $$y_i$$-coordinates approximate the distances measured by the laser scanner. The minimal standard deviations of the measurements distorted by systematic effects for the $$y_i$$-coordinates are computed by the Monte Carlo estimate of Sect. 6. The minimum of these minimal standard deviations is found in the grid of points by the machine learning algorithm and used to judge the outcome. Two results are given in Sect. 7. They differ by the widths of the grid and show that only for precise applications the systematic effects of the laser scanner have to be taken care of. Instead of assuming a standard deviation for the systematic effects from prior information as mentioned in Sect. 1, the $$x_i$$-, $$y_i$$-, $$z_i$$-coordinates are repeatedly measured by the laser scanner. However, there are too few repetitions to fulfill the conditions of the multivariate model of Sect. 2 for all measured coordinates. The variances of the measurements plus systematic effects computed by the Monte Carlo estimate of Sect. 6 can therefore be obtained for a restricted number of points only. This number is computed by random variates. For two numbers, the variations of the standard deviations of the $$y_i$$-coordinates, the variations of the standard deviations of the $$x_i$$-, $$y_i$$-, $$z_i$$-coordinates from the multivariate model, the variations of the standard deviations of the systematic effects and the variations of the confidence intervals are presented. The repeated measurements define time series whose auto- and cross-correlation functions are applied as correlations for the systematic effects of the measurements. The ergodicity of the time series is shown.

The concept of reliability was introduced into geodesy by Baarda (A testing procedure for use in geodetic networks. Publications on Geodesy, vol. 2. Netherlands Geodetic Commission, Delft, 1968). It gives a measure for the ability of a parameter estimation to detect outliers and leads in case of one outlier to the MDB, the minimal detectable bias or outlier. The MDB depends on the non-centrality parameter of the \(\chi ^2\)-distribution, as the variance factor of the linear model is assumed to be known, on the size of the outlier test of an individual observation which is set to 0.001 and on the power of the test which is generally chosen to be 0.80. Starting from an estimated variance factor, the \(F\)-distribution is applied here. Furthermore, the size of the test of the individual observation is a function of the number of outliers to keep the size of the test of all observations constant, say 0.05. The power of the test is set to 0.80. The MDBs for multiple outliers are derived here under these assumptions. The method is applied to the reconstruction of a bell-shaped surface measured by a laser scanner. The MDBs are introduced as outliers for the alternative hypotheses of the outlier tests. A Monte Carlo method reveals that due to the way of introducing the outliers, the false null hypotheses cannot be rejected on the average with a power of 0.80 if the MDBs are not enlarged by a factor.

For deriving the robust estimation by the EM (expectation maximization) algorithm for a model, which is more general than the linear model, the nonlinear Gauss Helmert (GH) model is chosen. It contains the errors-in-variables model as a special case. The nonlinear GH model is difficult to handle because of the linearization and the Gauss Newton iterations. Approximate values for the observations have to be introduced for the linearization. Robust estimates by the EM algorithm based on the variance-inflation model and the mean-shift model have been derived for the linear model in case of homoscedasticity. To derive these two EM algorithms for the GH model, different variances are introduced for the observations and the expectations of the measurements defined by the linear model are replaced by the ones of the GH model. The two robust methods are applied to fit by the GH model a polynomial surface of second degree to the measured three-dimensional coordinates of a laser scanner. This results in detecting more outliers than by the linear model.

A mixture of normal distributions is assumed for the observations of a linear model. The first component of the mixture represents the measurements without gross errors, while each of the remaining components gives the distribution for an outlier. Missing data are introduced to deliver the information as to which observation belongs to which component. The unknown location parameters and the unknown scale parameter of the linear model are estimated by the EM algorithm, which is iteratively applied. The E (expectation) step of the algorithm determines the expected value of the likelihood function given the observations and the current estimate of the unknown parameters, while the M (maximization) step computes new estimates by maximizing the expectation of the likelihood function. In comparison to Huber’s M-estimation, the EM algorithm does not only identify outliers by introducing small weights for large residuals but also estimates the outliers. They can be corrected by the parameters of the linear model freed from the distortions by gross errors. Monte Carlo methods with random variates from the normal distribution then give expectations, variances, covariances and confidence regions for functions of the parameters estimated by taking care of the outliers. The method is demonstrated by the analysis of measurements with gross errors of a laser scanner.

GOCO02S is a combined satellite-only geopotential model, regularized from degrees 180 to 250 of the expansion into spherical harmonics. To investigate the start of the regularization, the normal equations of GOCO02S have been used to compute additional geopotential models by regularizations beginning at degrees 160, 200, 220 and with no regularization. Three different methods are applied to determine where to start the regularization. The simplest one considers the decrease of the degree variances of the not regularized solution. The second one tests for the same solution the hypothesis that the square root of the degree variance is equal to the value computed by the estimated harmonic coefficients. If the hypothesis has to be rejected for a certain degree, the error degree variance is so large that the estimated harmonic coefficients cannot be trusted anymore so that the regularization has to start at that degree. The third method uses the density anomalies by which the disturbing potential is caused resulting from the geopotential model. The density anomalies are well suited to visualize the effects of the higher degree harmonics. In contrast to the base functions of the harmonic coefficients with global support, the density anomalies are expressed by a B-spline surface with local support. Multi-scale representations were applied and the hypotheses tested that the wavelet coefficients are equal to zero. Accepting the hypotheses means that nonsignificant wavelet coefficients were determined which lead to nonsignificant density anomalies. By comparing these anomalies for different regularizations, the degree where to start the regularization is determined. It turns out that beginning the regularization at degree 180, as was done for GOCO02S, is a correct choice.

Digital Images with 3D Geometry from Data Compression by Multi-scale Representations of B-Spline SurfacesTo build up a 3D (three-dimensional) model of the surface of an object, the heights of points on the surface are measured, for instance, by a laser scanner. The intensities of the reflected laser beam of the points can be used to visualize the 3D model as range image. It is proposed here to fit a two-dimensional B-spline surface to the measured heights and intensities by the lofting method. To fully use the geometric information of the laser scanning, points on the fitted surface with their intensities are computed with a density higher than that of the measurements. This gives a 3D model of high resolution which is visualized by the intensities of the points on the B-spline surface. For a realistic view of the 3D model, the coordinates of a digital photo of the object are transformed to the coordinate system of the 3D model so that the points get the colors of the digital image. To efficiently compute and store the 3D model, data compression is applied. It is derived from the multi-scale representation of the dense grid of points on the B-spline surface. The proposed method is demonstrated for an example.

N-dimensional B-spline surface estimated by lofting for locally improving IRIN-dimensional surfaces are defined by the tensor product of B-spline basis functions. To estimate the unknown control points of these B-spline surfaces, the lofting method also called skinning method by cross-sectional curve fits is applied. It is shown by an analytical proof and numerically confirmed by the example of a four-dimensional surface that the results of the lofting method agree with the ones of the simultaneous estimation of the unknown control points. The numerical complexity for estimating vn control points by the lofting method is O(vn+1) while it results in O(v3n) for the simultaneous estimation. It is also shown that a B-spline surface estimated by a simultaneous estimation can be extended to higher dimensions by the lofting method, thus saving computer time.An application of this method is the local improvement of the International Reference Ionosphere (IRI), e.g. by the slant total electron content (STEC) obtained by dual-frequency observations of the Global Navigation Satellite System (GNSS). Three-dimensional B-spline surfaces at different time epochs have to be determined by the simultaneous estimation of the control points for this improvement. A four-dimensional representation in space and time of the electron density of the ionosphere is desirable. It can be obtained by the lofting method. This takes less computer time than determining the four-dimensional surface solely by a simultaneous estimation.

Quantities like tropospheric zenith delays or station coordinates are repeatedly measured at permanent VLBI or GPS stations so that time series for the quantities at each station are obtained. The covariances of these quantities can be estimated in a multivariate linear model. The covariances are needed for computing uncertainties of results derived from these quantities. The covariance matrix for many permanent stations becomes large, the need for simplifying it may therefore arise under the condition that the uncertainties of derived results still agree. This is accomplished by assuming that the different time series of a quantity like the station height for each permanent station can be combined to obtain one time series. The covariance matrix then follows from the estimates of the auto- and cross-covariance functions of the combined time series. A further approximation is found, if compactly supported covariance functions are fitted to an estimated autocovariance function in order to obtain a covariance matrix which is representative of different kinds of measurements. The simplification of a covariance matrix estimated in a multivariate model is investigated here for the coordinates of points of a grid measured repeatedly by a laserscanner. The approximations are checked by determining the uncertainty of the sum of distances to the points of the grid. To obtain a realistic value for this uncertainty, the covariances of the measured coordinates have to be considered. Three different setups of measurements are analyzed and a covariance matrix is found which is representative for all three setups. Covariance matrices for the measurements of laserscanners can therefore be determined in advance without estimating them for each application.

For reverse engineering, nonuniform rational B-spline surfaces are fitted to measured coordinates of points. Two methods are often applied. The first one consists of simultaneously estimating the unknown control points of the surface. The second method is a special case of the lofting or skinning method where cross-sectional curves are not interpolated but fitted to the measurements, thus obtaining the control points. This approach is considered to give an approximate solution of the first one. However, it is shown here by an analytical proof and confirmed by a numerical example that both methods give identical results. Since the computational complexity of the first approach is the square of the second one, the simultaneous estimation of a large number of control points should be avoided.

A new method for dealing with systematic effects in laser scanning and visualizing them by confidence regions is derived. The standard deviations of the systematic effects are obtained by repeatedly measuring three-dimensional coordinates by the laser scanner. In addition, autocovariance and cross-covariance functions are computed by the repeated measurements and give the correlations of the systematic effects. The normal distribution for the measurements and the multivariate uniform distribution for the systematic effects are applied to generate random variates for the measurements and random variates for the measurements plus systematic effects. Monte Carlo estimates of the expectations and the covariance matrix of the measurements with systematic effects are computed. The densities for the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects are obtained by relative frequencies. They only depend on the size of the rectangular volume elements for which the densities are determined. The problem of sorting the densities is solved by sorting distances together with the densities. This allows a visualization of the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects.

Among the Markov chain Monte Carlo methods, the Gibbs sampler has the advantage that it samples from the conditional distributions for each unknown parameter, thus decomposing the sample space. In the case the conditional distributions are not tractable, the Gibbs sampler by means of sampling-importance-resampling is presented here. It uses the prior density function of a Bayesian analysis as the importance sampling distribution. This leads to a fast convergence of the Gibbs sampler as demonstrated by the smoothing with preserving the edges of 3D images of emission tomography.

The use of sampling-based Monte Carlo methods for the computation and propagation of large covariance matrices in geodetic applications is investigated. In particular, the so-called Gibbs sampler, and its use in deriving covariance matrices by Monte Carlo integration, and in linear and nonlinear error propagation studies, is discussed. Modifications of this technique are given which improve in efficiency in situations where estimated parameters are highly correlated and normal matrices appear as ill-conditioned. This is a situation frequently encountered in satellite gravity field modelling. A synthetic experiment, where covariance matrices for spherical harmonic coefficients are estimated and propagated to geoid height covariance matrices, is described. In this case, the generated samples correspond to random realizations of errors of a gravity field model.

To visualize the surface of an object, laser scanners determine the rectangular coordinates of points of a grid on the surface of the object in a local coordinate system. Vertical angles, horizontal angles and distances of a polar coordinate system are measured with the scanning. Outliers generally occur as gross errors in the distances. It is therefore investigated here whether rectangular or polar coordinates are better suited for the detection of outliers. The parameters of a surface represented by a polynomial are estimated in the nonlinear Gauss Helmert (GH) model and in a linear model. Rectangular and polar coordinates are used, and it is shown that the results for both coordinate systems are identical. It turns out that the linear model is sufficient to estimate the parameters of the polynomial surface. Outliers are therefore identified in the linear model by the expectation maximization (EM) algorithm for the variance-inflation model and are confirmed by the EM algorithm for the mean-shift model. Again, rectangular and polar coordinates are used. The same outliers are identified in both coordinate systems.

The global positioning system (GPS) model is distinctive in the way that the unknown parameters are not only real-valued, the baseline coordinates, but also integers, the phase ambiguities. The GPS model therefore leads to a mixed integer–real-valued estimation problem. Common solutions are the float solution, which ignores the ambiguities being integers, or the fixed solution, where the ambiguities are estimated as integers and then are fixed. Confidence regions, so-called HPD (highest posterior density) regions, for the GPS baselines are derived by Bayesian statistics. They take care of the integer character of the phase ambiguities but still consider them as unknown parameters. Estimating these confidence regions leads to a numerical integration problem which is solved by Monte Carlo methods. This is computationally expensive so that approximations of the confidence regions are also developed. In an example it is shown that for a high confidence level the confidence region consists of more than one region.

A method for splitting sea surface height measurements from satellite altimetry into geoid undulations and sea surface topography is presented. The method is based on a combination of the information from altimeter data and a dynamic sea surface height model. The model consists of geoid undulations and a quasi-geostrophic model for expressing the sea surface topography. The goal is the estimation of those values of the parameters of the sea surface height model that provide a least-squares fit of the model to the data. The solution is accomplished by the adjoint method which makes use of the adjoint model for computing the gradient of the cost function of the least-squares adjustment and an optimization algorithm for obtaining improved parameters. The estimation is applied to the North Atlantic. ERS-1 altimeter data of the year 1993 are used. The resulting geoid agrees well with the geoid of the EGM96 gravity model.

A robust Kalman filter is derived for rank deficient observation models. The datum for the Kalman filter is introduced at the zero epoch by the choice of a generalized inverse. The robust filter is obtained by Bayesian statistics and by applying a robust M-estimate. Outliers are not only looked for in the observations but also in the updated parameters. The ability of the robust Kalman filter to detect outliers is demonstrated by an example.

Best invariant quadratic unbiased estimates (BIQUE) of the variance and covariance components for a nonlinear Gauss Helmert (GH) model are derived. To detect outliers, the expectation maximization (EM) algorithm based on the variance-inflation model and the mean-shift model is applied, which results in an iterative reweighting least squares. Each step of the iterations for the EM algorithm therefore includes first the iterations for linearizing the GH model and then the iterations for estimating the variance components. The method is applied to fit a surface in three-dimensional space to the three coordinates of points measured, for instance, by a laser scanner. The surface is represented by a polynomial of second degree and the variance components of the three coordinates are estimated. Outliers are detected by the EM algorithm based on the variance-inflation model and identified by the EM algorithm for the mean-shift model.

In three basic points Bayesian statistics differs from traditional statistics. First Bayesian statistics is founded on Bayes’theorem. By this theorem unknown parameters are estimated, confidence regions for the unknown parameters are established and hypotheses for the parameters are tested. Furthermore, Bayesian statistics extends the notion of probability by defining the probability for statements or propositions. The probability is a measure for the plausibility of a statement. Finally, the unknown parameters of Bayesian statistics are random variables. But nevertheless, the unknown parameters can represent constants. There are numerous applications of Bayesian statistics for the analysis of geodetic data, some of them are pointed out.

Different types of present or future satellite data have to be combined by applying appropriate weighting for the determination of the gravity field of the Earth, for instance GPS observations for CHAMP with satellite to satellite tracking for the coming mission GRACE as well as gradiometer measurements for GOCE. In addition, the estimate of the geopotential has to be smoothed or regularized because of the inversion problem. It is proposed to solve these two tasks by Bayesian inference on variance components. The estimates of the variance components are computed by a stochastic estimator of the traces of matrices connected with the inverse of the matrix of normal equations, thus leading to a new method for determining variance components for large linear systems. The posterior density function for the variance components, weighting factors and regularization parameters are given in order to compute the confidence intervals for these quantities. Test computations with simulated gradiometer observations for GOCE and satellite to satellite tracking for GRACE show the validity of the approach.

Im Kapitel 2.2.8 war bereits angedeutet worden, das mit der Kenntnis der Posteriori-Dichte fur die unbekannten Parameter aus dem Bayes-Theorem die unbekannten Parameter geschatzt werden konnen, das Bereiche anzugeben sind, in denen sie mit vorgegebener Wahrscheinlichkeit liegen, und das Hypothesen fur die unbekannten Parameter gepruft werden konnen. Diese Verfahren werden im folgenden behandelt, wobei mit stetigen Zufallsvektoren gearbeitet wird. Die dabei auftretenden Integrale zur Berechnung der Wahrscheinlichkeiten, der Randverteilungen und der Erwartungswerte sind fur diskrete Zufallsvektoren entsprechend durch Summationen zur Ermittlung der Wahrscheinlichkeiten nach (2.69), der Randverteilungen nach (2.85) und der Erwartungswerte nach (2.140) zu ersetzen.

Im folgenden werden zwei Modelle behandelt, die aus dem linearen Modell entwickelt werden, namlich das Modell der Pradiktion und Filterung und das lineare Modell mit unbekannten Varianz- und Kovarianzkomponenten. Auserdem wird auf die Mustererkennung und auf Entscheidungen in Systemen mit Unsicherheiten durch Bayes-Netze eingegangen.

Chapter 3 dealt with the estimation of the unknown parameters. By means of the variances and covariances of the estimators, measures of the dispersion of the estimators about their expected values and measures of the linear dependency of the estimators were determined With respect to the variances and covariances, additional information shall be obtained in the following by estimating for several parameters the region and for one parameter the interval in which the values of the parameters can be expected with a given probability. This problem is called an interval estimation.

The statistical inference on parameters will be derived in linear models. A linear relation can be treated in a compact and lucid form by vectors and matrices, so that in the following, the definitions and theorems of linear algebra are introduced which will be needed later. The methods of vector spaces will also be discussed. They allow one to use geometric conceptions even then, when the spaces being used are of higher dimensions than the three-dimensional space we are familiar with. Finally, generalized inverses are discussed, by which one can easily change models of full rank for the estimation of parameters to models which are not full rank.

Satellite altimeter data contains information about the heights of the geoid above the reference ellipsoid, the geoid undulations, the heights of the sea surface above the geoid, the sea surface topography and the radial component of the orbital error. The deviation of the sea surface from the geoid is mainly caused by currents and tidal effects. For the simultaneous estimation of all unknown parameters, based on a modified Kalman filter developed from Bayesian statistics, it is necessary to introduce prior information. A simplification of the Navier-Stokes-equations in form of a quasi-geostrophic model is used to represent the sea surface topography. Information on the orbital error is obtained by a crossing point analysis.

A method is outlined for analysing altimeter data in order to improve the geoid over the oceans. It assumes that the height of the sea surface above a reference ellipsoid, derived from the altimetric measurement by an orbit computation, consists of the geoid undulations, the height of the sea surface above the geoid and the radial component of the orbital error. The height of the sea surface above the geoid is expressed by modelling the oceanic currents. To keep the computational burden for such an approach manageable, a quasi-geostrophic model with two layers is chosen, and the area of the analysis of the data is restricted to a region of an ocean. The orbital errors are determined by a crossing point analysis, and knowledge on the geoid from satellite tracking and surface gravity data is introduced as prior information for a Bayesian estimation. It leads to a modified Kalman filter for the simultaneous estimation of all parameters. Test results for computing the sea surface topography with respect to the geoid by means of the quasi-geostrophic model are presented.