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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.

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.

The plate-screw interface of an angular stable plate osteosynthesis is very rigid. So far, all attempts to decrease the stiffness of locked plating construct, e.g. the bridged plate technique, decrease primarily the bending stiffness. Thus, the interfragmentary motion increases only on the far cortical side by bending the plate. To solve this problem, the dynamic locking screw (DLS) was developed.Comparison tests were performed with locking screws (LS) and DLS. Axial stiffness, bending stiffness and interfragmentary motion were compared. For measurements, we used a simplified transverse fracture model, consisting of POM C and an 11-hole LCP3.5 with a fracture gap of 3 mm. Three-dimensional fracture motion was detected using an optical measurement device (PONTOS 5 M/GOM) consisting of two CCD cameras (2,448 x 2,048 pixel) observing passive markers.The DLS reduced the axial stiffness by approximately 16% while increasing the interfragmentary motion at the near cortical side significantly from 282 microm (LS) to 423 microm (DLS) applying an axial load of 150 N.The use of DLS reduces the stiffness of the plate-screw interface and thus increases the interfragmentary motion at the near cortical side without altering the advantages of angular stability and the strength.

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.

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 genus Homodiaetus Eigenmann & Ward, 1907 is revised and four species are recognized. Its distribution is restricted to southeastern South America, from Uruguay to Paraguay river at west to the coastal drainages of Rio de Janeiro State, Brazil. Homodiaetus is currently distinguished from other genus of Stegophilinae by the combination of the following characters: origin of ventral-fin at midlength between the snout tip and the caudal-fin origin; opercle with three or more odontodes; and gill membranes confluent with the istmus. Homodiaetus anisitsi Eigenmann & Ward, 1907, is diagnosed by the caudal-fin with black middle rays, margin of upper and lower procurrent caudal-fin rays with dark stripes extending to the caudal-fin, and 3-6 opercular odontodes; H. passarellii (Ribeiro, 1944) with 6-7 opercular odontodes, 21-24 lower procurrent caudal-fin rays and 23-26 upper procurrent caudal-fin rays; H. banguela sp. nov. with 9 opercular odontodes, 17-19 lower procurrent caudal-fin rays, 17-22 upper procurrent caudal-fin rays, reduction of fourth pharyngobranchial with only three teeth and untoothed fifth ceratobranchial; and H. graciosa sp. nov. with 5-6 dentary rows, 7-9 opercular odontodes and 16-23 upper procurrent caudal-fin rays.

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.

Summary The representation of the Earth's gravity field as the potential of a simple layer distributed over the surface of the Earth is determined by combining satellite observations and gravity anomalies. Density values of the simple layer for 192 surface elements are computed and converted into harmonic coefficients up to the 15th degree and order. These co-efficients are used to determine surface density values referred to a reference ellipsoid with the flattening of an earth in hydrostatic equilibrium. The geophysical implications of these values are outlined.

Ten weeks of Doppler tracking of the 5 satellites for which data are available at the National Space Science Data Center have been analyzed to determine the coordinates of 27 tracking stations and the gravity field of the earth represented by the potential of a simple layer. Density values of this layer for 104 surface elements have been computed in a least-squares adjustment and transformed into harmonic coefficients up to the eleventh degree and order. Comparisons with other solutions show good agreement. The results for the equatorial radius of the earth, its flattening, and its gravity at the equator are 6,378,156 meters, 1/298.255, and 978,028.8 mgal, respectively.

Instead of the expansion in spherical harmonics currently used in satellite geodesy, the geopotential in this analysis is represented by the potential of a simple layer distributed over the surface of the earth. Density values of this layer for 48 surface elements have been determined from Baker-Nunn camera observations of 4 satellites for 5 weeks. The solution closely approximates the one for the 1966 Smithsonian Institution Standard Earth and Anderle's solution (1967) based on Doppler data. The rms discrepancy in geoidal heights between those two solutions and the solution of this analysis amounts to ±15.5 m for both cases. Existing gravity anomalies are combined with the solution obtained from the satellite observations. The combination does not markedly differ from the satellite solution because the determination of the geopotential from satellite data is stronger.