Integrated time-space modelling based on correlated measurements

This project refers to the study of deformation processes with high spatial and time resolution. The two main objectives are the development of a space-continuous cinematic deformation model for the representation of these processes and the determination of optimum survey configurations. This initiative is motivated by the availability in engineering geodesy of new types of sensors with high spatial resolution and the outage of a consistent methodology for the detection and the modelling of deformations based on measurements performed with these sensors. The discretisation of the monitored object with these new measuring techniques leads to a large number of measured points. However, the points are highly-correlated and not reproducible in case of repeated measurements. Thus, the use of traditional, point-oriented methods for deformation analysis is not straightforward. The research proposal is fundamental and important for the entire activity field of engineering geodesy. The first part of the project aims the time-continuous and areal modelling of deformations. The used methodology is the collocation, which includes for this purpose a spatial and a time component. The deterministic part in space- domain is based on freeform geometric elements. The focus is laid on rational functions. This kind of geometric modelling is very flexible and allows for the representation of a large number of objects. In the time domain the deterministic part is the identity or the linear model. The collocation includes also a multi-dimensional stochastic field that accounts for existing spatial and time correlations between the measured points. This necessitates the identification of theoretically founded and empirically validated covariance functions. In a first step the measurements are processed after a complete measuring epoch. By developing a sequential processing method in a second step one accounts for possible deformations of the object during the observation time. The abovementioned deformation model is non-linear and non-additive. These properties are related to the second part of the research project that aims the development of a variance-based sensitivity analysis method for non- additive models in case of correlated measurements. Thus, correct statements regarding the influence of measurements on the parameters of the deformation model and on the parameters of the survey configuration can be obtained. The study of stochastic relations by means of synthetic error models and the formulation of criteria matrices enable the forming of groups of influence factors such that correlations can be eliminated. The non- additivity of the model will be handled by numerical analysis. The identification of conditions and models for which these approaches hold will also be treated in this part of the project.

The project IMKAD aims to describe geometrical changes of measuring objects based on laser scan data. Such an areal deformation analysis includes the modelling of the acquired point cloud by means of continuous functions in order to reduce the amount of data by encoding the geometrical information in the functions parameters. Due to their high flexibility, parametric free form surfaces like B-spline or Bézier surfaces have been proven to be particularly suitable in order to model point clouds. However, their high flexibility requires the determination of a variety of parameter types such as the number of control points, which directly defines the surfaces complexity, and the allocation of appropriate surface parameters, which localise the respective observation onto the surface. During the first project phase, the focus was on the development of appropriate methods solving these problems. Parallel to the development of the functional point cloud modelling, measurement deviations were investigated and stochastically modelled by the project partner of the institute of engineering geodesy Stuttgart (IIGS). The resulting stochastic model was integrated into the estimation of free form surfaces in terms of a synthetic variance-covariance-matrix. The occurrence of the estimated surfaces systematic biases indicated the need of a refinement of the stochastic model. Based on the functional description of point clouds which are acquired at different points in time two models to describe geometrical changes of the measuring objects were developed during the second project phase. The first model exclusively describes rigid body movements (rotations, translations and scaling). In the project it could be demonstrated that, under certain circumstances, the rigid body movement can directly be retrieved in the free form surfaces control points, allowing the application of classical point based methods on continuous and areal descriptions of point clouds. The second model uses a least squares collocation in order to describe the measuring objects distortions. The non-distorted object is modelled by means of deterministic free form surfaces, whereas the occurring changes are interpreted and modelled as a stochastic signal.