Geodetic measurements rely on high-resolution sensors, but produce data sets with many observations which may contain outliers and correlated deviations. This paper proposes a powerful solution using Bayesian inference. The observed data is modeled as a multivariate time series with a stationary autoregressive (VAR) process and multivariate t-distribution for white noise. Bayes’ theorem integrates prior knowledge. Parameters, including functional, VAR coefficients, scaling, and degree of freedom of the t-distribution, are estimated with Markov Chain Monte Carlo using a Metropolis-within-Gibbs algorithm.

In this contribution, a vector-autoregressive (VAR) process with multivariate t-distributed random deviations is incorporated into the Gauss-Helmert model (GHM), resulting in an innovative adjustment model. This model is versatile since it allows for a wide range of functional models, unknown forms of auto- and cross-correlations, and outlier patterns. Subsequently, a computationally convenient iteratively reweighted least squares method based on an expectation maximization algorithm is derived in order to estimate the parameters of the functional model, the unknown coefficients of the VAR process, the cofactor matrix, and the degree of freedom of the t-distribution. The proposed method is validated in terms of its estimation bias and convergence behavior by means of a Monte Carlo simulation based on a GHM of a circle in two dimensions. The methodology is applied in two different fields of application within engineering geodesy: In the first scenario, the offset and linear drift of a noisy accelerometer are estimated based on a Gauss-Markov model with VAR and multivariate t-distributed errors, as a special case of the proposed GHM. In the second scenario real laser tracker measurements with outliers are adjusted to estimate the parameters of a sphere employing the proposed GHM with VAR and multivariate t-distributed errors. For both scenarios the estimated parameters of the fitted VAR model and multivariate t-distribution are analyzed for evidence of auto- or cross-correlations and deviation from a normal distribution regarding the measurement noise.

The detection of deformation is one of the major tasks in surveying engineering. It is meaningful only if the statistical significance of the distortions is correctly investigated, which often underlies a parametric modelization of the object under consideration. So-called regression B-spline approximation can be performed for point clouds of terrestrial laser scanners, allowing the setting of a specific congruence test based on the B-spline surfaces. Such tests are known to be strongly influenced by the underlying stochastic model chosen for the observation errors. The latter has to be correctly specified, which includes accounting for heteroscedasticity and correlations. In this contribution, we justify and make use of a parametric correlation model called the Matern model to approximate the variance covariance matrix (VCM) of the residuals by performing their empirical mode decomposition. The VCM obtained is integrated into the computation of the congruence test statistics for a more trustworthy test decision. Using a real case study, we estimate the distribution of the test statistics with a bootstrap approach, where no parametric assumptions are made about the underlying population that generated the random sample. This procedure allows us to assess the impact of neglecting correlations on the critical value of the congruence test, highlighting their importance.

The choice of an appropriate metric is mandatory to perform deformation analysis between two point clouds (PC)—the distance has to be trustworthy and, simultaneously, robust against measurement noise, which may be correlated and heteroscedastic. The Hausdorff distance (HD) or its averaged derivation (AHD) are widely used to compute local distances between two PC and are implemented in nearly all commercial software. Unfortunately, they are affected by measurement noise, particularly when correlations are present. In this contribution, we focus on terrestrial laser scanner (TLS) observations and assess the impact of neglecting correlations on the distance computation when a mathematical approximation is performed. The results of the simulations are extended to real observations from a bridge under load. Highly accurate laser tracker (LT) measurements were available for this experiment: they allow the comparison of the HD and AHD between two raw PC or between their mathematical approximations regarding reference values. Based on these results, we determine which distance is better suited in the case of heteroscedastic and correlated TLS observations for local deformation analysis. Finally, we set up a novel bootstrap testing procedure for this distance when the PC are approximated with B-spline surfaces.

Today, short- and long-term structural health monitoring (SHM) of bridge infrastructures and their safe, reliable and cost-effective maintenance has received considerable attention. From a surveying or civil engineer's point of view, vibration-based SHM can be conducted by inspecting the changes in the global dynamic behaviour of a structure, such as natural frequencies (i.e. eigenfrequencies), mode shapes (i.e. eigenforms) and modal damping, which are known as modal parameters. This research work aims to propose a robust and automatic vibration analysis procedure that is so-called robust time domain modal parameter identification (RT-MPI) technique. It is novel in the sense of automatic and reliable identification of initial eigenfrequencies even closely spaced ones as well as robustly and accurately estimating the modal parameters of a bridge structure using low numbers of cost-effective micro-electro-mechanical systems (MEMS) accelerometers. To estimate amplitude, frequency, phase shift and damping ratio coefficients, an observation model consisting of: (1) a damped harmonic oscillation model, (2) an autoregressive model of coloured measurement noise and (3) a stochastic model in the form of the heavy-tailed family of scaled t-distributions is employed and jointly adjusted by means of a generalised expectation maximisation algorithm. Multiple MEMS as part of a geo-sensor network were mounted at different positions of a bridge structure which is precalculated by means of a finite element model (FEM) analysis. At the end, the estimated eigenfrequencies and eigenforms are compared and validated by the estimated parameters obtained from acceleration measurements of high-end accelerometers of type PCB ICP quartz, velocity measurements from a geophone and the FEM analysis. Additionally, the estimated eigenfrequencies and modal damping are compared with a well-known covariance driven stochastic subspace identification approach, which reveals the superiority of our proposed approach. We performed an experiment in two case studies with simulated data and real applications of a footbridge structure and a synthetic bridge. The results show that MEMS accelerometers are suitable for detecting all occurring eigenfrequencies depending on a sampling frequency specified. Moreover, the vibration analysis procedure demonstrates that amplitudes can be estimated in submillimetre range accuracy, frequencies with an accuracy better than 0.1 Hz and damping ratio coefficients with an accuracy better than 0.1 and 0.2 % for modal and system damping, respectively.

In this contribution, we extend the Gauss-Helmert model (GHM) with t-distributed errors (previously established by K.R. Koch) by including autoregressive (AR) random deviations. This model allows us to take into account unknown forms of colored noise as well as heavy-tailed white noise components within observed time series. We show that this GHM can be adjusted in principle through constrained maximum likelihood (ML) estimation, and also conveniently via an expectation maximization (EM) algorithm. The resulting estimator is self-tuning in the sense that the tuning constant, which occurs here as the degree of freedom of the underlying scaled t-distribution and which controls the thickness of the tails of that distribution’s probability distribution function, is adapted optimally to the actual data characteristics. We use this model and algorithm to adjust 2D measurements of a circle within a closed-loop Monte Carlo simulation and subsequently within an application involving GNSS measurements.

The detection of deformation is one of the major tasks in surveying engineering. It is meaningful only if the statistical significance of the distortions is correctly investigated, which often underlies a parametric modelization of the object under consideration. So-called regression B-spline approximation can be performed for point clouds of terrestrial laser scanners, allowing the setting of a specific congruence test based on the B-spline surfaces. Such tests are known to be strongly influenced by the underlying stochastic model chosen for the observation errors. The latter has to be correctly specified, which includes accounting for heteroscedasticity and correlations. In this contribution, we justify and make use of a parametric correlation model called the Matérn model to approximate the variance covariance matrix (VCM) of the residuals by performing their empirical mode decomposition. The VCM obtained is integrated into the computation of the congruence test statistics for a more trustworthy test decision. Using a real case study, we estimate the distribution of the test statistics with a bootstrap approach, where no parametric assumptions are made about the underlying population that generated the random sample. This procedure allows us to assess the impact of neglecting correlations on the critical value of the congruence test, highlighting their importance.

The iteratively reweighted least-squares approach to self-tuning robust adjustment of parameters in linear regression models with autoregressive (AR) and t-distributed random errors, previously established in Kargoll et al. (in J Geod 92(3):271–297, 2018. https://doi.org/10.1007/s00190-017-1062-6), is extended to multivariate approaches. Multivariate models are used to describe the behavior of multiple observables measured contemporaneously. The proposed approaches allow for the modeling of both auto- and cross-correlations through a vector-autoregressive (VAR) process, where the components of the white-noise input vector are modeled at every time instance either as stochastically independent t-distributed (herein called “stochastic model A”) or as multivariate t-distributed random variables (herein called “stochastic model B”). Both stochastic models are complementary in the sense that the former allows for group-specific degrees of freedom (df) of the t-distributions (thus, sensor-component-specific tail or outlier characteristics) but not for correlations within each white-noise vector, whereas the latter allows for such correlations but not for different dfs. Within the observation equations, nonlinear (differentiable) regression models are generally allowed for. Two different generalized expectation maximization (GEM) algorithms are derived to estimate the regression model parameters jointly with the VAR coefficients, the variance components (in case of stochastic model A) or the cofactor matrix (for stochastic model B), and the df(s). To enable the validation of the fitted VAR model and the selection of the best model order, the multivariate portmanteau test and Akaike’s information criterion are applied. The performance of the algorithms and of the white noise test is evaluated by means of Monte Carlo simulations. Furthermore, the suitability of one of the proposed models and the corresponding GEM algorithm is investigated within a case study involving the multivariate modeling and adjustment of time-series data at four GPS stations in the EUREF Permanent Network (EPN).

Terrestrial laser scanners (TLS) are powerful instruments that can be employed for deformation monitoring due to their high precision and spatial resolution in capturing 3D point clouds. Deformation detections from scatter point clouds can be based on different comparison methods, among which the geometry-based method is one of the most popular. Compared with approximating surfaces with predetermined geometric primitives, such as plane or sphere, the B-splines surface approximation offers a great flexibility and can be used to fit nearly every object scanned with TLS. However, a variance-covariance matrix (VCM) of the observations involved in approximating the scattered points to B-spline surfaces impact the results of a congruency test, which is the uniformly most powerful invariant (UMPI) test for discriminating between the null hypothesis of zero deformation and its alternative hypotheses. Consequently, simplified stochastic models may weaken the UMPI property. Based on Monte Carlo simulations, the impact of the heteroscedasticity and mathematical correlations often neglected in B-splines approximation are investigated. These correlations are specific in approximating TLS measurements when the raw measurements are transformed into Cartesian coordinates. The rates of rejecting the null hypothesis in a congruency test is employed to reflect the impact of unspecified VCMs on the power of the congruency test. The rejection rates are not sensitive to the simplification of the stochastic models, in the larger deformation area with higher point accuracy, while they are obviously influenced in the smaller deformation area with unfavourable geometries, i. e. larger uncertainties. A threshold ratio of estimated differences to the relative standard deviation highlights whereas the results of congruency test are reliable when using simplified VCMs. It is concluded that the simplification of the stochastic model has a significant impact on the power of the congruency test, especially in the smaller deformation area with larger uncertainties.

Abstract. In this study, the feasibility of Micro-Electro-Mechanical System (MEMS) accelerometers and an image-assisted total station (IATS) for short- and long-term deformation monitoring of bridge structures is investigated. The MEMS sensors of type BNO055 from Bosch as part of a geo-sensor network are mounted at different positions of the bridge structure. In order to degrade the impact of systematic errors on the acceleration measurements, the deterministic calibration parameters are determined for fixed positions using a KUKA youBot in a climate chamber over certain temperature ranges. The measured acceleration data, with a sampling frequency of 100 Hz, yields accurate estimates of the modal parameters over short time intervals but suffer from accuracy degradation for absolute position estimates with time. To overcome this problem, video frames of a passive target, attached in the vicinity of one of the MEMS sensors, are captured from an embedded on-axis telescope camera of the IATS of type Leica Nova MS50 MultiStation with a practical sampling frequency of 10 Hz. To identify the modal parameters such as eigenfrequencies and modal damping for both acceleration and displacement time series, a damped harmonic oscillation model is employed together with an autoregressive (AR) model of coloured measurement noise. The AR model is solved by means of a generalized expectation maximization (GEM) algorithm. Subsequently, the estimated model parameters from the IATS are used for coordinate updates of the MEMS sensor within a Kalman filter approach. The experiment was performed for a synthetic bridge and the analysis shows an accuracy level of sub-millimetre for amplitudes and much better than 0.1 Hz for the frequencies.

In the last two decades, the integration of a terrestrial laser scanner (TLS) and digital photogrammetry, besides other sensors integration, has received considerable attention for deformation monitoring of natural or man-made structures. Typically, a TLS is used for an area-based deformation analysis. A high-resolution digital camera may be attached on top of the TLS to increase the accuracy and completeness of deformation analysis by optimally combining points or line features extracted both from three-dimensional (3D) point clouds and captured images at different epochs of time. For this purpose, the external calibration parameters between the TLS and digital camera needs to be determined precisely. The camera calibration and internal TLS calibration are commonly carried out in advance in the laboratory environments. The focus of this research is to highly accurately and robustly estimate the external calibration parameters between the fused sensors using signalised target points. The observables are the image measurements, the 3D point clouds, and the horizontal angle reading of a TLS. In addition, laser tracker observations are used for the purpose of validation. The functional models are determined based on the space resection in photogrammetry using the collinearity condition equations, the 3D Helmert transformation and the constraint equation, which are solved in a rigorous bundle adjustment procedure. Three different adjustment procedures are developed and implemented: (1) an expectation maximization (EM) algorithm to solve a Gauss-Helmert model (GHM) with grouped t-distributed random deviations, (2) a novel EM algorithm to solve a corresponding quasi-Gauss-Markov model (qGMM) with t-distributed pseudo-misclosures, and (3) a classical least-squares procedure to solve the GHM with variance components and outlier removal. The comparison of the results demonstrates the precise, reliable, accurate and robust estimation of the parameters in particular by the second and third procedures in comparison to the first one. In addition, the results show that the second procedure is computationally more efficient than the other two.

The terrestrial laser scanning technology is increasingly applied in the deformation monitoring of tunnel structures. However, outliers and data gaps in the terrestrial laser scanning point cloud data have a deteriorating effect on the model reconstruction. A traditional remedy is to delete the outliers in advance of the approximation, which could be time- and labor-consuming for large-scale structures. This research focuses on an outlier-resistant and intelligent method for B-spline approximation with a rank (R)-based estimator, and applies to tunnel measurements. The control points of the B-spline model are estimated specifically by means of the R-estimator based on Wilcoxon scores. A comparative study is carried out on rank-based and ordinary least squares methods, where the Hausdorff distance is adopted to analyze quantitatively for the different settings of control point number of B-spline approximation. It is concluded that the proposed method for tunnel profile modeling is robust against outliers and data gaps, computationally convenient, and it does not need to determine extra tuning constants.

A stochastic process can be represented and analysed by four different quantities in the time and frequency domain: (1) the process itself, (2) its autocovariance function, (3) the spectral representation of the stochastic process and (4) its spectral distribution or the spectral density function, if it exits. These quantities and their relationships can be clearly represented by the “Magic Square”, where the quantities build the corners of this square and the connecting lines indicate the transformations into each other.

In this contribution, a procedure for deciding, whether the oscillation of a surveyed structure is damped or not, is proposed. For this purpose, two bootstrap tests under fairly general assumptions regarding autocorrelation and outlier-affliction of the random deviations (“measurement errors”) are suggested. These tests are derived from an observation model consisting of (1) a parametric oscillation model based on trigonometric functions, (2) a parametric auto-correlation model in the form of an autoregressive process, and (3) a parametric stochastic model in terms of the heavy-tailed family of scaled t-distributions. These three levels, which generalize current observation models for oscillating structures, are jointly expressed as a likelihood function and jointly adjusted by means of a generalized expectation maximization algorithm. Closed-loop Monte Carlo simulations are performed to validate the bootstrap tests. Visual inspection of models fitted by standard least-squares techniques are shown to be insufficient to detect a small significant damped oscillation. Furthermore, the tests are applied to a controlled experiment in a laboratory environment. The oscillation was generated by means of a portable shaker vibration calibrator and measured by a reference accelerometer and a low-cost accelerometer.

Automatically modeling and intelligently monitoring composite tunnel structures is of great significance considering the development of composite and diverse tunnel construction materials. Therefore, how to efficiently analyze the deformation of all kinds of tunnel structures with its expanding application is becoming increasingly important. In this paper, we analyze the over- and under-fitting problems under different point cloud profiles to generate an automatic and robust B-spline model which is a suitable method for high accuracy approximation capable of optimizing the composite tunnel model by adjusting the parameters. The innovation of this research is that we combine the maximum likelihood function and the degree of freedom to obtain the optimal parameters model of composite tunnels, which are validated by the various profiles.

In this contribution, a robust Bayesian approach to adjusting a nonlinear regression model with t-distributed errors is presented. In this approach the calculation of the posterior model parameters is feasible without linearisation of the functional model. Furthermore, the integration of prior model parameters in the form of any family of prior distributions is demonstrated. Since the posterior density is then generally non-conjugated, Monte Carlo methods are used to solve for the posterior numerically. The desired parameters are approximated by means of Markov chain Monte Carlo using Gibbs samplers and Metropolis-Hastings algorithms. The result of the presented approach is analysed by means of a closed-loop simulation and a real world application involving GNSS observations with synthetic outliers.

In this paper, we intend to test whether the random deviations of an observed regression time series with unknown regression coefficients can be described by a covariance-stationary autoregressive (AR) process, or whether an AR process with time-variable (say, linearly changing) coefficients should be set up. To account for possibly present multiple outliers, the white noise components of the AR process are assumed to follow a scaled (Student) t-distribution with unknown scale factor and degree of freedom. As a consequence of this distributional assumption and the nonlinearity of the estimator, the distribution of the test statistic is analytically intractable. To solve this challenging testing problem, we propose a Monte Carlo (MC) bootstrap approach, in which all unknown model parameters and their joint covariance matrix are estimated by an expectation maximization algorithm. We determine and analyze the power function of this bootstrap test via a closed-loop MC simulation. We also demonstrate the application of this test to a real accelerometer dataset within a vibration experiment, where the initial measurement phase is characterized by transient oscillations and modeled by a time-variable AR process.

B-spline surfaces possess attractive properties such as a high degree of continuity, which is important for computing curvature. Since the local support of the basis functions allows to control the shape of the estimated surface, they are increasingly used in the field of geodesy, where their main application is the fitting of surfaces to, e.g., 3D point clouds obtained from terrestrial laser scanners (TLS). By comparing different epochs, deformation can be detected more easily for nearly all kind of objects by using test statistics such as the congruency test. The concept of surface approximation is similar to a regression problem where the model is the surface representation and the data are the sampled points on the surface. Consequently, besides improving the functional model with new strategies to determine the knot vector, the stochastic model of the underlying observations has to be correctly specified. Otherwise, biases in test statistics are unavoidable and compromise the detection of small deformations when the congruency test is used. Unfortunately, measurements from TLS are not as simple as point clouds for Computer Aided Graphics: the raw observations are not directly Cartesian coordinates but polar coordinates, i.e. range and angles, which additionally possess different variances. A transformation from polar to Cartesian coordinates is mandatory to determine the weighting functions or control points of the B-splines approximation by a least-squares adjustment. Mathematical correlations are thus introduced in the already heteroscedastic transformed observations. As they lead to fully populated variance covariance matrices, they remain mostly neglected and an oversimplified stochastic model is used assuming homoscedasticity and independency of the transformed observations. In this contribution, the impact of neglecting mathematical correlations in deformation analysis with the congruency test will be studied. It will be shown that these correlations can eventually be reduced to an inflation variance factor, which allows for their simplified handling in matrix products. In a case study with real data from a bridge under loading, the impact of neglecting them will be investigated, highlighting in which cases considering mathematical correlations in the congruency test is necessary or not for a trustworthy deformation detection. Heteroscedasticity will be taken into consideration using an intensity-based model accounting for geometry and object properties.

In this paper, we propose a new technique-called Ellipsoidal and Gaussian Kalman filter-for state estimation of discrete-time nonlinear systems in situations when for some parts of uncertainty, we know the probability distributions, while for other parts of uncertainty, we only know the bounds (but we do not know the corresponding probabilities). Similarly to the usual Kalman filter, our algorithm is iterative: on each iteration, we first predict the state at the next moment of time, and then we use measurement results to correct the corresponding estimates. On each correction step, we solve a convex optimization problem to find the optimal estimate for the system's state (and the optimal ellipsoid for describing the systems's uncertainty). Testing our algorithm on several highly nonlinear problems has shown that the new algorithm performs the extended Kalman filter technique better-the state estimation technique usually applied to such nonlinear problems.