Bayesian adaptive robust adjustment of multivariate geodetic measurement processeswith data gaps and nonstationary colored noise

现代大地测量传感器经常产生多个空间时间序列，其中包含大量的测量值、大量的异常值以及数据间隙和随机误差，其特征是相当大的自相关和交叉相关(即有色噪声)。鉴于目前的大地测量数据分析工具无法完全解决这些不利因素，我们打算发展与调整程序有关的经典统计和贝叶斯统计，以便从这种时空测量系列中可靠和有效地估计参数模型。为了促进稳健性、统计和计算效率的同时，我们一方面采用了期望最大化(EM)原则。这使得能够估算数据间隙并同时自适应地估计函数模型的参数、向量自回归滑动平均(VARMA)有色噪声模型的系数、以及潜在误差分布的形状参数。后者由多变量、比例(学生)t分布定义，并涉及数据自适应自由度和比例因子。通过估计这些量，使概率密度函数的形状特别是尾部特征适应于数据中存在的实际误差和离群值特征。在接下来的工作步骤中，我们还将允许动态更改函数模型和噪声模型的参数。最后，我们研究了基于平均场变分贝叶斯和马尔可夫链蒙特卡罗(MCMC)技术的贝叶斯过程，它允许将关于函数模型、Varma模型和潜在t分布的参数的先验信息纳入自适应稳健平差。由于调整产生关于所有未知模型参数的详细概率信息，例如，我们还将能够严格地测试关于假定的误差分布、关于可疑的自相关/互相关模式以及关于这种模式的时间可变性的假设。我们将静态版本的一般观测模型和估计程序应用于基于大地数据集的平差问题，该平差问题源于静态多传感器系统的地理参考。他们的参考传感器可以是3D定位传感器，如GNSS设备或测速仪。将动力版本应用于加载某拱桥的试验数据。由于这些方法预期具有很高的灵活性和效率，我们预计它们也适用于其他类型的大地测量传感器数据，例如在卫星大地测量中获得的数据。

Modern geodetic sensors often produce multiple spatial time series which contain huge numbers of measurements, numerous outliers as well as data gaps, and random errors that are characterized by considerable auto- andcross-correlations (i.e., colored noise). In view of these adversities, which cannot be resolved by current geodetic data analysis tools in their entirety, we intend to develop both classical and Bayesian statistics in connection with adjustment procedures that allow for a robust and efficient estimation of parametric models from such spatio-temporal measurement series. To facilitate simultaneous robustness and statistical as well as computational efficiency, we employ on the one hand the principle of expectation maximization (EM). This enables an imputation of the data gaps and concurrently an adaptive estimation of the parameters of the functional model, of the coefficients of a vector autoregressive moving-average (VARMA) colored noise model, and of the shape parameters of the underlying error distribution. The latter is defined by a multivariate, scaled (Student) t-distribution and involves a data-adaptable degree of freedom and scale factor. By estimating these quantities, the shape and in particular the tail characteristics of the probability densityfunction is adapted to the actual error and outlier characteristics present in the data. In a subsequent work step, we will also allow for dynamic changes of the parameters of the functional and of the noise model. Finally, we investigate Bayesian procedures based on Mean-Field Variational Bayes and Markov Chain Monte Carlo (MCMC) techniques, which allow for the incorporation of prior information regarding the parameters of the functional model, of the VARMA model and of the underlying t-distribution into the adaptive robust adjustment. Since the adjustment yields detailed probabilistic information regarding all of the unknown model parameters, we will for instance also be able to rigorously test hypotheses about the assumed error distribution, about suspected auto-/cross-correlation patterns, and about the time-variability of such patterns. We apply the static version of the general observation model and estimation procedure to adjustment problems based on geodetic data sets stemming from geo-referencing of static multi-sensor systems. Their referencing sensors can be 3D positioning sensors, like GNSS equipment or tacheometer. The dynamic version is applied to loading test data stemming from an arch bridge. Due to the anticipated high level of flexibility and efficiency of the methods, we expect them to be applicable also to other types of geodetic sensor data, as obtained e.g. in satellite geodesy.