Covariance function modeling is an essential part of stochastic methodology. Many processes in geodetic applications have rather complex, often oscillating covariance functions, where it is difficult to find corresponding analytical functions for modeling. This paper aims to give the methodological foundations for an advanced covariance modeling and elaborates a set of generic base functions which can be used for flexible covariance modeling. In particular, we provide a straightforward procedure and guidelines for a generic approach to the fitting of oscillating covariance functions to an empirical sequence of covariances. The underlying methodology is developed based on the well known properties of autoregressive processes in time series. The surprising simplicity of the proposed covariance model is that it corresponds to a finite sum of covariance functions of second-order Gauss–Markov (SOGM) processes. Furthermore, the great benefit is that the method is automated to a great extent and directly results in the appropriate model. A manual decision for a set of components is not required. Notably, the numerical method can be easily extended to ARMA-processes, which results in the same linear system of equations. Although the underlying mathematical methodology is extensively complex, the results can be obtained from a simple and straightforward numerical method.

In time series analysis auto regressive (AR) modelling of zero mean data is widely used for system identification, signal decorrelation, detection of outliers and forecasting. An AR process of order p is uniquely defined by r coefficients and the variance of the noise. The roots of the characteristic polynomial can be used as an alternative parametrization of the coefficients, which is used to construct a continuous covariance function of the AR processes or to verify that the AR processes are stationary. In this contribution we propose an approach to estimate an AR process of time varying coefficients (TVAR process). In the literature, roots are evaluated at discrete times, rather than a continuous function like we have for time varying systems. By introducing the assumption that the movement of the roots are linear functions in time, stationarity for all possible epochs in the time domain is easy to accomplish. We will illustrate how this assumption leads to TVAR coefficients where the k-th coefficient is a polynomial of order k with further restrictions on the parameters of the coefficients. At first we study how to estimate TVAR process parameters by using a Least Squares approach in general. As any AR process can be rewritten as a combination of AR processes of order two with two complex conjugated roots and AR processes of order one, we limit our investigations to these orders. Higher order TVAR processes are computed by successively estimating TVAR processes of orders one or two. Based on a simulation, we will demonstrate the advantages of a time varying model and compare them to the stationary time stable model. In addition, we will give a method to identify time series, for which the model of the TVAR processes with linear roots is suitable.

In time series analyses, covariance modeling is an essential part of stochastic methods such as prediction or filtering. For practical use, general families of covariance functions with large flexibilities are necessary to model complex correlations structures such as negative correlations. Thus, families of covariance functions should be as versatile as possible by including a high variety of basis functions. Another drawback of some common covariance models is that they can be parameterized in a way such that they do not allow all parameters to vary. In this work, we elaborate on the affiliation of several established covariance functions such as exponential, Matérn-type, and damped oscillating functions to the general class of covariance functions defined by autoregressive moving average (ARMA) processes. Furthermore, we present advanced limit cases that also belong to this class and enable a higher variability of the shape parameters and, consequently, the representable covariance functions. For prediction tasks in applications with spatial data, the covariance function must be positive semi-definite in the respective domain. We provide conditions for the shape parameters that need to be fulfilled for positive semi-definiteness of the covariance function in higher input dimensions.

Through inverse modeling and adjustment techniques, the geodesists try to derive mathematical models from their measurements to get a better understanding of various processes in the system Earth. Sophisticated deterministic and stochastic models are developed to achieve the best possible reflection of reality and the remaining uncertainty.The main focus of this article is on the further development of stochastic model representations, with the capability to switch from the usual assumption of homogeneous (time-stationary) to inhomogeneous (time-variable) stochastic models. To accomplish this we build up and extend a methodical framework to connect the filter and the covariance approach represented by time-variable autoregressive processes (AR) and time-variable (inhomogeneous) covariance models for least squares collocation.We apply these time-variable covariance models to describe the temporal component of a spatio-temporal point stack of surface displacements derived from a DInSAR-SBAS analysis of the ERS1 and ERS2 missions from the Lower-Rhine Embayment in North Rhine-Westphalia. The construction of a time-variable spatio-temporal covariance model allows to use the least squares collocation approach to predict the vertical movements at any location and at any time. Furthermore, a report on the uncertainty of the prediction is provided.

In time series analyses, the auto-regressive (AR) modelling of zero mean data is widely used for system identification, signal decorrelation, detection of outliers and forecasting. An AR process of order p is uniquely defined by p coefficients and the variance in the noise. The roots of the characteristic polynomial can be used as an alternative parametrization of the coefficients, which can be used to construct a continuous covariance function of the AR process or to verify that the AR process is stationary. In a previous study, we introduced an AR process of time variable coefficients (TVAR process) in which the movement of the roots was specified as a polynomial of order one. Until now, this method was analytically derived only for TVAR processes of orders one and two. Thus, higher-level processes had to be assembled by the successive estimation of these process orders. In this contribution, the analytical solution for a TVAR(3) process is derived and compared to the successive estimation using a TVAR(1) and TVAR(2) process. We will apply the proposed approach to a GNSS time series and compare the best-fit TVAR(3) process with the best-fit composition of TVAR(2) and TVAR(1) process.

In this paper, we want to find a continuous function fitting through the discrete covariance sequence generated by a stationary AR process. This function can be determined as soon as the Yule–Walker equations are found. The procedure consists of two steps. At first the inverse zeros of the characteristic polynomial of the AR process must be fixed. The second step is based on the fact that an AR process can also be seen as a difference equation. By solving this difference equation, it is possible to determine a class of functions from which a candidate for a continuous covariance function can be determined. To analyze if this function is applicable as a positive definite covariance function, it is analyzed mathematically in view of the power spectral density compared to the characteristics of the power spectral density for the discrete covariances. Then it is shown that this function is positive semi-definite. At the end, a simulation of a stationary AR(3) process is elaborated to illustrate the derived properties.