After it was found that the gravity gradients observed by the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite could be significantly improved by an advanced calibration, a reprocessing project for the entire mission data set was initiated by ESA and performed by the GOCE High-level processing facility (GOCE HPF). One part of the activity was delivering the gravity field solutions, where the improved level 1b and level 2 data serve as an input for global gravity field recovery. One well-established approach for the analysis of GOCE observations is the so-called time-wise approach. Basic characteristics of the GOCE time-wise solutions is that only GOCE observations are included to remain independent of any other gravity field observables and that emphasis is put on the stochastic modeling of the observations’ uncertainties. As a consequence, the time-wise solutions provide a GOCE-only model and a realistic uncertainty description of the model in terms of the full covariance matrix of the model coefficients. Within this contribution, we review the GOCE time-wise approach and discuss the impact of the improved data and modeling applied in the computation of the new GO_CONS_EGM_TIM_RL06 solution. The model reflects the Earth’s static gravity field as observed by the GOCE satellite during its operation. As nearly all global gravity field models, it is represented as a spherical harmonic expansion, with maximum degree 300. The characteristics of the model and the contributing data are presented, and the internal consistency is demonstrated. The updated solution nicely meets the official GOCE mission requirements with a global mean accuracy of about 2 cm in terms of geoid height and 0.6 mGal in terms of gravity anomalies at ESA’s target spatial resolution of 100 km. Compared to its RL05 predecessor, three kinds of improvements are shown, i.e., (1) the mean global accuracy increases by 10–25%, (2) a more realistic uncertainty description and (3) a local reduction of systematic errors in the order of centimeters.

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.

In time series analysis, signal covariance modeling is an essential part of stochastic methods like prediction or filtering. In geodetic applications, covariance functions are rarely treated as true compactly supported functions although large amounts of data would approve such. Covariance models for complex correlation shapes are also rare. Ideally, general families of covariance functions with a large flexibility are desirable to model complex correlations structures like negative correlations. In this paper, we derive isotropic finite covariance functions that are parametrized in a way that positive definiteness is guaranteed. These are based on cutoff polynomials which are derived from operations such as autoconvolution and autocorrelation. Next to the compact support, the resulting autocovariance models share the advantages of (a) positive definiteness by design, (b) extensibility to arbitrary orders and (c) extensive flexibility by employing multiple tunable shape parameters. All these realize various correlation shapes such as negative correlations (the so-called hole effect) and several oscillations. The methodological concepts are derived for homogeneous and isotropic random fields in $$\mathbb{R}^d$$ R d . The family of covariance functions is then derived for one-dimensional applications. A data example demonstrates the covariance modeling approach using stationary time-series data.