In geodetic applications, stochastic processes are often used in the context of least squares collocation for prediction and filtering or in the context of least squares estimation for efficient decorrelation of signals. The biggest advantage of collocation method is that the covariance function, which serves as basis function, can be derived directly from the characteristics of the signals, meaning it is learned directly from the data. However, the major disadvantage is that memory requirements increase quadratically and the numerical complexity increases with sixth order of the number of data points. Especially with multidimensional data sets, large amounts of data are usually expected. To reduce numerical complexity, various strategies are employed, e.g., utilizing finite covariance functions and filter techniques for gridded data. But when processing spatio-temporal models, data points are often irregularly distributed in space, whereas repeated measurements are taken at fixed epochs. Specifically for such kind of data sets, an efficient approach is proposed. Key idea is the utilization of separable covariance functions, where the covariance matrices are represented as a Kronecker product. This enables a successive solution in space and time and reduces the numerical complexity drastically. This strategy is used to evaluate a spatio-temporal point stack of surface displacements for the period 1992 to 2000, derived from a DInSAR-SBAS analysis of the ERS1 and ERS2 missions in the Lower-Rhine Bay in North Rhine-Westphalia by least squares collocation.

Deriving trend functions from observed time series is one of the main tasks in the field of signal processing. Separating the observation into a deterministic function, a stochastic residual signal and a stochastic noise component using dedicated model representations enables further study of these individual components, e.g. screening for climate signals. Whereas the deterministic part is modelled as a linear combination of basis functions, the use of autoregressive processes to model the noise and signal is proposed. Within an iterative estimation scheme, the uncertainty information of the observed variables is properly modelled and carefully propagated to the resulting parameters. This enables the use of statistical testing and Least-Squares Collocation in further investigations of the separated signal components. In this study, the proposed iterative procedure is applied to relatively short total water storage time series derived from measurements of the satellite mission GRACE. The trend in total water storage is, for instance, relevant for climate studies, identifying regions getting drier or wetter. Accounting for the aforementioned covariance information based on autoregressive processes allows the use of Hypothesis tests to identify regions with significant trends. On the contrary, the smoothed stochastic signal components are required to identify extreme/anomalous events like floods and droughts in the observed time series. Additionally, to improve the estimation of the stochastic signal, data from numerical models are used to estimate the process characteristics.

We analyze the feasibility of so-called radial surface currents as derived from Sentinel-1 Wave mode Doppler shift observations to support the geodetic estimation of the mean dynamic topography (MDT). Broadly defined as subtracting the geoid from the mean sea surface, this separation problem suffers from data inhomogeneities and usually requires strong prior assumptions about the MDT’s smoothness. Recent advancements in calibration and current retrieval methodology for Sentinel-1 data give reason to assess potential gains from this observation type, which can be linked to the gradient of the MDT under geostrophic approximation. Due to current lack of long time-series, we synthesize 10 years of observations from daily surface currents grids and include these in a parametric least-squares adjustment of the MDT. Our results utilizing the synthetic data in place of smoothness constraints are promising, showing 2cm RMS agreement with a state-of-the-art MDT model.

The geodetic estimation of the mean dynamic ocean topography (MDT) as the difference between the mean of the sea surface and the geoid remains, despite the simple relation, still a difficult task. Mainly, the spectral inconsistency between the available altimetric sea surface height (SSH) observations and the geoid information causes problems in the separation process of the spatially and temporally averaged SSH into geoid and MDT. This is aggravated by the accuracy characteristics of the satellite derived geoid information, as it is only sufficiently accurate for a resolution of about 100 km. To enable the direct use of along-track altimetric SSH observations, we apply a parametric approach, where a C^1 -smooth finite element space is used to model the MDT and spherical harmonics to model the geoid. Combining observation equations for altimetric SSH observations with gravity field normal equations assembled from dedicated gravity field missions in a least-squares adjustment, allows for a joint estimation of both – i.e. the MDT and an improved geoid. In order to enable temporal averaging and to obtain a proper spatial resolution, satellite altimetry missions with an exact repeat cycle are combined with geodetic missions. Whereas the temporal averaging for the exact repeat missions is implicitly performed due to the regular temporal sampling, aliasing is introduced for the geodetic missions, because of the missing repeat characteristics. In this contribution, we will summarise the used approach and introduce the coestimation of long-term temporal sea level variations. It is studied how the additional spatio-temporal model component, i.e. linear trends and seasonal signals, reduces the aliasing problem and influences the estimate of the MDT and the geoid.

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.

The mean sea surface (MSS) is an important reference surface for oceanographic or geodetic applications such as sea level studies or the geodetic determination of the steady-state ocean circulation. Models of the MSS are derived from averaged along-track radar altimetry which provides instantaneous measurements of the sea surface heights (SSH). SSH observations corrected for tides and other physical signals and can be modeled as the sum of the MSS and sea level anomalies (SLA) which describe the temporal variability of the ocean. The typical MSS products are defined as grids of heights at a specific reference epoch and result from spatial and temporal prediction and filtering of the along-track SSH observations, whereas SLA products are computed with respect to an MSS model and are also defined as e.g. daily or averaged monthly grids. In this contribution a one-step least-squares approach is used to estimate a continuous spatio-temporal model of the MSS and filtered SLAs from along-track altimetric SSH measurements using C^1 -smooth finite element spaces for the spatial representation. The finite elements are defined on triangulations with different edge lengths and, thus, different spatial resolutions for MSS and SLA modeling. To model the temporal ocean variability finite B-Splines base functions are combined with the spatial finite elements to construct a spatio-temporal model. This contribution presents a concept to adapt the triangulations to the spatial characteristics of the signal of the MSS and SLA in a study region south of Africa. Least-squares residuals are studied to detect areas which show unmodeled spatial signal. These serve as input for the refinement of the triangulation. The results show that the residuals are indeed a good indicator for unmodeled signal, but as they are significantly influenced by unmodeled temporal signals as well, the refinement has only a small local impact on the obtained MSS and SLA models.

The mean sea surface (MSS) has an important role, both, in the calculation of the mean dynamic topography and in the area of sea-level change as a reference surface. This paper presents a new approach to estimate a continuous spatiotemporal MSS from along-track altimetric sea surface height measurements. A parametric function continuously defined in the spatial as well as temporal domain is constructed from a $$C^1$$ -smooth finite element space to represent the MSS. Least-squares observation equations are set up, to estimate the unknown scaling coefficients from the sea surface height measurements as collected by altimetric exact repeat missions and geodetic missions. An advantage of the proposed method is that the surface is represented by an analytic model and the unknown parameters can be physically interpreted. Whereas the static component of the function represents the MSS, the temporal component is used to absorb the ocean variability. Within this initial study, 10 years of satellite altimetry over the period 2010–2019 are analyzed in a small study region around the Agulhas Current. To obtain the best possible data coverage, all missions available via AVISO, which provide data in the study region and period, are included, i.e., geodetic missions and mission phases as well as exact repeat missions. Besides the static MSS, the temporal component, which is co-estimated to absorb the dominating ocean variability, is modeled with different basis functions to study their performance. On the one hand, global basis functions considering a linear trend and periodic functions are compared with B-Spline basis functions. The comparison of the static component to the global CNES_CLS15 MSS shows a reasonable agreement with a root-mean-square error in the range of 1–4 cm for the well-suited model configurations. To validate the modeling approach and the different analyzed configurations, the temporal model component is compared to gridded sea-level anomaly products. Although it is not (yet) a target quantity, the analysis can serve as a quality check of the MSS and the proposed modeling approach as well. It is shown that in regions with relatively low ocean variability the combination of a linear trend with an annual period is well suited to model the dominant temporal signal, whereas it is not sufficient in regions with strong ocean variability, e.g., close to the Agulhas Current. In those regions, the scenario which utilizes B-Splines in the temporal domain performs significantly better. In general, it is demonstrated that the proposed approach can be an alternative to the well-established MSS estimation procedures.

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 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 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.

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.

GOCO06s is the latest satellite-only global gravity field model computed by the GOCO (Gravity Observation Combination) project. It is based on over a billion observations acquired over 15 years from 19 satellites with different complementary observation principles. This combination of different measurement techniques is key in providing consistently high accuracy and best possible spatial resolution of the Earth's gravity field. The motivation for the new release was the availability of reprocessed observation data for the Gravity Recovery and Climate Experiment (GRACE) and Gravity field and steady-state Ocean Circulation Explorer (GOCE), updated background models, and substantial improvements in the processing chains of the individual contributions. Due to the long observation period, the model consists not only of a static gravity field, but comprises additionally modeled temporal variations. These are represented by time-variable spherical harmonic coefficients, using a deterministic model for a regularized trend and annual oscillation. The main focus within the GOCO combination process is on the proper handling of the stochastic behavior of the input data. Appropriate noise modeling for the observations used results in realistic accuracy information for the derived gravity field solution. This accuracy information, represented by the full variance–covariance matrix, is extremely useful for further combination with, for example, terrestrial gravity data and is published together with the solution. The primary model data consisting of potential coefficients representing Earth's static gravity field, together with secular and annual variations, are available on the International Centre for Global Earth Models (http://icgem.gfz-potsdam.de/, last access: 11 June 2020). This data set is identified with the following DOI: https://doi.org/10.5880/ICGEM.2019.002 (Kvas et al., 2019b). Supplementary material consisting of the full variance–covariance matrix of the static potential coefficients and estimated co-seismic mass changes is available at https://ifg.tugraz.at/GOCO (last access: 11 June 2020).

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.

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.

Digitale Sensoren liefern Zeitreihen von gleichabständigen und oft stark korrelierten Messungen. Eine strenge Auswertung, dieser zumeist umfangreichen Datensätze, in einem Kollokationsansatz stellt eine große Herausforderung dar. Standardverfahren sind nicht in der Lage solch große Systeme zu bewältigen. In diesem Artikel werden zwei unterschiedliche Verfahren – der Kovarianzansatz und der Filteransatz – einander gegenübergestellt, um deren Potential für den Einsatz bei sehr großen Datenmengen zu diskutieren. Im Kovarianzansatz erfolgt die Modellierung der Korrelationen durch Kovarianzfunktionen. Die Dekorrelation der Messungen erfolgt über die Faktorisierung der Kovarianzmatrizen, wofür unterschiedliche Varianten der Cholesky Faktorisierung untersucht werden. Dünn besetzte Systeme hervorgerufen durch den Einsatz von finiten Kovarianzfunktionen, regelmäßige Toeplitz Systeme resultierend aus der regelmäßigen Abtastung, aber auch die Auswirkungen von lokalen Datenverlusten (Datenlöcher) werden untersucht und maßgeschneiderte Zerlegungsverfahren entwickelt. Der Filteransatz bietet hingegen kaum Möglichkeiten flexibel auf Eigenheiten der Daten einzugehen. Für regelmäßige Abtastung von sehr langen Zeitreihen, ist die Dekorrelation über Filter extrem effizient. Durch die Modellierung der Kor- relationen durch finite, autoregressive Prozesse (AR-Prozesse) werden kausale, nichtrekursive, finite Filter aufgebaut, die unendlich ausgedehnte Meßreihen effizient dekorrelieren können. Datenlöcher, aber auch die Initialisierung des Filterprozesses bewirken aber einen Datenverlust, der speziell bei großen Korrelationslängen durchaus dramatisch sein kann. Messserien aus der Praxis bestehen aber aus endlich vielen Messungen, enthalten Unregelmäßigkeiten und Datenlöcher. Während der Kovarianzansatz ohne Probleme auf diese Daten anwendbar ist, sind beim effizienten Filteransatz spezielle Vorkehrungen (Näherungen) notwendig. Durch den Einsatz von rekursiv definierten Kovarianzsequenzen kann jedoch ein Übergang zwischen den beiden Ansätzen hergestellt werden. Die vollbesetzte Kovarianzmatrix kann mit Hilfe einer speziellen Variante der Cholesky Inversion – rekursives rückwärts Rändern – in eine dünnbesetzte Matrix zerlegt werden, die weitgehend eine Band-Toeplitz-Struktur aufweist, die dem kausalen, nichtrekursiven Filter im Filteransatz entspricht. Durch die Kombination der beiden Ansätze ist es somit möglich einen zeitvariablen, kausalen, nicht rekursiven Filter zu entwickeln, der endliche und leicht unregelmäßige Datensätze streng und effizient dekorrelieren kann.

Many geodetic measurements which are automatically gathered by sensors can be interpreted as a time series. For instance, measurements collected by a satellite platform along the satellite’s track can be seen as a time series along the orbit. Special treatment is required if the time series is contaminated by outliers or non-stationarities, summarized as ‘suspicious data’, stemming from sensor noise variations or changes in environment. Furthermore, the collected measurements are often – for instance due to the sensor design – correlated along the track.