Theoretical Geodesy Group

Design of frequency selective filters for non-equispaced data

2024-03-26T11:38:37Z

Design of frequency selective filters for non-equispaced data Schuh, Wolf-Dieter; Franken, Jessica; Brockmann, Jan Martin; Esch, Christina; Köhler, Joël; Gutjahr, Karlheinz Modern sensors and satellite missions deliver huge data sets and long time series of observations. In the most cases equispaced time series are collected. Filtering of time series is a standard task in data analysis. Therefore a large variety of methods is already established to deal with equispaced time series. Discrete digital filters can be individually designed in the time as well as in the frequency domain. Often a priori information in the frequency domain is used to extract the signal of interest from the remaining part. These well established strategies presume equispaced samples to get access to the frequency domain behaviour. In this article we focus our attention on non-equispaced time series, which often appear in satellite geodesy and remote sensing. Again we assume that a special behaviour in the frequency domain characterizes the desired part of the signal. To extract exactly this part of the signal we construct frequency limited base functions, which are able to provide a strict cut-off in the frequency domain. A linear combination of these base functions can be applied in an approximation procedure. Thus, it is possible to extract the desired band-limited signal also for non-equispaced data sets. Our article is focused on the construction of these frequency limited base functions. Starting with piecewise given polynomial base functions with finite support (Splines), we construct by the inversion of an infinite band Toeplitz system tailored base functions, which are mutually independent (sampling splines). These sampling splines can be transformed into the spectral domain and because of the finite support of the original base functions a closed form representation in the spectral domain is possible (finite sums). Detailed studies on the filter characteristic in the frequency domain allows then a simple access to design low pass filters for non-equispaced data with a predefined passband region. It can be seen that the order of the splines controls the transition width as well as the stopband attenuation. Some practical examples demonstrate the capability of this approach in practice.

Modeling of inhomogeneous spatio-temporal signals by least squares collocation

2024-03-26T11:33:32Z

Modeling of inhomogeneous spatio-temporal signals by least squares collocation Schuh, Wolf-Dieter; Korte, Johannes; Schubert, Till; Brockmann, Jan Martin 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. While deterministic modeling has been improved by much effort, there are still serious weaknesses in the applied stochastic models and representations. Especially in the collocation approach a remove-restore technique is often used to, on the one hand, guarantee homogeneity and, on the other, get better access to the different frequency contents, which are often hidden in the empirical covariance sequence. The main focus is on the further development of stochastic model representations, which can reflect the full signal content and have 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 extent a methodical framework to connect the filter and the covariance approach represented by autoregressive processes (AR) and least squares collocation. The term time-variable is often used differently. Following (Priestley, 1989, Sec. 6.1), time- variable processes can be subdivided into models with a deterministic trend (polynomial or seasonal) and explosive AR processes with roots of the characteristic polynomial outside the unit circle. Here we want to study an other type of non-stationary processes, where the coefficients of an AR-process are variable in time, but does not violate the condition for the roots inside the unit circle. In particular, we examine the time-variable AR-process of first order (AR(1)-process) and put special focus on the inhomogeneity of the first and second central moment, which represents inhomogeneuous covariance relations. We apply these inhomogeneuous covariances to model the temporal component of a spatio-temporal point stack derived from a DInSAR-SBAS analysis of the ERS1 and ERS2 missions. The test region is the Lower-Rhine Embayment in North Rhine-Westphalia, Germany with the still active open-cast mines Garzweiler, Hambach and Inden and the already closed coal mines Sophie-Jacoba in the mining region Erkelenz and Emil Mayrisch in the mining region Aachen. The construction of a time-variable spatio-temporal covariance model allows to use the least squares collocation approach to estimate the vertical movements at any place and at any time and provide a report on the uncertainty of this estimation.

Coestimating long-term temporal signals to reduce the aliasing effect in parametric geodetic mean dynamic topography estimation

2022-06-20T13:00:18Z

Coestimating long-term temporal signals to reduce the aliasing effect in parametric geodetic mean dynamic topography estimation Brockmann, Jan Martin; Borlinghaus, Moritz; Neyers, Christian; Schuh, Wolf-Dieter 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 the along-track altimetric SSH observations together with a proper stochastic model, we apply a parametric approach, where a C1-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. To allow for 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 altimetry missions is implicitly performed by the adjustment due to the regular temporal sampling, aliasing is introduced for the geodetic missions, because of the missing (or at least very long) repeat characteristics. In this contribution, we will summarize the parametric approach used, with a focus on co-estimation of long-term temporal sea level variations. Regularization strategies are applied to derive stable and smooth estimates. It is studied how the additional spatio-temporal model component, e.g. linear trends and seasonal signals, reduces the aliasing problem and influences the estimate of the geodetic MDT.

Towards the Development of a Continuous Spatio-Temporal Finite Element Based Representation of the Mean Sea Surface

2022-02-02T17:00:18Z

Towards the Development of a Continuous Spatio-Temporal Finite Element Based Representation of the Mean Sea Surface Borlinghaus, Moritz; Neyers, Christian; Brockmann, Jan Martin The mean sea surface 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 spatio-temporal mean sea surface 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 C¹-smooth finite element space to represent the mean sea surface. 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 mean sea surface, the temporal component is used to absorb the ocean variability.
Within a proof-of-concept study 10 years of satellite altimetry from CryoSat–2 and Jason 1–3 over the period 2010 to 2019 are used and analyzed in two study regions with different spatial resolutions. Besides the static mean sea surface, the temporal component which is estimated covers a linear trend and the annual period. The comparison of the static component to the global CNES_CLS15 MSS shows a reasonable agreement with a root mean square error below 10 cm over the entire North Atlantic. But still, systematic differences occur especially in regions with a high temporal variability. Comparisons of the temporal component with gridded sea level anomaly products show a good agreement in areas of low ocean variability, but highlights that in regions of large ocean variability the temporal basis function chosen in this initial study must be further investigated. In general, it is demonstrated that the proposed approach can be an alternative to the well established mean sea surface estimation procedures.