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Kollokation nach kleinsten Quadraten bei nichtstationären stochastischen Prozessen --- NonStopLSC

Fachliche Zuordnung Geodäsie, Photogrammetrie, Fernerkundung, Geoinformatik, Kartographie
Förderung Förderung von 2019 bis 2023
Projektkennung Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 435703911
 
Erstellungsjahr 2024

Zusammenfassung der Projektergebnisse

By using autonomous sensors to record geodetic data sets, more and more information can be obtained from the data sets to be evaluated. In many cases, different effects in the data are superimposed on each other and separating this information is not easy. Non-parametric methods are often used for analysis and evaluation, although stochastic modeling is often preferred. Different approaches are possible here: access via covariance functions and Least Squares Collocation (LSC) or digital filters. The flexibility of the covariance approach with respect to any sample distribution and finite sequences is often compensated by an inadequate modeling of the process characteristics with empirical covariances. Instead access via autoregressive processes (AR) allows to describe the entire signal behavior, not only for stationary processes but also for timevariable systems. The focus of this project is on stationary and nonstationary stochastic univariate processes. We build up and extent a methodical framework to connect the filter and the covariance approach. The majority of the signal content is captured by modeling AR processes and fed into the collocation model through the transition to covariance functions. Timevariable changes in the time series are also transferred to timevariable covariance functions and thus the stationary collocation method is extended to deal with timevariable characteristics. First, the transition from discrete-time covariance sequences to continuous-time covariance functions is studied. Starting from AR processes, the transition to a homogeneous linear difference equation with constant coefficients is achieved using the Yule-Walker equations. The linearly independent basis functions of the general solution of the differential equation are analyzed and examined for their suitability as covariance functions. From the Fourier transform of the basis functions for single and multiple zeros of the characteristic polynomial, areas of validity can be derived where the positive definiteness of the basis functions is guaranteed. This defines a family of continuous covariance functions that can map discrete-time AR processes. Using a least squares adjustment with inequality restrictions, the special solution of the differential equations can now be calculated from the discrete covariance sequence and introduced as a covariance model into collocation. In order to better understand timevariable processes, the transition between the timevariable coefficients of the AR processes (TVAR) and the movements of the roots of the characteristic polynomial is examined in more detail. By restricting to linear root movements, an estimation method for the parameters can be developed, ensuring local stationarity for the entire time domain. Corresponding conditions for TVAR(1), TVAR(2) and TVAR(3) processes are derived. For higher process orders, cascaded filters are investigated. For the TVAR(1) process, the transition to continuous covariance functions is carried out and an analytical representation for the timevariable covariance function is developed. In addition to numerical simulations, the developed methods are specifically applied to GNSS time series of water level observations and to the evaluation of surface movements from a DInSAR point stack.

Projektbezogene Publikationen (Auswahl)

 
 

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