Adjustment models for multivariate geodetic time series with vector-autoregressive errors

Boris Kargoll,Hamza Alkhatib,Alexander Dorndorf,Mohammad Omidalizarandi,Jens-André Paffenholz

doi:10.1515/jag-2021-0013

Abstract

Abstract In this contribution, a vector-autoregressive (VAR) process with multivariate t-distributed random deviations is incorporated into the Gauss-Helmert model (GHM), resulting in an innovative adjustment model. This model is versatile since it allows for a wide range of functional models, unknown forms of auto- and cross-correlations, and outlier patterns. Subsequently, a computationally convenient iteratively reweighted least squares method based on an expectation maximization algorithm is derived in order to estimate the parameters of the functional model, the unknown coefficients of the VAR process, the cofactor matrix, and the degree of freedom of the t-distribution. The proposed method is validated in terms of its estimation bias and convergence behavior by means of a Monte Carlo simulation based on a GHM of a circle in two dimensions. The methodology is applied in two different fields of application within engineering geodesy: In the first scenario, the offset and linear drift of a noisy accelerometer are estimated based on a Gauss-Markov model with VAR and multivariate t-distributed errors, as a special case of the proposed GHM. In the second scenario real laser tracker measurements with outliers are adjusted to estimate the parameters of a sphere employing the proposed GHM with VAR and multivariate t-distributed errors. For both scenarios the estimated parameters of the fitted VAR model and multivariate t-distribution are analyzed for evidence of auto- or cross-correlations and deviation from a normal distribution regarding the measurement noise.

Highlights

Geodetic observation models of surveyed phenomena often include unknown quantities in terms of parameters to be estimated by adjustment computations
This model consists of condition equations and of a stochastic model
We applied and investigated, in the context of multivariate time series, a more manageable type of stochastic model defined by a vector autoregressive (VAR) process

Summary

Introduction

Geodetic observation models of surveyed phenomena often include unknown quantities in terms of parameters to be estimated by adjustment computations. The most general structure of observation model is described by nonlinear condition equations in which multiple observations and unknown parameters may be linked to each other [1] This general case of adjustment calculus was called GaussHelmert model (GHM) by Wolf [2]. G., the errors-in-variables (EIV) model adjusted by the method of total least squares [3] or by its various generalizations [4, 5, 6, 7] Another special case of the GHM is the Gauss-Markov model (GMM), in which the observations are still linked to functional model parameters but occur in separate equations [8, 9, 10]. Since adjustment calculus has been elaborated on the basis of matrix calculus, variance-covariance information regarding the observables is usually arranged as a matrix, called the “variance-covariance matrix” (VCM)

Results

Discussion

Conclusion