Startseite The stochastic model for Global Navigation Satellite Systems and terrestrial laser scanning observations: A proposal to account for correlations in least squares adjustment
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The stochastic model for Global Navigation Satellite Systems and terrestrial laser scanning observations: A proposal to account for correlations in least squares adjustment

  • Gael Kermarrec ORCID logo EMAIL logo , Ingo Neumann , Hamza Alkhatib und Steffen Schön
Veröffentlicht/Copyright: 24. Januar 2019
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Abstract

The best unbiased estimates of unknown parameters in linear models have the smallest expected mean-squared errors as long as the residuals are weighted with their true variance–covariance matrix. As this condition is rarely met in real applications, the least-squares (LS) estimator is less trustworthy and the parameter precision is often overoptimistic, particularly when correlations are neglected. A careful description of the physical and mathematical relationships between the observations is, thus, necessary to reach a realistic solution and unbiased test statistics. Global Navigation Satellite Systems and terrestrial laser scanners (TLS) measurements show similarities and can be both processed in LS adjustments, either for positioning or deformation analysis. Thus, a parallel between stochastic models for Global Navigation Satellite Systems observations proposed previously in the case of correlations and functions for TLS range measurements based on intensity values can be drawn. This comparison paves the way for a simplified way to account for correlations for a use in LS adjustment.

The first order Gauss Markov process is often used to model the correlations of physical processes. The smoothness of the process is 1/2, i. e. the correlation function is not mean-square differentiable at the origin [58], so that the function decreases rapidly at the origin with time (or distance). It is, thus, a short memory process. The cofactor matrix reads:

QAR(1)=1ρρ2ρn1ρnρ1ρρn2ρn1ρ2ρ1ρn3ρn2ρn11ρρnρn1ρ1

An explicit inverse QAR(1)1 exists [52], provided that the autocorrelation coefficient ρ is known or estimated.

(10)QAR(1)1=11ρ21ρ000ρ1+ρ2ρ000ρ1+ρ20000001+ρ2ρ000ρ1

This greatly simplifies the computation of the equivalence matrix, since the sum of the elements can be computed directly, leading to:

QAR(1)ˍEQUI1=11ρ2×1ρ00000(1ρ)200000(1ρ)2000000(1ρ)2000001ρ

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Received: 2018-06-01
Accepted: 2019-01-08
Published Online: 2019-01-24
Published in Print: 2019-04-26

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