NATIONAL REPORT OF THE FEDERAL REPUBLIC OF ... - IAG Office

H. Kutterer, W.-D. Schuh: Quality Measures and Control (Stochastic and Non-Stochastic Methods of Data Evaluation) 161
Typical time series in engineering geodesy comprise
instationary components caused by varying external forces.
An extension of standard modeling and analysis techniques
has been developed by NEUNER and KUTTERER (2006) and
NEUNER (2007). It is based on Wavelet transforms and
statistical tests and focuses on irregularities of mean and
variance of the time series.
A compilation of filtering and related techniques for
applications in geodesy was given by TEUSCH (2004).
EICHHORN (2005) developed an adaptive Kalman filter for
dynamic structural models and applied it to the modeling
of the deformations of a steel cylinder induced by heat flow.
In geometric space-geodetic applications the stochastic
models of the observations are typically formulated in a
straightforward way. Actually, only diagonal matrices are
used for the original observations. In publications such as
BISCHOFF et al. (2005, 2006) and HOWIND (2006) a procedure
is presented which allows formulating and empirically
deriving a stochastic model for GPS observations which
is rigorously based on filtering techniques and statistical
tests for the identification of homoscedastic sequences in
the observation residual time series. SCHÖN and BRUNNER
(2006) present a physically meaningful approach for the
modeling of correlations of GPS phase observations.
TESMER (2004) and TESMER and KUTTERER (2004) used
the MINQUE approach for the estimation of variance and
covariance components of VLBI data.
Model Misspecification and Hypothesis Testing
Many geodetic testing problems concerning parametric
hypotheses may be formulated within the framework of
testing the validity of a set of linear constraints imposed
to a linear Gauss-Markov model. It is then usually argued
that a reasonable test statistic should be based on the ratio
of the variance factor estimated from the constraints and
the variance factor estimated under the unconstrained
Gauss-Markov model. Although this procedure is computationally
convenient and intuitively sound, no rigorous
attempt has been made yet to establish optimality with
respect to its power function. Another shortcoming of
current geodetic theory has been so far that no rigorous but
convenient approach exists for tackling testing problems
concerning, for instance, parameters within the weight
matrix.
To address these problems, it was proven in KARGOLL
(2007) that under the assumption of normally distributed
observation various geodetic standard tests, such as
Baarda’s or Pope’s test for outliers, multivariate significance
tests or tests concerning the specification of the a
priori variance factor, are uniformly most powerful (UMP)
within the class of invariant tests. The main characteristic
of an invariant test lies in the fact that its power function
exhibits certain symmetries with respect to the parameter
domain, which is a reasonable assumption as long as no
information about the parameters is available a priori. UMP
invariant tests were also shown to be generally equivalent
to likelihood ratio tests and Rao’s Score tests. The latter
have the advantage that they do not require the parameter
estimates under the unconstrained model, which is con-
venient if the constraints set parameter values equal to zero.
It was shown that the outlier tests mentioned above, being
functions of the residuals of a constrained Gauss-Markov
model, are in fact particular cases of Rao’s Score test, and
that also other standard tests may be easily transformed into
that form.
Finally, testing problems concerning parameters within the
weight matrix such as autoregressive correlation parameters
or overlapping variance components were addressed. It was
shown that, although strictly optimal tests do not exist in
that case, corresponding tests based on Rao’s Score statistic
are reasonable and computationally convenient diagnostic
tools for deciding whether such parameters are significant
or not, without requiring the estimation thereof. The current
thesis by KARGOLL (2007) concluded with the derivation
of the Jarque-Bera test of normality as another application
of Rao’s Score test, which is useful to check the validity
of the normality assumption presupposed in the aforementioned
tests.
Numerical Simulation – Monte Carlo Methods
The Gibbs sampler of the Markov Chain Monte Carlo
methods was applied to compute large covariance matrices
and to propagate them to the estimated parameters (GUND-
LICH et al., 2003). Covariance matrices of quantities obtained
by linear and nonlinear transformations of estimated
parameters can be directly obtained by this method without
determining the covariance matrix of the estimated parameters
thus saving a considerable amount of computation
time. The Gibbs sampler is well suited for parallel computing
so that this algorithm for computing covariance
matrices was implemented on a parallel computer (KOCH
et al., 2004). The method was applied to determine the
maximum degree of harmonic coefficients in a geopotential
model by hypothesis tests. Random variates for the harmonic
coefficients were nonlinearly transformed to random
values of quantities used for the hypothesis tests (KOCH,
2005). The Gibbs sampler was also used for the Bayesian
reconstruction of digital three-dimensional images of
computer tomography. Since the posterior density function
for the intensities of the voxels was intractable, the Gibbs
sampler by means of sampling-importance-resampling was
applied (KOCH, 2005, 2006, 2007a). A review of the
Markov Chain Monte Carlo methods, the Gibbs sampler
and the sampling-importance-resampling algorithm can be
found in KOCH (2007b).
The approach of GUNDLICH et al. (2003) requires a fully
populated normal equation matrix, which is not available
in iterative solvers. As a solution to this problem, an
alternative way to compute the variance-covariance information
by Monte Carlo integration was presented in
ALKHATIB and SCHUH (2007). The proposed variancecovariance
estimation procedure is flexible and may be
integrated into many types of solvers such as sparse solvers,
parallel direct solvers or iterative solvers. These algorithms
were applied in ALKHATIB (2007) to simulated GOCE data,
where Satellite Gravity Gradiometry (SGG) and Satellite-to-
Satellite Tracking (SST) data observations are combined
for recovering the Earth's gravity field.