Hello Krzysztof,

1. If I well understand, the global congruence test for the detection of datum point displacement(s) in JAG3D is based on the following quadratic form: omega = vT*P*v

Yes, you are right. The functional model used in JAG3D is given in your Eq. 1. It is a normal network adjustment. The residuals of the (joint) model are used for the global test. This global test checks the compatibility of the functional (and the stochastic) model - as usual in a normal network adjustment.

If datum points are unstable, the functional model is misspecified, and - depending on the critical value - the test is rejected. The reason of the rejected test cannot be identified, and this step is called detection.

Let us briefly review the "normal" outlier test statistic, i.e.,

$T_{prio} = \frac{\mathbf{\nabla^T Q_{\nabla\nabla}^{-} \nabla}}{m \sigma_0^2}$

where $\mathbf{Q_{\nabla\nabla}} = ( \mathbf{B^T P Q_{vv} P B} )^{-}$ and $\nabla = \mathbf{Q_{\nabla\nabla} B^TPv}$. Matrix $\mathbf{B}$ is the design matrix of the extended model, and $m = rg(\mathbf{Q_{\nabla\nabla}})$. We agree that this test statistic based on the likelihood ratio test, and Baards w-test is a special case of $T_{prio}$.

If we set $\mathbf{B = I}$, we obtain

$\mathbf{B^T P Q_{vv} P B} = \mathbf{P Q_{vv} P}$,

and $\mathbf{Q_{\nabla\nabla}} = ( \mathbf{P Q_{vv} P} )^{-}$ as well as $\nabla = -( \mathbf{P Q_{vv} P} )^{-} \mathbf{Pv}$.

Inserting these expresions yields the test statistic

$T_{prio} = \frac{\mathbf{v^T P} ( \mathbf{P Q_{vv} P} )^{-} ( \mathbf{P Q_{vv} P} ) ( \mathbf{P Q_{vv} P} )^{-} \mathbf{Pv} }{m \sigma_0^2} = \frac{\mathbf{v^T P} ( \mathbf{P Q_{vv} P} )^{-} \mathbf{Pv} }{m \sigma_0^2}$

Since $\mathbf{v} = -\mathbf{Q_{vv}Pl}$, where $\mathbf{l}$ is the observation vector, the test statistic reads

$T_{prio}= \frac{\mathbf{l^T P Q_{vv} P l}}{m \sigma_0^2} = \frac{\mathbf{v^T P v}}{m \sigma_0^2}$

because $\mathbf{Q_{vv}P Q_{vv}P = Q_{vv}P}$ and $m = f = n-u+d$

$T_{prio}$ is the global test (based on the likelihood ratio test). Your suggested test based on the likelihood ratio test, thats right, but this is also the case for the global test used in the network adjustment.

In other words, the observations are not here completely free as in the model from Eq. (2). In consequence, if the group of congruence/datum points included unstable (non-detected) point(s), the variance factor – which is estimated from the congruence model – would be biased.

Yes, you are right. But: Do you really test/check the object points without having a stable reference network (datum points)? I don't think so. Usually, one checks for a stable reference (sub-)network (a stable datum) first. If such a stable network is identified(!), object points or other deformation parameters are evaluated afterwards. In this case, the variance factor is (almost) unbiased.

Such a solution is also presented in literature known to me.

Yes, I know this literature; but the model implemented in JAG3D is also known in the geodetic literature, cf. Jäger et al (2005). By the way, your suggested/preferred model is also given in the textbook written by Jäger et al., pp. 270ff.

Kind regards
Micha