Thank you very much for your time and help.

You are welcome. Let me know, if you like to compare results or something like that; and how I can support your investigations.

Have a nice weekend
Micha

All is clear and reasonable.
Thank you very much for your time and help.

Best regards,
Krzysztof

- In my opinion, please correct me if I am wrong, we may not test observations against possible blunders and points against possible displacements at the same stage.

Yes, I fully agree!

We should first test observations against blunders, based on the unconstrained model given by me in Eq. (2).

Yes, I agree. Estimating both epochs individually to check the data and to adapt the stochastic model is recommended.

Only then, we may test datum points against displacements, based, e.g., on the congruence/constrained model (implicit hypothesis method).

Yes, I agree. On this stage, we should test the datum points using explicit or implicit hypothesis tests or a similar technique.

In other words, we should perform these two testing procedures separately. Otherwise, possible blunder(s) and displacement(s) will mix and it will be difficult to identify and adapt them.

Yes, I fully agree. However, the test statistic

$T_{prio} = \frac{\mathbf{\nabla^T Q_{\nabla\nabla}^{-1} \nabla}}{m \sigma_0^2}$

used for detecting/identifying blunders is equivalent to the test statistic used for the point test. The only difference is the matrix $\mathbf{B}$, i.e., the (assumed) misspecified functional model of the null model. Misspecifying contains blunders, point shifts, missing parameters (e.g. zero point parameter of the EDM), a wrong functional equation etc. In my opinion, we only test against misspecifying - nothing more. Since powerful local tests are always performed in JAG3D, usually, I evaluate only these test statistics instead of the (weak) global test - in outlier detection as well as in deformation analysis.

- Please correct me if I am wrong, the global congruence test for the detection of datum point displacement(s) based on: vT*P*v - (viT*Pi*vi + vjT*Pj*vj) is more powerful, i.e., a given displacement will be detectable with a higher probability than by the test based on vT*P*v.

Yes, this should be correct. However, your suggested implicit test is not implemented in JAG3D, but powerful local tests are provided, which are specified as explicit hypothesis tests. The golden rule is: If a global test is rejected, we can conclude that something is wrong. Usually, we start with the localisation of the problem at this state. On the other hand, if the global test is not rejected, we cannot draw the conclusion that everything is fine; and again, we start with the (local) analysis.

Kind regards
Micha

But please let me put forward a few thoughts yet.

- In my opinion, please correct me if I am wrong, we may not test observations against possible blunders and points against possible displacements at the same stage. We should first test observations against blunders, based on the unconstrained model given by me in Eq. (2). Only then, we may test datum points against displacements, based, e.g., on the congruence/constrained model (implicit hypothesis method). In other words, we should perform these two testing procedures separately. Otherwise, possible blunder(s) and displacement(s) will mix and it will be difficult to identify and adapt them.

- Please correct me if I am wrong, the global congruence test for the detection of datum point displacement(s) based on: vT*P*v - (viT*Pi*vi + vjT*Pj*vj) is more powerful, i.e., a given displacement will be detectable with a higher probability than by the test based on vT*P*v. It is due to the number of degrees of freedom of global test statistic. The test statistic which is based on vT*P*v - (viT*Pi*vi + vjT*Pj*vj) has fewer degrees of freedom and the test is more powerful.

- You have written: "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."
Yes, we have unstable datum points in practice and we accept this fact. Test (including detection test) does not have a success rate = 100% (it is not possible). It is always a lower rate and it is called test power. Please note that the probability of Type II error (beta) is, in fact, the probability that some point displacement(s) will not be detected by test. In other words, if we have some true value of displacement, we can calculate beta, or conversely, the power of test (1 - beta). Of course, the higher the true value of displacement is, the lower beta is.
For example, if the beta is 20% for some displacement value, it means that 20 times out of 100 times we won't detect this displacement by test. In consequence, this displacement will be in the congruent/datum part of a model.

Best regards,
Krzysztof

PS. Unfortunately, I do not have a work of Jäger et al. (2005) and, hence, I cannot refer to this book.

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