Thank you so much, this is exactly the kind of concrete, verifiable criterion I was looking for. The post-fit residual / redundancy contribution check, together with the standardized test statistics (T_prio/T_post), gives me a clear and practical way to validate whether a σ is actually acting as a proper quasi-fixed constraint, instead of relying on an arbitrary epsilon. And the laser-tracker/tape-measure analogy is a great way to put it, it really clarified the underlying logic for me.

Regarding the third point, please don't worry at all, I completely understand and I don't take it as brushing me off in the slightest. You've been extremely generous with your time and thoroughness throughout this thread. The remaining question is a domain/instrumentation matter to resolve by myself, not something left unsolved in JAG3D's model.

Thank you again for taking the time to think through all of this in such dept.

All the best,
Jan

Is there a sensible way to judge whether a given σ is "small enough" across observation types with different units/physical dimensions – for example, by comparing standardized residuals or weight contributions after a first adjustment run, rather than the raw σ values themselves? Or would you simply convert everything to a common measure (e.g. via the resulting positional effect) before comparing?

There are several criteria that can be used to check the strength of the pseudo-observation. If an observation is (quasi-)fixed, the related post-fit residual must be close to zero. Since precise measurements verify inaccurate ones (a laser tracker verifies the tape measure, but not the other way around), precise measurements exhibit only minimal redundancy. Thus, the redundancy should be very small. A (quasi-)fixed observation is (nearly) a perfect observation and is interpreted as the true value in a statistical sense. Thus, test statistics such as normalized residual tests (e.g. Tprio/Tpost) should be close to zero. Just evaluate e.g. the post-fit residual and adapt the corresponding weight - if required.

Attached two pics for better understanding the network.

Thank you. Although I now have a general idea of how the network is configured. However, I still can't quite picture the whole thing clearly enough (in my mind) to give a concrete answer to your third question from the last post. The implications aren't entirely clear to me. I don't have any 'best practices'-tips in this field - that's my problem (it is not a matter of the least-squares method). To get an appropriate answer to your last question, you should consult someone else who is more familiar with a gyro-theodolite. Don't get me wrong. I don't want to brush you off, and I'm happy to keep answering any other questions you have about JAG3D.

All the best
Micha

Thank you very much for taking the time to answer in such detail – the relative-scale criterion for choosing σ (small relative to the other observation uncertainties, rather than an absolute epsilon) is exactly the kind of practical guidance I was missing, and the danger-circle analogy makes the geometric-stability point very clear.

One follow-up, if you don't mind: the relative criterion is straightforward when comparing observations of the same type (e.g. distance vs. distance, as in your example). In my case the network mixes angles/directions (radians) with distances (meters) and, on top, a gyro azimuth that I want to treat as "almost fixed." Is there a sensible way to judge whether a given σ is "small enough" across observation types with different units/physical dimensions – for example, by comparing standardized residuals or weight contributions after a first adjustment run, rather than the raw σ values themselves? Or would you simply convert everything to a common measure (e.g. via the resulting positional effect) before comparing?

Attached two pics for better understanding the network.

Thanks again for your help!

I have to admit that I wasn't able to fully grasp the depth of the problem. My answer will therefore be somewhat general. In JAG3D, the a priori dispersion matrix of the observations must be positive definite. Thus, a zero variance (or zero standard deviation) is not allowed. To enforce a specific configuration, small variances (high weights) must be introduced to the stochastic model. To fix a specific measurement, its variance must be small compared to the variances of the other measurements - what matters is the relative ratio, not the absolute value. For example, if you want to fix a distance and all distances have an uncertainty of 10 mm, a standard deviation of 0.1 mm is sufficient to fix that distance adequately.

1) σ of the gyro direction observation → 0 (essentially a "hard constraint"):
- Does this effectively only fix the azimuth relationship between the start and end point, while both points otherwise remain free (translation, connection to the rest of the network)?

Yes, the points remain free (as long as the point type is not set as a e.g. reference point).

- Is there a known numerical lower bound for σ below which the normal equation system becomes unstable/singular?

No. Not only the magnitude of the value, but also the network configuration is critical to numerical stability. Think of a so-called danger circle during a three-point resection.

2) σ as a real, finite value (derived from an uncertainty budget combining several components):
- During the adjustment, how is the "tension" from this one direction observation distributed across the other observations connected to the start/end points? Does it depend purely on the weight/redundancy of the overall configuration, or is there any preferential treatment of the start point vs. the end point?

In general, the start and end points of a measurement are treated as equivalent; no distinction is made during the adjustment. Therefore, the configuration, the quality of the observations, and the weighting strategy are important for estimating the position of a point (regardless of whether the point is a station or a target point).

3) For the specific use case (gyro azimuth lines along a tunnel drive, some with and some without a forced connection to known points): from your perspective, what would be the methodically cleanest way to introduce such an observation – a highly weighted pseudo-observation with a very small σ, a real observation with a realistic σ, or rather a true constraint equation/datum point construction instead of a weighted observation?

I find it difficult to visualize this configuration well enough to make a precise statement. There may be exceptions (for example, if I need an orthogonality condition specified by an object), but generally speaking, I tend to assign a suitable measurement uncertainty to measurements. Your case may be an exception, but at this time I cannot reliably evaluate it.

Kind regards
Micha

first of all, apologies about this long post.

I'm working on an application that inserts a gyrotheodolite-derived azimuth (final result including its combined standard uncertainty u_Az) as an additional direction observation into an existing JAG3D project (observation group type "Direction", with the orientation unknown disabled so that the value acts as an absolute azimuth). The goal is to simulate how much a network (or a tunnel traverse chain) is tensioned or relaxed by this single additional observation.

What I'm not sure about is how JAG3D should, internally, most sensibly handle the two limiting cases:

1) σ of the gyro direction observation → 0 (essentially a "hard constraint"):
- Does this effectively only fix the azimuth relationship between the start and end point, while both points otherwise remain free (translation, connection to the rest of the network)? Or is there, in practice/in the source code, a different treatment (e.g. one of the two points effectively behaving like a datum point)?
- Is there a known numerical lower bound for σ below which the normal equation system becomes unstable/singular? I'm currently only guarding this with a small floor (e.g. 1e-12), but I'm not sure if that matches the approach you'd recommend.

2) σ as a real, finite value (derived from an uncertainty budget combining several components):
- During the adjustment, how is the "tension" from this one direction observation distributed across the other observations connected to the start/end points? Does it depend purely on the weight/redundancy of the overall configuration, or is there any preferential treatment of the start point vs. the end point?

3) For the specific use case (gyro azimuth lines along a tunnel drive, some with and some without a forced connection to known points): from your perspective, what would be the methodically cleanest way to introduce such an observation – a highly weighted pseudo-observation with a very small σ, a real observation with a realistic σ, or rather a true constraint equation/datum point construction instead of a weighted observation?

Thank you very much in advance for your input!

Best regards