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Message:

> Hello,
> 
> 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 [i]free[/i] (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 [i]danger circle[/i] 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

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