Question 6 of 7: Network Post-analysis — Confidence Ellipses and Ellipsoids

Question 6: Network Post-analysis — Confidence Ellipses and Ellipsoids (20 marks — a 8, b 8, c 4)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A least-squares network adjustment yielding a covariance matrix for each station, from which error ellipses (2-D) or ellipsoids (3-D) are formed.

Find. (a) the definition and confidence level of the standard error ellipse/ellipsoid; (b) the meaning of out-of-context vs in-context (simultaneous) regions; (c) how to scale the standard ellipse to 95%.

(a) Standard error ellipse / ellipsoid. After adjustment, each station has a \(2\times2\) (or \(3\times3\)) covariance sub-matrix. The standard error ellipse is the region whose semi-axes are the square roots of the eigenvalues of that covariance matrix, oriented along its eigenvectors; the semi-major axis \(a\) lies in the direction of maximum positional uncertainty and the semi-minor axis \(b\) in the direction of minimum uncertainty. It is the “\(k=1\)” (one-sigma) ellipse. The standard error ellipsoid is the 3-D analogue, with three semi-axes from the eigenvalues of the \(3\times3\) covariance matrix. The confidence level they define is not the familiar 1-D 68% value: in 2-D the standard ellipse encloses only about 39.4% probability (\(1-e^{-1/2}\)), and in 3-D the standard ellipsoid encloses about 19.9%. These low percentages are exactly why a scaled (e.g. 95%) region is normally reported.

(b) Out-of-context vs in-context (simultaneous). An out-of-context (or one-at-a-time) ellipse describes the uncertainty of a single point considered in isolation, ignoring that many points were estimated together. An in-context (or simultaneous) ellipse accounts for the fact that we are making a confidence statement about all the network’s points (or parameters) at once: to keep the joint confidence at the stated level, the region must be enlarged. The in-context region is therefore always larger than the out-of-context one for the same confidence, because guarding against error in any of many parameters simultaneously demands a bigger multiplier: with the variance factor known, the out-of-context factor uses \(\chi^2\) with 2 (or 3) degrees of freedom, whereas the in-context factor uses \(\chi^2\) with degrees of freedom equal to the total number of coordinates \(u\) in the network (for an estimated variance factor the \(\chi^2\) value is replaced by the corresponding \(F\)-based value).

(c) From standard to 95% ellipse. The shape and orientation stay the same; only the size scales. Multiply both semi-axes of the standard ellipse by the factor \(c\), where \(c^2\) is the 95% quantile of the \(\chi^2\) distribution with 2 degrees of freedom:

$$c = \sqrt{\chi^2_{2,\,0.95}} = \sqrt{5.991} = \boxed{2.448}.$$

So the 95% error ellipse has semi-axes \(a_{95}=2.448\,a\) and \(b_{95}=2.448\,b\). For example, a standard ellipse with \(a = 12.0\ \text{mm}\), \(b = 7.0\ \text{mm}\) becomes \(a_{95}=29.4\ \text{mm}\), \(b_{95}=17.1\ \text{mm}\) at 95% confidence. (For a 3-D ellipsoid the corresponding factor is \(\sqrt{\chi^2_{3,\,0.95}}=\sqrt{7.815}=2.796\).) These are out-of-context factors with a known a-priori variance factor; with the a-posteriori variance factor \(\hat\sigma_0^2\) (\(\nu\) degrees of freedom) the factor becomes \(\sqrt{2F_{2,\nu,0.95}}\), and the simultaneous (in-context) 95% region uses \(\sqrt{\chi^2_{u,0.95}}\) over all \(u\) network coordinates — e.g. 5.61 for a 10-station 2-D network (\(u=20\)).