Network Post-analysis — Confidence Ellipses and Ellipsoids: 18-Geom-A3 Geodesy and Positioning, December 2018, Q6

Question 6: Network Post-analysis — Confidence Ellipses and Ellipsoids (20 marks — a 12, b 8)

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) how to scale the standard ellipse to 95%.

Standard (k = 1) error ellipse from the station covariance matrix: semi-axes a, b are the square roots of the eigenvalues, oriented along the eigenvectors (orientation θ). It encloses about 39.4% probability in 2-D.

(a) Standard error ellipse / ellipsoid. After adjustment, each station has a $2\times2$ (or $3\times3$) covariance sub-matrix $\Sigma$. The standard error ellipse is the region whose semi-axes are the square roots of the eigenvalues of $\Sigma$, 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) From standard to 95% ellipse. The shape and orientation stay the same; only the size scales.

Approach. Multiply both semi-axes by $c=\sqrt{\chi^2_{2,0.95}}$, the square root of the 95% quantile of the chi-square distribution with 2 degrees of freedom.

Find the scale factor. The 95% quantile of $\chi^2$ with 2 dof is $5.991$, so $$c=\sqrt{\chi^2_{2,\,0.95}}=\sqrt{5.991}=\boxed{2.448}.$$
Scale the axes. The 95% error ellipse has $a_{95}=2.448\,a$ and $b_{95}=2.448\,b$. For example a standard ellipse with $a=14.0\ \text{mm}$, $b=8.0\ \text{mm}$ becomes $a_{95}=34.3\ \text{mm}$, $b_{95}=19.6\ \text{mm}$.

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 (one-point-at-a-time) factors for a known a-priori variance factor. If the a-posteriori variance factor $\hat\sigma_0^2$ (with $df$ degrees of freedom) is used to scale the covariance matrix, the factor becomes $\sqrt{2F_{2,\,df,\,0.95}}$ instead. A simultaneous (in-context) region that must hold for all network points at once uses the larger factor $\sqrt{\chi^2_{u,\,0.95}}$, with $u$ the total number of network coordinates.

Region	Enclosed probability / scale factor
Standard error ellipse (2-D, $k=1$)	39.4% ($1-e^{-1/2}$)
Standard error ellipsoid (3-D, $k=1$)	19.9%
Standard → 95% ellipse (2-D)	× $\sqrt{\chi^2_{2,0.95}} = 2.448$
Standard → 95% ellipsoid (3-D)	× $\sqrt{\chi^2_{3,0.95}} = 2.796$

Concept & Theory

Topic. Post-adjustment quality assessment: error ellipses/ellipsoids and their confidence scaling.

Key relations. Semi-axes $=\sqrt{\text{eigenvalues of }\Sigma}$; 2-D standard confidence $=1-e^{-1/2}=0.394$; scale to $(1-\alpha)$: $c=\sqrt{\chi^2_{k,\,1-\alpha}}$ ($k=2$ ellipse, $k=3$ ellipsoid).

Why this works. The covariance matrix defines a quadratic form whose level sets are ellipses/ellipsoids; the probability enclosed follows the $\chi^2$ law with degrees of freedom equal to the dimension. Scaling the semi-axes by $\sqrt{\chi^2}$ moves to any desired confidence without changing shape or orientation.

Common pitfall. Assuming the standard ellipse is 68% confidence (that is the 1-D value); it is only 39% in 2-D. Also confusing out-of-context with in-context (simultaneous) factors when reporting network reliability.

Source. Mikhail & Gracie, Analysis and Adjustment of Survey Measurements; Ghilani, Adjustment Computations, error-ellipse chapter; Vaníček & Krakiwsky, Geodesy: The Concepts, ch. 12.

Region	Enclosed probability / scale factor
Standard error ellipse (2-D, \(k=1\))	39.4% (\(1-e^{-1/2}\))
Standard error ellipsoid (3-D, \(k=1\))	19.9%
Standard → 95% ellipse (2-D)	× \(\sqrt{\chi^2_{2,0.95}} = 2.448\)
Standard → 95% ellipsoid (3-D)	× \(\sqrt{\chi^2_{3,0.95}} = 2.796\)

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

Question 6: Network Post-analysis — Confidence Ellipses and Ellipsoids (20 marks — a 12, b 8)