Self-assessment for: Measures of precision in two and three dimensions

These self-assessment problems accompany our in-class discussions on Measures of precision in two and three dimensions.

Problem 1

What is the confidence level corresponding to the point error ellipse we saw in class?

Problem 2

Imagine you do a survey and a subsequent adjustment to estimate the coordinates of a point, \mathbf{x}=\begin{bmatrix}N E\end{bmatrix}^T.

And that your least squares estimate yields the following covariance matrix of the adjusted unknowns in units of m^2:

    \begin{equation*} \mathbf{C}_{\hat{\mathbf{x}}} = \begin{bmatrix} 2.07035x10^-5 & -6.52486x10^-6 \\ -6.52486x10^-6 & 3.67365x10^-5 \end{bmatrix} \end{equation*}

a) Assuming a normal distribution, would the result of the survey be acceptable if the specification on each of the coordinates was 1.00 cm at a 95% confidence level?

b) Would the result be acceptable if the specification on the semi-major axis of the point error ellipse was 1.00 cm at a 95% confidence level?

c) Can you calculate a relative error ellipse in this case? If not, why not? If so, what are its defining parameters?

Problem 3

Under the assumptions we discussed in class, would you rather have:

a) A measurement system with CEP = 2.5 m or SEP = 3.0 m?

b) A measurement system with SEP = 2.0 mm or RMS = 1.66 mm?

Problem 4

How is an error ellipse like a confidence region? Explain this in words and with a figure.

Problem 5

a) Draw any two coordinates and the line between them in a 2D planar N, E coordinate system

b) Draw and label point error ellipses on your figure from above (the values and orientation don’t matter here – I just want to see that you understand how and where to put them in a general sense)

c) Draw and label a standard relative error ellipse on the same figure (the values and orientation don’t matter again here for the same reason, although the placement of its centre does)

d) To your figure add and label the relative error ellipse corresponding to a confidence level of 99.7% and make sure it is labeled and pretty accurately to scale and of the right orientation (I may measure things with a ruler to ensure you’ve understood this one)

Problem 6

Given the following covariance matrix of two stations in a 2D network, \mathbf{x} = \begin{bmatrix}E_1 & N_1 & E_2 & N_2\end{bmatrix}^T, compute the semi-major and semi-minor axes and orientation of the relative error ellipse between the two stations at the 95% confidence level:

    \begin{equation*} \mathbf{C}_{\hat{\mathbf{x}}} = \begin{bmatrix} 121 & 32 & 9 & 8 \\ 32 & 100 & 16 & 10 \\ 9 & 16 & 100 & 80 \\ 8 & 10 & 80 & 121 \end{bmatrix} \end{equation*}

This has units of mm^2.

The following schedules are for Alex’s in-class students:

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