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The following are some recommended practical self-assessment questions for the lesson called You’ve Already Been Propagating Errors. Here’s What You Need to Know. They’re intended for you to work through to test your own knowledge of the key concepts we covered there.

Question 1

Imagine you’re given the situation shown below where the azimuths (clockwise directions from north) of points 2, 3, and 4 are each measured at point 1, and found to have the values and standard deviations shown in the table.

azimuth measurement standard deviation
(arc seconds)
az_{12} 45^o14'42.360" 4.1
az_{13} 92^o6'48.816" 6.8
az_{14} 151^o13'17.220" 7.2

Compute the angles \theta_{23} and \theta_{24} in DMS, and use the concept of propagating errors to calculate their variance-covariance matrix.


1. This is similar to the example I gave in the video introducing this stuff.

2. I assume you’re already familiar with angles expressed in degree-minutes-seconds (DMS) but just in case, you can check this out.

Question 2

In the video found here, I showed you the basics of error propagation and how, when implementing the parametric least squares equations, you are also actually propagation uncertainties with this equation:

    \begin{equation*} \underset{u\times u}{\mathbf{C}_{\hat{\mathbf{x}}}} =\mathbf{C}_{\hat{\boldsymbol{\delta}}} =(\mathbf{A}^\mathsf{T}\mathbf{C}_{\mathbf{l}}^{-1}\mathbf{A})^{-1} \end{equation*}

a) Explain in words what this equation is for and why it’s important

b) If you had a reliable model of the uncertainties in your observations, \mathbf{C}_{\mathbf{l}}, and some reasonable approximate values of your desired parameters, \mathbf{x}^{0}, would it be possible to compute \mathbf{C}_{\hat{\mathbf{x}}} without going into the field at all, i.e. without collecting a single measurement?

c) How do you think the finding / your answer in b) might be useful in practice, e.g. when designing a survey or system that will be used for some kind of geospatial estimation application? On balance, I’m asking why this equation and the notion of propagation uncertainties matter.

Question 3

In an earlier topic, we looked at the concept of and equations for computing standard errors. And back then I told you you’d have enough knowledge by the end of our time together to derive a few of the equations for standard error. Well, now’s that time. And you do.

a) Use what you know now about error propagation to prove that SE_{range} = s_x\sqrt{2}}

b) Use what you know now about error propagation to prove that SE_{mean} = \frac{s_x}{\sqrt{n}}}

This is pretty straightforward, but here are a few hints:

1. In both of these cases, you can assume that the errors in one sampled value are not correlated to the errors in any other.

2. It helps to recall and start with the key equations for range and mean, i.e.:

range = x_{max}-x_{min}

    \[\bar{x} = \sum_{i=1}^{n} \frac{x_i}{n}\]

3. And to note that both of these are just linear expressions of some output in terms of some input, a reality that can be expressed in the more general form \mathbf{y}=\mathbf{L}\mathbf{x}.

4. Because it follows that \mathbf{C}_\mathbf{y} =\mathbf{L}\mathbf{C}_\mathbf{x}\mathbf{L}^\mathsf{T}, it’ll help if you solve for \mathbf{C}_{range} and \mathbf{C}_{\bar{x}} and if you give some thought to the relationship between what’s in those matrices and what is a standard error.

If you’re one of my students, then you’re expected to answer this on your own and submit according to the directions provided in class, i.e. you don’t  need to submit them through this website. Don’t forget that our TA and I are both here to help you in the associated lab (and/or tutorial) sessions.

Aim to provide succinct answers to these applications questions in the same document you created for the conceptual self-assessment questions.

I’d also like you to submit an Excel spreadsheet along with that document. This will likely require a bit of thoughtful organization on your part. For example, It would work well to put the answers to both of the above application problems onto their own sheet / tab within your spreadsheet, and name it accordingly. And then you can refer to that specific tab from your written document.

When you answer other applied question sets in future topics, you should do the calculations on separate sheets / tabs, also appropriately named. And refer to them where required to show your work. This way, you’ll have all of your Excel work in one nicely indexed place and will only need to hand in a single spreadsheet as an appendix to each set of self-assessments.

You can click through to other self-assessments or lessons (if any) using the button below, and return here whenever you wish.