In the first part of this lab you developed and linearized the model needed to estimate both the coordinates of point 5 and the misalignment error \boldsymbol{\varepsilon}. Then in the second part you did a preanalysis, reduced the measurements, and took a look at blunder detection. Then in the third part you estimated the unknown parameters and assessed whether or not to scale the variance-covariance matrices. Now let’s wrap things up.

Step 1: Check the precision of your coordinate against the spec again

If you did have to scale the variance-covariance matrices in Step 2 of the third part of this lab, then I would like you to check the precision of the coordinates of point 5 against the spec given earlier. This would mean using the scaled \hat{\mathbf{C}}_{\hat{x}} to calculate the semi-major axis at a 95% confidence level again.

Is your estimate of the unknown parameters any different? Does the network still meet spec in your estimation?

Reflect upon and document the situation in your report.

Step 2: Finish assessing the precision

You just looked again at the semi-major axis of the error ellipse of the unknown coordinate.

Now I want you to calculate the semi-minor axis (also at a 95% confidence level), as well as the angle \beta.

Document the results in your report and provide a sketch of the error ellipse to scale and oriented properly. Essentially, in doing this you will have diagonalized the variance-covariance matrix of your estimated parameters, a process also called eigenvalue decomposition or principle component analysis. Use your results to comment on this and why it’s important to do it in the case of a survey network.

Also calculate an appropriate measure of the precision of the estimated misalignment error in ppm at a 95% confidence level.

Step 3: Assess the reliability of your network

Use the matrix equation we saw in class to calculate the total reliability of your network and the individual redundancy and absorption numbers.

Document the results of this in your report.

Step 4: Studying the reliability for another look at network design

And answer the following questions:

1. Which measurement do the redundancy numbers suggest is the “most important” to your adjusted solution?

2. Which is the least important?

3. Even with having to estimate the misalignment error in your network, this survey had two more measurements than unknowns. What would the semi-major axis of your unknown coordinate look like if you removed the two “least important” measurements? Did those measurements make a big difference?

  • Note: Although it’ll mess with the redundancy numbers, a very quick way to do this is to re-run your adjustment using a very low weight (high standard deviation) for the measurements in question.

4. If you could go back in time and “undo” two measurements and replace them with two others in order to strengthen the network design, what would you recommend? (You don’t need to do a full pre-analysis here – I’m just looking for your intuition based on the geometry of the situation and what the measures of precision and reliability are telling you.)

Document your answers to these questions in your report.

Step 5: Finish and hand in your lab

We’ve covered a lot of ground in this lab with a (relatively) quick little example of a network. I want you to do two things to conclude your report:

1.  Write a brief conclusion and recommendation to “your boss”, recalling the challenge you set out for yourself at the outset of the lab to simultaneously estimate the coordinates of point 5 and the misalignment error, \boldsymbol{\varepsilon}. You don’t need to rewrite the quality measures. Just refer to your answers to other questions already found in your report.

2. I’m going to suggest that you will revisit what you did in this lab and the material we covered in this course many times in the future, e.g. from your coursework next semester to when you’re working in jobs in fields ranging from classical land surveying, to photogrammetry, to GPS positioning, to modern navigation. I spent a long time designing it to provide you with a practical “How does that work again?” kind of reference. So to conclude this lab, please write your future self a brief list of the most important things to remember when designing and adjusting a network by least squares. Think of this partly as things to do, partly as things not to forget, and partly as the lessons you learned while doing it.

Please also submit appropriate evidence of your implementation, e.g. the spreadsheet you created, your Matlab code, or your C++ project.