Bonus: Adjusting our trilateration network

In Lab 3 you designed a networking by carrying out a preanalysis.

In this part of the lab I want you to use the measurements I provide below and carry out a parametric least squares adjustment in order to determine the coordinates of the new stations 3, 4, 5, and 6.

This will require you to implement the appropriate geometric reductions in C++ functions within your library, and reduce the measurements before using them.

Step 1: Adjust it using all of the measurements

The table below contains all of the possible measurements as obtained and partially reduced by the field crew. (You can assume they’ve made all the usual corrections except geometric reductions you saw earlier in this lab and you can assume there are no gross errors.) Use these measurements to adjust the network.

(Note that in reality we wouldn’t go and measure all of these, especially if we’d done the work in a preanalysis to determine which you didn’t need to do. But I’m asking you to do this here in order to illustrate the power of that earlier preanalysis.)

Here are the slope distances, not yet geometrically reduced to the grid in which you need to do the adjustment:

Leg Measured slope distance (m)
13 5221.275
14 5126.307
15 8566.372
16 3128.920
23 11219.111
24 8054.121
25 5525.338
26 8794.685
34 4452.887
36 2442.202
45 5782.213
46 3031.119
56 8259.969

They were measured between the following points with these instrument heights, ellipsoidal heights, and approximate grid coordinates obtained from GPS as the measurements were made:

Point \Delta\textit{\textbf{h}} (m) \textit{\textbf{h}} (m) \textit{\textbf{N}} (m) \textit{\textbf{E}} (m)
1 1.441 2321.123
2 1.733 2435.774
3 0.883 2645.234 5651657 630162
4 1.442 2042.114 5656041 630641
5 1.673 2081.355 5661800 630176
6 1.231 2439.832 5653667 628799


1. If the field crew hadn’t supplied the above approximate coordinates for points 3-5, you could have used your approximate coordinates obtained from the map in Lab 3.

2. The coordinates of points 1 and 2 are known from Lab 3 and so weren’t approximated in the field

3. You can again assume that the distance measurements have the following standard deviation:

    \begin{equation*} \sigma_{l}\text{ = }\pm\text{ ( }4\text{ mm }+2\text{ ppm }\text{ ) } \end{equation*}

4. And you can otherwise treat everything the same as in Lab 3.

Carry out  the adjustment, being sure to compute and provide:

  1. The adjusted observations
  2. The residuals
  3. The corrections to the approximate parameters
  4. The final estimated parameters
  5. The elements of the variance-covariance matrix of the estimated parameters
  6. The a-posteriori variance factor
  7. The 95% point and relative confidence region parameters

And keep in mind that you will likely need to iterate two or three times because of the non-linear nature of the model.

Step 2: Compare the results to your preanalysis

Compare the predicted results from your preanalysis in Lab 3 with the actual results obtained here. Do all of the actual confidence regions meet the specifications of the survey? Are you happy with the a-posteriori variance factor?

Step 3: Repeat the adjustment using the design from your preanalysis in Lab 3

Now repeat Step 1 above using only the legs you said were needed in Lab 3. (And be sure to make clear in your report which are those legs.)

Then compare the results to those you obtained with the full data set. In particular, I’m interested in how the confidence ellipse parameters compare in this case to the earlier case. And whether the specifications are still met by the design.

Step 4: Prepare and submit your report and a suitable summary of your input and output

Be sure to answer all of the questions asked here and provide both your code and the relevant input and output files.

Step 5: Draw some conclusions

As well as addressing the specific questions I’ve asked, I’d like you to comment in your submission on your overall understanding and takeaways from doing the preanalysis in Lab 3 and then implementing its recommendations here in comparison to the full network. Show me what you’ve learned from all of this.

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