The first step is to develop and linearize the model needed for the problem at hand, i.e. estimating both the coordinates of point 5 and the misalignment error \boldsymbol{\varepsilon}.

Step 1: Develop the functional model (or observation equations)

For the given situation, derive and write out the functional model \mathbf{l}_{true} - \mathbf{F}(\mathbf{x}) = \mathbf{0}.

Step 2: Linearize the model

Linearize your model and write it in the form \mathbf{A}\boldsymbol{\delta} - \mathbf{e} + \mathbf{w} =\mathbf{0}.

Step 3: Implement your linear model

Develop a tool (Excel, Matlab, or C++) that implements the linearized model. You will be able to fill the design matrix \mathbf{A} given the following approximate values of the parameters.

For the row of \mathbf{A} corresponding to the misalignment error, be sure to account for the constants \delta\alpha_{AB} and \delta\alpha_{AC} as outlined here and as you did in Lab 2. You will need to do this.

You can also set things up for calculating the misclosure vector \mathbf{w}, although you’ll have to wait to finalize that until after you’ve reduced the distance measurements in Part 2 of this lab.

The published coordinates of points 1 and 2 are given in the following table:

Point \textit{\textbf{H}} (m) \textit{\textbf{N}} (m) \phi \lambda
1 1223.882 -16.21 N 51^o06^{'}4.979794^{"}  W 114^o06^{'}9.600068^{"}
2 1207.456 -16.21 N 51^o05^{'}52.00185^{"}  W 114^o06^{'}5.817410^{"}

 

where \Delta\textit{\textbf{h}} means the instrument heights.

The approximate coordinate of point 5 was measured using GPS to yield:

Point \textit{\textbf{h}} (m) \phi \lambda
5 1175.0 N 51^o06^{'}8.733861^{"}  W 114^o05^{'}45.017170^{"}

 

All coordinates are given in WGS84 and you can use this online tool to convert these coordinates to the UTM grid in which you’ve done your modeling and will do your adjustment. And be sure to carry enough decimal places to carry the precision through converting from DMS and to DD and to UTM.

You can approximate the misalignment error \boldsymbol{\varepsilon} as 0.0 ppm.

And the measurements are as follows:

Measurement Value Units Notes
d_{15} 493.6633 m no geometric reductions done
d_{25} 657.2884 m no geometric reductions done
\theta_{12@5} 38^o17^{'}37.7848039^{"} DMS geometric reductions already done
\theta_{15@2} 48^o27^{'}44.645938^{"} DMS geometric reductions already done
\theta_{52@1} 93^o15^{'}45.3450736^{"} DMS geometric reductions already done

Step 4: Document your results for your submission

I would like to see all of your work for this, as well as evidence of your implementation (the spreadsheet, m-file, or C++ project).

Note: I gave you some pretty detailed guidance on the modeling and linearization of this in class (and on D2L). While it’s true that those are a big help in Step 1 and 2 here, I do want you to work it all through for yourself. Write it up for yourself. And make sure you could do the same yourself for any of the systematic errors we studied as part of the measurement model.

Move to Part 2

Now you’re ready to begin the next steps on preanalysis, reductions, and gross error detection.