Example 8

Goal

Your goal in this example is to linearize the functional model for azimuth.

Situation

We first considered the measurement of azimuth in Example 3 where we imagined an observed azimuth from unknown station A to unknown station B in a planar 2D coordinate system, and we derived the following functional model:

    \begin{equation*} tan(\alpha_{AB})=\dfrac{E_B-E_A}{N_B-N_A} \end{equation*}

or:

    \begin{equation*} \alpha_{AB} - tan^{-1}(\dfrac{E_B-E_A}{N_B-N_A}) = 0 \end{equation*}

This situation is shown below.

It turns out that we need to handle things more carefully that the above simple equation will allow in cases where point B does not lie northeast of point A, as suggested by the following figure.

Problem

So, building on this example, I want you to consider the more general situation shown in the figure below. That is, the case where azimuth is given by:

    \begin{equation*} azimuth_{AB} = \alpha_{AB} + \Delta\alpha \end{equation*}

or:

    \begin{equation*} azimuth_{AB} = tan^{-1}(\dfrac{E_B-E_A}{N_B-N_A}) + \Delta\alpha \end{equation*}

And I want you derive the linearized functional model for this example in the form \mathbf{A}\boldsymbol{\delta} - \mathbf{e} + \mathbf{w} = 0.

I want you to be as explicit in doing so as we were in Example 6, e.g. by: sketching the situation; stating what is given; stating what is required; showing your work and which equations you use; and putting boxes around your final answers.

You can / should carry the term \Delta\alpha throughout.

And to keep things simple, you may assume we only measure the azimuth once.