Chapter 16, Problem 16.25P

To determine

Construct A and L matrices.

A= $\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$

L= $\left[\begin{array}{c}1,164,499.387\\ -4,660,600.879\\ 4,182,158.213\\ 1,164,499.400\\ -4,660,600.884\\ 4,182,158.214\\ 1,168,947.849\\ -4,650,766.255\\ 4,191,851.271\\ 1,168,947.841\\ -4,650,766.249\\ 4,191,851.263\end{array}\right]$

Given that

Baseline vector are given as:

Explanation:

Writing observation equation for X coordinate for bonnieto ray:

   $X_{B-R}=X_B+\Delta X_{B-R}+v_1$

WHERE

X_B= X coordinate of the station bonnie = $1,161,121.599m$

ΔX_B-R= X component in baseline vector = $\mathrm{3.377.788}m$

ʋ₁= residual

   $\begin{array}{l}{X}_{B-R}=1,161,121.599+\mathrm{3.377.788}+v_1\hfill \\ {}_{ } =1,164,499.387+{\text{ v}}_1\hfill \end{array}$

Writing observation equation for Y coordinate for Bonnie to Ray:

   $\begin{array}{l}{Y}_{B-R}=Y_B+\Delta Y_{B-R}+v_2\hfill \\ WHERE\hfill \\ Y_B=YcoordinateofthestationBONNIE=-4,655,872.977\hfill \\ \Delta Y_{B-R}=Ycomponentinbaselinevector=-4,727.902\hfill \\ v_2=residual\hfill \\ Y_{B-R}=-4,655,872.977+-4,727.902+v_2\hfill \\ {}_{ } =-4,660,600.879+{\text{ v}}_2\hfill \end{array}$

Writing observation equation for Z coordinate for Bonnie to Ray:

   $\begin{array}{l}{Z}_{B-R}=X_B+\Delta Z_{B-R}+v_3\hfill \\ WHERE\hfill \\ Z_B=Zcoordinateofthestationbonnie=4,188,330.232m\hfill \\ \Delta Z_{B-R}=Zcomponentinbaselinevector=-6,172.019m\hfill \\ v_3=residual\hfill \\ Z_{J-T}=4,188,330.232-6,172.019+v_3\hfill \\ {}_{ } =4,182,158.213+{\text{ v}}_3\hfill \end{array}$

Writing observation equation for X coordinate for tom to herb:

   $\begin{array}{l}{X}_{T-H}=X_T+\Delta X_{T-H}+v_4\hfill \\ WHERE\hfill \\ X_T=XcoordinateofthestationTom=1,176,398.558m\hfill \\ \Delta X_{T-H}=Xcomponentinbaselinevector=-7,450.717m\hfill \\ v_4=residual\hfill \\ X_{T-H}=1,176,398.558-7,450.717+v_4\hfill \\ {}_{ } =1,168,947.841+{\text{ v}}_4\hfill \end{array}$

Writing observation equation for y coordinate for tom to herb:

   $\begin{array}{l}{\begin{array}{l}{Y}_{T-H}=Y_T+\Delta Y_{T-H}+v_5\hfill \\ WHERE\hfill \\ Y_T=Ycoordinateofthestationtom=-4,653,039.613m\hfill \\ \Delta Y_{T-H}=Ycomponentinbaselinevector=2,273.364m\hfill \\ v_5=residual\hfill \\ Y_{A-T}=-4,653,039.613+2,273.364+{\text{ v}}_5\hfill \end{array}}_{ } \\ =-4,650,766.249+{\text{ v}}_5\end{array}$

Writing observation equation for Z coordinate for tom to herb

   $\begin{array}{l}{Z}_{T-H}=Z_T+\Delta Z_{T-H}+v_6\hfill \\ WHERE\hfill \\ Z_T=Zcoordinateofthestationtom=4,187,198.360m\hfill \\ \Delta Z_{T-H}=Zcomponentinbaselinevector=4652.903m\hfill \\ v_6=residual\hfill \\ Z_{T-H}=4,187,198.360m+4652.903+v_6\hfill \\ {}_{ } =4,191,851.263+{\text{ v}}_6\hfill \end{array}$

Writing observation equations for X coordinate for bonnie to herb:

  

Writing observation equation for Y coordinates for bonnie to herb:

Y_B-H= Y_B+ ΔY_B-H+ʋ₈WHERE

Y_B= Y coordinate of the station Tom = -4,655,872.977m

ΔY_B-H= Y component in baseline vector = 5,106.722m

ʋ₈= residual

X_A-T=4,655,872.977+5,106.722+ʋ₈ = -4,650,766.255+ ʋ₈

Writing observation equation for Z coordinates for bonnie to herb:

   $\begin{array}{l}{\begin{array}{l}{Z}_{B-H}=Z_B+\Delta Z_{B-H}+v_9\hfill \\ Where,\hfill \\ Z_B=ZcoordinateofthestationTom=4,188,330.232m\hfill \\ \Delta Z_{B-H}=Zcomponentinbaselinevector=3,521.039m\hfill \\ \hfill \\ v_9=residual\hfill \\ Z_{B-H}=4,188,330.232+3,521.039+v_9\hfill \end{array}}_{ } \\ =4,191,851.271+{\text{ v}}_9\end{array}$

Writing observation equation for X coordinate for tom to ray:

   $\begin{array}{l}{X}_{T-R}=X_T+\Delta X_{T-R}+v_{10}\hfill \\ WHERE\hfill \\ X_T=XcoordinateofthestationTom=1,176,398.558m\hfill \\ \Delta X_{T-R}=Xcomponentinbaselinevector=-11,899.158m\hfill \\ v_{10}=residual\hfill \\ X_{T-R}=1,176,398.558-11,899.15+v_{10}\hfill \\ {}_{ } =1,164,499.400+{\text{ v}}_{10}\hfill \end{array}$

Writing observation equation for Y coordinate for tom to ray:

   $\begin{array}{l}{Y}_{T-R}=Y_T+\Delta Y_{T-R}+v_{11}\hfill \\ WHERE\hfill \\ Y_T=YcoordinateofthestationTom=-4,653,039.613m\hfill \\ \Delta Y_{T-R}=Ycomponentinbaselinevector=-7,561.271m\hfill \\ v_{11}=residual\hfill \\ Y_{T-R}=1,176,398.558-11,899.15+v_{11}\hfill \\ {}_{ } =-4,660,600.884+{\text{ v}}_{11}\hfill \end{array}$

Writing observation equation for Z coordinate for tom to ray:

   $\begin{array}{l}{Z}_{T-R}=Z_T+\Delta Z_{T-R}+v_{12}\hfill \\ WHERE\hfill \\ Z_T=YcoordinateofthestationTom=4,187,198.360m\hfill \\ \Delta Z_{T-R}=Ycomponentinbaselinevector=-5,040.146m\hfill \\ v_{12}=residual\hfill \\ Z_{A-T}=4,187,198.360-5,040.146+v_{12}=-4,182,158.214+{\text{ v}}_{12}\hfill \\ \hfill \end{array}$

A = $\left[\begin{array}{cccccc}XB-R& 0& 0& 0& 0& 0\\ 0& YB-R& 0& 0& 0& 0\\ 0& 0& ZB-R& 0& 0& 0\\ 0& 0& 0& XT-R& 0& 0\\ 0& 0& 0& 0& YT-R& 0\\ 0& 0& 0& 0& 0& ZT-R\\ XB-H& 0& 0& 0& 0& 0\\ 0& YB-H& 0& 0& 0& 0\\ 0& 0& ZB-H& 0& 0& 0\\ 0& 0& 0& XT-H& 0& 0\\ 0& 0& 0& 0& YT-H& 0\\ 0& 0& 0& 0& 0& ZT-H\end{array}\right]$

A= $\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$

Construct L matrix:

L = $\left[\begin{array}{c}1,164,499.387\\ -4,660,600.879\\ 4,182,158.213\\ 1,164,499.400\\ -4,660,600.884\\ 4,182,158.214\\ 1,168,947.849\\ -4,650,766.255\\ 4,191,851.271\\ 1,168,947.841\\ -4,650,766.249\\ 4,191,851.263\end{array}\right]$

Conclusion:

A= $\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$

L= $\left[\begin{array}{c}1,164,499.387\\ -4,660,600.879\\ 4,182,158.213\\ 1,164,499.400\\ -4,660,600.884\\ 4,182,158.214\\ 1,168,947.849\\ -4,650,766.255\\ 4,191,851.271\\ 1,168,947.841\\ -4,650,766.249\\ 4,191,851.263\end{array}\right]$