Linearized Observation Equation for an Angle: 18-Geom-A2 Adjustment of Observations, December 2018, Q7

Question 7: Linearized Observation Equation for an Angle (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Observed clockwise angle $\angle GAB=107^\circ29'40''$ at instrument station A, from backsight G to foresight B, with approximate coordinates $G(578.741,1103.826)$, $A(415.273,929.868)$, $B(507.934,764.652)$.

Find. The angle observation equation linearized about these coordinates — its coefficients (in $''$/m) on the six coordinate corrections and its constant term.

Figure 7.1 — Angle $\angle GAB$ at instrument A, opening clockwise from backsight G to foresight B; the observation equation is the difference of the two azimuth equations A→B and A→G.

Approach. An angle is the difference of two azimuths, $\angle GAB=\text{Az}_{AB}-\text{Az}_{AG}$; differentiate each azimuth with respect to the station coordinates (the standard azimuth partials) and combine, giving coefficients in radians per metre that are scaled by $\rho''=206{,}265''$ to arc-seconds.

Approximate azimuths and distances. $\text{Az}_{AG}=\tan^{-1}\!\big(\tfrac{\Delta X}{\Delta Y}\big)=43^\circ13'10''$ ($D_{AG}=238.711$ m); $\text{Az}_{AB}=150^\circ42'51''$ ($D_{AB}=189.426$ m).
Computed angle and misclosure. $$\angle GAB_0=\text{Az}_{AB}-\text{Az}_{AG}=107^\circ29'41.5'',$$ $$k=\angle_{\text{obs}}-\angle_{\text{computed}}=107^\circ29'40''-107^\circ29'41.5''=\boxed{-1.5''}.$$
Azimuth partials. For a line I→J, $\partial\text{Az}/\partial X_J=\Delta Y_{IJ}/D_{IJ}^2$, $\partial\text{Az}/\partial Y_J=-\Delta X_{IJ}/D_{IJ}^2$ (and equal-and-opposite at I). Applying these to $\text{Az}_{AB}$ (foresight, $+$) and $\text{Az}_{AG}$ (backsight, $-$) and multiplying by $\rho''$ gives the coefficients (arc-seconds per metre of coordinate shift): $$\begin{aligned} &\text{G: } dX_G=-629.7,\ dY_G=+591.7\\ &\text{A: } dX_A=+1579.4,\ dY_A=-59.1\\ &\text{B: } dX_B=-949.7,\ dY_B=-532.6 \end{aligned}$$ (as a check, the three $dX$ coefficients sum to zero, as do the three $dY$ — the angle is invariant to a rigid shift of the whole figure).
Assemble the observation equation. Writing $v$ for the angular residual ($''$), $$\boxed{\begin{aligned} -629.7\,dX_G+591.7\,dY_G+1579.4\,dX_A-59.1\,dY_A\\ -949.7\,dX_B-532.6\,dY_B=-1.5''+v \end{aligned}}$$ which is one row of the design matrix for this angle (coordinate corrections in metres, coefficients in $''$/m).

Coefficient ($''$/m)	$dX_G$	$dY_G$	$dX_A$	$dY_A$	$dX_B$	$dY_B$	const $k$
Value	−629.7	+591.7	+1579.4	−59.1	−949.7	−532.6	−1.5″

Concept & Theory

Topic. Linearization of an angle observation for coordinate (parametric) adjustment.

Key relations. $\angle=\text{Az}_{\text{FS}}-\text{Az}_{\text{BS}}$; azimuth partials $\partial\text{Az}_{IJ}/\partial X_J=\Delta Y/D^2$, $\partial\text{Az}_{IJ}/\partial Y_J=-\Delta X/D^2$; scale to seconds with $\rho''=206265''$; constant $k=\angle_{\text{obs}}-\angle_{\text{computed}}$.

Why this works. Each azimuth’s sensitivity to a coordinate shift is a rotation rate: moving the target perpendicular to the line changes the direction fastest (hence $\Delta Y/D^2$ and $-\Delta X/D^2$), and the effect scales as $1/D$ — near points swing the line more than distant ones. Subtracting the two azimuth rows gives the angle row; because both rows contain the instrument station A, its coefficients are the algebraic sum of the two contributions, which is why the $dX_A$ term is the largest here.

Common pitfall. Forgetting the $\rho''$ scaling (leaving the coefficients in rad/m), or mis-assigning the instrument station’s partials — A appears in both azimuths, so its coefficient is $(\text{from-}AB)-(\text{from-}AG)$, not either one alone.

Source. Ghilani, Adjustment Computations §15 (adjustment of angle observations) — the angle observation equation and its coefficients.

Question 7 of 7: Linearized Observation Equation for an Angle

Question 7: Linearized Observation Equation for an Angle (20 marks)