Question

As shown in the following figure, let $d_1, d_2, d_3$ denote three uncorrelated clockwise directions, observed at point $P$ with equal standard errors for each direction, i.e., $\sigma_{d_1} = \sigma_{d_2} = \sigma_{d_3} = \pm \sqrt{2}''$. Let $\alpha_1$ and $\alpha_2$ be two included angles formed by these three directions. The covariance matrix (in arcsecond$^2$) for these included angles will be given as: 
 

Here, the covariance matrix would typically be presented in a LaTeX format, but for now we are using a placeholder.

• A. \[ \begin{pmatrix} 4 & 2 \\ 2 & 4 \end{pmatrix} \]
• B. $\begin{bmatrix} 4 & -2 \\ -2 & 4 \end{bmatrix}$
• C. $\begin{bmatrix} -4 & 2 \\ 2 & -4 \end{bmatrix}$
• D. $\begin{bmatrix} 4 & 2 \\ -2 & 4 \end{bmatrix}$

Hint:

For included angles built from independent directions, each angle’s variance is the sum of the two direction variances; adjacent included angles share exactly one direction, giving a covariance of \(-\sigma^2\).

Answer 1

The Correct Option is B

Let the observed directions (random variables) $\theta_1,\theta_2,\theta_3$ with \[ \operatorname{Var}(\theta_i)=\sigma^2, \sigma^2=(\sqrt{2})^2=2\ \text{(arcsec$^2$)}, \operatorname{Cov}(\theta_i,\theta_j)=0\ (i\neq j). \] Define the included angles (clockwise) as \[ \alpha_1=\theta_1-\theta_2,\qquad \alpha_2=\theta_2-\theta_3. \] Then \[ \operatorname{Var}(\alpha_1)=\operatorname{Var}(\theta_1-\theta_2)=\sigma^2+\sigma^2=2\sigma^2=4, \] \[ \operatorname{Var}(\alpha_2)=\operatorname{Var}(\theta_2-\theta_3)=\sigma^2+\sigma^2=2\sigma^2=4, \] \[ \operatorname{Cov}(\alpha_1,\alpha_2)=\operatorname{Cov}(\theta_1-\theta_2,\ \theta_2-\theta_3) =0-0-\sigma^2+0=-\sigma^2=-2. \] Therefore the covariance matrix is \[ \boxed{\begin{bmatrix}4 & -2 \\ -2 & 4\end{bmatrix}} \] (in arcsecond$^2$).