Question

Two adjacent angles \( A \) and \( B \) have been observed with the following mean values and correlation matrix \( \rho \):

\[ \bar{A} = 10^\circ 20'10'' \pm 10'', \quad \bar{B} = 25^\circ 35'15'' \pm 20'' \]
\[ \rho = \begin{bmatrix} 1.0 & 0.6 \\ 0.6 & 1.0 \end{bmatrix} \]

The standard deviation of the sum of the estimated angles \( A + B \) will be 2cm arcseconds (rounded off to 2 decimal places).

Hint:

When adding correlated quantities, always include the covariance term: \[ \sigma^2_{X+Y} = \sigma_X^2 + \sigma_Y^2 + 2 \cdot \rho_{XY} \cdot \sigma_X \cdot \sigma_Y \] This is essential in geodetic and error propagation calculations.

Answer 1

We are asked to find the standard deviation of the sum \( A + B \), given individual standard deviations and their correlation:

Let
\( \sigma_A = 10'' \),
\( \sigma_B = 20'' \),
\( \rho_{AB} = 0.6 \)

Using the formula for the variance of the sum of two correlated variables:

\[ \sigma^2_{A+B} = \sigma_A^2 + \sigma_B^2 + 2 \cdot \rho_{AB} \cdot \sigma_A \cdot \sigma_B \]
Substitute values:

\[ \sigma^2_{A+B} = 10^2 + 20^2 + 2 \cdot 0.6 \cdot 10 \cdot 20 = 100 + 400 + 240 = 740 \]
\[ \sigma_{A+B} = \sqrt{740} \approx 27.20'' \]