Question

For a profile given in the figure in the form of three steps A, B and C, the following information is available:
Height of step A ($H_A$) with respect to a reference line = 10 m (known and error free)
Difference in height between step A and step B ($h_1$) = 5 m $\pm$ 2 mm
Difference in height between step B and step C ($h_2$) = 8 m $\pm$ 3 mm
$H_B$ and $H_C$ are the unknown heights of step B and C, respectively. The step B is higher than step C and D.
The coefficient of correlation, $\rho_{h_1 h_2}$, between the height differences = 0.25
The coefficient of correlation between estimated heights of points B and C ($\rho_{H_B H_C}$) will be ............................. (Rounded off to 2 decimal places).

Hint:

When heights are built from correlated differences, express the heights as linear functions of the measurements and propagate: $\operatorname{Var}(aX+bY)=a^2\sigma_X^2+b^2\sigma_Y^2+2ab\,\operatorname{Cov}(X,Y)$ and $\rho_{XY}=\dfrac{\operatorname{Cov}(X,Y)}{\sigma_X\sigma_Y}$. Adding constants (known $H_A$) does not affect covariance or correlation.

Answer 1

Let the random (measurement) errors in $h_1$ and $h_2$ be zero-mean with standard deviations \[ \sigma_{h_1}=2\ \text{mm},\qquad \sigma_{h_2}=3\ \text{mm},\qquad \rho_{h_1h_2}=0.25, \] and $\operatorname{Cov}(h_1,h_2)=\rho_{h_1h_2}\sigma_{h_1}\sigma_{h_2}$.
Because $H_A$ is known exactly, \[ H_B=H_A + h_1,\qquad H_C=H_B - h_2 = H_A + h_1 - h_2. \] Hence \[ \operatorname{Var}(H_B)=\sigma_{h_1}^2, \] \[ \operatorname{Var}(H_C)=\operatorname{Var}(h_1-h_2) =\sigma_{h_1}^2+\sigma_{h_2}^2-2\,\operatorname{Cov}(h_1,h_2), \] \[ \operatorname{Cov}(H_B,H_C)=\operatorname{Cov}(h_1,\ h_1-h_2) =\sigma_{h_1}^2-\operatorname{Cov}(h_1,h_2). \] Therefore the correlation coefficient between $H_B$ and $H_C$ is \[ \rho_{H_BH_C}=\frac{\sigma_{h_1}^2-\rho_{h_1h_2}\sigma_{h_1}\sigma_{h_2}} {\sqrt{\sigma_{h_1}^2\left(\sigma_{h_1}^2+\sigma_{h_2}^2-2\rho_{h_1h_2}\sigma_{h_1}\sigma_{h_2}\right)}}. \] Substitute $\sigma_{h_1}=2,\ \sigma_{h_2}=3,\ \rho_{h_1h_2}=0.25$ (units cancel): \[ \rho_{H_BH_C}=\frac{2^2-0.25. 2. 3}{\sqrt{2^2\left(2^2+3^2-2. 0.25. 2. 3\right)}} =\frac{2.5}{\sqrt{40}} \approx 0.3953 \;⇒\; \boxed{0.40}. \]