Calculation routines for inner reliability

General calculation principles

When calculating largest remaining error, we use the blunder error detection routines directly. For redundancy, we see that it is used as an inner reliability measure alone and is also implicitly included in minimum detectable error. We will show that the blunder error detection routine can also be used to calculate redundancy.

The redundancy \((r)\) indicates, in the uncorrelated case, what proportion of the blunder error appears in the adjustment corrections.

The redundancy lies between 1 and 0 and depends only on the network geometry. Formula for redundancy: \(r\) = diagonal elements in the matrix \(Q_{vv}P\). In correlated cases, the same formula is used, but a verbal description of the magnitude becomes more difficult.

The redundancy \((r)\) can be easily calculated using the same technique as in blunder error detection. We assume that all statistically detectable blunder errors have been removed from the observation material, and form extra columns in the normal equation system as in blunder error detection.

In the simplest case, with independent distance measurements, we have:

\(a^2 = p_i - e_{i^T} \cdot P \cdot A \cdot (R^{-1})^T \cdot R^{-1} \cdot A^T \cdot P \cdot e_i\)

\(= p_{i^2} \cdot ( p_{i^-1} - e_{i^T} \cdot A \cdot (R{^-1})^T \cdot R^{-1} \cdot A^T \cdot e_i )\)

\(= p_{i^2} \cdot e_{i^T} \cdot Q_{vv} \cdot e_i = p_i r_i\)

Where:

• \(R^{-1} \cdot A^T \cdot P \cdot e_i\) : the reduced extra column
• \(e_i\) : unit vector for observation no. i

This gives us:

\(r_i = \frac{a^2}{p_i}\)

For angle measurements, when orientation elements must be considered, the vector \(e_i\) and matrices \(P\) and \(A\) are expanded with extra coefficients for the Schreiber equations, but the calculation principle remains the same.