Inner Reliability¶
What is Inner Reliability?¶
From the "Maps and Geodata" standard, we find that "...inner reliability refers to how well the observations in a system mutually control each other, or in other words, how (gross) errors in an observation are reflected in the corresponding adjustment correction."
This definition also encompasses Gemini Terrain's measures of inner reliability. Several measures of inner reliability are available in Gemini Terrain. All are described below:
- Largest remaining gross error
- Redundancy
- Maximum relative gross error
Largest remaining gross error¶
Consider that we have tested the observation material for gross errors and removed detectable errors. The interesting aspect is the magnitude of the remaining errors. These are errors that in the gross error detection are too small or too uncertainly determined for the observation to be rejected.
For each of these remaining errors, we calculate a confidence interval:
\([\hat\nabla-m_0T_{p,\alpha/2}, \hat\nabla+m_0T_{p,\alpha/2} ]\)
The outer edge of this interval is the largest remaining gross error \(\nabla_0\).
In the analysis list, this measure will be listed together with the mentioned inner reliability measures, under the designation largest remaining gross error. The denomination depends on whether it is a distance or angle observation, and is therefore of little interest in terms of relative reliability between observation types.
Redundancy¶
In the uncorrelated case, redundancy indicates what proportion of the gross error appears in the adjustment corrections. It then lies between 0 and 1 and depends only on the geometry of the network. Observations with redundancy close to 1 will consequently receive large adjustment corrections if they contain gross errors. The possibilities for detecting gross errors are therefore large.
In the correlated case, we use the general formula that redundancy is the diagonal elements in the matrix \(Q_{vv}P\) where:
\(Q_{vv} = P^{-1} - (A \cdot N^{-1} \cdot A^T)\)
where \(P\) is the weight matrix. This applies, for example, when we have satellite observations included in the calculation. We can then get both negative redundancy and redundancy greater than 1.
Maximum relative gross error¶
Maximum relative gross error is the largest remaining gross error in multiples of the observation's standard deviation. If the value is, for example, 2, we can at a given significance level assert that the observation does not contain gross errors larger than its standard deviation × 2.
Gemini Terrain's measure of inner reliability now becomes the largest remaining gross error in multiples of the observation's standard deviation:
\(\frac{\nabla_0}{m_i}\)
Documentation of Baarda Values¶
Minimum detectable error - Baarda¶
This inner reliability measure can also be listed on the result output if desired. This value is closely related to Baarda's method for gross error detection, called data-snooping. It is based on standardized adjustment corrections \(w_i\), (the adjustment correction in multiples of its standard deviation), which will be standard normally distributed if we assume normally distributed observations. If we now assume independent observations, it can be shown that the estimated error is:
\(\hat\nabla=w_i\frac{\sigma_i}{\sqrt r_i}\)
and that the assumed standard deviation of this quantity becomes:
\(\sigma_{\hat\nabla}=\frac{\sigma_i}{\sqrt r_i}\)
Where:
- \(\hat\nabla\) = estimated error
- \(r_i\) = redundancy of observation i
- \(\sigma_i\) = assumed standard deviation of observation i
If we now introduce that the expectation of this error's absolute value is \(0.8\sigma_\nabla\) and form a 5% confidence interval around the expectation, we get:
\(0.8\sigma_{\hat\nabla}+N_{2.5\%}\sigma_{\hat\nabla}\)
Inserting values for \(N_{2.5\%}\) (normal distribution table) gives us:
\(2.8\sigma_{\hat\nabla}=2.8\frac{\sigma_i}{\sqrt r_i}\)
and we call this value the minimum detectable error.
Along with the minimum detectable error, we list the error's effect on the adjustment result. We do this by multiplying the error by \((1 - r_i)\) where \(r_i\) is the redundancy.
Free or constrained adjustment?¶
Inner reliability is closely related to the detection of gross errors and the rejection of corresponding observations. It indicates how well the observations in the network control each other. For maximum gross error, the largest remaining gross error must be estimated. To prevent constraints in the basis from influencing this calculation, free adjustment must be used. If we know with high certainty that the basis is constraint-free, for example, the new control network, it can be justified to estimate gross errors after constrained adjustment. All inner reliability measures must refer to the same adjustment, so we get the rule:
Note
Inner reliability is based on the adjustment used when evaluating observation rejection (gross error detection).