This document discusses reliability analysis in adjustment computation. It defines reliability as the sensitivity of adjustment approaches to undetectable errors and their influence on results. Internal reliability refers to the maximum undetectable error from a technique, assessed using gross error lower bounds. External reliability relates to undiscovered errors' maximum effect on adjustment results. The document provides equations for evaluating internal and external reliability and describes desirable properties like uniformly distributed redundancy and small ratios of observations to redundancies or unknowns to redundancies.

1. Reliability analysis in
adjustment
computation
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2. Reliability
 Difference between reliability and quality.
 Reliability can be defined as the sensitivity of
the approaches as well as the influence of
undetectable errors on the results of the
adjustment.
 Type l error and type ll error.
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3. Reliability
Internal
External
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4. Internal Reliability
 The internal reliability refers to the maximum undetectable error resulting from
using that technique.
 The internal reliability is assessed by the lower values of gross error.
 The expressions for the evaluation of these lower bounds is given by:
…(1)
 Now since,
 Where,
 Thus,
where, ΔV is the influence of
the gross error on the observations.
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5.  From the properties of idempotent matrix
 If rᵢ are the diagonal elements
 Now equation (1) could be written as:
…(2)
 Where, λ₀ is non- centrality parameter at boundary (i.e. at α₀ and β₀)
 The reliability coefficient hlᵢ = 𝜆0/𝑟ᵢ show the sensitivity of the test.
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6.  For Global reliability measure:
 Equation (2) becomes:
 Experience has shown that for ordinary networks r/n =0.50. Using above
equation for α₀ = 0.001 and β₀ = 0.80 therefore, λ₀ = 17.
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7. Results
 In most of the networks a gross error smaller than 5.8σlᵢ will not be detected.
 Large rᵢ values imply that the gross error in an observation will be more
clearly reflected in the corresponding residual vᵢ, and consequently easily
revealed through the testing procedure.
 If the redundancy is uniformly distributed in the network, then all rᵢ's and
therefore all lᵢ's, are practically the same.
 It is desirable to have small values of n/r.
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rᵢ Undetectable
gross error

8. External Reliability
 External reliability relates to the maximum effect of possible undiscovered
observational gross errors on the results of the adjustment.
 This influence is given by:
…(3)
 The above expression require a large number of computations since for each
estimated parameter there are n different components, when testing under
conventional hypotheses, one for each hypothesis.
 Hence, Baarda and De Haus (Baarda, 1976, 1979) proposed the new
standardized general variate λᵢ,₀.
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9. …(4)
 Now since,
 substituting σ₀²
…(5)
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10.  Assuming that only one gross error affects the observations, equation (5)
yields:
…(6)
 Considering also the expression for the internal reliability (2) and further
simplifying, equation (3) could be written as:
<
 Thus,
 For global measure,
 Where, u : number of desired unknown parameter, and
r : total redundancy. 10

11. Results
 It is desirable to have λᵢ,₀ approximately constant for all i's so that the ability
of detecting gross errors is the same in every part of the network.
 It is desirable to have small values of u/r which would produce lower bounds
for ₀x, thus better reliability for the network.
 The effect of non detectable gross error lᵢ on the results of the adjustment
can be as much as times the computed precision σxp of the desired
parameters.
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12. Conclusion
 The internal reliability mainly indicated the sensitivity and controllability of
the network to the presence of outliers.
 The external reliability is the measure of how unknown parameters are
affected due to the presence of the blunder or outlier.
 The reliability number indicated the degree to which a particular observation
is controlled by others.
 The reliability based on Baarda’s method worked well for the uncorrelated
case under the assumption of single outlier only.
 The n/r and u/r ratio should be as small as possible for better internal and
external reliability respectively.
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13. Reference



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Kavouras, M., & University of New Brunswick. Dept. of Surveying Engineering. (1985). On the
detection of outliers and the determination of reliability in geodetic networks. University of New
Brunswick.
Ghilani, C. D. (2010). Adjustment computations: spatial data analysis. John Wiley & Sons, pp. 435-
436.
Baarda, W. (1967). Statistical concepts in geodesy (Vol. 2). Rijkscommissie voor Geodesie

14. Thank You
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