Preprocessing of Survey Data (PSD). Theoretical error models
For directly observed geodetic quantities to be usable in further computations, they must first be processed into relevant derived quantities. Doing so is trivial, however the process of determining their theoretical standard deviations (which are required for determining weights in adjustments and misclosure tolerances in traverses) is a bit more involved.
Standard error propagation
Error propagation in linear quantities, derived from TS measurements (elevation differences, horizontal distances)
Let's first define the following variables:
- radian to gon conversion constant
- vertical pointing error
- constant EDM error
- linear EDM error [ppm]
The following formula is used in case the errors in the station's and signal's heights are reduced per sighting (i.e. treated as random errors):
For elevation differences, the first two error sources come from instrument heights, which are directly present in the computation process for them; for horizontal distances, they come from the centering errors. All terms have coefficients, determined using standard error propagation, however for the first two, they are equal to 1 (i.e. the terms are linear). The latter two terms have coefficients , which are defined later on.
However those errors should logically occur only once per independent setup or placement. For example, when doing trig. levelling, making multiple sightings from the same setup would reduce only the errors in the zenith angle and the sloped distance, but not the error in the station's height. Considering this, we can build a more suitable error propagation model:
Where standard deviations marked with an overline are reduced using standard error propagation:
The final error is then all the per-setup errors reduced .In elevation differences, this also includes the errors in differential leveling.
Errors are used to find the weighed average value of each quantity throughout the different stages of error reduction. For any quantity , with different values and standard deviations, he following formula is used:
This is how errors propagate for elevation differences and horizontal distances in case the reduction models for the station and signal errors are set to "per setup/signal".
Trigonometric leveling
The following formula is used in computing differences in elevation using TS observations:
where:
is the station's height
is the signal's height
is the sloped distance measured
is the zenith angle measured
is the curvature correction (only applied if explicitly enabled)
Using standard error propagation, the theoretical standard deviation coefficients would be computed as such, in case curvature corrections are disabled:
Otherwise, when curvature corrections are applied:
Differential levelling
Elevation differences in differential levelling are computed as:
Where the s are corrected rod readings and are calculated as:
Where in turn and are the collimation and curvature corrections, applied only if explicitly enabled:
Where is the horizontal sighting distance and is the collimation error, set as one of the instrument's parameters. The following error sources are defined for differential levelling:
Both the reading and the plumbing error scale with the sight distance. The plumbing error is caused by imperfect levelling of the instrument. The combined error for the level setup is computed as:
Horizontal distances
The following formula is used in computing horizontal distances, measured with total stations.
where:
is the station's centering error
is the signal's centering error
and the other terms are as previously defined.
Analogically to trig. leveling, the term coefficients are defined as:
Error propagation in angular quantities, derived from TS measurements (horizontal directions and angles)
Important! Geolyth currently does not recognize directions measured using the repeated sightings method, and using them may result in incorrectly averaged directions. Only 2-face measurements are supported currently. During this module, only the horizontal directions are averaged. You will get horizontal angle data only after running PPN, as prior to this the program has no way to determine exactly which angles should be used in further processing.
- TS horizontal pointing error
Similarly to linear quantities, you may specify the way centering errors are reduced. In case both station and signal errors are set to be reduced as random errors (per sighting), the following formulae are used:
However, as explained prior, this error model is not ideal. In case both centering errors are treated as systematic (per setup), the computation process becomes:
The station centering error is then only applied for the full horizontal angle, not the separate directions. Let be occupied by a station, from which directions and are measured, the standard deviation for the angle is calculated using:
In case an angle is computed using different angles from different setups, its standard deviation is:
This concludes this module's functionality. After running it, you will be able to run PLN, ALN