Correction Codes¶
Correction codes are used to tell Gemini Terrain which corrections the program should apply to the observations.
The dialogs for this can be found under Settings - Gemini Terrain.
Usually, all horizontal calculations should be performed in the map plane and all corrections must thus be applied (code 0). Old observations are often manually corrected for certain correction types. This also applies to some total stations. By using other codes (than 0), one can avoid introducing the error of applying these corrections again. If the measurements are already corrected to the map plane, code 1 is used and no corrections are applied by Gemini Terrain.
In height, the geoid is the reference plane.
Which corrections Gemini Terrain applies to the observations depends on the correction code that is set (to the station).
Correction Types¶
Measured distances can be applied with the following corrections:
- Calibration values for instrument
- Metrological corrections if pressure and temperature have been entered for the station
- Refraction
- Slope distance to horizontal distance, if horizontal is not given for the distance
- Height above sea level
- From chord length to arc length on the ellipsoid
- Map projection corrections
Measured directions can be applied with the following corrections:
- Plumb line deviation
- Map projection corrections
Measured vertical angles can be applied with the following corrections:
- Earth curvature and refraction
- Instrument and target height
Below you will find the formulas used for the various corrections.
Distances¶
Instrument and Calibration Corrections¶
Electronic Distance Meter and Total Station¶
-
Addition constant:
\(D_{corrected} = D_{measured} + dA\)
-
Scale factor:
\(D_{corrected} = D_{measured} + \left( \frac{D_{measured} \cdot dSC}{1000000} \right)\)
-
Pressure and temperature:
\(D_{corrected} = D_{measured} + 0.0001056 \cdot D_{measured} \left( \frac{H_0}{273 \cdot T_0} - \frac{H}{273+T} \right)\)
Where:
\(D\) : Measured distance
\(dA\) : Addition constant in meters
\(dSC\) : Scale factor in ppm
\(H_0\) : Calibration pressure
\(T_0\) : Calibration temperature
\(H\) : Measured pressure
\(T\) : Measured temperature
Measuring Tape¶
-
Temperature:
\(D_{corrected} = (0.0000115 \cdot D_{measured} \cdot (T - T_0))\)
-
Tension:
\(D_{corrected} = D_{measured} + \left( \frac{D_{measured} \cdot (ST - ST_0)}{E \cdot A} \right)\)
-
Sag:
\(D_{corrected} = D_{measured} - \left( \frac{P^2 \cdot D_{measured}^3}{24 \cdot ST^2} \right) \cdot \sin^2 z\)
Where:
0.0000115: Temperature expansion coefficient for steel
\(D\) : Measured distance
\(T_0\) : Calibration temperature
\(T\) : Measured temperature
\(ST_0\) : Calibration tension
\(ST\) : Measured tension
\(E\) : Modulus of elasticity
\(A\) : Cross section
\(P\) : Weight per meter
\(z\) : Zenith distance
Refraction¶
\(D_{corrected} = \frac{2 \cdot R_m}{K} \arcsin \left( \frac{D \cdot k}{2 \cdot R_m} \right)\)
Where:
\(R_m\) : Radius of curvature at the midpoint of the line, calculated according to Euler's formula.
Slope Distance to Horizontal Distance¶
\(S_{hm}^2 = S_{slope}^2 - dH^2\)
Where:
\(S_{slope}\) : Observed slope distance corrected for pressure and temperature.
\(dH\) : Height difference derived from H1 and H2. If these are not available for the points, dH is calculated from measured zenith distance.
\(S_{hm}\) : Horizontal distance at mean height for start and end points, including any instrument height
Distance Reduction for Height¶
\(S_{cor} = S_{hm} - \left(\frac{S_{hm} \cdot H_m}{R_m + H_m}\right)\)
Where:
\(R_m\) : Radius of curvature at the midpoint of the line, calculated according to Euler's formula.
\(H_m\) : Mean height derived from H1 and H2, which includes geoid height and any instrument height.
\(S_{hm}\) : Horizontal distance at mean height for start and end points, including any instrument height.
\(S_{cor}\) : Chord distance
Chord to Arc¶
\(S_{ell} = 2 \cdot R_m \cdot \arcsin \left(\frac{S_{cor}}{2 \cdot R_m}\right)\)
Where:
\(S_{ell}\) : Ellipsoidal distance
\(R_m\) : Radius of curvature at the midpoint of the line, calculated according to Euler's formula
\(S_{cor}\) : Chord distance
Gauss-Kruger Projection (Transverse Mercator)¶
\(S_{corr} = S_{ell} + \frac{S_{corr}}{6 \cdot R_s^2}(y_1^2+y_1 \cdot y_2 + y_2^2)\)
Iterated twice for long distances
\(y_1\) and \(y_2\) Map projection coordinates (East) to station and target.
\(R_s\) Mean radius of curvature for the foot point of the line's midpoint.
If the distance is over 10,000 m and the East coordinate is over 70,000 m from the tangent meridian, the formula is extended with a third-degree term.
Lambert Conformal Conic Projection¶
\(S_{corr} = S_{ell} + \frac{S_{corr}}{6 \cdot R_s^2}(x_1^2 + x_1 \cdot x_2 + x_2^2)\)
Iterated twice for long distances
\(x_1\) and \(x_2\) : Map projection coordinates (North) to station and target.
\(R_s\) : Mean radius of curvature for the foot point of the line's midpoint.
If the distance is over 10,000 m and the X coordinate is over 70,000 m from the tangent meridian, the formula is extended with a third-degree term.
Conformal Stereographic Projection¶
\(S_{corr} = S_{ell} + \frac{S_{corr}}{12 \cdot R_s^2}(x_1^2+x_1 \cdot x_2 +x_2^2 + y_1^2 +y_1 \cdot y_2 + y_2^2)\)
\(x_1,x_2,y_1,y_2\) : Map projection coordinates to station and target.
\(R_s\) : Mean radius of curvature for the foot point of the line's midpoint.
Iterated twice for long distances
Note
Independent of the projection type, we can have a scale factor. It can be local and caused by a scale deformation in the network, or be part of the map projection definition. An example of the latter is the factor of 0.9996 (-400ppm) in the UTM projection.
Directions¶
Corrections for Plumb Line Deviation¶
\(\alpha_{ell} = \alpha_{obs} + \frac{(L_{east} \cdot \cos \phi - L_{north} \cdot \sin \phi)}{\tan(z)}\)
\(\alpha\) : Direction
\(L_{east}\) : Plumb line deviation in eastward direction
\(L_{north}\) : Plumb line deviation in northward direction
\(\phi\) : Azimuth
\(z\) : Zenith distance
Gauss-Kruger Projection (Transverse Mercator)¶
\(\alpha_{corr} = \alpha_{ell} - \frac{x_2-x_1}{6R_r^2} ( 2y_1+y_2)\)
\(x_1,x_2,y_1,y_2\) : Map projection coordinates to station and target.
\(R_r\) : Mean radius of curvature for the foot point of the nearest third point on the line.
\(\alpha\) : Direction
If the distance is over 10,000 m and the East coordinate is over 70,000 m, the formula is extended with a third-degree term.
Lambert Conformal Conic Projection¶
\(\alpha_{corr} = \alpha_{ell} - \frac{y_2 -y_1}{6R_r^2} ( 2x_1+x_2)\)
\(x_1,x_2,y_1,y_2\) : Map projection coordinates to station and target.
\(R_r\) : Mean radius of curvature for the foot point of the nearest third point on the line
\(\alpha\) : Direction
If distance is over 10,000 m and X coordinate is over 70,000 m, the formula is extended with a third-degree term.
Conformal Stereographic Projection¶
\(\alpha_{corr} = \alpha_{ell} - \frac{1}{4 \cdot R_r^2} \cdot ( x_1 \cdot (y_2-y_1)-y_1 \cdot (x_2-x_1))\)
\(x_1,x_2,y_1,y_2\) : Map projection coordinates to station and target.
\(R_r\) : Mean radius of curvature for the foot point of the nearest third point on the line
\(\alpha\) : Direction
Vertical Angles¶
Correction for Earth Curvature and Refraction¶
\(Z_{plane} = Z_{obs} - \frac{S_{ell}(1-k)}{2R_m}\)
\(S_{ell}\) : Ellipsoidal distance
k: Refraction coefficient
\(R_m\) : Radius of curvature at the midpoint of the line, calculated according to Euler's formula
Correction for Instrument and Target Height¶
\(Z_{corr} = \arctan \left[ \frac{H_{dist}}{H_{dist} \cdot \cot(Z_{plane}) + i - s} \right]\)
\(H_{dist}\) : Distance at terrain height calculated from the coordinates.
The following corrections are applied:
- Map projection
- General scale factor
- Height above sea level