A Method for the Precise Coordinate Determination of an Inaccessible Location
Abstract
:1. Introduction
- The problem: to obtain precise geocentric coordinates of inaccessible locations for photogrammetry or LiDAR survey.
- A method is proposed based on the directional intersection of the inaccessible locations using a total station from two stations with measured GNSS coordinates.
- Solution: the non-linear directional intersection observation equations are formed.
- The equally weighted least-squares solution of the non-linear observational equations (LMA) is calculated.
- Linearisation of observation equations and formulation of the Gauss–Helmert model (GHM) is carried out.
- A solution of the GHM (the weighted least-squares method) is obtained.
- Field experiments are carried out. The results of the LMA and GHM are compared.
2. Materials and Methods
2.1. Method
2.1.1. Basic Relationships
2.1.2. Least-Squares Solution of the Intersection Equations
2.1.3. Differential Relationships
2.1.4. LSQ Adjustment of the Intersection Based on the Gauss–Helmert Model
2.2. Experimental Design
2.3. Data
- Both experiments used a Leica Nova MS50 total station to measure the angles and distances (s, α, β). The heights of the instrument (i) and reflector (j) were also measured. The measurements obtained during the first experiment are listed in Table 1. The symbol ”A” denotes an inaccessible point, while 1 and 2 denote sites of the total stations from which measurements were carried out.
- The geocentric coordinates (X, Y, Z) of points 1 and 2 were measured using the Leica Viva GS08plus SmartAntenna GNSS receiver. The ASG-EUPOS service (https://www.asgeupos.pl/index.php, accessed on 25 September 2023) provided the RTK correction. The geocentric GNSS coordinates are listed in Table 2.
- The north-south ξ and east-west η components of the deflection of the vertical axis of the instrument, relative to the normal to ellipsoid, can be computed from an Earth Gravity Model, for example, EGM 2008 (https://www.usna.edu/Users/oceano/pguth/md_help/html/egm96.htm, accessed on 25 September 2023). The computations are based on the fundamental relations of the physical geodesy:ξ = Φ − φ,η = (Λ − λ) cos φwhereφ and λ are the latitude and longitude obtained from GNSS, andΦ and Λ are the astronomical latitude and longitude, respectively.
3. Results
3.1. Experiment 1
3.1.1. The Levenberg–Marquardt Algorithm (LMA) Adjustment
3.1.2. The Gauss–Helmert Model (GHM)
3.2. Experiment 2
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Observation | s (m) | α (gon) | β (gon) | i (m) | j (m) |
---|---|---|---|---|---|
1 to 2 | 37.121 | 0.0489 | 100.1286 | 1.611 | 1.500 |
1 to A | 43.571 | 339.2618 | 65.1532 | 1.611 | 2.150 |
2 to 1 | 37.124 | 71.5203 | 100.2894 | 1.635 | 1.500 |
2 to A | 40.953 | 141.2695 | 62.7610 | 1.635 | 2.150 |
Std Dev. | 0.006 | 0.001 | 0.001 | 0.001 | 0.001 |
Coordinate | 1 | 2 | A | Std Dev. |
---|---|---|---|---|
X (m) | 3,835,779.346 | 3,835,758.231 | 3,835,763.321 | 0.008 |
Y (m) | 1,177,321.994 | 1,177,351.033 | 1,177,324.809 | 0.008 |
Z (m) | 4,941,536.189 | 4,941,545.624 | 4,941,576.310 | 0.008 |
Deflection of Vertical | 1 | 2 | Std Dev. |
---|---|---|---|
ξ (″) | 5.9926 | 5.9852 | 1 |
η (″) | 6.2033 | 6.1967 | 1 |
Observation | s (m) | α (gon) | β (gon) | i (m) | j (m) |
---|---|---|---|---|---|
3 to 4 | 33.007 | 31.8735 | 100.4359 | 1.685 | 1.500 |
3 to B | 40.753 | 107.2129 | 63.2411 | 1.685 | 1.900 |
3 to C | 43.474 | 94.5445 | 65.8200 | 1.685 | 1.900 |
3 to D | 47.413 | 85.4091 | 68.9707 | 1.685 | 1.900 |
4 to 3 | 33.009 | 225.3364 | 100.2523 | 1.657 | 1.500 |
4 to B | 43.612 | 161.3899 | 65.8203 | 1.657 | 1.900 |
4 to C | 40.228 | 149.3560 | 62.6006 | 1.657 | 1.900 |
4 to D | 38.690 | 135.5829 | 60.9652 | 1.657 | 1.900 |
Std Dev. | 0.006 | 0.001 | 0.001 | 0.001 | 0.001 |
Point | 3 | 4 | B | C | D | Std Dev. |
---|---|---|---|---|---|---|
X (m) | 3,835,762.327 | 3,835,780.698 | 3,835,764.596 | 3,835,769.196 | 3,835,773.170 | 0.008 |
Y (m) | 1,177,338.201 | 1,177,311.976 | 1,177,313.716 | 1,177,307.830 | 1,177,302.003 | 0.008 |
Z (m) | 4,941,545.590 | 4,941,537.594 | 4,941,577.938 | 4,941,575.760 | 4,941,574.056 | 0.008 |
Unknown | s(1–A) | s(2–A) | Σ1 | Σ2 |
---|---|---|---|---|
43.572 m | 40.974 m | 73.4638 gon | 201.9942 gon | |
Coordinates of A | ||||
X (m) | 3,835,763.325 | |||
Y (m) | 1,177,324.803 | |||
Z (m) | 4,941,576.312 |
Unknown | s(1–A) | s(2–A) | Σ1 | Σ2 |
---|---|---|---|---|
43.576 m | 40.966 m | 73.4693 gon | 201.9980 gon | |
Coordinates of A | ||||
X (m) | 3,835,763.322 | |||
Y (m) | 1,177,324.807 | |||
Z (m) | 4,941,576.311 |
Model | dX (m) | dy (m) | dz (m) |
---|---|---|---|
LMA | −0.004 | 0.007 | −0.002 |
GHM | 0.002 | −0.002 | 0.001 |
Target | δX (m) | δY (m) | δZ (m) | (m) |
---|---|---|---|---|
B | −0.006 | −0.001 | 0.006 | 0.008 |
C | 0.003 | −0.003 | 0.009 | 0.010 |
D | −0.003 | −0.013 | −0.003 | 0.014 |
RMSE | 0.004 | 0.008 | 0.006 | - |
Coordinates of A | |
---|---|
X (m) | 3,835,763.324 |
Y (m) | 1,177,324.807 |
Z (m) | 4,941,576.311 |
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Osada, E.; Owczarek-Wesołowska, M.; Karsznia, K.; Becek, K.; Muszyński, Z. A Method for the Precise Coordinate Determination of an Inaccessible Location. Sensors 2023, 23, 8199. https://doi.org/10.3390/s23198199
Osada E, Owczarek-Wesołowska M, Karsznia K, Becek K, Muszyński Z. A Method for the Precise Coordinate Determination of an Inaccessible Location. Sensors. 2023; 23(19):8199. https://doi.org/10.3390/s23198199
Chicago/Turabian StyleOsada, Edward, Magdalena Owczarek-Wesołowska, Krzysztof Karsznia, Kazimierz Becek, and Zbigniew Muszyński. 2023. "A Method for the Precise Coordinate Determination of an Inaccessible Location" Sensors 23, no. 19: 8199. https://doi.org/10.3390/s23198199
APA StyleOsada, E., Owczarek-Wesołowska, M., Karsznia, K., Becek, K., & Muszyński, Z. (2023). A Method for the Precise Coordinate Determination of an Inaccessible Location. Sensors, 23(19), 8199. https://doi.org/10.3390/s23198199