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Locally robust Msplit estimation

  • Patrycja Wyszkowska ORCID logo EMAIL logo and Robert Duchnowski ORCID logo
Published/Copyright: August 22, 2024
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Journal of Applied Geodesy
From the journal Journal of Applied Geodesy Volume 19 Issue 2

Abstract

Processing measurement data is an essential part of surveying engineering. One can list several methods in such a context: least squares estimation, M-estimation, R-estimation, etc. Some methods were developed by surveyors, e.g., the Danish method, IGG scheme, or Msplit estimation. The last method is, in fact, a class of estimation procedures dedicated to different problems. As a new approach to processing data, Msplit estimation is still being developed and improved. That paper concerns the local robustness of Msplit estimation and introduces a new Msplit estimation variant that is less sensitive to local outliers. Such a property seems important, especially in big data processing, such as observations from Light Detection and Ranging systems. The new variant modifies the squared Msplit estimation (SMS estimation) by implementing the adapted Tukey weight function, hence its acronym SMSTL estimation. The basic theoretical and empirical analyses, which were performed for the univariate model using, among others, the appropriate measures of robustness, confirmed the expected property of the method. The further tests, based on simulated as well as real data, show that the new method might overperform other Msplit estimation variants and classical methods for the chosen types of observation sets.

Keywords: Msplit estimation; local robustness; robust estimation; terrestrial laser scanning; airborne laser scanning
Corresponding author: Patrycja Wyszkowska, Department of Geodesy, Institute of Geodesy and Civil Engineering, Faculty of Geoengineering, University of Warmia and Mazury in Olsztyn, Oczapowskiego 1, 10-719, Olsztyn, Poland, E-mail: 

Funding source: Department of Geodesy, Institute of Geodesy and Civil Engineering, Faculty of Geoengineering, University of Warmia and Mazury in Olsztyn

Award Identifier / Grant number: 29.610.001-110

  1. Research ethics: Not applicable.

  2. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Competing interests: The authors state no conflict of interest.

  4. Research funding: This work was supported by the Department of Geodesy, Institute of Geodesy and Civil Engineering, Faculty of Geoengineering, University of Warmia and Mazury in Olsztyn [statutory research no. 29.610.001-110].

  5. Data availability: The raw data can be obtained on request from the corresponding author.

Appendix

The components of objective, influence, and weight functions of SMSTL estimation

(23) ρ v i ( 1 ) , v i ( 2 ) = v i ( 1 ) 2 v i ( 2 ) 2     for v i ( 1 ) < k 1 − c 1 k 2 2 − v i ( 1 ) 2 3 6 + c 2 v i ( 2 ) + Δ X ( 1,2 ) 2 − k 2 2 3 6 + C     for v i ( 1 ) ≥ k 1 ∧ v i ( 1 ) < k 2 C     for v i ( 1 ) ≥ k 2 ∧ v i ( 2 ) ≤ − k 2 − c 2 k 2 2 − v i ( 1 ) − Δ X ( 1,2 ) 2 3 6 − c 1 k 2 2 − v i ( 2 ) 2 3 6 + C     for v i ( 2 ) > − k 2 ∧ v i ( 2 ) ≤ − k 1 v i ( 1 ) 2 v i ( 2 ) 2     for v i ( 2 ) > − k 1
(24) ψ ( 1 ) v i ( 1 ) , v i ( 2 ) = 2 v i ( 1 ) v i ( 2 ) 2     for v i ( 1 ) < k 1 c 1 v i ( 1 ) k 2 2 − v i ( 1 ) 2 2     for v i ( 1 ) ≥ k 1 ∧ v i ( 1 ) < k 2 0     for v i ( 1 ) ≥ k 2 ∧ v i ( 2 ) ≤ − k 2 c 2 v i ( 1 ) − Δ X ( 1,2 ) k 2 2 − v i ( 1 ) − Δ X ( 1,2 ) 2 2     for v i ( 2 ) > − k 2 ∧ v i ( 2 ) ≤ − k 1 2 v i ( 1 ) v i ( 2 ) 2     for v i ( 2 ) > − k 1 ψ ( 2 ) v i ( 1 ) , v i ( 2 ) = 2 v i ( 1 ) 2 v i ( 2 )     for v i ( 1 ) < k 1 c 2 v i ( 2 ) + Δ X ( 1,2 ) k 2 2 − v i ( 2 ) + Δ X ( 1,2 ) 2 2     for v i ( 1 ) ≥ k 1 ∧ v i ( 1 ) < k 2 0     for v i ( 1 ) ≥ k 2 ∧ v i ( 2 ) ≤ − k 2 c 1 v i ( 2 ) k 2 2 − v i ( 2 ) 2 2     for v i ( 2 ) > − k 2 ∧ v i ( 2 ) ≤ − k 1 2 v i ( 1 ) 2 v i ( 2 )     for v i ( 2 ) > − k 1
(25) w ( 1 ) v i ( 1 ) , v i ( 2 ) = v i ( 2 ) 2     for v i ( 1 ) < k 1 c 1 k 2 2 − v i ( 1 ) 2 2 2     for v i ( 1 ) ≥ k 1 ∧ v i ( 1 ) < k 2 0     for v i ( 1 ) ≥ k 2 ∧ v i ( 2 ) ≤ − k 2 c 2 v i ( 1 ) − Δ X ( 1,2 ) k 2 2 − v i ( 1 ) − Δ X ( 1,2 ) 2 2 2 v i ( 1 )     for v i ( 2 ) > − k 2 ∧ v i ( 2 ) ≤ − k 1 v i ( 2 ) 2     for v i ( 2 ) > − k 1 w ( 2 ) v i ( 1 ) , v i ( 2 ) = v i ( 1 ) 2     for v i ( 1 ) < k 1 c 2 v i ( 2 ) + Δ X ( 1,2 ) k 2 2 − v i ( 2 ) + Δ X ( 1,2 ) 2 2 2 v i ( 2 )     for v i ( 1 ) ≥ k 1 ∧ v i ( 1 ) < k 2 0     for v i ( 1 ) ≥ k 2 ∧ v i ( 2 ) ≤ − k 2 c 1 k 2 2 − v i ( 2 ) 2 2 2     for v i ( 2 ) > − k 2 ∧ v i ( 2 ) ≤ − k 1 v i ( 1 ) 2     for v i ( 2 ) > − k 1

where: C = k 1 2 k 1 − Δ X ( 1,2 ) 2 + k 1 − Δ X ( 1,2 ) 2 k 2 2 − k 1 2 3 + k 1 k 1 − Δ X ( 1,2 ) k 2 2 − k 1 2 3 , c 1 = 2 k 1 − Δ X ( 1,2 ) 2 k 2 2 − k 1 2 2 , c 2 = 2 k 1 k 1 − Δ X ( 1,2 ) k 2 2 − k 1 2 2 , ΔX(1,2) = X(2) − X(1) is shift between parameter versions.

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Received: 2024-02-26
Accepted: 2024-08-03
Published Online: 2024-08-22
Published in Print: 2025-04-28

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Locally robust Msplit estimation
  4. Extending geodetic networks for geo-monitoring by supervised point cloud matching
  5. Evaluation and homogenization of a marine gravity database from shipborne and satellite altimetry-derived gravity data over the coastal region of Nigeria
  6. Modelling geoid height errors for local areas based on data of global models
  7. Unmanned aerial vehicle-based aerial survey of mines in Shanxi Province based on image data
  8. Ionospheric TEC and its irregularities over Egypt: a comprehensive study of spatial and temporal variations using GOCE satellite data
  9. Monitoring of volcanic precursors using satellite data: the case of Taftan volcano in Iran
  10. Modeling of temperature deformations on the Dnister HPP dam (Ukraine)
  11. Impact of temporal resolution in global ionospheric models on satellite positioning during low and high solar activity years of solar cycle 24
  12. Comparative performance of PPP software packages in atmospheric delay estimation using GNSS data
  13. Assessment and fitting of high/ultra resolution global geopotential models using GNSS/levelling over Egypt
  14. An efficient ‘P1’ algorithm for dual mixed-integer least-squares problems with scalar real-valued parameters
  15. Spatio-temporal trajectory alignment for trajectory evaluation
  16. Monitoring of networked RTK reference stations for coordinate reference system realization and maintenance – case study of the Czech Republic
  17. Crustal deformation in East of Cairo, Egypt, induced by rapid urbanization, as seen from remote sensing and GNSS data
https://doi.org/10.1515/jag-2024-0023
Keywords for this article
Msplit estimation; local robustness; robust estimation; terrestrial laser scanning; airborne laser scanning

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Locally robust Msplit estimation
  4. Extending geodetic networks for geo-monitoring by supervised point cloud matching
  5. Evaluation and homogenization of a marine gravity database from shipborne and satellite altimetry-derived gravity data over the coastal region of Nigeria
  6. Modelling geoid height errors for local areas based on data of global models
  7. Unmanned aerial vehicle-based aerial survey of mines in Shanxi Province based on image data
  8. Ionospheric TEC and its irregularities over Egypt: a comprehensive study of spatial and temporal variations using GOCE satellite data
  9. Monitoring of volcanic precursors using satellite data: the case of Taftan volcano in Iran
  10. Modeling of temperature deformations on the Dnister HPP dam (Ukraine)
  11. Impact of temporal resolution in global ionospheric models on satellite positioning during low and high solar activity years of solar cycle 24
  12. Comparative performance of PPP software packages in atmospheric delay estimation using GNSS data
  13. Assessment and fitting of high/ultra resolution global geopotential models using GNSS/levelling over Egypt
  14. An efficient ‘P1’ algorithm for dual mixed-integer least-squares problems with scalar real-valued parameters
  15. Spatio-temporal trajectory alignment for trajectory evaluation
  16. Monitoring of networked RTK reference stations for coordinate reference system realization and maintenance – case study of the Czech Republic
  17. Crustal deformation in East of Cairo, Egypt, induced by rapid urbanization, as seen from remote sensing and GNSS data
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