Ill-posed analysis and regularized estimation for seafloor geodetic acoustic single-difference positioning

Area of Science:

• Geodesy
• Oceanography
• Geophysics

Background:

• Seafloor geodetic acoustic positioning accuracy is significantly impacted by long-term systematic errors.
• The single-difference positioning method is crucial for mitigating these systematic errors.
• Ill-posed problems, arising from poor observation geometry and collinearity, challenge single-difference positioning accuracy.

Purpose of the Study:

• To analyze the spatial distribution of condition numbers in single-difference positioning under various track and heave conditions.
• To investigate how observation structure optimization affects the ill-posed degree of single-difference positioning.
• To propose and validate a novel method (L-curve Liu type) for addressing the unavoidable ill-posed problems in single-difference positioning.

Main Methods:

• Analysis of condition numbers for single-difference positioning across different tracks and heaves.
• Optimization of observation structures to reduce collinearity and ill-posedness.
• Development and application of the L-curve Liu type method, utilizing the maximum curvature point of the L-curve (mean square error vs. residual two norm) for parameter and estimation optimization.

Main Results:

• Circular tracks with equidistant measurements exacerbate collinearity, increasing the ill-posed degree.
• Optimizing observation structure can effectively reduce the ill-posedness of single-difference positioning.
• The proposed L-curve Liu type method successfully reduces the impact of ill-posed problems.

Conclusions:

• Single-difference positioning is effective in eliminating systematic errors for seafloor geodetic acoustic positioning.
• The L-curve Liu type method provides a robust solution for the ill-posed nature of single-difference positioning.
• Experimental validation in the South China Sea and Panglong Lake confirms the efficacy of the proposed method.