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

The single-difference positioning method could eliminate the systematic error of long periods, which is one of the major factors affecting the seafloor geodetic acoustic positioning accuracy. Due to the poor observation geometry in short observation time, there is collinearity in the coefficient matrix. Therefore, a small observation error may lead to a large error in the least square solution, which is the ill-posed problem of single-difference positioning. The spatial distribution of the condition numbers of single-difference positioning with different tracks and heaves is analyzed. The circular tracks with almost equal distance measurements will cause serious complex collinearity, and the optimization of observation structure can reduce the ill-posed degree of single-difference positioning. For the unavoidable ill-posed problem of single-difference positioning, L-curve Liu type method is proposed. By calculating the maximum curvature point of L-curve composed of mean square error and residual two norm, the parameter optimization and estimation optimization of Liu-type estimation are realized. Through the experiments in Panglong Lake and South China Sea, it is verified that the single-difference positioning helps to eliminate the effect of systematic errors, and the proposed L-curve Liu type method can reduce the effect of ill-posed problem of single-difference positioning.