The Liu-Type Estimator Based on Parameter Optimization and its Application in SBAS Deformation Model Inversion

Abstract

A situation in which an image is combined with multiple images to form interferometric pairs is often observed in small baseline subset-interferometric synthetic aperture radar (SBAS-InSAR) deformation inversion, and this situation leads to a near linear correlation between the column vectors of the model design matrix. The Liu-type estimator introduces the parameters $k$ and $d$ into the normal equation to reduce the condition number of the design matrix and to improve the fitting properties. As the parameter $k$ is mainly used to reduce ill-posed problems of the design matrix, the value of $k$ is not limited. However, the value of $k$ , as determined by existing methods, is usually too large or too small. Since the calculation of the mean square error involves true values, the parameter $d$ is often affected by errors in the estimation results, which leads to the decreased accuracy of Liu-type estimation results. To determine the optimal value of $d$ , an iterative Liu-type estimator is proposed to eliminate errors. Then, the $L$ -curve optimization method and iterative Liu-type estimator are combined to achieve the optimal $k$ . The reliability and accuracy of the methods are analyzed through SBAS-InSAR deformation experiments. The experimental results show that after using the $L$ -curve method and an iterative operation to optimize $k$ and $d$ , the accuracy of the Liu-type estimator based on parameter optimization is clearly improved compared with that of the ridge estimator and the Liu-type estimator.

Keywords

• Liu-type estimator
• iterative method
• L-curve
• ill-posed problem
• biased estimator
• least squares estimator