Research On The Solution Of ?-posed Problems And Application In SBAS-InSAR Deformation Inversion

SBAS-InSAR(Small Baseline Subset Interferometric Synthetic Aperture Radar)is an interferometry technology developed on the basis of DInSAR(Differential Interferometric Synthetic Aperture Radar).It inherits the advantages of DInSAR in wide range,all-weather,fast real-time,and reduce the data quantities and improve data utilization,which has been an important technique for monitoring earthquake,landslide,debris flow and other geological disasters.However,in the practical application of surface deformation monitoring by SBAS-InSAR,the ill-posed problems in the solution of deformation model is existed,which seriously affected the inversion accuracy and reliability of deformation information.The ill-posed problem is the problem of instability of solutions caused by the influence of the error in observation model.It is widely existed in the surveying data processing and has the greater influence,which has always been the research hotspot in geodesy.The research status at current home and abroad of solving method of ill-posed problems and SBAS-InSAR deformation model inversion is systematically reviewed in this paper.According to existing problem in the classical solving method of ill-posed problem in geodesy field,the new theories and methods of biased estimator are analyzed,studied and improved by combined the actual situation of SBAS-InSAR deformation model inversion.The main contents and innovations are as follows:(1)The Characteristic Value Correction Iteration Method Based on L-curveBased on the proof that the characteristic value correction iteration method finally converges to least squares estimator,it is found that the iteration termination condition which based on the difference between adjacent iteration results is less than the threshold is not conducive to the optimal selected of estimation results.Therefore,the characteristic value correction iteration method based on L-curve is proposed.That is,by drawing the L curve of iterative progress,the estimation results of characteristic value correction iteration method with stable estimation and small residual is determined.Through numerical example and SBAS-InSAR deformation model inversion experiments,it is verified that the novel method is better than classical characteristic value correction iteration method in the improvement degree of ill-posed problem.However,the experiments show that although the classical characteristic value correction iteration method and novel method can improve least squares estimator,but it is not uniformly better than ridge estimator.The relationship between characteristic value correction iteration method and damped least squares estimator make it still have research value in solving method of ill-posed problem.(2)Regularization method based on singular value correction of minimum mean square errorBy adding constrains to ill-posed problem,the regularization method transforms the ill-posed problem to a less ill-posed problem or well-posed problem to solve.The key is the determination of regularization parameter and regularization matrix.In the existing method that the regularization matrix is constructed by right singular vector of smaller singular value,the variance is depressed and introduced biases is reduced by correcting singular values.As the determination criterion of smaller singular values is not clear,then the novel method that using minimum mean square error to determine boundary of smaller singular values satisfying the optimal estimation results is proposed.Through numerical example and SBAS-InSAR experiments,it is proved that the change of determination criterion of smaller singular values can affect the improvement degree of ill-posed problem.The regularization method based on singular value correction of minimum mean square error has a more rigorous theoretical basis,and the estimation results are stable and reliable.(3)Regularization method based on singular value correction of exponential functionThe stable functional added in the regularization method is usually defined as a nonnegative definite regularization matrix.The aim of regularization matrix constructed by the existing method is usually to satisfy the selective correction of singular value,and the nonnegative definite property of matrix is ignored.In order to satisfy the nonnegative definite property of regularization matrix,a regularization method based on singular value correction of exponential function is proposed.In this method,the standard deviation component of each parameter estimation corrected by regularization parameter is used as the independent variable.Though the selective correction of singular values,the emphasis correction of smaller singular value is realized and correction degree of larger singular value is reduced.Though numerical example and SBAS-InSAR deformation model inversion experiments,the effectiveness of regularization method based on singular value correction of exponential function is verified,and the estimation results are more superior than ridge estimator.(4)Determination of optimal parameter d in Liu type estimatorBy introducing double parameter of a and d,Liu type estimator uses parameter ? to reduce the ill-posedness of coefficient matrix and uses parameter d to improve fitting property of estimation results.Hence,the optimal value determination of parameters in Liu type estimator is crucial to improve the accuracy of estimation results.When parameter d is determined based on principle of minimum mean square error,the accurate estimation results and unit weight mean square error are generally required.But the estimation results and unit weight mean square error of least squares estimator affected by ill-posedness have been seriously distorted.Hence,the method that using iterative calculation to eliminate the error influence of initial value of estimation results on parameter d is proposed,and the convergence of the iteration method is proved by rigorous theoretical derivation.Though numerical example and SBAS-InSAR deformation model inversion experiments,it is proved that the iterative calculation can realize gradual correction of parameter d and estimation results.(5)Determination of optimal parameter ? in Liu type estimatorParameter ? in Liu type estimator is usually determined by empirical formula of condition number judging the degree of ill-posed,that is,if the condition number is less than 100,the degree of ill-posed problem is considered as unserious.According to the principle,the parameter ? is adjusted to make the condition number of coefficient matrix as 100.However,the theoretical basis of parameter determination method based on empirical formula is not strict enough,and it has great uncertainty on the improvement degree of ill-posed problem.Therefore,the optimal value determination method of parameter ? based on L curve is proposed.By calculating estimation results of different parameter ? in a certain range and then drawing L curve,the inflection point of L curve with stable estimation result and smaller residual is determined to obtain the optimal value of parameter a and estimation results.Through numerical example and SBAS-InSAR deformation model inversion experiments,the effectiveness of optimal value determination method of ? based on L curve is proved.