Expand Research On Regularization Method Of Ill-posed Equations In Surveying And Mapping

Ill-posed often exists in the measurement data processing,and the impact is large,which will reduce the reliability of parameter estimation.The morbidity is reflected in the singular value of the coefficient matrix of the observation equation which is small or even close to zero,which causes the variance of the parameter estimation to be greatly expanded by the small singular value,resulting in the decrease of the estimation accuracy.Based on the regularization method for dealing with ill-posed problems,this paper has improved the related methods.The main research work and results of the thesis are as follows:In the parameter solving process under the ill-posed of the coefficient matrix,reasonable selection of regularization parameters and regularization matrix can improve the reliability of parameter estimation.Aiming at the problem of how the regularization matrix is constructed,this paper proposes a new regularization matrix construction method.A symmetric matrix is constructed by the eigenvectors corresponding to the smaller singular values of the normal matrix,and the diagonal matrix is constructed by using the main diagonal elements of the matrix,and then combined with the unit matrix to obtain a new regularization matrix.The effectiveness and feasibility of the proposed algorithm are verified by an example.The ridge estimation is a special case of the regularization method.When the stable parameter estimation is obtained,the estimation variance is reduced,the deviation is increased,and the overall variance reduction is greater than the deviation introduction amount.The ridge estimation is usually not able to be calculated in a single calculation to minimize the mean square error.In order to obtain a smaller mean square error,multiple ridge estimation calculations can be performed.This paper derives the ridge estimation iteration method.The ridge estimation parameter estimate is brought into the adjustment model,the observation vector is updated,and the ridge estimation method is used to solve the parameters again.This iteration is used to calculate the variance and deviation for each iteration,and terminate when the mean square error reaches a minimum or converges.The ridge estimation iterative methodis validated by an example,the results show the effectiveness of the method.Ill-posed is a small eigenvalue of the normal matrix.It is more reasonable to target the modified singular value.The regularized matrix of the targeted correction is constructed by the eigenvector corresponding to the eigenvector corresponding to the smaller eigenvalue of the normal matrix.According to the characteristics of the targeted regularization matrix,it is applied in two aspects: First,in the iterative calculation of the ill-posed total least squares regularization method,the coefficient matrix is constantly changing.To target the correction matrix,the targeting regularity The matrix should also change.Aiming at the problem of targeting matrix change,this paper deduces two ill-posed total least squares singular value correction methods.By finding the new coefficient matrix,we then find the target regularization matrix,then iteratively calculate the parameter estimation,and use the example.Experiments show that the method has certain advantages.Second,the spectral correction iterative method is to correct all the spectra of the normal matrix and iteratively calculate the result.The result is unbiased estimation,and the ill state is several spectral singularities of the normal matrix,so there is a problem of spectral redundancy correction.Aiming at this problem,this paper deduces the target spectrum correction iterative method,and uses the regular regularization matrix to correct the singular spectrum of the matrix.The validity and feasibility of the method are verified by an example.In the process of acquisition of measurement data,there are often uncertainties,which will affect the parameter estimation results.The solution method of uncertainty adjustment model can effectively improve the validity and reliability of parameter estimation.When the coefficient matrix error of the observation equation is close to zero singular value,the regularization method can effectively suppress the influence of the ill-posed state of the observation equation on the parameter estimation results.When the uncertainty adjustment model is ill-posed,it is more seriously affected by the coefficient matrix error and the error of the observation value.In this paper,the regularization method is applied to the ill-posed uncertainty adjustment model,and the iterative algorithm is derived to improve the stability of the solution.By examples,the results show the effectiveness and feasibility of the new method.