﻿ 小波稳健最小二乘估计的轨道误差校正算法及其在DEM中的应用
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 大地测量与地球动力学  2020, Vol. 40 Issue (12): 1273-1276,1289  DOI: 10.14075/j.jgg.2020.12.013

### 引用本文

HE Yonghong, JIN Pengwei. Orbit Error Correction Algorithm Based on Wavelet Robust Least Square Estimation and Its Application in DEM[J]. Journal of Geodesy and Geodynamics, 2020, 40(12): 1273-1276,1289.

### Foundation support

National Natural Science Foundation of China, No.41531068, 41671356; Natural Science Foundation of Hunan Province, No.2020JJ4031; Projects of Education Science Planning of Hunan Province, No.XJK19CGD051.

### Corresponding author

JIN Pengwei, lecturer, majors in surveying and mapping and geotechnical engineering, E-mail:65450015@qq.com.

### 第一作者简介

HE Yonghong, PhD, associate professor, majors in InSAR data processing and application, E-mail:365022968@qq.com.

### 文章历史

1. 湖南科技学院土木与环境工程学院，湖南省永州市杨梓塘路130号，425199

1 小波稳健最小二乘估计的轨道误差校正算法 1.1 基于小波理论的InSAR差分干涉相位误差分析

 ${\varphi _{{\rm{diff}}}} = {\varphi _{{\rm{topo}}}} + {\varphi _{{\rm{orbit}}}} + {\varphi _{{\rm{atm}}}} + {\varphi _{{\rm{noi}}}}$ (1)

1.2 算法实现

 ${\phi ^{{\rm{orbit}}}}\left( {x, y} \right) = a + bx + cy + dxy + e{x^2} + f{y^2}$ (2)

 $\mathit{\boldsymbol{L}} = \mathit{\boldsymbol{BX}} + \mathit{\boldsymbol{\Delta }}$ (3)

 $\mathit{\boldsymbol{\Delta }} = \mathit{\boldsymbol{BX}} - \mathit{\boldsymbol{L}}$ (4)

 $\mathit{\boldsymbol{X}} = {\left( {{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}}} \right)^{ - 1}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PL}}$ (5)

 $\omega _k^j = \frac{1}{{\left| {{\bf{\Delta }}_k^j} \right| + u}}, k = 1, 2, \cdots , n;j = 1, 2, \cdots , m$ (6)

 $\begin{array}{l} \mathit{\boldsymbol{P}}_{ii}^j = \mathit{\boldsymbol{P}}_{ii}^{j - 1} \cdot \omega _i^j\\ {\mathit{\boldsymbol{P}}^j} = {\rm{diag}}\left[ {P_1^j{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} P_2^j{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} P_n^i} \right]\\ {{\mathit{\boldsymbol{\hat X}}}^j} = {\left( {{\mathit{\boldsymbol{B}}^\rm{T}}{\mathit{\boldsymbol{P}}^j}\mathit{\boldsymbol{B}}} \right)^{ - 1}}{\mathit{\boldsymbol{B}}^\rm{T}}{\mathit{\boldsymbol{P}}^j}\mathit{\boldsymbol{L}}\\ {\mathit{\boldsymbol{\Delta }}^j} = \mathit{\boldsymbol{B}}{{\mathit{\boldsymbol{\hat X}}}^j} - \mathit{\boldsymbol{L}} \end{array}$ (7)

 $\max \left\{ {\left| {\frac{{{\mathit{\boldsymbol{X}}^j} - {\mathit{\boldsymbol{X}}^{j - 1}}}}{{{\mathit{\boldsymbol{X}}^{j - 1}}}}} \right|} \right\} ＜ \delta$ (8)

2 实验结果与分析 2.1 模拟数据实验结果与分析

 图 1 模拟实验数据 Fig. 1 Simulation experiment data

 图 2 不同方法估计的轨道误差 Fig. 2 Orbital error estimated by different methods
2.2 实测数据验证

 图 3 义马地区实测数据及小波分解结果 Fig. 3 Measured data and wavelet decomposition results in Yima area

 图 4 不同方法估计的轨道误差与改正结果比较 Fig. 4 Comparison of orbital errors and correction results by different methods

 图 5 局部地区不同方法改正后生成的DEM结果对比图 Fig. 5 Comparison of DEM results after correction by different methods in local area
3 结语

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Orbit Error Correction Algorithm Based on Wavelet Robust Least Square Estimation and Its Application in DEM
HE Yonghong1     JIN Pengwei1
1. School of Civil and Environmental Engineering, Hunan University of Science and Engineering, 130 Yangzitang Road, Yongzhou 425199, China
Abstract: One problem in using the quadratic polynomial model for removing orbital errors is that it is necessary to assume the distribution properties of other interference phase. So, a method based on wavelet multi-scale analysis is proposed to remove orbital errors. Based on long wavelength and low frequency characteristics of orbital error phase, the method filters the phase with shortest wavelength compared to the orbital error in different scale spaces. Then, the robust least square method is used to estimate the parameters of the quadratic polynomial model in order to reduce the influence of residual terrain error phase on orbital errors polynomial fitting. The results show that the corrected interferograms contain less trend error, with better removal effect and higher reliability.
Key words: quadratic polynomial; multi-scale analysis; InSAR; orbital error