﻿ ﻿ 复数域最小二乘平差及其在PolInSAR植被高反演中的应用
 文章快速检索 高级检索

Criterion of Complex Least Squares Adjustment and Its Application in Tree Height Inversion with PolInSAR Data
ZHU JianjunJIE QinghuaZUO Tingying,WANG Changcheng,XIE Jian
School of Geosciences and Info-Physics, Central South University, Changsha 410083, China
First author: ZHU Jianjun(1962—), male, professor, PhD supervisor, majors in surveying adjustment and data processing.E-mail：zjj@mail.csu.edu.cn
Abstract:At present, data processing methods involving complex observations are mainly step-by-step or direct solver based on the observation process which cannot consider observation errors, redundant observation and so on. For this situation, this paper introduces least squares methods of complex data processing and tries to extend surveying adjustments from the real number space to the complex number space. Meanwhile, the two adjustment criteria in complex domain are compared quantitatively. In order to understand effectiveness of complex least squares, the tree height inversion from PolInSAR data is taken as an example. We firstly establish complex adjustment function model and stochastic model for PolInSAR tree height inversion and apply complex least squares method to estimate tree height. The results show that the complex least squares approach is reliable and better than other classic tree height retrieval methods. Besides, the method is simple and easy to realize.
Key words: surveying adjustment     complex least squares     polarimetric interferometric SAR (PolInSAR)     tree height inversion     three-stage algorithm

1 引 言

2 复数域最小二乘

(1) 第1种方法是以复数观测值残差的实部和虚部的平方和分别最小作为平差准则[12, 13]。即将观测方程分解为实部和虚部，利用传统的最小二乘分别求解未知参数的实部和虚部，然后合并参数实部和虚部的估计值从而得到最终的未知参数的估计值。下式即为分解后的观测方程

(2) 第2种方法是以复数观测值残差的模的平方和最小作为平差准则[12, 13, 14]。由于数学上模的平方等于实部的平方加上虚部的平方，故模的平方和最小意味着要保证复数观测值的实部残差平方和和虚部残差平方和的总和最小。这种准则同时兼顾了观测值的实部和虚部的误差，其表达式为

3 复数域平差准则对比

 平差准则 解算方法 Y 的中误差 Y 的均方根误差 残差的实部和虚部的平方和分别最小 直接解法 12.47 10.96 迭代解法 12.47 10.96 残差的模的平方和最小 直接解法 1.31 0.25 迭代解法 1.31 0.25

4 PolInSAR植被高反演的复数最小二乘算法

4.1 复数最小二乘植被高反演算法

4.1.1 复数最小二乘植被高反演算法的函数模型

4.1.2复数最小二乘植被高反演算法的随机模型

N代表参与平差的复相干性个数。式(17)即为的复数域最小二乘算法的随机模型。

4.1.3 复数最小二乘植被高反演算法的平差策略

4.2 植被高反演质量对比方法

4.2.1 DEM差分算法

DEM差分算法的基本思想是利用两种相位中心高度分别接近于树冠和树底的极化方式的复相干性γwV和γwS。假设γwV对应的地体幅度比μwV为0，代入到 RVOG 模型可以求得地表相位的估计值0，将接近于树冠的γwV与地表相位0的相位差作为植被高对应的干涉相位，最后根据相位高程转换关系将相位差转换为植被高V。具体算法表达式如下[19]

4.2.2 三阶段算法

5 试验结果及分析

5.1 数据介绍

 图 1 模拟的植被场景图(a)和落叶林模型(b) Fig. 1 The simulated forest scene(a) and deciduous forest model(b)

 平台高度/m 中心频率/GHz 入射角 垂直基线/m 水平基线/m 方位向分辨率/m 斜距向分辨率 植被类型 植被高度/m 3 674.235 1.3 30° -6.1 10.6 1 0.5 m 落叶林 10

 图 2 L波段数据HH功率图(a)、HV功率图(b)、VV功率图(c) Fig. 2 L-band data power diagrams corresponding to HH(a),HV(b),VV(c)

5.2 植被高反演结果及分析

 图 3 L波段极化干涉数据植被高反演结果 Fig. 3 Tree height estimation based on L-band PolInSAR data

 图 4 两种方法得到的植被高结果统计直方图 Fig. 4 Histograms of tree height results corresponding to two methods

 植被高反演方法 植被高真值 植被高均值 平均误差 均方根误差 DEM差分算法 10 4.17 5.82 5.88 三阶段算法 10 8.92 1.15 1.34 复数最小二乘 10 9.17 0.91 1.07
6 结 论

(1) 对于目前复数域数据处理时常用的两种平差准则，采用残差的模的平方和最小的平差准则得到的参数估计结果比采用采用残差的实部和虚部的平方和分别最小的平差准则得到的参数估计值更为准确，精度更高。

(2) 采用复数域最小二乘的方法来处理极化干涉SAR植被高反演所得的植被高精度高，明显优于DEM差分算法的精度，略优于三阶段算法的精度，并且复数域最小二乘计算还具有表达直观、编程简单、能进行精度评定的等优点。

(3) 对于PolInSAR植被高反演的复数域最小二乘，观测值按式(18)定权是可行的。

PolInSAR植被高反演模型只是复数平差模型的一个范例，实际上在大地测量的其他领域广泛存在。目前在实数域最小二乘平差模型中已形成了一整套成熟的参数估计和精度评定的理论，而复数域测量相应的计算方法和精度评定尚无系统的理论与方法。因此，把局限于实数域的经典最小二乘平差理论拓展到复数域，系统研究复数域平差模型的理论与方法，并根据复数本身的特性添加新的元素，拟建立起一整套复数非线性最小二乘的参数估计和精度评定的理论是今后研究的主要方向。

 [1] TREUHAFT R N, MADSEN S N, MOGHADDAM M, et al. Vegetation Characteristics and Underlying Topography from Interferolnetric Radar[J]. Radio Science, 1996, 31(6): 1449-1485. [2] CLOUDE S R, PAPATHANASSIOU K P. Polarimetric SAR Interferometry[J]. IEEE Transactions on Geoscience and Remote Sensing, 1998, 36(5): 1551-1565. [3] TREUHAFT R N, SIQUEIRA P R. Vertical Structure of Vegetated Land Surfaces from Interferometric and Polarimetric Radar[J]. Radio Science, 2000, 35(1): 141-177. [4] PAPATHANASSIOU K P, CLOUDE S R. Single-baseline Polarimetric SAR Interferometry[J]. IEEE Transactions on Geoscience and Remote Sensing, 2001, 39(11): 2352-2363. [5] CLOUDE S R, PAPATHANASSIOU K P. Three-stage Inversion Process for Polarimetric SAR Interferometry[J]. IEE Proceedings: Radar, Sonar and Navigation, 2003, 150(3): 125-134. [6] GILBOA G, SOCHEN N. Image Enhancement and Denoising by Complex Diffusion Processes[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004, 26(8): 1020-1036. [7] GRIGOLI F, CESCA S, DAHM T, KRIEGER L. A Complex Linear Least-squares Method to Derive Relative and Absolute Orientations of Seismic Sensors[J].Geophysical Journal International, 2012, 188(3): 1243-1254. [8] OPPENHEIM A V, LIM J S. The Importance of Phase in Signals[J]. Proceedings of the IEEE, 1981, 69(5): 529- 541. [9] TAEIGHAT A, SAYED A H. Least Mean-phase Adaptive Filters with Application to Communications Systems[J]. IEEE Signal Processing Letters, 2012, 11(2): 220-223. [10] DANILO P M, SU L G, KAZUYUKI A. Sequential Data Fusion via Vector Spaces: Fusion of Heterogeneous Data in the Complex Domain[J]. The Journal of VLSI Signal Processing, 2007, 48(1): 99-108. [11] THOMAS P. Navigation Signal Processing for GNSS Software Receivers[M]. Boston: Artech House Publishers, 2010. [12] GU Xiangqian, KANG Hongwen, CAO Hongxing. The Least-square Method in Complex Number Domain[J]. Progress in Natural Science, 2006, 16(1): 49-54. (谷湘潜, 康红文, 曹洪兴. 复数域内的最小二乘法[J]. 自然科学进展, 2006, 16(1): 49-54.). [13] LI Mengxia, CHEN Zhong. The Modification of Least Square Method(LSM) in the Complex Field[J]. Journal of Yangtze University: Natural Science Edition, 2008, 5(3): 7-8. (李梦霞, 陈忠. 复数域内最小二乘法的一种改进[J] . 长江大学学报: 自然科学版, 2008, 5(3): 7-8.). [14] MILLER K S. Complex Linear Least Squares[J]. SIAM Review, 1973, 15(4): 706-726. [15] WU Yirong, HONG Wen, WANG Yanping. The Current Status and Implications of Polarimetric SAR Interferometry[J]. Journal of Electronics & Information Technology, 2007, 29(5): 1258-1262. (吴一戎, 洪文, 王彦平. 极化干涉SAR的研究现状与启示[J]. 电子信息学报, 2007, 29(5): 1258-1262.) . [16] LI Tingwei, LIANG Diannong, ZHU Jubo. A Review of Inversion of the Forest Height by Polarimetric Interferometric SAR[J]. Remote Sensing Information, 2009, 3: 85-90. (李廷伟, 梁甸农, 朱炬波. 极化干涉SAR森林高度反演综述[J]. 遥感信息, 2009(3): 85-90.) . [17] YANG Lei, ZHAO Yongjun, WANG Zhigang. Polarimetric Interferometric SAR Data Analysis Based on TLS-ESPRIT of Joint Estimation of Phase and Power[J]. Acta Geodaetica et Cartographica Sinica, 2007, 36(2): 163-168. (杨磊, 赵拥军, 王志刚. 基于功率和相位联合估计TLS-ESPRIT算法的极化干涉SAR数据分析[J]. 测绘学报, 2007, 36(2): 163-168.) . [18] TAN Lulu, YANG Libo, YANG Ruliang. Investigation of Tree Height Retrieval with Polarimetric SAR Interferometry Based on ESPRIT Algorithm[J]. Acta Geodaetica et Cartographica Sinica, 2011, 40(3): 296-300. (谈璐璐, 杨立波, 杨汝良. 基于ESPRIT算法的极化干涉SAR植被高度反演研究[J]. 测绘学报, 2011, 40(3): 296-300.) . [19] LUO Huanming. Models and Methods of Extracting Forest Structure Information by Polarimetric SAR Interferometry[D]. Chengdu: University of Electronic Science and Technology of China, 2011. (罗环敏. 基于极化干涉SAR的森林结构信息提取模型与方法[D] . 成都:电子科技大学,2011.) . [20] WANG Xinzhou. Non-linear Model Parameter Estimation Theory and Application[M]. Wuhan: Wuhan University Press, 2002. (王新洲. 非线性模型参数估计理论与应用[M]. 武汉:武汉大学出版社, 2002.) . [21] SEYMOUR M S, CUMMING I G. Maximum Likelihood Estimation for SAR Interferometry[C]//Proceedings of International IGARSS '94: Surface and Atmospheric Remote Sensing: Technologies, Data Analysis and Interpretation. New York: IEEE, 1994, 4: 2272-2275. [22] FREEMAN A, DURDEN S L. A Three-component Scattering Model for Polarimetric SAR Data[J]. IEEE Transactions on Geoscience and Remote Sensing, 1998, 36(3): 963-973. [23] WILLIAMS M L. PolSARproSim Design Document and Algorithm Specification(u1.0)[R/OL]. (2006-12-01)[2012-03-20]. http://earth.eo.esa.int/polsarpro/Manuals/PolSARproSim_Design.pdf.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.
0007

0

#### 文章信息

ZHU Jianjun, XIE Qinghua, ZUO Tingying, et al.

Criterion of Complex Least Squares Adjustment and Its Application in Tree Height Inversion with PolInSAR Data

Acta Geodaetica et Cartographica Sinica,2014,43(1):45-51.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0007