﻿ 带球约束的平差算法及应用
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 大地测量与地球动力学  2020, Vol. 40 Issue (1): 71-76  DOI: 10.14075/j.jgg.2020.01.014

### 引用本文

ZUO Tingying, CHEN Bangju, SONG Yingchun. Adjustment Algorithm with Spherical Constraint and Application[J]. Journal of Geodesy and Geodynamics, 2020, 40(1): 71-76.

### Foundation support

National Natural Science Foundation of China, No.41674009,41574006,41674012..

### Corresponding author

CHEN Bangju, postgraduate,majors in surveying data processing,E-mail: 651527323@qq.com..

### 第一作者简介

ZUO Tingying,PhD,professor,majors in surveying data processing, E-mail: 791554659@qq.com..

### 文章历史

1. 中南大学地球科学与信息物理学院，长沙市麓山南路932号，410083;
2. 有色金属成矿预测与地质环境监测教育部重点实验室，长沙市麓山南路932号，410083

1 球约束平差模型与岭估计

 $\mathit{\boldsymbol{L}} = \mathit{\boldsymbol{AX}} + \mathit{\boldsymbol{e}}\$ (1)

 $\left| {{\mathit{\boldsymbol{X}}_i}} \right| \le {r_i},i = 1, 2, \cdots , n$ (2)

 $\left\{ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{L}} = \mathit{\boldsymbol{AX}} + \mathit{\boldsymbol{e}}}\\ {{\rm{s}}.{\rm{t}}.\;\;\left| {{\mathit{\boldsymbol{X}}_i}} \right| \le {r_i}} \end{array}} \right.$ (3)

 $\left\{ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{L}} = \mathit{\boldsymbol{AX}} + \mathit{\boldsymbol{e}}}\\ {{\rm{s}}.{\rm{t}}.\;\;{\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{SX}} \le {r^2}} \end{array}} \right.$ (4)

 $\left\{ {\begin{array}{*{20}{c}} {{\rm{min}}f\left( \mathit{\boldsymbol{X}} \right) = {{\left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{AX}}} \right)}^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{AX}}} \right)}\\ {{\rm{s}}.{\rm{t}}.\;\;\;\;{\mathit{\boldsymbol{X}}^{\rm{T}}}{\mathit{\boldsymbol{SX}}} \le {r^2}} \end{array}} \right.$ (5)

 $\left\{ {\begin{array}{*{20}{c}} {{\rm{min}}f\left(\mathit{\boldsymbol{X}} \right) = \frac{1}{2}{\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{NX}} + {\mathit{\boldsymbol{c}}^{\rm{T}}}\mathit{\boldsymbol{X}}}\\ {{\rm{s}}.{\rm{t}}.\;\;\;{\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{SX}} \le {r^2}} \end{array}} \right.$ (6)

 $\begin{array}{l}\;\;\;\;\;\;\; \mathit{\boldsymbol{F}}\left( {\mathit{\boldsymbol{X}},\lambda } \right) = \\ \frac{1}{2}{\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{NX}} + {\mathit{\boldsymbol{c}}^{\rm{T}}}\mathit{\boldsymbol{X}} + \frac{1}{2}\lambda \left( {{\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{SX}} - {r^2}} \right) \end{array}$ (7)

 $\left\{ {\begin{array}{*{20}{c}} {\left( {\mathit{\boldsymbol{N}} + \lambda \mathit{\boldsymbol{S}}} \right)\mathit{\boldsymbol{X}} = - {\mathit{\boldsymbol{c}}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{X}}^{\rm{T}}}S\mathit{\boldsymbol{X}} = {r^2}} \end{array}} \right.$ (8)

 $\begin{array}{l} {{\mathit{\boldsymbol{\hat X}}}_{{\rm{GLS}}}} = - {(\mathit{\boldsymbol{N + }}\lambda S)^{ - 1}}{\mathit{\boldsymbol{c}}^{\rm{T}}}=\\ \;\;\;\;\;\;{({\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{S}})^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}} \end{array}$ (9)

 $({\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}} + \lambda \mathit{\boldsymbol{S}}){{\mathit{\boldsymbol{\hat X}}}_{{\rm{GLS}}}} = {\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}}$

 $\lambda \mathit{\boldsymbol{S}}{{\mathit{\boldsymbol{\hat X}}}_{{\rm{GLS}}}} = {\mathit{\boldsymbol{A}}^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat X}}}_{{\rm{GLS}}}}} \right)$

 $\lambda = \frac{1}{{{r^2}}}\mathit{\boldsymbol{\hat X}}_{{\rm{GLS}}}^{\rm{T}}{\mathit{\boldsymbol{A}}^{\rm{T}}}{{\mathit{\boldsymbol{\hat e}}}_{{\rm{GLS}}}}$ (10)

 $\begin{array}{l} \lambda = \frac{1}{{{r^2}}}\mathit{\boldsymbol{\hat X}}_{{\rm{LS}}}^{\rm{T}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat X}}}_{{\rm{LS}}}}} \right) = \\ \frac{1}{{{r^2}}}\left( {\mathit{\boldsymbol{\hat X}}_{{\rm{LS}}}^{\rm{T}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{\hat X}}_{{\rm{LS}}}^{\rm{T}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat X}}}_{{\rm{LS}}}}} \right) = 0 \end{array}$

2 球约束平差的迭代算法

 $\begin{array}{l} \frac{1}{2}{\left( {\mathit{\boldsymbol{X}} - {\mathit{\boldsymbol{X}}_k}} \right)^{\rm{T}}}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right)\left( {\mathit{\boldsymbol{X}} - {\mathit{\boldsymbol{X}}_k}} \right) = \\ \frac{1}{2}{\mathit{\boldsymbol{X}}^{\rm{T}}}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right)\mathit{\boldsymbol{X}} - {\mathit{\boldsymbol{X}}^{\rm{T}}}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} + \\ \;\;\;\;\;\;\;\frac{1}{2}\mathit{\boldsymbol{X}}_k^{\rm{T}}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} \ge 0 \end{array}$

 $\begin{array}{l} - \frac{1}{2}{\mathit{\boldsymbol{X}}^{\rm{T}}}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right)\mathit{\boldsymbol{X}} \le - {\mathit{\boldsymbol{X}}^{\rm{T}}}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} + \\ \;\;\;\;\;\frac{1}{2}\mathit{\boldsymbol{X}}_k^{\rm{T}}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} \end{array}$

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;f\left( \mathit{\boldsymbol{X}} \right) = \frac{1}{2}{\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{N}}\mathit{\boldsymbol{X}} + {\mathit{\boldsymbol{c}}^{\rm{T}}}\mathit{\boldsymbol{X}} = \\ \;\;\;\;\;\;\frac{1}{2}\theta {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{S}}\mathit{\boldsymbol{X}} - {\mathit{\boldsymbol{X}}^{\rm{T}}}\left[ {\frac{1}{2}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right)\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{c}}} \right] \le \\ \;\;\;\;\;\;\;\;\frac{1}{2}\theta {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{SX}} - {\mathit{\boldsymbol{X}}^{\rm{T}}}\left[ {\left( \theta {\mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right] + \\ \frac{1}{2} \mathit{\boldsymbol{X}}_k^{\rm{T}}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} = \mathit{g}\left( \mathit{\boldsymbol{X}} \right) + \frac{1}{2}\mathit{\boldsymbol{X}}_k^{\rm{T}}\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} \end{array}$

 $\left\{ {\begin{array}{*{20}{c}} {{\rm{min}}\;\;g\left( \mathit{\boldsymbol{X}} \right) = \frac{1}{2}\theta {\mathit{\boldsymbol{X}}^{\rm{T}}}{\mathit{\boldsymbol{SX}}} - {\mathit{\boldsymbol{X}}^{\rm{T}}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}\\ {{\rm{s}}.{\rm{t}}.\;\;\;\;\;\;{\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{SX}} \le {r^2}} \end{array}} \right.$ (11)

g(X)关于X的导数等于0，有：

 $\theta \mathit{\boldsymbol{SX}} - \left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right] = 0$

 ${\mathit{\boldsymbol{X}}_{k + 1}} = \left\{ {\begin{array}{*{20}{c}} {\frac{{{\mathit{\boldsymbol{S}}^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}}{\theta },}\\ {\left\| {\frac{{{\mathit{\boldsymbol{S}}^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}}{\theta }} \right\| \le r}\\ {\frac{{{\mathit{\boldsymbol{S}}^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}}{{\left\| {{S^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]} \right\|}}r,}\\ {\frac{{\left\| {{\mathit{\boldsymbol{S}}^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]} \right\|}}{\theta } > r} \end{array}} \right.$ (12)

 $\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} \theta \mathit{\boldsymbol{S}}{\mathit{\boldsymbol{X}}_{k + 1}} - \left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right] + \\ {\lambda _{k + 1}}\mathit{\boldsymbol{S}}{\mathit{\boldsymbol{X}}_{k + 1}} = 0 \end{array}\\ {{\lambda _{k + 1}}\left( {\mathit{\boldsymbol{X}}_{k + 1}^{\rm{T}}\mathit{\boldsymbol{S}}{\mathit{\boldsymbol{X}}_{k + 1}} - {r^2}} \right) = 0} \end{array}} \right.$ (13)

 $\theta {r^2} - \mathit{\boldsymbol{X}}_{k + 1}^{\rm{T}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right] + {\lambda _{k + 1}}{r^2} = 0$

 ${\lambda _{k + 1}} = \frac{{{{\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}^{\rm{T}}}{\mathit{\boldsymbol{X}}_{k + 1}} - \theta {r^2}}}{{{r^2}}}$

 $\begin{array}{l} \;\;\;\;\;\;\;{\lambda ^{\rm{*}}} = \mathop {\lim }\limits_{k \to \infty } {\lambda _{k + 1}} = \\ \frac{{{{\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}^{\rm{T}}}{\mathit{\boldsymbol{X}}_{k + 1}} - \theta {r^2}}}{{{r^2}}} = \\ \;\;\;\;\;\;\frac{{ - {{({\mathit{\boldsymbol{X}}^*})}^{\rm{T}}}\mathit{\boldsymbol{N}}{\mathit{\boldsymbol{X}}^*} - {\mathit{\boldsymbol{c}}^{\rm{T}}}{\mathit{\boldsymbol{X}}^*}}}{{{r^2}}} \end{array}$

 ${\lambda ^{\rm{*}}} = \left\{ {\begin{array}{*{20}{c}} {{\lambda ^{\rm{*}}},{{({\mathit{\boldsymbol{X}}^*})}^{\rm{T}}}\mathit{\boldsymbol{S}}{\mathit{\boldsymbol{X}}^*} = {r^2}}\\ {0,{{({\mathit{\boldsymbol{X}}^*})}^{\rm{T}}}\mathit{\boldsymbol{S}}{\mathit{\boldsymbol{X}}^*} < {r^2}} \end{array}} \right.$

 $\left\{ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{N}}{\mathit{\boldsymbol{X}}^*} + \mathit{\boldsymbol{c}} + {\lambda ^{\rm{*}}}{\mathit{\boldsymbol{X}}^*} = 0}\\ {\frac{1}{2}{\lambda ^{\rm{*}}}\left( {{{({\mathit{\boldsymbol{X}}^*})}^{\rm{T}}}\mathit{\boldsymbol{S}}{\mathit{\boldsymbol{X}}^*} - {r^2}} \right) = 0} \end{array}} \right.$ (14)

1) 利用先验信息确定球约束半径r

2)${\mathit{\boldsymbol{X}}_0} \in \left\{ {\mathit{\boldsymbol{X}} \in {R^n}:\left\| \mathit{\boldsymbol{X}} \right\| \le r} \right\}$，让θ=(λ2+η)/μ(η≥0)，k=0，ε>0；

3) $\begin{array}{l} {\mathit{\boldsymbol{X}}_{k + 1}} = \\ \left\{ {\begin{array}{*{20}{\mathit{\boldsymbol{c}}}} {\frac{{{\mathit{\boldsymbol{S}}^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}}{\theta },}\\ {\left\| {\frac{{{\mathit{\boldsymbol{S}}^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}}{\theta }} \right\| \le r}\\ {\frac{{{\mathit{\boldsymbol{S}}^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}}{{\left\| {{\mathit{\boldsymbol{S}}^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]} \right\|}}r,}\\ {\left\| {\frac{{{\mathit{\boldsymbol{S}}^{ - 1}}\left[ {\left( {\theta \mathit{\boldsymbol{S}} - \mathit{\boldsymbol{N}}} \right){\mathit{\boldsymbol{X}}_k} - \mathit{\boldsymbol{c}}} \right]}}{\theta }} \right\| > r} \end{array}} \right. \end{array}$

4) 若$\left\| {{\mathit{\boldsymbol{X}}_{k + 1}} - {\mathit{\boldsymbol{X}}_k}} \right\| \le \varepsilon$，停止，输出X*= Xk+1；否则k=k+1，转步骤1)。

3 算例分析 3.1 数值模拟实验

 $\left[ {\begin{array}{*{20}{c}} 1&{\frac{1}{2}}&{\frac{1}{3}}&{\frac{1}{4}}\\ {\frac{1}{2}}&{\frac{1}{3}}&{\frac{1}{4}}&{\frac{1}{5}}\\ {\frac{1}{3}}&{\frac{1}{4}}&{\frac{1}{5}}&{\frac{1}{6}}\\ {\frac{1}{4}}&{\frac{1}{5}}&{\frac{1}{6}}&{\frac{1}{7}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}}\\ {{x_3}}\\ {{x_4}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\frac{{25}}{{12}}}\\ {\frac{{77}}{{60}}}\\ {\frac{{19}}{{20}}}\\ {\frac{{319}}{{420}}} \end{array}} \right]$

3.2 病态测边网实验 3.2.1 算例1

 图 1 测边网 Fig. 1 Distance-measuring triangulation network

 $\begin{array}{l} {\mathit{\boldsymbol{A}}_1} = \\ \left[ {\begin{array}{*{20}{c}} {0.7434}&{ - 0.6689}&0&0&0&0&0&0\\ {0.9952}&{0.0975}&0&0&0&0&0&0\\ {0.6457}&{0.7636}&{ - 0.6457}&{ - 0.7636}&0&0&0&0\\ 0&0&{0.0183}&{ - 0.9998}&0&0&0&0\\ 0&0&{0.8575}&{ - 0.5145}&{ - 0.8575}&{0.5145}&0&0\\ 0&0&0&0&{ - 0.8848}&{ - 0.4660}&0&0\\ 0&0&0&0&{0.3904}&{ - 0.9206}&{ - 0.3904}&{0.9206}\\ 0&0&0&0&0&0&{ - 0.8708}&{0.4917}\\ 0&0&0&0&0&0&{ - 0.9823}&{ - 0.1874} \end{array}} \right]\\ ,{\mathit{\boldsymbol{L}}_1} = \left[ {\begin{array}{*{20}{c}} {1.4955}\\ { - 0.7877}\\ { - 2.5966}\\ {0.5178}\\ {3.7201}\\ {0.2180}\\ { - 6.2599}\\ { - 3.3892}\\ { - 1.7623} \end{array}} \right] \end{array}$

3.2.2 算例2

 $\begin{array}{l} {\mathit{\boldsymbol{A}}_1} = \\ \left[ {\begin{array}{*{20}{c}} {0.9958}&{0.0920}&0&0&0&0&0&0\\ {0.9952}&{0.0975}&0&0&0&0&0&0\\ {0.6457}&{0.7636}&{ - 0.6457}&{ - 0.7636}&0&0&0&0\\ 0&0&{0.0183}&{ - 0.9998}&0&0&0&0\\ 0&0&{0.8575}&{ - 0.5145}&{ - 0.8575}&{0.5145}&0&0\\ 0&0&0&0&{ - 0.8848}&{ - 0.4660}&0&0\\ 0&0&0&0&{0.3904}&{ - 0.9206}&{ - 0.3904}&{0.9206}\\ 0&0&0&0&0&0&{ - 0.8708}&{0.4917}\\ 0&0&0&0&0&0&{ - 0.8710}&{0.4913} \end{array}} \right]\\ ,{\mathit{\boldsymbol{L}}_2} = \left[ {\begin{array}{*{20}{c}} { - 0.7351}\\ { - 0.7877}\\ { - 2.5966}\\ {0.5178}\\ {3.7201}\\ {0.2180}\\ { - 6.2599}\\ { - 3.3892}\\ { - 3.3113} \end{array}} \right] \end{array}$

4 结语

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Adjustment Algorithm with Spherical Constraint and Application
ZUO Tingying1,2     CHEN Bangju1,2     SONG Yingchun1,2
1. School of Geosciences and Info-Physics, Central South University, 932 South-Lushan Road, Changsha 410083, China;
2. Key Laboratory of Mineralization Prediction and Geological Environmental Monitoring, Ministry of Education, 932 South-Lushan Road, Changsha 410083, China
Abstract: Aiming at solving the ill-conditioned problem caused by inadequate observation, inadequate utilization of parameter physical information and prior information in measurement data processing, this paper transforms the measurement adjustment problem into a convex quadratic programming problem by constraining the parameters with prior information, and proposes an adjustment model with spherical constraints. Based on the optimization theory and Kuhn-Tucker condition, the adjustment problem under spherical constraints is studied, and a solution method is proposed for the model. The results of numerical simulation and practical application of trilateration network show that this method has obvious better estimation in dealing with ill-conditioned problems, and can be widely used in geodetic ill-conditioned data processing.
Key words: spherical constraints; least squares; morbid problems; Kuhn-Tucker condition; prior information