GPS精密定位的数学模型、数值算法及可靠性理论
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摘要
针对GPS双差相位精密定位的若干理论和应用问题,本文从数学模型、数值算法和可靠性理论三个方面进行了系统和深入的研究。综合起来,本文的主要工作和重要结论如下:
1.数学模型方面:
对于GPS精密静态定位,介绍了在双差模式下的函数模型和随机模型,构建了一个适宜于编程的通用法方程式。通过对不同类型观测值的权比因子是否设置为零,来达到这些类型观测值的选择使用的目的,以便得到各种类型的基线位置参数解。另外,还推导出了各类基线位置参数解之间的理论换算关系公式,以及各类基线位置参数解精度之间的关系公式。
对于GPS精密动态定位,分别给出了在单历元方式和OTF方式下的函数模型和随机模型,以及适宜于编程的通用法方程式。同样,通过对不同类型观测值的权比因子是否设置为零,来达到这些类型观测值的选择使用的目的,以便得到各种类型的位置参数解。
在GPS精密静态定位中法方程式的结构很有规律性,主要表现为方程中系数阵的各子块对各历元进行求和,因此在编制程序时只需要根据各历元的误差方程式和相应的随机模型组成法方程,再对各历元法方程系数阵的子块求和,来组成多个历元总的法方程式。这样可大大降低了各计算矩阵在计算机上运行和存储的阶数,极大地提高了计算效率。特别是在当历元数很大时,这种效果尤为明显。
对于GPS精密静态定位,在只用相位观测值进行平差的情况下,当采样间隔较短且观测历元数较少时,由于系数矩阵变化甚小,基线浮点解的精度很差。因此,当不顾及模糊度的整数性质时,应尽量用足够历元的数据来求出基线的浮点解,以保证其精度。而在只用精码观测值和同时用相位观测值与精码观测值的情况下,不存在这种现象。
2.模糊度的求解算法方面:
在介绍整数最小二乘方法、LAMBDA方法、直接凑整方法和Bootstrapping方法的同时,深入剖析了LAMBDA方法的结构。通过实例试算表明:LAMBDA方法的搜索效率明显高于整数最小二乘方法,它是整数最小二乘方法的一个发展。
由于在双差模式下短观测时段的GPS模糊度具有很强的相关性,如果直接应用整数最小二乘方法或LAMBDA方法将得不到正确的模糊度整数解,因此必须事先对原始的双差模糊度进行降相关的可容许整数变换。本论文介绍了Teunissen提出的二维整数变换方法和S.Han与C.Rizos提出的多维整数变换方法。这两种整数变换方法都是可容许的降相关的变换方法。实例计算表明:原始的双差模糊度通过降相关的可容许整数变换后,再使用整数最小二乘方法或LAMBDA方法不仅可提高求解模糊度整数解的正确性,而且还能够提高模糊度整数解的搜索速度。
Focused on several theoretical and applying problems on GPS precise positioning, this thesis makes a systemic and deep research on three aspects: mathematical model, numerical algorithm and reliability. Summarized speaking, the main research contents and important conclusions in this thesis are given as follows:1. Mathematical model sectionBased on the double-difference mode for GPS precise static positioning, the corresponding function model and stochastic model are given, and a general normal equation to suit programming in computer is constructed in this section. Through whether let the scale factors equal zero, the classes of observable can used selectively, hence many classes of baseline position parameter solution are obtained. In addition, the transformation formulas among many classes of baseline position parameter solution is deduced, and the relation formulas among precisions of many classes of baseline position parameter solution is also deduced in this section. For GPS precise kinematic/mobile positioning, the function models and stochastic models in both single epoch way and OTF (on-the-flying) way are respectively given, and the general normal equations to suit programming in computer are constructed in this section. Through whether let the scale factors equal zero, the classes of observable can used selectively, hence many classes of baseline position parameter solution are obtained.In GPS precise static positioning, the structure of normal equation exists obvious rule that each sub-block in coefficient matrix is summed up through each epoch ° This rule is very useful for programming.In the case of only using phase observable for GPS precise static positioning, when the sampling interval is very short and the epoch number is very small, the precision of baseline float solution is very poor since the coefficient matrix changes very little. Hence, when the property that ambiguities are integer is not considered, we should use the data of enough epochs to resolve the baseline float solution for guaranteeing its precision. But in the case of only using P-code observable or using both phase observable and P-code observable, there is not this kind of phenomena.2. Algorithm section for resolving ambiguity:After the integer least-square method, the LAMBDA method, the direct rounding method and the Bootstrapping method are introduced, the structure of the LAMBDA method is deeply analyzed. It is demonstrated by examples that the efficiency of LAMBDA method is apparently higher than that of integer least-square method. May to say, it is a developed integer least-square
method.Since there is strong correlation between GPS ambiguities of short observation spans based on t he d ouble-difference m ode, t he c orrect a mbiguity integer s olution w ill n ot b e obtained i f LAMBDA m ethod o r i nteger 1 east-square method i s u sed d irectly. T herefore t he decorrelated and admissible integer transformation for the original ambiguity vector must be performed beforehand. The two-dimensional integer transformation method and the multi-dimensional integer transformation method are introduced in this section. Both these two methods are decorrelated and admissible. It is showed by examples that: After the original ambiguities are transformed through using the decorrelated and admissible integer transformation methods, the integer least-square method or LAMBDA method can improve not only the correctness of ambiguity integer solution but also the searching efficiency of ambiguity integer solution.Bootstrapping method is such a method that resolves ambiguity integer solution without searching. It is an approximate algorithm of LAMBDA method too. It is demonstrated by examples that: when the observation epoch number is very small, the correct ambiguity integer solution cannot be obtained through using Bootstrapping method even though the decorrelated and admissible integer transformation is performed for original ambiguities. Hence this method does not fit GPS precise positioning on short observation spans.hi order to decrease the correlation of high-dimension ambiguities, the paired Cholesky integer transformation method is presented in this section. It is proved theoretically that this paired Cholesky integer transformation method is decorrelated and admissible. It is demonstrated by examples that the multi-time paired Cholesky integer transformation method is better than other methods for decreasing the correlation of high-dimension ambiguities.Through using the back sequent conditional LS ambiguity technique, another form of LAMBDA method—RLAMBDA method—is given and another form of Bootstrapping method—Return Bootstrapping method—is given too in this section. Since both LAMBDA method a nd RLAMBDA m ethod a re b ased o n t he i nteger 1 east-square m ethod, t hat i s t o s ay, their principles are identical, their ambiguity integer solutions are the same and their search efficiency is equivalent too. Return bootstrapping approach is an approximate algorithm of RLAMBDA method. When the observation epoch number is very small, the baseline results through using this Return bootstrapping approach is very poor, hence this approach cannot be used in GPS precise positioning on short observation spans.3. Reliability theory section of ambiguityOn the basis of the original definition for the admissible integer estimation given by Teunissen, a new severer definition is presented in this section. Based on this new definition, the pull-in regions of the direct rounding estimator, the bootstrapping estimator, the return bootstrapping estimator, and the integer least-square estimator (include the LAMBDA estimator and the RLAMBDA estimator) are given. At the same time, the concept of the ambiguity success
rate is introduced based on statistics theory in this section.After the easy-to-compute formula on ambiguity success rate of the Bootstrapping approach is introduced, the easy-to-compute formula on ambiguity success rate of the return bootstrapping approach is deduced. For the ambiguity success rate of the direct rounding approach, the upper and lower bound calculation formula presented by Teunissen is introduced, moreover the new upper bound calculation formula is deduced. In addition, after the upper and lower bound calculation formula on the ambiguity success rate of integer least-square method is introduced in this section, the new lower bound calculation formula is given, hi order to resolve directly the ambiguity success rate of the direct rounding approach, and the integer least-square method, the simulation approach on ambiguity success rate is also introduced in this section.For the expectation and variance of both the integer ambiguity solution and the baseline fixed-solution, their calculation formulas are introduced in this section. When the 'float' ambiguity solution is unbiased, the integer ambiguity solutions of the direct rounding approach, the bootstrapping approach, the return bootstrapping approach, and the integer least-square method all are unbiased. It is stated that these several methods all are unbiased estimation approach on integer ambiguity solutions. In addition, as long as both the 'float' ambiguity solution and the 'float' baseline position parameter solution are unbiased, the fixed baseline position parameter solution obtained through using these several approaches is also unbiased.The probability density function of fixed baseline position parameter solutions is discussed in this section. This probability density function has a symmetric property and a multi-peak property. At the same time, the probability mass function of which the fixed baseline position parameter solution belongs to the confidence space is given.After the joint probability density function between the 'float' ambiguity solution and the fixed ambiguity solution is introduced in this section, the conditional probability density function of the integer ambiguity solution is gotten according to this joint probability density function. It is very interesting that the conditional expectation of the integer ambiguity solution is just equal to the admissible integer estimator of this ambiguity vector. In order to discuss the property of the ambiguity residual error vector, the joint probability density function between the integer ambiguity solution and the ambiguity residual error vector is also introduced in this section. According to this joint probability density function, the probability density function of the ambiguity residual vector is obtained.At last, in order to discuss the joint statistic property between the ambiguity solution and the position parameter solution, their two joint probability density functions are introduced in this section. According to these two joint density functions, various joint probability density functions between the ambiguity solution and the position parameter solution are obtained, and various conditional probability density functions of baseline position parameter vectors are also obtained.
1.数学模型方面:
对于GPS精密静态定位,介绍了在双差模式下的函数模型和随机模型,构建了一个适宜于编程的通用法方程式。通过对不同类型观测值的权比因子是否设置为零,来达到这些类型观测值的选择使用的目的,以便得到各种类型的基线位置参数解。另外,还推导出了各类基线位置参数解之间的理论换算关系公式,以及各类基线位置参数解精度之间的关系公式。
对于GPS精密动态定位,分别给出了在单历元方式和OTF方式下的函数模型和随机模型,以及适宜于编程的通用法方程式。同样,通过对不同类型观测值的权比因子是否设置为零,来达到这些类型观测值的选择使用的目的,以便得到各种类型的位置参数解。
在GPS精密静态定位中法方程式的结构很有规律性,主要表现为方程中系数阵的各子块对各历元进行求和,因此在编制程序时只需要根据各历元的误差方程式和相应的随机模型组成法方程,再对各历元法方程系数阵的子块求和,来组成多个历元总的法方程式。这样可大大降低了各计算矩阵在计算机上运行和存储的阶数,极大地提高了计算效率。特别是在当历元数很大时,这种效果尤为明显。
对于GPS精密静态定位,在只用相位观测值进行平差的情况下,当采样间隔较短且观测历元数较少时,由于系数矩阵变化甚小,基线浮点解的精度很差。因此,当不顾及模糊度的整数性质时,应尽量用足够历元的数据来求出基线的浮点解,以保证其精度。而在只用精码观测值和同时用相位观测值与精码观测值的情况下,不存在这种现象。
2.模糊度的求解算法方面:
在介绍整数最小二乘方法、LAMBDA方法、直接凑整方法和Bootstrapping方法的同时,深入剖析了LAMBDA方法的结构。通过实例试算表明:LAMBDA方法的搜索效率明显高于整数最小二乘方法,它是整数最小二乘方法的一个发展。
由于在双差模式下短观测时段的GPS模糊度具有很强的相关性,如果直接应用整数最小二乘方法或LAMBDA方法将得不到正确的模糊度整数解,因此必须事先对原始的双差模糊度进行降相关的可容许整数变换。本论文介绍了Teunissen提出的二维整数变换方法和S.Han与C.Rizos提出的多维整数变换方法。这两种整数变换方法都是可容许的降相关的变换方法。实例计算表明:原始的双差模糊度通过降相关的可容许整数变换后,再使用整数最小二乘方法或LAMBDA方法不仅可提高求解模糊度整数解的正确性,而且还能够提高模糊度整数解的搜索速度。
Focused on several theoretical and applying problems on GPS precise positioning, this thesis makes a systemic and deep research on three aspects: mathematical model, numerical algorithm and reliability. Summarized speaking, the main research contents and important conclusions in this thesis are given as follows:1. Mathematical model sectionBased on the double-difference mode for GPS precise static positioning, the corresponding function model and stochastic model are given, and a general normal equation to suit programming in computer is constructed in this section. Through whether let the scale factors equal zero, the classes of observable can used selectively, hence many classes of baseline position parameter solution are obtained. In addition, the transformation formulas among many classes of baseline position parameter solution is deduced, and the relation formulas among precisions of many classes of baseline position parameter solution is also deduced in this section. For GPS precise kinematic/mobile positioning, the function models and stochastic models in both single epoch way and OTF (on-the-flying) way are respectively given, and the general normal equations to suit programming in computer are constructed in this section. Through whether let the scale factors equal zero, the classes of observable can used selectively, hence many classes of baseline position parameter solution are obtained.In GPS precise static positioning, the structure of normal equation exists obvious rule that each sub-block in coefficient matrix is summed up through each epoch ° This rule is very useful for programming.In the case of only using phase observable for GPS precise static positioning, when the sampling interval is very short and the epoch number is very small, the precision of baseline float solution is very poor since the coefficient matrix changes very little. Hence, when the property that ambiguities are integer is not considered, we should use the data of enough epochs to resolve the baseline float solution for guaranteeing its precision. But in the case of only using P-code observable or using both phase observable and P-code observable, there is not this kind of phenomena.2. Algorithm section for resolving ambiguity:After the integer least-square method, the LAMBDA method, the direct rounding method and the Bootstrapping method are introduced, the structure of the LAMBDA method is deeply analyzed. It is demonstrated by examples that the efficiency of LAMBDA method is apparently higher than that of integer least-square method. May to say, it is a developed integer least-square
method.Since there is strong correlation between GPS ambiguities of short observation spans based on t he d ouble-difference m ode, t he c orrect a mbiguity integer s olution w ill n ot b e obtained i f LAMBDA m ethod o r i nteger 1 east-square method i s u sed d irectly. T herefore t he decorrelated and admissible integer transformation for the original ambiguity vector must be performed beforehand. The two-dimensional integer transformation method and the multi-dimensional integer transformation method are introduced in this section. Both these two methods are decorrelated and admissible. It is showed by examples that: After the original ambiguities are transformed through using the decorrelated and admissible integer transformation methods, the integer least-square method or LAMBDA method can improve not only the correctness of ambiguity integer solution but also the searching efficiency of ambiguity integer solution.Bootstrapping method is such a method that resolves ambiguity integer solution without searching. It is an approximate algorithm of LAMBDA method too. It is demonstrated by examples that: when the observation epoch number is very small, the correct ambiguity integer solution cannot be obtained through using Bootstrapping method even though the decorrelated and admissible integer transformation is performed for original ambiguities. Hence this method does not fit GPS precise positioning on short observation spans.hi order to decrease the correlation of high-dimension ambiguities, the paired Cholesky integer transformation method is presented in this section. It is proved theoretically that this paired Cholesky integer transformation method is decorrelated and admissible. It is demonstrated by examples that the multi-time paired Cholesky integer transformation method is better than other methods for decreasing the correlation of high-dimension ambiguities.Through using the back sequent conditional LS ambiguity technique, another form of LAMBDA method—RLAMBDA method—is given and another form of Bootstrapping method—Return Bootstrapping method—is given too in this section. Since both LAMBDA method a nd RLAMBDA m ethod a re b ased o n t he i nteger 1 east-square m ethod, t hat i s t o s ay, their principles are identical, their ambiguity integer solutions are the same and their search efficiency is equivalent too. Return bootstrapping approach is an approximate algorithm of RLAMBDA method. When the observation epoch number is very small, the baseline results through using this Return bootstrapping approach is very poor, hence this approach cannot be used in GPS precise positioning on short observation spans.3. Reliability theory section of ambiguityOn the basis of the original definition for the admissible integer estimation given by Teunissen, a new severer definition is presented in this section. Based on this new definition, the pull-in regions of the direct rounding estimator, the bootstrapping estimator, the return bootstrapping estimator, and the integer least-square estimator (include the LAMBDA estimator and the RLAMBDA estimator) are given. At the same time, the concept of the ambiguity success
rate is introduced based on statistics theory in this section.After the easy-to-compute formula on ambiguity success rate of the Bootstrapping approach is introduced, the easy-to-compute formula on ambiguity success rate of the return bootstrapping approach is deduced. For the ambiguity success rate of the direct rounding approach, the upper and lower bound calculation formula presented by Teunissen is introduced, moreover the new upper bound calculation formula is deduced. In addition, after the upper and lower bound calculation formula on the ambiguity success rate of integer least-square method is introduced in this section, the new lower bound calculation formula is given, hi order to resolve directly the ambiguity success rate of the direct rounding approach, and the integer least-square method, the simulation approach on ambiguity success rate is also introduced in this section.For the expectation and variance of both the integer ambiguity solution and the baseline fixed-solution, their calculation formulas are introduced in this section. When the 'float' ambiguity solution is unbiased, the integer ambiguity solutions of the direct rounding approach, the bootstrapping approach, the return bootstrapping approach, and the integer least-square method all are unbiased. It is stated that these several methods all are unbiased estimation approach on integer ambiguity solutions. In addition, as long as both the 'float' ambiguity solution and the 'float' baseline position parameter solution are unbiased, the fixed baseline position parameter solution obtained through using these several approaches is also unbiased.The probability density function of fixed baseline position parameter solutions is discussed in this section. This probability density function has a symmetric property and a multi-peak property. At the same time, the probability mass function of which the fixed baseline position parameter solution belongs to the confidence space is given.After the joint probability density function between the 'float' ambiguity solution and the fixed ambiguity solution is introduced in this section, the conditional probability density function of the integer ambiguity solution is gotten according to this joint probability density function. It is very interesting that the conditional expectation of the integer ambiguity solution is just equal to the admissible integer estimator of this ambiguity vector. In order to discuss the property of the ambiguity residual error vector, the joint probability density function between the integer ambiguity solution and the ambiguity residual error vector is also introduced in this section. According to this joint probability density function, the probability density function of the ambiguity residual vector is obtained.At last, in order to discuss the joint statistic property between the ambiguity solution and the position parameter solution, their two joint probability density functions are introduced in this section. According to these two joint density functions, various joint probability density functions between the ambiguity solution and the position parameter solution are obtained, and various conditional probability density functions of baseline position parameter vectors are also obtained.
引文
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