尽管目前方差-协方差分量估计的方法众多^[1-12]，但大多数方法都假定每类观测值仅含1个方差分量或仅估计单个方差分量。刘长建等^[4]提出扩展的Helmert型方差分量估计，但其公式推导是基于间接平差的。本文基于概括函数模型，从每类观测值仅含1个方差分量扩展到每类观测值含有多个方差分量出发，推导出扩展的Helmert型方差分量估计的通用公式，并给出其特殊情况和简化公式。

1 概括函数模型

概括平差函数模型是间接平差、附有限制条件的间接平差、条件平差和附有参数的条件平差的统一，其一般的函数模型为^[3]：

$ \left\{ {\begin{array}{*{20}{l}} {\mathop F\limits_{c \times 1} (\mathit{\boldsymbol{\hat L}},\mathit{\boldsymbol{\hat X}}) = 0}\\ {\mathop \varPhi \limits_{s \times 1} (\mathit{\boldsymbol{\hat X}}) = 0} \end{array}} \right. $

(1)

式中，L为观测值向量，X为参数向量，c为一般条件方程个数，s为限制条件方程个数。如果式(1)是非线性的，按泰勒公式展开为线性形式，则概括函数模型的线性形式为：

$ \underset{c\times n}{\mathit{\boldsymbol{A}}}\,\underset{n\times 1}{\mathit{\boldsymbol{V}}}\,+\underset{c\times u}{\mathit{\boldsymbol{B}}}\,\underset{u\times 1}{\mathit{\boldsymbol{x}}}\,+\underset{c\times 1}{\mathit{\boldsymbol{W}}}\,=\underset{c\times 1}{\mathop{\mathbf{0}}}\, $

(2)

$ \underset{s\times u}{\mathit{\boldsymbol{C}}}\,\underset{u\times 1}{\mathit{\boldsymbol{\hat{x}}}}\,+\underset{s\times 1}{\mathop{{{\mathit{\boldsymbol{W}}}_{x}}}}\,=\underset{s\times 1}{\mathop{\mathbf{0}}}\, $

(3)

式中，n为观测值个数，u为参数个数，且满足r+u=s+c，r为多余观测数，系数矩阵的秩分别为R(A)=c，R(B)=u，R(C)=s。设随机模型为：

$ \underset{n\times n}{\mathop{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}}}\,=\sigma _{0}^{2}\mathit{\boldsymbol{Q}}=\sigma _{0}^{2}{{\mathit{\boldsymbol{P}}}^{-1}} $

(4)

式中，σ₀²为单位权方差，Q和P分别为观测值的协因数阵和权阵。根据最小二乘原理，在V^TPV=min的条件下构造条件极值方程为：

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} = {\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}} - 2{\mathit{\boldsymbol{K}}^{\rm{T}}}(\mathit{\boldsymbol{AV}} + \mathit{\boldsymbol{B\hat x}} + \mathit{\boldsymbol{W}}) - }\\ {2\mathit{\boldsymbol{K}}_s^{\rm{T}}(\mathit{\boldsymbol{C\hat x}} + {\mathit{\boldsymbol{W}}_x})} \end{array} $

(5)

式中，K和K_s分别对应于式(2)和式(3)的联系数。将式(5)分别对V和$\mathit{\boldsymbol{\hat x}}$取偏导并令其为0，再和式(2)及式(3)联立可得：

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat x}} = - (\mathit{\boldsymbol{N}}_{BB}^{ - 1} - \mathit{\boldsymbol{N}}_{BB}^{ - 1}{\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{CC}^{ - 1}\mathit{\boldsymbol{CN}}_{BB}^{ - 1}){\mathit{\boldsymbol{W}}_e} - }\\ {\mathit{\boldsymbol{N}}_{BB}^{ - 1}{\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{CC}^{ - 1}{\mathit{\boldsymbol{W}}_x}} \end{array} $

(6)

$ \mathit{\boldsymbol{V}} = - \mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{AA}^{ - 1}(\mathit{\boldsymbol{W}} + \mathit{\boldsymbol{B\hat x}}) $

(7)

式中，

$ \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{N}}_{AA}} = \mathit{\boldsymbol{AQ}}{\mathit{\boldsymbol{A}}^{\rm{T}}},{\mathit{\boldsymbol{N}}_{BB}} = {\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{AA}^{ - 1}\mathit{\boldsymbol{B}},{\mathit{\boldsymbol{W}}_e} = }\\ {{\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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{AA}^{ - 1}\mathit{\boldsymbol{W}},{\mathit{\boldsymbol{N}}_{CC}} = \mathit{\boldsymbol{CN}}_{BB}^{ - 1}{\mathit{\boldsymbol{C}}^{\rm{T}}}} \end{array} $

(8)

联系数K的求解详见文献[13]，参数向量$\mathit{\boldsymbol{\hat x}}$和常数项W的协因数阵为：

$ {{\mathit{\boldsymbol{Q}}_K} = \mathit{\boldsymbol{N}}_{AA}^{ - 1} - \mathit{\boldsymbol{N}}_{AA}^{ - 1}\mathit{\boldsymbol{B}}{\mathit{\boldsymbol{Q}}_{\hat x\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{AA}^{ - 1}} $

(9)

$ {{\mathit{\boldsymbol{Q}}_{\hat x\hat x}} = \mathit{\boldsymbol{N}}_{BB}^{ - 1} - \mathit{\boldsymbol{N}}_{BB}^{ - 1}{\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{CC}^{ - 1}\mathit{\boldsymbol{CN}}_{BB}^{ - 1}} $

(10)

$ {\mathit{\boldsymbol{Q}}_W} = \mathit{\boldsymbol{AQ}}{\mathit{\boldsymbol{A}}^{\rm{T}}} = {\mathit{\boldsymbol{N}}_{AA}} $

(11)

应用协方差传播定律，由式(7)可得V的方差阵为：

$ {\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_V} = \mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{A}}^{\rm{T}}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_w}{\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{AQ}} $

(12)

式中，Σ_W为常数项W的方差阵，且由式(11)可知：

$ {\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W} = \sigma _0^2{\mathit{\boldsymbol{Q}}_W} = \sigma _0^2\mathit{\boldsymbol{AQ}}{\mathit{\boldsymbol{A}}^{\rm{T}}} = \mathit{\boldsymbol{A}}\sigma _0^2\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{A}}^{\rm{T}}} = \mathit{\boldsymbol{A \boldsymbol{\varSigma} }}{\mathit{\boldsymbol{A}}^{\rm{T}}} $

(13)

根据文献[1]中二次型的期望公式E(x^TAx)=tr(AΣ_xx)+μ_x^TAμ_x，顾及E(V)=0可得：

$ \begin{array}{*{20}{l}} {{\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} {\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} {\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} {\kern 1pt} {\kern 1pt} E({\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}}) = {\rm{tr }}(\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_V}) = }\\ {{\rm{ tr}} ({\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W}) = {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}})\sigma _0^2} \end{array} $

(14)

2 扩展的Helmert型方差分量通用公式

假设推导的函数模型和式(2)、(3)一致，但不同于式(4)的随机模型，将式(4)的随机模型根据文献[13]扩展到一般的随机模型：

$ \mathit{\boldsymbol{ \boldsymbol{\varSigma} }} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_1}}&{}&{}&{}\\ {}&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_k}} \end{array}} \right] $

(15)

假设观测值向量L含有k类独立的观测值L₁, L₂，…，L_k，即

$ \mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} {\mathop {{\mathit{\boldsymbol{L}}_1}}\limits_{{n_1} \times 1} }\\ {\mathop {{\mathit{\boldsymbol{L}}_2}}\limits_{{n_2} \times 1} }\\ \vdots \\ {\mathop {{\mathit{\boldsymbol{L}}_k}}\limits_{{n_k} \times 1} } \end{array}} \right],\mathit{\boldsymbol{V}} = \left[ {\begin{array}{*{20}{c}} {\mathop {{\mathit{\boldsymbol{V}}_1}}\limits_{{n_1} \times 1} }\\ {\mathop {{\mathit{\boldsymbol{V}}_2}}\limits_{{n_2} \times 1} }\\ \vdots \\ {\mathop {{\mathit{\boldsymbol{V}}_k}}\limits_{{n_k} \times 1} } \end{array}} \right],\mathit{\boldsymbol{A}} = \left[ {\mathop {{\mathit{\boldsymbol{A}}_1}}\limits_{c \times {n_1}} ,\mathop {{\mathit{\boldsymbol{A}}_2}}\limits_{c \times {n_2}} , \cdots ,\mathop {{\mathit{\boldsymbol{A}}_k}}\limits_{c \times {n_k}} } \right] $

(16)

Σ₁, Σ₂, …Σ_k为各类观测值的方差阵，且假设每类观测值含有多个方差分量，即

$ \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_1} = \sigma _{11}^2{\mathit{\boldsymbol{Q}}_{11}} + \sigma _{12}^2{\mathit{\boldsymbol{Q}}_{12}} + \cdots + \sigma _{1{n_1}}^2{\mathit{\boldsymbol{Q}}_{1{n_1}}}}\\ {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2} = \sigma _{21}^2{\mathit{\boldsymbol{Q}}_{21}} + \sigma _{22}^2{\mathit{\boldsymbol{Q}}_{22}} + \cdots + \sigma _{2{n_2}}^2{\mathit{\boldsymbol{Q}}_{2{n_2}}}}\\ {{\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} {\kern 1pt} \ldots }\\ {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_k} = \sigma _{k1}^2{\mathit{\boldsymbol{Q}}_{k1}} + \sigma _{k2}^2{\mathit{\boldsymbol{Q}}_{k2}} + \cdots + \sigma _{k{n_k}}^2{\mathit{\boldsymbol{Q}}_{k{n_k}}}} \end{array}} \right. $

(17)

式中，σ_ij²为待求的方差分量，Q_ij为已知的协因数阵，σ_ij²和Q_ij中i为第i类观测值，j为第i类观测值中第j个方差分量，n₁, n₂, …n_k为第k类观测值的方差分量的个数。则式(2)可写为：

$ {\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{V}}_1} + {\mathit{\boldsymbol{A}}_2}{\mathit{\boldsymbol{V}}_2} + \cdots + {\mathit{\boldsymbol{A}}_k}{\mathit{\boldsymbol{V}}_k} + \mathit{\boldsymbol{B\hat x}} + \mathit{\boldsymbol{W}} = 0 $

(18)

令待求的σ_ij²初值为1，则式(15)可变为：

$ \begin{array}{l} {\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_0} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{10}}}&{}&{}&{}\\ {}&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{20}}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{k0}}} \end{array}} \right] = \mathit{\boldsymbol{Q}} = \\ {\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} {\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} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_1}}&{}&{}&{}\\ {}&{{\mathit{\boldsymbol{Q}}_2}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{{\mathit{\boldsymbol{Q}}_k}} \end{array}} \right] \end{array} $

(19)

式中，

$ \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_1} = {\mathit{\boldsymbol{Q}}_{11}} + {\mathit{\boldsymbol{Q}}_{12}} + \cdots + {\mathit{\boldsymbol{Q}}_{1{n_1}}}}\\ {{\mathit{\boldsymbol{Q}}_2} = {\mathit{\boldsymbol{Q}}_{21}} + {\mathit{\boldsymbol{Q}}_{22}} + \cdots + {\mathit{\boldsymbol{Q}}_{2{n_2}}}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots }\\ {{\mathit{\boldsymbol{Q}}_k} = {\mathit{\boldsymbol{Q}}_{k1}} + {\mathit{\boldsymbol{Q}}_{k2}} + \cdots + {\mathit{\boldsymbol{Q}}_{k{n_k}}}} \end{array}} \right. $

(20)

再令

$ \mathit{\boldsymbol{P}} = {\mathit{\boldsymbol{Q}}^{ - 1}} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{P}}_1}}&{}&{}&{}\\ {}&{{\mathit{\boldsymbol{P}}_2}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{{\mathit{\boldsymbol{P}}_k}} \end{array}} \right] $

(21)

式中，P_i=(Q_i1+Q_i2+…+Q_ini)^-1, i=1, 2, …k，P为已知的初始权阵。再令

$ \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{P}}_{i1}} = {\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{Q}}_{i1}}{\mathit{\boldsymbol{P}}_i}}\\ {{\mathit{\boldsymbol{P}}_{i2}} = {\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{Q}}_{i2}}{\mathit{\boldsymbol{P}}_i}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ldots }\\ {{\mathit{\boldsymbol{P}}_{i{n_i}}} = {\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{Q}}_{i{n_i}}}{\mathit{\boldsymbol{P}}_i}} \end{array},i = 1,2, \cdots k} \right. $

(22)

则有：

$ \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{P}}_1} = {\mathit{\boldsymbol{P}}_{11}} + {\mathit{\boldsymbol{P}}_{12}} + \cdots + {\mathit{\boldsymbol{P}}_{1{n_1}}}}\\ {{\mathit{\boldsymbol{P}}_2} = {\mathit{\boldsymbol{P}}_{21}} + {\mathit{\boldsymbol{P}}_{22}} + \cdots + {\mathit{\boldsymbol{P}}_{2{n_2}}}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots }\\ {{\mathit{\boldsymbol{P}}_k} = {\mathit{\boldsymbol{P}}_{k1}} + {\mathit{\boldsymbol{P}}_{k2}} + \cdots + {\mathit{\boldsymbol{P}}_{k{n_k}}}} \end{array}} \right. $

(23)

将式(16)、(19)、(20)代入N_AA=AQA^T可得：

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{N}}_{AA}} = \sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{A}}_1}} {\mathit{\boldsymbol{Q}}_{1i}}\mathit{\boldsymbol{A}}_1^{\rm{T}} + \sum\limits_{j = 1}^{{n_2}} {{\mathit{\boldsymbol{A}}_2}} {\mathit{\boldsymbol{Q}}_{2j}}\mathit{\boldsymbol{A}}_2^{\rm{T}} + \cdots }\\ { + \sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{A}}_k}} {\mathit{\boldsymbol{Q}}_{kh}}\mathit{\boldsymbol{A}}_k^{\rm{T}}} \end{array} $

(24)

而Σ_W=AΣA^T，顾及式(16)、(17)有：

$ \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W} = \sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{A}}_1}} \sigma _{1i}^2{\mathit{\boldsymbol{Q}}_{1i}}\mathit{\boldsymbol{A}}_1^{\rm{T}} + \sum\limits_{j = 1}^{{n_2}} {{\mathit{\boldsymbol{A}}_2}} \sigma _{2j}^2{\mathit{\boldsymbol{Q}}_{2j}}\mathit{\boldsymbol{A}}_2^{\rm{T}} + \cdots }\\ {{\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} {\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} {\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} {\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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{A}}_k}} \sigma _{kh}^2{\mathit{\boldsymbol{Q}}_{kh}}\mathit{\boldsymbol{A}}_k^{\rm{T}}} \end{array} $

(25)

令

$ \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{Q}}_{1i}}\mathit{\boldsymbol{A}}_1^{\rm{T}} = {\mathit{\boldsymbol{N}}_{1i}},i = 1,2, \cdots ,{n_1}}\\ {{\mathit{\boldsymbol{A}}_2}{\mathit{\boldsymbol{Q}}_{2j}}\mathit{\boldsymbol{A}}_2^{\rm{T}} = {\mathit{\boldsymbol{N}}_{2j}},j = 1,2, \cdots ,{n_2}}\\ {{\kern 1pt} {\kern 1pt} \cdots }\\ {{\mathit{\boldsymbol{A}}_k}{\mathit{\boldsymbol{Q}}_{kh}}\mathit{\boldsymbol{A}}_k^{\rm{T}} = {\mathit{\boldsymbol{N}}_{kh}},h = 1,2, \cdots ,{n_k}} \end{array}} \right. $

(26)

$ \begin{array}{*{20}{c}} {\sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{A}}_1}} {\mathit{\boldsymbol{Q}}_{1i}}\mathit{\boldsymbol{A}}_1^{\rm{T}} = {\mathit{\boldsymbol{N}}_1},\sum\limits_{j = 1}^{{n_2}} {{\mathit{\boldsymbol{A}}_2}} {\mathit{\boldsymbol{Q}}_{2j}}\mathit{\boldsymbol{A}}_2^{\rm{T}} = }\\ {{\mathit{\boldsymbol{N}}_2}, \cdots ,\sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{A}}_k}} {\mathit{\boldsymbol{Q}}_{kh}}\mathit{\boldsymbol{A}}_k^{\rm{T}} = {\mathit{\boldsymbol{N}}_k}} \end{array} $

(27)

则式(24)、(25)可改写为：

$ {\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W} = \sum\limits_{m = 1}^{{n_1}} {\sigma _{1m}^2} {\mathit{\boldsymbol{N}}_{1m}} + \sum\limits_{p = 1}^{{n_2}} {\sigma _{2p}^2} {\mathit{\boldsymbol{N}}_{2p}} + \cdots + \sum\limits_{q = 1}^{{n_k}} {\sigma _{kq}^2} {\mathit{\boldsymbol{N}}_{kq}} $

(28)

$ {\mathit{\boldsymbol{N}}_{AA}} = \sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{N}}_{1i}}} + \sum\limits_{j = 1}^{{n_2}} {{\mathit{\boldsymbol{N}}_{2j}}} + \cdots + \sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{N}}_{kh}}} $

(29)

式(16)、(21)、(22)代入式(14)左端V^TPV可得：

$ \begin{array}{l} {\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}} = \sum\limits_{l = 1}^k {\mathit{\boldsymbol{V}}_l^{\rm{T}}} {\mathit{\boldsymbol{P}}_l}{\mathit{\boldsymbol{V}}_l} = \sum\limits_{i = 1}^{{n_i}} {\mathit{\boldsymbol{V}}_1^{\rm{T}}} {\mathit{\boldsymbol{P}}_{1i}}{\mathit{\boldsymbol{V}}_1} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{j = 1}^{{n_2}} {\mathit{\boldsymbol{V}}_2^{\rm{T}}} {\mathit{\boldsymbol{P}}_{2j}}{\mathit{\boldsymbol{V}}_2} + \cdots + \sum\limits_{h = 1}^{{n_k}} {\mathit{\boldsymbol{V}}_k^{\rm{T}}} {\mathit{\boldsymbol{P}}_{kh}}{\mathit{\boldsymbol{V}}_k} \end{array} $

(30)

式(28)、(29)代入式(14)右端tr(Q_KN_AAQ_KΣ_W)可得：

$ \begin{array}{l} {\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} {\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} {\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} {\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} {\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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W}) = \\ {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}(\sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{N}}_{1i}}} + \cdots + \sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{N}}_{kh}}} ){\mathit{\boldsymbol{Q}}_K}(\sum\limits_{m = 1}^{{n_1}} {\sigma _{1m}^2} {\mathit{\boldsymbol{N}}_{1m}} + \\ {\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} {\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} {\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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {\sum\limits_{p = 1}^{{n_2}} {\sigma _{2p}^2} {\mathit{\boldsymbol{N}}_{2p}} + \cdots + \sum\limits_{q = 1}^{{n_k}} {\sigma _{kq}^2} {\mathit{\boldsymbol{N}}_{kq}})} \right) \end{array} $

(31)

将式(14)展开，去掉数学期望并将σ_ij²写成估值$\hat \sigma _{ij}^2$ (i=1, 2, …, k, j=1, 2, …, n_j)，并令

$ \begin{array}{*{20}{c}} {\mathop {\hat \theta }\limits_{({n_1} + {n_2} + \cdots + {n_k}) \times 1} = }\\ {{{[\hat \sigma _{11}^2,\hat \sigma _{12}^2, \cdots ,\hat \sigma _{1{n_1}}^2, \cdots \cdots ,\hat \sigma _{k1}^2,\hat \sigma _{k2}^2, \cdots ,\hat \sigma _{k{n_k}}^2]}^{\rm{T}}}} \end{array} $

(32)

$ \begin{array}{*{20}{c}} {\mathop f\limits_{({n_1} + {n_2} + \cdots + {n_k}) \times 1} = [\mathit{\boldsymbol{V}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_{11}}{\mathit{\boldsymbol{V}}_1},\mathit{\boldsymbol{V}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_{12}}{\mathit{\boldsymbol{V}}_1}, \cdots ,}\\ {\mathit{\boldsymbol{V}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_{1{n_1}}}{\mathit{\boldsymbol{V}}_1}, \cdots ,\mathit{\boldsymbol{V}}_k^{\rm{T}}{\mathit{\boldsymbol{P}}_{k1}}{\mathit{\boldsymbol{V}}_k},\mathit{\boldsymbol{V}}_k^{\rm{T}}{\mathit{\boldsymbol{P}}_{k2}}{\mathit{\boldsymbol{V}}_k}, \cdots ,\mathit{\boldsymbol{V}}_k^{\rm{T}}{\mathit{\boldsymbol{P}}_{k{n_k}}}{\mathit{\boldsymbol{V}}_k}{]^{\rm{T}}}} \end{array} $

(33)

$ \begin{array}{*{35}{l}} \underset{({{n}_{1}}+{{n}_{2}}+\cdots +{{n}_{k}})\times 1}{\mathop{{{\mathit{\boldsymbol{H}}}_{ij}}}}\,=\left[ \begin{matrix} \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{11}} \right) \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{12}} \right) \\ \cdots \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{1{{n}_{1}}}} \right) \\ \cdots \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{k1}} \right) \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{k2}} \right) \\ \ldots \\ \cdots \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{k{{n}_{k}}}} \right) \\ \end{matrix} \right], \\ \underset{({{n}_{1}}+{{n}_{2}}+\cdots +{{n}_{k}})\times ({{n}_{1}}+{{n}_{2}}+\cdots +{{n}_{k}})}{\mathop{\mathit{\boldsymbol{H}}}}\,= \\ \mathit{\boldsymbol{H}}_{ij}^{\text{T}},i=1,2,\cdots ,k;j=1,2,\cdots ,{{n}_{j}} \\ \end{array} $

(34)

将式(32)~(34)代入式(14)可得：

$ \mathit{\boldsymbol{H\hat \theta }} = \mathit{\boldsymbol{f}} $

(35)

式(35)即为扩展的Helmert型方差分量估计的通用公式。显然H为对称阵，且当H可逆时，式(35)的解为$\mathit{\boldsymbol{\hat \theta = }}{\mathit{\boldsymbol{H}}^{ - 1}}\mathit{\boldsymbol{f}}$；当H不可逆时，式(35)存在唯一的伪逆解$\mathit{\boldsymbol{\hat \theta = }}{\mathit{\boldsymbol{H}}^ + }\mathit{\boldsymbol{f}}$。将式(35)的各行元素求和，顾及式(27)、(29)可得相应的简化公式为：

$ \begin{array}{l} \sigma _{ij}^2 = \frac{{\mathit{\boldsymbol{V}}_i^{\rm{T}}{\mathit{\boldsymbol{P}}_{ij}}{\mathit{\boldsymbol{V}}_i}}}{{ {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{ij}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}})}},\\ i = 1,2, \cdots ,k;j = 1,2, \cdots ,{n_j} \end{array} $

(36)

若每类观测值仅含1个方差分量，则式(35)和(36)分别对应文献[5]中Helmert型方差分量估计通用公式的式(36)和(43)。

3 特殊情况

当式(2)、(3)中的系数阵B=C= 0时，即为条件平差的函数模型；当系数阵C= 0时，即为附有参数的条件平差的函数模型；当式(2)、(3)中的系数阵A=-I，C= 0 (I为单位阵)时，即为间接平差的函数模型；当系数阵A=-I时，即为附有限制条件的间接平差的函数模型。对于不同的平差模型，式(35)、(36)的计算不同之处仅在于Q_K而已。这里以A=-I，C= 0为例，推导间接平差扩展的Helmert型方差分量估计，其他平差方法扩展的Helmert型方差分量估计可类似得到，不再赘述。

当A=-I，C= 0时，式(2)、(3)可变为：

$ \left\{ {\begin{array}{*{20}{l}} {\mathop {{\mathit{\boldsymbol{V}}_1}}\limits_{{n_1} \times 1} = {\mathit{\boldsymbol{B}}_1}\mathit{\boldsymbol{\hat x}} - {\mathit{\boldsymbol{W}}_1}}\\ {\mathop {{\mathit{\boldsymbol{V}}_2}}\limits_{{n_2} \times 1} = {\mathit{\boldsymbol{B}}_2}\mathit{\boldsymbol{\hat x}} - {\mathit{\boldsymbol{W}}_2}}\\ \cdots \\ {\mathop {{\mathit{\boldsymbol{V}}_{k - 1}}}\limits_{{n_{k - 1}} \times 1} = {\mathit{\boldsymbol{B}}_{k - 1}}\mathit{\boldsymbol{\hat x}} - {\mathit{\boldsymbol{W}}_{k - 1}}}\\ {\mathop {{\mathit{\boldsymbol{V}}_k}}\limits_{{n_k} \times 1} = {\mathit{\boldsymbol{B}}_k}\mathit{\boldsymbol{\hat x}} - {\mathit{\boldsymbol{W}}_k}}\\ \end{array}} \right. $

(37)

而

$ {{\mathit{\boldsymbol{Q}}_{\hat x}} = {{({\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}})}^{ - 1}} = \mathit{\boldsymbol{N}}_{BB}^{ - 1}} $

(38)

$ {{\mathit{\boldsymbol{Q}}_K} = \mathit{\boldsymbol{P}} - \mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{Q}}_{\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P}}} $

(39)

顾及N_AA=AQA^T=Q，此时式(14)变为：

$ E({\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}}) = {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{Q}})\sigma _0^2 $

(40)

将式(38)、(39)代入式(40)得：

$ {\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{Q}} = \mathit{\boldsymbol{PQ}} - 2\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{Q}}_{\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}} + \mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{Q}}_{\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{Q}}_{\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}} $

(41)

顾及N_ij=Q_ij, i=1, 2, …, k; j=1, 2, …, n_j，并将式(20)、(22)代入式(41)，考虑求迹运算的循环性质，再一起代入式(14)，去掉期望便得到文献[4]中的式(15)和(16)。由此说明，文献[4]中基于间接平差推导扩展的Helmert型方差分量估计是本文中式(35)和(36)的特例。

4 结语

本文针对现行的方差分量估计方法大多假设每类观测值中仅含1个方差分量的情况，基于概括函数模型和Helmert型方差分量估计的原理，推导了估计每类观测值含有多个方差分量的通用公式、相应的简化公式及一些特殊情况。同时，随着观测手段的丰富与多样化，往往同类观测值受到多种因素影响，即同类观测值可含有多个方差分量，该公式必将得到越来越多的使用。通用公式的推导不仅可以提高数据处理的精度，也可以使扩展的Helmert型方差分量估计在其他平差方法中的使用得到统一，是对方差分量估计理论的又一补充。