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Adjustment of geodetic measurements with mixed multiplicative and additive random errors

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Abstract

Adjustment has been based on the assumption that random errors of measurements are added to functional models. In geodetic practice, we know that accuracy formulae of modern geodetic measurements often consist of two parts: one proportional to the measured quantity and the other constant. From the statistical point of view, such measurements are of mixed multiplicative and additive random errors. However, almost no adjustment has been developed to strictly address geodetic data contaminated by mixed multiplicative and additive random errors from the statistical point of view. We systematically develop adjustment methods for geodetic data contaminated with multiplicative and additive errors. More precisely, we discuss the ordinary least squares (LS) and weighted LS methods and extend the bias-corrected weighted LS method of Xu and Shimada (Commun Stat B29:83–96, 2000) to the case of mixed multiplicative and additive random errors. The first order approximation of accuracy for all these three methods is derived. We derive the biases of weighted LS estimates. The three methods are then demonstrated and compared with a synthetic example of surface interpolation. The bias-corrected weighted LS estimate is unbiased up to the second order approximation and is of the best accuracy. Although the LS method can warrant an unbiased estimate for a linear model with multiplicative and additive errors, it is less accurate and always produces a very poor estimate of the variance of unit weight.

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Acknowledgments

This work is partially supported by the National Foundation of Natural Science of China (No. 41204006) and a Grant-in-aid for Scientific Research (C25400449). The authors thank, with gratitude, the three reviewers for their very constructive and very detailed comments, which lead to a significant improvement and clarification of the paper.

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Authors and Affiliations

  1. Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto, 611-0011, Japan

    Peiliang Xu

  2. School of Geomatics, Xi’an University of Science and Technology, Xi’an, People’s Republic of China

    Yun Shi

  3. School of Land Science and Geomatics, China University of Geosciences, Beijing, People’s Republic of China

    Junhuan Peng

  4. GNSS Research Center, Wuhan University, Wuhan, 430071, People’s Republic of China

    Jingnan Liu & Chuang Shi

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  1. Peiliang Xu
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  2. Yun Shi
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  3. Junhuan Peng
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  4. Jingnan Liu
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  5. Chuang Shi
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Corresponding author

Correspondence to Yun Shi.

Appendix: Proof of Lemma 1

Appendix: Proof of Lemma 1

The proof of Lemma 1 consists of two steps. The first step is to use the first two terms on the left hand side of (28) to derive (29) of Lemma 1, and the second step is to prove:

$$\begin{aligned}&E\{\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}( \mathbf{D}_{ab}\varvec{\Sigma }_m \mathbf{D}_{a\beta } + \mathbf{D}_{a\beta }\varvec{\Sigma }_m\mathbf{D}_{ab}) \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\}\nonumber \\&\quad -E\{\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}( \mathbf{D}_{ab}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\nonumber \\&\quad + \mathbf{D}_{a\beta }\varvec{\Sigma }_m\mathbf{D}_{ab}) \varvec{\Sigma }_y^{-1}\mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\} = \mathbf{0}. \end{aligned}$$
(56)

Without loss of generality, let us first assume that (56) holds true. As a result, applying the mathematical expectation operation to (28) yields

$$\begin{aligned} \mathbf{N}E(\mathbf{b}_{\beta }) - E(\mathbf{G}_2\varvec{\epsilon }_y) = \mathbf{0}, \end{aligned}$$

or equivalently,

$$\begin{aligned} \mathbf{N}\mathbf{b}(\hat{\varvec{\beta }}) = E(\mathbf{G}_2\varvec{\epsilon }_y), \end{aligned}$$
(57)

where \(\mathbf{b}(\hat{\varvec{\beta }})\) denotes \(E(\mathbf{b}_{\beta })\) and stands for the bias of the WLS estimate of \(\varvec{\beta }\). The term on the right hand side of (57), i.e., \(E(\mathbf{G}_2\varvec{\epsilon }_y)\), can be computed as follows:

$$\begin{aligned} E(\mathbf{G}_2\varvec{\epsilon }_y)&= E\left\{ \left[ \begin{array}{l} \varvec{\epsilon }_y^\mathrm{T} \varvec{\Sigma }_y^{-1} \mathbf{D}_{ae_1}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1} \\ \varvec{\epsilon }_y^\mathrm{T} \varvec{\Sigma }_y^{-1} \mathbf{D}_{ae_2}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1} \\ \vdots \\ \varvec{\epsilon }_y^\mathrm{T} \varvec{\Sigma }_y^{-1} \mathbf{D}_{ae_t}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1} \end{array} \right] \varvec{\epsilon }_y \right\} \nonumber \\&= \left[ \begin{array}{l} E( \varvec{\epsilon }_y^\mathrm{T} \varvec{\Sigma }_y^{-1} \mathbf{D}_{ae_1}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ E(\varvec{\epsilon }_y^\mathrm{T} \varvec{\Sigma }_y^{-1} \mathbf{D}_{ae_2}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ \vdots \\ E(\varvec{\epsilon }_y^\mathrm{T} \varvec{\Sigma }_y^{-1} \mathbf{D}_{ae_t}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \end{array} \right] \nonumber \\&= \left[ \begin{array}{l} \mathrm{tr}\{ E( \varvec{\epsilon }_y\varvec{\epsilon }_y^\mathrm{T} )\varvec{\Sigma }_y^{-1} \mathbf{D}_{ae_1}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1} \} \\ \mathrm{tr}\{ E(\varvec{\epsilon }_y\varvec{\epsilon }_y^\mathrm{T} )\varvec{\Sigma }_y^{-1} \mathbf{D}_{ae_2}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1} \} \\ \vdots \\ \mathrm{tr}\{ E(\varvec{\epsilon }_y\varvec{\epsilon }_y^\mathrm{T} )\varvec{\Sigma }_y^{-1} \mathbf{D}_{ae_t}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1} \} \end{array} \right] \nonumber \\&= \left[ \begin{array}{l} \mathrm{tr}\{ \mathbf{D}_{ae_1}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1} \} \\ \mathrm{tr}\{ \mathbf{D}_{ae_2}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1} \} \\ \vdots \\ \mathrm{tr}\{ \mathbf{D}_{ae_t}\varvec{\Sigma }_m \mathbf{D}_{a\beta }\varvec{\Sigma }_y^{-1} \} \end{array} \right] = \mathbf{g}_2, \end{aligned}$$
(58)

because \(E( \varvec{\epsilon }_y\varvec{\epsilon }_y^\mathrm{T})=\varvec{\Sigma }_y\). By inserting (58) into (57), we can readily obtain

$$\begin{aligned} \mathbf{b}(\hat{\varvec{\beta }}) =\mathbf{N}^{-1}\mathbf{g}_2, \end{aligned}$$
(59)

which is exactly (29) of Lemma 1.

Now we have to prove that the second step, i.e., (56), is indeed true. The proof of (56) can be further simplified to prove:

$$\begin{aligned}&E\{\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1} \mathbf{D}_{a\beta }\varvec{\Sigma }_m\mathbf{D}_{ab} \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\}\nonumber \\&\quad -E\{\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1} \mathbf{D}_{a\beta }\varvec{\Sigma }_m\mathbf{D}_{ab} \varvec{\Sigma }_y^{-1}\mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\} = \mathbf{0}, \end{aligned}$$
(60)

because the same method of proof can be used to prove

$$\begin{aligned}&E\{\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1} \mathbf{D}_{ab}\varvec{\Sigma }_m \mathbf{D}_{a\beta } \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\} \nonumber \\&\quad -E\{\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1} \mathbf{D}_{ab}\varvec{\Sigma }_m \mathbf{D}_{a\beta } \varvec{\Sigma }_y^{-1}\mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\} = \mathbf{0}. \end{aligned}$$

Actually, (60) can be rewritten as follows:

$$\begin{aligned}&\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1} \mathbf{D}_{a\beta }\varvec{\Sigma }_m \left[ \begin{array}{l} E(\mathbf{a}_1\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_1 \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ E(\mathbf{a}_2\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_2 \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ \vdots \\ E(\mathbf{a}_n\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_n \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \end{array} \right] \nonumber \\&\quad = \mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1} \mathbf{D}_{a\beta }\varvec{\Sigma }_m \left[ \begin{array}{l} E(\mathbf{a}_1\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_1 \varvec{\Sigma }_y^{-1} \mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ E(\mathbf{a}_2\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_2 \varvec{\Sigma }_y^{-1}\mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ \vdots \\ E(\mathbf{a}_n\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_n \varvec{\Sigma }_y^{-1} \mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \end{array} \right] ,\nonumber \\ \end{aligned}$$
(61)

where \(\mathbf{e}_i\) is the ith row vector of the natural basis of dimension \(n\). Because the left and right hand sides of (61) have the same coefficient matrix \(\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1} \mathbf{D}_{a\beta }\varvec{\Sigma }_m\), the proof of (61) can be further simplified to prove

$$\begin{aligned}&\left[ \begin{array}{l} E(\mathbf{a}_1\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_1 \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ E(\mathbf{a}_2\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_2 \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ \vdots \\ E(\mathbf{a}_n\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_n \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \end{array} \right] \nonumber \\&\quad = \left[ \begin{array}{l} E(\mathbf{a}_1\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_1 \varvec{\Sigma }_y^{-1} \mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ E(\mathbf{a}_2\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_2 \varvec{\Sigma }_y^{-1}\mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \\ \vdots \\ E(\mathbf{a}_n\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_n \varvec{\Sigma }_y^{-1} \mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \end{array} \right] . \end{aligned}$$
(62)

In fact, for the ith element on the left side of (62), we have

$$\begin{aligned}&E(\mathbf{a}_i\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_i^\mathrm{T} \varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y)\nonumber \\&\quad = E(\mathbf{a}_i\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y \varvec{\epsilon }_y^\mathrm{T} \varvec{\Sigma }_y^{-1}\mathbf{e}_i) \nonumber \\&\quad = \mathbf{a}_i\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}E(\varvec{\epsilon }_y \varvec{\epsilon }_y^\mathrm{T}) \varvec{\Sigma }_y^{-1}\mathbf{e}_i \nonumber \\&\quad = \mathbf{a}_i\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\mathbf{e}_i^\mathrm{T}. \end{aligned}$$
(63)

In the similar manner, for the ith element on the right side of (62), we have

$$\begin{aligned}&E(\mathbf{a}_i\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y\mathbf{e}_i \varvec{\Sigma }_y^{-1} \mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y) \nonumber \\&\quad = E(\mathbf{a}_i\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\varvec{\epsilon }_y \varvec{\epsilon }_y^\mathrm{T}\varvec{\Sigma }_y^{-1}\mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T} \varvec{\Sigma }_y^{-1}\mathbf{e}_i^\mathrm{T}) \nonumber \\&\quad = E(\mathbf{a}_i\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\mathbf{A}\mathbf{N}^{-1}\mathbf{A}^\mathrm{T} \varvec{\Sigma }_y^{-1}\mathbf{e}_i^\mathrm{T}) \nonumber \\&\quad = \mathbf{a}_i\mathbf{N}^{-1}\mathbf{A}^\mathrm{T}\varvec{\Sigma }_y^{-1}\mathbf{e}_i^\mathrm{T}. \end{aligned}$$
(64)

By comparing (63) with (64), we immediately conclude that (62) is true. As a result, we complete the proof of Lemma 1.

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Xu, P., Shi, Y., Peng, J. et al. Adjustment of geodetic measurements with mixed multiplicative and additive random errors. J Geod 87, 629–643 (2013). https://doi.org/10.1007/s00190-013-0635-2

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