A total least squares solution for geodetic datum transformations

Abstract

In this contribution, the symmetrical total least squares adjustment for 3D datum transformations is classified as quasi indirect errors adjustment (QIEA). QIEA is a traditional geodetic adjustment category invented by Wolf (Ausgleichungsrechnung nach der Methode der kleinsten Quadrate, 1968), which is specifically used for quasi nonlinear models. The form of the QIEA objective function contains the information of the functional model, and presents an unconstrained minimization problem referring simply to the transformation parameters. Based on QIEA, a solution is presented through a quasi-Newton approach, specially, the Broyden–Fletcher–Goldfarb–Shanno method. In order to justify the solutions of the QIEA, three validation conditions are proposed to check the correctness of the symmetrical treatment by comparison between the transformation and its reverse transformation. Finally, the applicability of the proposed algorithm was tested in a deformation monitoring task.

Appendices

Appendix 1

The matrix analysis property in Grafarend and Schaffrin (1993) shows for the first derivative w.r.t. the parameter vector is as follows

$$\begin{aligned} \mathbf{J}_1 =\frac{\partial \mathbf{f}\left( {{\varvec{\upxi }},\mathbf{v}} \right) }{\partial {\varvec{\upxi }}^{T}}=\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {\frac{\partial \mathbf{f}}{\partial \alpha _1 }}&{} {\frac{\partial \mathbf{f}}{\partial \alpha _2 }}&{} {\frac{\partial \mathbf{f}}{\partial \alpha _3 }}&{} {\frac{\partial \mathbf{f}}{\partial \mu }}&{} {\frac{\partial \mathbf{f}}{\partial \Delta x}}&{} {\frac{\partial \mathbf{f}}{\partial \Delta y}}&{} {\frac{\partial \mathbf{f}}{\partial \Delta z}} \\ \end{array} }} \right] \end{aligned}$$

(29)

with

$$\begin{aligned}&\displaystyle \frac{\partial \mathbf{f}\left( {{\varvec{\upxi }},\mathbf{v}} \right) }{\partial \alpha _i }=vec\left( {\left( {\mathbf{A}+\mathbf{V}_\mathbf{A} } \right) \mu {\dot{\mathbf{M}}}_i} \right) \quad i=1,2,3\end{aligned}$$

(30)

$$\begin{aligned}&\displaystyle \frac{\partial \mathbf{f}\left( {{\varvec{\upxi }},\mathbf{v}} \right) }{\partial \mu }=vec\left( {\left( {\mathbf{A}+\mathbf{V}_\mathbf{A} } \right) \mathbf{M}} \right) \end{aligned}$$

(31)

$$\begin{aligned}&\displaystyle \frac{\partial \mathbf{f}\left( {{\varvec{\upxi }},\mathbf{v}} \right) }{\partial \left[ {{\begin{array}{l@{\quad }l@{\quad }l} {\Delta x}&{} {\Delta y}&{} {\Delta z} \\ \end{array} }} \right] }=\mathbf{I}_3 \otimes \mathbf{1}_n \end{aligned}$$

(32)

where

$$\begin{aligned} {\dot{\mathbf{M}}}_1&= \mathbf{M}_3 \mathbf{M}_2 \frac{\partial \mathbf{M}_1}{\partial \alpha _1 } \nonumber \\&= \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {\cos \alpha _3 }&{} {\sin \alpha _3 }&{} 0 \\ {-\sin \alpha _3 }&{} {\cos \alpha _3 }&{} 0 \\ 0&{} 0&{} 1 \\ \end{array} }} \right] \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {\cos \alpha _2 }&{} 0&{} {-\sin \alpha _2 } \\ 0&{} 1&{} 0 \\ {\sin \alpha _2 }&{} 0&{} {\cos \alpha _2 } \\ \end{array} }} \right] \left[ {{\begin{array}{c@{\quad }c@{\quad }c} 0&{} 0&{} 0 \\ 0&{} {-\sin \alpha _1 }&{} {\cos \alpha _1 } \\ 0&{} {-\cos \alpha _1 }&{} {-\sin \alpha _1 } \\ \end{array} }} \right] \qquad \end{aligned}$$

(33)

$$\begin{aligned} {\dot{\mathbf{M}}}_2&= \mathbf{M}_3 \frac{\partial \mathbf{M}_2}{\partial \alpha _2 }\mathbf{M}_1 \nonumber \\&= \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {\cos \alpha _3 }&{} {\sin \alpha _3 }&{} 0 \\ {-\sin \alpha _3 }&{} {\cos \alpha _3 }&{} 0 \\ 0&{} 0&{} 1 \\ \end{array} }} \right] \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {-\sin \alpha _2 }&{} 0&{} {-\cos \alpha _2 } \\ 0&{} 0&{} 0 \\ {\cos \alpha _2 }&{} 0&{} {-\sin \alpha _2 } \\ \end{array} }} \right] \left[ {{\begin{array}{c@{\quad }c@{\quad }c} 1&{} 0&{} 0 \\ 0&{} {\cos \alpha _1 }&{} {\sin \alpha _1 } \\ 0&{} {-\sin \alpha _1 }&{} {\cos \alpha _1 } \\ \end{array} }} \right] \qquad \end{aligned}$$

(34)

$$\begin{aligned} {\dot{\mathbf{M}}}_3&= \frac{\partial \mathbf{M}_3}{\partial \alpha _3 }\mathbf{M}_2 \mathbf{M}_1 \nonumber \\&= \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {-\sin \alpha _3 }&{} {\cos \alpha _3 }&{} 0 \\ {-\cos \alpha _3 }&{} {-\sin \alpha _3 }&{} 0 \\ 0&{} 0&{} 0 \\ \end{array} }} \right] \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {\cos \alpha _2 }&{} 0&{} {-\sin \alpha _2 } \\ 0&{} 1&{} 0 \\ {\sin \alpha _2 }&{} 0&{} {\cos \alpha _2 } \\ \end{array} }} \right] \left[ {{\begin{array}{c@{\quad }c@{\quad }c} 1&{} 0&{} 0 \\ 0&{} {\cos \alpha _1 }&{} {\sin \alpha _1 } \\ 0&{} {-\sin \alpha _1 }&{} {\cos \alpha _1 } \\ \end{array} }} \right] \qquad \end{aligned}$$

(35)

Thus, the Jacobian matirx $\mathbf{J}_1$ is written as the vector function $\mathbf{J}_1 :=\mathbf{J}_1 \left( {{\varvec{\upxi }},\mathbf{V}_\mathbf{A} } \right)$ which relates to the parameter vector ${\varvec{\upxi }}$ and the correction matrix $\mathbf{V}_\mathbf{A}$.

Similarly, the Jacobian matrix $\mathbf{J}_2$ is given as

$$\begin{aligned} \mathbf{J}_2 := \mathbf{J}_2 \left( {\varvec{\upxi }} \right) =\frac{\partial \mathbf{f}}{\partial \mathbf{l}^{T}}&= \left[ {{\begin{array}{l@{\quad }l} \displaystyle {\frac{\partial \mathbf{f}}{\partial vec^{T}\left( \mathbf{A} \right) }}&{} \displaystyle {\frac{\partial \mathbf{f}}{\partial vec^{T}\left( \mathbf{Y} \right) }} \\ \end{array} }} \right] \nonumber \\&= \left[ {{\begin{array}{ll} {\left( {\mu \mathbf{M}} \right) ^{T}\otimes \mathbf{I}_n }&{} {-\mathbf{I}_3 \otimes \mathbf{I}_n } \\ \end{array} }} \right] =\left[ {{\begin{array}{l@{\quad }l} {\left( {\mu \mathbf{M}} \right) ^{T}}&{} {-\mathbf{I}_3 } \\ \end{array} }} \right] \otimes \mathbf{I}_n \end{aligned}$$

(36)

due to $vec\left( {\mathbf{I}_n \left( {\mathbf{A}+\mathbf{V}_\mathbf{A} } \right) \mu \mathbf{M}} \right) =\left( {\left( {\mu \mathbf{M}} \right) ^{T}\otimes \mathbf{I}_n } \right) vec\left( {\mathbf{A}+\mathbf{V}_\mathbf{A} } \right)$ using the matrix property $vec\left( {\mathbf{ABC}} \right) =\left( {\mathbf{C}^{T}\otimes \mathbf{A}} \right) vec\left( \mathbf{B} \right)$ (see Koch 1999, p. 41). The Jacobian matrix $\mathbf{J}_2$ only relates to the parameter vector ${\varvec{\upxi }}$.

Appendix 2

The matrix analysis property (differentiation of a scalar function w.r.t. a vector) can be used for the first derivative of the objective function (22) w.r.t. the parameter vector

$$\begin{aligned} \frac{\partial f\left( {\varvec{\upxi }} \right) }{\partial {\varvec{\upxi }}}=\left[ {{\begin{array}{c} \displaystyle {\frac{\partial f\left( {\varvec{\upxi }} \right) }{\partial \upxi _1 }} \\ {...} \\ \displaystyle {\frac{\partial f\left( {\varvec{\upxi }} \right) }{\partial \upxi _k }} \\ {...} \\ \displaystyle {\frac{\partial f\left( {\varvec{\upxi }} \right) }{\partial \upxi _7 }} \\ \end{array} }} \right] \end{aligned}$$

(37)

Here, the parameter vector ${\varvec{\upxi }}=\left[ {\upxi _i } \right]$ is given by its elements, and $f\left( {\varvec{\upxi }} \right) =\mathbf{r}^{T}\mathbf{Q}_{\mathbf{rr}}^{-1} \mathbf{r}$.

The first derivative w.r.t. the parameter vector can be extended in three parts according to the product rule (the well-known Leibniz’s Law) as follows:

$$\begin{aligned} \frac{\partial \mathbf{r}^{T}\mathbf{Q}_{\mathbf{rr}}^{-1} \mathbf{r}}{\partial \upxi _k }&= \frac{\partial \mathbf{r}^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r}}{\partial \upxi _k } \nonumber \\&= \frac{\partial \mathbf{r}^{T}}{\partial \upxi _k }\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r}+\mathbf{r}^{T}\frac{\partial \left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}}{\partial \upxi _k }\mathbf{r}+\mathbf{r}^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\frac{\partial \mathbf{r}}{\partial \upxi _k } \nonumber \\&= 2\frac{\partial \mathbf{r}^{T}}{\partial \upxi _k }\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r}+\mathbf{r}^{T}\frac{\partial \left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}}{\partial \upxi _k }\mathbf{r} \end{aligned}$$

(38)

The last row of the above equation is valid because $\mathbf{r}^{T}\left( {{\hat{\mathbf{J}}}_2 \mathbf{Q}_{\mathbf{ll}} {\hat{\mathbf{J}}}_2^T } \right) ^{-1}\mathbf{r}$ is a scalar.

The first term of the above equation can be calculated as

$$\begin{aligned} \frac{\partial \mathbf{r}^{T}}{\partial {\hat{\varvec{\upxi }}}}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r}=-\left( {\mathbf{J}_\mathbf{A} } \right) ^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r} \end{aligned}$$

(39)

where

$$\begin{aligned} \mathbf{J}_\mathbf{A}=\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {vec\left( {\mathbf{A}\mu {\dot{\mathbf{M}}}_1 } \right) }&{} {vec\left( {\mathbf{A}\mu {\dot{\mathbf{M}}}_2 } \right) }&{} {vec\left( {\mathbf{A}\mu {\dot{\mathbf{M}}}_3} \right) }&{} {vec\left( {\mathbf{AM}} \right) }&{} {\mathbf{I}_3 \otimes \mathbf{1}_n } \\ \end{array} }} \right] \end{aligned}$$

(40)

The Jacobian matrix ${\hat{\mathbf{J}}}_\mathbf{A}$ is derived by a strategy wherein the Jacobian matrix $\mathbf{J}_1$ is obtained (Appendix 1).

Then, the second term can be solved with $\frac{\partial \mathbf{A}^{-1}}{\partial \upxi _k }=-\mathbf{A}^{-1}\frac{\partial \mathbf{A}}{\partial \upxi _k }\mathbf{A}^{-1}$ (see Grafarend and Schaffrin 1993) as

$$\begin{aligned} \mathbf{r}^{T}\frac{\partial \left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}}{\partial \upxi _k }\mathbf{r}&= -\mathbf{r}^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\frac{\partial \left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) }{\partial \upxi _k }\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r} \nonumber \\&= -2\mathbf{r}^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \frac{\partial \left( {\left[ {{\begin{array}{l} {\left( {\mu \mathbf{M}} \right) \otimes \mathbf{I}_n } \\ {-\mathbf{I}_3 \otimes \mathbf{I}_n } \\ \end{array} }} \right] } \right) }{\partial \upxi _k }\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r} \nonumber \\&= -2\left( {\frac{\partial \left( {\left[ {{\begin{array}{l@{\quad }l} {\left( {\mu \mathbf{M}} \right) ^{T}}&{} {-\mathbf{I}_3 } \\ \end{array} }} \right] \otimes \mathbf{I}_n } \right) }{\partial \upxi _k }\mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T \left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r}} \right) ^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r} \nonumber \\&= -2\left( {\left( {\frac{\partial \left( {\mu \mathbf{M}} \right) ^{T}}{\partial \upxi _k }\otimes \mathbf{I}_n } \right) \left[ {{\begin{array}{l@{\quad }l} {\mathbf{Q}_{\mathbf{AA}} }&{} {\mathbf{Q}_{\mathbf{AY}} } \\ \end{array} }} \right] \mathbf{J}_2^T \left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r}} \right) ^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r} \nonumber \\&= -2vec^{T}\left( {\mathbf{A}^{*}\frac{\partial \left( {\mu \mathbf{M}} \right) }{\partial \upxi _k }} \right) \left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r} \end{aligned}$$

(41)

with

$$\begin{aligned} \mathbf{A}^{*}=\mathbf{A}^{*}\left( {\varvec{\upxi }} \right) =-Ivec_{n\times u} \left( {\left[ {{\begin{array}{l@{\quad }l} {\mathbf{Q}_{\mathbf{AA}} }&{} {\mathbf{Q}_{\mathbf{AY}} } \\ \end{array} }} \right] \mathbf{J}_2^T \left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r}} \right) \end{aligned}$$

(42)

where the operator $Ivec_{n\times u}$ is the opposite of the ‘vec’ operator and reshapes the vector as the assigned matrix form. The last row of the above equation is due to the matrix property $vec\left( {\mathbf{ABC}} \right) =\left( {\mathbf{C}^{T}\otimes \mathbf{A}} \right) vec\left( \mathbf{B} \right)$. Actually, ${\hat{\mathbf{A}}}^{*}$ is identical to the estimated correction matrix obtained by Eq. (20), but it relates now only to the parameter and is therefore correctly applied since there is no correction matrix in the QIEA.

For the whole parameter vector, the first derivative can be obtained by a combination of the above Eqs. (39) and (41) in Appendix 2:

$$\begin{aligned} \frac{\partial f\left( {\varvec{\upxi }} \right) }{\partial {\varvec{\upxi }}}&= -2\left( {\mathbf{J}_\mathbf{A}} \right) ^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r}\nonumber \\&-2\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {vec\left( {\mathbf{A}^{*}\mu {\dot{\mathbf{M}}}_1} \right) }&{} {vec\left( {\mathbf{A}^{*}\mu {\dot{\mathbf{M}}}_2} \right) }&{} {vec\left( {\mathbf{A}^{*}\mu {\dot{\mathbf{M}}}_3} \right) }&{} {vec\left( {\mathbf{A}^{*}\mathbf{M}} \right) }&{} \mathbf{0} \\ \end{array} }} \right] ^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r} \nonumber \\&= -2\left[ {{\begin{array}{c} {vec^{T}\left( {\left( {\mathbf{A}+\mathbf{A}^{*}} \right) \mu {\dot{\mathbf{M}}}_1} \right) } \\ {vec^{T}\left( {\left( {\mathbf{A}+\mathbf{A}^{*}} \right) \mu {\dot{\mathbf{M}}}_2} \right) } \\ {vec^{T}\left( {\left( {\mathbf{A}+\mathbf{A}^{*}} \right) \mu {\dot{\mathbf{M}}}_3} \right) } \\ {vec^{T}\left( {\left( {\mathbf{A}+\mathbf{A}^{*}} \right) \mathbf{M}} \right) } \\ {\mathbf{I}_3 \otimes \mathbf{1}_n^T } \\ \end{array} }} \right] \left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r}=-2\left( {\mathbf{J}_1} \right) ^{T}\left( {\mathbf{J}_2 \mathbf{Q}_{\mathbf{ll}} \mathbf{J}_2^T } \right) ^{-1}\mathbf{r} \end{aligned}$$

(43)