A linearised adjustment model is built for the deformation analysis. Linearisation of all equations of system (10) is done with implicit differentiation relative to the observed vectors **a**_{1}, **b**_{i} (*i* = 2, . . ., *p*), **z**_{d} and **z**_{f}, and the unknown parameter vectors **f**, **c** and ∇.

The first and last two equations of system (10) are linearised as:
$$\{\begin{array}{ll}E\left\{{\underset{\_}{\Delta a}}_{1}\right\}\hfill & ={P}_{1}\Delta c\mathrm{,}\hfill \\ \Delta {z}_{f}\hfill & ={\left(\frac{\partial {\zeta}_{f}}{\partial f}\right)}_{0}\Delta f\mathrm{,}\hfill \\ \Delta {z}_{d}\hfill & ={\left(\frac{\partial {\zeta}_{d}}{\partial c}\right)}_{0}\Delta c+{\left(\frac{\partial {\zeta}_{d}}{\partial \nabla}\right)}_{0}\Delta \nabla .\hfill \end{array}$$(15)

We define for later use:
$$\begin{array}{c}{Z}_{f}={\left(\frac{\partial {\zeta}_{f}}{\partial f}\right)}_{0}\mathrm{,}\\ {Z}_{d}={\left(\frac{\partial {\zeta}_{d}}{\partial c}\right)}_{0}\mathrm{,}{Z}_{\nabla}={\left(\frac{\partial {\zeta}_{d}}{\partial \nabla}\right)}_{0}\mathrm{.}\end{array}$$(16)

The partial derivatives of the vectors ζ_{f} and ζ_{d} with respect to the vectors **f**, **c** and ∇ are matrices. The parentheses with zero (.)_{0} indicate that approximate values of the parameters have to be used to get the values in the matrices.

For the equations with **b**_{i} (*i* = 2, . . ., *p*) in system (10) the linearised equations are:
$${B}_{i}E\left\{{\underset{\_}{\Delta b}}_{i}\right\}+{F}_{i}\Delta {f}_{i}={P}_{i}\Delta c\mathrm{,}$$(17)

with the matrices **B**_{i} defined as follows:
$${B}_{i}={B}_{\mathrm{2,1}}{B}_{\mathrm{3,2}}\cdots {B}_{i\mathrm{,}i-1},$$(18)

and with ( *j* = 2, . . ., *i *– 1):
$$\begin{array}{c}{B}_{j\mathrm{,}j-1}={\left(\frac{\partial {\phi}_{j\mathrm{,}j-1}}{\partial {\phi}_{j+\mathrm{1,}j}}\right)}_{0}={\left(\frac{\partial {\phi}_{j\mathrm{,}j-1}}{\partial {b}_{i}^{(j)}}\right)}_{0}\mathrm{,}\\ {B}_{i\mathrm{,}i-1}={\left(\frac{\partial {\phi}_{i\mathrm{,}i-1}}{\partial {b}_{i}^{(i)}}\right)}_{0}\mathrm{.}\end{array}$$(19)

**F**_{i} is defined for *i* = 2, . . . *p* as follows:
$${F}_{i}=\left({F}_{\mathrm{2,1}}\mathrm{,}\cdots \mathrm{,}{F}_{i\mathrm{,}i-1}\mathrm{,}0\mathrm{,}\cdots \mathrm{,}0\right)\mathrm{,}$$(20)

with (*p*-*i*) matrices **0** of zeros, which have the same number of rows as **F**_{2,1 }, and the partioning of **F**_{i} in columns in accordance with the partitioning of ∆**f**:
$$\Delta f=\left(\Delta {f}_{\mathrm{2,1}}\mathrm{,}\cdots \mathrm{,}\Delta {f}_{i\mathrm{,}i-1}\mathrm{,}\Delta {f}_{i+\mathrm{1,}i}\cdots \mathrm{,}\Delta {f}_{p\mathrm{,}p-1}\right).$$(21)

For **F**_{i, i–1} (*i* = 2, . . ., *p*) we have:
$${F}_{i\mathrm{,}i-1}={B}_{\mathrm{2,1}}{B}_{\mathrm{3,2}}\cdots {B}_{i-\mathrm{2,}i-1}{\left(\frac{\partial {\phi}_{i\mathrm{,}i-1}}{\partial {f}_{i\mathrm{,}i-1}}\right)}_{0}\mathrm{.}$$(22)

Matrix **B**_{i, i–1} for an affine transformation is given by [8] as follows:
$${B}_{i\mathrm{,}i-1}=\left(\begin{array}{ccc}{a}_{11}^{0}I& {a}_{12}^{0}I& {a}_{13}^{0}I\\ {a}_{21}^{0}I& {a}_{22}^{0}I& {a}_{23}^{0}I\\ {a}_{31}^{0}I& {a}_{32}^{0}I& {a}_{33}^{0}I\end{array}\right)\mathrm{,}$$(23)

with ${a}_{ij}^{0}\left(i,j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right)$ the approximate values of *a*_{ij} and **I** the (*n* × *n*) unit matrix and *n* the amount of points in **b**_{i}.

As explained in section 5.3.4, we can take ${a}_{ij}^{0}={\delta}_{ij}$, which results in a unit matrix for **B**_{i, i–1}, from which follows, see equations (18) and (22):
$$\begin{array}{c}{B}_{i}=I\mathrm{,}\\ {F}_{i\mathrm{,}i-1}={\left(\frac{\partial {\phi}_{i\mathrm{,}i-1}}{\partial {f}_{i\mathrm{,}i-1}}\right)}_{0}\mathrm{.}\end{array}$$(24)

Matrix **F**_{i, i–1} for an affine transformation is given by [8] as follows:
$${F}_{i\mathrm{,}i-1}=\left(\begin{array}{cccc}{\beta}_{i}& 0& 0& {\epsilon}_{1}\\ 0& {\beta}_{i}& 0& {\epsilon}_{2}\\ 0& 0& {\beta}_{i}& {\epsilon}_{3}\end{array}\right)\mathrm{,}$$(25)

where *β*_{i}, *ε*_{1}, *ε*_{2}, *ε*_{3} and **0** are all (*n* × 3) matrices, as follows:

*β*_{i} = (**u**_{0}, **v**_{0}, **w**_{0}); **u**_{0}, **v**_{0}, **w**_{0} are approximate values of **u**, **v**, **w**, (the *x*, *y*, z coordinates in **b**_{i}), which can be transformed to make the barycentre the origin.
$$\begin{array}{l}{\epsilon}_{1}=\left(\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ \vdots & \vdots & \vdots \\ 1& 0& 0\end{array}\right)\mathrm{,}\end{array}$$

*ε*_{2} and *ε*_{3} are analogous matrices as *ε*_{1} with ones in the second, resp. third column, **0** is the (*n* × 3) zero matrix.

We define **F**_{1} as the null matrix **0** and put it together with the **F**_{i}, *i* = 2 . . . *p* of equation (20) into matrix **F**. Analogously we take all **P**_{i} together in a matrix **P**:
$$\begin{array}{c}F={\left({F}_{1}\mathrm{,}\dots \mathrm{,}{F}_{p}\right)}^{T}\\ P={\left({P}_{1}\mathrm{,}\dots \mathrm{,}{P}_{p}\right)}^{T}\end{array}$$(26)

We define vector ∆**b** as:
$$\Delta b={\left(\Delta {a}_{1}\mathrm{,}\Delta {b}_{2}\mathrm{,}\dots \mathrm{,}\Delta {b}_{p}\right)}^{T}\mathrm{.}$$(27)

We can now formulate the linearised equivalent of system (10):
$$E\left\{\left(\begin{array}{c}\underset{\_}{\Delta b}\\ \Delta {z}_{f}\\ \Delta {z}_{d}\end{array}\right)\right\}=\left(\begin{array}{ccc}P& -F& 0\\ 0& {Z}_{f}& 0\\ {Z}_{d}& 0& {Z}_{\nabla}\end{array}\right)\left(\begin{array}{c}\Delta c\\ \Delta f\\ \Delta \nabla \end{array}\right)\mathrm{.}$$(28)

The covariance matrix of the observation vector on the left-hand side consists of the covariance matrices of **a**_{1} and **b**_{i}, *i *= 2, . . .,*p,* as described in section 5.1, approximately transformed as described in section 5.3.4, and zero matrices for the remainder if no correlation between the epochs is assumed (which is, however, not necessary to solve the model).

The model takes each epoch as a separate geodetic network: each point has a different point number for each epoch, for example point *A* is called *A*_{1} in epoch 1, *A*_{2} in epoch 2, etc. The hypothesis that no deformation has occured is formulated by stating that
$$\{\begin{array}{c}0={x}_{{A}_{2}}-{x}_{{A}_{1}}\mathrm{,}\\ 0={y}_{{A}_{2}}-{y}_{{A}_{1}}\mathrm{,}\\ 0={z}_{{A}_{2}}-{z}_{{A}_{1}}\mathrm{,}\\ \text{etc}\text{.}\end{array}$$(29)

The separate geodetic networks are linked together in this way. Equation (29) gives the nonstochastic observation equations (the zeros constitute together vector **z**_{d} and have a standard deviation of zero). The number of rows of matrix **Z**_{d} is three times the number of points. In each row there are zeros and one 1 and one −1 for respectively the coordinate of epoch 2 and epoch 1 (which are separate unknowns in the parameter vector). There are no parameters ∇ and no matrix **Z**_{∇}.

Let us now assume that a deformation is present for point *A*. Let it be a linear movement for which we write:
$$\begin{array}{c}0={x}_{{A}_{2}}-{x}_{{A}_{1}}+{a}_{x}{t}_{12}\mathrm{,}\\ 0={y}_{{A}_{2}}-{y}_{{A}_{1}}+{a}_{y}{t}_{12}\mathrm{,}\\ 0={z}_{{A}_{2}}-{z}_{{A}_{1}}+{a}_{z}{t}_{12}\end{array}$$(30)

The *a*_{x}*, a*_{y}*, a*_{z} are unknown parameters, which enter the parameter vector ∇, and for which a least squares estimate is determined in the adjustment. *t*_{12} is the time interval between epoch 1 and 2. The matrix **Z**_{∇} is in this case a matrix with three columns and three elements *t*_{12} on the rows of the three nonstochastic observations mentioned, and with zeros on all other positions.

We can also leave *a*_{x}*, a*_{y}*, a*_{z} out of the adjustment. Then the last column of the coefficient matrix of equation (28) disappears. The null hypothesis states now that there is no deformation. We test for a linear movement by using **Z**_{∇} in the test statistic of equation (34).

Generally the transformation between epoch *i* and *i – *1 is a similarity or congruence, not an affine transformation. Matrix **F**_{i} is constructed according to equation (20) from matrices **F**_{i, i –1} as given in equation (22) for the affine transformation. Matrix **Z**_{f} is the matrix that describes the constraints for a congruence or similarity transformation. The coefficient matrix of equation (13) or (14) is used to construct matrix **Z**_{f}.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.