Abstract
This paper gives a linearised adjustment model for the affine, similarity and congruence transformations in 3D that is easily extendable with other parameters to describe deformations. The model considers all coordinates stochastic. Full positive semi-definite covariance matrices and correlation between epochs can be handled. The determination of transformation parameters between two or more coordinate sets, determined by geodetic monitoring measurements, can be handled as a least squares adjustment problem. It can be solved without linearisation of the functional model, if it concerns an affine, similarity or congruence transformation in one-, two- or three-dimensional space. If the functional model describes more than such a transformation, it is hardly ever possible to find a direct solution for the transformation parameters. Linearisation of the functional model and applying least squares formulas is then an appropriate mode of working. The adjustment model is given as a model of observation equations with constraints on the parameters. The starting point is the affine transformation, whose parameters are constrained to get the parameters of the similarity or congruence transformation. In this way the use of Euler angles is avoided. Because the model is linearised, iteration is necessary to get the final solution. In each iteration step approximate coordinates are necessary that fulfil the constraints. For the affine transformation it is easy to get approximate coordinates. For the similarity and congruence transformation the approximate coordinates have to comply to constraints. To achieve this, use is made of the singular value decomposition of the rotation matrix. To show the effectiveness of the proposed adjustment model total station measurements in two epochs of monitored buildings are analysed. Coordinate sets with full, rank deficient covariance matrices are determined from the measurements and adjusted with the proposed model. Testing the adjustment for deformations results in detection of the simulated deformations.
Highlights
Geodetic deformation analysis is about analysing the geometric changes of objects on, above or under the earth’s surface or changes of this surface itself
Determining the transformation parameters between two or more coordinate sets is a problem that can be solved without linearisation of the functional model, if the transformation is one, two- or threedimensional and an affine, similarity or congruence transformation
This paper proposes a solution with a linearised model, which is elaborated upon in the section ‘Linearised adjustment model’
Summary
Geodetic deformation analysis is about analysing the geometric changes of objects on, above or under the earth’s surface or changes of this surface itself. Determining the transformation parameters between two or more coordinate sets is a problem that can be solved without linearisation of the functional model, if the transformation is one-, two- or threedimensional and an affine, similarity or congruence transformation. In this paper it is shown that it is very well possible to construct an adjustment model, where all coordinates are considered stochastic and the coefficient matrix does not contain stochastic elements This makes application of the total least squares method unnecessary. The transformation parameters in (2) can be solved by linearising the equation and using the standard least squares algorithm The advantage of this approach is the possibility of solving adjustment models, for which direct solutions are not known. A solution for handling a positive semidefinite cofactor matrix of the observations is given for the reduced general model by Teunissen et al (1987) for the similarity transformation. Matrix B has the following structure in the case of an affine transformation
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