On the deformation analysis of point fields

Authors: Hiddo Velsink

Publish Date: 2015/07/14

Volume: 89, Issue: 11, Pages: 1071-1087

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Abstract

A new approach to determine a multipoint deformation of the earth’s surface or objects upon it represented by point fields measured in two epochs is presented The problem of determining which points have been deformed is not approached by testing pointbypoint but by formulating alternative hypotheses that test if one two or more subsets of points have been deformed each subset in its own way The method is based on the least squares connection adjustment defines alternative hypotheses and searches the best one by testing a large amount of them If the best hypothesis is found a least squares estimation of the deformations is provided The test results of the presented method are invariant under changes of the Ssystems in which the point coordinates are defined The results of a numerical test of the method applied to a simulated network are given In designing a geodetic deformation network minimal detectable deformations can be computed belonging to likely deformation patterns The proposed method leads to a reconsideration of the duality of reference and object points A comparison with the method of testing confidence ellipsoids is made The relevance of the difference between geometric and physical interpretations of deformations and the consequences of the presented method for future developments are discussedDeformation analysis is done in many fields of human activity eg the production of gas and oil civil and construction engineering water management industrial installations and because of threats of natural phenomena such as land slides An often applied method is to determine coordinates one two or threedimensional of points that are representative of the earth’s surface or the object that is or may be subject to deformation This object can be a civil engineering work a building a dam an industrial storage tank part of the earths’ surface etc The object is represented by point coordinates If the coordinates are acquired by geodetic means the point field is called in this paper a geodetic network The deformation analysis looks at the changes of the coordinates in the course of timeModern technology offers many possibilities to produce coordinates eg total station measurements levelling GPS terrestrial photogrammetry and 3D laser scanning terrestrial or airborne Also hydrological techniques such as networks of transponders on the seafloor can be considered hereThe approach to compare coordinates is appropriate as it is generally natural to describe deformations in their terms Where in the past the amount of acquired coordinates was often limited to a few tens or hundreds modern techniques can deliver streams of coordinates almost continuously both in time and in spaceThe proposed method The method proposed in this paper is not based on an analysis of displacement vectors but on testing the results of connection adjustments and can test for more deformed points simultaneously It is invariant under changes of the chosen Ssystems the reference systems of both the coordinates and their covariance matrix also known as the geodetic datums The concept of an Ssystem is introduced by Baarda 1973 and generalised by Teunissen 1985 p 41The proposed method elaborates on the theory in Teunissen 2006b Teunissen et al 1987a Velsink 1998a and applies it to the geometric analysis of a geodetic network that has been measured in two epochs The least squares adjustment of the coordinates resulting from the measurements at each epoch and the detection specification and quantification of existing deformations are treated The application of the theory of testing multidimensional alternative hypotheses of Teunissen 2006b p 71ff an extension of the testing of onedimensional alternative hypotheses of Baarda 1968b is shown The need to perform Stransformations during the testing process is avoided by inserting transformation parameters into the adjustment modelThe proposed method follows an approach of formulating alternative hypotheses that allow for complex hypotheses By testing large amounts of multidimensional tests it is possible to find the points that have been deformed most likely without the need to have prior information about the deformations The method is capable of giving least squares estimates of the deformations Moreover it can compute the minimal detectable deformations ie the size of the deformations specified by an alternative hypothesis that can be found with a specified probability by testing the hypothesis It is an important tool for designing a geodetic network for deformation analysis

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