J. Appl. Geodesy 2017; 11(2): 77–87 Stephanie Kauker* and Volker Schwieger A synthetic covariance matrix for monitoring by terrestrial laser scanning DOI 10.1515/jag-2016-0026 Received January 30, 2017; accepted February 14, 2017 Abstract: Modelling correlations within laser scanning point clouds can be achieved by using synthetic covari- ance matrices. These are based on the elementary error model which contains diferent groups of correlations: non-correlating, functional correlating and stochastic cor- relating. By applying the elementary error model on ter- restrial laser scanning several groups of error sources should be considered: instrumental, atmospheric and ob- ject based. This contribution presents calculations for the Leica HDS 7000. The determined variances and the spa- tial correlations of the points are estimated and discussed. Hereby, the mean standard deviation of the point cloud is up to 0.6 mm and the mean correlation is about 0.6 with respect to 5 m scanning range. The change of these numer- ical values compared to previous publications as Kauker and Schwieger [17] is mainly caused by the complete con- sideration of the object related error sources. Keywords: Synthetic covariance matrix, elementary error model, terrestrial laser scanning 1 Introduction Terrestrial laser scanning has become a common tool for deformation and displacement measurements in the geodetic feld. In order to guarantee the quality of mea- surements and to get realistic deformation and analysis re- sults, it is essential to be aware of all error sources and their impact on the measurements. A synthetic covariance ma- trix for terrestrial laser scanning is necessary in order to model the impacts of the main error sources on the vari- ances and the covariances and correlations respectively within point clouds. These correlations describe stochastic relations among several measurements assuming multi- dimensional normal distributed measurements. The co- variance matrix can be modelled by applying the elemen- *Corresponding author: Stephanie Kauker, Institute of Engineering Geodesy, University of Stuttgart, Stuttgart, Germany, e-mail: stephanie.kauker@ingeo.uni-stuttgart.de Volker Schwieger, Institute of Engineering Geodesy, University of Stuttgart, Stuttgart, Germany, e-mail: volker.schwieger@ingeo.uni-stuttgart.de tary error model [29]. Based on this model these impacts must be classifed into diferent correlation groups frst. Since Schwieger [29] introduced stochastic correlations as a third group of correlations, non-correlating, functional correlating and stochastic correlating elementary errors can be diferentiated. 1 Each group requires an infuenc- ing matrix and a covariance matrix. In [18] a similar pro- cedure based on the Guide to the Expression of Uncer- tainty in Measurement [13, 12] is developed. The main dif- ference to this contribution is the introduction of non- linear relationships between infuencing errors and mea- sured point clouds and the error propagation via Monte Carlo Simulation instead using the law of error propaga- tion. Koch [18, 19] did not model correlations among the error sources, but showed the infuence of correlations among the observations on estimated parameters. In this contribution the infuencing matrix contains appropriate impacts of the elementary errors on the point coordinates of the point clouds. For defning the covari- ance matrix, the variances and, if stochastic correlating errors are considered, the covariances of the elementary errors have to be known. These can be defned by using empirical investigations or manufacturers’ information, among others. Next, the elementary error model has to be applied on all the error sources which afect observa- tions. In contradiction to e.g. Zogg [38] these include in- strumental, atmospheric and object based errors only. The reason for not considering the confguration errors as an independent error source is the fact that the confgura- tion itself does not produce errors but infuences the other error sources like the object-based errors, e.g. a changed incidence angle or a changed distance to the object will change the infuence of the material characteristics on the measurements. Additionally the relative geometry be- tween scanner and object infuences the efect of the in- strumental errors e.g. the collimation axis error. The in- strumental error group comprises, e.g. zero point error, collimation axis error, vertical collimation error and tum- bling error. Furthermore, the atmospheric group consists of air temperature, air pressure and partial water vapour pressure. Regarding objects, the impact of errors, such as angle of incidence and refectivity, should be taken into ac- 1 The terms “non-correlating”, “functional correlating” and “stochastic correlating” are already established in literature.