Uncertainties estimation in surveying measurands: application to lengths, perimeters and areas

Abstract

The present paper develops a series of methods for the estimation of uncertainty when measuring certain measurands of interest in surveying practice, such as points elevation given a planimetric position within a triangle mesh, 2D and 3D lengths (including perimeters enclosures), 2D areas (horizontal surfaces) and 3D areas (natural surfaces). The basis for the proposed methodology is the law of propagation of variance–covariance, which, applied to the corresponding model for each measurand, allows calculating the resulting uncertainty from known measurement errors. The methods are tested first in a small example, with a limited number of measurement points, and then in two real-life measurements. In addition, the proposed methods have been incorporated to commercial software used in the field of surveying engineering and focused on the creation of digital terrain models. The aim of this evolution is, firstly, to comply with the guidelines of the BIPM (Bureau International des Poids et Mesures), as the international reference agency in the field of metrology, in relation to the determination and expression of uncertainty; and secondly, to improve the quality of the measurement by indicating the uncertainty associated with a given level of confidence. The conceptual and mathematical developments for the uncertainty estimation in the aforementioned cases were conducted by researchers from the AssIST group at the University of Oviedo, eventually resulting in several different mathematical algorithms implemented in the form of MATLAB code. Based on these prototypes, technicians incorporated the referred functionality to commercial software, developed in C++. As a result of this collaboration, in early 2016 a new version of this commercial software was made available, which will be the first, as far as the authors are aware, that incorporates the possibility of estimating the uncertainty for a given level of confidence when computing the aforementioned surveying measurands.

Uncertainties estimation in surveying measurands: application to volumes

“…Building upon the previously published work related to the determination of uncertainty associated with points, lengths, and areas [7,17], this paper has developed specific propagation models for two commonly used volume calculation methods: the grid method and the average end area method. Furthermore, the corresponding algorithms for the calculation of volume estimation and uncertainty have been proposed.…”

“…In order to calculate the elements of the Jacobian matrix, the mathematical model that relates source measurands or observables to the target measurands (RM) has to be known. In agreement with this generic uncertainty propagation law, previous research by the authors has discussed and achieved the propagation of uncertainty when lengths and areas are the target measurands [7]. Furthermore, the developed algorithms have been incorporated into commercial software and are now available to practitioners in the field [17].…”

“…(1st step) In the RM Ia, the height (z P ) corresponding to the vertex (P) of each cell on the regular grid, for both surfaces, is computed based on the position coordinates from the triangle whose horizontal projection contains the vertex. The model for the calculation of the height of a point within a triangle has already been discussed by the authors [7] and it depends on whether the plane defined by the three vertices of the triangle (T 1 , T 2 , T 3 ) is close to a vertical plane or not (equation (3)). The two possible scenarios are described below:…”

“…(3rd step) RM IIIb sequentially computes the areas of the triangles defined by each of three regularised intersection points, one of them on one surface and the remaining two, on the other. The formulation used is Heron's formula, whose uncertainty propagation has been discussed in depth in [7].…”

“…Hence, provided that enough points are measured and that they have been appropriately selected, it is possible to create a DTM that constitutes a reliable representation of reality by using computer-aided design (CAD) or other specific surveying software. Finally, based on the DTM, calculations of lengths, surfaces [7] and volumes can be conducted. In the case of linear infrastructures in civil engineering, i.e.…”