The paper presents a coherent, original theory and an effective algorithm to calculate the surface area of a geodesic polygon, i.e. the area on an ellipsoid of revolution limited by geodesics, connecting pairs of subsequent vertices of known geographical coordinates. Using the cylindrical coordinate system and introducing an equatorial geodesic triangle, the novel strict recursive formulas were derived for the longitude on the ellipsoid and the distance along the geodesic and the area of a geodesic polygon. The area is determined by constant coefficients, not the values of finite series. The orthodromic properties of geodesics, which led to the detection of bifurcation lines composed of the double solutions of the inverse geodetic problem were analysed. The novel algorithm presented, integrating the problem with geodesic polygon area calculation, is based on the newly proposed parameterisation with two alternative working unknowns and a new formula to obtain the initial value of the geodesic C parameter. It allows solutions for all cases, including near-antipodal or near-vertices, and offers two solutions if bifurcation occurs. The algorithm presented was tested and verified using numerical examples from known publications. The numerical tests proved the accuracy of 1m2 of the computed geodesic area using standard IEEE 754 Double Floating Point PC arithmetic, for geodesic lengths up to ten thousand km.

Absolute horizontal displacements are an important element of dam safety level assessment. Appropriate design of measurement network is a prerequisite for the acquisition of displacement values that meet the reliability requirements. A network of this kind, apart from ensuring the required precision of displacement determination, should be characterised by reliability allowing for elimination of gross errors in the results of geodetic surveys. This study aims to propose a method to improve reliability characteristic of surveying network used for horizontal displacement identification in Zatonie dam. The desired effect (increase in the network’s reliability) is obtained by the authors in two stages. The first stage concerns expansion of the existing network by addition of three free stations. As the obtained effect did not prove to be satisfactory, in the second stage so called observation accuracy harmonisation was carried out, which optimally utilises the reliability potential of the measurement construction. In order to successfully carry out the harmonisation, a modification to the procedure’s algorithm had to be introduced. A design of a network ensuring detection of a gross error in any given observation was obtained as the result of the performed actions.

Appropriate precision and low cost are the basic conditions that have to be fulfilled by a project of a geodetic network. Reliability, translating into the ability to detect gross errors in the observations and higher certainty of the obtained point position, is an important network characteristic. The principal way to provide appropriate network reliability is to acquire a suitably large number of redundant observations. Optimisation of the observation accuracy harmonisation procedure allowing for the acquisition of an appropriate level of reliability through modification of the observation a priori standard deviations is the focus of this study. Parameterisation of the accuracy harmonisation is proposed. Furthermore, the influence of the individual parameter operation on the effectiveness of the harmonisation procedure is tested. Based on the results of the tests an optimal set of harmonisation parameters which guarantees the maximal efficiency of the harmonisation algorithm is proposed.

A geodesic survey of an existing route requires one to determine the approximation curve by means of optimization using the total least squares method (TLSM). The objective function of the LSM was found to be a square of the Mahalanobis distance in the adjustment fi eld ν. In approximation tasks, the Mahalanobis distance is the distance from a survey point to the desired curve. In the case of linear regression, this distance is codirectional with a coordinate axis; in orthogonal regression, it is codirectional with the normal line to the curve. Accepting the Mahalanobis distance from the survey point as a quasi-observation allows us to conduct adjustment using a numerically exact parametric procedure.Analysis of the potential application of splines under the NURBS (non-uniform rational B-spline) industrial standard with respect to route approximation has identifi ed two issues: a lack of the value of the localizing parameter for a given survey point and the use of vector parameters that defi ne the shape of the curve. The value of the localizing parameter was determined by projecting the survey point onto the curve. This projection, together with the aforementioned Mahalanobis distance, splits the position vector of the curve into two orthogonal constituents within the local coordinate system of the curve. A similar system corresponds to points that form the control polygonal chain and allows us to fi nd their position with the help of a scalar variable that determines the shape of the curve by moving a knot toward the normal line.

An optimally designed geodetic network is characterised by an appropriate level of precision and the lowest possible setup cost. Reliability, translating into the ability to detect blunders in the observations and higher certainty of the obtained point positions, is an important network characteristic. The principal way to provide appropriate network reliability is to acquire a suitably large number of redundant observations. This approach, however, faces limitations resulting from the extra cost. This paper analyses the possibility of providing appropriate reliability parameters for networks with moderate redundancy. A common problem in such cases are dependencies between observations preventing the acquisition of the required reliability index for each of the individual observation. The authors propose a methodology to analyse dependencies between observations aiming to determine the possibility of acquiring the optimal reliability indices for each individual observation or groups of observations. The suggested network structure analysis procedures were illustrated with numerical examples.