Journal of Surveying Engineering

This paper reconsiders the two-step analysis method in an attempt to show how correct processing of both steps will ensure results identical to those obtained when using the one-step method and moreover may result in improved variance-covariance estimation of the deformation parameters. The approach for solving a vector of parameters indirectly from the measurement vector by using a vector of pseudomeasurements is referred to as a two-step analysis. In deformation networks, the first step is processing the geodetic measurements sequentially without modeling the variation in the network geometry. The results of this first step are sets of network point coordinates pertaining to the specific epochs of the measurements. In the second step the variation in network geometry is modeled. The result of this step is a set of deformation model parameters. It is argued that a proper processing of the two-step method will ensure a correct estimation of the variance-covariance matrix of the deformation parameters, which is a critical value in the overall deformation analysis scheme.

Estimating curve parameters using observed data is of fundamental importance to surveyors. In 1999 Easa converted the $L_1$ estimation of parabolic curve parameters into a linear programming (LP) problem, which enabled a rapid solution by LINDO. However, this approach requires LP formulation and associated software that may not be easily inaccessible to all surveyors, while its efficiency declines when handling the nonlinear global problem with unknown start and end points $(x_1$ and $x_2)$ of the curve. Easa sought the global solution by repeating the LP procedure over various combinations of $x_1$ and $x_2$ manually, using a step size of 5 m for each variable. With this approach, one faces a trade-off between labor and accuracy. As an alternative, a convenient spreadsheet method suitable for $L_1\text{,}$ $L_2\text{,}$ or “mini-max” optimization is developed here that reduces the labor and formulation involved and requires no LP language. The new approach is further automated by Visual Basic for Applications programming to solve the global problem, allowing a fine step size of 0.1 m to be used. The vast data set generated leads to an accurate 3D picture that reveals the true behavior of the global solution, which cannot be captured when a step size of 5 m is used.

The total-least-squares approach is a relatively new adjustment method of estimating parameters in linear models that include error in all variables. Specifically, given an overdetermined set of linear equations $\mathbf{y}\approx \mathbf{A}\mathbf{\xi }$ where y is the observation vector, A is a positive defined data matrix, and ξ is the vector of unknown parameters, the total-least-squares problem is concerned with estimating ξ providing that the number of observations n is larger than the number of parameters to be estimated and given that both the observation vector y and the data matrix A are subjected to errors and need to be adjusted. This model is different from the classical least-squares model where only the observation vector y is subjected to errors. This paper starts with a brief summary of the least-squares approach and then explains how one can modify the approach to include error in all variables using the generalized least-squares technique. Then the total-least-squares problem is presented along with its formulas and the procedures used to solve it. Finally, the total-least-squares approach is used to determine the trend in a spatial point process.

When using state plane coordinates or performing other geodetic computations, the ratio $r/(r+h)$ is the elevation factor commonly used to reduce a horizontal distance to its sea-level or ellipsoidal equivalent. In most cases, an approximation is used for the Earth’s radius while the value of the elevation is usually much better known. This is as it should be, but the question examined in this paper is, What is an acceptable approximation for the Earth’s radius and how accurately must the elevation be known to assure sufficient precision in the answer? An equation that can be used to answer these questions is derived using concepts of error propagation.

A complete datum transformation between two geodetic coordinate systems requires seven parameters: three translations, three rotations, and one scale factor. When a datum transformation is needed, these parameters, which are generally unknown beforehand, are computed using the common points whose coordinates are available in both systems. Therefore, the accuracies of the computed parameters depend on the accuracies of the coordinates of the common points. Accordingly, the accuracies of the point coordinates transformed by these parameters will depend on the accuracies of the parameters, along with the accuracies of the coordinates before transformation. For this reason, one should investigate whether the figure condition of the computed parameters is satisfied for all the transformation area in which the parameters will be involved. In the cases where the figure condition could not be satisfied for all the transformation area, the design of the network constituted by common points should be revised, either by adding well-distributed new common points to the network, or by dividing the transformation area into parts for which the figure conditions would be satisfied.

Road length is part of the geometry of the roadway network. Its measurement is critical to all road inventory databases. One approach to obtaining it is to drive cars equipped with a distance measurement instrument (DMI) along roads to measure mileages. This method provides accurate measurements, but it is expensive and time consuming. This paper proposes an alternative way to acquire actual road length, which is currently under consideration by the geographic information system (GIS) unit of the North Carolina Department of Transportation. The emphasis of this study was to determine the accuracy of the proposed approach. The proposed approach employs GIS application programs written in ARC macro language to calculate the actual length (surface length) along the sloped surface of highway centerlines based on elevation data and the road network geometry. This was done for all interstate highways in North Carolina. The calculated GIS results were compared with DMI measurements, which is the most accurate approach presently available to NCDOT. Three filters were applied to remove suspect road segments that have significant errors which are irrelevant to the proposed approach. All remaining segments were grouped by slope and length to evaluate the impact of slope and length on the accuracy. Frequency analysis and root mean square error were determined for all groups. It was found that the proposed method is a technically feasible method with reasonable accuracy. The study also revealed that where there are errors, they occur primarily for road segments with relatively high slopes, short lengths, or both. The meaning of this is that GIS and digital elevation model technologies can be combined and used in lieu of DMI measurements, thus reducing resource demands.