A 3-D similarity transformation is frequently used to convert GPS-WGS84-based coordinates to those in a local datum using a set of control points with coordinate values in both systems. In this application, the GaussMarkov (GM) model is often employed to represent the problem, and a least-squares approach is used to compute the parameters within the mathematical model. However, the Gauss-Markov model considers the source coordinates in the data matrix (A) as fixed or error-free; this is an imprecise assumption since these coordinates are also measured quantities and include random errors. The errorsin-variables (EIV) model assumes that all the variables in the mathematical model are contaminated by random errors. This model may be solved using the relatively new total least-squares (TLS) estimation technique, introduced in 1980 by Golub and Van Loan. In this paper, the similarity transformation problem is analyzed with respect to the EIV model, and a novel algorithm is described to obtain the transformation parameters. It is proved that even with the EIV model, a closed form Procrustes approach can be employed to obtain the rotation matrix and translation parameters. The transformation scale may be calculated by solving the proper quadratic equation. A numerical example and a practical case study are presented to test this new algorithm and compare the EIV and the GM models.
Coastal erosion presents a serious problem throughout U.S. coastal areas. The OhioGeological Survey estimates that more than 3,200 acres of Ohio's Lake Erie shore have been lost to erosion since the 1870s, resulting in economic losses exceeding tens of millions of dollars per year. This article presents research results of a project that monitors shoreline erosion using high-resolution imagery and examines erosion causes. Spatial modeling and analysis methods are applied to the project area along the south shore of Lake Erie. The shoreline is represented as a dynamically-segmented linear model that is linked to a large amount of data describing shoreline changes. A new method computes an instantaneous shoreline using a digital water level model, a coastal terrain model, and bathymetric data. This method provides an algorithm for deriving the Mean-Lower Low Water (MLLW) and the Mean High Water (MHW) shorelines that are essential to navigation charts. The results describe a part of our effort towards a coastal spatial information infrastructure to support management and decision-making in the dynamic coastal environment.Erosion is de ned as the gradual wearing away of the earth's surface by the action of natural forces of wind and water. Lake Erie, the great body of fresh water forming Ohio's north coast, is the fourth largest of the ve Great Lakes (Figure 1). Lake Erie provides an unlimited fresh water supply to communities along its shore and is an unmatched recreational and sport-shing area. It provides signi cant quantities of sand and gravel for construction. On the other hand, Lake Erie is also a dynamic body of water noted for the ferocity of its storm waves and the havoc they wreak along the lakeshore (MacKey and Guy 1994; Highman 1997). Over much of this region, erosion rates have been less than 1 m per year. However, local rates may exceed 2 m per year (Highman 1997). Even where rates are slow, the highly developed nature of the coast makes recession a serious property-damage problem. Furthermore, about 95% of Ohio's Lake Erie shore is eroding and thus nearly 2,500 structures are within 15 m of destruction. Figure 2 shows a stretch of eroded Lake
Proper incorporation of linear and quadratic constraints is critical in estimating parameters from a system of equations. These constraints may be used to avoid a trivial solution, to mitigate biases, to guarantee the stability of the estimation, to impose a certain "natural" structure on the system involved, and to incorporate prior knowledge about the system. The Total Least-Squares (TLS) approach as applied to the Errors-InVariables (EIV) model is the proper method to treat problems where all the data are affected by random errors. A set of efficient algorithms has been developed previously to solve the TLS problem, and a few procedures have been proposed to treat TLS problems with linear constraints and TLS problems with a quadratic constraint. In this contribution, a new algorithm is presented to solve TLS problems with both linear and quadratic constraints. The new algorithm is developed using the Euler-Lagrange theorem while following an optimization process that minimizes a target function. Two numerical examples are employed to demonstrate the use of the new approach in a geodetic setting.
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