Journal of Surveying Engineering

This paper summarizes the majority of classic formulae of weighted total least-squares (WTLS) solutions for errors-in-variables (EIV) model and classified them into six categories. Next, the existing approximate posterior precision estimation methods of the WTLS solution are categorized and summarized. Based on the criterion of the orders of Taylor expansion, the current posterior precision estimation methods can be divided into first-order approximate (FOA) and second-order approximate (SOA). From the perspective of formulae, the derivation, comparison, and analysis of FOAs are placed emphasis in theory. Except for the existing FOA methods, four FOA-typed methods from the major variants of WTLS formulations are derived based on error propagation law, which enriched the posterior precision evaluation method of the WTLS solution. Two experimental examples, namely straight-line fitting and three-dimensional (3D) affine transformation, are used to evaluate the cons and pros of these posterior precision estimation methods, and their representation capability are investigated and discussed, with some suggestions being offered.

This paper is a case study and historical account of nineteenth-century surveying of the Texas and Pacific Railway Eighty-Mile Reserve, and the consequences from ignoring curvature of the earth allowances in large surveys extending beyond the horizon. The methods used in the survey of the Eighty-Mile Reserve in Texas are compared to the way surveyors of the Public Lands of the United States carried out their work by making allowances for the curvature of the earth. When Texas became a republic in 1836, it was rich in land extending as far north as present-day Wyoming and Colorado, but financially broke. The first governmental agency in the Republic of Texas was the General Land Office and its instructions in the survey of the Texas public domain required boundary corners to be square, when practicable. The first land grants in Texas were very large, with no formal system of surveying to make allowances for the curvature of the earth. Consequently in later years, large errors were found, creating boundary misclosures, overlaps, gaps, and huge excesses in acreage over what was patented; and vacancies of unsurveyed, unsold land were not listed in the General Land Office as permanent school fund lands. When Texas was annexed by the United States in 1845, it still had enormous outstanding debts. By 1850, the State of Texas sold 67-million-acres (37,113,938 ha) of its public domain to the United States in order to clear its public debts. Ignoring curvature of the earth corrections in surveys continued until around 1930 when prominent surveyors, geodesists, and engineers convinced the Texas General Land Office to embrace a Texas coordinate system that would project a survey onto a flat map and accurately represent the surface of the earth in the least distorted way by creating the state plane coordinate system of Texas.