Abstract
Modern observation technology has verified that measurement errors can be proportional to the true values of measurements such as GPS, VLBI baselines and LiDAR. Observational models of this type are called multiplicative error models. This paper is to extend the work of Xu and Shimada published in 2000 on multiplicative error models to analytical error analysis of quantities of practical interest and estimates of the variance of unit weight. We analytically derive the variance-covariance matrices of the three least squares (LS) adjustments, the adjusted measurements and the corrections of measurements in multiplicative error models. For quality evaluation, we construct five estimators for the variance of unit weight in association of the three LS adjustment methods. Although LiDAR measurements are contaminated with multiplicative random errors, LiDAR-based digital elevation models (DEM) have been constructed as if they were of additive random errors. We will simulate a model landslide, which is assumed to be surveyed with LiDAR, and investigate the effect of LiDAR-type multiplicative error measurements on DEM construction and its effect on the estimate of landslide mass volume from the constructed DEM.
Highlights
Theory and methods of adjustment have been developed, both with the advance of measurement technology and with our deepened understanding of measurement errors
This section is to serve two major purposes through numerical simulations: (i) to investigate the effect of light detection and ranging (LiDAR)-type measurements on digital elevation models (DEM) construction; and (ii) to collectively use the error analysis and the estimates of the variance of unit weight given in the previous two sections to estimate the errors of the volume of landslide mass, since a precise estimate of the volume of a landslide from the constructed DEM can be important in practical hazard evaluation
Adjustment has been founded on the basis of observational models with additive random errors to process data in geoscience
Summary
Theory and methods of adjustment have been developed, both with the advance of measurement technology and with our deepened understanding of measurement errors. From this point of view, adjustment theory and methods, as developed on the basis of the model (1) (with only additive errors), cannot serve as a solid theoretical foundation to process measurements that are collected on the model (3) and/or (4). Wedderburn [27], Xu and Shimada [33] did not assume any pdf for the measurements with multiplicative errors Instead, they directly started with the least squares (LS) method and derived the bias-corrected LS method for parameter estimation in the model (4). One of the purposes of this paper is to extend the methods of DEM construction to the case in which measurements are contaminated with multiplicative random errors and investigate their effect on the estimate of volume computed from LiDAR-type DEM, which can be important in practical hazard evaluation of landslides.
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