Assessment of ship position probability in the geometrical figure of position lines errors

Abstract

Accurate and reliable determination of the ship position coordinates is one of the most important tasks for ensuring the safety of navigation at sea. A significant condition that contributes to the effective solution of this task is the use of redundancy in measurements of navigation parameters. The implementation of this redundancy allows the navigator or automatic navigation aid to control the trueness of measurements; eliminate gross and systematic errors; reduce the effect of random errors; apply various methods of analysis and algorithms for processing information in order to obtain the most accurate and reliable ship position. Adequate redundancy of navigation measurements makes it possible to obtain asymptotically unbiased estimates of coordinates, as well as to solve other navigation tasks based on redundant data. The results of a study on the probability of finding the true coordinates of the ship’s position in a geometric figure formed by random measurement errors are presented. It has been shown in the study that a single redundancy of measurements is not enough to achieve the required level of probability. Elements of classical combinatorics based on the Bernoulli scheme and including Newton’s binomial and Pascal’s triangle as tools are used as a mathematical apparatus for the study. These tools allow us to effectively analyze combinations of mathematical signs, such as plus and minus, of random measurement errors, reducing them into binary sequences, which are described by binomial coefficients. It has been argued in the paper that the low level of probability of finding the true values of the coordinates in the geometrical figure of measurement errors is a strong signal that any method of averaging the measurement results can lead to asymptotically biased estimates of the ship position coordinates. This is analogous to the influence of systematic errors on the coordinates within a certain time interval. From the point of view of navigational safety, a low probability can be qualified as a near-miss situation, which can lead to incorrect control of the ship position. The probabilistic criteria for the necessary redundancy of measurements to approach asymptotically unbiased estimates of ship position are presented in the paper.