## 2 Materials and Methods  
A Model of Positioning Accuracy and Availability in Hydrographic Surveys According to the IHO Standards  
In order to carry out an assessment of the possibility for applying particular positioning methods (DGPS (Differential Global Positioning System), EGNOS (European Geostationary Navigation Overlay Service), and multi-GNSS) in hydrography, a statistical model was proposed that enables the determination of the compliance (or non-compliance) of these three systems with the requirements set out in the IHO S-44 standard [5] and specified by the orders. The model was based on the theory of renewal (repair) process reliability, where the system's operation and failure statistics are referred to as life and failure times. Let us consider a positioning system that determines a position with the error \(\delta_{n}\) as a function of time, and for which four maximum allowable positioning error values corresponding to the minimum accuracy requirements set out for four IHO orders--special, 1a, 1b and 2--were specified. Figure 2 (the upper diagram) shows a curve that presents the position error value as a function of time conducted by any given positioning system whose assessment in terms of the possibility for use in hydrography should be carried out by assigning an IHO order to it. Let us note that in the presented diagram, since the position error value changes as a function of time, at the beginning it ensures an accuracy of more than 20 m, which prevents it from complying with any of the IHO order requirements. After some time, the position error decreases to just under 2 m, which means that the system can be used for a certain time in all orders, to later reach a value of approximately 8 m, such that it only complies with the requirements set out for order 2. In the diagram, the maximum allowable positioning error values for particular orders: special (green color), 1a/1b (blue color), and 2 (red color) are also plotted. Under the position error diagram, as a function of time, three diagrams are presented in which the system's operational status in terms of particular IHO order minimum requirements are shown. From this perspective, a system can have two statuses: fitness (life) designated with a binary value of 1, and unfitness (failure) designated with a binary value of 0. The system's status is determined using the relationship between the current position error and the maximum allowable positioning error value assigned to a particular IHO order. Three colored diagrams show the system's operational status in relation to four IHO orders: special, 1a/1b (jointly), and 2. Thus, the positioning process was transformed to a two-status stationary renewal process. In this way, a transition was made from a random variable, i.e., the position error (a classical approach used in navigation) to a reliability system in which the life and failure times have become random variables. In order to determine whether the system has either life or failure status, let us introduce a variable \(U\) that corresponds to the maximum allowable positioning error value for four IHO orders, which is to be expressed as follows:  
\[U=\left\{ \begin{array}{c}2\;m\;\{p=0.95\;\;\mbox{for special order}\\ 5\;m\;\{p=0.95\;+\;5\%\;\mbox{of depth for order 1a/1b}\\ 20\;m\;\{p=0.95\;+\;10\%\;\mbox{of depth for order 2}\end{array} \right. \tag{1}\]  
To present the process concerned in a mathematical form, let us assume that the determination of a position's coordinates is an alternating renewal process, according to the general reliability theory; therefore, it can be assigned two statuses as a function of time: life, i.e., a status in which the position error is smaller than the arbitrarily established value corresponding to the requirements of particular IHO orders, which is to be expressed as \(\delta_{n}\;\leq\;U\) for the subsequent determinations of \(n\;=\;1\), \(2\), \(\ldots\), and the failure time in which an opposite relationship \(\delta_{n}\;>\;U\) occurs.