Safe time intervals

Safe time interval on each kilometer position

Safe time intervals are time windows that a ship can be at a kilometer position with the given speed and course. The start and end points of these intervals are the times that the chance of touching the bottom is exactly equal to the maximum allowed criterion.

Safety criterion with uncertainty

The safety criterion can be a minimum under keel clearance scheme (for instance a minimum of 17.5% UKC before / after squat reduction) or based on a maximum probability. Because we do not know the exact conditions, but only know predictions of the conditions, we simulate a range of possible conditions on one point as much times as needed to get a good notion of the safety at that time and location. For one time / location PROTIDE therefore calculates a lot of UKCs and chances:

1. Estimate possible water levels -> N(mean, standard deviation)
2. Estimate possible swell heights -> N(mean, standard deviation)
3. Repeat a number of iterations:
4. Draw water level from N(mean, standard deviation)
5. Get channel depth
6. Get bottom elevation
7. Get channel heading (angle)
8. Calculate under keel clearance
9. Calculate squat with OCTOPUS
10. Draw swell height from N(mean, standard deviation)
11. Draw possible random vertical motion (from scenarios calculated by OCTOPUS)
12. Calculate spectral moment and bottom touch frequency
13. Calculate P(bottom touch) as if the frequency would hold during a whole channel passage

The result is a distribution of possible UKC values (with or without squat reduction) and bottom touch probabilities. We assume that each distribution is normally distributed and calculate the mean and standard deviation of each distribution. With these figures we choose the 95% worst chance or UKC to compare with the safety criterion. This way we only make a 5% error that we overestimate the UKC or underestimate the probability of touching the channel bottom. The figure below shows an example of the possible distribution of X (the chance of touching the bottom) and the chance that X is smaller then k (the safety criterion). This example applies to the P(bottom touch) only, because for the UKC we need the lower 95% part.

Root finding algorithm

PROTIDE uses a root finding algorithm to investigate each kilometer position. The root finding algorithm finds the times that the chance of touching the bottom is exactly equal to the safety criterion (again, this can be measured in P(bottom touch) or just a minimum UKC criterion). These roots indicate the times where safe shifts to not safe or the other way around. Therefore between these roots, on the "safe" side, we find time intervals that it is safe to be at that kilometer position.