Bottom-touch probability

The probability that a ship touches the channel bottom is equal to the probability that the vertical downward movement due to waves of the most critical point of the ship exceeds the keel clearance. The keel clearance is defined as:

Keel clearance = channel depth + water level - draft - squat

Consider a ship that sails from points A to B through a channel. In advance it is not known when or where a bottom touch will occur, but imagine that the following is true:

• The mean frequency of bottom touches is constant in time, then the expected number of bottom touches during a transit is equal to the transit time times the mean frequency of bottom touches.
• The bottom touches per transit are independent random variables: one bottom touch tells nothing about the others.

Expected number of bottom touches during a transit = mean frequency of bottom touches * passage time

The expected number of bottom touches during a transit, then proves be Poisson distributed. The mean frequency of bottom touches is equal to the mean frequency of vertical ship motions that are bigger than the keel clearance. Based on the formula of Rice this is given by (m₀ is the 0th spectral moment of ship motions):

Mean frequency of bottom touches = (1 / mean period of ship motion) * e(-(keel clearance2)/(2 * m0))

In this formula m₀ can be determined from the significant vertical ship motion. Like the distribution of the incoming waves it is assumed that also the vertical motions of a ship are Raleigh distributed. Therefore the following relation is true:

m0 = (1/16) * (significant vertical ship motion)2

Because the expected number of bottom touches is Poisson distributed, we can calculate the probability of one or more bottom touches during a transit via:

P(number of bottom touches > 0) = 1 - enumber of bottom touches

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