The most probable maximum (MPM) is the extreme value statistic commonly used in the offshore industry. The extreme value of vessel motions, structural response, and environment are often expressed using the MPM. For a Gaussian process, the MPM is a function of the root-mean square and the zero-crossing rate of the process. Accurate estimates of the MPM may be obtained in frequency domain from spectral moments of the known power spectral density. If the MPM is to be estimated from the time-series of a random process, either from measurements or from simulations, the time series data should be of long enough duration, sampled at an adequate rate, and have an ensemble of multiple realizations. This is not the case when measured data is recorded for an insufficient duration, or one wants to make decisions (requiring an estimate of the MPM) in real-time based on observing the data only for a short duration. Sometimes, the instrumentation system may not be properly designed to measure the dynamic vessel motions with a fine sampling rate, or it may be a legacy instrumentation system. The question then becomes whether the short-duration and/or the undersampled data is useful at all, or if some useful information (i.e., an estimate of MPM) can be extracted, and if yes, what is the accuracy and uncertainty of such estimates.
In this paper, a procedure for estimation of the MPM from the short-time maxima, i.e., the maximum value from a time series of short duration (say, 10 or 30 minutes), is presented. For this purpose pitch data is simulated from the vessel RAOs (response amplitude operators). Factors to convert the short-time maxima to the MPM are computed for various non-exceedance levels. It is shown that the factors estimated from simulation can also be obtained from the theory of extremes of a Gaussian process. Afterwards, estimation of the MPM from the short-time maxima is explored for an undersampled process; however, undersampled data must not be used and only the adequately sampled data should be utilized. It is found that the undersampled data can be somewhat useful and factors to convert the short-time maxima to the MPM can be derived for an associated non-exceedance level. However, compared to the adequately sampled data, the factors for the undersampled data are less useful since they depend on more variables and have more uncertainty. While the vessel pitch data was the focus of this paper, the results and conclusions are valid for any adequately sampled narrow-banded Gaussian process.
Efficient Gaussian Process-Based Modelling and Prediction of Image Time Series
