Another area of interest in the oil and gas industry is that of offshore platform de- sign. Offshore platforms are used to extract hydrocarbons from reservoirs located below the sea: they must often be operated in extremely challenging ocean environ- ments, and so it is crucial that they are designed to withstand the worst conditions that they could experience in a given area.
This need to design to withstand harsh environments must be offset against the cost of doing so: companies must be run so that they are profitable, and so oil will only be extracted from reservoirs where it is economical to do so. One of the main factors in determining the economic viability of extraction from a given reservoir is the cost of the necessary equipment: therefore, reducing the cost of designing safe platforms increases the profit that can be made through operating offshore fields, and opens up the possibility of exploration in areas which were previously deemed economically inviable.
Much work has been done on modelling ocean waves in the past, and many compa- nies employ specialists who develop models that use atmospheric and ocean condi- tions to simulate the characteristics of waves. These ocean simulators are tuned by running them as hindcasts, meaning that they are run using historic atmospheric data as inputs in order to re-create conditions that were actually observed; if re- liable forecasts for future atmospheric behaviour are available, then a simulator can be run to predict the worst waves that may occur in the future, and offshore structures can be designed against these. In situations where suitable models or atmospheric forecasts are not available, then designs are chosen by multiplying the
1.3. Extreme ocean wave modelling 15 worst historically-observed waves by industry-standard safety factors. These safety factors are typically conservative, and so while the resulting designs are safe, they are wasteful in many situations.
There has been much recent interest in developing models which can handle uncer- tainties about future waves when generating designs, and can represent the effects of storm parameters on the worst observed waves; in the following section, we consider the way in which this modelling is usually carried out, and the sources of uncertainty that need to be considered.
1.3.1
Modelling
Ocean behaviour is recorded in terms of sea states; these are periods of time (typ- ically 1 or 3 hours) in which ocean characteristics (e.g. wave height, wave period, power spectrum) are believed to be roughly stationary. Within these sea states, ocean behaviour is recorded in terms of a number of summary statistics, for exam- ple:
the significant wave height HS is defined as 4 times the standard deviation of
the ocean surface elevation over a given sea state;
the peak wave period TP is the period corresponding to the peak spectral
frequency during a sea state;
When modelling for platform design, we are interested in the extremes of ocean behaviour. The extreme behaviour of a process is typically characterised in one of two standard ways:
Block maxima: under this approach, the data is divided into temporal blocks (in the case of sea states, typically days, months), and the maximum of the process is taken within each block;
Peaks over threshold: here, a threshold is set, and the extremes are defined as the maxima of the process within any continuous period during which it exceeds this threshold.
For ocean sea states, the maxima are most commonly defined in terms of peaks over threshold; any period during which a characteristic exceeds the threshold is classed as a storm, and the largest within the storm is known as the storm-peak value of that characteristic.
In each of these cases, it is possible to derive the limiting distribution of the extreme values (see, for example, Jonathan and Ewans [2013], Coles [2001]) for general dis- tributions; in the Block maxima case, the extremes follow a generalized extreme value (GEV) distribution (for large enough blocks), and in the peaks over threshold case, they are distributed according to a Generalized Pareto (GP) distribution (for
large enough thresholds). The GP distribution for HS at a particular setting of some
inputs θ (e.g. location, time) is
p (HS(θ) |HS(θ) > µ (θ) , ξ (θ) , σ (θ) ) = 1 σ (θ) 1 + ξ (θ) (HS(θ) − µ (θ) ) σ (θ) −(1+ξ(θ)1 ) . (1.3.3) This distribution for the peaks (over threshold µ (θ) ) is characterised by a shape parameter ξ (.) and a scale parameter σ (.) . When fitting such an extreme value model, the choice of a suitable threshold µ (.) is critical. The fitting of such a covariate-dependent extreme value model is described in, for example, Randell et al. [2015] and Randell et al. [2016], and the paper by Jones et al. [2016] compares different parametrisations of the model. Figure 1.3 (taken from Randell et al. [2015]) shows storm-peak significant wave heights for a location in the Makassar Strait between the islands of Borneo and Sulawesi in Indonesia, observed during the period from August 1956 to July 2012; it is clear from this plot that both of the covariates
(direction of arrival and season) have a systematic effect on the distribution of HS(.) .
Since interest lies in the extreme quantiles of the ocean characteristics, a large quantity of data is needed in order to obtain a large enough number of exceedences of a high enough threshold for a model fit. Observation of the real ocean typically only provides a limited amount of data for a small number of locations, and so a common approach is to generate the data for the fitting of the extreme value model using an ocean simulator. To begin with, the ocean simulator is run as a hindcast, using historical atmospheric information to predict historically observed
1.3. Extreme ocean wave modelling 17 wave characteristics; the simulator is then tuned to give the best possible match to the sea state characteristics that were actually observed. Once it has been tuned, the simulator is then run using a forecast of future atmospheric characteristics, predicting future wave behaviour; this simulated wave data is then used to fit a covariate-dependent extreme value model to the storm-peak characteristics.
Sources of uncertainty In this problem, the key sources of uncertainty which
must be handled are as follows:
the ocean simulator used to model the historic wave data will typically have a number of non-physical parameters which must be selected so as to give the best representation of the real ocean. We wish to use the observed waves to learn about the range of appropriate settings;
we then wish to use our uncertainty about these parameters to work out our corresponding uncertainty specification for the simulator at this unknown ‘best’ setting for all of the other storms that we are modelling;
since the simulator is a simplified representation of the ocean, there will always be aspects of the real wave behaviour that it fails to capture; we want to use the data that we did observe on the system to learn about the structure of the difference between the simulator and the real ocean, and then compute the implications of this uncertainty for the simulator predictions at all other points;
when using the simulator to represent the real ocean, uncertainty about the wind field generating the waves explains a substantial amount of the uncer- tainty about the waves themselves;
once we have worked out our uncertainty about the wave characteristics at all prediction locations, we aggregate this information and use it to fit an extreme value model of the form (1.3.3); in doing so, we must compute the implications
of our uncertainty about HS(θ) for our ability to estimate the components of
Figure 1.3: Storm-peak significant wave height (black markers) for a location in the Makassar Strait between August 1956 and July 2012, plotted as a function of direction of arrival (top panel) and season (bottom panel); taken from Randell et al. [2015]. The dashed grey lines correspond to storm trajectories.