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Sea Level Measurement and Interpretationsea levels. It begins with the classical method of AnnualExtremes, which first appeared in the early 1960s andcontinued to be developed for some time thereafter.Following this, the Joint Probability Method, which wasdeveloped in the late 1970s, is considered. This makesmore efficient use of data by incorporating our extensiveknowledge of the tides and storm surges, whichare the two main components of sea level, as a part ofthe estimation procedure.More recent work on the Annual Exceedance Method isdiscussed, followed by a revision of the Joint ProbabilityMethod to correct its deficiencies in areas where the sealevel is dominated by the meteorological surge component.Finally, very recent work on the spatial estimationof extremes is mentioned. References are given at eachstage so that the reader can examine any of the methodsin greater depth. Although extreme high sea levelsare considered, results for extreme low sea levels can beobtained in an analogous way.2.8.2 The Annual Maximum Method (AMM)This is the classical general method of analysis ofextremes having been applied to sea level estimationsince 1963 (Lennon, 1963; Suthons, 1963). It is basedon a result from probabilistic extreme value theorywhich states: if X 1 ,... X n is a sequence of independentand identically distributed random variables, thenmax(X 1 ,... X n ), suitably linearly normalized, convergesas n ∞ , to a random variable with a distributionfunction which is one of the so called extremevaluedistributions. The general case is known as theGeneralized Extreme Value (GEV) distribution. Animportant special case is the Gumbel distribution.The Annual Maximum Method takes the GEV to bethe distribution function of the maximum sea level ina year. Therefore, for a place of interest, the annualmaximum for each year is extracted from hourly observationsand is used as data to estimate the parametersof the distribution that they follow. From the estimateddistribution one can obtain the sea level correspondingto a chosen ‘Return Period’. In practice, return periodsof 50, 100 and 1,000 years are common. The basicmethod assumes that there is no trend in the data, butit can be extended to deal with those cases where atrend is present.A recent extension of the annual maximum methodinvolves using probabilistic extreme value theory toobtain the asymptotic joint distribution of a fixed number(r) of the largest independent extreme values, forexample the five largest in each year. Essentially theapproach is the same as above except that more relevantdata are included in the analysis thereby improvingthe estimation. Care must be taken to ensure that thenumber of annual maxima ‘r’ is not excessive, such thatthe lower extremes fall outside the tail of the extremevalue distribution.This method of estimating sea level extremes is highlyinefficient in its use of data, since it extracts very fewvalues from each yearly record. This is particularly importantwhen the sea level record is short, since it yieldsreturn level estimates with unacceptably large standarderrors. In addition, it makes no use of our knowledge ofthe sea level and storm surge processes. However, theadvantage of annual maxima methods is that they donot require knowledge of tide–surge interaction whichcan sometimes be a significant feature of the data.Consequently the methods are relatively straightforwardto apply.2.8.3 The Joint Probabilities Method (JPM)This method of analysis was introduced to exploit ourknowledge of the tide in short data sets to which theannual maxima method could not be applied (Pughand Vassie, 1979). At any time, the observed sea level,after averaging out surface waves, has three components:mean sea level, tidal level and meteorologicallyinduced sea level. The latter is usually referred to as astorm surge. Using standard methods, the first two ofthese components can be removed from the sea levelsequence leaving the surge sequence, which is just thetime-series of non-tidal residuals. For simplicity these areassumed to be stationary. Because the tidal sequenceis deterministic, the probability distribution for all tidallevels can be generated from tidal predictions. Thisdistribution can be accurately approximated using 18.6years of predictions.The probability distribution of hourly sea levels can beobtained either directly using an empirical estimate orby combining the tidal and surge probability densityfunctions (pdf). The latter is preferable, as it smoothesand extrapolates the former. However the nature ofthe combination of the pdf’s depends on whetherthere is dependence between the tide and surgesequences. Initially, consider the case in which theyare independent.By combining the pdf’s of tide and surge, the distributionfunction of hourly (instantaneous) sea levels is obtained.From this, the distribution function of the annual maximais required. If hourly values were independent, whichis approximately the case where the tide dominates theregime, then this is straightforward.The method has been widely applied. It makes betteruse of the data and of our extensive knowledge of thetides, and accounts for surges that could have occurredon high tide but by chance did not. Most successfulapplications have been to sites which have several yearsof hourly records (>10 years) and where the site is tidallydominant, i.e. where the tidal range is large in comparisonto the surge amplitude. Least successful applicationshave been to sites with both short lengths of data andwhere the site is surge dominant.8IOC <strong>Manual</strong>s and Guides No 14 vol IV
Sea Level Measurement and Interpretation2.8.4 The Revised Joint Probabilities Method(RJPM)Particular emphasis was given to two principalimprovements that make the revised method morewidely applicable than the original joint probabilitiesmethod (Tawn et al., 1989). It was principally directedat sites where the storm surge was responsiblefor a respectable proportion of the sea level and toimprove the estimation procedure for sites whereless than 10 years of data were available.The first issue was that of converting the hourly distributioninto annual return periods. It is clear thateach hourly value of sea level is not independent ofits predecessor or successor. Of the 8,760 hourly valuesin a year, it is necessary to determine the effectivenumber of independent observations per year. Thiswas done through an Extremal Index which is derivedfrom the mean overtopping time of a level for eachindependent storm which exceeds that level. In factthe Extremal Index can be shown to be a constantin the region of the extremes. Because large valuestend to cluster as storms, it should be expected thatthe Extremal Index >1; for example, in the NorthSea, it is 1.4. This effectively reduces the number ofindependent observations from 8,760 to 8,760/1.4.If the site is tidally dominant then the Extremal Indexis considerably smaller than if the site is surge dominant.The immediate advantages of this modificationare: firstly, that no assumption about the localdependence of the process is required; secondly, thatthe conversion from the hourly distribution to annualmaxima is invariant to sampling frequency.The second modification enabled probabilities forlevels beyond the existing range of the surge datato be obtained, in addition to providing smoothingfor the tail of the empirical distribution. The methodis based on the idea of using a fixed number ofindependent extreme surge values from each year toestimate probabilities of extreme surges. The procedureinvolves two important steps. Firstly, the identificationof independent extreme surges. Secondly,the selection of a suitable number of independentextreme surges from each year of data, perhaps fiveper year. Using these surge data, estimates can bemade of the parameters of the distribution of theannual maximum surge (Smith, 1986).When interaction is present, the level of the tideaffects the distribution of the surge. In particular, thetail of the surge pdf depends on the correspondingtidal level. Thus the convolution of tide and surgecan be adapted so that the surge parameters arefunctions of tidal level. This formulation also enablesstatistical tests of independence to be performed.2.8.5 The Exceedance Probability Method(EPM)An alternative method of obtaining extreme sealevel estimates from short data sets is called theexceedance probability method (EPM) (Middletonet al., 1986; Hamon et al., 1989). The EPM, like theRJPM, involves combining the tide and surge distributionsand accounting for dependence in the sealevel sequence. The approach differs in the way thatit handles extreme surges. The EPM uses results forcontinuous time processes and makes assumptionsabout the joint distribution of the surge and its derivative.Improvement is achieved by allowing flexibilityin the surge tail through the use of a contaminatednormal distribution.2.8.6 Spatial Estimation of ExtremesExtreme sea levels along a coastline are typically generatedby the same physical mechanisms, so the parametersthat describe the distribution are likely to bespatially coherent. Models that describe the separateconstituents of the sea level are best suited to exploitingthis spatial coherence, as the individual parametersshould change smoothly along a coastline.The joint distribution of annual maxima over severaldata sites can be modelled using a multivariateextreme-value distribution (Tawn, 1992). Changes ineach of the parameters of the distribution, over sites,can be modelled to be consistent with the propertiesof the underlying generating process identified fromthe RJPM. The main advantage of the spatial methodis that it can utilize data sites with extensive sea levelrecords and augment these with data from sites withshorter records of a few years.Using the ideas for extremes of dependent sequences,this can be related to the distribution functionof hourly surge levels, and then the empirical surgedensity function can be replaced by the adjusteddensity. Using the adjusted density function, theconvolution can be performed to combine the tidaland surge distributions to obtain the hourly sea leveldistribution and hence the return periods can becalculated for different levels.IOC <strong>Manual</strong>s and Guides No 14 vol IV9
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