NIVERSITYOSignificantValuesSouthampton.Significant value is the average (or mean) ofthe highest 1/3 peakvalues, i.e. if we measured 99 peak values, the mean of the highest 33peakvaluesUsually denoted by the index1/3,egh/3 significant wave height,aparameter usedto classifymeasured wavedata.:Based onRayleigh'sdistributionhi/3=4Vmo,wheremo:Mean Squarevalue of wave elevation.Another statistic is the i/1o average,being theaverage of thehighest10% peak values, denoted by the index 1/10, e.g. hi/10Rayleigh'sPDFJIVERSITYOSouthamptonProbabilityofexceedanceUsingRayleigh'sPDF,probabilityofapeak valuebeinggreaterthan Zo3XP[=>z0]=-dexp2mo2mo二.22010= 0-(-)exp=exp2mo2mo=1- P[z≤z0]6
6 11 Significant Values • Significant value is the average (or mean) of the highest 1/3 peak values, i.e. if we measured 99 peak values, the mean of the highest 33 peak values • Usually denoted by the index 1/3, e.g. h1/3: significant wave height, a parameter used to classify measured wave data. • Based on Rayleigh’s distribution h1/3=4√m0, where m0: Mean Square value of wave elevation • Another statistic is the 1/10 average, being the average of the highest 10% peak values, denoted by the index 1/10, e.g. h1/10 12 Rayleigh’s PDF Probability of exceedance • Using Rayleigh’s PDF, probability of a peak value being greater than z0 1 [ ] 2 exp 2 0 ( ) exp 2 exp 2 [ ] exp 0 0 2 0 0 2 0 0 2 0 2 0 0 0 P z z m z m z m z m z P z z d z z
IVERSITYORandom Process-BasicsSouthamptonX(t), X(2)(t), X(3)(t) etc are a.setofmeasurements (orXa(t)observations)outofaninfinitenumberofpossiblemeasurements-waveelevation,motion,bendingmomentetcX()(t) etc are realisations of the.X(a)(t)random processx(t)StationaryRandomprocess:probabilitydistribution (andstatistics)arenot affectedbyaX(3)(t)translation in time;i.e.statisticsof random variablesX(t=t,)andX(t=t,),across the realisationsare same.tErgodicRandomprocess:Stationaryand statistics (orexpectations)areequal totemporal averages alonga single13realisationUNIVERSITYOFErgodicRandomProcess-BasicsSouthampton.Mean or Expectedvalueusingtemporal averagealong a realisation1 T/2 X()()dt=(X(a)()E[X(0)]=limT-→ TT/2MeanSquarevalue,takenalongarealisation1 T/2x 0=mxa=(i(t)-T/2Autocorrelationfunction:temporal averageoftheproduct oftherandomprocessattimestandt+tRxx(t)=(X(t)X(t + t))such thatRxx (0)=(x2(t)7
7 13 Random Process - Basics • X(1)(t), X(2)(t), X(3)(t) etc are a set of measurements (or observations) out of an infinite number of possible measurements – wave elevation, motion, bending moment etc. • X(1)(t) etc are realisations of the random process X(t) • Stationary Random process: probability distribution (and statistics) are not affected by a translation in time; i.e. statistics of random variables X(t=t1) and X(t=t2), across the realisations are same. • Ergodic Random process: Stationary and statistics (or expectations) are equal to temporal averages along a single realisation X(1)(t) X(2)(t) X(3)(t) t1 t t t t2 14 Ergodic Random Process - Basics • Mean or Expected value using temporal average along a realisation • Mean Square value, taken along a realisation • Autocorrelation function: temporal average of the product of the random process at times t and t+τ ( ) ( ) 1 [ ( )] lim (1) (1) / 2 / 2 X t dt X t T E X t T T T ( ) ( ) 1 [ ( )] lim 2 (1) 2 (1) / 2 / 2 2 X t dt X t T E X t T T T (0) ( ) such that ( ) ( ) ( ) 2 R X t R X t X t XX XX