Statistical assessment of annual patterns in coastal extreme wave conditions

Coastal Processes 111 39 Statistical assessment of annual patterns in coastal extreme wave conditions J. L. Vega1, J. González2, G. Rodríguez' 1 Departamento de Fí sica .. Universidad de Las Palmas de Gran Canaria, Spain 2Plataforma Oceánica de Canarias, Spain Abstract The annual cycle in extreme values of significant wave height is examined by transforming the timing ofthe storm peaks in a circular variable, with the aim of taking advantage of the many tests devised to explore uniformity on the circle. The use of four different but complementary uniformity tests makes possible a robust assessment of the annual cycle statistical significance. Seasonality of storms in a long time series of significant wave heights, measured in a coastal zone, is examined. The presence of a seasonal pattem is statistically beyond doubt. Keyword s: wave storms, seasonality, circular statistics, circular uniformity. 1 Introduction Many geophysical processes may exhibit periodic cycles along time, such as diurna!, weekly, monthly, and annual variations. In particular, geophysical variables may exhibit marked periodic behaviour on annual time scales, mainly induced by the solar cycle and commonly known as seasonal variations. lnformation on the temporal behaviour of severe wave conditions, and more specifically on the mean annual pattem of these extreme conditions, is required in many applications such as, for example, coastal zone management, coastal morphodynamics, and coastal engineering infrastructure operations. Furthermore, confidence on the existence of a seasonal periodicity in longterm time series of significant wave height, H s, the most common parameter used to characterize the sea state severity, is very important when using time series models to represent the stochastic evolution of such parameter (e.g., WIT Transactions on Ecology and th e Enviro nm e nt , Vol 169, www.witpress.com, ISSN 1743-354 1 (on-line) doi: 10.2495/C PI 30041 2013 WIT Press 40 Coastal Processes 111 Athanassoulis and Stefanakos [ 1 ]; Guedes Soares et al. [2]). Additionally, statistical methods of extreme value analysis used for the derivation of design events, by extrapolating the data outside the range of observations, commonly assume that data are independent and identically distributed. The assumption of independence is generally well satisfied by using an adequate data selection procedure which ensures the independence of representative values of successive storms. Nevertheless, the presence of seasonal variability in the selected extreme values violates the assumption of data identically distributed. lt seems natural that if processes underlying wave climate show a seasonal variation their extremes do as well. However, such cycles in extremes have not received much attention, as the conventional method of extreme wave height analysis, the commonly named annual maxima technique, does not require their explicit modelling. The i11creasing adoption of the peaks over threshold approach, which demands the inclusion of annual cycles, has stimulated the interest in considering the seasonal effects in the study of extreme wave events. Thus, many studies (e.g., Morton et al. [3]; Méndez et al . [4]; Jonathan and Ewans [5]; Mackay et al. [6]) have evidenced the importance of accounting for seasonality, as a covariate when estimating extreme wave conditions by means of the peaks over threshold method, or removing its effects, by using sorne appropriated approach, such as analysing the data in discrete seasons (e.g. Morton et al. [3]), to allow the assumption of data identically distributed to be met. The identification of the annual cycle in a long-term series of Hs is relatively easy, such as is observed in Figure 1, where the average year constructed from a time series of Hs three-hourly sampled is shown. Contrarily, the presence ofthis cycle in extreme values is commonly doubtful and not easily assessable. This paper aims to evaluate in a consistent way the statistical significance of possible seasonal variations present in series of extreme values derived by considering storms separated by several days and declustering values corresponding to the same storm. For this, the time of occurrence of individual values of Hs representing each storm is considered as a circular variable, so that it can be properly described in tenns of directional statistical tools. The paper is structured as follows. Principal characteristics of the significant wave height time series used in the study, as well as the statistical methodology used to assess the existence of seasonal variations in the timing of wave storms along the year are introduced in section 2. The results of applying the statistical approaches are discussed in section 3. Conclusions are summarised in section 4. 2 Data and methodology 2.1 Data set The experimental data set used in the study is a time series ofthree-hourly values of significant wave heights derived from wave measurements made in the north coast of Gran Canaria island, ata place of coordinates (28º 8.5'N, 15º 27.5'W) and a water depth of about 40 m. Measurements cover a period of fifteen years, WlT Transactions on Ecology and the Environment, Vol 16 9, O 2013 WIT Press www.witpress.com, ISSN 17 43-3541 (on-line) Coas tal Processes 111 4 1 from 1987 to 2001. Each year includes 365 days. The records of 29 February are omitted in leap years. The mean year obtained by averaging the fifteen values measured at the same time at each of the fifteen years is shown in Figure 1 and reveals, on average, a more or less clear annual cycle. The time of each storm event has been extracted from the Hs series by selecting the value and timing associated to the peak sea state during a storm, defined as a period of relative severe wave conditions that satisfy the requirement of independence. The approach used to extract the peaks representing the storms was suggested by Simiu and Heckert [7] and considers that peaks of different storms are separated at least by 48 hours. This methodology has been applied by González et al. [8] to characterize the intensity of storms in the same series by using the peaks over threshold approach. :e 0.6 c "<.~ ~~ ~ .:;;,,:;. c ' ~( .1 ~:;.. <:- ,,, ¿. >-, >"' ..... "' o ;:,.e ' ...: ,~ ~ ' ' '"':' ...: ' :- , <::J ' ...: ' Figure 1: Three-hourly mean value of the Hs time series recorded in the period ( 1987-2001 ). 2.2 Statistical assessment of seasonality In many practica! situations the examined variable is a direction in a twodimensional plane. In such a case, the sample space consists of points on a unit circle and conventional statistics is not applicable. Circular statistics is the special branch of statistics developed for the proper analysis of this kind of random variable, in which probability distributions are characterised by their cyclic nature. That is , circular statistics includes methods to study random variables that have a cyclic behaviour (Mardia and Jupp [9]). In this context, it is interesting to note that any circular temporal measure can be translated into an gles. Thus, for example, it is possible to consider the day ofthe year at which a storm occurs as a circular random variable. For this, the day of the year, d, must be converted to a angular value, B, in radians, by WIT Transactions on Ecology a nd th e Enviro nm ent, Vol 169, © 20 13 WIT Pr ess www.witpress. co m, ISSN 1743-354 1 {o n-line) 42 Coas tal Processes 111 () =~d 365 (1) A very common question in circular statistics is whether a data sample is , or not, distributed uniformly around the circle. This means that the uniform distribution is usually considered as the null hypothesis. Then, to know if wave storms ata given coastal region are uniformly distributed through the year (null hypothesis), or there is one or severa! time periods during which storms are more frequent (altemative hypothesis), is necessary to know if the time of occurrence ofwave storms along the year follows a uniform distribution f(()) = 2... 2rr O$() $ 2rr (2) There are multiple tests to answer this question, which differ in their efficiency to detect certain departures from uniformity. Note that uniformity, also referred as randomness or isotropy, represents the situation in which probability is spread out unifonnly on the circumference ofa circle. Four commonly used tests to assess the uniformity of circular variables are applied in the present study. The fundamentals of these tests are briefly introduced bellow. A detailed description ofthese and other tests can be found in Fisher [ 1 O]. 2.2.1 The chi-squared test The chi-square test is the most frequently used test to assess the goodness of fit of the empirical distribution of a data set to a theoretical model. To apply the chisquare test the circle is divided in twelve sectors, k= 12 , of 30° each one, and the observed, o ;, and expected, e;, frequencies of storms in the i-th month computed. The Chi-square statistic is given by T = ¿!<_ (o¡-e¡)2 !-1 e¡ (3) Null hypothesis (uniformity) is rejected if T exceeds the corresponding critica! value for k-1 degrees of freedom. Critica! values for severa! confidence levels, a, are given in Fisher [10]. lt is worth ofmentioning that this test is robust for unimodal and multimodal samples. 2.2.2 The Rayleigh test The Rayleigh test is based in the estimated value of the resultant vector length R, given by R = ..Jc 2 + 52 where and 1 ¿k e = Ñ i=i n¡ cos ei S = ~ L~=l n¡ sin()¡ WIT Trans ac1i ons on Ecology and lh e Environrnenl. Vol 169. www.wilpress.com, IS SN 1743-354 1 (o n-line) (4) (5) (6) 20 13 WI T Press Coastal Proce sses 111 43 and 0¡ is the date at the centre of each monthly bin i, expressed in radians. The confidence level P associated with the mean resultant length R is given by p = e -Z [l + 2z-z 2 _ 24z-132z 2 +76z 3 -9z 4] 4n 2ssn2 (7) The Rayleigh test is considered a powerful test only if is possible to assume that the population distribution does not have more than one mode. Furthennore, it is important to remark that, it assumes sampling from a von Mises distribution. Note that a value of R= I indicates that ali stonns occur on the same calendar day in all years. However, a value of R=O that the probability of occurrence of stonns is the same for any calendar day. The latter is exactly true only if the distribution does not have more than one mode. 2.2.3 Kuiper test This test is an altemative to de chi-square one based on the cumulative distribution function. The basic idea of this test is that the observed and the theoretical distributions should closely resemble one another if the sample has been drawn from the assumed circular unifonn distribution. The main step in the approach to assess unifonnity is to compute the deviations between the unifonn and empirical cumulative distributions. The following statistics are defined o+ = max{Fn(B) - F(B)} o- = max{F(B) -Fn(B)} (8) where F,, ( ()) and F (()) are the sample and the unifonn cumulative distributions. The sum of D' and 0-values define the Kuiper test statistic (9) or even best, V V = Vn ( n 112 + 0.155 + ~~ 2 1 !) ( 10) The unifonnity hypothesis is rejected if the test statistic, V, exceeds the critica! values tabulated in Fisher [ 1 O]. An important aspect to remark is that the Kuiper test is specially indicated in case of multimodal distributions. 2.2.4 The modified Kolmogorov-Smirnov test Freedman [ 11] suggested a modification of the classical Kolmogorov-Smimov test to examine seasonality in data. This non-parametric methodology removes sorne drawbacks existing with the conventional one. The hypothetical (unifonn) cumulative distribution is a step function denoted by F(t) =ti 12 , where t is the rank of each month of the year. The sample cumulative distribution is al so a step function denoted by FN ( t) =k/N, where k is the number of events that have occurred during ali months 5 1. The test statistic, T, is given by VN = max(FN(t) - F(t)) + lmin(FN(t) - F(t))I ; 1 ~ t ~ 12 (11) WIT Transactions on Ecology and the Enviro nm e nl , Vol 169, www.witpress.com, ISSN 1743-354 1 (onlin e) 20 13 WlT Press 44 Coastal Proce ss es 111 The distribution of T does not follow any specified distribution, but has been empirically evaluated by means of Monte Cario simulations, and is tabulated in Freedman [11]. 3 Results and discussion Representation of circular data in polar coordinates is commonly useful. This requires the specification of the angle, 0, and a distance, r, and provides a method of uniquely defining the location of data points in the circle. This representation method has the advantage of clearly separating directional and distance (intensity) information. Polar plots of the storm events for six different thresholds of significant wave height are represented in Figure 2. The threshold varíes from 2 meters until 4.5 meters with increments of 0.5 meters, from the left upper comer to right and down. The number of storms considered for each threshold is indicated in the second column ofTable 1. lt can be observed that the non-uniformity of the time of occurrence of wave storms becomes more and more clear as the wave height threshold imposed to define the extreme events is increased. Thus, while for a threshold of 2 meters storms occur in any period of the year, for a threshold of 4 meters the stormy period reduces to autumn and winter. Tables 1-4 include the results of the Chi-square, Rayleigh, Kuiper, and modified Kolmogorov-Smirnov test of uniformity. Each table includes a first column indicating the wave height threshold, a column with the corresponding test statistic value and the critica! value for a confidence level of a=O.O 1. The last column indicates the acceptance or rejection of the null hypothesis of uniformity. Ali the tests used reject the uniformity, accept the seasona li ty, ofthe wave storm occurrence for any threshold. lt should be noted that the tests used to assess randomness are mutually complementary. Thus, while Chi-square test is robust for unimodal and multimodal samples, Rayleigh test is considered a powerful test ifthe population distribution is unimodal, Kuiper and modified Kolmogorov-Smirnov tests are specially indicated in case of multimodal distributions. Then the rejection of uniformity by ali the tests demonstrate statistically significant trend of wave storms to cluster during a given period of the year. In other words, the use of various types oftests , parametric and non-parametric, adequate for unimodal and for multimodal distributions, evidences without doubts the existence of a cyclic annual pattem in the timing ofwave storms for the studied zone. 4 Conclusions Results derived from the application offour tests ofuniformity clearly reveal the presence of an annual cycle in the time of occurrence of wave storms in the coastal area examined. This seasonal pattern is independent of the significant wave height threshold considered. WIT Transactions on Ecology and the Environment, Vol 169, www.witpress.com, ISSN 1743-354 1 (on-line) 20 13 WIT Press Nov Oc t Scp Nov Oc1 Scp O\" Oct Se p Figure 2: Coas tal Proc esses 111 45 J an Jan D cc Fcb Dt:c Feb /\p r May /\ug .lun Jul Jan Jan Dec Fcb Dt.:c · ·.· . ... .. . Mar Nm ,. •• ••• • f\.1 ar . ·' . .. . ~·.¡··· "' .. : .. ~ ... · . ··":· ··· ?\F cb ".... . . ... 1. .. -: Aug D t>c Aug •• 11. ~ ~__....,._........_..----; Apr o 2 .: ... 4 6 r v1ay Jun Jul Jan Fe b · ·.· .... .. Ma r Oc 1 Sc p Nov H Apr Oc1 o 2 4 6 Scp .lun .l ul 11 1 \pr o 2 • .4 6 v1a y Aug Jun Jul Jan Dt:c Fcb ··.· Mar 11 , Apr o 2 4 6 May A ug .l un .l ul Polar plots of the time of occurrence in the year and the severity of wave storms for different threshold values. Upper left 2.0 m, upper right 2.5 m, middle left 3.0 m, middle right 3.5 m, lower left 4.0 m, and lower right 4.5 m. WIT Transac1ion on Ecology a nd lhe Environmenl, Vol 169, www. wi tpress. co m, ISSN 1743-354 1 (on-line) 2013 WIT Press 46 Coastal Processes 111 Table 1: Table 2: Results of the Chi-square test for assessing uniformity in the time of occurrence of wave storms. ;@ ;;, Chi-sqll8re test Hs(m) T N 1 T 1 !~ (a= 0, 01) 1 Unlformlly > 2,0 355 11 7, 6648 24.725 Rejected > 2,1 309 133 , 0583 24,725 Re¡ecled > 2,2 270 138 , 2667 24,725 Re ie cted > 2,3 238 1 30 . 6723 24,725 Rejected > 2,4 210 125 , 6571 24,725 R eiect ed > 2,5 186 120,709 7 24 , 725 Re]Bcled > 2,6 168 116, 71 43 24,725 Rejected > 2,7 15 3 1 12 , 4002 24,725 R eiected > 2,8 132 100,9091 24 , 725 Rejected > 2,9 116 99 ,3 793 2 4,725 R ejected > 3,0 104 96 , 0769 24,725 Rejected > 3,1 85 91 , 0471 24 , 725 Re¡ecled > 3,2 77 BB , 3506 24,725 Rejected > J,3 74 84 , 2703 24 , 725 Rejected > 3,4 64 75 , 8750 24 , 725 R e¡ecled > 3,5 55 64 , 3455 24,725 Rejected > 3,6 47 62 , 0213 24,725 Reiected > 3,7 45 56 , 6000 24.725 R ejected > 3,8 38 45, 3684 24 , 725 Re¡ecled > 3,9 31 56 , 0968 24,725 Reiected > 4,0 30 52 , 4000 24,725 Rejec!ed Results of the Rayleigh test for assessing uniformity in the time of occurrence of wave storms. Rayteigh Test Hs(m) 1 R 1 &(a =0 .01) 1 Unlformlly > 2,0 0, 3989 0.4035 Rejected !o 2, 1 0.4566 0. 4618 Rejected >2.2 0, 497 3 0, 5030 R ejOCt ed > 2.J 0,51 18 0, 5177 Re¡ected > 2,4 0. 5307 0, 5368 Rejected > 2,5 0. 5509 0. 5573 Re¡ec ! ed > 2.6 0.5646 0, 57 11 Rejected > 2.7 0, 5811 0, 5878 Re¡ected > 2.8 0, 5928 0, 5997 Re¡ected >2.9 0, 6220 0, 6291 Rejected > 3,0 0, 6431 0, 6505 Re¡ecled > 3, 1 0. 6723 06800 ReJecled > 3,2 0. 6933 0.7063 Rej€cied > 3.3 0, 6959 0, 7039 ReJOC!ed > 3.4 0,7 073 0, 7154 Rejected > 3,5 0.7025 0.710C Re¡ecled > 3,6 0,7 4 11 0,7496 Re¡ecl ed > 3,7 0.73 24 0,7408 Rejected > 3,8 0.7133 0.7215 Re]OCted > 3.9 0,7 930 0, 8022 Re¡ected > 4.0 0,7868 0, 7958 Re¡ected WJT Transaclíons on Eco lo gy and lhe Envíronmenl, Vol 16 9, © 2013 WIT Press www.witpress.com, ISSN 1 743 -354 1 (on-line) Table 3: Table 4: Coastal Processes llI 4 7 Results of the Kuiper test for assessing uniformity m the time of occurrence of wave storms. Kuiper Test HS(m) 1 V 1 V.~ (a= 0, 01) 1 Unlformlly >20 5 2 461 2 00 Re¡ecle<:I > 2 1 5 5000 200i Ra¡eded >22 5, 6769 2001 Re¡ected > 2.3 5, 5663 2 00 1 Re¡ected > 24 5,4 874 2 001 Re¡ected > 2 5 5,41 00 2001 Re¡ected > 26 5, 2613 2001 Re¡ecled > 2 7 5, l(f,J4 2001 Re¡ected > 2,8 4,78ó l 2001 Re ¡ ected > 28 4J1ó2 2001 Re¡ected > 3,0 5. 5727 2 001 Re ¡ ecl ed > 3 1 4 3132 2 001 Re¡ecled > 3.2 4 2B56 2 001 Re¡ecled >33 4 i3J7 2 001 Re¡ected >34 4 0 448 2 001 Re¡ected > 3.5 5, 6588 2001 Re ¡ ecled >36 3 7 719 2 ())1 Re¡ecled > 3 7 3, 6585 2 001 Re¡ected > 3.8 3, 33 73 2 001 Re¡ ected > 3 . ~ 3,51 40 2 001 e¡ected > 4,0 3, 447 1 2001 Re¡ecled Results of the modified Kolmogorov-Smimov test for assessing uniform ity in the time of occurrence of wave storms. Modified Kotmogorov-Smirnov Test HS(m) Max Mln YA vn•(n)•0.5 Unlformtty > 2,0 o 1455 0,10 77 0, 2533 4 .7723 Re¡ect ed > 2.1 0. 1845 O. 109 2 o 2937 5. 1526 Re ¡e cl ed > 2,2 0, 2000 0, 115 7 o 31 67 5. 20 34 R e¡ ect ed > 2.3 0. 2080 0, 1155 03235 4, 991 2 Re¡ect ed > 2,4 0, 20 71 0,1 357 0, 3429 4. 9635 Re ¡ect ed >2 5 0. 20 70 o 14 78 o 3548 4, 8394 Re ¡ect ed > 2,5 0. 2024 0,1 60 7 0, 3531 4 7C'B3 Re ¡ect ed > 2,7 0. 2141 o 1552 o 3Q9J 4, 5678 Re ¡ecl ed > 2,8 o 2"'1 8 0, 1439 0,37 88 4,3519 Re¡ected >2& 0. 241 4 o 1638 o 4052 .t . 3638 Re¡ected > 3,0 o 23 08 0, 1827 o 41 35 4 2 65 Re¡ected >31 0. 2451 o 1863 o 4314 3, 9771 Re ¡ected > 3.2 o 2825 0,17 10 04535 3,97 91 Re,ect ed > 3.3 02905 0. 15 77 04482 3, 8555 Re ¡ect •d > 3,4 0, 2969 0, 15!5 0, 4583 3, 666 7 Re /0'::t ed >35 0, 350J 0, 106 04561 3,382 2 Re¡ected > 3,6 0, 36 70 0,1 312 0 ,4 982 3, 415 7 Re¡ected > 3 7 0. 3500 1 444 0494 4 J.3153 Re¡ected > 3.8 o 2763 0, 2018 0,4781 2 94 70 Re¡e..~ ed >3 9 0. 3172 o 252 7 05699 3 ,1 7 30 Re ¡eci ed > 4.0 0. 3CoQO 0, 266 7 o 5667 3 .1 038 Reiecled WIT Transac ti ons on Ecology a nd the Environmenl, Vol 169, O 201 3 WIT Press www.witprcss.com, ISSN 1743-3541 (on-line)