Wave Analysis

Short Term and Long Term Analysis

Introduction

Short term analysis:

com

analyse of wave that occurs within one wave train vis

o.os

eano

bs.c

wave train Long term analysis

f

ww

w.a

v

derivation of statistical distributions that cover many years 0 01

0.02

0.03

0.04

l (m

)

many years

-0.03

-0.02

-0.01

0

0.01

Wat

er L

evel

-0.040 5 10 15 20 25 30

Time (sec)

Wind Wave correlationWind Wave correlationSignificant wave height and wind speed are shown inwind speed are shown in July 2007, units are metres and metres per second respectively. These figuresrespectively. These figures highlight the relationship between wind speed and significant wave height: the g gfaster the wind, the highest the waves.The lowest waves (dark blue) are mainly in tropical and subtropical oceans, where the smallest wind speed are recorded.

www.aviso.oceanobs.com/en/applications/atmosphere-wind-and-waves/wind-and-waves/seasonal-variations/index.html

Why collect wave data? Monitoring of coastal processes

such as beach erosion and sediment transport.

Baseline design statistics for gcoastal projects.

Operational assistance in coastal Operational assistance in coastal construction projects.

Monitoring of severe weather Monitoring of severe weather conditions.O hi h Oceanographic research.

(Manly Hydraulic Lab : http://mhl.nsw.gov.au/)

How to measured wave? Use wave recorder : i.e:

Wave staffst e.g: www.oceansensorsystem.com

Wave rider / wave buoy e.g: www.datawell.nlg

Pressure sensor e.g : www.civiltek.com

S t llit i Satellite images e.g : GFO (Geosat Follow On)

Place wave recorder in deepwater (>0 5L) Place wave recorder in deepwater (>0.5L) Record wave height, period and direction (duration 15 60) Links : http://cdip.ucsd.edu, http://mhl.nsw.gov.au/, etcp p p g Assignment 2 : Find more info on wave recorder (product detail, vendors, measurement

and analysis methods, etc)

Typical wave recorder www.datawell.nlTypical wave recorder

TRITON-ADVwww.sontek.com

www.civiltek.com

www.oceansensorsystem.com

http://www.scienceprog.com/wp-content/uploads/2007i/Ocean_embedded/wave_heigh_measurement.jpg

AWACwww.nortek.com

Typical recorded samplesTypical recorded samplesFile created by: WAVELOG on 12/11/2006 1:53:28 AMDevice: Model 730W_CTPSerial No: 17528File Type: WAVE STATSOperating Mode: Wave BurstContract Reference: NoneSite Reference: NoneLocation Latitude: NoneLocation Longitude: NoneSetup by: NoneUser Info: NO SITE INFORMATIONPressure Calibration: P = 1:RPT200-1791,3500,35,958161,1,040897T t C lib ti T 12 +3 7484500 10 +1 7482800 03 +1 0768100 01Temperature Calibration: T = 12 +3.7484500e-10 +1.7482800e-03 +1.0768100e-01Conductivity Calibration: C = 13 +3.8447599e-13 -9.6605406e-09 +3.0098500e-03 -7.8249698e-02File Number: 0Sample Rate: 0.4Density: 1025.97Gravity: 9.807Instrument Height: 0Trigger Level: 0Trigger Level: 0Wave Data Save Trigger Level: 0Tide Burst Duration (secs): 600Tide Cycle Time (secs): 1200Wave Burst Duration (secs): 5119.99992370606Wave Cycle Time (secs): 1200Speed of sound formula Chen & MilleroPressure formula: 1essu e o u aDate Time Hs Tp Hmax E

m(U) Secs m(U) N.m(U)^2.m^-39/12/2006 12/9/06 23:16 0.0134 0.0210 0.11252249/12/2006 12/9/06 23:36 0.0224 0.0352 0.31571639/12/2006 23:56:15 0.0139 0.0218 0.1213013

10/12/2006 0:16:15 0.0117 0.0184 8.65E-0210/12/2006 0:36:15 19647.13 meragukan10/12/2006 0:56:15 0.0165 4.413793 0.0258 0.17047610/12/2006 1:16:15 0.0168 4.129032 0.0263 0.176645210/12/2006 1:36:15 4.129032 11155.82 meragukan10/12/2006 1:56:15 0.0144 4.129032 0.0226 0.130291110/12/2006 2:16:15 0.0125 4.129032 0.0196 9.84E-0210/12/2006 2:36:15 0.0046 0.0073 1.35E-0210/12/2006 2:56:15 0.0136 4.129032 0.0213 0.115540210/12/2006 3 16 15 0 0134 3 657143 0 0210 0 112791410/12/2006 3:16:15 0.0134 3.657143 0.0210 0.112791410/12/2006 3:36:15 0.0128 3.878788 0.0201 0.103352910/12/2006 3:56:15 0.0215 42.66667 0.0338 0.290728910/12/2006 4:56:15 0.0111 3.878788 0.0174 7.75E-0210/12/2006 5:16:15 0.0111 3.878788 0.0174 7.71E-02

Triton Webinar

Typical Analysis ResultsTypical Analysis Results

www.sontek.comwww.datawell.nl

Terms Ensemble of Three Realizations Terms Realization

Representative record at 00.01

0.02

0.03

0.04

evel

(m) 1

Representative record at particular time range

Ensemble -0.04-0.03

-0.02

-0.01

0

0 5 10 15 20 25 30

Wat

er L

e

Compilation of several realization

E h bl h

Tim e (sec)

0 02

0.03

0.04

2 Each ensemble has parameters such as mean, standard deviation,

-0.03

-0.02

-0.01

0

0.01

0.02

Wat

er L

evel

(m)

skewness, kurtosis, etc -0.040 5 10 15 20 25 30

T im e (sec)

0 04

-0.01

0

0.01

0.02

0.03

0.04

ater

Lev

el (m

) 3

-0.04

-0.03

-0.02

0 5 10 15 20 25 30T im e (sec)

Wa

Terms0.01

0.02

0.03

0.04

vel (

m)

a

Stationary none of the ensembles -0.04

-0.03

-0.02

-0.01

0

0 5 10 15 20 25 30

Wat

er L

ev

none of the ensembles parameter (z) vary in timeexp: 2a = 2b = 2c = b =

Tim e (sec)

0 02

0.03

0.04

ba b c3a = 3b = 3c4a = 4b = 4c-0.03

-0.02

-0.01

0

0.01

0.02

Wat

er L

evel

(m)

ErgodicTime-0.04

0 5 10 15 20 25 30T im e (sec)

0 04( ) ( ) ( )k k b kz z z + + -0 .01

0

0.01

0.02

0.03

0.04

ater

Lev

el (m

) c( ) ( ) ( )

31 1

k a k b k ck

K N

z z zz

harus samaz z

+ += -0 .04

-0 .03

-0 .02

0 5 10 15 20 25 30T im e (sec)

Wa( )

1 1k k j

k jz z

K N = ==

Short Term Analysis

Height of waves () are random Impossible to predict exact value of at any time

or location Probability that has a certain value is called

PDF (Probability Density Function), p().( y y ), p() p() can be described by normal distribution. 2

2 22

1

1 1( ) exp ,22

N

jj

pN

= = = j

Short Term Analysis contd

Assumed that H = 2max, then PDF for H(P b bilit th t H h t i l )(Probability that H has a certain value)

2 2H H Rayleigh Distribution2 22 2( ) exp4 8

H Hp H =

Short Term Analysis contd

The Cumulative Distribution Function (CDF) of wave height: probability that any individual wwave height H isheight: probability that any individual wwave height H is less than a specified wave height H

2 2 2H

2 2 2

2 2 20

( ' ) exp 1 exp4 8 8

H H H HP H H dH < = =

The Probability of Exceedance : the probability that any individual wave of height H is greater than a specified

individual wave of height H is greater than a specified wave height H :

2H

2

2( ' ) 1 ( ' ) exp 8HQ H H P H H

> = < =

Short Term Analysis contd

0.9

1

P

0.7

0.8

Q

PCumulative Distribution Function

0.4

0.5

0.6

H).p

(H),

P,

(H).p(H)

Rayleigh Distribution

0 1

0.2

0.3

(H

Q

Probability of

0

0.1

0 1 2 3 4 5 6 7 8H/Sigma

yExceedance

H/Sigma

Short Term Analysis contd

Wave height with Probability of Exceedance Q:

2 18 ( ln ) 2 2lnQH Q Q = = ( )Q Q Q

2 21 NjN

= 1

jjN =

Short Term Analysis contdShort Term Analysis cont d

Short Term Analysis contd Example 3.1 Calculation of Short-Term Wave Heights Example 3.1 Calculation of Short Term Wave Heights

To analyze a wave record it must be stationary. Hence, it is normal to record waves for relatively short time durations (10 to 20to record waves for relatively short time durations (10 to 20 minutes). A longer record would not be stationary because wind and water level variations would change the waves. Thus it is usual to record, for example, 15 minutes every three hours. It is

b tl d th t th 15 i d i t ti fsubsequently assumed that the 15 min. record is representative of the complete three hour recording interval.

S th l i f h d i ld Suppose the analysis of such a record yieldsmandT 0.1sec10 ==

We want to calculate significant wave height Hs, average wave height , average of the highest 1% of the waves , and the maximum wave height in the record. 01.0

HH

SShort Term Analysis contd

Long Term Analysis

Purpose :To organise wave height dataTo extrapolate data set to extreme values of

wave height occuring at low probability of exceedance

MethodsStatistical Analysis of grouped wave datay g pExtreme Value Analysis from ordered data

Grouped wave data

Nov 01 - Nov 30, 19834

4.5

(2)

(4)

(5)

3

3.5

ht (m

)

Ht = 1.5 m

(2)

(1)

(5)

1 5

2

2.5

ve H

eigh

(3)(6)

(8)(7)

0 5

1

1.5

Wav

0

0.5

0 100 200 300 400 500 600 700 800Time (hrs from Nov 01, 0:00, 1983)

Grouped wave data contd

D i h l f bl b i dData in the left table obtained from 34,9 years of record.

= number of data points / yr= 282306 / 34.9 = 8089

= 2738 / 34.9 = 78.45

Grouped wave data contd( ' )P P H H= ( )

1019( ' 1.75) 0.37227381019 549( ' 2.00) 0.573

2738

P P H H

P H

P H

= =

+ = =738

1019 549 382( ' 2.25) 0.7122738

( ' ) 1

P H

Q Q H H P

+ + = == > =

Curveline is difficult to interpolate! transformed into straightline

Curve TransformationCumulative Distribution Function

Normal Distribution

Normal Probability Distribution Transferred to: Transferred to:

Log-Normal Probability Distribution

Log Normal Distribution

Gumbel Distribution

Gumbel Distribution

Gumbel DistributionWeibul Distribution

Weibull Distribution

y=A.x + B

Distribution Models

sH = standard deviation, H h i htH = mean wave height and = Weibull and Gumbel Parameter = lower limit of H = threshold value in a Peak over Threshold data set

Probability Table

z = -3.4 P = 3.37 x 10-4z = +3.4P =1 - 3.37 x 10-4

Normal Distribution Curvez 3.4 P 1 3.37 x 10 4

= 0.999663

In Excel : NORMINVP = 3.37 x 10-4

z =0z = -3.4

(1) (2) (3) (4) (5) (6) (7) (8) (9)(1) (2) (3) (4) (5) (6) (7) (8) (9) H N P Q z ln H G W

=0.8 W

=1.3 1.75 1019 0.372 0.628 -0.326 0.560 0.012 0.384 0.555 2 00 549 0 573 0 427 0 183 0 693 0 584 0 816 0 8832.00 549 0.573 0.427 0.183 0.693 0.584 0.816 0.883 2.25 382 0.712 0.288 0.560 0.811 1.081 1.316 1.184 2.50 254 0.805 0.195 0.859 0.916 1.528 1.848 1.459 2.75 174 0.869 0.131 1.119 1.012 1.959 2.421 1.723 3 00 113 0 910 0 090 1 339 1 099 2 359 2 996 1 9643.00 113 0.910 0.090 1.339 1.099 2.359 2.996 1.964 3.25 81 0.939 0.061 1.550 1.179 2.772 3.627 2.210 3.50 60 0.961 0.039 1.766 1.253 3.232 4.366 2.477 3.75 40 0.976 0.024 1.976 1.322 3.713 5.176 2.750 4.00 27 0.986 0.014 2.190 1.386 4.244 6.105 3.044 4.25 19 0.993 0.007 2.442 1.447 4.916 7.326 3.406 4.50 10 0.996 0.004 2.683 1.504 5.611 8.638 3.769 4.75 4 0.998 0.002 2.849 1.558 6.122 9.632 4.031 5.00 2 0.99854 0.00146 2.976 1.609 6.528 10.436 4.234 5.25 1 0.99890 0.00110 3.063 1.658 6.816 11.014 4.377 5.50 2 0.99963 0.00037 3.378 1.705 7.915 13.276 4.910 5 75 0 0 99963 0 00037 3 378 1 749 7 915 13 276 4 9105.75 0 0.99963 0.00037 3.378 1.749 7.915 13.276 4.910 6.00 1 1.00000 0.000 Total 2738

X in Distribution Models : [1], [6] Y in Distribution Models : [5], [7], [8], [9]

Extrapolation

The Exceedence Probability of one event in TRyears :

Extreme Value Analysis

If only few major events are known

Limited number of extreme events Limited number of extreme events

Method

Rank the data in decreasing order Calculate Probability of Exceedence (Q)

1ci i = ranking of the data point

21cNciQ +=

i = ranking of the data point N = total number of points c1 c2 = constants for unbiased plotting position c1, c2 = constants for unbiased plotting position

Calculate Probability (P) Calculate Reduced Variate (z W G) Calculate Reduced Variate (z, W, G)

Constant for Unbiased PlottingConstant for Unbiased Plotting

Distribution c1 c2Normal 0.375 0.375Log Normal 0.250 0.125gGumbel 0.440 0.120Weibull 0 2 + 0 27 0 2 + 0 23Weibull 0.2 + 0.27 0.2 + 0.23

Distribution Models

sH = standard deviation, H h i htH = mean wave height and = Weibull and Gumbel Parameter = lower limit of H = threshold value in a Peak over Threshold data set

Probability Table

z = -3.4 P = 3.37 x 10-4z = +3.4 P = 1- 3.37 x 10-4

Normal Distribution Curve

= 0.999663

In Excel : NORMINVP = 3.37 x 10-4

Example: =NORMIV(J38,0,1)

z =0z = -3.4

1ciQ 1

1

i H Q W i H Q W

21cNciQ +=

1ln

=

QW

i H Q W i H Q W 1 5.95 0.010 6.675 23 4.22 0.505 0.621 2 5.38 0.033 4.642 24 4.21 0.527 0.572 3 5.26 0.055 3.775 25 4.20 0.550 0.526 4 5 03 0 078 3 227 26 4 20 0 572 0 4824 5.03 0.078 3.227 26 4.20 0.572 0.482 5 4.82 0.100 2.832 27 4.17 0.595 0.441 6 4.75 0.123 2.524 28 4.17 0.617 0.402 7 4.71 0.145 2.274 29 4.16 0.640 0.365 8 4.68 0.168 2.064 30 4.16 0.662 0.3308 4.68 0.168 2.064 30 4.16 0.662 0.330 9 4.63 0.190 1.884 31 4.14 0.685 0.297

10 4.54 0.213 1.727 32 4.14 0.707 0.266 11 4.49 0.235 1.588 33 4.13 0.730 0.236 12 4.43 0.258 1.463 34 4.09 0.752 0.208 13 4.40 0.280 1.351 35 4.09 0.775 0.182 14 4.38 0.303 1.250 36 4.08 0.797 0.156 15 4.36 0.325 1.157 37 4.07 0.820 0.133 16 4.35 0.348 1.071 38 4.07 0.842 0.111 17 4.34 0.370 0.993 39 4.06 0.865 0.090 18 4.33 0.393 0.920 40 4.05 0.887 0.071 19 4.29 0.415 0.852 41 4.04 0.910 0.053 20 4.25 0.437 0.788 42 4.04 0.932 0.036 21 4.24 0.460 0.729 43 4.03 0.954 0.022 22 4.23 0.482 0.673 44 4.01 0.977 0.009

Plot Extreme Distribution

7.0

8.0

- W

y=A.x + BA = 3 395

y = 3.395x - 13.704R2 = 0.9954 0

5.0

6.0

d Va

riate

- A = 3.395B = - 13.704

2.0

3.0

4.0

l Red

uced

1 0

0.0

1.0

Wei

bull

-1.04 4.5 5 5.5 6 6.5

Wave Height - H (m)

Weibull Distribution for Ordered Data Set (=0.8).

Plot Extreme Distribution

2.0

2.5y=A.x + BA 5 270

y = 5.270x - 7.180R2 0 9790 5

1.0

1.5

aria

te -

z

A = 5.270

B = - 7.180

R2 = 0.979

-0.5

0.0

0.5

educ

ed V

a

-1.5

-1.0

Re

-2.01.4 1.5 1.6 1.7 1.8 1.9

ln H

Log-Normal Distribution for Ordered Data Set

Plot Extreme Distribution

2 211 8 5914 0

5.0

G

y=A.x + BA 2 211y = 2.211x - 8.591

R2 = 0.992

2 0

3.0

4.0

Var

iate

- A = 2.211B = - 8.591

1.0

2.0

Red

uced

-1.0

0.0

Gum

bel

-2.04 4.5 5 5.5 6 6.5

Wave Height - H (m)

Gumbel Distribution for Ordered Data Set

Distribution Models

sH = standard deviation, H h i htH = mean wave height and = Weibull and Gumbel Parameter = lower limit of H = threshold value in a Peak over Threshold data set

Return Period Prediction

Ret rn Period (Yrs)Weibull

Return Period (Yrs) N 20 50 100 200 44 1.26 0.80 0.29 3.97 5.22 5.68 6.05 6.43

Return Period (Yrs) N 20 50 100 200

GumbelN 20 50 100 20044 1.26 0.45 3.87 5.31 5.73 6.04 6.36

Return Period (Yrs) N Hl s 20 50 100 200

Log NormalN Hln s 20 50 100 200 44 1.26 1.36 0.19 5.44 5.86 6.16 6.45