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NDBC Technical Document 96-01

Nondirectional andDirectional WaveData AnalysisProcedures

Stennis Space CenterJanuary 1996

U.S. DEPARTMENT OF COMMERCE

National Oceanic and Atmospheric Administration

National Data Buoy Center

96-002(1)

96-002(1)

NDBC Technical Document 96-01

Nondirectional andDirectional WaveData AnalysisProcedures

Marshall D. Earle, Neptune Sciences, Inc.150 Cleveland Avenue, Slidell, Louisiana 70458January 1996

U.S. DEPARTMENT OF COMMERCERonald Brown, Secretary

National Oceanic and Atmospheric AdministrationDr. D. James Baker

National Data Buoy CenterJerry C. McCall, Director

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iii96-002(1)

TABLE OF CONTENTSPAGE

1.0 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2.0 PURPOSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3.0 DATA ANALYSIS PROCEDURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 DOCUMENTATION APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 MATHEMATICAL BACKGROUND AND DATA ANALYSIS THEORY . . . . . . . . . . . . . 2

3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.2 Types of Data and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.3 Data Segmenting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2.4 Mean and Trend Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2.5 Spectral Leakage Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2.6 Covariance Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2.7 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.8 Spectra and Cross-Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.9 Spectra Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.10 Directional and Nondirectional Wave Spectra . . . . . . . . . . . . . . . . . . . . . . . . 83.2.11 Wave Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.12 NDBC Buoy Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.12.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.12.2 Nondirectional Wave Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.12.3 Directional Wave Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.12.4 Spectral Encoding and Decoding . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 ANALYSIS OF DATA FROM INDIVIDUAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . 173.3.1 Types of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.2 GSBP WDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.3 DACT WA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.4 DACT DWA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.5 DACT DWA-MO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.6 VEEP WA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.7 WPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.8 Additional Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.0 ARCHIVED RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.0 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.0 ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.0 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

APPENDIX A — DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

APPENDIX B — HULL-MOORING PHASE ANGLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

TABLES

Table 1. Wave Measurement System Data Collection Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 18Table 2. GSBP WDA Data Analysis Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Table 3. DACT WA Data Analysis Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Table 4. DACT DWA Data Analysis Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Table 5. DACT DWA-MO Data Analysis Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Table 6. VEEP WA Data Analysis Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Table 7. WPM Data Analysis Steps1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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1.0 INTRODUCTION

The National Data Buoy Center (NDBC), acomponent of the National Oceanic andAtmospheric Administration's (NOAA) NationalWeather Service (NWS), maintains a network ofdata buoys to monitor oceanographic andmeteorological data off U.S. coasts including theGreat Lakes. NDBC's nondirectional and directionalwave measurements are of high interest to usersbecause of the importance of waves for marineoperations including boating and shipping. NDBC'swave data and climatological information derivedfrom the data are also used for a variety ofengineering and scientific applications.

NDBC began its wave measurement activities inthe mid-1970's. Since then, NDBC has developed,tested, and deployed a series of buoy wavemeasurement systems. Types of systems for whichdata analysis is described in this report have beendeployed since 1979. These systems are generallyrecognized as representing the evolving state-of-the-art in buoy wave measurement systems. AsNDBC's newer systems have been developed, theyhave provided increasingly more comprehensivewave data. NDBC's data buoys relay waveinformation to shore by satellite in real time.Transmission of raw time-series data is not feasiblewith today's satellite message lengths. Thus, muchof the data analysis is performed onboard usingpre-programmed microprocessors. Key analysisresults as well as Data Quality Assurance (DQA)information are relayed to shore. Additional dataprocessing and quality control are performedonshore. Several aspects of NDBC's wave dataanalysis are unique to NDBC because of onboardanalysis and NDBC's detailed consideration of buoyhull-mooring response functions.

This NDBC Technical Document is derived fromcontract deliverable CSC/ATD-91-C-002 which wascompleted in its original form in May 1994. Whileminor editorial changes have been made, thecontent has not been changed. Therefore, anyimprovements or changes in NDBC wavemeasurement systems and data analysisprocedures since May 1994 are not included in thisreport. Appendix A contains definitions andacronyms used in this report.

2.0 PURPOSE

This report's primary purpose is to documentNDBC's wave data analysis procedures in a single

publication for users of NDBC's wave data andresults based on these data. NDBC's methods foranalysis of nondirectional and directional wave datahave been described over many years in paperspublished in the refereed literature, paperspresented at professional society meetings, reports,and internal documents and memoranda. Althoughthese descriptions were correct at the time of theirpreparation, the methods have evolved over timeso that no one document describes NDBC's wavedata analysis procedures. In addition, concisedescriptions with mathematical background areneeded to facilitate technical understanding byNDBC's wave information users.

The U.S. Army Corps of Engineers CoastalEngineering Research Center (CERC) is a majorNDBC sponsor and user. CERC is documentingwave data analysis procedures of organizationsthat provide wave data and analysis results to theirField Wave Gaging Program (FWGP) with theexception of NDBC. Partly because of onboarddata analysis with satellite transmission of resultsand consideration of buoy hull-mooring responsefunctions, NDBC's wave data processing issignificantly different than that of the otherorganizations participating in the FWGP. TheFWGP's wave measurements also are made withdifferent types of sensors (e.g., fixed in situ sensorssuch as pressure sensors and wave orbital velocitysensors). This report's secondary purpose is toaccompany and parallel CERC's documentation tosatisfy CERC's documentation needs for theFWGP.

3.0 DATA ANALYSIS PROCEDURES

3.1 DOCUMENTATION APPROACH

Previous papers, reports, memoranda, and otherpertinent documentation dating from the mid-1970'sto the present were identified and obtained. Thisinformation was reviewed to develop an outline ofNDBC's wave data analysis procedures. Appropriateinform at ion was then extrac ted andincorporated into this report.

The first part of this section provides themathematical background and theory which servesas the scientific basis for NDBC's wave dataanalysis procedures. This information is describedfirst because much of it applies to several NDBCwave measurement systems. Analysis of data fromindividual systems is described next. A later section

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briefly describes archived results that are analysisoutputs.

This report emphasizes NDBC's wave data analysistechniques, rather than wave data collection,instrumentation, or applications. Aspects of wavedata collection are not described except as theydirectly relate to wave data analysis. The mostpertinent references pertaining to analysis that areavailable in the open literature or through NDBCare referenced. Many of these references alsoprovide further detail about NDBC's data collectionand instrumentation. There are numerousreferences that describe applications (e.g.,descriptions of wave conditions during particularevents of meteorological or oceanographic interest)of NDBC's wave data. Unless they provideadditional information about analysis proceduresbeyond that provided in other references, theseapplications oriented references are not referencedin this report.

Because this report accompanies and parallelsCERC's documentation of wave data analysisprocedures used by the FWGP, this report's formatis similar to CERC's corresponding report. Wherewave analysis theory is the same, text for the tworeports is identical or similar.

3.2 MATHEMATICAL BACKGROUND

AND DATA ANALYSIS THEORY

3.2.1 Overview

NDBC's wave data analysis involves application ofaccepted time-series analysis and spectral analysistechniques to time-series measurements of buoymotion. Hull-mooring response function corrections,methods for obtaining buoy angular motions frommagnetic field measurements, corrections for use ofhull-fixed accelerometers on some systems, andencoding-decoding procedures for relay ofinformation by satellite involve techniquesdeveloped by NDBC. Following parts of this sectionprovide the mathematical background and theorywhich applies to analysis of data from NDBC'swave measurement systems. Some techniques,such as calculation of confidence intervals, are notroutinely used by NDBC but may be applied byusers of NDBC's results. Information in this sectionis written to generally apply to analysis of buoy-collected wave data as much as possible so that itwill have application to other buoy wavemeasurements. Thus, specific symbols andmanners in which equations are written may be

different than in some noted references. In severalcases, equations can be re-written or combined indifferent ways to yield the same results. Becausethis section is generally applicable to analysis ofbuoy-collected wave data, information in it does nothave to be repeated in later descriptions ofparticular NDBC wave measurement systems. Thestyle of presenting mathematical background andtheory also parallels the corresponding CERCdocumentation.

3.2.2 Types of Data and Assumptions

Measured nondirectional wave data time seriesconsist of digitized data from a single-axisaccelerometer with its measurement axisperpendicular to the deck of the buoy within whichit is mounted. If NDBC's buoys were perfect slope-following buoys, the accelerometer axis would beperpendicular to the wave surface.

Measured directional wave data time series consistof digitized data representing one of the followingtypes of data sets:

(1) buoy nearly vertical acceleration, pitch, and rollmeasured using a Datawell Hippy 40 sensor,incorporating a nearly vertical stabilizedplatform, and buoy azimuth obtained frommeasurements of the earth's magnetic fieldwith a hull-fixed magnetometer. On somesystems, nearly vertical displacement timeseries are obtained by electronic doubleintegration of nearly vertical acceleration timeseries.

(2) buoy acceleration from a single-axisaccelerometer with its measurement axisperpendicular to the deck of the buoy withinwhich it is mounted as well as buoy pitch, roll,and azimuth obtained from measurements ofthe earth's magnetic field with a hull-fixedmagnetometer. Systems that provide this typeof data set are sometimes calledMagnetometer Only (MO) systems toemphasize that they do not use a DatawellHippy 40 sensor.

For nondirectional wave measurement systems,spectra are calculated directly from measuredacceleration time series. For directional wavemeasurement systems, buoy pitch and roll timeseries are resolved into earth-fixed east-west andnorth-south buoy slope time series using buoyazimuth. Spectra and cross-spectra are calculated

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from these slope time series and acceleration (ordisplacement) time series. Parameters derivedfrom the spectra and cross-spectra providenondirectional and directional wave information,including directional spectra.

Spectral analysis assumes that the measured timeseries represent stationary random processes.Ocean waves can be considered as a randomprocess. For example, two wave records that arecollected simultaneously a few wavelengths apartwould have similar, but not identical, results due tostatistical variability. Similarly, two wave recordsthat are collected at slightly different times, evenwhen wave conditions are stationary, would havesimilar, but not identical, results. For the purpose ofdocumenting NDBC's wave data analysis, arandom process is assumed simply to be one thatmust be analyzed and described statistically. Astationary process is one for which actual (true)values of statistical information (e.g., significantwave height or wave spectra) are time invariant.Wave conditions are not truly stationary. Thus,wave data analysis is applied to time series withrecord lengths over which conditions usually changerelatively little with time.

In applying statistical concepts to ocean waves, thesea surface is assumed to be represented by asuperposition of small amplitude linear waves withdifferent amplitudes, frequencies, and directions.Statistically, individual sinusoidal wave componentsare assumed to have random phase angles (0o to360o). To describe the frequency distribution ofthese wave components, wave spectra areassumed to be narrow. That is, the wavecomponents have frequencies within a reasonablynarrow range. Except for high wave conditions, theassumption of small amplitude linear wave theoryis usually realistic for intermediate and deep waterwaves. It is not generally true for shallow waterwaves, but NDBC's buoys are seldom in shallowwater from a wave theory viewpoint. Regardless,this assumption is well known to provide suitableresults for most purposes. The narrow spectrumassumption, except in some cases of swell, isalmost never strictly true but also provides suitableresults for most purposes. Even though theseassumptions are not always valid, they provide atheoretical basis for wave data analysis proceduresand help establish a framework for a consistentapproach toward analysis.

Kinsman (1965) provides useful philosophical andintuitive descriptions of these concepts. Earle and

Bishop (1984) describe wave analysis proceduresinvolving statistics in an introductory manner.Several papers by Longuet-Higgins (e.g., 1952,1957, 1980) are among the most useful paperswhich describe statistical aspects of ocean waves.Longuet-Higgins, et al. (1963) provide a classicdescription of directional wave data analysis that isused for buoy directional wave measurements.Donelan and Pierson (1983) provide statisticalresults that are particularly useful for significantwave height. Dean and Dalrymple (1984) discusstheoretical and practical aspects of waves withoutemphasis on statistics.

Time-series analysis and spectral analysistechniques that are applied to measured buoymotion data are described in the following parts ofthis section.

3.2.3 Data Segmenting

A measured time series can be analyzed as asingle record or as a number of data segments.Data segmenting with overlapping segmentsdecreases statistical uncertainties (i.e., confidenceintervals). Data segmenting also increases spectralleakage since, for shorter record lengths in eachsegment, fewer Fourier frequencies are used torepresent actual wave frequencies. However, formost wave data applications, spectral leakageeffects are small compared to spectral confidenceinterval sizes.

Several earlier NDBC wave measurement systemsuse data segmenting, while NDBC's newestsystem, the Wave Processing Module (WPM), doesnot. Data segmenting is based on ensembleaveraging of J spectral estimates from 50 percentoverlapping data segments following proceduresadapted from Welch (1967). Estimates from50 percent overlapping segments produce betterstatistical properties for a given frequencyresolution than do estimates from the commonlyused band-averaging method (Carter, et al., 1973)without data segmenting. A measured time series,x(n)t), with N data points digitized at a timeinterval, )t, is divided into J segments of length L.Each segment is defined as x(j,n)t), where jrepresents the segment number, and x indicatesthat values within each segment are the same asthe ones that were in the equivalent part of theoriginal time series. Following are definitions of thesegments

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1 x(1,n) t) ' x(n) t), n ' 0...,L&1

2 x(2,n) t) ' x(n) t), n 'L

2,...,

3L

2& 1

J x(J,n) t) ' x(n) t), n ' (J&1)L

2,...,

(J%1)L

2&1

J '

2 N&L

2

L

W(n) t) '1

21&cos

2Bn

L,0# n # L&1

xw (j,n) t) ' x(j,n) t) ( W(n) t)

where

The number of segments affects confidenceintervals which are discussed later. Not usingsegmenting is equivalent to using one segmentcontaining all data points (L = N, J = 1).

3.2.4 Mean and Trend Removal

Measured buoy motion time series (acceleration,pitch, and roll) as well as calculated east-west andnorth-south buoy slopes (obtained from pitch androll with consideration of buoy azimuth) have small,often near-zero, time-averaged means. Non-zeromean values of pitch and roll may occur due towind and/or current effects on a buoy's hull andsuperstructure. Means of several parameters arecalculated as DQA checks. Means are alsoremoved from buoy acceleration and slope timeseries before further analysis (data segmenting andspectral analysis) to obtain wave spectra.

Seiches, tides, and other long-period waterelevation changes produce negligibly small trendsin the buoy motion time series that are measured.Thus, trends are not calculated or removed.

3.2.5 Spectral Leakage Reduction

Spectral leakage occurs when measured wave datatime series, which represent contributions fromwave components with a nearly continuousdistribution of frequencies, are represented by thefinite number of frequencies that are used forcovariance calculations or Fourier transforms.Mathematically, a data record or segment has beenconvolved with a boxcar window. The data beginabruptly, extend for a number of samples, and thenend abruptly. The boxcar window has the

advantage that all data samples have equal weight.However, the discontinuities at the beginning andthe end introduce side lobes in later Fouriertransforms. The side lobes allow energy to leakfrom the specific frequency at which a spectralestimate is being computed to higher and lowerfrequencies. How to reduce spectral leakage isdiscussed in many time-series analysis textbooks(e.g., Bendat and Piersol, 1971, 1980, 1986; Otnesand Enochson, 1978) and leads to somedisagreement in the wave measurementcommunity.

Wave data are frequently analyzed without spectralleakage reduction. This approach is followed by theWPM. The rationale is that leakage effects areusually small for wave parameters, such assignificant wave height and peak wave period, eventhough spectra may differ somewhat from thosethat would be obtained with use of leakagereduction techniques. Moreover, effects of leakageon spectra are generally far less than spectralconfidence interval sizes. Not reducing leakagealso eliminates the need for later variancecorrections.

A measured wave data time series that has beenprocessed by covariance methods or Fouriertransforms provides estimates of wave componentcontributions at a finite number of frequencies.Different NDBC wave measurement systems thatapply leakage reduction techniques do so in thetime domain or the frequency domain. Thesesystems use Hanning because it is the mostcommonly used method for reducing spectralleakage (e.g., Bendat and Piersol, 1971, 1980,1986; Otnes and Enochson, 1978).

For time domain Hanning, a cosine bell taper overeach data segment is used. The cosine bell windowis given by

where L is the record or data segment length.

The data are multiplied by the tapering function

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c 'F

Fw

F '1

L&1jL&1

n'0

x(j,n) t)2&

jL&1

n'0

x(j,n) t)

2

L

1/2

Fw '1

L&1jL&1

n'0

xw (j,n) t)2&

jL&1

n'0

xw(j,n) t)

2

L

1/2

Sw&xy(m)f) ' 0.25Sxy((m&1))f)

% 0.5 Sxy(m)f)

% 0.25Sxy((m%1))f)

Sw&xy(M)f) ' 0.5Sxy(M&1)f)%0.5 Sxy(M)f)

c 'F

Fw

F ' Sxx1/2

Fw ' Sxx &w1/2.

Rj'1

N&jjN&j

n'1

anan%j&amean2

Saa

(m)f)'2)t R0%2jM&1

j'1

Rjcos

Bjm

M% &1

kRM

where the subscript w indicates that the data havebeen windowed.

Because of Hanning, the windowed data varianceis less than the unwindowed data variance.Frequency domain corrections are made bymultiplying spectral and cross-spectral densities bypairs of standard deviation ratios for each involvedtime series. A standard deviation ratio is given by

where

in which x and xw indicate the time series beforeand after windowing respectively.

Leakage reduction in the frequency domaininvolves weighting frequency components within awindow which is moved through all analysisfrequencies. Because the variance of the data isreduced, a subsequent variance correction is alsomade.

For frequency domain Hanning, spectral estimatesare smoothed using the following equations.

where Sxy is a cross-spectral density (CSD)estimate for time series, x and y, M is the numberof frequencies, )f is the frequency interval, thesubscript w indicates frequency domain windowing,and m = 1, 2, ... (M-1). CSD estimates are definedin the Fourier transform section. If x=y, Sxy

becomes a power spectral density (PSD) estimate,Sxx.

The standard deviation ratios for frequency domaincorrections can be written as

where

As for corrections based on time domaincalculations, windowed spectral densities and CSDsare multiplied by pairs of standard deviation ratiosfor each involved time series. For example, for buoydirectional wave measurements, standarddeviations may be for acceleration (ordisplacement), east-west slope, or north-southslope time series.

3.2.6 Covariance Calculations

The earliest system described in this report is theGeneral Service Buoy Payload (GSBP) Wave DataAnalyzer (WDA) system. This system is the onlydescribed system that determines wave spectra(nondirectional) by covariance (autocorrelation)techniques rather than by use of Fast FourierTransforms (FFT). Spectral calculations viacovariance techniques follow standard procedures(e.g., Bendat and Piersol, 1971, 1986).

For a buoy acceleration record, a covariance(autocorrelation) function for M lags is obtainedfrom

where j is the lag number (1, 2, ... M), an indicatesthe nth acceleration value, N is the number of datapoints per record, and amean is the meanacceleration.

Acceleration spectra are computed from

Spectral estimates can be obtained at frequencies,m)f, where m ranges from 1 to M and )f is givenby

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)f 'fNyquist

M

fNyquist '1

2)t

X(j,m)f) ' )tjL&1

n'0

x(j,n)t)e&i

2Bmn

L

m ' 0,1,2, ...,L

2L even

m ' 0,1,2, ...,L&1

2L odd

Re X j,m)f ')tjL&1

n'0

x j,n)t cos2Bmn

L

Im X j,m)f '&)tjL&1

n'0

x j,n)t sin2Bmn

L

) f '1

L) t

fNyquist '1

2) t

Sxx (j,m) f) 'X ((j,m) f)X(j,m) f)

L) t

'*X(j,m) f)*2

L) t

Sxy(j,m) f) 'X ((j,m) f)Y(j,m) f)

L) t

' Cxy (j,m) f) & iQxy (j,m) f)

and the Nyquist frequency is given by

where )t is the time between data points. TheNyquist frequency is the highest resolvablefrequency in a digitized time series. Spectral energyat frequencies above fNyquist incorrectly appears asspectral energy at lower frequencies.

The GSBP WDA calculates spectral estimates onlyup to m = (2/3)M so that spectra are cutoff at afrequency corresponding to the half-powerfrequency of the onboard low-pass antialiasinganalog filter. Further GSBP WDA data analysisdetails are provided in a following section.

3.2.7 Fourier Transforms

Fourier transforms are calculated by FFT algorithmsthat were tested with known input data. Tests ofsystem hardware also include tests with knownanalog inputs in place of actual sensor signals.Algorithms themselves were implemented indifferent computer languages, for different onboardmicroprocessors, and for different wavemeasurement systems. Thus, the algorithmsthemselves are not identical.

An FFT is a discrete Fourier transform thatprovides the following frequency domainrepresentation, X, of a measured time series, x (orxw with use of a window).

where

The real and imaginary parts of X are given by

Spectral estimates are obtained at Fourierfrequencies, m)f, where the interval betweenfrequencies is given by

Utilized FFT algorithms require that the number ofdata points be a power of two for computationalefficiency so that L is even. The frequencycorresponding to m = L/2 is the Nyquist frequencygiven by

3.2.8 Spectra and Cross-Spectra

PSD estimates for the jth segment are given by

where X* is the complex conjugate of X.

CSD estimates for the jth segment are given by

where Cxy is the co-spectral density (co-spectrum),Qxy is the quadrature spectral density (quadraturespectrum), and X and Y are frequency domainrepresentations of the time series x and y. Theseequations have units of spectral density which arethe multiplied units of each time series divided byHz. Cxy and Qxy can be written as

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Cxy(j,m)f) 'Re[X] Re[Y] % Im[X]Im[Y]

L) t

Qxy(j,m)f) 'Im[X] Re[Y] & Re[X]Im[Y]

L) t

Sxx(m) f) '1

Jj

J

j'1

Sxx(j,m) f)

Cxy(m) f) '1

Jj

J

j'1

Cxy(j,m) f)

Qxy(m) f) '1

Jj

J

j'1

Qxy(j,m) f)

Sxx(f)EDF

P2 EDF,(1.0&")

2

,Sxx(f)EDF

P2 EDF,(1.0%")

2

where the arguments, (j,m)f), of X and Y are notshown for brevity.

Final spectral estimates are obtained by averagingthe results for all segments to obtain

With the definition of Cxy, the equation for Sxx is notneeded. The co-spectra, Cxx and Cyy, are the sameas the power spectra, Sxx and Syy. In most followingsections, the argument, m)f, is dropped and f isused to indicate frequency.

If data segmenting is not used, one segment (J = 1)contains all of the data points in the original timeseries and spectral estimates are at individualFourier frequencies, m)f, with L = N (the totalnumber of data points in the measured time series).Spectral estimates are then band-averaged overgroups of consecutive Fourier frequencies toincrease the degrees of freedom and the statisticalconfidence.

Conventional wave data analysis with band-averaging often uses frequency bandwidths withinwhich one more or one less Fourier frequency mayfall for adjacent bands. This situation occurs whenbandwidths are not multiples of the Fourierfrequency interval. Thus, adjacent bands may haveslightly different degrees of freedom and confidenceintervals. For NDBC wave measurement systemsthat employ band-averaging, Fourier frequencies fallprecisely on boundaries between frequencybands. Spectral and cross-spectral estimates at

boundaries are spread equally into adjacentfrequency bands when band-averaging isperformed.

Frequency-dependent effects caused by themeasurement systems (e.g., hull-mooring, sensor,electronic filtering) are corrected for at this point inthe analysis. Non-zero frequency-dependent phaseshifts have no effect on calculation of nondirectionalwave spectra but may affect directional wavespectra. Nonunity frequency-dependent responseamplitude operators affect both nondirectional anddirectional wave spectra. A following sectiondescribes hull-mooring response functioncorrections which are an important and uniqueaspect of NDBC's wave data analysis.

3.2.9 Spectra Confidence Intervals

Calculated spectra are estimates of the actualspectra. Degrees of freedom describe the number ofindependent variables which determine thestatistical uncertainty of the estimates.

Following standard wave data analysis practice,confidence intervals are defined for power spectra(PSDs), but not for co-spectra and quadrature-spectra (CSDs). Confidence intervals are notprovided by NDBC, but may be calculated by usersof NDBC's wave information.

There is 100 " percent confidence that the actualvalue of a spectral estimate is within the followingconfidence interval

where i2 are percentage points of a chi-squareprobability distribution and EDF are the equivalentdegrees of freedom. As earlier noted, Sxx = Cxx.Ninety percent confidence intervals (" = 0.90) areoften used.

For data processed by Fourier transforms withoutdata segmenting, the degrees of freedom for afrequency band are given by twice the number ofFourier frequencies within the band. Forconsistency, degrees of freedom are referred to asequivalent degrees of freedom to correspond toterminology used for segmented data. Withoutsegmenting, the equivalent degrees of freedom aregiven by

896-002(1)

EDF ' 2nb

EDF '2J

1%0.4(J&1)

J

DF '2N

M

S(f,2) ' C11(f) D(f,2)

S(f) ' m

2B

0

S(f,2) ' C11(f)

S(f,2) 'ao

2% j

2

n'1

[ancos(n2) % bnsin(n2)]

where nb is the number of Fourier frequencies inthe band. In general, nb equals the bandwidthdivided by the Fourier frequency interval.

For data processed by Fourier transforms with datasegmenting, the equivalent degrees of freedom,EDF, for J segments, is given by

where J is the number of overlapping segments.The value 0.4 in the above equation is anapproximation, but is adequate for segment lengths($100 data points) used by NDBC.

For data processed by covariance techniques, theequivalent degrees of freedom are given by

where N is the number of data points and M is thenumber of covariance lags. The degrees offreedom when data are processed by covariancetechniques appear to be larger than the valuebased on the frequency separation interval becauseonly every other spectral estimate is independent(e.g., Bendat and Piersol, 1971).

3.2.10 Directional and Nondirectional Wave

Spectra

A directional wave spectrum provides thedistribution of wave elevation variance as a functionof both wave frequency, f, and wave direction, 2. Adirectional wave spectrum can be written as

where C11 is the nondirectional wave spectrum(which could be determined from a wave elevationtime series if such a time series were available)and D is a directional spreading function.Integration of a directional wave spectrum over alldirections (0 to 2B) provides the correspondingnondirectional spectrum given by

Following NDBC's conventions, subscripts aredefined as follows:

1 = wave elevation (also called displacement)which would correspond to buoy heave aftersensor and buoy hull-mooring responsecorrections

2 = east-west wave slope which would correspondto buoy tilt in this direction after sensor andbuoy hull-mooring response corrections

3 = north-south wave slope which wouldcorrespond to buoy tilt in this direction aftersensor and buoy hull-mooring responsecorrections

The CERC wave data analysis documentation notedearlier and some papers in the scientificliterature use (x, y, and z) subscripts in place of (1,2, and 3) subscripts. NDBC's wave measurementsystems usually measure buoy acceleration ratherthan buoy heave. A superscript m indicates aspectral or cross-spectral estimate based on ameasured time series. Estimates based onbuoy acceleration are converted to estimates basedon wave elevation (displacement) duringsubsequent analysis steps as later described.

Directional spectra are estimated using a directionalFourier series approach originally developed byLonguet-Higgins, et al. (1963). Since itsdevelopment, this approach has been describedand used by many others (e.g., Earle and Bishop,1984; Steele, at al., 1985; Steele, et al., 1992). Ityields directional analysis coefficients that are partof the World Meteorological Organization (WMO)FM65-IX WAVEOB code for reporting spectralwave information (W orld MeteorologicalOrganization, 1988).

The directional Fourier series approach providesthe directional Fourier coefficients, an and bn, in thefollowing Fourier series

which can also be written as

996-002(1)

S(f,2) ' C11(f) D(f,2)

C11 ' B ao

D(f,2)'1

B

1

2%r1cos[2&21]%r2cos[2(2&22)]

r1 '1

ao

a21 % b

21

1

2

r2 '1

ao

a22 % b

22

1

2

21 ' tan&1b1

a1

22 '1

2tan&1

b2

a2

(2B f)2' (gk) tanh(kd)

ao 'C11

B

a1 'Q12

kB

b1 'Q13

kB

a2 '(C22 & C33)

k 2B

b2 '2 C23

k 2B

in which

and the directional spreading function is given by

with

The ambiguity of B for 22 is resolved by choosingthe value that is closest to 21. Numerator anddenominator signs in the inverse tangent functionsdetermine the quadrants of 21 and 22. Theparameter 21 is called mean wave direction, and theparameter 22 is called principal wave direction.These directional parameters (r1, 21, r2, and 22) aremore commonly considered as analysis results thanthe parameters (a1, b1, a2, and b2) originallydeveloped by Longuet-Higgins, et al. (1963). Thelatter parameters are often used as an intermediatecalculation step.

Equations for calculation of the directional-spectrumparameters from buoy wave measurements follow.Calculated directional spectra have units of waveelevation variance/(Hz z- radians). Directionalspectra are thus spectral densities in terms of bothfrequency, Hz, and direction, radians. Co-spectraand quadrature spectra estimates would becorrected for sensor and buoy hull-mooringresponses before use of these equations. As seen

when considerations that are specific to NDBC'sdata processing are described, NDBC combinesresponse corrections, conversion from spectralestimates involving acceleration to those involvingdisplacement, and conversion to the directionalparameters (r1, 21, r2, and 22) into a few equationsthat can be applied without many intermediatesteps.

In the following equations, wavenumber, k, isrelated to frequency, f, by the dispersionrelationship for linear waves given by

where g is acceleration due to gravity, and d ismean water depth during the measurements. For agiven water depth, this equation can be solved for kby iterative methods, or, as later described forNDBC's wave data analysis, k can be estimatedfrom elevation and slope spectra.

For a buoy that measures heave, pitch, roll, andbuoy azimuth (heading) the Longuet-Higginsdirectional parameters are given by

Buoy azimuth is used to convert buoy pitch and rollvalues into buoy east-west and north-south slopes.

For a buoy that measures nearly verticalacceleration, pitch, roll, and azimuth, theparameters are given by

1096-002(1)

ao 'C11

(2Bf )4B

a1 ' &Q12

(2Bf )2 kB

b1 ' &Q13

(2Bf )2 kB

a2 '(C22 & C33)

k 2B

b2 '2 C23

k 2B

" '3B

2& 2

Hmo ' 4.0 mo

where the subscript 1 temporarily indicatesacceleration which is considered positive upwardand negative downward. Signs for the parametersinvolving quadrature spectra are often written aseither positive or negative to consider outputs ofparticular accelerometers used to make themeasurements. For these equations, wave slopesare positive if the slopes tilt upward toward the east(for east-west slopes) or toward the north (fornorth-south slopes).

The direction convention for these equations is thescientific convention (e.g., Longuet-Higgins, et al.,1963) which is the direction toward which wavestravel measured counterclockwise from the x axis(usually east, subscript 2, in NDBC's notation).Theoretically, this is the direction of thewavenumber vector for a particular wavecomponent.

NDBC's direction convention is used by marinersand marine forecasters. It also corresponds to thatof the WMO FM65-IX WAVEOB code (WorldMeteorological Organization, 1988). An NDBCwave direction is the direction from which wavescome measured clockwise from true north. If the xaxis is positive toward the east, an NDBC wavedirection, ", is related to a scientific conventionwave direction, 2, by

This method for estimating a directional wavespectrum is described mathematically as a

directional convolution of a weighting function withthe actual directional spectrum. For the utilizedD(f,2) parameters, the half-power width of theweighting function is 88o. This width is sometimescalled the directional resolution. Longuet-Higgins,et al. (1963) provide a weighting of the directionalFourier coefficients to prevent unrealistic negativevalues of D(f,2) for directions far from 21, but thisapproach is not used by NDBC because itincreases the half-power width to 130o. A directionalspreading function that is determined by thedescribed procedure is a smoothed version of thetrue directional spreading function. Even so,estimated directional spectra are useful. Separatedirections of sea and swell can usually be identifiedsince sea and swell often occur at differentfrequencies.

Higher resolution techniques, most notablyMaximum Entropy Methods (MEM) and MaximumLikelihood Methods (MLM), have been developed.These techniques involve the same cross-spectralvalues and thus could be used. NDBC hasinvestigated versions of these methods (Earle,1993). However, because they are not as widelyused, have several nonstandardized versions, andmay provide erroneous directional informationunless they are carefully applied, they are notpresently used by NDBC except for researchpurposes. Benoit (1994) summarizes and comparesmany higher resolution techniques.

3.2.11 Wave Parameters

NDBC calculates several widely used waveparameters. As an example of use of waveparameters, wave climatologies may be based onstatistical analysis of wave parameters on amonthly, seasonal, or annual basis. As a secondexample, engineering calculations of extreme waveconditions may be based on statistical analysis ofwave parameters near the times of highest wavesduring high wave events.

Significant wave height, Hmo, is calculated from thewave elevation variance which is also the zeromoment, mo, of a nondirectional wave spectrumusing

where mo is computed from

1196-002(1)

mo ' jNb

n'1

C11(fn) dfn

TDF '

2 jN

n'1

C11(fn)

2

jN

n'1

C11(fn)2

Hm0

TDF

P2 TDF,(1.0&")

2

1

2 ,

Hm0

TDF

P2 TDF,(1.0%")

2

1

2

P2(TDF,0.05)•TDF 1&2

9TDF%1.645

2

9TDF

1

2

3

P2(TDF,0.95)•TDF 1&2

9TDF&1.645

2

9TDF

1

2

3

Tp '1

fp

Tav 'mo

m1

in which the summation is over all frequency bands(centered at fn) of the nondirectional spectrum anddfn is the bandwidth of the nth band. NDBC's newestwave measurement system, the WPM, usesfrequency bands that are more narrow at lowfrequencies than at high frequencies.

The theoretical significant wave height, H1/3, is theaverage height of the highest one-third waves in awave record. An assumption for approximatingsignificant wave height by Hmo is that wave spectraare narrow-banded (e.g., Longuet-Higgins, 1952).Although this assumption is not strictly valid foractual waves, considerable wave datarepresentative of different wave conditions showthat this method for determining significant waveheight is suitable for nearly all purposes. Finitespectral width is the likely explanation fordifferences between H1/3 and Hmo with Hmo valuestypically being about 5 to 10 percent greater thanH1/3 values (Longuet-Higgins, 1980).

NDBC does not calculate significant wave heightconfidence intervals, but these can be estimated byusers based on statistical approaches forestimating time-series variance confidence intervals(e.g., Bendat and Piersol, 1980; Donelan andPierson, 1983). Confidence intervals depend on thetotal degrees of freedom, TDF, which in turndepend on spectral width. When a spectrum hasbeen determined, the total degrees of freedom canbe estimated from

The 100 " percent confidence interval for Hmo isgiven by

where i2 values are obtained from chi-squareprobability distribution tables. Ninety percentintervals (" = 0.90) are generally about -10 to+15 percent below and above calculated Hmo

values. These confidence intervals are smaller thanthose for individual spectral density values becausethe variance is based on information over allfrequencies. The following equations provideaccurate results for 90 percent (" = 0.90)confidence intervals when TDF exceeds thirty whichit usually does for waves.

Peak, or dominant, period is the periodcorresponding to the center frequency of the C11

(nondirectional spectrum) spectral frequency bandwith maximum spectral density. That is, peak periodis the reciprocal of the frequency, fp (peakfrequency), for which spectral wave energy densityis a maximum. It is representative of the higherwaves that occurred during the wave record.Confidence intervals for peak period are notdetermined and there is no generally acceptedmethod for doing so. Peak period, Tp, is given by

Average and zero-crossing periods are used lessoften than peak period, but provide importantinformation for some applications. Average waveperiod is not calculated by NDBC, but can becalculated from nondirectional spectra by thefollowing equation

1296-002(1)

Tzero 'mo

m2

1

2

mi ' jnb

n'1

fniC11(fn) dfn i ' 0,1,2

PTF ' R hHR sHR faR fn 2

where Tav is average period. NDBC calculates zero-crossing wave period from the following equation

where Tzero is zero-crossing period. The spectralmoments (mo, m1, and m2) are given by

Tzero is called zero-crossing period because itclosely approximates the time domain mean periodwhich would be obtained from zero-crossinganalysis of a wave elevation record. For thisreason, NDBC and others sometimes refer to Tzero

as mean wave period. Average period and zero-crossing period each represent typical waveperiods rather than periods of higher waves whichare better represented by peak period.

Among those involved in wave data collection andanalysis, there are differences of opinion about thebest spectral-width and directional-widthparameters. Calculation of these parameters is leftto more specialized users of NDBC's waveinformation. Spectral- and directional-widthparameters can be calculated from nondirectionaland directional spectra respectively.

Many applications of NDBC's data are based on asignificant wave height, Hmo, peak wave period, Tp,and a representative wave direction for each datarecord rather than the spectrum. Mean wavedirection, 21 using NDBC's direction convention, atthe spectral peak is most often considered as thebest direction to use as a representative direction. Itis the mean wave direction of the nondirectionalspectrum frequency band with maximum spectraldensity, C11.

NDBC also calculates a wave direction called peak,or dominant, wave direction. This direction is thedirection with the most wave energy in a directionalspectrum, S(f,"), where NDBC's directionconvention is used. NDBC typically computes S(f,")at 5o intervals to estimate peak, or dominant, wavedirection.

3.2.12 NDBC Buoy Considerations

3.2.12.1 Overview

This section documents aspects of NDBC's wavedata analysis that are not simple applications of theprovided mathematical background and theory. Akey unique aspect for determination of NDBC'snondirectional spectra is use of a Power TransferFunction (PTF) to consider sensor, hull-mooring,and data collection filter responses. For directionalspectra, methods for obtaining buoy azimuth, aswell as pitch and roll for some wave measurementsystems, from measurements of the earth'smagnetic field have been developed and applied byNDBC. Directional spectra also require carefuldetermination and application of both the PTF andphase responses associated with eachmeasurement system. Accurate calculation ofdirectional spectra require that frequency-dependent response information be determined foreach measurement time period (i.e., set ofsimultaneous wave records) at each measurementlocation. To reduce quantities of information relayedto shore by satellite message, while preservingaccuracies of relayed information, NDBC developedand uses special encoding-decoding procedures.

3.2.12.2 Nondirectional Wave Spectra

A PTF converts acceleration spectra that aretransmitted to shore to displacement spectra andcorrects for frequency-dependent responses of thebuoy hull and its mooring, the acceleration sensor,and any filters (analog or digital) that were appliedto the data. The DACT Directional Wave Analyzer(DWA) now utilizes output from a nearly verticallystabilized accelerometer, but formerly utilized anelectronically double-integrated output from thisaccelerometer. In the latter case, the PTFconsidered the response of the double integrationelectronics and did not convert from acceleration todisplacement spectra. The PTF is given by

where

RhH = hull-mooring (h) amplitude response forheave (H)

RsH = sensor (s) amplitude response for heave (H)Rfa = analog antialiasing filter (fa), if used,

amplitude responseRfn = digital (numerical) filter (fn), if used,

amplitude response

1396-002(1)

C11 ' Cm

11&NC /PTF Cm

11$NC

C11 ' 0 Cm

11#NC

Bi'bi0%Bez bi1sin(P)&bi2cos(P)sin(R) %

Bey6 &bi2cos(R) sin(A)%

%bi1cos(P)%bi2sin(P)sin(R) cos(A)>

sin(A) 'S

D

cos(A) 'C

D

S'[b21cos(P)%b22sin(P)sin(R)][B1&b10]&

[b11cos(P)&b12sin(P)sin(R)][B2&b20]&

Bez)sin(R)

C'[b22(B1&b10)&b12(B2&b20)]

cos(R)&Bez)sin(P)cos(R)

D ' Bey)cos(P)cos(R)

) ' b11b22 & b12b21

A ' tan&1 sin(A)

cos(A)

RhH includes the factor, radian frequency raised tothe fourth power, for conversion of accelerationspectra to displacement spectra, or the doubleintegration electronics response. NDBC maintainstables of PTF component values for each wavemeasurement system and buoy. Steele, et al.(1985, 1992) also discuss obtaining these values.

The equation for calculation of nondirectional wavespectra is given by

where NC is a frequency-dependent noisecorrection function. Empirical equations for valuesof NC were determined originally using lowfrequencies (e.g., 0.03 Hz) where real wave energydoes not occur (Steele and Earle, 1979; Steele,et al., 1985). Use of fixed accelerometers on somewave measurement systems was demonstrated tobe the main cause of noise (Earle and Bush, 1982;Earle, et al., 1984) and provided better formulationsfor empirical noise corrections (Lang, 1987) whensuch systems are used. For systems with nearlyvertically stabilized accelerometers, Wang andChaffin (1993) developed empirical NC values tocorrect for instrument-related low-frequencyacceleration spectra noise.

3.2.12.3 Directional Wave Spectra

Directional wave spectra involve the additionalmeasurements of buoy azimuth, pitch, and roll.Buoy azimuth (heading of bow axis) is determinedonboard a buoy from measurements of the earth'smagnetic field along axes aligned in the buoy-fixedbow and starboard directions.

The bow (i=1) and starboard (i=2) components ofthe magnetic field vector measured by amagnetometer within a buoy hull are given by

where A, P, and R are, respectively, buoy azimuth,buoy pitch, and buoy roll angles. NDBC'sconventions are that azimuth of the buoy bow ispositive clockwise from magnetic north, pitch ispositive for upward bow motion, and roll is positive

for downward motion of the starboard side of thebuoy. Bey and Bez are, respectively, the horizontaland vertical components of the local earth magneticfield. The other constants correct for the residual(b10, b20) and induced (b11, b22, b12, b21) hullmagnetic fields and can be determined for aparticular hull by proven methods (Steele and Lau,1986; Remond and Teng, 1990).

The equations for Bi (i=1, 2) are two equations intwo unknowns, sin(A) and cos(A). Solutions aregiven by

in which

Azimuth relative to magnetic north is obtained from

A variation (Steele, 1990) of this approach (Steeleand Earle, 1991) is used by NDBC for systems firstemploying the MO method for obtaining buoyazimuth, pitch, and roll. Pitch and roll are obtainedfrom high-frequency parts of measured magneticfield components and used with low-frequencyparts of measured magnetic field components toprovide azimuth. Using total magnetic fieldcomponents in the azimuth part of the calculationsin theory provides some improvement. Differences

1496-002(1)

sin(Atrue)'sin(A)cos(VAR)%cos(A)sin(VAR)

cos(Atrue)'cos(A)cos(VAR)&sin(A)sin(VAR)

zx'sin(Atrue)sin(P)

cos(P)&

cos(Atrue)sin(R)

cos(P)cos(R)

zy'cos(Atrue)sin(P)

cos(P)%

sin(Atrue)sin(R)

cos(P)cos(R)

bi1sin(P)&bi2sin(R)'B

)

i

Bez

sin(P)'b22B

)

1&b12B)

2

)Bez

sin(R)'b21B

)

1&b11B)

2

)Bez

n ' nsH% nh

are small because buoy azimuth motions aremainly at low frequencies.

The azimuth quadrant is determined by the signs ofsin(A) and cos(A). Magnetic azimuth is converted totrue azimuth by use of the magnetic variation angleat the measurement location. Azimuth componentsrelative to true north are obtained from

where Atrue is bow heading relative to true north, Ais bow heading relative to magnetic north, and VARis the magnetic variation angle.

For each data point, azimuth is used to convertpitch and roll to buoy slopes in east-west andnorth-south directions, zx and zy respectively,relative to true north using

NDBC's directional wave measurement systemsobtain buoy pitch and roll in one of two ways. ADatawell Hippy 40 sensor provides direct outputs ofpitch (actually sin(P)) and roll (actually sin(R)cos(P))relative to a nearly vertically stabilized platformwithin the sensor. NDBC also developed atechnique to obtain pitch and roll frommeasurements of magnetic field components madewith a magnetometer (Steele, 1990; Steele andEarle, 1991; Wang, et al., 1994) that is rigidlymounted within a buoy hull. Because the secondmethod utilizes magnetometer data and, indirectly,accelerometer data, and these sensors are onNDBC's directional wave measurement systemsanyway, the second method is often referred to asan MO method.

Data show that hull inertia and a wind fin on mostbuoy superstructures inhibit buoy azimuth motionsat wave frequencies. To obtain buoy pitch and roll,time series of B1 and B2 are first digitally high-passfiltered with the filter cutoff frequency just below the

lowest frequency where nonnegligible wave energyappears in the nondirectional spectrum which iscalculated from acceleration data as earlierdescribed. Considering that variations in sin(A) andcos(A) are negligible above the cutoff frequencyand making small angle approximations for P andR provides

where BNi represents the high-pass filtered magneticfield data. Based on pitch and roll data fromDatawell Hippy sensors, the small angleapproximations are generally realistic. Theseequations for i=1, 2 are two equations in twounknowns, sin(P) and sin(R). Solutions are givenby

As noted by Steele and Earle (1991), these pitchand roll values could differ from true wave-producedpitch and roll values if forces due to winds orcurrents on the hull or mooring also causenonnegligible pitch and roll.

NDBC's directional wave measurement systemsdetermine wave direction information from cross-spectra between buoy acceleration (ordisplacement) and east-west and north-south buoyslopes. Depending on the system, accelerationsmay be measured by a nearly vertically stabilizedaccelerometer or a fixed accelerometer with itsmeasurement axis perpendicular to the buoy deck.Cross-spectra are corrected for sensor and buoyhull-mooring effects as described in this section.Further mathematical detail is provided by Steele,et al. (1985, 1992).

Effects of a buoy hull, its mooring, and sensors areconsidered to shift wave elevation and slope cross-spectra by a frequency-dependent angle, n, toproduce hull-measured cross-spectra. The angle,n,is the sum of two angles,

1596-002(1)

nh' nhH

% nhS

Q12 ' R sH/R h /PTF Q)

12

Q13 ' R sH/R h /PTF Q)

13

R h' R hS/R hH

Q)

12 ' Qm

12cos(n) % Cm

12sin(n)

Q)

13 ' Qm

13cos(n) % Cm

13sin(n)

k 2C11 ' C22 % C33

kR h/R sH' C

m

22 % Cm

33 /Cm

11

1/2

r 1' Q

)

12

2% Q

)

13

2/ C

m

11 Cm

22 % Cm

33

1/2

"1 ' 3B/2 & tan&1 Q)

13,Q)

12

r2 ' Cm

22 & Cm

33

2% 2C

m

23

2 1/2/ C

m

22 % Cm

33

"2' 3B/2 &1

2tan&1 2C

m

23, Cm

22&Cm

33 %either 0

or B,

S f," ' C11 f D f,"

D f," ' 1/2 % r1cos" & "1 % r2cos" & "2 /B

C11s ' C22 % C33 /k 2

where

nsH = an acceleration (or displacement) sensor (s)

phase angle for heave (H)

and

in which

nhH = a hull-mooring (h) heave (H) phase angle

nhS = a hull-mooring (h) slope (S) phase angle

With a phase shift, n, measured co-spectra andquadrature spectra are related to true quadraturespectra, Q12 and Q13, by

where

in which

RhS = hull-mooring (h) amplitude response for buoypitch and roll (same amplitude response asbuoy slopes, S)

and

Steele, et al. (1985) use the following relationshipfrom linear wave theory

to show that

After considerable algebra, this equation combinedwith the quadrature spectra corrected for phaseshifts and the Longuet-Higgins directional Fourierseries formulation for directional spectra yield thefollowing forms for the directional parameters usedto describe directional wave spectra (Steele, et al.,1985, 1992).

whichever makes the angle between "1 and "2smaller.

Directional spectra are then calculated from

where C11(f) is the nondirectional spectrum, D(f,")is a spreading function, f is frequency, and " isdirection, clockwise relative to north, from whichwaves come. The spreading function can be writtenas a Fourier series,

The dispersion relationship is used to providewavenumber, k, and the nondirectional spectrum isalso estimated from the wave slope spectra by

This estimate, C11s, is used as an ad hoc DQA toolsince it should be similar to C11 for parts of spectrawith reasonable wave energy levels.

There are other equivalent ways that thesecalculations can be made and that provide identicalresults. For example, the Longuet-Higginsdirectional Fourier coefficients (a1, b1, a2, and b2)described in the background and theory sectioncould be determined explicitly with consideration ofresponse functions as an intermediate step andused in place of the directional parameters (r1, "1,r2, and "2). The forms of the equations finallyprovided by Steele, et al. (1992) for the directionalparameters are compact and well-suited forNDBC's operational use.

The angle, n, is a critical parameter for obtainingproper wave directions. If n was time-invariant, itcould be determined once for each type of wavemeasurement system. However, as environmentalconditions (e.g., winds, waves, and currents)change, NDBC's data show that phase angleschange somewhat because of environment-induced

1696-002(1)

nh' n & nsH

q ' tanh kd ' T2/gk

R h/q ' gR sH/T2 cm

22 % Cm

33 / cm

11

1/2

forces and tensions placed on an overall hull-mooring system. Considerably better wavedirections are obtained by calculating n as afunction of frequency for each wave record usinglinear wave theory.

According to linear wave theory, quadrature spectrabetween wave elevation and wave slopes are non-zero and co-spectra are zero for directions in whichwaves travel. NDBC's first approach for determiningn used this criteria with weighting based onquadrature spectra involving acceleration (ordisplacement) and both east-west and north-southslopes to consider that quadrature spectra involvingslopes perpendicular to wave directions are small.The theory involves considerable algebra and isfully described by Steele, et al. (1985). Thisapproach generally worked well, but occasionallyprovided mean wave directions in the wrongcompass quadrant.

During 1988-1989, an improved approach fordetermination of n was implemented. Thisapproach is based on the fact that quadraturespectra at each frequency have a maximum valuefor some direction. Steele, et al. (1992) perform acomplicated derivation that shows how n iscalculated with this criterion. This approach'sadvantage is that calculations are based on adirection in which substantial wave energy travelsso that n values are minimally contaminated by lowsignal to noise ratios. Steele, et al. (1992) showthat an ambiguity of B (180o) can be removedthrough use of an initial estimated n value that iswithin B/2 (90o) of the correct value. For each buoystation, NDBC determines initial values as afunction of frequency and adjusts the initial valuesif needed after the first data are acquired andprocessed. Procedures used since 1988-1989 forobtaining hull-mooring phase angles, n, aredescribed in Appendix B.

Because sensor responses (RsH,nsH) are knownand do not vary with time, these responses can beused to estimate hull-mooring effects (whichimplicitly include water depth effects on themooring), separate from sensor effects. nh, whichdepends only on hull-mooring responses, is givenby

which determines nh after n is calculated. To

separate hull-mooring and sensor effects for Rh, theparameter, q, is defined as

where k is wavenumber, d is mean water depthduring a measurement time period, and g isacceleration due to gravity. From small amplitudewave theory, q = 1 in deep water and ranges from 0to 1. Eliminating k between this equation and theequation for kRh/RsH yields

Because q is usually 1 or nearly 1 for NDBC'smeasurement locations, and Rh is order ofmagnitude 1 for NDBC's buoys, Rh/q has aconvenient magnitude.

Estimating (Rh/q) from data was reasonablysuccessful, but division of apparent noise in theslope co-spectra by small values of T

2 led tounsatisfactory results at low frequencies. Betterestimates are obtained by replacing (Cm

22,Cm33) in

the previous equation by noise-corrected values forfrequencies #0.18 Hz. These values are obtainedby subtraction of an empirical frequency-dependentnoise correction function based on (Cm

22,Cm3 3) at

0.03 Hz where there is not real wave energy(Steele, et al., 1992).

Frequency-dependent values of nh and Rh/q

quantify hull-mooring phase and amplituderesponses. As later noted, these parameters areincluded in information that is relayed to shore byNDBC's newest wave measurement system, theWPM.

3.2.12.4 Spectral Encoding and Decoding

Transmitting results by satellite requires minimizingtransmitted record lengths. For systems thattransmit spectral estimates, a nonlinear logarithmicencoding-decoding algorithm is used to do this,rather than a linear algorithm, so that small spectralvalues are accurately obtained in the presence oflarge values at other frequencies. Logarithmicencoding-decoding is also used for some othersatellite-relayed information that may have largeranges of values. Strictly speaking, this encoding-decoding negligibly affects analysis results sincevalues can be decoded with errors of less than±2 percent and usually less than ±1 percent . Theabsolute value of each spectral and cross-spectralestimate is normalized to the maximum absolutevalue (a different value for each type of estimate)and these ratios are encoded. For cross-spectral

1796-002(1)

Iencoded ' 0 value<value1

Iencoded ' "%$ log(value) value1#value<value2

Iencoded ' 2nbit&1 value$value2

" ' 1&$ log(value1)

$ '(2nbit

&2)

log value2 /value1

valuedecoded ' value1expIencoded&0.5

$

*max(%) ' 100value2

value1

C

&1

C ' 2 2nbit&2 &1

estimates that can be negative, signs areseparately transmitted. The encoding equation isgiven by

in which

where Iencoded is an encoded integer, value is theabsolute value to be encoded, nbit is the number ofencoding bits, value1 is the minimum value toencode, and value2 is the maximum value toencode. The maximum absolute value itself isrelayed either as a floating point number or islogarithmically encoded as described with manybits for high accuracy.

The decoding equation is given by

The separately transmitted signs (each requiringonly one bit) are applied to decoded values forencoded information that can be negative.

The maximum percentage error due to encodingand decoding is given by

where

3.3 ANALYSIS OF DATA FROM

INDIVIDUAL SYSTEMS

3.3.1 Types of Systems

NDBC applies aspects of the describedmathematical background and theory to analyzedata from the following wave measurement systemsthat are used at the time of this report.

(1) GSBP WDA

(2) DACT Wave Analyzer (WA)

(3) DACT DWA

(4) DACT DWA-MO

(5) VEEP WA

(6) WPM

where

GSBP = General Service Buoy PayloadDACT = Data Acquisition and Control TelemetryVEEP = Value Engineered Environmental Payload

Table 1 summarizes the most important datacollection parameters for these systems. Segmentlength is not applicable to the GSBP WDA becausecovariance calculations are made for entire datarecords. Segment length also is not applicable tothe WPM because entire data records are Fouriertransformed without segmenting. As noted with thetable, systems that do not use analog and/or digitalfilters rely on hull filtering of high-frequency wavesto avoid aliasing. This approach works wellbecause NDBC buoys are large compared to wavelengths of high-frequency waves that could causealiasing. These waves also have negligible energyfor most applications of NDBC wave information. A3-m-diameter discus buoy has become NDBC'sstandard buoy although data covered by this reporthave been collected by boat-shaped hullsapproximately 6 m in length (NOMAD buoys),10-m-diameter discus buoys, and 12-m-diameterdiscus buoys. Different sampling rates have beenused. The 1.28-Hz, 1.7066-Hz, and 2.56-Hz rateswere selected to place Fourier frequenciesprecisely on frequency band boundaries forsubsequent spectra and cross-spectra calculations.Differences in Table 1 result in negligibledifferences in analysis results for users of NDBCwave information.

1896-002(1)

SYSTEM GSBPWDA

DACT WA DACTDWA

DACTDWA-MO

VEEP WA WPM

SENSOR(S)

FIXEDACC.

FIXEDACC.

HIPPY 40AND3-AXISMAG.

FIXEDACC. AND3-AXISMAG.

FIXEDACC.

HIPPY 40 AND3-AXIS MAG. ORFIXED ACC. AND3-AXIS MAG.

A-TO-DBITS

8 12 12 12 4.5 DIGITDECIMAL

12

RECORDLENGTH

20 MIN(1200 S)

20 MIN(1200 S)

20 MIN(1200 S)

20 MIN(1200 S)

20 MIN(1200 S)

40 MIN(2400 S)

ANALOGFILTER

0.50 HZ 0.50 HZ NONE NONE 0.50 HZ NONE

SAMPLING RATE

1.50 HZ 2.56 HZ 2.00 HZ 2.00 HZ 1.28 HZ 1.7066 HZ

DIGITALFILTER

NONE NONE 0.39 HZ, 1HZSUBSAMPLING

0.39 HZ, 1HZSUBSAMPLING

NONE NONE

SEGMENTLENGTH

N/A 100 S 100 S 100 S 100 S N/A

System abbreviations are defined in the text and the following tables.

A-to-D = analog-to-digital conversion

acc. = accelerometer

mag. = magnetometer. Although a 3-axis magnetometer is used, only bow and starboard components of the measured magnetic field are usedduring data analysis.

Half-power frequency is given for low-pass analog filters.

Hippy 40 = Datawell Hippy 40 that can provide: (1) nearly vertical acceleration, pitch, and roll, or (2) nearly vertical displacement (from electronicdouble integration of acceleration), pitch, and roll.

Systems that do not use analog and/or digital filters rely on hull filtering of high-frequency waves to avoid aliasing.

Table 1. Wave Measurement System Data Collection Parameters

3.3.2 GSBP WDA

Data from these nondirectional wave measurementsystems were first collected operationally in 1979.Although some systems are still used, thesesystems are obsolete and are being phased out asnewer systems are deployed. Table 2 summarizesdata analysis steps in the order that they areperformed. Steele, et al. (1976), Magnavox (1979),and Steele and Earle (1979) provide further detailabout GSBP WDA wave measurement systems.

For each 20-minute acceleration record consistingof 1800 data points sampled at 1.50 Hz, anacceleration covariance (autocorrelation) functionfor M lags (75 for normal data collection) iscalculated by the onboard system. Covariancevalues are relayed to shore along with mean,minimum, and maximum acceleration values.

After DQA checks, acceleration spectra arecomputed onshore by direct summation ofcovariance values and cosine functions. Spectralestimates are obtained at frequencies, m)f, wherem ranges from 1 to 50 and )f = 0.01 Hz. The timebetween samples, )t, is 0.667 s and the Nyquistfrequency is 0.75 Hz. Spectral estimates arecalculated to 0.50 Hz (m = 50) rather than to0.75 Hz (m = 75) so that spectra are cutoff at thehalf-power frequency of the onboard low-passantialiasing analog filter. Frequency domainHanning with variance correction is used to reducespectral leakage. Acceleration spectra are thencorrected for electronic noise and low-frequencynoise due to use of a hull-fixed accelerometer. Buoyhull-mooring and onboard low-pass analog filterresponse functions are applied. Acceleration spectraare converted to displacement spectra by divisionby radian frequency to the fourth power. Thenumber of degrees of freedom (also calledequivalent degrees of freedom, EDF, to correspond

1996-002(1)

Calculation of mean, minimum, and maximum acceleration for entire 20-min (1200-s) record sampled at 1.5 Hz

Time domain calculation of acceleration covariance (autocorrelation) function for M lags incorporating mean removal. M = 75 (normal datacollection), M = 150 (test data collection)

Transmission of acceleration covariances (normally 75), mean, minimum, and maximum values to shore

DQAParity checksComplete message checksMinimum or maximum acceleration out of range

Direct calculation of acceleration spectra from covariances using cosine functions. Nyquist frequency = 0.75 Hz. Spectra cutoff at 0.50Hz corresponding to half-power frequency of onboard low-pass analog filter.

Spectral leakage reduction (Hanning) in frequency domain

Correction of acceleration spectra for electronic noise and noise due to use of fixed accelerometer by subtraction of empirical low-frequency noise correction function. Before approximately 1984, low-frequency noise correction described by Steele and Earle (1979)used. After approximately 1984, low-frequency noise correction described by Earle, et al. (1984) used.

Conversion of acceleration spectra to displacement (elevation) spectra including use of frequency-dependent hull-mooring transferfunction, correction for onboard low-pass analog filter, and division by radian frequency to the fourth power. Frequency bandwidth = 0.01Hz, degrees of freedom = 48 for 75 covariances

Calculation of wave parameters from displacement spectra

Key References: Steele, et al. (1976)Steele and Earle (1979)Earle, et al. (1984)Steele and Mettlach (1994)

Table 2. GSBP WDA Data Analysis Steps

to terminology when data segmenting is used) is 48considering that adjacent spectral estimates are notindependent as noted in the section describingspectral confidence intervals.

The main difference between GSBP WDAs andlater nondirectional wave measurement systems isuse of the covariance technique instead of directFourier transforms for onboard calculation ofspectral estimates.

3.3.3 DACT WA

DACT WA data were first collected operationally in1984. Table 3 summarizes data analysis steps in theorder that they are performed. DACT WAs arenondirectional wave measurement systems. DACTdirectional wave measurement systems aredescribed in the next section.

Each 20-minute buoy acceleration record consistingof 3072 data points is sampled at 2.56 Hz. Datasegmenting with 50 percent overlapping segmentsis used to decrease spectral estimate uncertainties(i.e., confidence intervals). As acceleration data areacquired, the data are stored in two groups of 256data points each so that data points in the first half

of the second group are the same as those in thesecond half of the first group. This approach permitsanalyzing one 256-data-point group as data tocomplete the next group are acquired. Twenty-three256-point records with 50 percent overlap andlengths of 100 s are analyzed. For each segment,the pre-window variance is calculated, minimumand maximums are found, the mean is calculatedand removed, Hanning is performed, and thepostwindow variance is calculated. Next, an FFTprovides frequency domain information from 0.00 to1.28 Hz (Nyquist frequency) with a Fourierfrequency separation of 0.01 Hz, but only spectralestimates for frequencies to 0.40 Hz are calculated.These estimates are corrected for windowing andaveraged over the segments. Consideringsegmenting, spectral estimates have 33 equivalentdegrees of freedom. Minimum and maximumaccelerations for the 20-minute record aredetermined from segment minimums andmaximums.

Acceleration spectra are normalized to themaximum spectral density (i.e., at the peakfrequency), logarithmically encoded, and relayed toshore. Minimum and maximum accelerations arealso relayed and used onshore for DQA. The

2096-002(1)

Segmenting of record (as data acquired with analysis after each segment is complete) into 23 50-percent overlapping segments withlengths of 100 s (256 data points sampled at 2.56 Hz)

Calculation of mean, minimum, and maximum acceleration, mean removal for each segment

Spectral leakage reduction (Hanning) in time domain

FFT

Correction for window use and spectral calculations

Averaging of spectra over segments. Nyquist frequency = 1.28 Hz. Spectra cutoff at 0.40 Hz slightly less than half-power frequency,0.5 Hz, of onboard low-pass analog filter.

Acceleration spectra logarithmic encoding, finding overall minimum and maximum acceleration

Transmission of acceleration spectra, minimum, and maximum values to shore

DQAParity checksComplete message checksMinimum or maximum acceleration out of range

Correction of acceleration spectra for electronic noise and noise due to use of fixed accelerometer by subtraction of empirical low-frequency noise correction function described by Lang (1987).

Conversion of acceleration spectra to displacement (elevation) spectra including use of frequency-dependent hull-mooring transferfunction, correction for onboard low-pass analog filter, and division by radian frequency to the fourth power. Frequency bandwidth = 0.01Hz, equivalent degrees of freedom considering segmenting = 33

Calculation of wave parameters from displacement spectra

Key References: Magnavox (1986)Lang (1987)Steele and Mettlach (1994)

Table 3. DACT WA Data Analysis Steps

spectra are corrected for electronic noise andlow-frequency noise due to use of a hull-fixedaccelerometer. Buoy hull-mooring and onboard low-pass analog filter response functions are applied.Acceleration spectra are converted to displacementspectra by division by radian frequency to thefourth power. Onshore processing is similar to thatfor GSBP WDA data. DACT WA systems arecapable of performing onboard noise and responsefunction corrections as well as conversion ofacceleration spectra to displacement spectra ifappropriate noise and power transfer functions arepre-stored in the systems, but these systems arenot operated in this manner.

3.3.4 DACT DWA

Operational DACT DWA data were first collected in1986. Table 4 summarizes data analysis steps in theorder that they are performed. Steele, et al.(1992) provide detailed descriptions of mostaspects of DACT DWA data collection andprocessing.

Each 20-minute buoy acceleration (ordisplacement) record consisting of 2400 data pointsis sampled at 2.00 Hz. East-west and north-south

buoy slopes are determined from buoy azimuth,pitch, and roll. Acceleration (or displacement) dataas well as slope data are digitally low-pass filteredand subsampled at 1.00 Hz to provide 1200 datapoints. Mean, minimum, and maximum values ofacceleration (or displacement) and the slopes arecalculated and subtracted from each data point ofeach time series. Twenty-three 100-point recordswith 50 percent overlap and lengths of 100 s areanalyzed. For each segment, the pre-windowvariance is calculated, the mean is calculated andremoved, and Hanning is performed. An FFT thenprovides frequency domain information from 0.00 to0.50 Hz (Nyquist frequency for subsampled data)with a Fourier frequency separation of 0.01 Hz.Postwindow variances are calculated from thespectra and corrections for windowing are made.Spectral estimates are averaged over the segmentsso that there are 33 equivalent degrees of freedom.

Spectra and cross-spectra are normalized to theirrespectiv e maximum absolute values,logarithmically encoded, and relayed to shore alongwith the absolute maximum value. Sign informationalso is relayed for cross-spectra that may havenegative values. Maximum encoding-decodingerrors are less than ±1 percent. Nondirectional

2196-002(1)

Calculation of azimuth from pitch, roll, and measured bow and starboard magnetic field components. Acceleration, pitch, and rollprovided by Datawell Hippy 40 sensor.

Conversion of pitch and roll to earth-fixed east-west and north-south wave slopes

Digital low-pass filtering of acceleration (or displacement1) and slope data available at 2-Hz sampling rate and subsampling at 1 Hz

Calculation of mean, minimum, and maximum acceleration (or displacement1) and slope components for entire 20-min (1200-s) record

Segmenting of acceleration (or displacement1) and slope component data into 23 50-percent overlapping segments with lengths of100 s (100 data points)

Mean removal for acceleration (or displacement1) and slope components for each segment

Spectral leakage reduction (Hanning) in time domain for acceleration (or displacement1) and slope components

FFT

Correction for window use (acceleration or displacement and slope components)

Spectral and cross-spectral calculations

Averaging of spectral estimates over segments. Nyquist frequency = 0.50 Hz. Directional spectra cutoff at 0.35 Hz slightly less thanhalf-power frequency, 0.39 Hz, of onboard low-pass analog filter. Nondirectional spectra cutoff at 0.40 Hz.

Transmission of cross-spectra and other information to shore. Other information includes mean, minimum, and maximum values ofbuoy acceleration (or displacement1), buoy pitch, and buoy roll as well as maximum buoy combined pitch and roll tilt angle, original (firstdata point) buoy bow azimuth (heading), and maximum clockwise and counterclockwise deviations from original azimuth.

DQAParity checksComplete message checksParameter minimums or maximums out of range

Correction of acceleration spectra (usually transmitted rather than displacement spectra) for electronic and sensor-related noise usingan empirical noise correction function (Wang and Chaffin, 1993).

Conversion of acceleration spectra to displacement (elevation) spectra (unless displacement originally used) by division by radianfrequency to the fourth power, use of frequency-dependent hull-mooring transfer function, correction for onboard low-pass digital filter.Frequency bandwidth = 0.01 Hz, equivalent degrees of freedom considering segmenting = 33

Calculation of directional wave spectra including corrections for hull-mooring amplitude and phase responses that affect directionalinformation. Techniques described by Steele, et al. (1985) followed before 1988-1989. Techniques described by Steele, et al. (1990)and Steele, et al. (1992) followed after 1988-1989.

Calculation of wave parameters from displacement and directional spectra

Key References: Steele, et al. (1985)Tullock and Lau (1986)Steele, et al. (1990)Steele, et al. (1992)Wang and Chaffin (1993)Steele and Mettlach (1994)

-----------1 The Datawell Hippy 40 onboard sensor can provide either nearly vertically stabilized acceleration or displacement obtained byelectronic double integration of acceleration. Either output has been used for particular buoy installations. Buoy pitch and roll are alsoprovided.

Table 4. DACT DWA Data Analysis Steps

2296-002(1)

spectra (Cm11) are relayed to shore over the

frequency range 0.03 Hz to 0.40 Hz. Directionalspectra parameters (Cm

22, Cm33, Cm

23, Qm12, Cm

12, Qm13,

Cm13) are relayed over the frequency range 0.03 Hz

to 0.35 Hz. Qm23, which is theoretically zero, is not

determined by DACT DWA systems. Nondirectionalacceleration spectra (Cm

11) are corrected forelectronic noise and sensor-related noise using anempirical correction developed by Wang andChaffin (1993). Buoy hull-mooring and onboardlow-pass analog filter response functions are appliedto spectra and cross-spectra through use of the PTFand phase angles described in the theory andbackground section. Acceleration spectra areconverted to displacement spectra by division byradian frequency to the fourth power.

Various information referred to as "housekeeping"information pertains to system setup and onshoreDQA. This information includes mean, minimum,and maximum values of buoy acceleration (ordisplacement), buoy pitch, and buoy roll as well asmaximum buoy combined pitch and roll tilt angle,original (first data point) buoy bow azimuth(heading), and maximum clockwise and counter-clockwise deviations from original azimuth.

3.3.5 DACT DWA-MO

Operational DACT DWA-MO data were firstcollected operationally in 1991. Table 5summarizes data analysis steps in the order thatthey are performed.

These systems are essentially identical to DACTDWA systems except that a fixed accelerometer isused in place of a Datawell Hippy 40 sensor andthat pitch, roll, and azimuth are all calculated fromthe magnetometer data. Low-frequency noisecorrections described by Lang (1987) are used tocorrect for effects of using a fixed accelerometer.Steele (1990) and Steele, et al. (1992) providedetailed descriptions of most aspects of DACTDWA-MO data collection and processing.

3.3.6 VEEP WA

Data from these nondirectional wave measurementsystems were first collected operationally in 1989.Table 6 summarizes data analysis steps in theorder that they are performed.

From a data analysis point of view, these systemsare essentially implementations of DACT WA

procedures on VEEP hardware. DACT systemswere developed in the early 1980's and VEEPsystems were developed in the mid- to late 1980's.The primary difference between DACT WAs andVEEP WAs is that the sampling rates are 2.56 Hzand 1.28 Hz respectively. VEEP WAs thus requireless memory for processing data through useof twenty-three 50-percent overlapping 100-ssegments.

3.3.7 WPM

The WPM represents a major increase in NDBC'swave data analysis capabilities. The WPM consistsof onboard software and a powerful microprocessordedicated to processing wave data that passeswave data analysis results to other onboardsystems for data relay via satellite. The WPM isbeing initially used with VEEP systems. However,it is a software-hardware module that can also beused with future systems that control overall buoyfunctions including satellite data transmission.Earlier onboard software was programmed inlanguages that were not widely used and that werefairly specific to buoy payload microprocessors. Toprovide flexible onboard software that can bemodified or expanded readily, WPM software wasprogrammed in C language (Earle and Eckard,1989). The actual code that operates on a buoycontains some nonstandard C aspects because ofthe utilized C compiler (Chaffin, et al., 1994).WPMs have been successfully field tested, but arenot yet operational (Chaffin, et al., 1994). With theflexibility provided by the onboard C code, changesfrom the following description are likely beforeWPMs become operational.

WPMs can analyze either nondirectional ordirectional wave data. Directional wave dataanalysis is described here since nondirectionalanalysis is a subset of directional analysis.Nondirectional analysis uses only buoy accelerationdata. Table 7 summarizes data analysis steps inthe order that they are performed. WPM dataanalysis aspects are further described by Earle andEckard (1989), Chaffin, et al. (1992), and Chaffin, etal. (1994). In 1989, the prototype WPM softwarewas called the Wave Record Analyzer (WRA)software.

Data from a magnetometer and either a DatawellHippy 40 or a fixed accelerometer can be analyzed.If a Hippy 40 sensor is used, it provides buoynearly vertical acceleration, pitch, and roll. Buoyazimuth is obtained from the magnetometer data.

2396-002(1)

Calculation of pitch and roll from high-frequency parts of measured bow and starboard magnetic field components

Calculation of azimuth from pitch, roll, and low-frequency parts of measured bow and starboard magnetic field components

Conversion of pitch and roll to earth-fixed east-west and north-south wave slopes

Digital low-pass filtering of acceleration and slope data available at 2-Hz sampling rate and subsampling at 1 Hz. Acceleration providedby fixed accelerometer.

Calculation of mean, minimum, and maximum acceleration and slope components for entire 20-min (1200-s) record

Segmenting of acceleration and slope component data into 23 50-percent overlapping segments with lengths of 100 s (100 data points)

Mean removal for acceleration and slope components for each segment

Spectral leakage reduction (Hanning) in time domain for acceleration and slope components

FFT

Correction for window use (acceleration and slope components)

Spectral and cross-spectral calculations

Averaging of spectral estimates over segments. Nyquist frequency = 0.50 Hz. Directional spectra cutoff at 0.35 Hz slightly less thanhalf-power frequency, 0.39 Hz, of onboard low-pass analog filter. Nondirectional spectra cutoff at 0.40 Hz.

Transmission of cross-spectra and other information to shore. Other information includes mean, minimum, and maximum values ofbuoy acceleration, buoy pitch, and buoy roll as well as maximum buoy combined pitch and roll tilt angle, original (first data point) buoybow azimuth (heading), and maximum clockwise and counterclockwise deviations from original azimuth.

DQAParity checksComplete message checksParameter minimums or maximums out of range

Correction of acceleration, spectra for electronic noise using a noise threshold following approach of Steele, et al. (1985)

Conversion of acceleration spectra to displacement (elevation) spectra by division by radian frequency to the fourth power, use offrequency-dependent hull-mooring transfer function, correction for onboard low-pass digital filter. Frequency bandwidth = 0.01 Hz,equivalent degrees of freedom considering segmenting = 33

Calculation of directional wave spectra including corrections for hull-mooring amplitude and phase responses that affect directionalinformation. Techniques described by Steele, et al. (1985) followed before 1988-1989. Techniques described by Steele, et al. (1990)and Steele, et al. (1992) followed after 1988-1989.

Calculation of wave parameters from displacement and directional spectra

Key References: Steele (1990)Steele, et al. (1992)Steele and Mettlach (1994)

Table 5. DACT DWA-MO Data Analysis Steps

2496-002(1)

Segmenting of record (as data acquired with analysis after each segment is complete) into 23 50-percent overlapping segments withlengths of 100 s (128 data points sampled at 1.28 Hz)

Calculation of mean, minimum, and maximum acceleration, mean removal for each segment

Spectral leakage reduction (Hanning) in time domain

FFT

Correction for window use and spectral calculations

Averaging of spectra over segments. Nyquist frequency = 0.64 Hz. Spectra cutoff at 0.40 Hz slightly less than half-power frequency,0.5 Hz, of onboard low-pass analog filter.

Acceleration spectra logarithmic encoding, finding overall minimum and maximum acceleration

Transmission of acceleration spectra, minimum, and maximum values to shore

DQAParity checksComplete message checksMinimum or maximum acceleration out of range

Correction of acceleration spectra for electronic noise and noise due to use of fixed accelerometer by subtraction of empirical low-frequency noise correction function described by Lang (1987).

Conversion of acceleration spectra to displacement (elevation) spectra including use of frequency-dependent hull-mooring transferfunction, correction for onboard low-pass analog filter, and division by radian frequency to the fourth power. Frequency bandwidth = 0.01Hz, equivalent degrees of freedom considering segmenting = 33

Calculation of wave parameters from displacement spectra

Key References: Lang (1987)Chaffin, et al. (1992)Steele and Mettlach (1994)

Table 6. VEEP WA Data Analysis Steps

If a fixed accelerometer is used, the MO techniqueis used to obtain buoy azimuth, pitch, and roll. Inthe applied MO technique (Steele and Earle, 1991),pitch and roll are obtained from high-frequencyparts of measured magnetic field components andused with the total measured components toprovide azimuth. As for other directional wavemeasurement systems, east-west and north-southbuoy slopes are determined from buoy azimuth,pitch, and roll.

Forty minutes of data (4096 data points) areacquired at a sampling rate of 1.7066 Hz.Acceleration and slope data are Fouriertransformed using FFTs for the full data record(4096 data points), the last 20 minutes (2048 datapoints) of the data record, and the last 10 minutes(1024 data points) of the data record. Spectra andcross-spectra are calculated without use of aleakage reduction window. Results based on longer

data records are used for lower frequencies toimprove frequency resolution while maintaining aconstant 24 equivalent degrees of freedom for eachfrequency band. Spectral estimates are obtained atresolutions of 0.005 Hz for frequency bandscentered between 0.0325 Hz and 0.0925 Hz (basedon 4096 data points), 0.01 Hz for frequency bandscentered between 0.1000 Hz and 0.3500 Hz (basedon 2048 data points), and 0.02 Hz for frequencybands centered between 0.3650 Hz and 0.4850 Hz(based on 2048 data points). Spectra and cross-spectra involving acceleration are converted tospectra and cross-spectra involving displacement atindividual Fourier frequencies before band-averaging rather than at the more widely spacedband center frequencies. To reduce the dynamicrange of spectral estimates, band-averaged spectraand cross-spectra are temporarily converted (laterreverse conversion performed onshore) toacceleration spectra using band center frequencies.

2596-002(1)

2696-002(1)

In MO modeCalculation of pitch and roll from high-frequency parts of measured bow and starboard magnetic field components

In Hippy 40 modePitch and roll provided by Hippy 40 onboard sensor

Calculation of azimuth from pitch, roll, and measured bow and starboard magnetic field components

Calculation of mean, minimum, and maximum acceleration, pitch, and roll for entire 40-min (2400-s) record sampled at 1.7066 Hz

Conversion of pitch and roll to earth-fixed east-west and north-south wave slopes

FFT of data for 4096, 2048, and 1024 data point sections each ending with the last data point of the original 4096 data point record.

Cross-spectral calculations using longer record lengths for lower frequencies to obtain finer frequency resolution. Nyquist frequency= 0.8533 Hz.

Transmission to shore of frequency-dependent parameters for calculation of nondirectional and directional spectra and otherinformation. Other information includes mean, minimum, and maximum values as well as standard deviations of buoy pitch, buoy roll,buoy combined pitch and roll tilt angle, bow and starboard magnetic field components, buoy bow azimuth (heading), and buoy east-westand north-south slopes. Also relayed are original (first data point) buoy bow azimuth (heading), maximum clockwise andcounterclockwise deviations from original azimuth, and mean buoy bow azimuth.

DQA is performed using the same procedures that are used for other NDBC directional wave measurement systems.

In MO modeCorrections of acceleration spectra for electronic noise and noise due to use of fixed accelerometers by subtraction ofempirical low-frequency noise correction function are not made but may be implemented later.

In Hippy 40 modeCorrections of acceleration spectra for electronic noise and sensor-related noise are not made but may be implemented later.

Conversion of acceleration spectra to displacement (elevation) spectra by division by radian frequency to the fourth power and use offrequency-dependent hull-mooring transfer function. Frequency bandwidth = 0.005 Hz between 0.03250 and 0.0925 Hz, bandwidth= 0.01 Hz between 0.0100 and 0.3500 Hz, bandwidth = 0.02 Hz between 0.3650 and 0.4850 Hz, degrees of freedom = 24 for allfrequency bands.

Calculation of directional wave spectra from transmitted parameters including corrections for hull-mooring amplitude and phaseresponses that affect directional information. Techniques described by Steele, et al. (1992).

Calculation of wave parameters from displacement and directional spectra

Key References: Earle and Eckard (1989)Steele, and Earle (1991)Steele, et al. (1992)Chaffin, et al. (1992)Chaffin, et al. (1994)Steele and Mettlach (1994)

-----------1 System is under development (initial field tests completed) and analysis details may change. System can also operate innondirectional mode if the only sensor is a fixed accelerometer.

Table 7. WPM Data Analysis Steps1

2796-002(1)

Acceleration spectra are normalized to themaximum value, logarithmically encoded, andrelayed to shore along with the maximum value.The maximum encoding-decoding error is less thantwo percent. As presently configured, a WPMcalculates directional-spectra parameters (r1, "1, r2,and "2) onboard. Values of r1 and r2 arelogarithmically encoded without normalization andrelayed. Values of "1 and "2 are linearly encodedand relayed. Maximum encoding-decoding errorsare less than 1 percent and 1o for r values and "values respectively. The response functions nh andRh/q are also calculated onboard for eachfrequency band. Values of nh are linearly encodedand relayed. Values of Rh/q are logarithmicallyencoded and relayed. Maximum encoding-decodingerrors are less than 0.1 percent and 1o for Rh/qvalues and nh values values respectively.

Nondirectional spectra (Cm11) and directional spectra

parameters (r1, "1, r2, and "2) relayed by WPMs arepart of the WMO WAVEOB code for reportingspectral wave information (World MeteorologicalOrganization, 1988). Following derivations bySteele, it has been shown that the standard cross-spectral parameters between wave displacementand slopes can be derived from the relayedinformation (Earle, 1993). With this capability andthe relayed information (including responseinformation), it is possible to re-perform responsecalculations and/or recalculate directional spectraonshore by other methods.

WPMs transmit considerable "housekeeping"information that pertains to system setup andonshore DQA. Information includes mean,minimum, and maximum values as well as standarddeviations of buoy pitch, buoy roll, buoy combinedpitch and roll tilt angle, bow and starboard magneticfield components, buoy bow azimuth (heading), andbuoy east-west and north-south slopes. Also relayedare original (first data point) buoy bow azimuth(head ing), m axi m um c l oc k w i s e andcounterclockwise deviations from original azimuth,and mean buoy bow azimuth.

3.3.8 Additional Information

Additional, more detailed, information for NDBC'swave measurement systems than that providedhere is not generally needed by NDBC's waveinformation users. Considerable additionalinformation is in the noted key references, butinformation in a given reference may not always

apply for times after the referenced document waswritten. Very detailed information, such as values ofspecific parameters (e.g., for response corrections,encoding-decoding, etc.) would almost never beneeded by users, and would have to be obtainedby NDBC inspection of databases and dataanalysis computer codes that are retained.

4.0 ARCHIVED RESULTS

Results of NDBC's wave data analysis are archivedat and made available to users through theNational Oceanographic Data Center (NODC), acomponent of the NOAA. Significant wave height,peak (dominant) wave period, and peak (dominant)wave direction parameters are also archived atNOAA's National Climatic Data Center (NCDC).Information is available in hardcopy, on magnetictape, and on CD-ROM. NDBC also maintains anarchive for its own use.

The most comprehensive information is availablefrom NODC. Two similar formats, type "191" andtype "291", have been used. Data that are nowbeing acquired are provided in type "291" format.This section summarizes the wave information thatcan be provided by NODC, but does not describedetails of these formats nor the "housekeeping",meteorological, and other information that is alsoprovided in these formats. Format specificationsheets can be obtained from NDBC or NODC.

NDBC wave data are acquired and analyzedhourly. At each measurement location for eachhour that data were collected and passed DQAchecks, the following information is retained andavailable through NODC. This information is alsoprovided on a regular basis to other major userssuch as CERC and the Minerals ManagementService (Department of Interior).

! Significant wave height, Hmo

! Mean (average) wave period, Tzero

! Peak (dominant) wave direction (mean direction,"1, at spectral peak)

! Peak (dominant) wave period, Tpeak

! Center frequency of each spectral band! Bandwidth of each spectral band! Nondirectional spectral density, C11, for each

band! The following cross-spectra for each band

C22, C33, C12, Q12, C13, Q13, C23, Q23 (somesystems), C22-C33

! The following directional Fourier coefficients foreach band — a1, b1, a2, b2

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! The following directional parameters for eachband — r1, "1, r2, "2

! Nondirectional spectral density, C11s, estimatedfrom wave slope spectra for each band

Although directional wave spectra are not directlyavailable through NODC, directional spectra may becalculated from available frequency-dependentparameters.

Less comprehensive results are provided in almostreal time every 3 hours to NWS marine forecasters,NDBC, and others, via NWS communication circuitsafter DQA and onshore processing by the NWSTelecommunications Gateway (NWSTG). Becauseof the almost real-time requirement to meetforecasting needs, NWSTG's DQA is not asextensive as that performed by NDBC forinformation that is permanently archived.Information is distributed by NWSTG in thefollowing two message formats: Automation of FieldOperations and Services (AFOS) spectral wavecode format and the WMO FM65-IX WAVEOB codefor reporting higher resolution spectral waveinformation (World Meteorological Organization,1988). An NDBC user's guide for real-timedirectional wave information (Earle, 1990) describesthe real-time information and the AFOS message indetail. NDBC is updating this user's guide to alsocover WAVEOB messages.

5.0 SUMMARY

Wave data analysis procedures used by NDBC aredocumented. Wave data analysis assumptions arenoted and appropriate mathematical backgroundand theory is provided. Procedures that can beapplied by users of NDBC's wave information toobtain additional useful information (e.g.,confidence intervals) are described. This reportsummarizes NDBC's wave data analysis proceduresin one document. Earlier documentation iscontained in numerous documents that wereprepared as NDBC's wave measurement systemsand procedures evolved since the mid-1970's. Thisreport was developed for users of NDBC's waveinformation, including CERC, but should also be avaluable resource as a general guide for usingbuoys to measure waves.

6.0 ACKNOWLEDGEMENTS

Kenneth E. Steele of NDBC initiated thisdocumentation effort and provided valuable adviceand assistance. Theodore Mettlach and David W.C.

Wang, both of Computer Sciences Corporation,provided various input information and helped trackdown previous wave data analysis details. DavidMcGehee, of CERC, earlier initiated thecorresponding CERC FWGP documentation effort.CERC's documentation provided some of themathematical background and theory material forthis report.

7.0 REFERENCES

Benoit, M., 1994, "Extensive comparison ofdirectional wave analysis methods from gaugearray data," Proceedings Ocean WaveMeasurement and Analysis, WAVES 93Symposium, New Orleans, LA, American Society ofCivil Engineers, New York, NY, pp. 740-754.

Bendat, J.S., and Piersol, A.G., 1971, 1986,Random Data: Analysis and Measurement

Procedures, John Wiley & Sons, New York, NY,407 pp.

Bendat, J.S., Piersol, A.G., 1980, Engineering

Applications of Correlation and Spectral Analysis,John Wiley & Sons, New York, NY.

Carter, G. C., Knapp, C.H., and Nuttall, A.H., 1973,"Estimation of the magnitude-squared coherencefunction via overlapped fast Fourier transformprocessing," IEEE Transactions on Audio and

Electroacoustics, vol. AU-21, No. 4.

Chaffin, J.N., Bell, W., and Teng, C.C., 1992,"Development of NDBC's Wave ProcessingModule," Proceedings of MTS 92, MTS,Washington, D.C. pp. 966-970.

Chaffin, J.N., Bell, W., O'Neil, K., and Teng, C.C.,1994, "Design, integration, and testing of the NDBCWave Processing Module," Proceedings Ocean

Wave Measurement and Analysis, WAVES 93Symposium, ASCE, New York, NY, pp. 277-286.

Childers, D., and Durling, A., 1975, Digital Filtering

and Signal Processing, West Publishing Company,St. Paul, MN.

Dean, R.G., and Dalrymple, R.A., 1984, Water

Wave Mechanics for Engineers and Scientists,Prentice-Hall, Englewood Cliffs, NJ.

Donelan, M., and Pierson, W.J., 1983, "Thesampling variability of estimates of spectra of wind-

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generated gravity waves," Journal of Geophysical

Research, vol. 88, pp. 4381-4392.

Earle, M.D., 1990, "NDBC real-time directionalwave information user's guide," National Data BuoyCenter, Stennis Space Center, MS, 24 pp. andappendices.

Earle, M.D., 1993, "Use of advanced methods ofestimating directional wave spectra from NDBCpitch-roll buoy data," Neptune Sciences Inc. reportfor National Data Buoy Center, Stennis SpaceCenter, MS, 24 pp. and appendices.

Earle, M.D., and Bush, K.A., 1982, "Strapped-downaccelerometer effects on NDBO wavemeasurements," Proceedings of OCEANS 82,

IEEE, New York, NY, pp. 838-848.

Earle, M.D., and Bishop, J.M., 1984, A Practical

Guide to Ocean Wave Measurement and Analysis,Endeco Inc., Marion, MA.

Earle, M.D., Steele, K.E., and Hsu, Y.H.L., 1984,"Wave spectra corrections for measurements withhull-fixed accelerometers," Proceedings ofOCEANS 84, IEEE, New York, NY, pp. 725-730.

Earle, M.D., and Eckard, J.D., Jr., 1989, "WaveRecord Analyzer (WRA) software," MEC SystemsCorp. report for Computer Sciences Corp. andNDBC, National Data Buoy Center, MS, 74 pp.appendices.

Kinsman, B., 1965, Wind Waves Their Generation

and Propagation on the Ocean Surface, Prentice-Hall, Englewood Cliffs, NJ.

Lang, N., 1987, "The empirical determination of anoise function for NDBC buoys with strapped-downaccelerometers," Proceedings of OCEANS 87,IEEE, New York, NY, pp. 225-228.

Longuet-Higgins, M.S., 1952, "On the statisticaldistribution of the heights of sea waves," Journal of

Marine Research, vol. 11, pp. 245-266.

Longuet-Higgins, M.S., 1957, "The statisticalanalysis of a random moving surface," Proceedings

of the Royal Society of London, 249A, pp. 321-387.

Longuet-Higgins, M.S., 1980, "On the distribution ofthe heights of sea waves: some effects ofnonlinearity and finite band width," Journal ofGeophysical Research, 85, pp. 1519-1523.

Longuet-Higgins, M.S., Cartwright, D.E., and Smith,N.D., 1963, "Observations of the directionalspectrum of sea waves using the motions of afloating buoy," Ocean Wave Spectra, Prentice-Hall,Englewood Cliffs, NJ.

Magnavox, 1976, "Technical manual (revised) forthe General Service Buoy Payload (GSBP),"Magnavox Government & Industrial Electronics Co.,Fort Wayne, IN, February 1979, revised August1979.

Magnavox, 1986, "Amendment no. 1 to thetechnical manual for Data Acquisition andTelemetry (DACT)," Magnavox Electronics SystemsCo., Fort Wayne, IN, 26 pp.

Otnes, R.K., and Enochson, L., 1978, Applied TimeSeries Analysis, Interscience, New York, NY.

Remond, F.X., and Teng, C.C., 1990, "Automaticdetermination of buoy hull magnetic constants,"Proceedings Marine Instrumentation 90, pp.151-157.

Steele, K.E., 1990, "Measurement of buoy azimuth,pitch, and roll angles using magnetic fieldcomponents," National Data Buoy Center, StennisSpace Center, MS, 16 pp.

Steele, K.E., Wolfgram, P.A., Trampus, A., andGraham, B.S., 1976, "An operational high resolutionwave data analyzer system for buoys," Proceedings

of OCEANS 76, IEEE, New York, NY.

Steele, K.E., and Earle, M.D., 1979, "The status ofdata produced by NDBO wave data analyzer(WDA) systems," Proceedings of OCEANS 79,IEEE, New York, NY.

Steele, K.E., Lau, J.C., and Hsu, Y.H.L., 1985,"Theory and application of calibration techniques foran NDBC directional wave measurements buoy,"Journal of Oceanic Engineering, IEEE, New York,NY, pp. 382-396.

Steele, K.E., and Lau, J.C., 1986, "Buoy azimuthmeasurements-corrections for residual and inducedmagnetism," Proceedings of Marine Data Systems

International Symposium, MDS 86, MTS,Washington, D.C., pp. 271-276.

Steele, K.E., Wang, D.W., Teng, C., and Lang,N.C., 1990, "Directional wave measurements with

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NDBC 3-meter discus buoys," National BuoyCenter, Stennis Space Center, MS, 35 pp.

Steele, K.E., and Earle, M.D., 1991, "Directionalocean wave spectra using buoy azimuth, pitch, androll derived from magnetic field components,"Journal of Oceanic Engineering, IEEE, New York,NY, pp. 427-433.

Steele, K.E., Teng C., and Wang, D.W.C., 1992,"Wave direction measurements using pitch-rollbuoys," Ocean Engineering, Vol. 19, No. 4,pp. 349-375.

Steele, K.E., and Mettlach, T., 1994, "NDBC wavedata - current and planned," Proceedings Ocean

Wave Measurement and Analysis, WAVES 93Symposium, ASCE, New York, NY, pp. 198-207.

Tullock, K.H., and Lau, J.C., 1986, "Directionalwave data acquisition and preprocessing on remotebuoy platforms," Proceedings of Marine DataSystems International Symposium, MDS 86, MTS,Washington, D.C., pp. 43-48.

Wang, D., and Chaffin, J., 1993, "Development ofa noise-correction algorithm for the NDBC buoydirectional wave measurement system,"Proceedings of MTS 93, MTS, Washington, D.C.,pp. 413-419.

Wang, D.W., Teng, C.C., and Ladner, R., 1994,"Buoy directional wave measurements usingmagnetic field components," Proceedings Ocean

Wave Measurement and Analysis, WAVES 93Symposium, ASCE, New York, NY, pp. 316-329.

Welch, P.D., 1967, "The use of fast Fouriertransform for the estimation of power spectra: amethod based on time averaging over short,modified periodograms," IEEE Transactions on

Audio and Electroacoustics, Vol. AU-15, p. 70-73.

World Meteorological Organization, 1988, Manual

on Codes, Vol. I, International Codes, FM 65-IX,WAVEOB, Geneva, Switzerland.

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APPENDIX A — DEFINITIONS

Azimuth: Buoys are considered to have two axeswhich form a plane parallel to the buoy deck. Oneaxis points toward the "bow" and the other axispoints toward the starboard (righthand) "side".Azimuth is the horizontal angular deviation of thebow axis measured clockwise from magnetic north(subsequently calculated relative to true north).Azimuth describes the horizontal orientation(rotation) of a buoy.

Confidence intervals: Spectra and wave parameters(e.g., significant wave height) have statisticaluncertainties due to the random nature of waves.For a calculated value of a given spectral density orwave parameter, confidence intervals are valuesless than and greater than the calculated valuewithin which there is a specific estimated probabilityfor the actual value to occur.

Cross Spectral-Density (CSD): The CSD representsthe variance of the in-phase components of twotime series (co-spectrum) and the variance of theout-of-phase components of two time series(quadrature spectrum). Phase informationcontained in CSD estimates helps determine wavedirectional information.

DACT: Data Acquisition and Control TelemetryNDBC onboard system.

Deep-water waves: Waves have differentcharacteristics in different water depths. Waterdepth classifications are based on the ratio of d/Lwhere d is water depth and L is wavelength. Wavesare considered to be deep-water waves when d/L> 1/2.

Directional spreading function: At a givenfrequency, the directional spreading functionprovides the distribution of wave elevation variancewith direction.

Directional wave spectrum: A directional wavespectrum describes the distribution of waveelevation variance as a function of both wavefrequency and wave direction. The distribution is forwave variance even though spectra are often calledenergy spectra. For a single sinusoidal wave, thevariance is 1/2 multiplied by wave amplitudesquared and is proportional to wave heightsquared. Units are those of energy density =amplitude2 per unit frequency per unit direction.

Units are usually m2/(Hz-degree) but may bem2/(Hz-radian). Integration of a directional wavespectrum over all directions provides thecorresponding nondirectional wave spectrum.

DWA: Directional Wave Analyzer NDBC onboardsystem.

GSBP: General Service Buoy Payload NDBConboard system.

Intermediate (transitional) water depth waves:Waves have different characteristics in differentwater depths. Water depth classifications are basedon the ratio of d/L where d is water depth and L iswavelength. Waves are considered to beintermediate, or transitional, waves when 1/25 < d/L< 1/2.

Leakage: Leakage occurs during spectral analysisbecause a finite number of analysis frequencies areused while measured time series that are analyzedhave contributions over a nearly continuous rangeof frequencies. Because a superposition of wavecomponents at analysis frequencies is used torepresent a time series with actual wavecomponents at other frequencies, variance mayappear at incorrect frequencies in a wavespectrum.

Mean wave direction: Mean wave direction is theaverage wave direction as a function of frequency.It is mathematically described by 21(f) or "1(f).

Measured time series: A sequence of digitizedmeasured values of a wave parameter is called ameasured time series. For computational efficiencyreasons, the number of data points is equal to 2raised to an integer power (e.g., 1024, 2048, or4096 data points). A measured time series is alsocalled a wave record.

MO Method: An NDBC technique for obtaining buoypitch, roll, and azimuth from measurements of thetwo components of the earth's magnetic fieldparallel to a buoy's deck without use of a pitch/rollsensor.

Nondirectional wave spectrum: A nondirectionalwave spectrum provides the distribution of waveelevation variance as a function of wave frequencyonly. The distribution is for wave variance even

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though spectra are often called energy spectra. Itcan be calculated by integrating a directional wavespectrum over all directions for each frequency.Units are those of spectral density which areamplitude2/Hz (e.g., m2/Hz). Also see directionalwave spectrum.

Nyquist frequency: The Nyquist frequency is thehighest resolvable frequency in a digitizedmeasured time series. Spectral energy atfrequencies above the Nyquist frequency appearsas spectral energy at lower frequencies. TheNyquist frequency is also called the foldingfrequency.

Peak (dominant) wave period: Peak, or dominant,wave period is the wave period corresponding to thecenter frequency of the frequency band with themaximum nondirectional spectral density. Peakwave period is also called the period of maximumwave energy.

Pitch: Buoys are considered to have two axes whichform a plane parallel to the buoy deck. One axispoints toward the "bow" and the other axis pointstoward the starboard (righthand) "side". Pitch is theangular deviation of the bow axis from thehorizontal (tilt of the bow axis).

Power Spectral Density (PSD): See spectraldensity.

Principal wave direction: Principal wave direction issimilar to mean wave direction, but it is calculatedfrom other directional Fourier series coefficients. Itis mathematically described by 22(f) or "2(f).

Random process: Measured wave characteristics(e.g., wave elevation, pressure, orbital velocities)represent a random process in which their valuesvary in a nondeterministic manner over acontinuous period of time. Thus, descriptions ofwave characteristics must involve statistics.

Roll: Buoys are considered to have two axes whichform a plane parallel to the buoy deck. One axispoints toward the "bow" and the other axis pointstoward the starboard (righthand) "side". Roll is theangular deviation of the starboard axis from thehorizontal (tilt of the starboard axis).

Sea: Sea consists of waves which are observed ormeasured within the region where the waves aregenerated by local winds.

Shallow-water waves: Waves have differentcharacteristics in different water depths. Waterdepth classifications are based on the ratio of d/Lwhere d is water depth and L is wavelength. Wavesare considered to be shallow-water waves whend/L < 1/25.

Significant wave height, Hmo: Significant waveheight, Hmo, can be estimated from the variance ofa wave elevation record assuming that thenondirectional spectrum is narrow. The variancecan be calculated directly from the record or byintegration of the spectrum as a function offrequency. Using the latter approach, Hmo is givenby 4(mo)

1/2 where mo is the zero moment ofnondirectional spectrum. During analysis, pressurespectra are converted to equivalent sea surface(elevation) spectra so that these calculations canbe made. Due to the narrow spectrum assumption,Hmo is usually slightly larger than significant waveheight, H1/3, calculated by zero-crossing analysis.

Significant wave height, H1/3: Significant waveheight, H1/3, is the average crest-to-trough height ofthe highest one-third waves in a wave record.Historically, H1/3 approximately corresponds to waveheights that are visually observed. If a wave recordis processed by zero-crossing analysis, H1/3 iscalculated by actual averaging of the highest-oneheights. Hmo is generally used in place of H1/3 sincespectral analysis is more often performed than zero-crossing analysis and numerical wave modelsprovide wave spectra rather than wave records.

Small amplitude linear wave theory: Severalaspects of wave data analysis assume that wavessatisfy small amplitude linear wave theory. Thistheory applies when ak, where a is wave amplitudeand k is wavenumber (2B/wavelength), is small.Wave amplitudes must also be small compared towater depth. Except for waves in very shallow waterand very large waves in deep water, smallamplitude linear wave theory is suitable for mostpractical applications.

Spectral density: For nondirectional wave spectra,spectral density is the wave elevation variance perunit frequency interval. For directional wavespectra, spectral density is the wave elevationvariance per unit frequency interval and per unitdirection interval. Values of a directional ornondirectional wave spectrum have units of spectraldensity. Integration of spectral density values overall frequencies (nondirectional spectrum) or over allfrequencies and directions (directional spectrum)

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mr ' jNb

n'1

(fn)rC11(fn) ) fn

(2B f)2' (gk) tanh(kd)

provides the total variance of wave elevation. Alsosee directional wave spectrum and nondirectionalwave spectrum.

Spectral Moments: Spectral moments are used forcalculations of several wave parameters from wavespectra. The rth moment of a wave elevationspectrum is given by

where )fn is the spectrum frequency band width, fn

is frequency, C11 is nondirectional-spectral density,and Nb is the number of frequency bands in thespectrum.

Stationary: Waves are considered stationary ifactual (true) statistical results describing the waves(e.g., probability distributions, spectra, mean values,etc.) do not change over the time period duringwhich a measured time series is collected. From astatistical viewpoint, this definition is one of stronglystationary. Actual waves are usually not trulystationary even though they are assumed to be sofor analysis.

Swell: Swell consists of waves that have travelledout of the region where they were generated by thewind. Swell tends to have longer periods, morenarrow spectra, and a more narrow spread of wavedirections than sea.

VEEP: Value Engineered Environmental Payloadonboard NDBC system.

WA: Wave Analyzer onboard NDBC system.

Wave amplitude: Wave amplitude is the magnitudeof the elevation of a wave from mean water level toa wave crest or trough. For a hypothetical sinewave, the crest and trough elevations are equal andwave amplitude is one-half of the crest to troughwave height. For actual waves, especially highwaves and waves in shallow water, crests arefurther above mean water level than troughs arebelow mean water level.

Wave crest: A wave crest is the highest part of awave. A wave trough occurs between each wavecrest.

Wave energy: For a single wave, wave energy perunit area of the sea surface is (1/2)Dga2 where D is

water density, g is acceleration due to gravity, anda is wave amplitude. The constant factor, Dg, isusually not considered so that energy or wavevariance is considered as (1/2) a2. Since wavecomponents have different frequencies anddirections, wave energy can be determined as afunction of frequency and direction.

Wave frequency: Wave frequency is 1/(waveperiod). Wave period has units of s so that wavefrequency has units of Hertz (Hz) which is the sameas cycles/s. Radian wave frequency (circularfrequency) is 2Bf where f is frequency in Hz.

Wave height: Wave height is the vertical distancebetween a wave crest and a wave trough. Waveheight is approximately twice the wave amplitude.However, for large waves and waves in shallowwater, wave crests may be considerably furtherabove mean sea level during the time of themeasurements than crests are below mean sealevel. When using a spectral analysis approach,wave heights are calculated from spectra using wellestablished wave statistical theory instead of beingcalculated directly from a measured wave elevationtime series.

Wavelength: Wavelength is the distance betweencorresponding points on a wave profile. It can be thedistance between crests, between troughs, orbetween zero-crossings (mean water levelcrossings during measurement time period).

Wavenumber: Wavenumber is defined as2B/wavelength. Wavenumber, k, and wavefrequency, f, are related by the dispersionrelationship given by

where g is acceleration due to gravity and d ismean water depth during a wave record.

Wave period: Wave period is the time betweencorresponding points on a wave profile passing ameasurement location. It can be the time betweencrests, between troughs, or between zero-crossings(mean sea level crossings). The distribution ofwave variance as a function of wave frequency(1/period) can be determined from spectral analysisso that tracking of individual wave periods fromwave profiles is not necessary when wave spectraare calculated.

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Wave trough: A wave trough is the lowest part of awave. A wave crest occurs between each wavetrough.

WDA: Wave Data Analyzer onboard NDBC system.

WPM: Wave Processing Module NDBC onboardsystem.

Zero-crossing wave period: Zero-crossing waveperiod is the average of the wave periods that

occur in a wave height time-series record where awave period is defined as the time interval betweenconsecutive crossings in the same direction of meansea level during a wave measurement time period.It is also statistically the same as dividing themeasurement time period by the number of waves.An estimate of zero-crossing wave period can becomputed from a nondirectional spectrum. Thisestimate is statistically the same as averaging allwave periods that occurred in the wave record.

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n ' nsH% nh

nh' nhH

% nhS

z2(a) ' z2(0)cos(a) & z3(0)sin(a)

z3(a) ' z2(0)cos(a) & z3(0)sin(a)

Cm

12(a) ' Cm

12(0)cos(a) & Cm

13(0)sin(a)

Qm

12(a) ' Qm

12(0)cos(a) & Qm

13(0)sin(a)

Cm

13(a) ' Cm

12(0)sin(a) % Cm

13(0)cos(a)

Qm

13(a) ' Qm

12(0)sin(a) % Qm

13(0)cos(a)

Q)

12(a)2' Q

m

12(a)2% C

m

12(a)2

Q)

13(a)2' Q

m

13(a)2% C

m

13(a)2

Q)

12(a)2' ½ W & Zcos 2a & b

Q)

13(a)2' ½ W & Zcos 2a & b

W ' Q)

13(0)2% Q

)

12(0)2

z ' X 2% Y 2 1/2

b ' tan&1(Y,X)

X ' Q)

13(0)2& Q

)

12(0)2

Y ' 2 Cm

12(0)Cm

13(0) % Qm

12(0)Qm

13(0)

a0 ' b/2

Cm

12(a0) ' Cm

12(0)sin(a0) % Cm

13(0)cos(a0)

Qm

13(a0) ' Qm

12(0)sin(a0) % Qm

13(0)cos(a0)

APPENDIX B — HULL-MOORING PHASE ANGLES

Effects of a buoy hull, its mooring, and sensors areconsidered to shift wave elevation and slope cross-spectra by a frequency-dependent angle, n, toproduce hull-measured cross-spectra. The angle,n,is the sum of two angles,

where

nsH = an acceleration (or displacement) sensor (s)

phase angle for heave (H)

and

in which

nhH = a hull-mooring (h) heave (H) phase angle

nhS = a hull-mooring (h) slope (S) phase angle

This appendix describes how n is determined foreach frequency from each wave measurementrecord following Steele, et al. (1992). Describedprocedures were implemented for NDBC'sdirectional wave measurement systems during1988-1989. The main text notes references thatdescribe earlier procedures.

The time-varying hull slopes in an earth-fixedframe of reference that has been rotatedclockwise by an angle "a" relative to the true(east(z2(0))/north(z3(0)) frame of reference can bewritten as

If both sides of these equations are transformed tothe frequency domain, and heave/slope co-spectraand quadrature spectra in the rotated frame ofreference are formed, the results are

in which spectra on the righthand sides are thosereported by a buoy. The following definitions

and the previous four spectra equations provide

where

Here, a comma separating numerator anddenominator in the argument of the arc tangentmeans that the signs of Y and X are consideredseparately to place the angle b in the correctquadrant. When a slash (/) is used to separatenumerator and denominator, the angle lies in thefirst or fourth quadrant.

and

QN13(a) has its largest magnitude(s) at either of twoangles, B apart, one of which is given by

Assuming that the best estimate of n each hour foreach frequency band can be obtained by using thevalues of Cm

13(a) and Qm13(a) corresponding to the

earth-fixed frame of reference yielding the largestvalue of QN13(a), these cross-spectra can be writtenas

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C)

13(a0) ' 0 ' &Qm

13(a0)sin(n) % Cm

13(a0)cos(n)

n ' tan&1 Cm

13(a0) / Qm

13(a0) % 0 or B

y ' &Qm

13(a0)sin n̄ % Cm

13(a0)cos n̄

x ' %Qm

13(a0)cos n̄ % Cm

13(a0)sin n̄

n ' n̄ % tan&1 y/x

If Cm13(a0) and Qm

13(a0) are used with

to determine n, the result is given by

To remove the B ambiguity, the n frame ofreference is first rotated by an angle n¯ (f), which isthe best available estimate of n for each frequency.This estimate is determined from hull-mooringmodels, or from previously measured data thathave been supported by hull-mooring models. From

Cm13(a0) and Qm

13(a0), the new co-spectra andquadrature spectra are given by

With these values and if the angle n¯ is within (B/2)of the correct angle, the correct angle is given by

The ratio (y/x) is computed first, so that n is forcedto lie within ±(B/2) degrees of n¯ , the best estimate.For each station with a directional wavemeasurement system, a frequency-dependent tableof n

¯ is maintained because of station-to-stationdifferences. The equation for n is used to estimaten for each wave record.