{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:43:31Z","timestamp":1760114611615,"version":"build-2065373602"},"reference-count":67,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T00:00:00Z","timestamp":1726617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia\u2014FCT","doi-asserted-by":"publisher","award":["CIRCNA. OCT.0300.2019_1801P.01023","SFRH\/BD\/143567\/2019","NA20OAR4320472"],"award-info":[{"award-number":["CIRCNA. OCT.0300.2019_1801P.01023","SFRH\/BD\/143567\/2019","NA20OAR4320472"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"name":"FCT","award":["CIRCNA. OCT.0300.2019_1801P.01023","SFRH\/BD\/143567\/2019","NA20OAR4320472"],"award-info":[{"award-number":["CIRCNA. OCT.0300.2019_1801P.01023","SFRH\/BD\/143567\/2019","NA20OAR4320472"]}]},{"name":"Cooperative Institute for Marine and Atmospheric Studies (CIMAS)","award":["CIRCNA. OCT.0300.2019_1801P.01023","SFRH\/BD\/143567\/2019","NA20OAR4320472"],"award-info":[{"award-number":["CIRCNA. OCT.0300.2019_1801P.01023","SFRH\/BD\/143567\/2019","NA20OAR4320472"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["JMSE"],"abstract":"<jats:p>The uncertainty in estimating the wave spectrum from the records of wave elevation by heave\u2013pitch\u2013roll buoys is studied, considering the effects of the estimation method and the spectral resolution adopted in the process. This investigation utilizes measurements from a wave buoy moored in deep water in the South Atlantic Ocean. First, the spectra are computed using the autocorrelation function and the direct Fourier method. Second, the spectral resolution is tested in terms of degrees of freedom. The degrees of freedom are varied, and the resulting spectra and integrated parameters are computed, showing significant variability. A simple and robust methodology for determining the wave spectrum is suggested, which involves calculating the average energy density in each frequency band. The results of this methodology reduce the variability of the estimated parameters, improving overall accuracy while preserving frequency resolution, which is crucial in complex sea states. Additionally, to demonstrate the feasibility of the implemented approach, the final spectrum is fitted using an empirical model ideal for that type of spectrum. Finally, the performance and the goodness of the fit process for the final averaged curve are checked by widely used statistical metrics, such as R2 = 0.97 and root mean square error = 0.49.<\/jats:p>","DOI":"10.3390\/jmse12091666","type":"journal-article","created":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T09:43:13Z","timestamp":1726652593000},"page":"1666","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Uncertainty of Wave Spectral Shape and Parameters Associated with the Spectral Estimation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2977-5806","authenticated-orcid":false,"given":"Guilherme","family":"Clarindo","sequence":"first","affiliation":[{"name":"Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9268-3552","authenticated-orcid":false,"given":"Ricardo M.","family":"Campos","sequence":"additional","affiliation":[{"name":"Cooperative Institute for Marine and Atmospheric Studies (CIMAS), University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149, USA"},{"name":"Atlantic Oceanographic and Meteorological Laboratory (AOML), National Oceanic and Atmospheric Administration, 4301 Rickenbacker Causeway, Miami, FL 33149, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8570-4263","authenticated-orcid":false,"given":"Carlos","family":"Guedes Soares","sequence":"additional","affiliation":[{"name":"Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/0167-4730(90)90052-Q","article-title":"Stochastic Models of Load Effects for the Primary Ship Structure","volume":"8","year":"1990","journal-title":"Struct. Saf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/S0951-8339(99)00025-8","article-title":"On the Uncertainty in Long Term Predictions of Wave Induced Loads in Ships","volume":"12","year":"1999","journal-title":"Mar. Struct."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/S0141-1187(02)00006-8","article-title":"Modelling Bivariate Distributions of Significant Wave Height and Mean Wave Period","volume":"24","author":"Ferreira","year":"2002","journal-title":"Appl. Ocean Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1016\/S0029-8018(98)00022-5","article-title":"Modelling the long-term distribution of significant wave height with the Beta and Gamma models","volume":"26","author":"Ferreira","year":"1999","journal-title":"Ocean. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"104260","DOI":"10.1016\/j.coastaleng.2022.104260","article-title":"Spectral wave energy period and peak period statistics concomitant with maximum significant wave heights","volume":"183","author":"Muraleedharan","year":"2023","journal-title":"Coast. Eng."},{"key":"ref_6","unstructured":"Guedes, S. (2018). Robust estimation and representation of climatic wave spectrum. Progress in Maritime Technology and Engineering\u2014Martech 2018, Taylor & Francis Group."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"104255","DOI":"10.1016\/j.coastaleng.2022.104255","article-title":"Climatic directional wave spectra in coastal sites","volume":"180","author":"Lucas","year":"2023","journal-title":"Coast. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4381","DOI":"10.1029\/JC088iC07p04381","article-title":"The sampling variability of estimates of spectra of wind-generated gravity waves","volume":"88","author":"Donelan","year":"1983","journal-title":"J. Geophys. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1061\/(ASCE)0733-950X(1985)111:4(603)","article-title":"Distortions associated with random sea simulators","volume":"111","author":"Medina","year":"1985","journal-title":"J. Waterw. Port Coast. Ocean Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1061\/(ASCE)0733-950X(1986)112:2(338)","article-title":"Probability distribution of spectral integrals","volume":"112","author":"Young","year":"1986","journal-title":"J. Waterw. Port Coast. Ocean Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1061\/(ASCE)0733-950X(1987)113:1(77)","article-title":"Bias of effective degrees of freedom of a spectrum","volume":"113","author":"Elgar","year":"1987","journal-title":"J. Waterw. Port Coast. Ocean Eng."},{"key":"ref_12","first-page":"509","article-title":"Directional spectra of wind-generated waves","volume":"315","author":"Donelan","year":"1985","journal-title":"Phil. Trans. Roy. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1115\/1.2828828","article-title":"Observations of the High Frequency Range of the Wave Spectrum","volume":"118","author":"Prevosto","year":"1996","journal-title":"J. Offshore Mech. Arct. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/S0141-1187(99)00007-3","article-title":"A Criterion for the Automatic Identification of Multimodal Sea Wave Spectra","volume":"21","author":"Rodriguez","year":"1999","journal-title":"Appl. Ocean Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/S0141-1187(99)00014-0","article-title":"Uncertainty in the Estimation of the Slope of the High Frequency Tail of Wave Spectra","volume":"21","author":"Rodriguez","year":"1999","journal-title":"Appl. Ocean Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/0167-4730(90)90010-M","article-title":"Uncertainties in data for the offshore environment","volume":"7","author":"Hagen","year":"1990","journal-title":"Struct. Saf."},{"key":"ref_17","unstructured":"Veritas, D.N. (2017). DNVGL-RPC205 recommended practice: Environmental conditions and environmental loads. Tech. Rep., 259."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.oceaneng.2014.05.002","article-title":"Some uncertainties associated with wind and waves description and their importance for engineering applications","volume":"86","author":"Ewans","year":"2014","journal-title":"Ocean Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"103217","DOI":"10.1016\/j.marstruc.2022.103217","article-title":"Uncertainties in long-term wave modelling","volume":"84","author":"Waseda","year":"2022","journal-title":"Mar. Struct."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1643","DOI":"10.1007\/s10236-014-0768-8","article-title":"Effect of intrinsic and sampling variability on wave parameters and wave statistics","volume":"64","author":"Magnusson","year":"2014","journal-title":"Ocean Dyn."},{"key":"ref_21","first-page":"10","article-title":"Uncertainty of the sea state parameters resulting from the methods of spectral estimation","volume":"26","author":"Machado","year":"1999","journal-title":"Ocean Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0378-3839(77)90004-7","article-title":"The stability of some currently used wave parameters","volume":"1","author":"Rye","year":"1977","journal-title":"Coast. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1363","DOI":"10.1029\/JC082i009p01363","article-title":"Statistical distribution of periods and heights of ocean waves","volume":"82","author":"Chakrabarti","year":"1977","journal-title":"J. Geophys. Res."},{"key":"ref_24","unstructured":"Arhan, M.F. (1979, January 12\u201323). Analysis of the variability of spectral estimation using long continuous wave records. Proceedings of the International Conference on Sea Climatology, Paris, France."},{"key":"ref_25","unstructured":"Cavani\u00e9, A.G. (1979, January 12\u201323). Evaluation of the standard error in the estimation of mean and significant wave heights as well as mean period from records of finite length. Proceedings of the International Conference on Sea Climatology, Paris, France."},{"key":"ref_26","unstructured":"Mansard, E.P.D., and Funke, E.R. (1986). On the statistical variability of wave parameters. Technical Report, TR-HY-015, NRCC, Hydraulics Laboratory."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1016\/0029-8018(95)00002-3","article-title":"The determination of confidence limits associated with estimates of the spectral peak frequency","volume":"22","author":"Young","year":"1995","journal-title":"Ocean Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/0029-8018(83)90015-X","article-title":"On the stability of the moments of the maximum entropy wind wave spectrum","volume":"10","year":"1983","journal-title":"Ocean Eng."},{"key":"ref_29","unstructured":"Rodr\u00edguez, G.R., Grisol\u00eda, D., and D\u00edaz, M. (1992, January 18\u201320). On the statistical variability of some spectral bandwidth and non-linearity parameters of wind generated gravity waves. Proceedings of the 6th International Symposium on Stochastic Hydraulics, Taipei, Taiwan."},{"key":"ref_30","unstructured":"Gomes, D., and Guedes Soares, C. (1997, January 13\u201317). Estimation of directional spectrum by maximum entropy method. Proceedings of the Offshore Mechanics and Arctic Engineering, Yokohama, Japan."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"103670","DOI":"10.1016\/j.coastaleng.2020.103670","article-title":"Numerical investigation of harbor oscillations induced by focused transient wave groups","volume":"158","author":"Gao","year":"2020","journal-title":"Coast. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"117941","DOI":"10.1016\/j.oceaneng.2024.117941","article-title":"Influences of Bragg reflection on harbor resonance triggered by irregular wave groups","volume":"305","author":"Gao","year":"2024","journal-title":"Ocean Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"102500","DOI":"10.1016\/j.apor.2020.102500","article-title":"Impact of heavy biofouling on a nearshore heave-pitch-roll wave buoy performance","volume":"107","author":"Campos","year":"2021","journal-title":"Appl. Ocean Res."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Ochi, M.K. (1998). Ocean Waves the Stochastic Approach, Cambridge University Press.","DOI":"10.1017\/CBO9780511529559"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Goda, Y. (2000). Random Seas and Design of Maritime Structures. Advanced Series on Ocean Engineering, World Scientific Publishing Co.","DOI":"10.1142\/3587"},{"key":"ref_36","first-page":"119","article-title":"Sea waves and quasi-determinism of rare events in random processes","volume":"76","author":"Boccotti","year":"1984","journal-title":"Proc. Acad. Ligure Di Sci. E Lett. Ser. VIII LXXVI"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/0029-8018(85)90014-9","article-title":"On the distribution of crest-to-trough wave heights","volume":"12","author":"Naess","year":"1985","journal-title":"Ocean Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/0141-1187(89)90024-2","article-title":"The statistical distribution of wave heights in a random seaway","volume":"11","author":"Vinje","year":"1989","journal-title":"Appl. Ocean Res."},{"key":"ref_39","unstructured":"Boccotti, P. (2000). Wave Mechanics for Ocean Engineering, Elsevier Science."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1061\/(ASCE)0733-950X(1990)116:6(686)","article-title":"Distribution of large wave heights","volume":"116","author":"Tayfun","year":"1990","journal-title":"J. Waterw. Port Coast. Ocean Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apor.2004.08.004","article-title":"Identification of the Components of Wave Spectra by Hilbert Huang Transform Method","volume":"26","author":"Veltcheva","year":"2004","journal-title":"Appl. Ocean Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1061\/9780872620490.013","article-title":"Confidence intervals for ocean wave spectra","volume":"1972","author":"Borgman","year":"1972","journal-title":"Proc. 13th Conf. Coast. Eng."},{"key":"ref_43","unstructured":"Priestley, M.B. (1981). Spectral Analysis and Time Series, Academic Press."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1098\/rspa.1956.0173","article-title":"The Statistical Distribution of the Maxima of a Random Function","volume":"237","author":"Cartwright","year":"1956","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2688","DOI":"10.1029\/JC080i018p02688","article-title":"On the Joint Distribution of Wave Periods and Amplitudes of Sea Waves","volume":"80","year":"1975","journal-title":"J. Geophys. Res."},{"key":"ref_46","first-page":"241","article-title":"On the joint distribution of wave periods and amplitudes in a random wave field","volume":"389","year":"1983","journal-title":"Proc. R. Soc."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1038\/161686a0","article-title":"Smoothing Periodograms from Time Series with Continuous Spectra","volume":"161","author":"Bartlett","year":"1948","journal-title":"Nature"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1109\/TAU.1967.1161901","article-title":"The use of fast Fourier transforms for the estimation of power spectra: A method based on time averaging over short, modified periodograms","volume":"15","author":"Welch","year":"1967","journal-title":"IEEE Trans. Audio Electroacoust."},{"key":"ref_49","first-page":"88","article-title":"Discussion on the symposium on autocorrelation in time series","volume":"8","author":"Daniell","year":"1946","journal-title":"J. Roy. Statist. Soc."},{"key":"ref_50","unstructured":"Kay, S.M. (1988). Modern Spectral Estimation, Prentice Hall."},{"key":"ref_51","unstructured":"Yuen, C.K., and Fraser, D.A. (1979). Digital Spectral Analysis, Pitman."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"5185","DOI":"10.1029\/JZ069i024p05181","article-title":"A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii","volume":"69","author":"Pierson","year":"1964","journal-title":"J. Geophys. Res."},{"key":"ref_53","first-page":"95","article-title":"Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP)","volume":"12","author":"Hasselmann","year":"1973","journal-title":"Dtsch. Hydro. Z."},{"key":"ref_54","first-page":"899","article-title":"On the methodology of selecting design wave height","volume":"21","author":"Goda","year":"1981","journal-title":"Coast. Eng. Proc."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1017\/S0022112085002221","article-title":"Spectral and physical properties of the equilibrium range in wind-generated waves","volume":"156","author":"Phillips","year":"1985","journal-title":"J. Fluid Mech."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.apor.2009.06.001","article-title":"Modelling of Multipeaked Directional Wave Spectra","volume":"31","author":"Boukhanovsky","year":"2009","journal-title":"Appl. Ocean Res."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/0029-8018(84)90019-2","article-title":"Representation of Double-Peaked Sea Wave Spectra","volume":"11","year":"1984","journal-title":"Ocean Eng."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1016\/j.oceaneng.2015.08.017","article-title":"On the modelling of swell spectra","volume":"108","author":"Lucas","year":"2015","journal-title":"Ocean Eng."},{"key":"ref_59","unstructured":"Titov, L.F. (1969). Wind-Driven Waves, Israel Program for scientific Translations."},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Thompson, E.F. (1980). Energy spectra in shallow US coastal waters. Tech. Pap., 80\u201382.","DOI":"10.5962\/bhl.title.47478"},{"key":"ref_61","unstructured":"Cumming, W.E., Bales, S.L., and Gentile, D.M. (1981, January 24\u201328). Hindcasting waves for engineering applications. Proceedings of the International Symposium on Hydrodynamics in Ocean Engineering, Trondheim, Norway."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1115\/1.2919981","article-title":"Spectral Modelling of Sea-States with Multiple Wave Systems","volume":"114","author":"Nolasco","year":"1992","journal-title":"J. Offshore Mech. Arct. Eng."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Ponce de Leon, S., and Guedes Soares, C. (2021). Numerical modelling of the effects of the Gulf Stream on the wave characteristics. J. Mar. Sci. Eng., 9.","DOI":"10.3390\/jmse9010042"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"112333","DOI":"10.1016\/j.oceaneng.2022.112333","article-title":"Numerical study of the effect of current on waves in the Agulhas Current Retroflection","volume":"264","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1115\/1.1576816","article-title":"Probability distributions of wave heigths and periods in measured sea-states from the Portuguese coast","volume":"125","author":"Carvalho","year":"2003","journal-title":"J. Offshore Mech. Arct. Eng."},{"key":"ref_66","unstructured":"Torsethaugen, K. (1996). Model for Double Peaked Wave Spectrum, SINTEF Civil and Environmental Engineering. Rep. No. STF22 A96204."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/0029-8018(91)90040-W","article-title":"On the Occurrence of Double Peaked Wave Spectra","volume":"18","year":"1991","journal-title":"Ocean Eng."}],"container-title":["Journal of Marine Science and Engineering"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2077-1312\/12\/9\/1666\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:58:32Z","timestamp":1760111912000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2077-1312\/12\/9\/1666"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,18]]},"references-count":67,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["jmse12091666"],"URL":"https:\/\/doi.org\/10.3390\/jmse12091666","relation":{},"ISSN":["2077-1312"],"issn-type":[{"type":"electronic","value":"2077-1312"}],"subject":[],"published":{"date-parts":[[2024,9,18]]}}}