Forecasts Of Significant Wave Height Research Articles

Model validation and applications of wave and current forecasts from the Hong Kong Observatory's Operational Marine Forecasting System

The Hong Kong Observatory has been running an Operational Marine Forecasting System (OMFS) adapted from the Regional Ocean Modelling System (ROMS) coupled with the WaveWatch III and SWAN wave models to provide wave, current and sea temperature forecasts up to 144 h twice a day since December 2021. To facilitate users’ interpretation of model forecasts of significant wave height and current speed in coastal predictions and open seas which are of particular significance in high wind situations, model forecasts were validated against moored buoy observations and wave recorder measurements near the shores of Hong Kong and drifting buoy data over the South China Sea, as well as Mercator Ocean model reanalysis in 2022. The validation results showed that the wave forecasts generally agreed well with the buoy observations with coefficient of determination (R2) of around 0.7 and root-mean-square error (RMSE) of less than 0.2 m up to 72 h ahead. The R2 for sea current forecasts ranged between 0.4 and 0.6, and the RMSE was around 8 to 11 cm/s in near shores up to T + 144 forecast hours. Validation against drifting buoy demonstrated that the trend of current forecasts generally agreed well with the measurements. RMSE of surface current forecasts over open seas ranged from 19 cm/s for 24-hour forecast to around 30 cm/s for 144-hour forecast when compared against Mercator Ocean reanalysis. Results from the current downscaling approach could serve as a benchmark reference for HKO to enhance OMFS in the future. In this paper, applications of model forecasts in the provision of marine weather services in Hong Kong are also introduced.

Multivariate GRU and LSTM models for wave forecasting and hindcasting in the southern Caspian Sea

Reliable wave forecasts and hindcasts are pivotal elements for a wide range of marine activities, coastal engineering applications, renewable energy and wave climate studies. This study explores capability of recurrent neural networks for wave hindcasting and multi-step wave forecasting in the southern Caspian Sea, a region with a limited observational data. Gated recurrent unit (GRU) and long short-term memory (LSTM) models are fed by buoy records for wave forecasting up to 6 h ahead and ERA5 reanalysis data as the model inputs for wave hindcasting. Special efforts are devoted to improve the accuracy of the models by including some additional features (e.g., wave age and swell) in the input layer. Different error measures are employed to evaluate performance of the models at two stations, Amirabad and Anzali. Using multivariate GRU and LSTM models, suitable wave forecasts and hindcasts are achieved for the locations. Although the model performance deteriorates with extending the forecasting time horizon, the forecasts of significant wave height (Hs) up to 6 h ahead are still satisfactory. The best hindcasting model yields a high linear correlation between hindcasted and measured Hs with R2 = 0.75 for Anzali and R2 = 0.83 for Amirabad. Finally, wave age and significant height of total swell (Hs-swell) can be considered as tentative input variables to improve efficiency of the models.

A deep-learning real-time bias correction method for significant wave height forecasts in the Western North Pacific

Significant wave height (SWH) is one of the most important parameters characterizing ocean waves, and accurate numerical ocean wave forecasting is crucial for coastal protection and shipping. However, due to the randomness and nonlinearity of the wind fields that generate ocean waves and the complex interaction between wave and wind fields, current forecasts of numerical ocean waves have biases. In this study, a spatiotemporal deep-learning method was employed to correct gridded SWH forecasts from the European Centre for Medium-Range Weather Forecasting System Integrated Forecast System Global Model (ECMWF-IFS). This method was built on the trajectory gated recurrent unit deep neural network, and it conducts real-time rolling correction for the 0–240-h SWH forecasts from ECMWF-IFS. The correction model is co-driven by wave and wind fields, providing better results than those based on wave fields alone. A novel pixel-switch loss function was developed. The pixel-switch loss function can dynamically fine-tune the pre-trained correction model, focusing on pixels with large biases in SWH forecasts. According to the seasonal characteristics of SWH, four correction models were constructed separately, for spring, summer, autumn, and winter. The experimental results show that, compared with the original ECMWF SWH predictions, the correction was most effective in spring, when the mean absolute error decreased by 12.972–46.237%. Although winter had the worst performance, the mean absolute error decreased by 13.794–38.953%. The corrected results improved the original ECMWF SWH forecasts under both normal and extreme weather conditions, indicating that our SWH correction model is robust and generalizable.

Short-term prediction of the significant wave height and average wave period based on the variational mode decomposition–temporal convolutional network–long short-term memory (VMD–TCN–LSTM) algorithm

Abstract. The present work proposes a prediction model of significant wave height (SWH) and average wave period (APD) based on variational mode decomposition (VMD), temporal convolutional networks (TCNs), and long short-term memory (LSTM) networks. The wave sequence features were obtained using VMD technology based on the wave data from the National Data Buoy Center. Then the SWH and APD prediction models were established using TCNs, LSTM, and Bayesian hyperparameter optimization. The VMD–TCN–LSTM model was compared with the VMD–LSTM (without TCN cells) and LSTM (without VMD and TCN cells) models. The VMD–TCN–LSTM model has significant superiority and shows robustness and generality in different buoy prediction experiments. In the 3 h wave forecasts, VMD primarily improved the model performance, while the TCN had less of an influence. In the 12, 24, and 48 h wave forecasts, both VMD and TCNs improved the model performance. The contribution of the TCN to the improvement of the prediction result determination coefficient gradually increased as the forecasting length increased. In the 48 h SWH forecasts, the VMD and TCN improved the determination coefficient by 132.5 % and 36.8 %, respectively. In the 48 h APD forecasts, the VMD and TCN improved the determination coefficient by 119.7 % and 40.9 %, respectively.

Significant wave height forecasts integrating ensemble empirical mode decomposition with sequence-to-sequence model

Significant wave height forecasts integrating ensemble empirical mode decomposition with sequence-to-sequence model

CLTS-Net: A More Accurate and Universal Method for the Long-Term Prediction of Significant Wave Height

Significant wave height (SWH) prediction plays an important role in marine engineering areas such as fishery, exploration, power generation, and ocean transportation. For long-term forecasting of a specific location, classical numerical model wave height forecasting methods often require detailed climatic data and incur considerable calculation costs, which are often impractical in emergencies. In addition, how to capture and use the dynamic correlation between multiple variables is also a major research challenge for multivariate SWH prediction. To explore a new method for predicting SWH, this paper proposes a deep neural network model for multivariate time series SWH prediction—namely, CLTS-Net. In this study, the sea surface wind and wave height in the ERA5 dataset of the relevant points P1, P2, and P3 from 2011 to 2018 were used as input information to train the model and evaluate the model’s SWH prediction performance. The results show that the correlation coefficients (R) of CLTS-Net are 0.99 and 0.99, respectively, in the 24 h and 48 h SWH forecasts at point P1 along the coast. Compared with the current mainstream artificial intelligence-based SWH solutions, it is much higher than ANN (0.79, 0.70), RNN (0.82, 0.83), LSTM (0.93, 0.91), and Bi-LSTM (0.95, 0.94). Point P3 is located in the deep sea. In the 24 h and 48 h SWH forecasts, the R of CLTS-Net is 0.97 and 0.98, respectively, which are much higher than ANN (0.71, 0.72), RNN (0.85, 0.78), LSTM (0.85, 0.78), and Bi-LSTM (0.93, 0.93). Especially in the 72 h SWH forecast, when other methods have too large errors and have lost their practical application value, the R of CLTS-Net at P1, P2, and P3 can still reach 0.81, 0.71, and 0.98. The results also show that CLTS-Net can capture the short-term and long-term dependencies of data, so as to accurately predict long-term SWH, and has wide applicability in different sea areas.

Influence of Hurricane Wind Field Variability on Real‐Time Forecast Simulations of the Coastal Environment

AbstractDynamic conditions occur in the coastal ocean during severe storms. Forecasting these conditions is challenging, and large‐scale numerical models require significant computing power. In this paper, we describe a real‐time modeling system (DUNEX‐RT), developed in support of the During Nearshore Event experiment (DUNEX) off the coast of North Carolina, United States of America. The model is run with wave, current, and water level boundary conditions from larger‐scale models, and provides 36‐h forecasts of significant wave height, depth‐averaged velocity, and water levels every 6‐h using Delft3D‐SWAN. Observations and forecasts run at different times are compared and communicated via an interactive website to verify model performance in real‐time and to visualize uncertainty from changing inputs. Here, we evaluate model sensitivity to inputs from seven different atmospheric hindcasts and two atmospheric forecasts for Hurricane Dorian in September 2019. The results emphasize the importance of accurate wind forcing, with significant differences observed between the output model results for different input atmospheric forcing models and forecasts produced at different times. The best results were achieved using atmospheric forcing from the high resolution rapid refresh model, and overall, DUNEX‐RT had low errors at 33 wave, water level, and current sites across the system. The model results for water levels and significant wave heights were also accurate over a longer period of 49 days. Overall, the good forecast skill achieved for the wide range of conditions over this time results suggest that this high‐resolution regional approach could be applied to forecast conditions in other coastal areas.

An Application of NEMOVAR for Regional Wave Model Data Assimilation

The benefits of applying data assimilation to a regional wave model for the northwest European continental shelf have been assessed. The assimilation method uses a 3D variational scheme commonly used in ocean model data assimilation, NEMOVAR, coupled to a regional configuration of WAVEWATCH III deployed on a Spherical Multiple-Cell grid with rotated pole. Significant wave height forecast skill was positively impacted over lead times of up to 12 hours, in both offshore and coastal areas of the model. Assimilation experiments tested the impacts of applying both satellite altimeter and in-situ platform observations of significant wave height as the state variable. Addition of in-situ data introduced an improvement in model skill versus assimilating altimeter measurements only, but these improvements were relatively short-lived in lead time (on the order of a few hours). Extending observation coverage into the coastal zone introduced a benefit for only the first 1-2 hours of forecast lead time at coastal sites. Although impacts on significant wave height forecasts were substantial at short lead times, the relatively short ‘system memory’ of the assimilation suggests that, in the case of the northwest European shelf region, the best application of wave data assimilation is in a rapidly cycling short range forecast system used to generate nowcasts, or for a reanalysis product.

Probabilistic access forecasting for improved offshore operations

Improving access is a priority in the offshore wind sector, driven by the opportunity to increase revenues, reduce costs, and improve safety at operational wind farms. This paper describes a novel method for producing probabilistic forecasts of safety-critical access conditions during crew transfers. Methods of generating density forecasts of significant wave height and peak wave period are developed and evaluated. It is found that boosted semi-parametric models outperform those estimated via maximum likelihood, as well as a non-parametric approach. Scenario forecasts of sea-state variables are generated and used as inputs to a data-driven vessel motion model, based on telemetry recorded during 700 crew transfers. This enables the production of probabilistic access forecasts of vessel motion during crew transfer up to 5 days ahead. The above methodology is implemented on a case study at a wind farm off the east coast of the UK.

Comparison of ECMWF Significant Wave Height Forecasts in the China Sea with Buoy Data

Abstract This paper focuses on a comprehensive comparison of the European Centre for Medium-Range Weather Forecasts (ECMWF) significant wave height (SWH) forecasts with buoy data in the China Sea, and analysis of accuracy characteristics varying with related variables (SWH, water depth, and distance from shore) and different scenarios (each month, different sea area, and typhoon- and cold air activity–induced waves). This is the first time that observations from the Chinese Ocean Monitoring Network have been used to verify ECMWF wave forecasts in the China Sea. Two years’ worth of data from 24 hydrometeorological buoys are used. The comparison shows good accuracy and predictive stability of SWH forecasts for the Chinese buoys, which is consistent with Korean and global buoys. However, the accuracy in the Bohai Sea and Taiwan Strait is worse than that in the South and East China Sea. SWH forecasts for QF206 in the southern Taiwan Strait have systematically underestimated observations, which may be mainly due to the coarse resolution of wave forecasts. Besides, forecasts underestimate observations when 1.5 m < SWH ≤ 6.5 m. The accuracy and predictive stability in spring and summer are worse than those in winter, especially in April, which is the worst in 12 months. The accuracy increases with water depth and distance from shore. During typhoon conditions, the accuracy is worse than during cold air conditions.

Visualisation of Probabilistic Access Forecasts for Offshore Operations

Access forecasting aims to predict the quality of transfer of maintenance technicians to/from vessels and the constituent offshore structures at the wind farm. This is highly dependent on sea state conditions as well as other environmental factors such as visibility. Typically, scheduling or dispatch decisions are made on the basis of deterministic forecasts of significant wave height, often coupled with service contracts where transfers are expected to be attempted below a threshold significant wave height. However, there is always uncertainty in a weather prediction which can be accounted for by probabilistic forecasts. The aim of this work is to explore visualisation ideas for transforming vessel specific access forecasts into an interpretable and intuitive decision-support tool. Three simple methods are proposed based on a score out of 10, classification of transfer conditions, and a threshold score. Methods for summarising access conditions for 2–5 days ahead are also developed. This new forecasting and visualisation capability has significant implications for marine coordinators and skippers who will be able to make better informed safety-critical decisions, with the potential for reductions in the cost-of-energy offshore.

Better Baltic Sea wave forecasts: improving resolution or introducing ensembles?

Abstract. The performance of short-range operational forecasts of significant wave height (SWH) in the Baltic Sea is evaluated. Forecasts produced by a base configuration are intercompared with forecasts from two improved configurations: one with improved horizontal and spectral resolution and one with ensembles representing uncertainties in the physics of the forcing wind field and the initial conditions of this field. Both of the improved forecast classes represent an almost equal increase in computational costs. Therefore, the intercomparison addresses the question of whether more computer resources would be more favorably spent on enhancing the spatial and spectral resolution or, alternatively, on introducing ensembles. The intercomparison is based on comparisons with hourly observations of significant wave height from seven observation sites in the Baltic Sea during the 3-year period from 2015 to 2017. We conclude that for most wave measurement sites, the introduction of ensembles enhances the overall performance of the forecasts, whereas increasing the horizontal and spectral resolution does not. These sites represent offshore conditions, in that they are well exposed from all directions, are a large distance from the nearest coast and in deep water. Therefore, there is the a priori expectation that a detailed shoreline and bathymetry will not have any impact. Only at one site do we find that increasing the horizontal and spectral resolution significantly improves the forecasts. This site is situated in nearshore conditions, close to land and a nearby island, and is therefore shielded from many directions. Consequently, this study concludes that to improve wave forecasts in offshore areas, ensembles should be introduced. For near shore areas, in comparison, the study suggests that additional computational resources should be used to increase the resolution.

Wave Probabilities Consistent with Official Tropical Cyclone Forecasts

Abstract Development of a 12-ft-seas significant wave height ensemble consistent with the official tropical cyclone intensity, track, and wind structure forecasts and their errors from the operational U.S. tropical cyclone forecast centers is described. To generate the significant wave height ensemble, a Monte Carlo wind speed probability algorithm that produces forecast ensemble members is used. These forecast ensemble members, each created from the official forecast and randomly sampled errors from historical official forecast errors, are then created immediately after the official forecast is completed. Of 1000 forecast ensemble members produced by the wind speed algorithm, 128 of them are selected and processed to produce wind input for an ocean surface wave model. The wave model is then run once per realization to produce 128 possible forecasts of significant wave height. Probabilities of significant wave height at critical thresholds can then be computed from the ocean surface wave model–generated significant wave heights. Evaluations of the ensemble are provided in terms of maximum significant wave height and radius of 12-ft significant wave height—two parameters of interest to both U.S. Navy meteorologists and U.S. Navy operators. Ensemble mean errors and biases of maximum significant wave height and radius of 12-ft significant wave height are found to be similar to those of a deterministic version of the same algorithm. Ensemble spreads capture most verifying maximum and radii of 12-ft significant wave heights.

Erik Vanem: Bayesian hierarchical space-time models with application to significant wave height

1 Forristall Ocean Engineering, Inc., 101 Chestnut St., Camden, ME 04843, USA rithms for estimating the necessary integrals has been crucial in these applications. The core of this book is the development of a space-time model for significant wave heights. The data come from a wave hindcast for the area of the North Atlantic between Iceland and theUnitedKingdom. Spatial dependence of the data is modeled using a Markov random field. The time evolution is modeled as a Markov chain. Because the main motivation of the study is to determine long-term trends in the data, linear and quadratic terms are explicitly included in the temporal model. Most of the priors are specified as simple Gaussian distributions. The integrals necessary for theBayesian update are calculated using Markov Chain Monte Carlo techniques. These techniques are explained in a detailedAppendix.Other Appendices describe extreme value modeling, Markov random fields, and sampling from a multi-normal distribution. These Appendices are the most useful part of the book. But it would have been helpful to novices tomore fully describe the path from the probability integrals to Bayesian simulations. The book beginswith a literature reviewof 215 references. The review is broad but not deep. Most of the references are described in a sentence or two, and the connections between references are not clear. For example, forecasts of significant wave height are mixed up with statistics of individual waves in a section on short-term models. Many variations of the model were run. As is generally true, changing the prior distributions had very little effect on the results. The basic model sums the time-independent, space-time, seasonal, and secular time components. Taking a log transform of the data leads to a multiplicative model. The author was unable to find a convincing argument to prefer one model over the others. But almost all of the models show an increasing trend of significant wave height over the 42years of hindcast data. The modeled increase in monthly maximum wave heights is about 70cm. The author is reluc-

Statistical post processes for the improvement of the results of numerical wave prediction models. A combination of Kolmogorov-Zurbenko and Kalman filters

A new mathematical technique for adaptation of the results of numerical wave prediction models to local conditions is proposed.The main aim is to reduce the systematic part of the prediction error in the direct model outputs by taking advantage of the availability of local measurements in the area of interest. The methodology is based on a combination of two different statistical tools: Kolmogorov-Zurbenko (KZ) and Kalman filters.The first smoothes the observation time series as well as that of model direct outputs in order to be comparable via a Kalman filter.This is not the case in general, since forecasted values are smoothed spatially and temporarily by the model itself while observations are point records where no smoothing procedure is applied.The direct application of a Kalman filter to such qualitatively different series may lead to serious instabilities of the method and discontinuities in the results. The proper utilisation of KZ-filters turn the two series into a compatible mode and, therefore, makes possible the exploitation of Kalman filters for the identification and subtraction of systematic errors.The proposed method was tested in an open sea area for significant wave height forecasts using the wave model WAM and six buoys as observational stations.

An Observational and High-Resolution Model Analysis of Gale Wind Events in the Gulf of California

Abstract The Tropical Analysis and Forecast Branch (TAFB) of the National Oceanic and Atmospheric Administration’s (NOAA’s) National Hurricane Center in Miami, Florida, provides high-seas forecasts to portions of the eastern Pacific Ocean, including the Gulf of California. These forecasts include wind velocity and significant wave height forecasts and are initiated by forecast winds of at least 20 kt (10.3 m s−1) or significant wave heights of at least 8 ft (2.4 m). The Gulf of California is a commonly traveled area, where winds are highly modulated by nearby terrain variations. This provides a unique forecast challenge, especially in the absence of regular surface observations. In October and November 2008, the NOAA R/V David Starr Jordan was stationed in the Gulf of California and occasionally reported gale force winds [34–47 kt (17.5–24.2 m s−1)], which operational models regularly missed. A ship log of these events provided the basis for determining mean and anomaly fields for a handful of meteorological variables, from which a conceptual model for the synoptic-scale environment supporting these events is presented. An index based on the mean sea level pressure (MSLP) difference between Ely, Nevada, and Yuma, Arizona, was developed to measure the potential for gales, which was found to be statistically significant in discriminating between “gale” and “marginal wind” events. The fifth-generation NCAR–Pennsylvania State University Mesoscale Model (MM5) is used to conduct doubly nested high-resolution simulations centered on the Gulf of California. These simulations appeared to resolve the gales better than traditional global model guidance, lending credence toward the need for high-resolution modeling in areas of highly variable terrain. Relatively small errors were found in MM5 output using the National Aeronautics and Space Administration (NASA) Quick Scatterometer (QuikSCAT) data as verification.

Making good weather-based decisions

Processes in good weather-based decision making: Understand the threat. While wind and rainfall threats are generally well understood, some other events—such as negative storm surge, long distance lightning strikes, or tsunami—are often not considered. Although these events are not as frequent, they can be devastating. Be aware of resources. An awareness of products, and making appropriate use of these products, contributes greatly to good decision making. The use of probabilities is growing in popularity; however, the uptake of these products has been limited because of the desire to make yes/no decisions. Understand the products. The meanings of products, and terms within the products, are not always well understood. This can be seen in cone of uncertainty TC forecasts and significant wave height forecasts where the potential risks are often underestimated. There is an evolving need to provide educational material and resources to decision makers. Understand the spread of possibilities. It can be easy to just look at a deterministic forecast—the most likely weather conditions—without considering alternative outcomes. The best strategy is to work with the weather service provider to determine the likelihood of the worst case scenario at any given time. Use good sources of information. Too often we hear that alternative opinions have been sourced from the internet. There can be very good reasons for having more than one service provider, but using unverified sources or raw model data can have calamitous results. Have well understood response procedures. When developing response plans, work with a skilled meteorologist to identify potential weaknesses. As an example, we see many strategies that rely on only observed TC characteristics rather than forecast characteristics. Make sure false alarm and probability of detection ratios of response plans are known. This can alleviate concern among senior management if they accept that a number of false alarm responses are a normal part of safe operations. Report the weather. Reporting onsite weather in a standardised manner has a tremendous impact on the overall skill of forecasts. Forecasts are usually tuned to match the prevailing site idiosyncrasies, especially in remote or terrain-influenced locations. Additionally, weather usually moves from point A to point B, so if there is a culture of sharing information then the community in general benefits.

Consensus Forecasts of Modeled Wave Parameters

Abstract The use of numerical guidance has become integral to the process of modern weather forecasting. Using various techniques, postprocessing of numerical model output has been shown to mitigate some of the deficiencies of these models, producing more accurate forecasts. The operational consensus forecast scheme uses past performance to bias-correct and combine numerical forecasts to produce an improved forecast at locations where recent observations are available. This technique was applied to forecasts of significant wave height (Hs), peak period (Tp), and 10-m wind speed (U10) from 10 numerical wave models, at 14 buoy sites located around North America. Results show the best forecast is achieved with a weighted average of bias-corrected components for both Hs and Tp, while a weighted average of linear-corrected components gives the best results for U10. For 24-h forecasts, improvements of 36%, 47%, and 31%, in root-mean-square-error values over the mean raw model components are achieved, or 14%, 22%, and 18% over the best individual model. Similar gains in forecast skill are retained out to 5 days. By reducing the number of models used in the construction of consensus forecasts, it is found that little forecast skill is gained beyond five or six model components, with the independence of these components, as well as individual component’s quality, being important considerations. It is noted that for Hs it is possible to beat the best individual model with a composite forecast of the worst four.

A new methodology for the extension of the impact of data assimilation on ocean wave prediction

It is a common fact that the majority of today's wave assimilation platforms have a limited, in time, ability of affecting the final wave prediction, especially that of long-period forecasting systems. This is mainly due to the fact that after “closing” the assimilation window, i.e., the time that the available observations are assimilated into the wave model, the latter continues to run without any external information. Therefore, if a systematic divergence from the observations occurs, only a limited portion of the forecasting period will be improved. A way of dealing with this drawback is proposed in this study: A combination of two different statistical tools—Kolmogorov–Zurbenko and Kalman filters—is employed so as to eliminate any systematic error of (a first run of) the wave model results. Then, the obtained forecasts are used as artificial observations that can be assimilated to a follow-up model simulation inside the forecasting period. The method was successfully applied to an open sea area (Pacific Ocean) for significant wave height forecasts using the wave model WAM and six different buoys as observational stations. The results were encouraging and led to the extension of the assimilation impact to the entire forecasting period as well as to a significant reduction of the forecast bias.

Consensus of Numerical Model Forecasts of Significant Wave Heights

Abstract The operational consensus forecast (OCF) scheme uses past performance to bias correct and combine numerical forecasts to produce an improved forecast at locations where recent observations are available. Here, OCF uses past observations and forecasts of significant wave height from five numerical wave models available in real time at the Australian Bureau of Meteorology. In addition to OCF, different adaptive weighting and forecast combination strategies are investigated. At deep-water sites (ocean depth > 25 m), all of the interpolated raw model forecasts outperformed 24-h persistence and, after bias correction, one model was clearly best. Significant improvements over raw model significant wave height forecasts were achieved by bias correction, linear-regression methods, and combination strategies. The best forecasts were obtained from a “composite of composites” in which models with highly correlated errors were combined before being included in the performance-weighted bias-corrected forecast. This technique slightly outperformed the linear-regression-corrected best model. At shallow-water sites (ocean depth < 25 m), all raw models perform poorly relative to the 24-h persistence. The composited, corrected forecasts significantly improved on raw model significant wave height forecasts but only slightly outperformed the 24-h persistence. The raw models generated unrealistically large biases that tended to be amplified with larger observed values of significant wave height.