Comparison of the ERA5 Wave Forecasting Dataset Against Buoy Record

Short waves can be described using several synthetic parameters [11, 19]: the most representative are the significant wave height H_s, the energy period T_e, the mean direction Dir_m and the directional width s. These parameters allow to describe each individual sea state fully. In particular, sea waves are largely studied via the two-dimensional wave spectrum S(ω,θ) [19], i.e. the distribution of energy over frequencies and directions. This parameter provides a complete statistical description of the waves and, from a theoretical point of view, can be obtained as the product of the spectrum of frequencies S(ω) [20] and directions D(θ) [21, 22]:

$S(\omega ,\theta )=S(\omega )D(\theta )$     (1)

As with any statistical distribution, the shape of a wave spectrum is usually expressed in terms of moments: the n-th order moment m_n, of the variance density S(ω) is defined by:

${{m}_{n}}=\int\limits_{0}^{\infty }{{{f}^{n}}S(\omega )d\omega }$      (2)

Starting from the general momentum m_n definition, and considering, in general, the spectrum, the synthetic parameters of interest can be obtained as:

${{H}_{s}}=4\sqrt{{{m}_{0}}}$      (3)

${{T}_{e}}=\frac{{{m}_{-1}}}{{{m}_{0}}}$      (4)

$Di{{r}_{m}}=\arctan \frac{\int\limits_{0}^{2\pi }{\int\limits_{{{\omega }_{\min }}}^{{{\omega }_{\max }}}{\sin \theta S(\omega ,\theta )d\omega d\theta }}}{\int\limits_{0}^{2\pi }{\int\limits_{{{\omega }_{\min }}}^{{{\omega }_{\max }}}{\cos \theta S(\omega ,\theta )d\omega d\theta }}}$      (5)

$s=\frac{2}{\int_{-\pi}^{\pi}\left[2 \sin \left(\frac{1}{2}\left(\operatorname{Dir}-\operatorname{Dir}_{m}(\omega)\right)\right)\right]\quad^{2} D(\theta) d \theta}-1$      (6)

Eq. (3), (4) and (5) are widely used; Eq. (6), on the other hand, is less known and has been obtained starting from the relationship that links directional spreading with directional width and considering the definition of directional spreading [11].

The present study highlights the differences concerning in-situ measured data and the opensource data available from the ECMWF portal. In particular, the considerations based upon the data comparison can be used to define how and when a category of source is better than another. To achieve this objective, the Pantelleria island has been chosen as the case study. The data obtained from the buoy installed near the North-West coast are compared with the ERA5 dataset [29].

4.1 Pantelleria island

The site of analysis is located near the North-West coast of the island of Pantelleria. It has a volcanic nature and emerged about 324,000 years ago [30]; the island consists of the emerged summit of an imposing submarine volcanic building and its maximum height is 836 m.

Pantelleria is located almost in the center of the Sicilian Channel, together with the Pelagie islands; this area is characterized by a trend of irregular bathymetric and variable depths. The Sicilian channel, in fact, has two deep thresholds, respectively 500 m towards the Tyrrhenian Sea and 600 m towards the Ionian Sea. Only in the central area are higher depths reached due to some depressions, such as the Pantelleria Basin, with a maximum depth of 1317 m. Pits therefore separate the island from the continental shelf.

Figure 1 shows the buoy installation site, on the North-West coast of the island of Pantelleria.

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Figure 1. Pantelleria area

4.2 Datawell Waverider DWR-G

The buoy used for recording wave information is a Datawell Waverider DWR-G: this instrument is equipped with a Global Positioning System (GPS) module that allows the measurement of directional waves. The buoy (Figure 2), whose hull is characterized by a spherical shape, can measure frequencies up to 0.01 Hz (corresponding to a wave period of 100 s), while the size of the buoy itself limits its high frequency response. Indeed, if the wavelength is less than the size of the buoy, the buoy is unable to follow the wave profile and the observed data is not representative of the wave condition.

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Figure 2. Datawell Waverider DWR-G

Furthermore, the low frequency force to which the system is subjected is determined by the hydrodynamic properties of the buoy itself in combination with its mooring.

Specifications are shown in Table 1, extracted from the Datawell Waverider DWR-G manual.

Table 1. Specifications of Datawell Waverider DWR-G

To measure the direction of the wave, the buoy is based on a disc shape positioned at the pivot point of its movement, which follows the two-dimensional shape of the wave. By monitoring the pitch and roll angles of the buoy and its lifting motion, it is possible to characterize the incident wave fully.

The Datawell Waverider DWR-G is equipped with a sensor package that measures the accelerations along the x, y, z axes, the magnetic field along these axes and the pitch and roll movements.

4.2.1 Buoy data processing

The buoy measures raw North, West and vertical displacements at a constant rate of 1.28 Hz. Such displacements are to be intended as excursions concerning the average position of the buoy, and are used to obtain processed data for the description of the sea state. In particular, the vertical displacement of the buoy is used to obtain a power spectral density (PSD) of the wave amplitude, showing what wave amplitudes occur at what frequencies. If also North and West displacements are included in the analysis, it is possible to obtain waves directional spectrum, by conducting the calculation of the Fourier series of the three signals, and then the PSD, as in the previous case.

The data format in which the data are acquired form the buoy is called the real-time format. It consists of long-existing hexadecimal vectors containing information about vertical, North and West displacements and also words forming the spectrum and system files. The real-time format is organized at four levels:

1. Vectors of 64 bits, containing real-time data together with cyclical data;

2. Blocks of 18 vectors assembling cyclical data with spectral and system data;

3. Spectrum file, constituted by the spectral data of 16 blocks;

4. System file, constituted by the system data of 16 blocks;

Each 64-bit vector is divided into three parts: the first contains cyclical data (regarding the system file or the spectrum file), the second contains the real-time displacement in vertical, North and West directions, and the third part contains information on transmission errors and corrections.

With respect to the timing of the data, each vector of 64 bits covers one acquisition (at the frequency of 1.28 Hz, or 0.78125 s); each block contains 18 vectors, equivalent to 14.0625 s of data acquisition; each file contains 16 blocks (or 288 vectors) with 225 s of data; finally, each full cycle contains 8 repeated files, or equivalently 30 minutes of data acquisition. It is worth to underline that in a full cycle are contained 30 minutes of displacement data for the buoy, while the spectrum data are invariant during the 30 min time span. Specifically, during half an hour, 8 spectra of 200 s data interval each are collected and averaged. At the end of the half hour, over which the calculations are executed, all spectral parameters are available. The spectrum is transmitted 8 times during the next half-hour for redundancy.

4.3 ECMWF

Numerical models, such as atmospheric and oceanic models from ECMWF (European Centre for Medium-Range Weather Forecasting), provide many environmental datasets. It was founded in 1975 with the aim of producing global medium-term weather forecasts by pooling various scientific and technological resources. The center therefore started from the goal of developing numerical models capable of making predictions and allowing free access to such data.

The ERA5 database [29] of ECMWF is among the most used databases in marine analysis and provides several time series: wave, wind, solar and much other information can be obtained. However, the spatial resolution corresponding is not suitable for accurate calculations: the ERA5 database provides wave parameters with a spatial resolution of 0.5°x0.5° (about 50 km x 50 km). In addition, the complicated orography of the Mediterranean Sea further reduces the accuracy of the wind parameters. Consequently, since waves are generated by the interaction between the wind and the sea surface, and since numerical models exploit this correlation, the ERA5 wave data are less accurate in the Mediterranean basin than in other locations.

Generally, ERA5 data are often used as input to calculate more accurate data by applying additional numerical models specific to coastal regions.

Synthetic wave parameters provided by ERA5 were used to compare measured data from the buoy with modeled data.

The synthetic parameters of the buoy were compared with those acquired from the ECMWF portal. The first difference that distinguishes them concerns the temporal resolution of the data: the information of the wave recordings is provided every half an hour, while the modelled data has an hourly resolution. Because of this difference, several analyzes were carried out. The comparison of the historical series was carried out by comparing all the available data. Similarly, the wave roses and the occurrences bivariate diagrams were calculated and compared considering the whole set of recorded and modelled data. Subsequently, the capabilities of the ECMWF model to reproduce sea states were investigated in greater detail: the data sets used were manipulated and the data corresponding to the gaps present in the recordings was discarded from the time series obtained from numerical modelling. As for the recorded data, they have been selected to have a single parameter every hour. Then, a homogeneous comparison of the hourly sea states was performed from the beginning of September 2021 to the end of February 2022.

6.1 Complete time series

Time series overlays of the significant wave height H_s (Figure 4), the energy period T_e (Figure 5), and the directional width s (Figure 6), concerning the two datasets under consideration, have been obtained. Visualization of the time series is helpful to appreciate the variability of the parameters as time passes. Significant wave height H_s obtained from numerical modelling manage to describe the wave climate with good detail. In fact, from the superposition shown in Figure 4 a similar trend is evident. However, the ERA5 data generally underestimate the recorded wave conditions.

Similarly, the temporal evolution of the energy period T_e is well represented by the ERA5 dataset, although usually, the values are lower (Figure 5). The only obvious exception is between mid-October and mid-November 2021, where the values obtained from the model overestimate the recorded data.

On the other hand, the directional width parameter s is not well calculated by the numerical model. The data obtained from the buoy have much higher variability than other data (Figure 6). This difference depends on the spatial resolution used by the numerical model. The directional width s describes the variability of the wave energy distribution in different directions; however, considering an extensive cell (0.5° x 0.5°) does not allow to obtain good information in terms of direction.

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Figure 4. Time series of significant wave height H_s

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Figure 5. Time series of energy period T_e

Figures 7 and 8 represent the wave rose, obtained from the buoy and ERA5 data, respectively. As expected, the roses are different: the polar plot obtained from the modelled data also predicts incoming waves from the South-East direction. This error depends on the spatial resolution of ERA5 data and on the island location: Pantelleria is located in the Sicilian Channel and this area is characterized by a wind that blows both from the Nort-West to South-East and viceversa. Since the wind generates the waves, also these have the corresponding directions. However, the site where the buoy has been installed is protected from the arrival of the southern waves: once they reach the island, they must circumnavigate it, changing direction. Depending on the application and the objectives, the direction of the waves can be a crucial parameter, and such significant errors may not be acceptable.

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Figure 6. Time series of directional width s

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Figure 7. Rose plot using buoy data

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Figure 8. Rose plot using ERA5 data

Occurrence scatters were identified to determine which sea states are most commonly occurring. Figure 9 represents the frequency of H_s-T_e pairs recorded by the buoy: the most frequently occurring sea states have a wave height H_s of about 0.5 m and an energy period T_e of nearly 4 s. The scatter obtained with the ERA5 data (Figure 10) shows similar characteristics despite a greater spread around the most occurring H_s-T_e pair is evident.

Since the significant wave height and energy period obtained from numerical modeling generally underestimate the real values, the wave power is also lower. Considering the values of the recorded data, the average wave power is 6.8 kW/m, while in the case of the ERA data it is 6.2 kW/m.

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Figure 9. Occurrences scatter using buoy data

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Figure 10. Occurrences scatter using ERA5 data

6.2 Matching time series

In order to compare the measured data with modelled ones, through scatter plots, homogenization of the data was necessary. Subsequently, both synchronous and asynchronous analyses were conducted to evaluate the simultaneity of the recorded sea states with the calculated ones.

6.2.1 Synchronous analysis

Synchronous analysis are conducted by comparing the homogenized hourly time series. The same number of data characterizes these new time series: each hourly sea state recorded by the buoy has been compared with that calculated by numerical modelling. Figures 11, 12 and 13 represent the scatters of the significant wave height H_s, the energy period T_e, and the directional spreading s, respectively. The bisector is represented in all scatters to facilitate the graphs' interpretation. The conformity of the data is described numerically by the value of the R²: high values, tending towards unity, indicate a good adherence between the data, low values, tending to zero, on the other hand, refer to poor conformity.

The significant wave height is well calculated by the model, in particular, for low values of H_s, the model tends to overestimate slightly, while for higher values, the model tends to underestimate the real ones. The conformity of the data is confirmed by the value of the R², which is approximately 0.9.

Concerning the energy period, the value of the R² is lower than that obtained for the significant wave height and it is equal to 0.67, indicating a more significant deviation between the recorded and modelled data. Figure 12, in fact, shows a cloud of points above the bisector due to the overestimation of the parameter during the period October-November 2021.

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Figure 11. Significant wave height time series

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Figure 12. Significant wave height time series

Figure 13 shows the correlation between the measured and modelled directional width values: a high dispersion around the bisector is evident, and the R² value confirms the poor fit of the data. As expected, in fact, the modelled data do not correctly describe reality.

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Figure 13. Significant wave height time series

6.2.2 Asynchronous analysis

Asynchronous analyses were conducted to verify that the differences from the synchronous analyses did not depend on homogenization. In particular, the new data sets were obtained considering the maximum values of H_s, T_e and s, recorded in time windows of 12 h. The new data was released from the reference time, and any time lags were ignored.

Figures 14, 15 and 16 show the scatter plot of the significant wave height H_s, the energy period T_e and the directional width s, respectively.

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Figure 14. Significant wave height time series

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Figure 15. Significant wave height time series

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Figure 16. Significant wave height time series

Both the points distribution and the values of R² are similar to what is determined by the synchronous analysis and this shows that there is no time lag of the sea states. In particular, the conformity of data from the model and in-situ instrumentation is confirmed by the R²value of about 0.9. As in the synchronous analysis, the model overestimates the lower wave heights and underestimates the higher ones. Regarding the energy period, a scattering of the cloud around the bisector is evident, and the model tends to overestimate the periods near 4s, referring to the October-November 2021 period. Figure 16 shows the correlation of directional width values, obtained by asynchronous analysis, between the measurements and the model. The scatter around the bisector shows a poor correlation, confirmed by the R² value of 0.09.