# An Experimental Assessment of Extreme Wave Evaluation by Integrating Model and Wave Buoy Data

^{*}

## Abstract

**:**

## 1. Introduction

_{R}(in the following: SWH(T

_{R})) deriving from synthetic data are compared and calibrated with the similar curves computed from buoy data in different locations. This provides a way of deriving SWH(T

_{R}) curves for sites where no experimental data are available.

_{R}) curves for such locations.

## 2. Methods

#### 2.1. Integrated Procedure

_{R}) function, which links SWH with its return time T

_{R}, are themselves randomly distributed and that the distribution of such parameters can be estimated by integrating the data from the model with those from the buoys in the area. A somewhat similar approach with rainfall data is reported in [31].

_{m}(T

_{R}, X, Y) is the significant wave height for the generic location X Y and for a given T

_{R}estimated from a wave spectral generation and propagation, it can be assumed that the “true” value of SWH

_{b}is given by the following Equation (1):

_{b}(T

_{R}, X, Y) = SWH

_{m}(T

_{R}, X, Y) + E(T

_{R}).

_{b}is “true” in a purely conventional sense, and it is assumed here to be the value computed from buoy recorded data, i.e., the value that would be used for design or research purposes if experimental data were available. Additionally, the procedure proposed here is independent from the particular form of extreme value distribution SWH(T

_{R}), which is adopted and fitted to the data: there are indeed many alternatives, and the relevant state of the research on the field has been briefly discussed in Section 1.

_{P}) = 1 − exp{−[(H

_{P}− B)/A]

^{k}},

_{P}are the peak values of SWH while A, B and k are the distribution parameters also respectively known as scale, position, and shape parameters. While A and B can be computed with the least square method, the shape parameter k, following the usual practise as indicated in [6,32], is chosen with the best fit criterion among the following 4 values (0.75; 1.00; 1.40 and 2.00).

_{R}) or any particular procedure to estimate its parameters, the only requirement is that such form and procedure should be uniform throughout the whole analysis.

_{R}(in years) is computed by making use of Equation (3):

_{R}) = B + A[ln(λT

_{R})]

^{1/k},

_{T}and the length n of the observation period expressed in years.

_{b}(T

_{R}) and with the model data series SWH

_{m}(T

_{R}); since however most of the times the buoy positions do not exactly coincide with model grid points, a spatial bi-linear interpolation procedure (co-location) is used, as described in [33].

_{R}, by taking into account the “true” values SWH

_{bi}(T

_{R}) computed at the locations i where the wave buoys are available. E

_{i}(T

_{R}), again for each location i is thus evaluated as:

_{i}(T

_{R}) = SWH

_{bi}(T

_{R}, X

_{i}, Y

_{i}) − SWH

_{mi}(T

_{R}, X

_{i}, Y

_{i}).

_{i}at location i is then given by Equation (5):

_{i}(T

_{R}) = E

_{i}(T

_{R})/SWH

_{mi}(T

_{R}).

_{R}) in the area can then be estimated as:

_{R}) as:

_{b}wave buoys considered in the region. Therefore, for a generic location t where no buoy data are available, an estimated SWH

_{st}of the “true” SWH

_{bt}, can be obtained from SWH

_{mt}(T

_{R}) by following Equation (8)

_{st}(T

_{R}) = SWH

_{mt}(T

_{R}) + µ(T

_{R}) × SWH

_{mt}(T

_{R}),

_{mt}(T

_{R}) is evaluated by using model recorded data at location t.

_{b}+ 1 EVPD, an easily standardized and commonly available algorithm, as opposed to a single one, as it would be needed for a conventional procedure based on model data only. The estimation of E

_{i}(T

_{R}), e

_{i}(T

_{R}) and SWH

_{st}(T

_{R}) is straightforward and can be carried out in a single EXCEL

^{®}file.

#### 2.2. Validation

_{b}(T

_{R}) with a given T

_{R}return time in order to compare it with the estimated value. It is worth recalling that, as stated above, “true” means the value that would be computed from an experimental time series.

_{b}buoys available, the procedure is applied by taking one of them to provide the “true” values at the test location t, while the remaining N

_{a}= N

_{b}− 1 series are used to estimate the relative error distribution e(T

_{R}) according to Equation (5). The procedure highlighted in Figure 1 can thus be applied N

_{b}times, each time choosing in rotation one of the available data series to be taken as test location. Such a methodology, normally called “jackknife” is well known and has been tested in various application [34,35].

_{b}available test locations a corrected estimated return time curve (SWH

_{st}) is thus obtained, and it can be compared with the “true” SWH

_{bt}curve. The following Figure 2 reports an example of the results for NOAA buoy 46014 located along the US Pacific Coast. Here N

_{b}= 8 and therefore N

_{a}= 7.

_{st}is much closer to the buoy value SWH

_{bt}than the model SWH

_{mt}, thus providing a better and safer evaluation of the sea state design value. The same computation can be carried out by rotating the test position among the N

_{b}available buoy locations: this kind of analysis is performed over three test areas described in Section 2.2.

- ERM = (Model − Buoy)/Buoy = (SWH
_{mt}− SWH_{bt})/SWH_{bt}; - ERS = (Integrated − Buoy)/Buoy = (SWH
_{st}− SWH_{bt})/SWH_{bt}; - Improvement = abs(ERM) − abs(ERS).

#### 2.3. Study Sites

- Gulf of Mexico and NW Atlantic 10 min (ecg_10m) for buoys located in Gulf of Mexico and along US Atlantic Coast;
- US West Coast 10 min (wc_10m) for buoys located along US Pacific Coast.

## 3. Results

_{R}) evaluated through the different procedures for each zone:

- SWH
_{bt}through wave meter data; - SWH
_{mt}through model data; - SWH
_{st}through integrated procedure proposed here.

#### 3.1. Pacific Coast

_{R}= 100 are very similar to those computed for T

_{R}= 50. This derives from the similarity of the SWH(T

_{R}) curves for high values of T

_{R}, as noted in Section 2.2 and as shown in Figure 2. The analysis of the results can therefore be limited to just one value of the return periods, i.e., T

_{R}= 50 years. Figure 4 compares the results of the integrated procedure versus the model for T

_{R}= 50 years and provides an insight into the behaviour of the errors.

^{2}between SWH

_{st}and the identity line reflects the dispersion of the error; obviously its average and its extreme values represent a net improvement over the simple model data.

#### 3.2. Atlantic Coast

_{bt}in Figure 5. The similarity between the results for T

_{R}= 100 and 50 years is also confirmed.

#### 3.3. Gulf of Mexico

_{R}considered the absolute values of ERS are greater than correspondent ERM values with consequent negative improvements (−2.74% and −4.04%); however, even in this case, as in all the others, the model results lead to underestimating the SWH

_{bt}(T

_{R}) while the integrated procedure overestimates it, certainly a safer error than the former one. The effectiveness of the integrated procedure in removing the bias of the model is thus confirmed again.

## 4. Discussion

_{R}) are the outcome of a statistical elaboration on extreme wave data, and as such they are unavoidably affected by random variations.

_{mt}values, computed through the simple use of model data are systematically affected by an error that is never lower in absolute value than 9.70%, and on average of the order of 20%. Besides, the errors are always negative, i.e., the SWH

_{mt}present a negative bias over the buoy values, so that using model results as design parameters would seriously put any coastal or offshore construction at risk.

_{R}) curves portray a similar behaviour for the high values of T

_{R}, which are of interest in practical applications.

_{b}wave buoy are available, requires downloading N

_{b}extra model data series as well as N

_{b}buoy data series. On the other hand, the computational effort is not particularly heavy since the extreme value fitting procedures, briefly recalled in Section 2.1, are nowadays fully standardized.

## 5. Conclusions

_{R}in a given location.

_{R}and that a considerable improvement in accuracy is gained by making use of this integrated procedure in place of using model data.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Example of results for test location on the site of buoy 46014 along US Pacific Coast: SWH

_{bt}is the curve computed from experimental data; SWH

_{mt}is the curve computed from model data; SWH

_{st}is the estimated curve, obtained through the integrated procedure.

**Figure 4.**Pacific Coast error analysis for T

_{R}= 50 years: SWH(T

_{R}) values computed through model data (red circles) and integrated procedure (green circles) are compared with SWH(T

_{R}) values computed through buoy data (solid blue line). R

^{2}between SWH

_{st}and the identity line 0.0076.

**Figure 5.**Atlantic Coast error analysis for T

_{R}= 50 years: SWH(T

_{R}) values computed through model data (red circles) and integrated procedure (green circles) are compared with SWH(T

_{R}) values computed through buoy data (solid blue line). R

^{2}between SWH

_{st}and the identity line 0.911.

**Figure 6.**Gulf of Mexico error analysis for T

_{R}= 50 years: SWH(T

_{R}) values computed through model data (red circles) and integrated procedure (green circles) are compared with SWH(T

_{R}) values computed through buoy data (solid blue line). R

^{2}between SWH

_{st}and the identity line 0.925.

Test Location | SWH_{bt} (m) | SWH_{mt} (m) | SWH_{st} (m) | ERM (%) | ERS (%) | Improvement (%) |
---|---|---|---|---|---|---|

46011 | 9.34 | 7.47 | 9.72 | −20.03 | 4.06 | 15.97 |

46012 | 9.53 | 8.05 | 10.55 | −15.53 | 10.72 | 4.81 |

46013 | 9.90 | 8.02 | 10.45 | −19.01 | 5.57 | 13.44 |

46014 | 10.86 | 8.56 | 11.12 | −21.25 | 2.41 | 18.73 |

46022 | 11.80 | 8.81 | 11.35 | −25.36 | −3.83 | 21.53 |

46027 | 10.87 | 7.87 | 10.09 | −27.60 | −7.14 | 20.46 |

46028 | 10.11 | 7.52 | 9.68 | −25.61 | −4.20 | 21.41 |

46042 | 10.34 | 7.62 | 9.80 | −26.29 | −5.21 | 21.09 |

Mean | 10.34 | 7.99 | 10.35 | −22.57 | 0.30 | 17.18 |

Test Location | SWH_{bt} (m) | SWH_{mt} (m) | SWH_{st} (m) | ERM (%) | ERS (%) | Improvement (%) |
---|---|---|---|---|---|---|

46011 | 9.73 | 7.78 | 10.16 | −20.04 | 4.42 | 15.62 |

46012 | 9.91 | 8.38 | 11.03 | −15.41 | 11.28 | 4.13 |

46013 | 10.30 | 8.35 | 10.92 | −18.96 | 6.02 | 12.94 |

46014 | 11.30 | 8.89 | 11.58 | −21.30 | 2.54 | 18.76 |

46022 | 12.32 | 9.14 | 11.81 | −25.77 | −4.10 | 21.67 |

46027 | 11.34 | 8.15 | 10.48 | −28.15 | −7.62 | 20.52 |

46028 | 10.53 | 7.79 | 10.06 | −25.99 | −4.43 | 21.57 |

46042 | 10.81 | 7.93 | 10.22 | −26.67 | −5.43 | 21.24 |

Mean | 10.78 | 8.30 | 10.78 | −22.79 | 0.34 | 17.06 |

Test Location | SWH_{bt} (m) | SWH_{mt} (m) | SWH_{st} (m) | ERM (%) | ERS (%) | Improvement (%) |
---|---|---|---|---|---|---|

44004 | 13.02 | 10.07 | 13.34 | −22.64 | 2.45 | 20.19 |

44005 | 9.91 | 7.55 | 9.98 | −23.80 | 0.70 | 23.09 |

44008 | 12.34 | 9.56 | 12.66 | −22.57 | 2.56 | 20.02 |

44009 | 9.45 | 6.65 | 8.69 | −29.60 | −8.05 | 21.55 |

44011 | 13.90 | 10.43 | 13.75 | −24.98 | −1.07 | 23.90 |

44013 | 10.04 | 7.05 | 9.21 | −29.77 | −8.31 | 21.46 |

44014 | 10.06 | 8.21 | 10.95 | −18.41 | 8.84 | 9.57 |

44025 | 9.02 | 7.14 | 9.48 | −20.87 | 5.12 | 15.76 |

Mean | 10.97 | 8.33 | 11.01 | −24.08 | 0.28 | 19.44 |

Test Location | SWH_{bt} (m) | SWH_{mt} (m) | SWH_{st} (m) | ERM (%) | ERS (%) | Improvement (%) |
---|---|---|---|---|---|---|

44004 | 13.75 | 10.62 | 14.14 | −22.77 | 2.80 | 19.97 |

44005 | 10.31 | 7.85 | 10.43 | −23.85 | 1.16 | 22.69 |

44008 | 13.12 | 10.12 | 13.46 | −22.90 | 2.59 | 20.31 |

44009 | 10.13 | 7.06 | 9.26 | −30.26 | −8.57 | 21.69 |

44011 | 15.04 | 11.23 | 14.87 | −25.35 | −1.12 | 24.23 |

44013 | 10.81 | 7.56 | 9.91 | −30.09 | −8.30 | 21.78 |

44014 | 10.72 | 8.71 | 11.67 | −18.82 | 8.78 | 10.04 |

44025 | 9.61 | 7.56 | 10.09 | −21.32 | 4.99 | 16.33 |

Mean | 11.69 | 8.84 | 11.73 | −24.42 | 0.29 | 19.68 |

Test Location | SWH_{bt} (m) | SWH_{mt} (m) | SWH_{st} (m) | ERM (%) | ERS (%) | Improvement (%) |
---|---|---|---|---|---|---|

42001 | 12.31 | 9.54 | 11.78 | −22.54 | −4.33 | 18.21 |

42002 | 9.44 | 8.47 | 10.67 | −10.30 | 13.04 | −2.74 |

42003 | 12.64 | 10.00 | 12.38 | −20.94 | −2.06 | 18.88 |

42019 | 6.77 | 5.79 | 7.25 | −14.43 | 7.18 | 7.25 |

42020 | 9.25 | 7.18 | 8.86 | −22.40 | −4.13 | 18.27 |

42036 | 8.80 | 7.45 | 9.31 | −15.41 | 5.78 | 9.63 |

42039 | 14.10 | 11.14 | 13.80 | −20.99 | −2.13 | 18.86 |

42040 | 19.21 | 14.12 | 17.29 | −26.50 | −9.96 | 16.54 |

Mean | 11.57 | 9.21 | 11.42 | −19.19 | 0.42 | 13.11 |

Test Location | SWH_{bt} (m) | SWH_{mt} (m) | SWH_{st} (m) | ERM (%) | ERS (%) | Improvement (%) |
---|---|---|---|---|---|---|

42001 | 13.78 | 10.60 | 13.06 | −23.09 | −5.24 | 17.85 |

42002 | 10.32 | 9.32 | 11.74 | −9.70 | 13.74 | −4.04 |

42003 | 14.14 | 11.13 | 13.77 | −21.24 | −2.62 | 18.62 |

42019 | 7.13 | 6.16 | 7.72 | −13.52 | 8.33 | 5.19 |

42020 | 10.16 | 7.94 | 9.81 | −21.86 | −3.49 | 18.36 |

42036 | 9.42 | 8.00 | 9.99 | −15.14 | 6.03 | 9.12 |

42039 | 15.83 | 12.47 | 15.41 | −21.25 | −2.63 | 18.62 |

42040 | 21.93 | 16.09 | 19.68 | −26.63 | −10.27 | 16.54 |

Mean | 12.84 | 10.21 | 12.65 | −19.05 | 0.48 | 12.53 |

**Table 7.**Mean values of SWH (buoy, model, and integrated) and of metric (ERM, ERS, and Improvement) defined in Section 2.2 for the three areas and two (50 and 100 years) return periods.

Zone | T_{R} (year) | SWH_{bt} (m) | SWH_{mt} (m) | SWH_{st} (m) | ERM (%) | ERS (%) | Improvement (%) |
---|---|---|---|---|---|---|---|

Pacific Cosat | 50 | 10.97 | 8.33 | 11.01 | −24.08 | 0.30 | 17.18 |

100 | 11.69 | 8.84 | 11.73 | −24.42 | 0.34 | 17.06 | |

Atlantic Coast | 50 | 10.34 | 7.99 | 10.35 | −22.57 | 0.28 | 19.44 |

100 | 10.78 | 8.30 | 10.78 | −22.79 | 0.29 | 19.63 | |

Gulf of Mexico | 50 | 11.57 | 9.21 | 11.42 | −19.19 | 0.42 | 13.11 |

100 | 12.84 | 10.21 | 12.65 | −19.05 | 0.48 | 12.53 |

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## Share and Cite

**MDPI and ACS Style**

Reale, F.; Dentale, F.; Furcolo, P.; Di Leo, A.; Pugliese Carratelli, E.
An Experimental Assessment of Extreme Wave Evaluation by Integrating Model and Wave Buoy Data. *Water* **2020**, *12*, 1201.
https://doi.org/10.3390/w12041201

**AMA Style**

Reale F, Dentale F, Furcolo P, Di Leo A, Pugliese Carratelli E.
An Experimental Assessment of Extreme Wave Evaluation by Integrating Model and Wave Buoy Data. *Water*. 2020; 12(4):1201.
https://doi.org/10.3390/w12041201

**Chicago/Turabian Style**

Reale, Ferdinando, Fabio Dentale, Pierluigi Furcolo, Angela Di Leo, and Eugenio Pugliese Carratelli.
2020. "An Experimental Assessment of Extreme Wave Evaluation by Integrating Model and Wave Buoy Data" *Water* 12, no. 4: 1201.
https://doi.org/10.3390/w12041201