# Influence of Sea State on Sea Surface Height Oscillation from Doppler Altimeter Measurements in the North Sea

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## Abstract

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_{t}. The same analysis carried out between ${\mathsf{\sigma}}_{\mathrm{h}}$ and the swell component of the wave spectrum shows a smaller correlation. In contrast, the correlation between the PLRM ${\mathsf{\sigma}}_{\mathrm{h}}$ and any component of the SWH spectrum has not been found to be significant. To provide an explanation of these results, the aliasing effect caused by the interaction between the sea wavelength and the altimeter resolution has been considered; a simple model has, therefore, been produced to simulate the dependence of the aliasing-derived, ${\mathsf{\sigma}}_{\mathrm{A}}$, on the sea wavelength. The alias/wavelength curve obtained helps to explain why—at least for the relatively low wavelength sea data considered—the wave direction and its wavelength have little or no influence on ${\mathsf{\sigma}}_{\mathrm{h}}$.

## 1. Introduction

## 2. Materials and Methods

_{t}), significant wave height of wind waves (SWH

_{w}), significant wave height of the swell (SWH

_{s}), their respective average periods (T

_{t}, T

_{w}, T

_{s}), and their directions.

_{t}, SWH

_{w}, and SWH

_{s}. SSSWH is the sum of the squares of the SWH of each event, computed separately for the total, wind, and swell components. This parameter was considered because the square of the SWH is proportional to the energy (En) of the wave field, and it is a good indicator of the sea state. Finally En/Entot, i.e., the ratio between the energy of each component and the total, is also considered.

_{t}, L

_{w}, and L

_{s}).

_{t}, for the “Total” data). The flight paths of the satellite are oriented at about 6° and 174°.

## 3. Results

_{t}, and the altimeter significant wave heights, SWH

_{SAR}and SWH

_{PLRM}. They were highly correlated, with a correlation of 0.89 for both types of altimeter data (Figure 3). This was expected as satellite measured SWH values are routinely assimilated into ECMWF analysis and re-analysis. Therefore, the quality of the data was excellent and this justifies the choice of the area and of the ECMWF model as a reference.

^{2}= 0.09). Moreover, PLRM error was, on average, larger than SAR error: 0.0157 versus 0.0096. This agrees, at least qualitatively, with previous results that have provided a larger PLRM ${\mathsf{\sigma}}_{\mathrm{h}}$ than SAR ${\mathsf{\sigma}}_{\mathrm{h}}$ (see, for instance, Figure 1a in [7]).

_{t}, and the SAR 1 Hz standard deviation, ${\mathsf{\sigma}}_{\mathrm{h}}$ (R

^{2}= 0.4). On the contrary, there was no correlation between either of the two components of the sea spectrum (SWH

_{s}and SWH

_{w}) and SAR or PLRM ${\mathsf{\sigma}}_{\mathrm{h}}$ (see Table 3).

_{t}and T

_{t}yield the same results as the respective simple regressions between ${\mathsf{\sigma}}_{\mathrm{h}}$ and SWH

_{t}, thus, confirming the lack of any simple correlation with the average wave period (Figure 6 and Table 5).

_{thr}, was applied to the direction of the events considered: Only events where |α| > α

_{thr}were considered. See Figure 7.

_{thr}is taken to be 40°, all sea directions whose propagation angle with the flight direction was less than 40° were excluded. The results are shown in Figure 8.

_{t}and ${\mathsf{\sigma}}_{\mathrm{h}}$ was clearly higher for the SAR altimeter data, compared to the same correlation for the PLRM data, this latter being practically negligible (R

^{2}= 0.15), while the former, independently from the threshold, was constantly about 0.40 except for a small decrease as α

_{thr}approached 90°: i.e., if only nearly perpendicular events are considered, the correlation decreases.

## 4. Discussion

_{t}, while the correlation was much lower if only the swell component was considered. No such effect exists for the PLRM data considered here.

_{m}), average wavelength period ™, and corresponding wavelength (L

_{m}), were considered. Indeed, in real life, the waves are far from being sinusoidal: Each sea state is normally made up of a number of components of a spectrum. The surface was sampled at 20 Hz, i.e., 20 samples (looks) over a length of 6960 m. Thus, the distance between samples was 6960/20, which is 348 m, and was comparable to the along-track theoretical resolution of DDA. Therefore, we assume that the along-track altimeter resolution was equal to the sample length at 20 Hz. The error, ${\mathsf{\sigma}}_{\mathrm{A}}$, which can be assumed as a proxy of the part of the ${\mathsf{\sigma}}_{\mathrm{h}}$ induced by the aliasing effect, was computed as the standard deviation of the simulated 20 Hz SSH:

_{m}and L

_{m}, as well as the angle, α, between the satellite flight path and the wave direction. Figure 10 shows the dependence on α and L

_{m}for a fixed wave height, H

_{m}= 3 m.

_{m}, did play a role, but even for the highest wavelength considered here (L

_{m}= 264 m, T

_{m}= 13 s) this was negligible. As it was shown above, very few of the events considered have an average wavelength, L

_{m}, that was over 156 m (T

_{m}= 10 s).

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A map of the North Sea. Study area is delimited by the black box and is between 53–59° of latitude and 4–9° of longitude.

**Figure 2.**Distribution of events end event wave energy according to direction (

**a**,

**c**) and wavelength (

**b**,

**d**) for the whole spectrum (

**a**,

**b**) and the swell component (

**c**,

**d**). Dashed lines show the approximate satellite flight direction.

**Figure 3.**Altimeter versus ECMWF co-located SWH values: (

**a**) Altimeter in SAR mode; and (

**b**) altimeter in PLRM mode.

**Figure 4.**Correlation between the standard deviation, ${\mathsf{\sigma}}_{\mathrm{h}}$, of SSH in SAR and PLRM mode.

**Figure 5.**Correlation between the standard deviation, ${\mathsf{\sigma}}_{\mathrm{h}}$, of SSH and SWH

_{s}, SWH

_{w}and SWH

_{t}: (

**a**,

**c**,

**e**) altimeter in SAR mode; (

**b**,

**d**,

**f**) altimeter in PLRM mode.

**Figure 6.**Multiple correlation between ${\mathsf{\sigma}}_{\mathrm{h}}$, SWH

_{t}, and T

_{t}both for the SAR (

**a**) and PLRM data (

**b**).

**Figure 7.**Effect of wave direction: Only events that fall outside the threshold angle α

_{thr}are considered.

**Figure 8.**Correlation coefficient between ${\sigma}_{\mathrm{h}}$ and SWH

_{t}as a function of the threshold angle, α

_{thr}.

**Figure 10.**Aliasing effect, ${\mathsf{\sigma}}_{\mathrm{A}}$, as a function of the angle, α, between the satellite flight path and the wave direction for four different average wavelengths, L

_{m}, and for a fixed wave height, H

_{m}= 3 m.

Parameters | SWH_{t} | SWH_{w} | SWH_{s} |
---|---|---|---|

ASWH (m) | 1.68 | 1.20 | 0.94 |

SSSWH (m^{2}) | 1.54 | 74,620 | 37,484 |

En/Entot | 1.00 | 0.66 | 0.33 |

**Table 2.**Average (A) and standard deviation (SD) of the three spectrum average wavelength components for the events in the ECMWF dataset.

Parameters | L_{t} | L_{w} | L_{s} |
---|---|---|---|

A (m) | 51.92 | 29.14 | 73.45 |

SD (m) | 27.83 | 22.43 | 38.90 |

**Table 3.**Correlation coefficients, R

^{2}, between ${\mathsf{\sigma}}_{\mathrm{h}}$ values for the SAR and PLRM mode and the three SWH components of the wave spectrum.

Altimeter Mode | Swell (SWH_{s}) | Wind (SWH_{w}) | Total (SWH_{t}) |
---|---|---|---|

SAR mode | 0.15 | 0.30 | 0.41 |

PLRM mode | 0.07 | 0.12 | 0.11 |

**Table 4.**Correlation coefficients, R

^{2}, between ${\mathsf{\sigma}}_{\mathrm{h}}$ values for the SAR and PLRM mode and the mean periods of the wave spectrum and of its swell and wind components.

Altimeter Mode | Swell (T_{s}) | Wind (T_{w}) | Total (T_{t}) |
---|---|---|---|

SAR mode | 0.25 | 0.26 | 0.24 |

PLRM mode | 0.11 | 0.12 | 0.11 |

**Table 5.**Multiple correlation coefficients, R

^{2}, between ${\mathsf{\sigma}}_{\mathrm{h}}$ values, SWH

_{t}, and T

_{t}of the total spectrum for the SAR and PLRM data.

Altimeter Mode | Total (SWH_{t}) |
---|---|

SAR mode | 0.41 |

PLRM mode | 0.16 |

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**MDPI and ACS Style**

Reale, F.; Dentale, F.; Carratelli, E.P.; Fenoglio-Marc, L. Influence of Sea State on Sea Surface Height Oscillation from Doppler Altimeter Measurements in the North Sea. *Remote Sens.* **2018**, *10*, 1100.
https://doi.org/10.3390/rs10071100

**AMA Style**

Reale F, Dentale F, Carratelli EP, Fenoglio-Marc L. Influence of Sea State on Sea Surface Height Oscillation from Doppler Altimeter Measurements in the North Sea. *Remote Sensing*. 2018; 10(7):1100.
https://doi.org/10.3390/rs10071100

**Chicago/Turabian Style**

Reale, Ferdinando, Fabio Dentale, Eugenio Pugliese Carratelli, and Luciana Fenoglio-Marc. 2018. "Influence of Sea State on Sea Surface Height Oscillation from Doppler Altimeter Measurements in the North Sea" *Remote Sensing* 10, no. 7: 1100.
https://doi.org/10.3390/rs10071100