^{1}

^{2}

^{*}

^{1}

^{2}

^{1}

^{2}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The determination of wave height by active satellite remote sensing, be it Synthetic Aperture Radar (SAR) or altimeter, has been a common practice for many years and is now imbedded on many meteorological and oceanographic forecasting systems. Despite their differences, all active sensors are based on the measurement of the Normalized Radar Cross Section (NRCS) of the sea surface, _{s}

Radar altimeters such as fitted in many satellite constellations (_{s} (Significant Wave Height) over the oceans for many years. The theory, as well as the practice of radar altimeter is fully consolidated, and it is reported in many classical works. Briefly summarizing, an electromagnetic impulse of very short duration (typical effective pulse duration after pulse compression technique is about 3 n∙s) is sent from the antenna to the sea surface and the reflected signal (waveform) after preliminary processing to correct errors of electromagnetic origin, is analyzed to extract the required parameters. A schematic waveform is illustrated in

Total reflected energy (segment _{0}

Calibration of the link between _{0}

Signal processing also provides the delay of the central point (epoch at middle height in _{s}_{s} from radar altimeters data originate from the so called Brown’s model and successive variants. According to such models, the only relevant information on the sea state is the statistical distribution

Under such assumptions, SSL derives immediately from the position of the

Things become more complicated if the effects of floating foam are taken into consideration. The mechanics of foam over the sea surface is still an active subject of research, and the terminology is not totally uniform. Following, for instance, Holthuijsen

Foam backscatters electromagnetic waves very differently from blue water, and in particular the reflection at nearly-vertical incidence angle, as it occurs in altimeters, is much lower. Floating foam is far from being uniformly distributed over the wave height, and the shape of the waveform is no longer related in a simple way to _{s} and the SSL; this latter, in particular is very important because the precision needed for the geophysical applications [

At present, however, a full reconstruction of all the phenomena involved is still barred by the complexity of the hydrodynamic aspects which limits the applicability of analytical approaches.

A numerical rather than analytical technique is certainly the best way to take into account the geometrically and physically complex problem of the spatial distribution of the backscattering coefficient over the sea surface; this comes, of course, at the price of a very heavy computational effort, which explains why only recently useful results have become available in this field. An early work along these lines was started by Pugliese Carratelli _{s} on the peak spectrum period. More recent results for both altimeter and SAR data can be found in [

In the present work a more complete algorithm is used. By starting from a pseudo-random realization of the sea state water height and by adding appropriate hypotheses on the dynamics and the positions of whitecaps and foam, radio altimeter waveforms are simulated. A description is reported in _{i}_{i}_{i}

The computation of _{f}_{w}

The example shown in this paper is an attempt to investigate into the effects of foam on such function. Fractional water covered surface

The evaluation of

Its distribution

the frequency and the location of foam birth (defined as “whitecaps” in the following);

the time decay of foam once it is generated (defined as “floating foam” in the following);

the movement of floating foam over the wave surface.

Ample research work has been carried out on these three aspects separately, and by appropriately making use of available results a solution can be found to the problem of defining the average

As for the formation of whitecaps, [

After its birth, floating foam decays, and this process has also been the object of much research work; to our purposes the results are summarized as in [^{*} and

For the purpose of present work the formation time of whitecaps ^{*} is negligible [

A connection between the thickness of the foam layer and the microwave properties is provided in [_{f}

The next step is the calculation of the vertical distribution _{f}_{f}_{f}

The calculations are carried out through a pseudo-code outlined in

Sea surface realizations

The trajectories of foam particles are then calculated from their birth to their disappearance by considering the surface velocity values and the time evolution laws described above.

_{f}

Such functions are in turn used in the procedure outlined in _{s}_{i}

As shown above, it is possible to simulate the effects of foam spatial distribution on the sea state parameters. According to the diagram outlined in

Applying the standard procedure to extract the sea parameters from the waveforms, two different set of values are thus obtained, one of which represents the conventional “constant backscattering” point of view, the other—presumably—more realistic. The best candidate for this kind of test is SSL, since the accuracy required for this parameter is very high:

This difference is one of the components of the so-called Sea State Bias (SSB) [_{s} and wind speed U_{10}.

Even though a proper calibration of the procedure cannot be carried out since no universally valid procedure to model SSB exists to date, it makes however sense to compare the results against a classical set of experimental data such as those provided by Melville

The order of magnitude is well reproduced, and what is more important FISSB is systematically lower—for low winds and wave heights—than the measured one, as it could be expected since other mechanisms which contribute to SSB and in particular the non-symmetry of water heights has not been taken onto account, since the surface model used here is inherently Gaussian. The difference tends to disappear for very strong seas and wind, but the dispersion of data and the various uncertainties of the procedure make the comparison arguable beyond a certain point. A consistent explanation of the effects of uneven distribution of foam over the wave has however been provided, and it has been shown to be a major effect.

A numerical flexible and comprehensive model has been presented to simulate the formation of the waveforms of satellite altimeters by taking into account the non-uniform distribution of the backscattering coefficient over the waves. The algorithm has been applied successfully to simulate the influence of whitecaps and floating foam, and results show that part of the SSB can be consistently attributed to these effects.

The flexible numerical model presented here is already useful to improve the understanding of SSB, and might eventually prove also a tool to investigate into the mechanism of whitecaps formation and floating foam movement. Future developments should involve the introduction of a non-linear sea surface model to correctly represent wave skewness. Simulated results should be compared with available altimeter wind speed calibration curves and—when available—with altimeter waveform data.

Work carried out within ESA-ESRIN Project CAT-1 No 1172: “Remote sensing of wave transformation”. The contribution of CUGRI (University Centre for Research on Major Hazard) is also gratefully acknowledged.

The main ideas leading to this work were equally due to the three authors. F. Reale carried out most of the computations.

The Authors declare no conflict of interest.

Typical altimeter waveform over sea surface. _{0}_{s}

Schematic simulation procedure for altimeter waveform: (

Simulation procedure for altimeter waveform: pseudo-code (comment in italic).

Time evolution foam-layer thickness (after [

Simulation procedure for fractional foam surface pseudo-code (comments in italic).

Examples of trajectories of foam particle. The particles move from left (red dots) to right (blue dots). Scale is distorted.

Typical vertical distribution of reflectivity vertical distribution _{f}

Procedure to compare the effects of the foam spatial distribution on sea state parameters.

Simulated altimeter waveform with foam (red curve) and without foam (blue curve) for two different sea state conditions: (_{s} = 6 m; (_{s} = 8 m.

Simulated Sea State Bias SSB as function of H_{s} (_{10} (_{s}.

Comparison between FISSB as function of Hs through simulated foam effect (present paper) and Melville

Some values of Foam-Induced Sea State Bias (FISSB) as function of H_{s} and wind speed U_{10.}

_{s} (m) |
_{p} (s) |
_{10} (m∙s^{−1}) |
_{s} | |||
---|---|---|---|---|---|---|

4 | 8.84 | 11.39 | 250 | 1.65 | 19 | 0.48% |

6 | 10.29 | 18.00 | 250 | 6.19 | 11 | 1.83% |

8 | 11.47 | 24.89 | 250 | 16.17 | 33 | 4.13% |

10 | 12.47 | 32.00 | 250 | 34.60 | 55 | 5.50% |