The Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter

Based on Longuet-Higgins\u27s theory of the probability distribution of wave amplitude and wave period and on some observations, a new probability density function (PDF) of ocean surface slopes is derived. It is f(zeta(x), zeta(y)) = n/2 pi(n - 1)sigma(u) sigma(c) x [1 + zeta(x)(2)/(n - 1)sigma(n)(2) + zeta(y)(2)/(n - 1)sigma(c)(2)](-(n + 2)/2) + skewness, where zeta(x) and zeta(y) are the slope components in upwind and crosswind directions, respectively; sigma(u)(2) and sigma(c)(2) are the corresponding mean-square slopes. The peakedness of slopes is generated by nonlinear wave-wave interactions in the range of gravity waves. The skewness of slopes is generated by nonlinear coupling between the short waves and the underlying long waves. The peakedness coefficient n of the detectable surface slopes is determined by both the spectral width of the gravity waves, and the ratio between the gravity wave mean-square slope and the detectable short wave mean-square slope. When n equals 10, the proposed PDF fits the Gram Charlier distribution, given by Cox and Munk, very well in the range of small slopes. When n --\u3e infinity, it is very close to the Gaussian distribution. Radar backscatter cross sections (RBCS), calculated from specular reflection theory using the new PDF of the C-band radar filtered surface slopes, are in keeping with empirically based ERS-1 C-band scatterometer models. In other words, the proposed PDF can be used successfully in the specular reflection theory to predict the RBCS in the range of incidence angles away from normal incidence. This suggests that the proposed PDF can be used to describe the distribution of surface slopes over the full range of slopes. This is an improvement over the Gaussian distribution and the Gram Charlier distribution. The comparison between the calculated RBCS and the ERS-1 C-band scatterometer models indicates that the peakedness coefficient n should be 5, for wind condition of U-10 less than or equal to 10 m s(-1). It is also found that the spectral width plays an important role on radar backscatter in the range of incidence angles less than 30 degrees

On alpha stable distribution of wind driven water surface wave slope

We propose a new formulation of the probability distribution function of wind driven water surface slope with an $\alpha$-stable distribution probability. The mathematical formulation of the probability distribution function is given under an integral formulation. Application to represent the probability of time slope data from laboratory experiments is carried out with satisfactory results. We compare also the $\alpha$-stable model of the water surface slopes with the Gram-Charlier development and the non-Gaussian model of Liu et al\cite{Liu}. Discussions and conclusions are conducted on the basis of the data fit results and the model analysis comparison.Comment: final version of the manuscript: 25 page

Technical background, chapter 3, part B

A description is given of the physics of electromagnetic scattering from the sea and a guideline is presented to relate an observable (such as the radar cross section) to the hydrodynamics or physical properties of the sea. As specific examples of the interdisciplinary science of electromagnetics and geophysical oceanography, the physics is discussed in connection with data provided by three instruments: namely, the scatterometer, the altimeter, and the imaging radar. The data provided by each instrument are discussed in context with specular point and Bragg scattering theories. Finally, the degrading effect of extraneous sources of noise is discussed as a limiting mechanism of the accuracy of the ocean surface measurement

Determination of mean surface position and sea state from the radar return of a short-pulse satellite altimeter

Using the specular point theory of scatter from a very rough surface, the average backscatter cross section per unit area per radar cell width is derived for a cell located at a given height above the mean sea surface. This result is then applied to predict the average radar cross section observed by a short-pulse altimeter as a function of time for two modes of operation: pulse-limited and beam-limited configurations. For a pulse-limited satellite altimeter, a family of curves is calculated showing the distortion of the leading edge of the receiver output signal as a function of sea state (i.e., wind speed). A signal processing scheme is discussed that permits an accurate determination of the mean surface position--even in high seas--and, as a by-product, the estimation of the significant seawave height (or wind speed above the surface). Comparison of these analytical results with experimental data for both pulse-limited and beam-limited operation lends credence to the model. Such a model should aid in the design of short-pulse altimeters for accurate determination of the geoid over the oceans, as well as for the use of such altimeters for orbital sea-state monitoring

Microwave backscattering theory and active remote sensing of the ocean surface

The status is reviewed of electromagnetic scattering theory relative to the interpretation of microwave remote sensing data acquired from spaceborne platforms over the ocean surface. Particular emphasis is given to the assumptions which are either implicit or explicit in the theory. The multiple scale scattering theory developed during this investigation is extended to non-Gaussian surface statistics. It is shown that the important statistic for the case is the probability density function of the small scale heights conditioned on the large scale slopes; this dependence may explain the anisotropic scattering measurements recently obtained with the AAFE Radscat. It is noted that present surface measurements are inadequate to verify or reject the existing scattering theories. Surface measurements are recommended for qualifying sensor data from radar altimeters and scatterometers. Additional scattering investigations are suggested for imaging type radars employing synthetically generated apertures

Radar imaging mechanism of marine sand waves at very low grazing angle illumination

The investigations carried out between 2002-2004 during several field experiments within the Op-erational radar and optical mapping in monitoring hydrodynamic, morphodynamic and environ-mental parameters for coastal management project (OROMA) aimed to improve the effectiveness of new monitoring technologies such as shipborne imaging radars in coastal waters. The coastal monitoring radar of the GKSS Research Centre, Geesthacht, Germany, is based on a Kelvin Hughes RSR 1000 X-band (9.42 GHz) VV polarized river radar and was mounted on board the research vessel Ludwig Prandtl during the experiments in the Lister Tief, a tidal inlet of the German Bight in the North Sea. The important progress realized in this investigation is the availability of calibrated X-band radar data. Another central point of the study is to demonstrate the applicability of the quasi-specular scattering theory in combination with the weak hydrodynamic interaction the-ory for the radar imaging mechanism of the sea bed. It is shown that specular point scattering con-tributes significantly to the normalized radar cross section (NRCS) modulation due to marine sand waves. According to the theory quasi-specular scattering can be applied for wind speeds Uw ≤ 8 m s-1. Measured and simulated NRCS modulations caused by flood and ebb tide oriented marine sand waves have been compared and agree fairly wel

Remote sensing of sea state by laser altimeters

The reflection of short laser pulses from the ocean surface was analyzed based on the specular point theory of scattering. The expressions for the averaged received signal, shot noise and speckle induced noise were derived for a direct detection system. It is found that the reflected laser pulses have an average shape closely related to the probability density function associated with the surface profile. This result is applied to estimate the mean sea level and significant wave height from the receiver output of the laser altimeter

Experimental measurement and theoretical modeling of microwave scattering and the structure of the sea surface influencing radar observations from space

The electromagnetic bias is an error present in radar altimetry of the ocean due to the non-uniform reflection from wave troughs and crests. A study of the electromagnetic bias became necessary to permit error reduction in mean sea level measurements of satellite radar altimeters. Satellite radar altimeters have been used to find the upper and lower bounds for the electromagnetic bias. This report will present a theory using physical optics scattering and an empirical model of the short wave modulation to predict the electromagnetic bias. The predicted electromagnetic bias will be compared to measurements at C and Ku bands

Altimetric system: Earth observing system. Volume 2h: Panel report

A rationale and recommendations for planning, implementing, and operating an altimetric system aboard the Earth observing system (Eos) spacecraft is provided. In keeping with the recommendations of the Eos Science and Mission Requirements Working Group, a complete altimetric system is defined that is capable of perpetuating the data set to be derived from TOPEX/Poseidon, enabling key scientific questions to be addressed. Since the scientific utility and technical maturity of spaceborne radar altimeters is well documented, the discussion is limited to highlighting those Eos-specific considerations that materially impact upon radar altimetric measurements

Feasibility study and design concept for an orbiting ice-penetrating radar sounder to characterize in three-dimensions the Europan ice mantle down to (and including) any ice/ocean interface

This report presents a radar sounding model based on the range of current working hypotheses for the nature of Europa's icy shell.Institute for Geophysic