# The SAR Image of Short Gravity Waves On a Long Gravity Wave

## Abstract

A SAR imaging model appropriate to oceanographic applications is derived, unifying fundamental models of hydrodynamics, rough surface scattering, and SAR imaging of time-variant scenes. The sea surface is a sinusoidal long gravity wave upon which short gravity waves propagate and are modified by the long wave in accordance with a recent theory of Phillips; the electromagnetic scattering is described by the two-scale approximation appropriate to long wave and short wave ensembles that are, respectively, smoothly varying and not too rough with respect to the radar wavelength. The resulting model, accurate to first order in the long wave slope, for the first time fundamentally characterizes the nonlinear hydrodynamic and scattering interactions of the long and short waves and their effect, along with temporal variation, on the SAR image. Of particular importance, the long wave enters (among other ways) as a phasemodulated waveform that, when filtered by the SAR system, can be, for large-amplitude long waves, the principal determinant of the image nature. The numerical analysis of the model is discussed and an approximation describing the image of a delimited scene area is derived and exemplified. (1) When the small waves are a range-directed ensemble and the long wave is azimuth directed, the latter’s temporal variation “blurs,” in azimuth, the image due, primarily, to the SAR system’s narrowband filtering of the aforementioned phase- modulated waveform and, secondarily, the nonlinear hydrodynamic interaction; it is shown that this “blurring” is, at higher long wave amplitudes, due to a quadratic phase error proportional to the phase velocity of the long wave. That part of the short wave ensemble allowed influential by SAR system (wavenumber) filtering is approximately nondispersive during the SAR azimuth integration time, its concerted effect being a rigid azimuth image translation, proportional to the short wave’s mean phase velocity. (2) More briefly treated, when the long wave is ranged directed and the short wave ensemble is simply a single range-directed sinusoid (“Bragg-matched”), the image is solely range variant and its nature is primarily determined by the aforementioned narrowband filtering effect, the secondary effects of nonlinear hydrodynamics and physical optics (i.e., surface slope) being evident. Therefore, the present model, as thus far elaborated, contradicts predictions of models based on the SAR response to a point scatterer in motion in accordance with the “orbital motion” of the long wave: e.g., no “azimuth bunching” attributed to such motion is observed.

## Keywords

Phase Velocity Short Wave Synthetic Aperture Radar Modulation Transfer Function Wavenumber Domain## Preview

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## References

- Alpers, W. R., and C. L. Rufenach (1979): The effect of orbital motions on synthetic aperture radar imagery of ocean waves.
*IEEE Trans. Antennas Propag.***AP-27**, 685–690.CrossRefGoogle Scholar - Alpers, W. R., D. B. Ross, and C. L. Rufenach (1981): On the detectability of ocean surface waves by real and synthetic aperture radars.
*J. Geophys. Res.***86**, 6481–6498.CrossRefGoogle Scholar - Bass, F. G., I. M. Fuks, A. I. Kalmykov, I. E. Ostrovsky, and A. D. Rosenberg (1968): Very high frequency radio wave scattering by a disturbed sea surface.
*IEEE Trans. Antennas Propag.***AP-16**, 554–568.CrossRefGoogle Scholar - Beal, R. C., P. S. DeLeonibus, and I. Katz (eds.) (1981):
*Spaceborne Synthetic Aperture Radar for Oceanography*Johns Hopkins Press, Baltimore.Google Scholar - Beckmann, P., and A. Spizzichino (1963):
*The Scattering of Electromagnetic Waves from Rough Surfaces*, Macmillan Co., New York.MATHGoogle Scholar - Harger, R. O. (1970):
*Synthetic Aperture Radar Systems*, Academic Press, New York.Google Scholar - Harger, R. O. (1980a): On SAR sea image prediction.
*lUCRM Symposium on Oceanography from Space*, Venice.Google Scholar - Harger, R. O. (1980b): The synthetic aperture radar image of time-variant scenes.
*Radio Sci*.**15**, 749–756.CrossRefGoogle Scholar - Harger, R. O. (1981): SAR ocean imaging mechanisms.
*Spaceborne Synthetic Aperture Radar for Oceanography*(R. C. Beal, P. S. Deleonibus, and I. Katz, eds.), Johns Hopkins Press, Baltimore.Google Scholar - Harger, R. O. (1983): A sea surface height estimator using synthetic aperture radar complex imagery.
*IEEE Trans. Oceanic Eng*.**OE-8**, 71–78.CrossRefGoogle Scholar - Harger, R. O. (1984): A fundamental model and efficient inference for SAR ocean imagery.
*IEEE Trans. Oceanic Eng*.**OE-9**, 260–276.Google Scholar - Jain, A. (1981): SAR imaging of ocean waves: Theory.
*IEEE Trans. Ocean. Eng*.**OE-6**, 130–139.CrossRefGoogle Scholar - Kinsman, B. (1965):
*Wind Waves*, Prentice-Hall, Englewood Cliffs, N.J.Google Scholar - Kitaigorodskii, S. A. (1981): Comments.
*Spaceborne Synthetic Aperture Radar for Oceanography*(R. C. Beal, P. S. DeLeonibus, and I. Katz, eds.), Johns Hopkins Press, Baltimore, 187.Google Scholar - Longuet-Higgins, M. S., and R. W. Stewart (1961): Changes in the form of short gravity waves on long waves and tidal currents.
*J. Fluid Mech*.**11**, 565–583.Google Scholar - Phillips, O. M. (1977):
*The Dynamics of the Upper Ocean*, Cambridge University Press, London. Phillips, O. M. (1981a).MATHGoogle Scholar - Phillips, O. M. The structure of short gravity waves on the ocean surface.
*Spaceborne Synthetic Aperture Radar for Oceanography*(R. C. Beal, P. S. DeLeonibus, and I. Katz, eds.), Johns Hopkins Press, Baltimore.Google Scholar - Phillips, O. M. (1981b): The dispersion of short wavelets in the presence of a dominant long wave.
*J. Fluid Mech*.**107**, 465–485.MATHCrossRefGoogle Scholar - Plant, W. J. (1977): Studies of backscattered sea return with a CW dual-frequency, X-band radar.
*IEEE Trans. Antennas Propag*.**AP-25**, 28–36.Google Scholar - Plant, W. J. (1981): The two-scale radar wave probe and SAR imagery of the ocean. U.S. Naval Research Laboratory Report (informally communicated).Google Scholar
- Shuchman, R. A., and J. S. Zelenka (1978): Processing of ocean wave data from a synthetic aperture radar.
*Boundary-Layer Meteorol*.**13**, 181–192.CrossRefGoogle Scholar - Swift, C. T., and L. R. Wilson (1979): Synthetic aperture radar imaging of ocean waves.
*IEEE Trans. Antennas Propag*.**AP-27**, 725–729.CrossRefGoogle Scholar - Valenzuela, G. R. (1968): Scattering of electromagnetic waves from a tilted, slightly rough surface.
*Radio Sci*.**3**, 1057–1066.Google Scholar - Valenzuela, G. R. (1980): An asymptotic formulation for SAR images of the dynamical ocean surface.
*Radio Scl***15**, 105–114.CrossRefGoogle Scholar - Wright, J. W. (1966): Backscattering from capillary waves with application to sea clutter.
*IEEE Trans*.*Antennas Propag*.**AP-14**, 749–754.CrossRefGoogle Scholar - Wright, J. W. (1968): A new model for sea clutter.
*IEEE Trans. Antennas Propag*.**AP-16**, 217–223.CrossRefGoogle Scholar - Zalkan, R., and L. Wentzel (eds.) (1981):
*Proceedings of a NORD A Workshop on Describing Ocean Phenomena Using Coherent Radars*. NORDA Technical Note 104, MayGoogle Scholar