An exact analytic representation of a regular or interferometric SAR
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 2, MARCH 1999
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An Exact Analytic Representation of a Regular or Interferometric SAR Image of Ocean Swell Gregory Zilman and Lev Shemer
Abstract—The problem of obtaining quantitative data on spatial ocean wave spectra from the images of the ocean surface by either regular SAR or along-track interferometric SAR (INSAR) is studied. The dominant mechanism which allows imaging of ocean waves by SAR/INSAR is the so-called velocity bunching. This mechanism is essentially nonlinear. The theoretical analysis of SAR/INSAR imagery of the ocean surface due to velocity bunching is performed, and nonlinear solutions of the SAR/INSAR images of monochromatic waves and of the spectra of these images are obtained. Analytic expressions are presented here which allow to simulate accurately both SAR and INSAR images of waves with arbitrary lengths, heights and propagation directions. It is demonstrated that a monochromatic wave expands in the SAR/INSAR images into an infinite number of harmonics. In addition to the nonlinearity parameters of SAR which is related to the velocity bunching mechanism, it is shown that for complex INSAR, the degree of nonlinearity depends also on separation time between the two antennas. The results of the present study indicate that in addition to the prevailing practice to consider the phase component of the INSAR image, an analysis of the imaginary part of the complex INSAR map of the ocean surface may provide some supplementary information, beneficial, in particular, for rough sea. Index Terms—Ocean waves, remote sensing, SAR/along track interferometric SAR image, sea surface.
I. INTRODUCTION
R
EMOTE sensing of the ocean surface by regular synthetic aperture radar (SAR) offers substantial advantages over alternative instruments due to its high resolution, broad coverage and independence of weather conditions [1]. Spatial wave spectra were successfully measured by SAR [2]–[6]. Goldstein and Zebker [7] and Goldstein et al. [8] demonstrated the possibility to use the along-track INSAR for measuring currents in the ocean by using the phase component of the complex INSAR image. Under certain simplifying assumptions the phase component is directly proportional to the line-of-sight component of the scattered velocity on the surface. Marom et al. [9]–[10] took the advantage of these features of INSAR, and applied this instrument to study the twodimensional ocean wave spectra. It was demonstrated that the quantitative information on ocean wave heights can indeed be retrieved from the INSAR phase component.
Manuscript received July 31, 1997; revised June 24, 1998. This work was supported by Israel Science Foundation Grant 253/96-3 The authors are with the Department of Fluid Mechanics and Heat Transfer, Tel-Aviv University, Tel-Aviv 69978, Israel. Publisher Item Identifier S 0196-2892(99)01978-6.
Theoretical analysis of the principles of the imaging of ocean waves by INSAR, performed by Shemer and Kit [11], indicates that the velocity bunching may lead to a substantial distortion of the phase component of the complex INSAR image. A further study demonstrated that the phase component of INSAR is only weakly affected by the real aperture radar (RAR) modulation [12]. On the other hand, under certain conditions the magnitudes of both INSAR and the regular SAR outputs can be modified notably by RAR modulation, complicating even further determination of wave heights. Plant and Zurk [13] applied quasilinear theory of SAR imagery of the ocean surface to simulate SAR image spectra for a number of SAR parameters and sea states. In many occasions, their simulations indicate that it is possible not only to extract from the SAR images of the ocean surface the directions of propagation of the dominant waves, but even to estimate the significant wave height. Bao et al. applied MonteCarlo method to simulate ocean waves images by INSAR [14]. One of the major mechanism which renders ocean waves visible in the SAR images is the so-called “velocity-bunching” [15]–[16]. However, SAR imaging mechanism can cause substantial distortion of the spectra and a notable deviation of the propagation direction of the dominant wave [13], [17]. The question therefore arises under which conditions it is possible to extract sufficiently representative information about the waves from the SAR (or INSAR) images. One step toward the solution of this problem is through a full nonlinear SAR/INSAR imaging theory to simulate the image of a monochromatic ocean wave. Hasselmann and Hasselmann [18] and Krogstag [19] derived a closed form nonlinear spectral transform resulting from velocity bunching. The Hasselmanns model can be used to simulate the SAR image (direct problem) and to estimate the wave spectrum from measurements (inverse problem). The spectra of the and that of the resulting image actual wave field are related through an essentially nonlinear mathematical operator (1) From the mathematical standpoint the ultimate goal of SAR/INSAR imagery of the ocean surface is to find a by inverting the nonlinear representative estimate operator (1). It should be taken into account that the structure is known only approximately, of the direct operator is contaminated by noise. Thus, the whereas the output inversion of the functional equation (1) may be impossible in a
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rigorous mathematical sense due to a salient feature of typical inverse problems, i.e., they are frequently ill-posed [20]. This means that the solution of an inverse problems may either be unstable with respect to small input errors, be non unique, or not exist at all. The principal difficulties related to the inversion of the SAR images of ocean waves were considered by Hasselmann and Hasselmann [18], who emphasized that the ill-posed problem can be solved by invoking an a priori information named “first-look spectrum.” In such a way, the solution of an ill-posed inverse problem is reduced to estimation of the wave spectrum parameters by minimizing an appropriately defined difference between the output spectrum and its prediction which takes into account the “first-look spectrum.” There are no formal mathematical rules for identifying a problem as ill-posed. However, if an analytic solution of the direct problem exists, it becomes possible to apply such a solution to estimate the sensitivity of the SAR/INSAR processing to the input parameters, and to use it also as an a priori information for solving the ill-posed inverse problem (1). To obtain an analytic solution, a single monochromatic wave can be used not only because of its practical use and the relative ease with which it can be analyzed, but also because the results of such a study bear important similarities to more general ocean wave fields with a narrow band spectrum, and in particular, for ocean swells. Shemer [21] presented an asymptotic approximation to the complex INSAR image of a monochromatic long and not too steep ocean wave, which accounts for the velocity bunching imaging mechanism. An analytic investigation of the SAR image spectrum was carried out by Lyzenga [22] who represented the temporal correlation function of reflectivity as a Fourier series where the coefficients of the series are expressed analytically. In the present work we also consider a monochromatic progressive wave and calculate the Fourier transform of SAR/INSAR images of such a wave in an explicit form. One of the goals of the work is to illustrate that even for such a simple case, the SAR/INSAR image of a monochromatic wave expands into an infinite number of harmonics which apparently illustrate that the corresponding inverse problem is essentially ill-posed and should be solved by an appropriate regularization as has been proposed in [18]. The derived exact representation of SAR/INSAR Fourier transform may help to delineate the limits of detectability of wave fields with a narrow spectrum and ocean swell by SAR and INSAR.
SAR/INSAR platform moves at a height along the axis in its positive direction with a constant velocity The aircraft in is equipped with two antennas separated by a distance the flight direction. The fore antenna serves as a transmitter, and both antennas serve as receivers. At each instant the so that coordinates of the antennas are two antennas arrive at the same location with the time shift Consider an additional orthogonal system of coordinates also fixed in space, such that the angle between the and is A plane monochromatic harmonic wave axes (an azimuthal wave propagates in moves along the axis 0). The instantaneous surface elevation the direction is defined by the wave amplitude wave number and as Within the wave frequency satisfy framework of the linear surface wave theory, and where is the acceleration the dispersion relation the radar line-of-sight of gravity. For an incidence angle of the free surface point radial velocity component is
where (2) and For modeling SAR/INSAR image of a monochromatic wave the -dependence along the lines of constant range is not essential and may be neglected. By generalizing the approach adopted in [23] to the along-track INSAR its output can be simplified and reduced to a single integral [11]
(3) where
II. PROBLEM FORMULATION The expression for the SAR/INSAR image for a monochromatic wave given in [11] is adopted here. It is assumed in [11] that the radar cross section per unit area is constant and, thus, effects of the titling and the hydrodynamic modulation are not considered. For a regular SAR the expression obtained in [11] is consistent with [23]. is A fixed in space orthogonal coordinate system coincides with the plane of undisselected. The plane axis is directed upward. The turbed free surface and the
Here is the radar wave number, is the wave is the SAR/INSAR integration time, is phase velocity, the scene characteristic coherence time, and a real constant factor affecting the absolute value of the image only.
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Applying the inverse Fourier transform to (16) gives
III. THE SAR/INSAR IMAGE IN THE FOURIER AND IN THE PHYSICAL SPACES is introduced and A new variable of integration (3) is rewritten in a form which is more convenient for the present purposes
(19)
(5)
is used now to define all indepenwhere the coefficient dent factors appearing in the process of derivation (19). The ( , 4) and are given by (2) and quantities (6)–(10). Spliting (19) into the real and the imaginary parts yields
(6)
where
(4)
where
(20)
(7) (8) (9)
(21)
(10) (11) Consider the Fourier transform of the integral (4) with respect to (12) Applying (12) to (4) and integrating with respect to
yields
(22) For a SAR,
see (8), and (20)–(21) yields (23)
(13) where for
where (14)
(24) and IV. NUMERICAL RESULTS The function [24]
can be expanded in Fourier series
(15) are the Bessel functions of the first where kind. Substitution of (15) into (14) and analytic integration with respect to involving Dirac -function gives for (13) (16) where (17) and (18) The analytic expressions (16)–(17) represent the wave-number spectrum of SAR/INSAR image of a monochromatic sea wave.
AND
DISCUSSION
The analytic expression (19) shows that a harmonic wave being processed by SAR/INSAR, with a wave number expands into an infinite number of spectral components with the azimuthal wave numbers where the dependence on the wave phase velocity can be considered as a generalization of the so-called scanning distortion [25]. The occurrence of higher harmonics in the images is an inevitable consequence of nonlinear SAR/INSAR mechanism. The 180 ambiguity results in two system of waves, propagating in the positive and in the negative directions of the axis. The amplitudes of the harmonics in the resulting spectra depend on the wave amplitude nonlinearly are nonlinear with respect since the Bessel functions to the argument and the number of the harmonic Due to the exponential factors (17) in the series (20)–(21) they converge rather fast. Practically it means that only a few terms in (20)–(21) have to be taken into account. For very long 0) with small amplitudes ( 0) it follows waves ( 0. In this case, the series (20)–(21) can from (2) that be truncated to the first term only (25)
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where the phase component represents the well known expression of the “ideal” INSAR output [11]. For small values the Bessel functions behaves as of the argument [27]. Thus, only the Bessel function is linear with respect to its argument. The representation (19) can be seen as linear with respect to the wave amplitude only if the argument of the Bessel function is small, so that the inequality (26) is satisfied. The velocity bunching parameter can be defined as a ratio to the distorted of the typical length scale For INSAR, the additional radar wave length nonlinearity parameter represents the ratio of another length to In view of (23)–(24) and (26), scale, i.e., the parameter defining the nonlinearity of SAR imaging mechThe nonlinearity anism can be represented as which results from velocity bunching, has been parameter obtained by many authors, e.g., [21], [22], and [26]. Similarly, recalling (17)–(18), for INSAR the nonlinearity parameter can be defined as
(a)
(27) Numerical calculations presented where below shows that for typical INSAR configurations The parameter (27) illustrates that for INSAR the nonlinearity depends not only on the velocity bunching, but also on the time delay due to antennas separation. Because of the 180 ambiguity, the nonlinearity parameter is different for waves propagating in the positive and in the negative directions of the axis. Since the Fourier coefficients in (20)–(21) appear as for long and not too steep a sum of Bessel functions waves the degree of nonlinearity practically depends mainly on For comparatively short the velocity bunching parameter or steep waves, however, the directional ambiguity results also in the variations of the velocity bunching for the two waves propagating in the opposite directions. For comparative calculations we define the INSAR parameter of nonlinearity as i.e.: a maximum value of The intrinsic properties of the nonlinear mechanism can be illustrated by considering several first harmonics of SAR grow monotonically image. The Bessel functions until they attain their corresponding first positive extremum at where 1.85), 3.05 1.5), 4.2 ( 1.4) and 5.4 ( ( 1.35) [27]. Thus, the SAR output also grows monotonically as long as the nonlinearity parameter For with the SAR relatively large values of nonlinearity parameter image can be distorted beyond of recognition, especially if the dominant first harmonic vanishes We illustrate the corresponding behavior of SAR/INSAR images and the importance of higher harmonics by a numerical example. A set of INSAR parameters is used which roughly corresponds to those in the experiments with the NASA/JPL 7000 m, 200 L-band airborne INSAR [9]–[10]: 45 , 25 m 0.05 s, 1 2.0 m/s, 0.1 s. s, and
(b)
(c) Fig. 1. SAR images of azimuthally propagating waves and their amplitude spectra. 200 m. (b) T0 2.0 s. (c) s 0.1 s.
=
=
=
Regular SAR output is studied first. Computations are per200 formed for an azimuthally-propagating waves with m, various wave amplitudes, and correspondingly for different Fig. 1(a) represents the values of nonlinearity parameter variation of the regular SAR image intensity as a function of along a constant the distorted phase of the imaged wave range line.
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(a)
(b) Fig. 2. INSAR image of an azimuthally propagating wave.
= 200 m, T0 = 2.0 s, and = 0.1 s.
Results of Fig. 1(a) indicate that the maximum values of the SAR output increase almost linearly with the wave amplitude, remains below some as long as the nonlinearity parameter limiting value of about 1.2. The values of the SAR output for the imaging conditions of Fig. 1 attain their maximum 1.4. For higher values for the nonlinearity parameter of nonlinearity parameter the contrast of the SAR image decreases. For relatively small values of the nonlinearity parameter, the Fourier coefficients of the SAR output decomposition 200 decay monotonically. For example, for wavelength 1.2 m ( 0.67) the dominant m and wave amplitude Fig. 1(b)], harmonic is stronger than the background level resulting in a strong contrast in the corresponding output of Fig. 1(a). On the other hand, the highly distorted SAR output 3.83 in Fig. 1(a) corresponds to the spectrum of for
the image represented in Fig. 1(c) which is characterized by a large value of the background in the absence of the first harmonic and notable second and third harmonics. The along-track INSAR images of the azimuthally propa200 m and identical imaging conditions gating wave with are presented in Fig. 2 for a number of wave amplitudes. The INSAR modulus [Fig. 2(a)] becomes distorted already 0.56, for waves with amplitude exceeding 1 m ( 2.52). Note that the regular SAR output reflects faithfully the imaged wave for much higher amplitudes [see Fig. 1(a)]. The phase component of INSAR, however, provides a relatively weakly distorted image of the ocean swell even for relatively high wave amplitudes [Fig 2(b)]. Since the spectrum of the imaginary part of the INSAR image does not include a DC component ( ), the contrast of the phase component of INSAR is stronger than that of the regular SAR.
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. INSAR image of an azimuthally propagating wave. Wave length
= 75
For extreme imaging conditions, both the modulus and the phase of INSAR can be strongly distorted. An example of such a strong distortion is given in Fig. 3 for an azimuthally propagating wave with 75 m and wave amplitude 1.1 m. The image of the INSAR modulus [Fig. 3(a)] incorporates a considerable second harmonic, while the phase of the image contains discontinuities due the fact that the absolute value of INSAR phase [Fig. 3(b)] exceeds 180 .
m. Amplitude aw
= 1.1 m. T0 = 1.0 s. s = 0.1 s.
The complex INSAR image is presented in Fig. 3(a) and (b) in a traditional way, i.e., as its modulus and the phase angle. Its real and imaginary parts are presented in Fig. 3(c) and (d). The modulus of the complex INSAR output [Fig. 3(a)] and its real part [Fig. 3(c)] are distorted due to the presence of a considerable second harmonics. The spectrum of the real part has a strong DC component which reduces the contrast. The imaginary part of the image also contains a considerable second harmonic [Fig. 3(f)], but compared to the
ZILMAN AND SHEMER: REGULAR OR INTERFEROMETRIC SAR IMAGE OF OCEAN SWELL
phase of the image [Fig. 3(b)], the shape of the imaginary part remains quite close to the “ideal output.” As noted above, the spectrum of the imaginary part of the INSAR image does not include a DC component which results in a strong contrast. It thus may be concluded that for some particular imaging conditions, especially when the phase of the output contains discontinuities, the analysis of an imaginary part of INSAR can add some essential information about the wave field. It should be emphasized that the numerical examples presented here do not cover the variety of the problem parameters which can affect the results. V. CONCLUSION 1) A new closed-form explicit representation of the SAR/INSAR image of ocean swell is derived in this paper. It provides a clearer insight into the imaging mechanism by illustrating how a single monochromatic wave is transformed into a nonsinusoidal function so that its spectral representation includes an infinite number of higher harmonics. The amplitudes of these harmonics depend on the degree of the nonlinearity of the SAR/INSAR mechanism. Two criteria of the nonlinearity are derived. For a regular SAR, the nonlinearity depends mainly on the value of velocity bunching coefficient ([21], [22], [26]). For the alongtrack INSAR the derived criterion of nonlinearity depends not only on the velocity bunching parameter, but also on the separation time between two antennas. 2) Higher harmonics in SAR/INSAR images constitute an unavoidable consequence of the inherent properties of the nonlinear nature of the SAR/INSAR imaging. The present study illustrates that due to the difficulties in distinguishing between the fundamental and higher harmonics, as well as due to the presence of measurement noise, the inverse problem of SAR/INSAR imaging should be considered as a ill-posed [20], [28]. One of the possible ways to regularize the ill-posed problem is by invoking a priory information as suggested by Hasselmann and Hasselmann [18]. 3) Under some conditions, in particular for comparatively short and steep waves INSAR nonlinearity can be more pronounced than that for a regular SAR resulting in stronger higher harmonics in images. When the phase of the INSAR contains discontinuities, the imaginary part of the complex INSAR image and its Fourier coefficients may provide useful supplementary information about the wave field. This may be particularly true for the imaging conditions when the absolute value of the INSAR phase can exceed 180 . REFERENCES [1] K. Hasselmann, R. K. Raney, W. J. Plant, W. Alpers, R. A. Shuchman, D. R. Lyzenga, C. L. Rufenach, and M. J. Tucker, “Theory of synthetic aperture radar ocean imaging: A MARSEN view,” J. Geophys. Res, vol. 90, pp. 4659–4686, 1985. [2] R. C. Beal, D. G. Tilley, and F. M. Monaldo, “Large- and smallscale spatial evolution of digitally processed ocean wave spectra from SEASAT synthetic aperture radar,” J. Geophys. Res., vol. 88, pp. 1761–1778, 1983.
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[3] R. C. Beal, F. M. Monaldo, D. G. Tilley, D. E. Irvine, E. J. Walsh, F. C. Jackson, D. W. Hancock III, D. E. Hines, R. N. Swift, F. I. Gonzales, D. R. Lyzenga, and L. F. Zambresky, “A comparison of SIR-B directional ocean wave spectra with aircraft scanning radar spectra,” Science, vol. 232, pp. 1531–1535, 1986. [4] P. W. Vachon, R. B. Olsen, C. E. Livingstone, and N. G. Freeman, “Airborne SAR imagery of ocean surface waves obtained during LEWEX: Some initial results,” IEEE Trans. Geosci. Remote Sensing, vol. 26, pp. 548–561, 1988. [5] T. Maklin and R. Cordey, “Ocean-wave imaging by synthetic-aperture radar: Results from SIR-B experiment in the N. E. Atlantic,” IEEE Trans. Geosci. Remote Sensing, vol. 27, pp. 28–35, 1989. [6] C. L. Rufenach, R. B. Olsen, R. A. Shuchman, and C. A. Russel, “Comparison of aircraft synthetic aperture radar and buoy spectra during the Norwegian continental shelf experiment of 1988,” J. Geophys. Res., vol. 10, pp. 10 423–10 441, 1991. [7] R. M. Goldstein and H. A. Zebker, “Interferometric radar measurement of ocean surface currents,” Nature, vol. 328, pp. 707–709, 1987. [8] R. M. Goldstein, T. P. Barnett, and H. A. Zebker, “Remote sensing of ocean currents,” Science, vol. 246, pp. 1282–1285, 1989. [9] M. Marom, R. M. Goldstein, E. B. Thornton, and L. Shemer, “Remote sensing of ocean wave spectra by interferometric synthetic aperture radar,” Nature, vol. 345, pp. 793–795, 1990. [10] M. Marom, L. Shermer, and E. B. Thornton, “Energy density directional spectra of near shore wave field measured by interferometric synthetic aperture radar,” J. Geophys. Res., vol. 96, pp. 22 125–22 134, 1991. [11] L. Shemer and E. Kit, “Simulation of an interferometric SAR imagery of an ocean system consisting of a current and a monochromatic wave,” J. Geophys. Res., vol. 96, pp. 22 063–22 074, 1991. [12] L. Shemer, “Interferometric SAR imagery of a monochromatic ocean wave in the presence of the real aperture radar modulation,” Int. J. Remote Sensing, vol. 14, pp. 3005–3019, 1993. [13] W. J. Plant and L. M. Zurk, “Dominant wave directions and significant wave heights from synthetic aperture radar imagery of the ocean,” J. Geophys. Res., vol. 102, pp. 3473–3482, 1997. [14] M. Bao, M., C. Br¨uning, and W. Alpers, “Simulation of ocean waves imaging by along-track interferometric synthetic aperture radar,” IEEE Trans. Geosci. Remote Sensing, vol. 35, pp. 618–631, 1997. [15] W. R. Alpers and C. L. Rufenach, “The effect of orbital motions on synthetic aperture radar imagery of ocean waves,” IEEE Trans. Antennas Propagat., vol. AP-27, pp. 685–690, 1979. [16] C. T. Swift and L. R. Wilson, “Synthetic aperture radar imaging of moving ocean waves,” IEEE Trans. Antennas Propagat., vol. AP-27, pp. 725–729, 1979. [17] C. Br¨uning C, W. Alpers, L. F. Zambersky, and D. G. Tilley, “Validation of a synthetic aperture radar ocean wave imaging theory by the shuttle imaging radar-B experiment over North sea,” J. Geophys. Res., vol. 93, pp. 15 403–15 425, 1988. [18] K. Hasselmann and S. Hasselmann, “On the nonlinear mapping of an ocean wave spectrum into a synthetic aperture radar image spectrum and its inversion,” J. Geophys. Res., vol. 96, pp. 10, 10 713–10 729, 1991. [19] H. E. Krogstad, “A simple derivation of Hasselmann’s nonlinear oceansynthetic aperture radar transform,” J. Geophys. Res., vol. 97, pp. 2,421–2,425, 1992. [20] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems. Washington, DC: Winston, 1977. [21] L. Shemer, “An analytical presentation of the monochromatic ocean wave image by a regular or an interferometric synthetic aperture radar,” IEEE Trans. Geosci. Remote Sensing, vol. 33, pp. 1008–1013, July 1995. [22] D. R. Lyzenga, “An analytic representation of the synthetic aperture radar image spectrum for ocean waves.” J. Geophys. Res., vol. 93, pp. 13 859–13 865, 1988. [23] D. P. Kasilingam and O. H. Shemdin, “Theory for synthetic aperture radar imaging of the ocean surface: With application to the tower ocean wave and radar dependence experiment on focus, resolution, and wave height spectra,” J. Geophys. Res., vol. 93, pp. 13 837–13 858, 1988. [24] H. Bateman and A. Erd´eyi, Higher Transcendental Function. New York: McGraw-Hill, 1954. [25] R. K. Raney and R. T. Lowry, “Oceanic wave imagery and wave spectra distortions by synthetic aperture radar imagery,” in Proc. 12th Int. Symp. Remote Sensing Environ., Manila, The Phillippines, 1978, pp. 683–702. [26] W. Alpers, “Monte Carlo simulations for studying the relationship between ocean wave and synthetic aperture radar image spectra,” J. Geophys. Res., vol. 88, pp. 1745–1759, 1989. [27] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Washington, DC: National Bureau of Standards, 1964. [28] P. Eykhoff, System Identification. Parameter and State Estimation. New York: Wiley, 1974.
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Gregory Zilman received the Dipl. Eng.Researcher degree in hydrodynamics and the Ph.D. degree in naval hydrodynamics, both from Leningrad (St. Petersburg) Shipbuilding Institute (LSI), Russia, in 1971 and 1978, respectively. He was employed at LSI from 1971 to 1990. He started as an Engineer-Researcher in the LSI towing tank. The academic positions of Senior Research Scientist and Associate Professor in the Department Theory of Ships and Hydrodynamics followed. In 1991, he joined the Department of Fluid Mechanics, Tel-Aviv University, Tel-Aviv, Israel, as a Senior Research Fellow and where he is now a Principal Research Fellow. His research interests in marine and naval hydrodynamics include wave motion, with emphasis on ship generated surface and internal waves, as well as microwave remote sensing of ship wakes.
Lev Shemer received the M.Sc. degree in physics from the Moscow Phys. Tech. Institute, Moscow, Russia, and the Ph.D. degree in mechanical engineering from Tel-Aviv University, Tel-Aviv, Israel, in 1970 and 1981, respectively. After a postdoctoral stay with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, he joined the faculty at the Department of Fluid Mechanics, TelAviv University, where he currently serves as a Professor of Fluid Mechanics. His main research interests are in experimental fluid mechanics, as well as in experimental and theoretical study of nonlinear water waves. During a sabbatical stay in 1989–1990, at Naval Postgraduate School, Monterey, CA, he became involved in studies of the ocean waves and currents using either regular or along-track interferometric SAR. Since 1990, his research activities have included various aspects of remote sensing of water surface.
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An Exact Analytic Representation of a Regular or Interferometric SAR Image of Ocean Swell Gregory Zilman and Lev Shemer
Abstract—The problem of obtaining quantitative data on spatial ocean wave spectra from the images of the ocean surface by either regular SAR or along-track interferometric SAR (INSAR) is studied. The dominant mechanism which allows imaging of ocean waves by SAR/INSAR is the so-called velocity bunching. This mechanism is essentially nonlinear. The theoretical analysis of SAR/INSAR imagery of the ocean surface due to velocity bunching is performed, and nonlinear solutions of the SAR/INSAR images of monochromatic waves and of the spectra of these images are obtained. Analytic expressions are presented here which allow to simulate accurately both SAR and INSAR images of waves with arbitrary lengths, heights and propagation directions. It is demonstrated that a monochromatic wave expands in the SAR/INSAR images into an infinite number of harmonics. In addition to the nonlinearity parameters of SAR which is related to the velocity bunching mechanism, it is shown that for complex INSAR, the degree of nonlinearity depends also on separation time between the two antennas. The results of the present study indicate that in addition to the prevailing practice to consider the phase component of the INSAR image, an analysis of the imaginary part of the complex INSAR map of the ocean surface may provide some supplementary information, beneficial, in particular, for rough sea. Index Terms—Ocean waves, remote sensing, SAR/along track interferometric SAR image, sea surface.
I. INTRODUCTION
R
EMOTE sensing of the ocean surface by regular synthetic aperture radar (SAR) offers substantial advantages over alternative instruments due to its high resolution, broad coverage and independence of weather conditions [1]. Spatial wave spectra were successfully measured by SAR [2]–[6]. Goldstein and Zebker [7] and Goldstein et al. [8] demonstrated the possibility to use the along-track INSAR for measuring currents in the ocean by using the phase component of the complex INSAR image. Under certain simplifying assumptions the phase component is directly proportional to the line-of-sight component of the scattered velocity on the surface. Marom et al. [9]–[10] took the advantage of these features of INSAR, and applied this instrument to study the twodimensional ocean wave spectra. It was demonstrated that the quantitative information on ocean wave heights can indeed be retrieved from the INSAR phase component.
Manuscript received July 31, 1997; revised June 24, 1998. This work was supported by Israel Science Foundation Grant 253/96-3 The authors are with the Department of Fluid Mechanics and Heat Transfer, Tel-Aviv University, Tel-Aviv 69978, Israel. Publisher Item Identifier S 0196-2892(99)01978-6.
Theoretical analysis of the principles of the imaging of ocean waves by INSAR, performed by Shemer and Kit [11], indicates that the velocity bunching may lead to a substantial distortion of the phase component of the complex INSAR image. A further study demonstrated that the phase component of INSAR is only weakly affected by the real aperture radar (RAR) modulation [12]. On the other hand, under certain conditions the magnitudes of both INSAR and the regular SAR outputs can be modified notably by RAR modulation, complicating even further determination of wave heights. Plant and Zurk [13] applied quasilinear theory of SAR imagery of the ocean surface to simulate SAR image spectra for a number of SAR parameters and sea states. In many occasions, their simulations indicate that it is possible not only to extract from the SAR images of the ocean surface the directions of propagation of the dominant waves, but even to estimate the significant wave height. Bao et al. applied MonteCarlo method to simulate ocean waves images by INSAR [14]. One of the major mechanism which renders ocean waves visible in the SAR images is the so-called “velocity-bunching” [15]–[16]. However, SAR imaging mechanism can cause substantial distortion of the spectra and a notable deviation of the propagation direction of the dominant wave [13], [17]. The question therefore arises under which conditions it is possible to extract sufficiently representative information about the waves from the SAR (or INSAR) images. One step toward the solution of this problem is through a full nonlinear SAR/INSAR imaging theory to simulate the image of a monochromatic ocean wave. Hasselmann and Hasselmann [18] and Krogstag [19] derived a closed form nonlinear spectral transform resulting from velocity bunching. The Hasselmanns model can be used to simulate the SAR image (direct problem) and to estimate the wave spectrum from measurements (inverse problem). The spectra of the and that of the resulting image actual wave field are related through an essentially nonlinear mathematical operator (1) From the mathematical standpoint the ultimate goal of SAR/INSAR imagery of the ocean surface is to find a by inverting the nonlinear representative estimate operator (1). It should be taken into account that the structure is known only approximately, of the direct operator is contaminated by noise. Thus, the whereas the output inversion of the functional equation (1) may be impossible in a
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rigorous mathematical sense due to a salient feature of typical inverse problems, i.e., they are frequently ill-posed [20]. This means that the solution of an inverse problems may either be unstable with respect to small input errors, be non unique, or not exist at all. The principal difficulties related to the inversion of the SAR images of ocean waves were considered by Hasselmann and Hasselmann [18], who emphasized that the ill-posed problem can be solved by invoking an a priori information named “first-look spectrum.” In such a way, the solution of an ill-posed inverse problem is reduced to estimation of the wave spectrum parameters by minimizing an appropriately defined difference between the output spectrum and its prediction which takes into account the “first-look spectrum.” There are no formal mathematical rules for identifying a problem as ill-posed. However, if an analytic solution of the direct problem exists, it becomes possible to apply such a solution to estimate the sensitivity of the SAR/INSAR processing to the input parameters, and to use it also as an a priori information for solving the ill-posed inverse problem (1). To obtain an analytic solution, a single monochromatic wave can be used not only because of its practical use and the relative ease with which it can be analyzed, but also because the results of such a study bear important similarities to more general ocean wave fields with a narrow band spectrum, and in particular, for ocean swells. Shemer [21] presented an asymptotic approximation to the complex INSAR image of a monochromatic long and not too steep ocean wave, which accounts for the velocity bunching imaging mechanism. An analytic investigation of the SAR image spectrum was carried out by Lyzenga [22] who represented the temporal correlation function of reflectivity as a Fourier series where the coefficients of the series are expressed analytically. In the present work we also consider a monochromatic progressive wave and calculate the Fourier transform of SAR/INSAR images of such a wave in an explicit form. One of the goals of the work is to illustrate that even for such a simple case, the SAR/INSAR image of a monochromatic wave expands into an infinite number of harmonics which apparently illustrate that the corresponding inverse problem is essentially ill-posed and should be solved by an appropriate regularization as has been proposed in [18]. The derived exact representation of SAR/INSAR Fourier transform may help to delineate the limits of detectability of wave fields with a narrow spectrum and ocean swell by SAR and INSAR.
SAR/INSAR platform moves at a height along the axis in its positive direction with a constant velocity The aircraft in is equipped with two antennas separated by a distance the flight direction. The fore antenna serves as a transmitter, and both antennas serve as receivers. At each instant the so that coordinates of the antennas are two antennas arrive at the same location with the time shift Consider an additional orthogonal system of coordinates also fixed in space, such that the angle between the and is A plane monochromatic harmonic wave axes (an azimuthal wave propagates in moves along the axis 0). The instantaneous surface elevation the direction is defined by the wave amplitude wave number and as Within the wave frequency satisfy framework of the linear surface wave theory, and where is the acceleration the dispersion relation the radar line-of-sight of gravity. For an incidence angle of the free surface point radial velocity component is
where (2) and For modeling SAR/INSAR image of a monochromatic wave the -dependence along the lines of constant range is not essential and may be neglected. By generalizing the approach adopted in [23] to the along-track INSAR its output can be simplified and reduced to a single integral [11]
(3) where
II. PROBLEM FORMULATION The expression for the SAR/INSAR image for a monochromatic wave given in [11] is adopted here. It is assumed in [11] that the radar cross section per unit area is constant and, thus, effects of the titling and the hydrodynamic modulation are not considered. For a regular SAR the expression obtained in [11] is consistent with [23]. is A fixed in space orthogonal coordinate system coincides with the plane of undisselected. The plane axis is directed upward. The turbed free surface and the
Here is the radar wave number, is the wave is the SAR/INSAR integration time, is phase velocity, the scene characteristic coherence time, and a real constant factor affecting the absolute value of the image only.
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Applying the inverse Fourier transform to (16) gives
III. THE SAR/INSAR IMAGE IN THE FOURIER AND IN THE PHYSICAL SPACES is introduced and A new variable of integration (3) is rewritten in a form which is more convenient for the present purposes
(19)
(5)
is used now to define all indepenwhere the coefficient dent factors appearing in the process of derivation (19). The ( , 4) and are given by (2) and quantities (6)–(10). Spliting (19) into the real and the imaginary parts yields
(6)
where
(4)
where
(20)
(7) (8) (9)
(21)
(10) (11) Consider the Fourier transform of the integral (4) with respect to (12) Applying (12) to (4) and integrating with respect to
yields
(22) For a SAR,
see (8), and (20)–(21) yields (23)
(13) where for
where (14)
(24) and IV. NUMERICAL RESULTS The function [24]
can be expanded in Fourier series
(15) are the Bessel functions of the first where kind. Substitution of (15) into (14) and analytic integration with respect to involving Dirac -function gives for (13) (16) where (17) and (18) The analytic expressions (16)–(17) represent the wave-number spectrum of SAR/INSAR image of a monochromatic sea wave.
AND
DISCUSSION
The analytic expression (19) shows that a harmonic wave being processed by SAR/INSAR, with a wave number expands into an infinite number of spectral components with the azimuthal wave numbers where the dependence on the wave phase velocity can be considered as a generalization of the so-called scanning distortion [25]. The occurrence of higher harmonics in the images is an inevitable consequence of nonlinear SAR/INSAR mechanism. The 180 ambiguity results in two system of waves, propagating in the positive and in the negative directions of the axis. The amplitudes of the harmonics in the resulting spectra depend on the wave amplitude nonlinearly are nonlinear with respect since the Bessel functions to the argument and the number of the harmonic Due to the exponential factors (17) in the series (20)–(21) they converge rather fast. Practically it means that only a few terms in (20)–(21) have to be taken into account. For very long 0) with small amplitudes ( 0) it follows waves ( 0. In this case, the series (20)–(21) can from (2) that be truncated to the first term only (25)
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where the phase component represents the well known expression of the “ideal” INSAR output [11]. For small values the Bessel functions behaves as of the argument [27]. Thus, only the Bessel function is linear with respect to its argument. The representation (19) can be seen as linear with respect to the wave amplitude only if the argument of the Bessel function is small, so that the inequality (26) is satisfied. The velocity bunching parameter can be defined as a ratio to the distorted of the typical length scale For INSAR, the additional radar wave length nonlinearity parameter represents the ratio of another length to In view of (23)–(24) and (26), scale, i.e., the parameter defining the nonlinearity of SAR imaging mechThe nonlinearity anism can be represented as which results from velocity bunching, has been parameter obtained by many authors, e.g., [21], [22], and [26]. Similarly, recalling (17)–(18), for INSAR the nonlinearity parameter can be defined as
(a)
(27) Numerical calculations presented where below shows that for typical INSAR configurations The parameter (27) illustrates that for INSAR the nonlinearity depends not only on the velocity bunching, but also on the time delay due to antennas separation. Because of the 180 ambiguity, the nonlinearity parameter is different for waves propagating in the positive and in the negative directions of the axis. Since the Fourier coefficients in (20)–(21) appear as for long and not too steep a sum of Bessel functions waves the degree of nonlinearity practically depends mainly on For comparatively short the velocity bunching parameter or steep waves, however, the directional ambiguity results also in the variations of the velocity bunching for the two waves propagating in the opposite directions. For comparative calculations we define the INSAR parameter of nonlinearity as i.e.: a maximum value of The intrinsic properties of the nonlinear mechanism can be illustrated by considering several first harmonics of SAR grow monotonically image. The Bessel functions until they attain their corresponding first positive extremum at where 1.85), 3.05 1.5), 4.2 ( 1.4) and 5.4 ( ( 1.35) [27]. Thus, the SAR output also grows monotonically as long as the nonlinearity parameter For with the SAR relatively large values of nonlinearity parameter image can be distorted beyond of recognition, especially if the dominant first harmonic vanishes We illustrate the corresponding behavior of SAR/INSAR images and the importance of higher harmonics by a numerical example. A set of INSAR parameters is used which roughly corresponds to those in the experiments with the NASA/JPL 7000 m, 200 L-band airborne INSAR [9]–[10]: 45 , 25 m 0.05 s, 1 2.0 m/s, 0.1 s. s, and
(b)
(c) Fig. 1. SAR images of azimuthally propagating waves and their amplitude spectra. 200 m. (b) T0 2.0 s. (c) s 0.1 s.
=
=
=
Regular SAR output is studied first. Computations are per200 formed for an azimuthally-propagating waves with m, various wave amplitudes, and correspondingly for different Fig. 1(a) represents the values of nonlinearity parameter variation of the regular SAR image intensity as a function of along a constant the distorted phase of the imaged wave range line.
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(a)
(b) Fig. 2. INSAR image of an azimuthally propagating wave.
= 200 m, T0 = 2.0 s, and = 0.1 s.
Results of Fig. 1(a) indicate that the maximum values of the SAR output increase almost linearly with the wave amplitude, remains below some as long as the nonlinearity parameter limiting value of about 1.2. The values of the SAR output for the imaging conditions of Fig. 1 attain their maximum 1.4. For higher values for the nonlinearity parameter of nonlinearity parameter the contrast of the SAR image decreases. For relatively small values of the nonlinearity parameter, the Fourier coefficients of the SAR output decomposition 200 decay monotonically. For example, for wavelength 1.2 m ( 0.67) the dominant m and wave amplitude Fig. 1(b)], harmonic is stronger than the background level resulting in a strong contrast in the corresponding output of Fig. 1(a). On the other hand, the highly distorted SAR output 3.83 in Fig. 1(a) corresponds to the spectrum of for
the image represented in Fig. 1(c) which is characterized by a large value of the background in the absence of the first harmonic and notable second and third harmonics. The along-track INSAR images of the azimuthally propa200 m and identical imaging conditions gating wave with are presented in Fig. 2 for a number of wave amplitudes. The INSAR modulus [Fig. 2(a)] becomes distorted already 0.56, for waves with amplitude exceeding 1 m ( 2.52). Note that the regular SAR output reflects faithfully the imaged wave for much higher amplitudes [see Fig. 1(a)]. The phase component of INSAR, however, provides a relatively weakly distorted image of the ocean swell even for relatively high wave amplitudes [Fig 2(b)]. Since the spectrum of the imaginary part of the INSAR image does not include a DC component ( ), the contrast of the phase component of INSAR is stronger than that of the regular SAR.
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. INSAR image of an azimuthally propagating wave. Wave length
= 75
For extreme imaging conditions, both the modulus and the phase of INSAR can be strongly distorted. An example of such a strong distortion is given in Fig. 3 for an azimuthally propagating wave with 75 m and wave amplitude 1.1 m. The image of the INSAR modulus [Fig. 3(a)] incorporates a considerable second harmonic, while the phase of the image contains discontinuities due the fact that the absolute value of INSAR phase [Fig. 3(b)] exceeds 180 .
m. Amplitude aw
= 1.1 m. T0 = 1.0 s. s = 0.1 s.
The complex INSAR image is presented in Fig. 3(a) and (b) in a traditional way, i.e., as its modulus and the phase angle. Its real and imaginary parts are presented in Fig. 3(c) and (d). The modulus of the complex INSAR output [Fig. 3(a)] and its real part [Fig. 3(c)] are distorted due to the presence of a considerable second harmonics. The spectrum of the real part has a strong DC component which reduces the contrast. The imaginary part of the image also contains a considerable second harmonic [Fig. 3(f)], but compared to the
ZILMAN AND SHEMER: REGULAR OR INTERFEROMETRIC SAR IMAGE OF OCEAN SWELL
phase of the image [Fig. 3(b)], the shape of the imaginary part remains quite close to the “ideal output.” As noted above, the spectrum of the imaginary part of the INSAR image does not include a DC component which results in a strong contrast. It thus may be concluded that for some particular imaging conditions, especially when the phase of the output contains discontinuities, the analysis of an imaginary part of INSAR can add some essential information about the wave field. It should be emphasized that the numerical examples presented here do not cover the variety of the problem parameters which can affect the results. V. CONCLUSION 1) A new closed-form explicit representation of the SAR/INSAR image of ocean swell is derived in this paper. It provides a clearer insight into the imaging mechanism by illustrating how a single monochromatic wave is transformed into a nonsinusoidal function so that its spectral representation includes an infinite number of higher harmonics. The amplitudes of these harmonics depend on the degree of the nonlinearity of the SAR/INSAR mechanism. Two criteria of the nonlinearity are derived. For a regular SAR, the nonlinearity depends mainly on the value of velocity bunching coefficient ([21], [22], [26]). For the alongtrack INSAR the derived criterion of nonlinearity depends not only on the velocity bunching parameter, but also on the separation time between two antennas. 2) Higher harmonics in SAR/INSAR images constitute an unavoidable consequence of the inherent properties of the nonlinear nature of the SAR/INSAR imaging. The present study illustrates that due to the difficulties in distinguishing between the fundamental and higher harmonics, as well as due to the presence of measurement noise, the inverse problem of SAR/INSAR imaging should be considered as a ill-posed [20], [28]. One of the possible ways to regularize the ill-posed problem is by invoking a priory information as suggested by Hasselmann and Hasselmann [18]. 3) Under some conditions, in particular for comparatively short and steep waves INSAR nonlinearity can be more pronounced than that for a regular SAR resulting in stronger higher harmonics in images. When the phase of the INSAR contains discontinuities, the imaginary part of the complex INSAR image and its Fourier coefficients may provide useful supplementary information about the wave field. This may be particularly true for the imaging conditions when the absolute value of the INSAR phase can exceed 180 . REFERENCES [1] K. Hasselmann, R. K. Raney, W. J. Plant, W. Alpers, R. A. Shuchman, D. R. Lyzenga, C. L. Rufenach, and M. J. Tucker, “Theory of synthetic aperture radar ocean imaging: A MARSEN view,” J. Geophys. Res, vol. 90, pp. 4659–4686, 1985. [2] R. C. Beal, D. G. Tilley, and F. M. Monaldo, “Large- and smallscale spatial evolution of digitally processed ocean wave spectra from SEASAT synthetic aperture radar,” J. Geophys. Res., vol. 88, pp. 1761–1778, 1983.
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[3] R. C. Beal, F. M. Monaldo, D. G. Tilley, D. E. Irvine, E. J. Walsh, F. C. Jackson, D. W. Hancock III, D. E. Hines, R. N. Swift, F. I. Gonzales, D. R. Lyzenga, and L. F. Zambresky, “A comparison of SIR-B directional ocean wave spectra with aircraft scanning radar spectra,” Science, vol. 232, pp. 1531–1535, 1986. [4] P. W. Vachon, R. B. Olsen, C. E. Livingstone, and N. G. Freeman, “Airborne SAR imagery of ocean surface waves obtained during LEWEX: Some initial results,” IEEE Trans. Geosci. Remote Sensing, vol. 26, pp. 548–561, 1988. [5] T. Maklin and R. Cordey, “Ocean-wave imaging by synthetic-aperture radar: Results from SIR-B experiment in the N. E. Atlantic,” IEEE Trans. Geosci. Remote Sensing, vol. 27, pp. 28–35, 1989. [6] C. L. Rufenach, R. B. Olsen, R. A. Shuchman, and C. A. Russel, “Comparison of aircraft synthetic aperture radar and buoy spectra during the Norwegian continental shelf experiment of 1988,” J. Geophys. Res., vol. 10, pp. 10 423–10 441, 1991. [7] R. M. Goldstein and H. A. Zebker, “Interferometric radar measurement of ocean surface currents,” Nature, vol. 328, pp. 707–709, 1987. [8] R. M. Goldstein, T. P. Barnett, and H. A. Zebker, “Remote sensing of ocean currents,” Science, vol. 246, pp. 1282–1285, 1989. [9] M. Marom, R. M. Goldstein, E. B. Thornton, and L. Shemer, “Remote sensing of ocean wave spectra by interferometric synthetic aperture radar,” Nature, vol. 345, pp. 793–795, 1990. [10] M. Marom, L. Shermer, and E. B. Thornton, “Energy density directional spectra of near shore wave field measured by interferometric synthetic aperture radar,” J. Geophys. Res., vol. 96, pp. 22 125–22 134, 1991. [11] L. Shemer and E. Kit, “Simulation of an interferometric SAR imagery of an ocean system consisting of a current and a monochromatic wave,” J. Geophys. Res., vol. 96, pp. 22 063–22 074, 1991. [12] L. Shemer, “Interferometric SAR imagery of a monochromatic ocean wave in the presence of the real aperture radar modulation,” Int. J. Remote Sensing, vol. 14, pp. 3005–3019, 1993. [13] W. J. Plant and L. M. Zurk, “Dominant wave directions and significant wave heights from synthetic aperture radar imagery of the ocean,” J. Geophys. Res., vol. 102, pp. 3473–3482, 1997. [14] M. Bao, M., C. Br¨uning, and W. Alpers, “Simulation of ocean waves imaging by along-track interferometric synthetic aperture radar,” IEEE Trans. Geosci. Remote Sensing, vol. 35, pp. 618–631, 1997. [15] W. R. Alpers and C. L. Rufenach, “The effect of orbital motions on synthetic aperture radar imagery of ocean waves,” IEEE Trans. Antennas Propagat., vol. AP-27, pp. 685–690, 1979. [16] C. T. Swift and L. R. Wilson, “Synthetic aperture radar imaging of moving ocean waves,” IEEE Trans. Antennas Propagat., vol. AP-27, pp. 725–729, 1979. [17] C. Br¨uning C, W. Alpers, L. F. Zambersky, and D. G. Tilley, “Validation of a synthetic aperture radar ocean wave imaging theory by the shuttle imaging radar-B experiment over North sea,” J. Geophys. Res., vol. 93, pp. 15 403–15 425, 1988. [18] K. Hasselmann and S. Hasselmann, “On the nonlinear mapping of an ocean wave spectrum into a synthetic aperture radar image spectrum and its inversion,” J. Geophys. Res., vol. 96, pp. 10, 10 713–10 729, 1991. [19] H. E. Krogstad, “A simple derivation of Hasselmann’s nonlinear oceansynthetic aperture radar transform,” J. Geophys. Res., vol. 97, pp. 2,421–2,425, 1992. [20] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems. Washington, DC: Winston, 1977. [21] L. Shemer, “An analytical presentation of the monochromatic ocean wave image by a regular or an interferometric synthetic aperture radar,” IEEE Trans. Geosci. Remote Sensing, vol. 33, pp. 1008–1013, July 1995. [22] D. R. Lyzenga, “An analytic representation of the synthetic aperture radar image spectrum for ocean waves.” J. Geophys. Res., vol. 93, pp. 13 859–13 865, 1988. [23] D. P. Kasilingam and O. H. Shemdin, “Theory for synthetic aperture radar imaging of the ocean surface: With application to the tower ocean wave and radar dependence experiment on focus, resolution, and wave height spectra,” J. Geophys. Res., vol. 93, pp. 13 837–13 858, 1988. [24] H. Bateman and A. Erd´eyi, Higher Transcendental Function. New York: McGraw-Hill, 1954. [25] R. K. Raney and R. T. Lowry, “Oceanic wave imagery and wave spectra distortions by synthetic aperture radar imagery,” in Proc. 12th Int. Symp. Remote Sensing Environ., Manila, The Phillippines, 1978, pp. 683–702. [26] W. Alpers, “Monte Carlo simulations for studying the relationship between ocean wave and synthetic aperture radar image spectra,” J. Geophys. Res., vol. 88, pp. 1745–1759, 1989. [27] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Washington, DC: National Bureau of Standards, 1964. [28] P. Eykhoff, System Identification. Parameter and State Estimation. New York: Wiley, 1974.
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Gregory Zilman received the Dipl. Eng.Researcher degree in hydrodynamics and the Ph.D. degree in naval hydrodynamics, both from Leningrad (St. Petersburg) Shipbuilding Institute (LSI), Russia, in 1971 and 1978, respectively. He was employed at LSI from 1971 to 1990. He started as an Engineer-Researcher in the LSI towing tank. The academic positions of Senior Research Scientist and Associate Professor in the Department Theory of Ships and Hydrodynamics followed. In 1991, he joined the Department of Fluid Mechanics, Tel-Aviv University, Tel-Aviv, Israel, as a Senior Research Fellow and where he is now a Principal Research Fellow. His research interests in marine and naval hydrodynamics include wave motion, with emphasis on ship generated surface and internal waves, as well as microwave remote sensing of ship wakes.
Lev Shemer received the M.Sc. degree in physics from the Moscow Phys. Tech. Institute, Moscow, Russia, and the Ph.D. degree in mechanical engineering from Tel-Aviv University, Tel-Aviv, Israel, in 1970 and 1981, respectively. After a postdoctoral stay with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, he joined the faculty at the Department of Fluid Mechanics, TelAviv University, where he currently serves as a Professor of Fluid Mechanics. His main research interests are in experimental fluid mechanics, as well as in experimental and theoretical study of nonlinear water waves. During a sabbatical stay in 1989–1990, at Naval Postgraduate School, Monterey, CA, he became involved in studies of the ocean waves and currents using either regular or along-track interferometric SAR. Since 1990, his research activities have included various aspects of remote sensing of water surface.
