Two-scale treatment of low-grazing-angle scattering from spilling breaker water waves



[1] A two-scale scattering analysis is applied to two series of profiles that represent the measured time evolution of the crests of spilling breaker waves that were mechanically generated and measured in a wave tank. The backscattering at microwave frequencies and low-grazing-angle (LGA) illumination were considered. A deterministic implementation of the perturbation theory that is the foundation of any two-scale scattering model (TSM) is used that allows an instantaneous comparison of TSM cross sections with reference “exact” cross sections found using a numerical moment-method-based electromagnetic approach. TSM proved unable to model the scattering from steep features that appear on the crest both before and after the onset of breaking, even in cases where the horizontal-to-vertical polarization ratios are as low as −10 dB. This shows that the “fast” scattering (with Doppler speeds greater than or equal to the velocity of the breaker) associated with these features is not due to Bragg scattering in the traditional sense in these cases despite the low polarization ratios. TSM is more accurate after the steep features subside and the turbulent “scar” roughness is primarily on the front face of the breaker. However, accuracy is lost as the wave propagates and the scar roughness moves to the crest top and back face of the wave. It appears that a scattering coefficient based on the spectral content of the roughness at the Bragg-resonant condition can provide only a rough estimate of the actual LGA scattering coefficient even after the major steep features have disappeared.

1. Introduction

[2] Breaking waves have been identified as a strong contributor to the microwave backscattering from the ocean surface at low-grazing-angle (LGA) illumination [Smith et al., 1996; Lee et al., 1995]. Breaking waves are thought to be responsible for brief bursts of backscattered power known as “sea spikes,” where the horizontally polarized backscatter (HH) can exceed the vertically polarized backscatter (VV) by as much as 10 dB. Specular (or specular-like) reflection from very steep features on the crest is also suggested as the cause of “fast-scatterer” signals that are characterized by Doppler shifts associated with the phase velocity of the breaker [Lee et al., 1995]. For this reason, most theoretical and numerical studies of the scattering from breakers have focused primarily on the steep features and the multipath interference that can give rise to the sea-spike phenomena [Wetzel, 1990; Trizna, 1997; Holliday et al., 1998; West, 1999a]. However, there are other mechanisms that are thought to be important in the back scattering from a breaking wave. Plant [1997] introduced a “bound-Bragg” model to explain the fast scattering signals. In this theory, it is assumed that small waves that are Bragg-resonant with the incident electromagnetic energy are bound to the crests of the long waves. Since the resonant waves are bound to the breaker, the scattering gives Doppler shifts associated with the breaker phase velocity. The slope of the long wave increases the local grazing angle of the incident illumination on the bound waves, raising the Bragg backscatter from these features above the background Bragg scattering from freely propagating roughness at HH through the well-known tilt modulation effect [see, e.g., Brown, 1978]. VV backscatter is much less affected by tilt modulation and so is not biased to fast scattering. Experimental results said to support this model were presented by Lee et al. [1999] and Lamont-Smith [2000].

[3] Another feature of a breaking wave that has been shown to contribute to significant backscatter is the turbulent “scar” (as labeled by Fuchs et al. [1999]) which remains on the surface after the initial breaking. With fairly energetic breaking, Fuchs et al. [1999] experimentally showed that at 6° grazing incidence (with respect to horizontal) the radar cross section of the scar slowly drops to the background levels as the scar decays, with HH/VV polarization ratios reducing to −10 dB and less. Polarization ratios this low are typically assumed to be due to Bragg resonance, perhaps with the dominant Bragg scatterers tilted toward the radar by the large-scale breaking wave [Lee et al., 1999; Smith et al., 1996; Lamont-Smith, 2000], and therefore describable by the two-scale (or composite-surface) model of backscattering [Lee et al., 1995]. Ericson et al. [1999] found that the two-scale model (TSM) is accurate at moderate grazing angles (45°) away from the breaking region of a stationary breaker.

[4] Ja et al. [2001] numerically examined the LGA (10°) backscattering from two spilling breakers that were mechanically generated in a wave tank. A high-speed digital imaging system that moved with the crest of the breaker gave a continuous measurement of the evolution of the crest profile. The backscatter was found using a numerical electromagnetic technique based on the moment method (MM). Since the temporal evolution of the crest profile was available, the numerical results could be used to find the time-dependent Doppler shift of the backscatter. It was found that the fast signals identified were always correlated with steep features (surface slopes well above unity) that appear on the wave before and immediately after the onset of breaking. Based on the steepness of the features it was argued that the backscattering should not be considered “Bragg” even when the HH/VV ratios were as low as −10 dB. However, even steep features can be decomposed into spectral components (with energy at the Bragg-resonant wave number) provided that they are single-valued, so the above argument alone is not conclusive. In this paper we more fully consider the Bragg-scattering hypothesis of crest backscattering by applying a complete two-scale analysis to the measured crests. A deterministic application of the two-scale perturbation theory, as presented by by Brown [1978] and Guissard and Sobieski [1987], is used to yield instantaneous scattering cross sections that can be directly compared with the MM cross sections. As the perturbation theory is the foundation of the two-scale scattering model [Burrows, 1972], this tests any two-scale model (TSM) implementation at its most fundamental level. The validity of TSM is examined at LGA when applied to both the steep features before and immediately after the initial breaking, as well as with the turbulent scar after the steep features have subsided.

2. Surfaces

[5] The surface profile histories of the two breaking waves considered here are shown in Figures 1a and 2a. The waves were generated mechanically by a dispersive focusing technique in a wave tank that is 14.8 m long, 1.22 wide and 1.0 m deep. The profiles were measured with an optical system consisting of a high-speed digital movie camera and a laser light sheet that were mounted on an instrument carriage that was set to travel along the tank with the crest of the breaking wave. The water was mixed with fluorescent dye. Image processing techniques were used to obtain the shape of the crest from each image of the movie. The figures were formed by stacking the sequence of crest profiles so that time increases vertically from profile to profile; there is a 4.24 ms time difference from profile to profile. (Profiles were actually available every 2.12 ms, but only half are plotted in the figure for clarity.) The horizontal axis of the figures is distance in the downstream direction in the reference frame fixed with respect to the crest (the forward face of the crest is on the left). Since the measurement system was moving at the phase velocity of the wave, profile features moving to the left with increasing time are moving faster than the phase velocity, while those moving to the right are moving slower than the phase velocity. Selected profiles from each history (with vertical and horizontal axis to scale) are shown in Figures 1b and 2b. A more detailed description of both the wave generation and optical measurement procedures is given by Ja et al. [2001].

Figure 1.

“Clean” spilling breaker. (a) Complete time history. (b) Specific profiles. (The 191 ms profile is plotted in absolute coordinates relative to mean water displacement. The subsequent profiles are adjusted vertically by multiples of 15 mm.)

Figure 2.

“Surfactant” spilling breaker. (a) Complete time history. (b) Specific profiles. (The 381 ms profile is plotted in absolute coordinates relative to mean water displacement. The subsequent profiles are adjusted vertically by multiples of 15 mm.)

[6] For the breaker in Figure 1, the water surface was relatively free of ambient surfactants, so this measured history is hereafter termed the “clean” breaker. The center frequency of the wave packet used to generate the breaker was 1.42 Hz, and the phase velocity of the crest was 0.945 m/s. The history is first characterized by a steepening crest, which then develops into a well-defined bulge at 210 ms. (The wave features such as the “bulge” and “toe” are identified in Figure 3.) The leading edge (toe) of the bulge travels slightly faster than the phase velocity of the wave. Capillary waves form upstream of the toe from 210 ms to 310 ms. The bulge then collapses at 370 ms, leaving the turbulent scar. No steep features appear on the surface after 420 ms. Ja et al. [2001] showed that the Doppler shift of the LGA, upwave-looking microwave (10 GHz and 20 GHz) backscatter while the bulge appears on the surface is associated with the phase velocity of the breaker. After the bulge collapses the strongest backscatter has Doppler shifts lower than that from the phase velocity. These signals were shown to be associated with the turbulent eddies shed from the crest after the onset of breaking that are not bound to the wave, but rather are fixed to the underlying fluid. A weaker response at a Doppler frequency significantly higher than that of the wave phase velocity was associated with the forward movement of the steep toe, centered at 420 ms. The VV backscatter was considerably stronger than HH at all times after the bulge collapsed.

Figure 3.

Features of the breaker crest.

[7] The second breaker, shown in Figure 2, was generated with liquid soap added to the water. This time history is hereafter referred to as the “surfactant” wave. Other than a slightly higher wave-maker amplitude, all other parameters of the wave generation were the same as those in the clean wave. The presence of the surfactant prevents the formation of parasitic capillaries as the bulge forms. A jet forms near the leading edge of the bulge at 360 ms, and then impacts with the smooth forward face of the wave at ∼420 ms. Smaller steep features continue to form on the crest after the initial jet impact, with additional overturnings at ∼570 ms and ∼740 ms. Turbulent eddies that are fixed to the underlying fluid also form during this time. In the surfactant wave, steep features are absent from the surface after 600 ms, other than the brief, very small overturning at 740 ms. Ja et al. [2001] showed that backscattering with high Doppler shifts were again correlated with the steep and overturning features, while smaller shifts could be correlated with the more slowly moving ripples. VV backscattering exceeded that at HH at all times after 600 ms, other than a very short time at 20 GHz associated with the final overturning at 740 ms.

3. Electromagnetic Treatment

3.1. MM/GTD “Exact” Results

[8] The reference “exact” backscattering calculations were performed using a technique based on the moment method (MM) that was extended using the geometrical theory of diffraction (GTD). Again, the details of the MM/GTD calculations were given by Ja et al. [2001] and are not repeated here, other than to say that the individual surface profiles must be extended to infinity to allow the application of the numerical technique. An example extended surface is given in Figure 4. The extensions are chosen so that they do not introduce artificial multipath scattering that can lead to large interference in the calculated backscattering [Trizna, 1997].

Figure 4.

Example of surface extended to allow the application of the reference MM/GTD scattering technique.

3.2. Two-Scale Model

[9] Application of TSM requires that the scattering surface be separated into electromagetically large- and small-scale components. The most common approach is to define a surface wave number cutoff (or threshold). The roughness energy below the cutoff is considered the large-scale roughness while the energy above the cutoff is the small-scale component. This approach is best suited to Monte Carlo simulations of scattering from random surfaces whose roughness is described by a linear spectrum. In this case, the small- and large-scale components can be be generated separately, in effect giving an idealized rectangular filter separation. A rectangular filter is not appropriate for treating the deterministic surfaces considered here since it could only be applied using a discrete Fourier transform, and the resulting surfaces would be affected by Gibbs phenomenon ringing. Instead, we rely upon a moving average applied in the spatial domain to do the scale separation. Multiple passes of a three-point moving average were applied to the discretely sampled surface to act as a low-pass filter. The resulting smoothed surface was then treated as the large-scale surface. The small-scale energy was found from the difference of the original (or composite) surface and the large-scale surface. The triangular weighting window used is

equation image

where the subscript m indicates the mth (evenly sampled) horizontal sample and the superscript n indicates the number passes of the window already applied. Averaging was not applied to the surface endpoints (i.e. y1n + 1 = y1n and yMn + 1 = yMn, where M is the total number of points in the discrete profile). Since numerous passes of the weighting window were applied, the resulting effective filter impulse response approaches a Gaussian function, as does the filter passband. Taking the Fourier transform of equation (1), the transfer function of single pass of the weighting window is

equation image

where K is the surface wave number and Δx is the spacing between the surface samples in the x direction. After N passes of the triangular window the final filter transfer function is

equation image

An effective cutoff wave number for the filter was defined as

equation image
equation image

This definition insures that the total area under the true transfer function is equal to the area under the transfer function of the equivalent rectangular low-pass filter defined by the threshold. (The upper limit of integration in equation (4) appears since the surface points are discretely sampled.) Note that due to the Gaussian shape of the filter passband this does not represent an absolute cutoff between large- and small-scale energy; some energy at wave numbers above the cutoff will be included in the large-scale surface, and some energy at wave numbers below the cutoff is included in the small-scale surface.

[10] The scale-separation procedure described can obviously only be applied to surfaces that are single valued and uniformly sampled in the horizontal dimension. As mentioned, the surfaces under consideration are multivalued at some points. To allow the surfaces to be treated as two-scale, the multivalued points were simply removed. The single-valued surfaces were then resampled to even spacing in the horizontal dimension. The surface was resampled at a horizontal spacing of λ/100 (0.3 mm at 10 GHz). The surface separation filtering was then performed using the multiple passes of the triangular window.

[11] For all the TSM results presented herein, the large-scale surfaces were filtered with a cutoff wave number of Kc = k/1.6, where k is the electromagnetic wave number, requiring 2000 passes of the three point weighting window with λ/100 sampling. This proved to consistently give the best results in the surfaces considered at the 10° grazing of interest. The impulse response of the filter and corresponding transfer function are shown in Figure 5. Also shown are the impulse response and transfer function of the equivalent rectangular low-pass filter at the cutoff Kc = k/1.6. The transfer function in Figure 5b shows that the roughness energy at the Bragg wave number (KB =1.97k at 10°) is fully filtered from the large-scale surface with the filter used. (The transfer function magnitude at KB is 0.005.) A sample separation is shown in Figure 6. It was taken from the clean wave profile at 530 ms using a cutoff of Kc = k/1.6 at 10 GHz. (Note that the actual surface extends to infinity to either side, but this is entirely a large scale section and so is not shown.) The small-scale roughness extends over approximately 60 mm, which is about 4 Bragg wavelengths at 10° grazing at 10 GHz and about 8 Bragg wavelengths at 20 GHz.

Figure 5.

Filter used to extract the large-scale roughness from the surfaces (solid line). Also shown is the equivalent rectangular low-pass filter with the same cutoff wave number (dotted line). (a) Impulse response. (b) Transfer function.

Figure 6.

Sample separated surface. The original, large-scale, and small-scale profiles are shown as the solid, dotted, and dot-dashed lines respectively.

[12] Once the scale separation was completed, the electromagnetic scattering from the surface was modeled using a two-scale approach that was initially developed by Brown [1978] and extended to finite-conductivity surfaces by Guissard and Sobieski [1987]. The same approach was previously applied to wind-roughened water surfaces by West [1999b]. The Kirchhoff approximation (KA) was first applied to the large-scale surface numerically, yielding a current that was then radiated to give the large-scale field. The small-scale field was then found by numerically integrating equation (8) of Guissard and Sobieski [1987]. Coherent addition of the Kirchhoff-approximation (KA) field associated with the large-scale surface and the small-perturbation method (SPM) fields associated with the small-scale surface yielded the two-scale field for the composite surface that can be compared directly to the moment-method derived field on a realization by realization basis. Shadowing was incorporated using idealized optical self-shadowing of the large-scale surface. Both the KA and SPM fields scattered from points where the large-scale surface was optically self-shadowed were set to zero.

4. Results

4.1. Preliminary Calculations

[13] We first consider the effects of the removal of the multivalued surface sections on the reference “exact” scattering. A test case is shown in Figure 7. This shows the upwave (right) looking 20 GHz backscattering at both VV and HH polarizations of both the original surfactant-wave history (the history that includes more multivalued sections) and the resulting single-valued time history. The illumination grazing angle is 10°, as it is in all LGA results shown here. The scattering cross sections are given as two-dimensional cross-sections in decibels relative to 1 m since the profiles were uniform in one dimension [Balanis, 1989], and 20 dB has been added to the VV cross sections for clarity. (An equivalent three-dimensional cross section in units of decibels relative to 1 m2 could be found by assuming a surface width and using equation (11–22e) of Balanis [1989].) Somewhat surprisingly, the surface modifications had very little effect on the backscatter. The maximum difference is only 1.5 dB, which occurs in the VV cross sections during the maximum jetting at 381 ms. There is virtually no difference after 450 ms. These conditions represent the worst case examined. The largest differences observed at lower frequencies with the surfactant wave and with the clean wave at any frequency were never more than 0.5 dB. All subsequent results in this paper show comparisons between the reference backscattering from the unmodified surfaces and TSM applied to the modified surfaces. However, this result shows that dramatic disagreement between the reference scattering and TSM does not result because of the surface modifications needed to remove multivalued sections.

Figure 7.

MM/GTD backscattering from the surfactant wave profile before (solid) and after (dashed) the removal of multivalued sections at 20 GHz and 10° grazing. 20 dB has been added to the VV response for clarity. (All scattering cross sections are given in decibels relative 1 m.)

[14] Both to test the implementation of the two-scale model and to provide a reference for benchmarking the accuracy of the model in the LGA cases, the backscattering from the clean wave at 20 GHz when looking upwind at the moderate grazing angle of 45° is now considered. This is within the region where TSM is considered accurate for scattering from the sea surface. The results are shown in Figure 8. A scale-separation-filter cutoff of Kc = k/1.1 was used at 45° grazing to insure that the low-pass filter had completely filtered the energy at the Bragg-resonant wave number (KB =1.4k) from the large-scale surface.) At vertical polarization TSM overpredicts the backscattering while the steep features are present on the surface (through 410 ms). After that, the agreement is quite good, except for small overpredictions (about 2 dB) at 675 ms and 695 ms. There is also a short region of relatively poor agreement centered at 585 ms. However, this appears at a small cross section time, so it does not contribute significantly to the average energy of the backscatter. At horizontal polarization the agreement actually appears to be quite good while the steep features are present. However, the results at both polarizations are extremely sensitive to the scale-separation cutoff used in this region, and small changes give very large changes in the backscattering. TSM tends to overpredict the peak cross sections slightly after the steep features subside, with the most serious being just under 2 dB at 615 ms. As at VV, the relative minimum at 585 ms is poorly predicted. Similar results were achieved at 10 GHz with the clean wave and with the surfactant wave when no steep features were present. This behavior is consistent with the findings of Ericson et al. [1999]. The slight HH inaccuracy in the absence of steep features also appeared with the surface modeled as perfectly conducting, and may result from the less than ideal scale-separation procedure used. Changing the cutoff wave number to Kc = k/1.6 slightly improved the HH accuracy after 410 ms, but at the expense of VV accuracy in the same time period. The lower cutoff also gave much worse agreement HH agreement prior to 410 ms. Kc = k/1.1 gave slightly better overall results in the time of interest (after 410 ms), so those results are shown here.

Figure 8.

Comparison of reference and two-scale-model 20 GHz backscattering from clean wave at 45° grazing. (a) Vertical polarization. (b) Horizontal polarization.

4.2. Clean Wave

[15] A comparison between the reference backscattering and the two-scale calculations at an illumination grazing angle of 10° and a frequency of 10 GHz is shown in Figure 9. Prior to 100 ms the wave crest is quite round, giving very small cross sections. The pixelization error of the profile measurements gives the noisy response at those times. Figure 9a shows the VV polarization comparison. TSM significantly overpredicts the backscattering while the steep features appear on the surface through 410 ms. After that, TSM tracks the oscillations in the backscatter, but sometimes overpredicts and sometimes underpredicts the reference cross sections. Only the peak at 460 ms is described to the same accuracy that was consistently achieved at vertical polarization in Figure 8a. The backscattering is consistently underpredicted after 620 ms.

Figure 9.

Comparison of reference and two-scale-model 10 GHz backscattering from clean wave at 10° grazing. (a) Vertical polarization. (b) Horizontal polarization.

[16] At horizontal polarization, Figure 9b, TSM severely underpredicts the cross sections while steep features appear on the surface prior to 410 ms. After that, the agreement is better through 650 ms, although not as good as that achieved in Figure 8b. It appears that the cross sections after 650 ms are overpredicted by TSM, but the cross sections are small enough at this time that pixelization noise is important. We note that relative minimums in the reference scattering are matched in the two-scale scattering (although often with limited accuracy), showing that they are not due to interference of multiple scattering from the wave features and front face, but rather due to the changes in the direct backscattering from the roughness itself. (Each relative maximum is actually correlated to the time when a turbulent eddy appears at the wave peak.)

[17] The calculations in Figure 9 are repeated in Figure 10 but with the frequency increased to 20 GHz. The two-scale model severely overpredicts the VV backscattering when steep features are present through 400 ms. Better accuracy is obtained after that, although the peak cross sections around 490 ms are overpredicted by about 4 dB. The peak values at the end of the record after 670 ms are underpredicted by 4 dB. At HH polarization TSM is surprisingly accurate with the relatively high cross sections through 400 ms, despite the steep features. (This does not indicate that the model should be considered valid under these conditions, as will be discussed in section 5.) After that, the mean cross section drops by approximately 15 dB. TSM overpredicts relative maxima in the scattering at several times in this range, sometimes by as much a 5 dB, while underpredicting at 520 ms.

Figure 10.

Comparison of reference and two-scale-model 20 GHz backscattering from clean wave at 10° grazing. (a) Vertical polarization. (b) Horizontal polarization.

4.3. Surfactant Wave

[18] The comparison between the two-scale model backscattering and the reference calculations when looking upwave at the surfactant wave at 10 GHz and 10° grazing are shown in Figure 11. Pixelization noise dominates through 300 ms. The bulge forms and evolves into a jet that impacts at 420 ms with this wave, and steep features remain on the surface through 600 ms. TSM consistently overpredicts the VV scattering quite seriously throughout this period. TSM becomes more accurate overall after the larger steep features disappear, although there is a continuous error ranging from +2 dB to a bit more than +3 dB from 620 ms through 700 ms. After that, the agreement is good. At horizontal polarization, TSM tends to underpredict the scattering through 570 ms, most dramatically at 540 ms where the underprediction is about 8 dB. After the steep features disappear at 600 ms the cross sections drop significantly. TSM gives good accuracy from 600 ms through 670 ms, and then tends to overpredict several dB from then on. Note that times after 600 ms where TSM appears accurate at one polarization are matched by inaccuracies at the other polarization.

Figure 11.

Comparison of reference and two-scale-model 10 GHz backscattering from surfactant wave at 10° grazing. (a) Vertical polarization. (b) Horizontal polarization.

[19] Figure 12 shows the results with the surfactant wave when the frequency was increased to 20 GHz. TSM dramatically overpredicts the VV scattering during the periods from 300 ms to 600 ms, and also from 680 ms to 740 ms. There are less severe overpredictions of the peaks at 640 ms and 670 ms. A small underprediction occurs from 750 ms to 780 ms. At horizontal polarization TSM sometimes overpredicts and sometimes underpredicts the scattering when the larger steep features are present from 350 ms to 600 ms, often with errors exceeding 3 dB. The mean level of the backscattering drops by about 15 dB after 600 ms when the steep features dissipate. Better accuracy is achieved in this time period, although there are short periods of poor agreement centered at 650 ms, 700 ms, and 740 ms. The brief peak in the reference response at 735 ms that is not matched in the TSM curve corresponds to the final, quite small overturning seen in Figure 2.

Figure 12.

Comparison of reference and two-scale-model 20 GHz backscattering from surfactant wave at 10° grazing. (a) Vertical polarization. (b) Horizontal polarization.

5. Discussion

[20] The two-scale model clearly gives poor accuracy at low-grazing-angle incidence when steep features appear on the surface (prior to 400 ms on the clean wave and 600 ms on the surfactant wave). The Doppler responses presented by Ja et al. [2001] showed that these features gave the “fast” signals equal to or larger than the wave phase velocity. This behavior is not surprising during the times when the bulges are formed on the waves and the polarization ratios are about −3 dB (10 GHz) or 0 dB (20 GHz). These ratios clearly indicate a non-Bragg mechanism so TSM is expected to be inaccurate (and even proved inaccurate in the same time period at a moderate grazing angle of 45°). Of more interest is the behavior after the initial breaking of the surfactant wave where smaller steep features are present. The 10 GHz HH/VV backscatter ratio from the surfactant wave (Figure 11) averages approximately −10 dB from 420 ms to 600 ms, which is small enough to perhaps suggest a Bragg-resonant scattering may be responsible [Lee et al., 1999; Lamont-Smith, 2000]. Ja et al. [2001] suggested that the steepness of the features themselves during this time indicated that the responses were non-Bragg. The poor accuracy of TSM during this time (particularly at VV) further supports this conclusion. The agreement in the same time period is even worse at 20 GHz (Figure 12). The HH/VV ratio is about −10 dB from about 500 ms to about 560 ms at 20 GHz, so again this might be mistaken to be a Bragg response. Reiterating a suggestion previously made by Ja et al. [2001], the fast scattering with Bragg-like polarization ratios that were observed by Lee et al. [1999] at low wind speeds may be due to steep features that are too small electromagnetically to give true specular reflection, rather than due to bound Bragg-resonant waves as they suggested. At higher wind speeds the breaking is more energetic, so the steep features will be larger and give HH/VV ratios closer to 0 dB, similar to the effect of increasing the frequency from 10 GHz to 20 GHz here.

[21] We should again mention that the TSM results are dependent upon the wave number cutoff used to separate the large- and small-scale roughness. Since the roughness has a continuous spectrum, lowering the cutoff moves roughness energy from the large-scale surface to the small-scale surface. The large-scale surface is treated using the Kirchhoff approximation, which gives nearly the same backscattering cross sections with the finite conductivity surfaces considered here (and gives identical cross sections with perfectly conducting surfaces [Ulaby et al., 1982]), so tends to underpredict nonspecular VV scattering and overpredict nonspecular HH scattering. On the other hand, the small-perturbation treatment of the small-scale surface gave much stronger responses at VV than at HH. Thus, lowering the separation cutoff tends to decrease the HH response and increase the VV response. The cutoff Kc = k/1.6 was chosen since it overall gave the best results at 10° after the steep features dissipate at both polarizations. At some times TSM appears to be quite accurate at one polarization but quite inaccurate at the other. When this occurs, the inaccuracy at one polarization indicates that the requirements of both the large-scale and small-scale surfaces are not simultaneously met. The apparent accuracy at the other polarization is simply coincidence, and should not be interpreted as confirming the validity of the model under those conditions. Good examples are the 20 GHz responses from the clean wave at both 10° and 45° grazing while the steep features exist from 250 ms to 400 ms, where the VV response is seriously overpredicted but HH is quite close to correct. TSM is obviously invalid with these electromagnetically large steep features. The results were extremely sensitive to the cutoff used at this time, and adjusting the cutoff to improve the VV agreement gave a correspondingly reduced accuracy at HH.

[22] The other question to be addressed is whether or not TSM can be used to model the scattering from the breaker scar after the steep features associated with the primary breaking dissipate. The HH/VV ratios at these times are typically −15 dB and below. We first consider the simpler case of the weak breaking on the clean wave after 420 ms. Comparing Figure 9 and Figure 10 to Figure 8, it is clear that TSM cannot be expected to yield the same accuracy at 10° grazing as at more moderate grazing. TSM sometimes overpredicts and sometimes underpredicts the scattering cross sections in this region. Moreover, the accuracy drops rapidly as the wave passes the scar and the roughness appears primarily on the top and back face of the wave. This is apparent in the clean-wave cross sections after 630 ms. Also, the poor VV accuracy in Figure 10 at 490 ms appears to be correlated with the large turbulent cell on the top of the wave at that time. Overall, there are only a few times at either frequency where accuracy approaching the benchmark of Figure 8 is achieved simultaneously at both polarizations at either frequency. The performance is similar with the more energetic breaking of the surfactant wave after 600 ms at 10 GHz in Figure 11, and much worse when the frequency is increased to 20 GHz with the surfactant wave in Figure 12.

[23] On the other hand, TSM at times does perform better at the lower grazing angle than might have been expected based on known limitations in the model when surface self-shadowing occurs. It appears that when the scar is primarily on the front face and no steep features appear the cross sections can be fairly accurate. This is most likely due to the tilt of the large-scale wave. The front-face tilt reduces the local incidence angle, moving it to a region where TSM is more valid. This behavior is consistent with the model proposed by Plant [1997] (although the roughness is not bound in this case). Obviously, the two surface profiles considered here are limited in scope, and it is not possible to predict which is more typical in open water. As the perturbation analysis given here is fundamental to any two-scale model implementation, the results suggest that the Bragg response from the scar roughness energy at the resonant wave number may be able to give a rough estimate of the true scattering coefficient, and will be more accurate if sufficient time has passed after the initial breaking that there are no steep features remaining but not so much time has passed that the scar has moved onto the top or back face of the wave. Still, the inherent limitations revealed even when treating the simplest case of the clean wave show that even under the best of conditions the scattering coefficient will be of lower accuracy than at moderate grazing. Care must be used when using TSM at the smaller grazing angles.

6. Conclusions

[24] The perturbation analysis that is the foundation of the two-scale scattering model has been applied to a time series of surface profiles measured from mechanically generated spilling breaking water waves. The modeled backscattering was deterministically compared to reference “exact” backscattering found using a moment-method-based numerical technique. As expected, TSM was unable to consistently predict the scattering from the very steep and/or multivalued features that form on the wave crest immediately before breaking when the HH/VV backscattering ratio approaches 0 dB. After the initial breaking the accuracy of TSM depended upon the nature of the roughness in the turbulent “scar” left behind on the crest. With the weak breaking of the clean surface, no steep features remained on the surface after the bulge collapsed. TSM gave a fairly good prediction of the backscattering while the scar roughness appeared primarily on the front face of the wave (although not nearly as good as that achieved at 45° grazing), but accuracy was quickly lost as the wave overtook the scar and the roughness moved to the top of the crest and the back face of the wave.

[25] With the more energetic breaking of the surfactant wave, steep, but small, features remain on the surface well after the initial breaking. TSM proved unable to predict the scattering from the surface at these times even when the average HH/VV backscattering ratio is as low −10 dB. The backscatter in this case is clearly non-Bragg despite the low HH/VV ratio. Accuracy improves significantly when the initial steep features subside, but even a very small overturning event late in the profile history yielded very poor results, particularly at 20 GHz. Overall, it appears that TSM can provide only a rough estimate of the scattering coefficient, even well after breaking when the largest steep features have dissipated. As previously suggested by Ja et al. [2001], no times where “fast” scattering from the surface (with Doppler shifts at least as high as that expected from the phase velocity of the breaker) could be accurately described by Bragg scattering in this data set.


[26] The authors would like to thank J. H. Duncan for providing the surface profiles used in the study and for his aid in preparing the experimental description. This work was supported by the U.S. Office of Naval Research Ship Structures and Systems S&T Division (grant N00014-00-1-0082; program officer, Ronald P. Radlinski).