SEARCH

SEARCH BY CITATION

Keywords:

  • radar scattering;
  • breaking waves;
  • low grazing angles

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup
  5. 3. Results
  6. 4. Numerical Simulations
  7. 5. Discussion and Conclusions
  8. 6. Summary
  9. References

[1] This paper describes the results of radar scattering experiments carried out at the wavetank facility at the University of Maryland, College Park. Spilling and plunging breakers with a water wavelength of approximately 80 cm were generated through dispersive focusing of a chirped wave packet, and were then imaged with a high-speed camera in conjunction with a laser sheet. Simultaneously, the radar backscatter generated by the breakers when viewed with an “up-wave” look direction was measured by an ultrawideband, dual-polarized, X-band radar with a range resolution of approximately 4 cm. The nominal grazing angle of the radar was 12°. In addition to providing both quantitative profiles of the evolving water surface and the corresponding ultrahigh resolution radar backscatter, this experimental setup also included a moving instrument carriage that allowed the sensors to follow the breakers throughout their entire evolution. Numerical scattering simulations that use the measured surface profiles as inputs were also conducted in order to further dissect the scattering mechanism. An analysis of the results shows that for the spilling breaker, over 90% of the horizontally (HH) polarized radar backscatter is generated during the initial stage of breaking by the small bulge near the wave crest. For vertical (VV) polarization, the crest bulge produces about 60% of the total backscattered energy. For both polarizations, the Doppler velocity associated with this energy is very close to that of the phase speed of the dominant wave in the water wave packet. For VV, the remainder of the backscattered energy is generated by the turbulent, postbreaking surface, and in fact a close correlation is observed between increases in the VV backscatter amplitude and the shedding of vortex ripples after the wave breaks. For the plunging breaker, the initial feature on the crest (an overturning jet) generates a lower percentage of the total backscattered energy. For the spilling breaker, agreement between the experimental and numerical results is good, particularly in the Doppler domain. The simulations also indicate uncertainty in the polarization ratio measurements that is related to multipath scattering. To the extent that they can be applied to the open ocean, these results can serve as a guide for the development of future breaker models.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup
  5. 3. Results
  6. 4. Numerical Simulations
  7. 5. Discussion and Conclusions
  8. 6. Summary
  9. References

[2] Radar backscatter from breaking water waves has long been the focus of radar remote sensing research, owing to the important role that breakers play in the interaction between the atmosphere and the ocean. Radar backscatter from breaking waves is also an important consideration for marine surveillance applications, since breakers and sharp-crested waves are associated with anomalous echoes, termed “sea spikes,” that complicate target detection at low grazing angles (LGA). Other reasons for the long history of breaking wave research include the elusive, transient nature of these ocean surface features and the fact that their evolution is governed by extremely nonlinear hydrodynamics.

[3] In order to make investigations of these nonlinear, dynamic features tractable, experiments are often conducted in laboratory wavetanks [Kwoh and Lake, 1984; Lee et al., 1998; Fuchs et al., 1999; Sletten and Trizna, 1994; Sletten and Wu, 1996; Dano et al., 2001a, 2001b; Ericson et al., 1999; Loewen and Melville, 1991; Rozenberg et al., 1995; Walker et al., 1996; Trizna et al., 1993, 1991; Ebuchi et al., 1993; Plant et al., 1999a, 1999b]. In the most recent investigation that addresses the LGA regime, spilling and plunging breakers were studied over a range of incidence and azimuth angles using several different radar systems and high-speed cameras [Dano et al., 2001a, 2001b] (hereinafter referred to as Dano), including a camera designed to investigate specular scattering. Fuchs et al. [1999] have also studied LGA backscatter from laboratory breakers using simultaneous radar and optical imaging systems. The waves in that study were plungers generated by the Benjamin-Feir instability, rather than by the dispersive-focusing technique used in Dano and in the experiment discussed in this paper. Lee et al. [1998] have also studied plunging breakers generated through the Benjamin-Feir mechanism. However, while simultaneous optical imagery was collected in both cases, it is of limited quantitative use due to the viewing angles used. Rozenberg et al. [1995] used wind forcing to generate waves for their LGA studies and characterized the surface using wire wave gauges. Trizna et al. [1991, 1993] also used wind forcing, and used a scanning laser to measure the surface slope over the region interrogated by their pulsed LGA radar system.

[4] The present work adds to this body of research in several ways. First, this investigation pays particular attention to spilling breakers [Duncan, 2001], a weaker form of breaker that may not be as dramatic as the more energetic plunger, but one that is quite likely more prevalent outside of the surf zone. Second, this investigation combines a high-speed video camera that provides highly quantitative surface profiles with an ultrahigh resolution, dual-polarized radar. Third, this study makes use of a moving instrument carriage that allows both sensors to follow the wave and collect an uninterrupted record of its entire space/time evolution. This combination of capabilities provides a means to make extremely detailed, simultaneous measurements of both the dynamic morphology of the breaking wave and the resultant radar backscatter. This study also includes numerical simulations that help dissect the scattering process. Finally, this investigation was conducted in the LGA regime, where important questions regarding the scattering mechanisms still remain.

[5] In this paper, these capabilities are used to determine the relative importance, from a radar scattering point-of-view, of the major features that form on or near the crest of the breaking waves generated in this study. To the extent that these wavetank breakers are representative of waves in the ocean, these results can guide the development of practical, accurate, radar scattering models.

[6] This paper is divided into 6 sections. Following this introduction is a description of both the optical and radar systems used in the experiment. The experimental results are presented in Section 3 and compared to the results of numerical simulations in Section 4. Section 5 contains a discussion of the findings as well as some conclusions, and the paper closes with a summary.

2. Experimental Setup

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup
  5. 3. Results
  6. 4. Numerical Simulations
  7. 5. Discussion and Conclusions
  8. 6. Summary
  9. References

2.1. U-MD Wavetank and Optical Imaging System

[7] The experiments discussed in this paper were performed at the wavetank facility in the Mechanical Engineering Department of the University of Maryland, College Park (U-MD). At the U-MD facility, described in detail in Duncan et al. [1999], waves are generated by means of a computer-controlled, wedge-type wavemaker mounted at one end of the 14.6 m long basin. A system of skimmers keeps the surface clear of unwanted surfactant. A moving instrument carriage is available to allow sensors to follow waves down the tank as they evolve in space and time. In the experiments discussed in this paper, both the optical imaging system and the radar were mounted on this carriage.

[8] A digital movie camera mounted on the side of the instrument carriage provides high-speed optical imagery of the evolving breakers. (Note that this digital camera replaces the older, analog system described in Duncan et al. [1999].) While capable of a frame rate of 1 KHz, the camera was operated in this study at 250 Hz in order to match the effective frame rate of the NRL radar system. In order to enhance the visibility of the air-water interface in the imagery, a thin light sheet generated by a Nd:YAG laser is used to excite rhodamine dye dissolved in the water, while an optical filter on the camera lens blocks out direct laser light. With this arrangement, the camera records a vertical, two-dimensional profile of the water surface located along the centerline of the tank. Variations in the water surface in the across-tank dimension, while sometimes detectable to the eye in the imagery, cannot be quantified by this technique. As discussed later in this paper, this two-dimensional representation of the surface is adequate in the case of spilling breakers, but is incomplete when used to describe the more complicated surfaces produced by plungers. The size of the area imaged by the camera is 18.5 × 18.5 cm.

[9] The breakers in this study were generated by dispersive focusing [Duncan et al., 1999]. In this technique, the wavemaker generates a chirped wave packet, with the short wavelength, slower moving components occurring first, followed by the longer wavelength, faster components. The packet then naturally coalesces into a breaker at a specified point along the axis of the tank. In this study, the center frequency of the packet was 1.42 Hz, with a corresponding phase speed of 94.5 cm/s. The instrument carriage velocity was set to this speed in order to allow the camera and radar to follow the breaker crest as it evolved. To generate the spilling breakers, an amplitude parameter (normalized by the nominal wavelength of the wave packet) of 0.062 was used, while the plunging breakers in this study were generated by increasing the amplitude parameter to 0.066. As determined from the optical images, the trough-to-crest height of the waves just prior to breaking was approximately 12.5 cm.

2.2. NRL Ultrawideband (UWB) Radar

[10] Various modified forms of the radar in this study have been used in a number of previous sea scatter investigations [Sletten and Wu, 1996; Sletten, 1998; Trizna et al., 1991, 1993], and a complete description of the system can be found in Sletten and Trizna [1994]. The radar is based on a time-domain reflectometer (TDR) module for a sampling oscilloscope, and owes its ultrahigh resolution capability to the fast rise time and precise timing of the TDR pulse. Equivalent time sampling is used to effectively sample the microwave echo at a rate in excess of 50 GHz. A 6–12 GHz microwave amplifier is used for pulse formation. The combination of the amplifier response and the TDR pulse rise time produces microwave pulses with a 3-dB pulse width of approximately 0.5 ns, providing a range resolution of approximately 4 cm. Alternatively, the short microwave echoes can be transformed into the frequency domain and calibrated to provide the frequency response of the scatterer [Sletten, 1994].

[11] In the present study, the system was operated in a dual-polarized mode in which the copolarized vertical (VV) and horizontal (HH) echoes were collected at a per-polarization pulse repetition frequency of 125 Hz. One complete scan of the 37.5 cm range swath of the radar was completed every 4 ms, with HH and VV echoes collected on alternate scans. The nominal range to the water surface from the two system antennas (one for transmit, one for receive) was 1.6 m, producing a bistatic angle of 9 degrees. The radar was mounted such that it viewed the breakers in an up-wave direction (i.e., the radar was located downstream of the waves and looked upstream, into the front face of the breakers).

[12] The output power of the radar system used in this investigation was relatively low. Due to the space and weight limitations of the instrument carriage, a small, solid-state amplifier was used to form the transmit pulse, as in previous laboratory experiments, rather than the high power traveling wave tube used more recently in the field [Sletten, 1998; Sletten et al., 1996]. As a result, signal-to-noise considerations made it necessary to locate the breaking wave close to the system antennas (1.6 m), which have a far-field range of approximately 3.5 m at 10 GHz.

[13] A radiometric calibration of the VV and HH signals was performed in the frequency domain using a sphere. This calibration was used to determine the absolute radar cross section of the backscatter presented in the frequency domain plots in Figures 6, 12b, and 13b. However, due to the fact that the frequency response of the breakers (i.e., their radar cross section as a function of frequency) varies with time and polarization [Sletten and Wu, 1996] (see Figure 6 below), the sphere was used only to provide an approximate, relative calibration between the VV and HH time domain echoes, such as those in Figures 2 and 9. Assigning an absolute radar cross section to the amplitude of the time-domain echo amplitude would be potentially misleading, since such a calibration would ignore variations in the frequency content of the backscatter.

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup
  5. 3. Results
  6. 4. Numerical Simulations
  7. 5. Discussion and Conclusions
  8. 6. Summary
  9. References

3.1. Spilling Breaker

[14] A series of optical images of the spilling breaker and plots of the corresponding range scans of the radar backscatter envelope are shown in Figures 1 and 2, respectively. The time interval between successive images and scans displayed in these figures is 40 ms, ten times the interval provided by the 250 Hz frame rate. As can be seen between 0.164 s and 0.244 s, the initial stage of the breaker evolution involves the formation and subsequent breakdown of a small “bulge” near the crest of the wave [Duncan et al., 1999]. The backscatter becomes detectable just as the bulge forms, and the peak HH echo occurs near 0.244 s, shortly after the wave begins to break. (As defined in this paper, “breaking” begins when the flow first becomes turbulent after the bulge begins to move down the wave face.) The HH radar echoes are compact and Gaussian-shaped, similar to the impulse response of the radar, implying the presence of an isolated scattering center. The VV echoes are generally impulse-like as well, although a few are asymmetric or multi-peaked and thus imply the presence of two or more closely spaced scatterers.

image

Figure 1. Image sequence depicting the temporal evolution of the morphology of a spilling breaker. These stills were extracted from a “movie” of the breaking wave, collected at a frame rate of 250 Hz. The area depicted in each image measures 18.5 × 18.5 cm.

Download figure to PowerPoint

image

Figure 2. Range scans of the radar backscatter envelope corresponding to each of the optical images shown in Figure 1. The vertical scale is linear, and the dashed, vertical lines indicate the locations of the left and right edges of the optical images. Upper plots: vertical polarization. Lower plots: horizontal polarization.

Download figure to PowerPoint

[15] A waterfall plot of the surface profiles extracted from the optical imagery is shown in Figure 3. The two most important features of this plot from the radar scattering viewpoint are the initial, steep bulge that becomes apparent near t = 0.2 s, and the multiple ridges that make up the scalloped pattern in the upper right quadrant of the figure. (Note that in this waterfall plot, the leftmost data point of each surface profile indicates the collection time for that particular profile.) The scalloped pattern is formed by vortex ripples that are shed from the crest and which have a propagation speed that is less than that of the crest and instrument carriage [Duncan et al., 1999; Qiao and Duncan, 2001; Lin and Rockwell, 1995]. The slower propagation speed gives these features a retrograde motion in the video imagery. The “humps” on the surface caused by these ripples can also be seen between 0.324 s and 0.524 s in Figure 1. The plot in Figure 3 also shows that between approximately t = 0.2 and 0.4 s, the leading edge of the broken surface near the crest moves somewhat faster than the crest itself. After 0.5 s, the leading edge slows down and drifts backwards relative to the instrument carriage. A geometrical analysis of these features for a similar wave can be found in Duncan et al. [1999].

image

Figure 3. Waterfall plot of the spilling breaker surface profile. Each profile was extracted from an optical image, such as those shown in Figure 1.

Download figure to PowerPoint

[16] The radar echo envelopes are displayed in a range-time format in Figures 4a and 4b. (For clarity, the HH and VV color scales are normalized separately. The HH peak is approximately 11 dB stronger than that for VV.) The HH backscatter is localized in a particular region of the space/time plot, which, after comparison with Figures 1 and 3, can be associated with the initial stage of breaking and the crest bulge. Integration of the squared values of the backscatter envelope shows that 90% of the total HH energy received by the radar is produced between 0.2 and 0.3 s, during which time the crest bulge forms and breaks down. This initial stage of breaking produces approximately 60% of the total backscattered energy for VV, while the remainder is produced over the next 0.7 s. As can be seen in both Figures 3 and 4a, until approximately t = 0.5 s, the surface features that generate the radar echoes are spawned from a point on the crest that moves at a velocity slightly greater than that of the carriage. After t = 0.5 s, these features are shed from a point that falls progressively farther behind the instrument carriage, causing the late-time “tail” in the VV plot in Figure 4a to drift rearward in range.

image

Figure 4. Range versus time plots of the radar echo envelope for the spilling breaker. (a) Vertical polarization. (b) Horizontal polarization. The grayscales in images (a) and (b) are normalized separately.

Download figure to PowerPoint

[17] The series of VV echoes in Figure 4a between t = 0.2 and t = 0.7 s correlates well with the scallop pattern in Figure 3. This can be seen in Figure 5, where a time (vertical) cut through Figure 4a at a range of 182 cm is plotted along with a corresponding vertical slice through the waterfall plot of the surface height in Figure 3. As shown in this comparison, each increase in the radar echo is associated with an increase in the surface height, which is in turn caused by the passing of a vortex ripple. The correlation is good, even out to t = 0.6 s. This correlation implies that for this weak breaker, the two-dimensional profile provided by the camera is a good representation of the three-dimensional surface seen by the radar. Close examination of the optical frames in Figure 1 also indicates that the turbulent cells appear to be relatively long-crested, although it is very difficult to obtain any quantitative measure of the lateral variation of the surface from these images.

image

Figure 5. Plots of the vertically polarized backscatter (solid line) and the surface height (dashed line) as a function of time at a range of 182 cm. The correlation between increases in surface height, caused by the generation of vortex ripples, and radar backscatter is evident.

Download figure to PowerPoint

[18] Figures 4a and 4b bear a strong resemblance to Figures 9a and 9b in Dano et al. [2001b], suggesting that the breakers studied in the two experiments were very similar. Much more structure is visible in the VV tail in Figure 4a, however, due mainly to the fact that no averaging was required to generate the plot. The data presented here were collected without interruption and describe the space/time evolution of a single breaker. Averaging was not necessary to properly characterize the backscatter evolution, since run-to-run variation for a given set of wavemaker parameters was not significant.

[19] As discussed in Sletten et al. [1996], the UWB data produced by the radar used in this investigation can either be used to provide high resolution range scans, as displayed in Figures 2 and 4, or it can be transformed into the frequency domain to provide an estimate of the frequency response of the scatterer. Figure 6 shows the radar cross section for the spiller, near the initial peak at t = 0.225 s, after transformation into the frequency domain. The VV and HH responses at this instant in time differ significantly. The VV response has a maximum near 11 GHz, whereas the HH response has its maximum near 8 GHz. A similar difference between the HH and VV frequency responses of breaking waves has been observed previously [Sletten and Trizna, 1994; Sletten and Wu, 1996; West and Sletten, 1997; Sletten, 1998]. In these earlier cases, the difference was attributed to multipath scattering between the front face of the wave and a distinct feature near the crest. In particular, the roughly complementary positions of the VV and HH maxima and minima, also observed in Figure 6, were attributed to the phase difference between the HH and VV reflection coefficients for water [Kerr, 1951]. As discussed in Section 4, numerical simulations also suggest the presence of multipath scattering in the current experiment.

image

Figure 6. Absolute radar cross section of the spilling breaker as a function of frequency near the backscatter peak at t = 0.225 s. (a) Vertical polarization. (b) Horizontal polarization.

Download figure to PowerPoint

[20] Time-resolved Doppler spectra at 10 GHz for the spilling breaker are displayed in Figure 7. These spectra were computed by first transforming each frame of the radar data into the frequency domain (data from one transformed frame are shown in Figure 6), and then selecting the 10 GHz sample from each transformed frame. A spectrum at a point in time was computed from this time series of complex samples by windowing the series about the given point with a 0.2 s Hanning window, zero-padding to 0.4 s, and performing an FFT. Subsequent spectra were then obtained by sliding the window forward in increments of 0.04 s, resulting in a 0.16 s overlap between successive spectra.

image

Figure 7. Time-resolved Doppler spectra for the spilling breaker at frequency of 10 GHz. (a) Vertical polarization. (b) Horizontal polarization. A frequency of 0 Hz corresponds to the velocity of the instrument carriage. The dashed lines indicate the migration speed of the echo envelope.

Download figure to PowerPoint

[21] The strong initial backscatter in Figure 7 for both VV and HH has a relative Doppler frequency near 0 Hz, a value that corresponds to the 94.5 cm/s speed of the crest bulge and the instrument carriage. (All the Doppler frequencies cited in this paper are relative to the moving instrument carriage, while all cited velocities are relative to the stationary lab. The conversion is given by the relationship fD = 2(vD − 94.5)/λ, where vD is the computed velocity in cm/s, λ is the radar wavelength in cm, and fD is the Doppler frequency in Hz.) Virtually all the HH energy is concentrated around this initial peak. After 0.3 s, the VV Doppler spectrum splits into two branches, one at or slightly above a frequency of 0 Hz, and another that steadily decreases in frequency over the next 0.4s to a value near −30 Hz (50 cm/s). The former is most likely produced by the turbulent remnant of the crest bulge that continues to move at or slightly faster than the wave phase velocity after the bulge collapses, as evidenced by the leading edge of the wave disturbance in Figure 3 between 0.25 and 0.5 s. The lower-frequency branch of the Doppler spectrum is produced by the vortex ripples and the turbulent “scar” into which they decay. The decreasing velocity of this branch is determined in part by the speed of the vortex ripples, which decreases with time since the ripples are bound to the crest to a diminishing degree as time progresses. (In Figure 3, note that successive ripples form scallop patterns with progressively lower initial space-time slopes, that is, with progressively lower initial velocities.) The orbital velocity of the underlying wave, which also decreases with time as the wave decays after breaking, also plays a role. For comparison, linear wave theory predicts an orbital velocity of −26 Hz (56 cm/s), using a frequency of 1.42 Hz and the initial crest-to-trough wave height of 12.5 cm.

[22] The dashed lines in Figure 7 indicate another measure of the backscatter velocity: the migration speed of the echo envelope. These curves were determined by tracking the position of the peak in the echo envelope, computing its velocity as a function of time over the breaker lifetime, and then converting this velocity to a 10 GHz Doppler frequency. In order to smooth this migration velocity curve, a polynomial was fit to the peak position as a function of time before computing the speed. Deviations between these curves and the Doppler spectra are an indication of multiple scatterers and/or nonspecular mechanisms. In this case, the results of this comparison are not unexpected. The migration speed curves pass through or near the initial Doppler peaks, and for VV, the curve lies between the two branches of the spectrum present after 0.3 s. More interesting differences occur in the case of the plunging breaker, as discussed in the next section.

3.2. Plunging Breaker

[23] A series of optical images of the plunging breaker and plots of the corresponding range scans of the radar backscatter envelope are shown in Figures 8 and 9. The surface evolution is considerably more energetic than in the case of the spiller. Close inspection of the complete sequence of optical imagery shows that a jet begins to form near the crest of the wave at approximately 0.15 s and impacts the front face of the wave near 0.28 s. The top side of the jet is visible in Figure 8 between 0.18 s and 0.26 s. Due to the slight downward viewing angle of the camera necessary to allow imaging of the mid-line of the tank, the underside of the jet cannot be imaged using this photographic technique. The impact of the jet on the front face generates a steep, splash-up structure, visible between 0.34 and 0.42 s, that collapses by 0.46 s onto a turbulent surface that has significant lateral structure, judging from a qualitative inspection of the optical imagery. The increased complexity of the surface relative to that of the spilling breaker produces a corresponding increase in the complexity of the radar range profiles. Most of the echoes in Figure 9, for both VV and HH, exhibit multiple peaks, indicating that several scattering centers are responsible for the backscatter. As the interior of the jet forms a small cavity with dimensions on the order of 1–2 cm, the possibility also exists that the multi-peaked echoes are caused by multiple reflections within the jet itself, a phenomenon referred to as “phantom binning” by some authors [Lee et al., 1997]. The strongest echo from this wave was the HH backscatter produced by the splash-up structure at 0.36 s.

image

Figure 8. Image sequence depicting the temporal evolution of the morphology of a plunging breaker. These stills were extracted from a “movie” of the breaking wave, collected at a frame rate of 250 Hz.

Download figure to PowerPoint

image

Figure 9. Range scans of the radar backscatter envelope corresponding to each of the optical images shown in Figure 8. The vertical scale is linear, and the dashed, vertical lines indicate the locations of the left and right edges of the optical images. Upper plots: vertical polarization. Lower plots: horizontal polarization.

Download figure to PowerPoint

[24] Range-time plots of the radar echo envelope for the plunging breaker are displayed in Figure 10. The region on the crest that produces the strongest backscatter initially moves at a velocity significantly faster than the dominant wave, as shown by the diagonal trajectory of the echo envelope between 0.2 and 0.6 s. Between 0.6 s and the point where the VV echo drops into the noise near 1.1 s, the echo envelope moves at a velocity that is slower than the dominant wave, and thus the envelope trajectory slopes from near-range to far-range. (A more quantitative discussion of the envelope migration speed appears later in this section.) The strongest VV echo is produced by the initial jet at 0.18 s, whereas the secondary splash-up produces the strongest HH echo at 0.36 s. As in Figure 4, the color scales in these two figures are normalized separately. The peak HH echo is again approximately 11 dB stronger than the peak VV echo.

image

Figure 10. Range versus time plots of the radar echo envelope for the plunging breaker. (a) Vertical polarization. (b) Horizontal polarization. The grayscales in images (a) and (b) are normalized separately.

Download figure to PowerPoint

[25] In the case of the spilling breaker, a close correlation exists between the crest bulge and the initial pulses of VV and HH backscatter, seen between 0.2 and 0.3 s in Figure 4. However, as shown in Figure 10 between 0.15 and 0.28 s, the relationship between the initial plunging jet and the backscatter it produces is rather complicated. The initial, strong pulse of VV backscatter in Figure 10a peaks near 0.18 s, then rapidly drops off into a null near 0.225 s. Two weaker VV echoes are generated after that null but before the jet impacts the surface at 0.28 s. Two HH echoes are produced by the jet, a weaker one centered near 0.22 and a stronger one near 0.25 that ends as the jet impacts the front face. Similar backscatter dynamics, in particular, the null in the VV echo near 0.225 that occurs as the jet is forming, have recently been observed in numerical simulations of the backscatter generated by the initial stage of a plunging breaker [Holliday et al., 1998; West, 2002]. As discussed by West, this behavior suggests the existence of an additional sea spike mechanism, one that does not involve multipath scattering between the crest and the front face of the wave.

[26] Integrating the squared values of the echo envelopes shows that for both HH and VV, approximately 35% of the total backscatter energy is produced by the time the jet impacts the front face. By 0.5 s, 90% of the HH backscatter has been produced, while for VV, the same percentage is not reached until 0.7 s.

[27] Time-resolved Doppler spectra for the plunging breaker at 10 GHz are shown in Figure 11. The velocity spread is significantly broader than for the spilling breaker. The initial VV backscatter has a Doppler frequency near 10 Hz (105 cm/s), while the initial peak in the HH spectra occurs near 25 Hz (130 cm/s). There is a time offset between the initial peaks for VV and HH as well, with the VV peak occurring approximately 0.07 s earlier. This same time offset can also be observed in Figure 10 between the first VV echo near 0.18 s and the strong HH echo at 0.25 s. Energy is present near 40 Hz (155 cm/s) for both polarizations when the splash-up feature is present between 0.3 and 0.5 s. As for the spilling breaker, the VV Doppler centroid decreases during the latter stage of breaking, reaching a minimum near −40 Hz (30 cm/s).

image

Figure 11. Time-resolved Doppler spectra for the plunging breaker at frequency of 10 GHz. (a) Vertical polarization. (b) Horizontal polarization. A frequency of 0 Hz corresponds to the velocity of the instrument carriage. The dashed lines indicate the migration speed of the echo envelope.

Download figure to PowerPoint

[28] As in Figure 7, the dashed lines in Figure 11 indicate the migration velocity of the echo envelope. To first order, the peaks in the Doppler spectra in Figure 11 tend to follow the migration speed curves, although the spectral spread about the curves is significant. At least after the jet impacts the front face and causes the surface to become highly three-dimensional, the spread in the Doppler spectra about the migration curves is in all likelihood due to the range of scatterer velocities that are present. However, significant deviations between the migration speed and the spectra exist while the jet is the only significant feature present, that is, between 0.15 and 0.28 s, before it impacts the surface. For HH, there is a maximum of 50 Hz (170 cm/s) in the envelope migration speed near 0.25 s, whereas the spectral peak at that time is near 25 Hz (130 cm/s). For VV, the difference at this point in time is even greater: the migration speed maximum is near 60 Hz (185 cm/s), but the primary spectral peak is located at only 5 Hz (100 cm/s). A secondary spectral peak for VV exists at this time near 20 Hz (125 cm/s). The source of this discrepancy is most likely the complicated scattering that can occur on an isolated, evolving jet [Holliday et al., 1998; West, 2002]. This complicated mechanism produces a phase progression in time (i.e., a Doppler velocity) that is only partly determined by the actual translation speed of the jet.

4. Numerical Simulations

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup
  5. 3. Results
  6. 4. Numerical Simulations
  7. 5. Discussion and Conclusions
  8. 6. Summary
  9. References

[29] A numerical technique has recently been used to investigate the backscatter generated by deterministic, breaking-wave surfaces [West et al., 1998]. This technique provides a means to dissect the scattering mechanism, and has been used successfully with both measured as well as simulated surfaces. In this paper, the scattering process is investigated further, using additional simulations based upon the measured surface profiles displayed in Figure 3.

[30] In these simulations, a hybrid Geometric Theory of Diffraction/Method of Moments (GTD/MOM) technique is used to calculate the backscatter. The simulated surfaces are formed by joining infinitely long extensions to the leading and trailing edges of the measured surface profiles shown in Figure 3. These extensions eliminate spurious reflections from the leading and trailing edges, and allow the moment method to be used at low grazing angles. The surfaces are assumed to be two-dimensional, i.e., they are assumed to extend infinitely, without variation, in the lateral dimension perpendicular to the plane of incidence. Plane wave illumination is used, and the calculations are performed in the frequency domain. Complete details of the method can be found in West et al. [1998].

4.1. Comparison With Experimental Results

[31] Figures 12a and 13a display the simulated scattering cross sections of the spilling breaker as a function of time at two frequencies, 8 and 10 GHz. The grazing angle for both simulations is 12 degrees. For comparison, the 8 and 10 GHz components of the experimental data are plotted in Figures 12b and 13b. Apart from a discrepancy in the relative values of the VV and HH cross sections (discussed below), the comparison is quite favorable. Both the numerical and experimental curves are characterized by a broad, initial peak in the backscatter near 0.225 s, produced by the initial crest bulge, followed by a series of shorter duration peaks, produced by the turbulent surface after breaking. At 8 GHz, the initial VV peak has a somewhat jagged, irregular shape, due to the destructive multipath that occurs at that frequency and time (see Figures 6 and 16 below). At 10 GHz, the initial VV peak is smoother. In both the numerical and experimental data, the peaks that occur after the crest bulge collapse fall off in intensity more rapidly for HH than for VV. These short duration peaks are more numerous in the numerical data, especially at 10 GHz, probably as a result of the fact that the numerical simulations assume that even the smallest undulations in the surface profile extend perfectly in the lateral dimension. This assumption results in an overestimate of the backscatter that they produce.

image

Figure 12. Absolute radar cross section (RCS) as a function of time at a frequency of 8 GHz for the spilling breaker. (a) Simulated RCS, using the measured crest profiles joined to a flat-water extension as inputs. (b) Measured RCS.

Download figure to PowerPoint

image

Figure 13. Absolute radar cross section (RCS) as a function of time at a frequency of 10 GHz for the spilling breaker. (a) Simulated RCS, using the measured crest profiles joined to a flat-water extension as inputs. (b) Measured RCS.

Download figure to PowerPoint

[32] Simulated Doppler spectra for the spilling breaker at 10 GHz are displayed in Figure 14. The comparison with the experimental data in Figure 7 is very good. Both the numerical and experimental plots show strong, initial peaks at the carriage velocity for both VV and HH, and a significant lower velocity tail for VV after breaking occurs. A 20 dB gray-scale is used in the experimental plot, while only 13 dB is displayed in the numerical data. This difference in the dynamic range of the experimental and numerical results can be attributed to multipath propagation (discussed below), but the idealized, two-dimensionality of the numerical simulations may also explain some of the difference. Due to the (incorrect) assumption that they extend infinitely in the lateral dimension, the small surface undulations that occur towards the end of the breaker lifetime generate disproportionately strong backscatter in the simulations, compared to the larger crest bulge undulations that occur earlier.

image

Figure 14. Simulated, time-resolved Doppler spectra for the spilling breaker at a frequency of 10 GHz. (a) Vertical polarization. (b) Horizontal polarization. A frequency of 0 Hz corresponds to the velocity of the instrument carriage.

Download figure to PowerPoint

[33] Simulated Doppler spectra for the plunging breaker at 10 GHz are displayed in Figure 15. While not as favorable as the corresponding comparison for the spiller, this simulated response still shares several important features with its experimental counterpart displayed in Figure 11. The HH energy in both the experimental and numerical plots occurs in a cluster between 0.2 and 0.5 s and between −10 and 40 Hz, and both VV plots show a fall-off to lower frequencies after 0.5 s. Significant VV energy between 0.4 and 0.5 s with a Doppler frequency in excess of 40 Hz is also accurately predicted. The numerical results also predict the greater spread in frequencies exhibited by the experimental plunger data relative to that for the spiller. But greater fidelity in these plunger simulations is precluded by inaccuracies in the measured surface profiles (not shown). As mentioned earlier, the underside of the initial jet was blocked from the camera's view, and the significant lateral structure of the surface present after the collapse of the secondary jet could not be captured by the two-dimensional, centerline profile provided by the laser sheet and camera. Three-dimensional surface measurements and simulations are necessary in order to reproduce the plunger's scattering behavior in greater detail.

image

Figure 15. Simulated, time-resolved Doppler spectra for the plunging breaker at a frequency of 10 GHz. (a) Vertical polarization. (b) Horizontal polarization. A frequency of 0 Hz corresponds to the velocity of the instrument carriage.

Download figure to PowerPoint

[34] The simulations suggest that multipath scattering is a major determinant of the backscatter intensity during the initial stage of breaking. This is illustrated in Figure 16, where the simulated backscatter is plotted as a function of frequency, at a particular instant in time, for three breaking wave profiles with different front-face extensions. The profile in the immediate vicinity of the crest is the same in all three cases, and is the measured crest profile for the spiller corresponding to the initial peak in the backscatter at t = 0.225 s. The three profiles differ in the location of a leading “flat water” extension that is smoothly connected to the front of the crest. The depth of this extension below the mean water surface for the three cases is 0.0, 1.5, or 3.0 cm. The complete profiles are shown in Figure 16a. This family of profiles is intended to approximate waves with a range of leading trough depths. (As the camera could only capture the wave geometry over the 18.5 cm centered on the crest, the depth and precise profile of the leading trough is not known.) Multipath is evidenced by the strong peaks and nulls seen in the three corresponding frequency responses (Figures 16b–16d). These maxima and minima are interference fringes that result from the coherent addition of direct backscatter from the crest and backscatter that also involves one or more “double-bounce” reflections off the front face extension. This figure also illustrates the sensitivity of the initial backscatter to the geometry of the front face. As shown, variation of the trough depth of only a few centimeters has a significant effect on the radar backscatter at a given frequency. This small change in geometry causes a significant change in the relative path lengths traveled by the various multipath components, and thus in their total, coherent sum. The figure also shows that the profile with a trough depth of 0 cm produces VV and HH responses (Figure 16b) that are qualitatively similar to those that were measured experimentally (Figure 6). The surfaces used to generate the simulated RCS plots shown in Figures 12a and 13a included a flat-water extension located at this depth.

image

Figure 16. Simulated frequency responses for three surface profiles with different front-face extensions but the same crest profile. (a) Surface profiles, with the front-face extension located at 0, 1.5, and 3.0 cm below the mean water level. (b–d) Simulated frequency responses for the three surfaces. (VV, solid line; HH, dashed line.)

Download figure to PowerPoint

[35] Other simulations (not shown) illustrate how multipath introduces sensitivity to the grazing angle, a sensitivity that is again a result of path length differences between the various multipath components. For example, at 10 degrees grazing, the simulated polarization ratio for the initial backscatter peak increases to approximately 10 dB from its value near 3 dB at a 12 degree grazing angle. Additional simulations also show that during the later stages of breaking, the backscatter is insensitive to the front face geometry, indicating that multipath is not an important mechanism once the wave has collapsed and a sharp crest supporting a distinct feature no longer exists. West and Ja [2002] and Ja et al. [2001] have determined that a small perturbation method predicts the backscatter from these breakers fairly well once the crest has collapsed, a result that corroborates Dano's results.

4.2. Uncertainties

[36] The peaks and nulls in Figures 6 and 16 imply that multipath is an important scattering mechanism during the initial stage of breaking. However, the numerical simulations also show that the sensitivity of the radar backscatter to the grazing angle and front face geometry of the wave, both byproducts of multipath scattering, make a quantitative comparison of the experimental and simulated polarization ratios difficult. The experimental measurement of the grazing angle could be in error by a few degrees, and as discussed above, the front face profile could not be measured. These uncertainties are in all likelihood the source of the polarization ratio discrepancies observed during the initial backscatter peaks in Figures 12 and 13.

[37] Operation of the radar in the near field of the antennas is another potential source of discrepancies between the measured and simulated results, one that is even more difficult to investigate quantitatively. The numerical results presented in this paper all assume plane wave (far-field) illumination. While still not a true near field calculation, the simulations used in this research can be modified to account for the amplitude of the antenna pattern and the finite distance between the radar and the breaking waves, but only if the antennas are modeled as point sources. As the actual antennas have an aperture width of 23 cm, a dimension that is a significant fraction of the distance to the breakers, (1.6 m), this point-source approximation is questionable. (In fact, it was found that modified simulations that incorporate the point-source antenna model predict that multipath scattering can occur only if the grazing angle is less than 7°, regardless of the position of the flat-water extension.) However, the good agreement between the measured and simulated responses shown in Figures 12 and 13, 7 and 14, and 11 and 15 implies that the simulations accurately account for the important physics and surface features. In addition, earlier near-field measurements made with the NRL radar have been successfully simulated using plane-wave illumination [West and Sletten, 1997]. Near-field effects in the experimental results presented in this paper are thus not expected to be significant.

5. Discussion and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup
  5. 3. Results
  6. 4. Numerical Simulations
  7. 5. Discussion and Conclusions
  8. 6. Summary
  9. References

[38] The results of this investigation have several implications for the modeling of low grazing angle radar backscatter from breaking waves. They suggest that for HH polarization and an up-wave look direction, the spilling breaker is well represented by the crest bulge alone. For HH, this feature accounts for approximately 90% of the backscatter produced by the wave, and the Doppler velocity of the HH backscatter matches the velocity of the crest. The basic HH model is potentially quite simple, as a discrete scatterer (for instance, a cylinder) moving at the phase velocity of the underlying wave exhibits these same characteristics. For VV, the vortex ripples also make a significant contribution to the total backscatter, and thus they must be accounted for. However, it appears that a small perturbation model in conjunction with an appropriate roughness spectrum may provide a reasonably accurate estimate of their contribution. The results of this experiment also corroborate earlier investigations that highlight the role of multipath in the determination of the polarization ratio and in the generation of sea spikes. However, the results indicate that the plunger will require a more complicated scattering model, as more features with a wider range of velocities contribute to its backscatter for both polarizations. The plunging breaker results also provide initial experimental evidence for an additional sea spike mechanism, one that does not involve multipath.

[39] While these conclusions can help guide the development of low grazing angle breaking wave models, they must be used with caution. First, only the up-wave look direction has been considered in this work. Second, small breaking waves in the ocean may not always resemble the wavetank breakers in this study. The generation mechanisms may be different, and this may result in differences in the wave morphology. This is a difficult issue to address, given the challenges posed by high-resolution measurement of open-ocean surface waves. However, the basic scattering description—the sensitivity of HH polarization to sharp, elevated features, and the additional contribution to VV from postbreaking surface roughness—should apply in any case. Multipath scattering between crest features and the front face is another characteristic that should be common to all breaking waves at low grazing angles, regardless of the details of their morphology. The results of Fuchs et al. [1999] support this assertion. Although the waves in that study were plunging breakers generated through the Benjamin-Feir mechanism, the scaling law that they observed for HH polarization can be derived from a simple multipath model and the assumption that the wave height scales linearly with wavelength [Fuchs and Tulin, 1998]. They do not observe a similar scaling law for VV polarization, most likely due to the significant role that distributed, Bragg-like scatter plays in that case. Of even more significance than differences in the generation mechanism may be the fact that the surface of an open ocean breaker is often covered with small-scale, wind-driven roughness. The wavetank breakers in this study were generated in the absence of wind. Efforts have begun to determine this effect and incorporate it into breaking wave models [Zhao and West, 2003].

[40] Multipath may also explain the difference between the low peak polarization ratios (HH/VV) observed by Dano (near 0 dB) for a spilling-plunging breaker, and the relatively high values reported in this paper (11 dB or more). In Dano et al. [2001b], the wave profile and grazing angle may have been such that the multipath interference produced nearly equal VV and HH radar cross sections during the initial stage of breaking, whereas in the current experiments, they combined in such a manner as to favor HH. However, a more likely explanation is that at the lower grazing angle used by Dano, the multipath reflection point on the front face was shadowed by a wave crest in front of the breaker, eliminating the interference necessary to raise the HH cross section above that for VV. As this scenario is likely to be common at very low grazing angles in the open ocean, it is important to explore additional sea spike mechanisms, such as the one investigated by Holliday et al. [1998] and West [2002], that do not involve multipath.

[41] A few additional comments regarding scattering mechanisms should be made. Distributed (Bragg) scattering is the most commonly cited mechanism by which the sea surface produces radar backscatter. Yet it might seem that this mechanism is not supported by an ultrawideband radar like the one in this study, given its extremely high range resolution and small footprint on the surface. From the time domain standpoint, Bragg scatter might appear to be precluded by virtue of the fact that the ultrashort pulse never illuminates more than a limited number of Bragg crests at any instant of time. From the frequency domain standpoint, one could correctly point out that even if Bragg scatter occurs at one frequency within the radar pulse bandwidth, a strong, ultrashort, time domain echo will not be produced, since this echo is the coherent sum of the responses over a much wider band of (generally) nonresonant frequencies. But nevertheless, evidence of Bragg scatter is still theoretically present in ultrashort time domain data whenever that scattering mechanism is at work. The resonant peaks will be evident in the Fourier transform of the time-domain data record, that is, in the frequency domain representation of that data, provided that the range extent of the record is long enough to cover the “patch” of Bragg scatterers on the surface, and provided that the dynamic range of the radar is sufficiently high. This is a result of the fact that the electromagnetic scattering is a linear process for which the superposition principle holds. However, due to practical limitations on the dynamic range and/or sensitivity of the ultrawideband radar, this distributed backscatter may not be detectable. Backscatter from the compact crest scatterers, features which can be fully contained within the short pulse, may be many orders of magnitude stronger than the distributed backscatter generated by any small patch of roughness that may also be present. The relatively weak distributed backscatter may be outside the limits of the system digitizer or lost in the system noise. But with a longer pulse, of course, the disparity between the Bragg and feature contributions to the total backscatter may lessen significantly, since the former may increase as the pulse widens. For investigations of the role Bragg scattering plays in the generation of radar backscatter from steep waves, see, for instance, the recent works of Plant et al. [1999a, 1999b].

6. Summary

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup
  5. 3. Results
  6. 4. Numerical Simulations
  7. 5. Discussion and Conclusions
  8. 6. Summary
  9. References

[42] This paper presents the results of laboratory experiments in which both the surface profile of spilling and plunging breakers and their LGA radar backscatter were measured simultaneously. The dynamic surface profiles were extracted from optical imagery collected by a high-speed camera, while the radar backscatter was measured by a dual-polarized, ultrawideband radar with a resolution of 4 cm. Both the camera and the radar were mounted on a moving instrument carriage that followed the waves throughout their evolution, allowing both sensors to record an uninterrupted record of the breakers' space/time evolution. The results indicate that for the spilling breaker with an up-wave look direction, the crest bulge present during the initial stage of breaking is responsible for over 90% of the HH backscatter. This feature generates approximately 60% of the VV backscatter, while the remaining 40% is produced by vortex ripples that are periodically shed by the crest after the bulge breaks down. The evolution of the plunging breaker morphology and backscatter were observed to be significantly more complicated, involving more surface features and greater transverse variability. Numerical simulations of the radar backscatter that use the measured surface profiles as inputs agree reasonably well with the experimental results, particularly for the spilling breaker in the Doppler domain.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup
  5. 3. Results
  6. 4. Numerical Simulations
  7. 5. Discussion and Conclusions
  8. 6. Summary
  9. References
  • Dano, E. B., D. R. Lyzenga, and M. Perlin, Radar backscatter from mechanically generated transient breaking waves—part I: Angle of incidence dependence and high resolution surface morphology, IEEE J. Oceanic Eng., 26(2), 2001a.
  • Dano, E. B., D. R. Lyzenga, G. Meadows, L. Meadows, H. V. Sumeren, and R. Onstott, Radar backscatter from mechanically generated transient breaking waves—part II: Azimuth and grazing angle dependence, IEEE J. Oceanic Eng., 26(2), 2001b.
  • Duncan, J. H., Spilling breakers, Ann. Rev. Fluid Mech., 33, 519547, 2001.
  • Duncan, J. H., H. Qiao, V. Philomin, and A. Wenz, Gentle spilling breakers: Crest profile evolution, J. Fluid Mech., 379, 191222, 1999.
  • Ebuchi, N., H. Kawamura, and Y. Toba, Physical processes of microwave backscattering from laboratory wind wave surfaces, J. Geophys. Res., 98(C8), 14,66914,681, 1993.
  • Ericson, E., D. R. Lyzenga, and D. T. Walker, Radar backscatter from stationary breaking waves, J. Geophys. Res., 104(C12), 29,67929,695, 1999.
  • Fuchs, J., and M. P. Tulin, Laboratory measurements of the joint dependence of radar frequency and water wavelength on sea spike returns, paper presented at International Geoscience and Remote Sensing Symposium (IGARSS '98), Inst. of Electr. and Electron. Eng., Seattle, Wash., 6 – 10 July 1998.
  • Fuchs, J., D. Regas, T. Waseda, S. Welch, and M. Tulin, Correlation of hydrodynamic features with LGA radar backscatter from breaking waves, IEEE Trans. Geosci. Remote Sens., 37(5), 1999.
  • Holliday, D., L. L. DeRaad, and G. J. St.-Cyr, Sea-spike backscatter from a steepening wave, IEEE Trans. Antennas Propag., 46(1), 1998.
  • Ja, S.-J., J. C. West, H. Qiao, and J. H. Duncan, Mechanisms of low-grazing-angle scattering from spilling-breaker water waves, Radio Sci., 36(5), 981998, 2001.
  • Kerr, D. E., (Ed.), Propagation of Short Radio Waves, McGraw-Hill, New York, 1951.
  • Kwoh, D. S., and B. M. Lake, A deterministic, coherent, and dual-polarized laboratory study of microwave backscattering from water waves, part I: Short gravity waves without wind, IEEE J. Oceanic Eng., 9(5), 1984.
  • Lee, P. H. Y., J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, H. R. Thompson, and R. Yee, Experiments on Bragg and non-Bragg scattering using single-frequency and chirped radars, Radio Sci., 32(5), 17251744, 1997.
  • Lee, P. H. Y., J. D. Barter, K. L. Beach, B. M. Lake, H. Rungaldier, H. R. Thompson, and R. Yee, Scattering from breaking gravity waves without wind, IEEE Trans. Antennas Propag., 46(1), 1998.
  • Lin, J.-C., and D. Rockwell, Evolution of a quasi-steady breaking wave, J. Fluid Mech., 302, 2943, 1995.
  • Loewen, M. R., and W. K. Melville, Microwave backscatter and acoustic radiation from breaking waves, J. Fluid Mech., 224, 601623, 1991.
  • Plant, W. J., W. C. Keller, V. Hesany, T. Hara, E. Bock, and M. A. Donelan, Bound waves and Bragg scattering in a wind-wave tank, J. Geophys. Res., 104(C2), 32433263, 1999a.
  • Plant, W. J., P. H. Dahl, and W. C. Keller, Microwave and acoustic scattering from parasitic capillary waves, J. Geophys. Res., 104(C11), 25,85325,866, 1999b.
  • Qiao, H., and J. H. Duncan, Gentle spilling breakers: Crest flow field evolution, J. Fluid Mech., 439, 5785, 2001.
  • Rozenberg, A. D., D. C. Quigley, and W. K. Melville, Laboratory study of polarized scattering by surface waves at grazing incidence: Part I—Wind waves, IEEE Trans. Geosci. Remote Sens., 33(4), 1995.
  • Sletten, M. A., Resolution of a phase ambiguity in a calibration procedure for polarimetric radar systems, IEEE Trans. Geosci. Remote Sens., 32(1), 1994.
  • Sletten, M. A., Multipath scattering in ultrawide-band radar sea spikes, IEEE Trans. Antennas Propag., 46(1), 1998.
  • Sletten, M. A., and D. B. Trizna, An ultrawideband, polarimetric radar for the study of sea scatter, IEEE Trans. Antennas Propag., 42(11), 1994.
  • Sletten, M. A., and J. Wu, Ultrawideband, polarimetric radar studies of breaking waves at low grazing angles, Radio Sci., 31(1), 181192, 1996.
  • Sletten, M. A., D. B. Trizna, and J. P. Hansen, Ultrawideband radar observations of multipath propagation over the sea surface, IEEE Trans. Antennas Propag., 44(5), 1996.
  • Trizna, D. B., J. P. Hansen, P. Hwang, and J. Wu, Laboratory studies of radar sea spikes at low grazing angles, J. Geophys. Res., 96(C7), 12,52912,537, 1991.
  • Trizna, D. B., J. P. Hansen, P. Hwang, and J. Wu, Ultrawideband radar studies of steep crested waves with scanning laser measurements of wave slope profiles, Dyn. Atmos. Oceans, 20, 3353, 1993.
  • Walker, D. T., D. R. Lyzenga, E. A. Ericson, and D. E. Lund, Radar backscatter and surface roughness measurements for stationary breaking waves, Proc. R. Soc. London, Ser. A, 452, 19531984, 1996.
  • West, J. C., Low-grazing-angle (LGA) sea-spike backscattering from plunging breakercrests, IEEE Trans. Geosci. Remote Sens., 40(2), 523526, 2002.
  • West, J. C., and S.-J. Ja, Two-scale treatment of low-grazing-angle scattering from spilling breaker water waves, Radio Sci., 37(4), 1054, doi:10.1029/2001RS002517, 2002.
  • West, J. C., and M. A. Sletten, Multipath EM scattering from breaking ocean waves at grazing incidence, Radio Sci., 32(4), 14551467, 1997.
  • West, J. C., J. M. Sturm, and J.-S. Ja, Low-grazing scattering from breaking water waves using an impedance boundary MM/GTD approach, IEEE Trans. Antennas Propag., 46(1), 93100, 1998.
  • Zhao, Z., and J. C. West, Electromagnetic modeling of multipath scattering from breaking water waves with rough faces, IEEE Trans. Geosci. Remote, in press, 2003.