## 1. Introduction

The mean relationship between sea wave slope statistics and near-surface ocean wind speed has long been established (Cox and Munk 1956), as has the tie between these optically derived wave slope variance measurements and the ocean radar backscatter measured by a microwave altimeter (Barrick 1968b; Jackson et al. 1992). However, it has also been supposed for decades that factors such as wind gustiness, currents, fetch, and long gravity wave dynamics will serve to significantly vary any such mean wind–wave correlation and its inversion using radio probing techniques (Barrick 1974). Understanding, detecting, and monitoring these dynamics using today's complement of spaceborne ocean remote sensors is a challenge requiring a blend of field measurements, modeling, and satellite data interpretation. Opportunities presented include the goal to improve wind stress estimation from space by enfolding the noted second-order effects in air–sea coupling formulations at global to local scales.

Short waves with *λ* < 10–20 cm are considered critical to microwave remote sensing of ocean winds. This is because these wavelets respond quickly and dynamically to increases in the local wind, and because the transmit wavelength of such sensors makes the radio probing of these waves possible. However, long-wave tilting of the short waves is also fundamental perturbation carried in the increasingly common two- or three-scale models for surface wave interaction and ocean wave remote sensing alike (e.g., Bass et al. 1968; Barrick 1968a; Valenzuela 1978; Longuet-Higgins 1982; Gotwols and Thompson 1994; Liu et al. 1997; Chapron et al. 2000; Plant 2002). Such models impose a separation between long and short waves, permitting separate characterization of their wind-dependent statistics, forcing, and dissipation as well as subsequent recombination to address possible interactions between the scales. Separation is also imposed because a substantial fraction of the long gravity wave field is often due to swell and sea generated by distant or turning winds, waves that are uncoupled and misaligned as far as the local wind is concerned. Such dynamics can impact how well a remote sensor inverts wind stress and short wave information.

The following work provides new field measurements related to long and short gravity wave slopes on the sea—both their covariance with respect to local wind forcing and with longer wave dynamics divorced from the local wind. Specific goals are to document the long-wave field using directly measured slope statistics and to then to illustrate and quantify the effect that this tilting field has upon an altimeter's nadir-view normalized radar backscatter cross section, *σ*°(*θ* = 0°), and its subsequent interpretation.

*σ*°: absolute radar calibration, long-wave impacts, and short-wave diffraction. These factors are central to

*σ*° use in wind speed and wave spectral estimations, and in attempts to gain new geophysical insight using altimeter observations at differing transmit frequencies (Jackson et al. 1992; Elfouhaily et al. 1998). This paper concentrates mainly on the first two issues. A first-order specular point scattering theory (Barrick 1968b) is considered adequate for the examination of the first two factors:where the numerator holds the frequency-dependent surface Fresnel reflectivity factor and mss′ is a radar-derived estimate of the surface wave slope variance, mss. One implicit assumption is that there is no a priori need to invoke an effective Fresnel coefficient in the derivation (cf. Chapron et al. 2000). This and further assumptions behind Eq. (1) are discussed in section 2d. This paper concentrates mainly on the first two issues above while the issues of diffraction and precise inference of high-wavenumber spectral information can be reviewed elsewhere (e.g., Jackson et al. 1992; Chapron and Vandemark 1996; Plant 2002).

Absolute *σ*° levels at the nadir view angle are addressed using a Ka-band (36 GHz) radar, in part to revisit past nadir-view Ka-band data (Masuko et al. 1986). Those data represent the only such data available at Ka-band and indicate an unexpectedly large 5–6-dB difference with respect to nadir-view X- and Ku-band observations. Recent results using an indirect absolute radar cross-section determination method (Walsh et al. 1998) suggest there may be a calibration error and this possibility will be assessed. Present results should also resemble those obtained using a satellite radar altimeter operating at 36 GHz. As such, they provide preliminary design data for proposed Ka-band altimeters such as AltiKa, a new mission being considered by the European Space Agency.

Regarding long-wave impacts, typically this gravity wave (*λ* > 30 cm) contribution to mss is assumed to be small *O*(20%–30%) and highly correlated with wind speed; thus deviation from a nominal wind-dependent mss due to long waves is often considered unlikely or negligible. Some recent work has shown the effect should not be neglected (e.g., Hwang and Shemdin 1988; Hesany et al. 2000). Other recent satellite altimeter work (Glazman and Greysukh 1993; Hwang et al. 1998; Gommenginger et al. 2002; Gourrion et al. 2002a,b) use proxies such as significant wave height *H*_{s} to demonstrate measurable long-wave variability impacts upon *σ*°. Still, observed ambiguities indicate that physical explanation and quantification remains somewhat uncertain (Gourrion et al. 2002a). For example, is *σ*° variation strictly geometrically controlled, that is, due to long-wave tilt field changes, or is it due to perturbation of the centimeter scale waves by long-wave hydrodynamics? This study looks at such questions by considering radar *σ*° data collected in concert with direct measurement of slope statistics for long-to-intermediate-scale gravity waves *O*(100–1 m). These longer-wave observations are derived using a three-laser slope measurement system, providing a quantitative check of the oft-cited oil-slickened surface slope estimates reported by Cox and Munk (1956). As such, the laser results also serve the more general purpose of documenting long-wave slope statistics relevant to satellite sensors working in frequencies from L band to the visible.

The platform used to acquire the coincident wave and wind measurements for this study was the National Oceanic and Atmospheric Administration (NOAA) LongEZ research aircraft. Data were collected over a 2-yr period as part of the Office of Naval Research Shoaling Waves Experiment (SHOWEX) and the National Aeronautics and Space Administration (NASA) Wave Profile Experiment (WAPEX). Following is an overview of the data collection, measurement techniques, and their validation. Laser-derived slope and Ka-band radar backscatter data are then presented and discussed in context of past observations. Analysis combining these tools helps to clarify, in quantitative terms, the impact of long waves on the detection of short waves and, hence, wind using remote sensors in regions such as the coastal zone.

## 2. Measurements and processing

Measurements to be discussed were collected over either the western Atlantic Ocean or the inland waters of Currituck Sound, always within 120 km of Duck, North Carolina. All flights occurred between 1997 and 1999, with most data collected in the month of November. Data come from 36 separate flights of the NOAA LongEZ covering a variety of wind and wave conditions. Aircraft flight patterns over the region varied to accomodate several complementary investigations (Sun et al. 2001; Vandemark et al. 2001; Mahrt et al. 2001; French et al. 2000). The composite coverage provides a dataset containing extensive sampling over open ocean, coastal, and inland water. For all data presented herein, the aircraft flew at an average altitude of 15 m above the surface. This is a unique vantage point that permits high-resolution and high-fidelity sensing of the sea surface and air above, much as for a fixed experimental platform. Efforts mentioned above describe the aircaft measurement systems and data processing used in this study; thus only details critical to this work are discussed below. All data have undergone extensive quality checks noted in the studies above. One focus here is on the wind dependence of wave slope statistics and radar backscatter. Typical time and spatial averaging windows for ocean wind measurement are 8–10 min and 5–15 km. A 5-km along-track aircraft extent is chosen for the averaging length scale in this study to retain spatial wind and wave dynamics inherent to the region (Sun et al. 2001).

### a. Micrometeorology

*u*∗) estimates are produced using the processed LongEZ (French et al. 2000) wind data and eddy covariance flux estimations (Sun et al. 2001). The main atmospheric derivative for this study is neutral-stability wind speed at 10 m above the surface,

*U*

_{10N}. Values are obtained from measured winds

*U*

_{m}at altitude

*z*usingwhere

*κ*is von Kármán's constant;

*L,*the Monin–Obukhov stability length, is computed using a bulk formulation based on Richardson's number Ri (Donelan 1974); and the stratification correction

*ψ*follows Paulson (1970) and Dyer (1974) for unstable and stable cases, respectively. The LongEZ also measures humidity and sea and air temperature scalars needed to derive Ri. Formulation of

*L*in terms of a bulk Ri estimate is done to avoid noise frequently introduced by the cubic

*u*∗ term needed when using direct eddy correlation estimates to derive L. While

*U*

_{10n}estimates obtained here may differ slighty from alternate methods, intercomparisons show variance to be less than 0.3 m s

^{−1}, a level of accuracy that is adequate for this study.

Several in situ meteorological platforms were overflown by the aircraft during these experiments. Direct comparison of aircraft winds with those from the National Data Buoy Center (NDBC) buoy 44014, located 80 km off shore, are shown in Fig. 1. Data shown represent measurements collected within 10 km and 30 min of each other. Aircraft samples are derived from 5-km segments along the flight track, while the buoy wind is the average over an 8-min time window; *U*_{10n} values shown for both the aircraft and NDBC buoy are derived using the scalar quantities, including *U*_{m}(*z*), measured by each platform. This bulk estimation facilitates comparison for this case where the buoy does not directly measure the fluxes. The Tropical Ocean–Global Atmosphere Comprehensive Ocean–Atmosphere Response Experiment bulk flux algorithm (Fairall et al. 1996) was applied to both buoy and aircraft data to derive *U*_{10n}. The rms difference between measures is 1.1 m s^{−1}, and close overall agreement is apparent over the available dynamic range.

A separate comparison is made using flights over the University of Miami's Air–Sea Interaction Sensor (ASIS; Graber et al. 2000) during SHOWEX in November of 1999. The maximum time and distance between surface and air measurements are 15 min and 10 km, respectively. ASIS data are derived from three separate platforms operating during SHOWEX, and all ASIS estimates, including eddy correlation fluxes, represent a 20-min averaging period. The middle panel of Fig. 1 provides intercomparison of *U*_{10n}, computed using eddy correlation *u*∗ and scalar estimates from both sensors, and following Eqs. (2) and (3). The bottom panel displays ASIS versus aircraft friction velocity estimates. The linear regression correlation coefficient for both of these ASIS intercomparisons exceeds 0.94. The rms difference between *U*_{10n}, measures is 1.0 m s^{−1}. Overall agreement between both the flux and wind estimates in Fig. 1 indicates the aircraft wind data are well calibrated.

### b. Laser altimeter system and surface slope statistics

The aircraft houses three downlooking laser altimeters, mounted to form an equilateral triangle in the horizontal plane with leg spacing of 0.96 m. This provides three simulataneous wave elevation measurements as the aircraft traverses over the surface. The sensor geometry and processing, including aircraft motion compensation, are discussed elsewhere (Sun et al. 2001; Vandemark et al. 2001). The synchronized sea elevation estimates are used to compute a surface wave slope vector at roughly 1-m-spaced horizontal intervals along the aircraft ground track. The 1-m horizontal spacing of the mounted lasers limits the wavelengths resolved to *λ* > 2 m. For a 5-km flight segment nominally 5000 individual slope measurements are made. The slope vector is given as **s** = *s* cos(*α*)**i** + *s* sin(*α*)**j**, where *α* is the azimuthal angle with respect to north, the slope magnitude is *s* = tan*β,* and *β* is the elevation angle from horizontal. The slope probability density function (pdf) for these intermediate to long-scale gravity waves is thus obtained (Vandemark et al. 1999) and the variance, skewness, and kurtosis directly computed for each data segment. Triply redundant wave elevation statistics, including *H*_{s} (defined as 4 times elevation rms), are obtained as well. Last, a transect wavenumber spectrum can also be computed from data along a given flight track. Note that this is an encounter spectrum and, given the aircraft's nominal speed of 50 m s^{−1}, a swell or sea true wavenumber will suffer significant Doppler shifts. While not the focus of this study, transect spectra have been utilized for the case of flight legs oriented along the wind and wave direction (Melville and Matusov 2002; Sun et al. 2001). Our preliminary inspection of the high-wavenumber portion of Doppler-corrected transect spectra for varied wind conditions supports the form *k*^{−3.0±0.1} down to the 2.5 rad m^{−1} cutoff in the equilibrium subrange (e.g., Banner et al. 1989). Future analysis intends to assess the level of variability for this high wavenumber subrange.

The fidelity of the laser-measured wave height and slope statistics depends upon precise compensation for the continual laser pointing angle changes due to aircraft motions. Figure 2 illustrates the quality of these corrections for the case where the aircraft performed large roll manuevers over nearly smooth inland water. Aircraft-estimated wind speed over the Currituck sound was 2.5 m s^{−1} for this data segment. After correction, results should indicate a nearly flat sea surface. Raw 50-Hz aircraft roll and laser-derived aircraft altitude, sea surface elevation, and slopes are displayed for a 2.5-km ground segment (∼2500 data points). A running estimate of mss_{l} is provided via a 200-m along-track sliding average for 〈*s*^{2}_{i}_{l} falls below 3.0 × 10^{−4}. Estimated *H*_{s} is 0.10 m and mss_{l} is 1.2 × 10^{−3}, both indicative of a surface that is 10–30 times as smooth as typical open-ocean conditions. Results from repeated inland water legs under such manuevers are used to estimate that open-ocean rms wave height and slope uncertainties typically fall below 3%. As critical, even for these uncharacteristically large aircraft motions, wave data exhibit little correlation with that variation.

Comparison of buoy and aircraft-derived *H*_{s} measurements provides further validation. Intercomparison data are extracted from the same ASIS and NDBC data subsets discussed in the wind comparisons above. Results indicate that the standard deviation between buoy and aircraft *H*_{s} estimates falls below 0.15 m and the linear regression coefficient exceeds 0.92 over the 60 samples in the intercomparison.

### c. Radar scatterometer

*λ*

_{0}= 8.3 mm) scatterometer, pointed directly normal to the surface, is used to collect reflected power measurements from the roughened sea surface as detailed in Vandemark et al. (2001). Measured receiver voltages are converted to normalized radar cross section using the radar equation:where

*P*

_{t}and

*P*

_{r}represent the receive and transmit power levels,

*G*the antenna gain, and

*λ*the radar wavelength,

*r*is the laser-derived distance from the sea to the antenna, and

*A*is the effective observed surface area. All parameters are accurately determined and cross-checked through rigorous calibrations. Absolute calibration of the microwave system is carried out by measurement of

*P*

_{r}off a series of calibrated point targets before and after each measurement campaign. The radar is also equipped with an internal calibration loop, continual transmit power monitoring, and internal temperature control to better than 2°C. External calibration results indicate an uncertainty of ±0.4 dB in the absolute level. Precision for a given estimate approaches ±0.2 dB for a given 5-km segment average in this study. This precision holds between successive campaign measurements as well as from point to point on a given day. This is a system with high horizontal resolution, having a 1-m surface footprint when flown at 10-m altitude. Data processing to provide the averaged

*σ*° over a given 5-km flight leg is discussed in the noted reference. The appendix describes a small

*σ*° correction required to adjust the wider beam antenna data collected here to that measured by a narrow-beam system such as a satellite altimeter.

### d. Radar-inferred slope estimates

*θ*is the incidence angle from vertical. In this study the view angle is

*θ*= 0°, the nominal configuration for the satellite altimeter and for the present aircraft data. Thus

*θ*dependence on the right-hand side vanishes to leave the leading fractional term, that is, Eq. (1);

*σ*° in this study refers to

*σ*

^{°}

_{Ka}

*R*(0°)

^{2}|) at Ka band is estimated to fall between 0.52 and 0.55 for the salinity and water temperatures encountered during our measurements (Stogryn 1997). This factor's 1%–3% variation is accounted for in presented mss

_{Ka}estimates based on this Stogryn model and aircraft-measured sea surface temperatures. Note that no effective Fresnel coefficient is imposed in the development. It has been traditional to invoke such a term in lieu of the true Fresnel factor to account for unexplained disagreements between observations in both the overall bias between observed

*σ*° and mss and in the wind dependence for these two variables (e.g., Valenzuela 1978). Chapron et al. (2000) argues that this tuning term in not necessary if one reexamines the radar and optical measurements to include the effects of slope pdf peakedness and present understanding of the absolute calibrations for Ku-band satellite datasets.

A prime notation is attached to all radar-derived mss estimates to explicitly tie these measurements to the approximations of Eq. (5). While mss′ may correlate well with an optically derived mss estimate, numerous uncertainties remain in quantitative comparisons that fall outside of this study's main objectives. These include the issues (see Jackson et al. 1992; Chapron et al. 2000; Plant 2002) of slope pdf non-Gaussianity, centimeter-scale wave steepness and diffraction of incident radiation, and wave directionality and spectral content at all scales. That said, it is asserted that Eq. (5) is adequate for the objectives at hand.

The optical scattering assumption implies the incident radiation wavelength is much shorter than all roughness lengths on the surface. For microwave probing of the ocean surface this is typically violated due to the presence of of steep gravity–capillary and capillary waves (Jackson et al. 1992). Thus, to first order, the slope variance estimate of Eq. (5) is considered as a partial integration, encompassing waves greater than 3*λ*_{0} (*k*_{f} = *k*_{0}/3 = 250 rad m^{−1}). This means wavelets longer than 2 cm for a Ka-band sensor.

*k*

_{l}≈ 3 rad m

^{−1}:This is the omnidirectional value representing slope integration over wavenumber and azimuth angle

*α.*The wavenumber vector is

**k**=

*k*cos(

*α*)

**i**+

*k*sin(

*α*)

**j**. The integrand represents the slope spectral density function where

*S*(

**k**) is the directional wave height spectrum. As discussed, each mss

_{l}estimate is directly obtained over ∼5000 measured

*s*

_{i}realizations.

*k*

_{l}to

*k*

_{f}isThe subscripts

*l*and

*h*are chosen to reflect the integration limits over low- and high-wavenumber portions of the spectrum respectively. Therefore

^{′}

_{h}

*λ*= 200 to 2 cm). Forthcoming data interpretation makes use of

^{′}

_{h}

## 3. Results

The mean wind dependence of surface estimates under presupposed open ocean wind and wave conditions are detailed first. Measurement locations are shown in Fig. 3. The water depth exceeds 20 m for all cases. Samples within 15 km of the coastline are excluded, as suggested by recent work on this coastline (Sun et al. 2001; Vickers et al. 2001), showing that the atmospheric boundary layer often contains strong land-induced turbulence advected 10–15 km from shore. The map also shows the location of the buoys mentioned in section 2.

This subset of aircraft data holds more than 2300 5-km samples gathered on 36 separate days over a 2-yr period. Figure 4 depicts the range and frequency of wind speeds observed along with the mean relation between *H*_{s} and the wind. The *U*_{10} distribution highlights that this set is weighted toward moderate wind observations, but a substantial number of high and low wind events are also present. Wave height variation with wind indicates that the sea state generally falls below 2.0 m and that, on average, *H*_{s} levels reside above a fully developed sea prediction (Sverdup and Munk 1947) for low winds and fall below this criterion for moderate to high winds. The observed levels of *H*_{s} fall well below globally observed mean values at any given wind (Gourrion et al. 2002b). This is not unexpected off this East Coast region where swell impacts are limited. All references to *U*_{10} from this point forward refer to the 10-m neutral stability value computed using Eq. (3).

### a. Wind dependence of long gravity wave properties

^{−1}. The error bars represent 95% confidence intervals assuming a random distribution usingwhere

*σ*

_{b}is the bin standard deviation and

*N*is the sample number within a bin. Also shown in all panels are the optically deduced predictions (Cox and Munk 1956, hereinafter referred to as CM56). These predictions are for the special case where the surface was altered using oil to limit the presence of short waves. This so-called slick surface was considered to contain waves only greater than about 30 cm, a limit that coincides well with the 2-m limit (

*k*

_{l}= 3) of the laser geometry used here. Note that the present observations are derived from an unaltered ocean surface.

Data in the top panel of Fig. 5 provide a clear indication of wind dependence for the longer wave slope variance, mss_{l}, Eq. (6). Recalling Fig. 4, one notes that error bar magnitude follows the observed wind histogram. There is a linear trend up to 8–10 m s^{−1} that falls about 10% below the CM56 model. At higher winds, the present data fall farther below their prediction, but do indicate a continued, but weaker increase with wind. The one standard deviation levels about the mean indicate 20%–30% variability, likely geophysical as discussed later.

*k*

_{l}= 3 rad m

^{−1}:where

*k*

_{p}

*g*

*U*

^{2}

_{10}

*B*=

*a*(1.0, 1.25, 1.5) and where

*a*= 4.6 × 10

^{−3}is prescribed by Phillips under the wind-driven asymptotic limit where a

*k*

^{−3}saturation subrange for the wave spectrum is established. At low

*U*

_{10}the average mss

_{l}lies considerably above zero in agreement with CM56 estimates. This suggests that, on average, the dataset carries a background wave field component at light winds that elevates mss

_{l}slightly above a solely wind-dependent level.

*γ*is defined aswhere

*μ*

_{i}is the

*i*th central moment of the univariate slope distribution. By definition,

*γ*is zero for the normal distribution. The positive-definite omnidirectional slope distribution for normally distributed slopes is the Rayleigh distribution for which

*γ*= 0.245. Data in Fig. 5b show that

*γ*always lies well above the Rayleigh value, indicating an excess in steep and shallow slopes. The measured peakedness is largest for lightest wind and there is a clear decrease with increasing wind up to 10 m s

^{−1}. Above this speed the magnitude rises again. CM56 also inferred non-Gaussian

*γ*levels, but their technique's fidelity in resolving steep waves and the resulting observations provided no evidence of wind dependence. The

*γ*level presented for CM56 is derived from their principal component Gram–Charlier slope fit parameters,

*C*

_{40}and

*C*

_{04}, and our subsequent omnidirectional simulation relating component slope statistics to those computed over the modulus |

**s**|. That simulation is based on the pdf generating function discussed in the appendix. Results indicate that CM56's

*γ*= 0.70 ± 0.35. CM56 also reported that this peakedness magnitude did not change substantially between the slick and clean surface observations.

While their *γ* estimate is included here, it is important to note that the CM56 technique involved indirect inferrence of the variance, skewness, and kurtosis under given assumptions and limitations. This included an inability to measure the true slope variance and the steepest waves, and truncation of their Gram–Charlier fit to fourth order. These are points specific to the CM56 study that should be considered in quantitative definition and use of the slope distribution (Tatarskii 2003; Chapron et al. 2000). In this case, the CM56 *γ* is given as reference but it is suggested that the uncertainty and mean level may exceed what is derived from their truncated fit coefficients.

A similar *γ* adjustment from one to two dimensions is applied to the wind-dependent findings of Shaw and Churnside [1997, their Eq. (17)]. The resulting fit is shown in Fig. 5b. In all cases, the present data and those of Shaw and Churnside (1997) lie above CM56. Close agreement in magnitude and wind dependence is observed between present data and Shaw and Churnside (1997). The latter results come from an optically based field study where the slope pdf was deduced over all wave scales, thus including wavelengths much shorter than those measured here. That study also involved a technique with angular resolution that far exceeded the CM56 study and permitted a Gram–Charlier expansion truncated at the eighth order.

Figure 5c provides the average estimate of the ratio between crosswind and alongwind slope variances obtained for these longer wave scales. Almost no asymmetry is observed for the lowest wind speed, but with much uncertainty. Otherwise, the data lie between 0.8 and 0.9, indicating weak asymmetry that varies little as the wind increases. No clear wind dependence emerges considering the given uncertainties. The CM56 surface slick model shows close agreement with the present results for winds above 5 m s^{−1}.

Last, computed slope skewness values (not shown) are small for both along- and crosswind estimates, never exceeding −0.1, where negative values assign the most likely slope as falling to the downwind side of the pdf. The mean value for both falls at −0.01 ± 0.20 and uncertainty is too high to assign significance to a small observed magnitude increase with wind seen in the alongwind measure. These findings are also consistent with that observed in CM56.

### b. Wind dependence of Ka-band σ°

Nadir-view radar backscatter versus *U*_{10} is shown in Fig. 6. As above, the data correspond to the open ocean sample locations shown in Fig. 3 and comprise more than 2000 observations. Figure 6a presents the individual measurements as well as the other available data for this frequency and viewing geometry as obtained by Masuko et al. (1986). There is a large difference between these two studies where the present data consistently lie 5 dB above the Masuko dataset.

Referring solely to the present dataset it is pointed out that the data generally fall ±1 dB about the mean bin-averaged result with a somewhat larger data spread occuring at moderate *U*_{10} where more observations and, hence, diverse weather patterns were sampled. While this study is primarily focused on the ensemble dependencies, one group of outlying data are noted. The samples having *σ*° = 10–11 dB and winds from 6 to 10 m s^{−1} were examined and found to represent data from three days where the swell was opposed to the wind direction and/or a cold-air outbreak was in progress over the coast. Such distinctions are being addressed in a paper to follow and are mentioned primarily to highlight the point that data variability observed in Fig. 6a is foremost physically driven rather than measurement noise.

*σ*

^{°}

_{Ka}

*b*

_{err}to delimit the estimate confidence interval. This result is then adjusted slightly downward at each wind bin based on the antenna pattern correction model discussed in the appendix. The adjusted result provides an improved estimate of

*σ*

^{°}

_{Ka}

*σ*

^{°}

_{Ku}

*U*

_{10}derived from an extensively validated Ocean Topography Experiment (TOPEX) satellite altimeter–derived model at Ku band [Gourrion et al. 2002b; Eqs. (3)–(7)] given as

*σ*

^{°}

_{Ku}

*f*

*U*

_{10}

*H*

_{s}

*H*

_{s}for a given wind bin are fed into Eq. (10) and averaged to produce the Ku-band geophysical model function (GMF) curve shown. Model-predicted

*σ*

^{°}

_{Ku}

*U*

_{10}but the added influence of

*H*

_{s}data gained through Eq. (10) should serve to adjust globally derived

*σ*

^{°}

_{Ku}

*H*

_{s}wave climate carried in the present aircraft dataset (see Fig. 4).

Comparison of the wind-dependence between Ku-band and Ka-band observations in Fig. 6b indicates close agreement. Below winds of 5 m s^{−1} the Ku-band results fall below those at Ka band by 1 dB at most; otherwise the data agree to within ±0.2 dB for winds of 5–15 m s^{−1}. According to Eq. (1), holding mss′ constant, Ka-band and Ku-band (14 GHz) results should differ by the ratio of their Fresnel coefficents (∼0.6 dB) where the Ka-band should fall below the Ku-band data. While the predicted −0.6 dB offset for Ka band is not consistently observed, absolute calibration of either system to a level better than ±0.5 dB is understood to be unlikely. This suggests that Ka-band *σ*° levels are consistent with prediction based upon a quasi-optical assumption and sensor calibration limitations. Thus results to consider here are that the Ka-band data agree closely with the Ku-band results in their magnitude and mean wind-dependent characteristics and that the magnitude of *σ*^{°}_{Ka}

Next, *σ*^{°}_{Ka}^{′}_{Ka}^{′}_{Ka}*U*_{10}. The present data follow a logarithmic fit quite well from 1 to 15 m s^{−1}, having the form ^{′}_{Ka}*U*_{10}).

Jackson et al. (1992), among others, provide extensive comparison between radar and optically derived mss estimates. The close agreement observed between present Ka-band and TOPEX altimeter Ku-band results suggests these new ^{′}_{Ka}*σ*° as for the satellite altimeter, whereas much of the recent aircraft work uses an alternate approach (e.g., Jackson et al. 1992; Walsh et al. 1998; Vandemark et al. 1997; Hesany et al. 2000) where the *σ*° variation with *θ* is measured. One then estimates mss′ via Eq. (5).

### c. Long gravity wave impacts

A point to recall is that this dataset, while regionally limited, is extensive in its sampling at wind speeds below 12 m s^{−1}. Flight data were acquired under differing synoptic conditions and in this section the observed variability in aircraft wave estimates at fixed wind speeds is examined. A main objective here is to quantify the extent of wind-independent long gravity wave variations and their impacts upon *σ*°, and equivalently on ^{′}_{Ka}_{l} and ^{′}_{Ka}*U*_{10} for each of these zones and the results given in Fig. 9. No wind bin holds fewer than five estimates. Note that the inland water was sampled less often than for the other cases. The unique perspective provided by the aircraft shows that for a fixed wind there is a clear overall increase in the roughness measured by both the radar and laser systems as one progresses from the sheltered to open waters. Results hold for *U*_{10} = 3–12 m s^{−1} in a fairly consistent quantitative manner where a factor of 2–3 increase in mss_{l} is observed from inland to open waters. Note that this average result represents the ensemble over varied on-, off-, and alongshore wind and wave situations, yet the general smooth-to-rough picture always holds across the inland-to-open sea transition. Changes in ^{′}_{Ka}*O*(20%–50%).

*k*

^{2}

*S*(

*k*)

*dk*varies significantly from shore to sea. The partitioning chosen here is perhaps less physically attractive than data division by other wave field descriptors such as wave age, fetch, or sea versus swell content. A depiction of the long-wave control on the overall slope variance (

^{′}

_{Ka}

^{′}

_{h}

^{′}

_{h}

*U*

_{10}

*U*

_{10}

^{−1}. As with

^{′}

_{Ka}

*U*

_{10}holds for

^{′}

_{h}

^{−1}. The wind-generated short waves appear to adhere to the same form over the diverse range of longer gravity wave tilting conditions. The overall convergence of the data suggests that, in an average sense, most sea-to-shore

^{′}

_{Ka}

*λ*exceeding the laser system's 2-m cutoff.

Results of Fig. 10 raise the question of whether the observed high-wavenumber signature is indeed solely dependent on *U*_{10}. One means to assess this question is to stratify ^{′}_{h}*U*_{10}) against the long-wave slope variance. Results for two disparate mss_{l} levels are provided in Fig. 11. Here data from both coastal ocean and open-ocean sampling were included in the averages to improve the statistical confidence. Estimated means do not differ substantially from those obtained using only the open-ocean data.

Recall that the mss_{l} estimator represents an azimuthally and spectrally integrated parameter and therefore serves as an average for the overall gravity wave steepness, albeit weighted toward shorter gravity waves. Referral to Fig. 9 indicates that the low and high mss_{l} values chosen for Fig. 11 represent nominal levels for light to moderate and high winds, respectively. Figure 11 ^{′}_{h}_{l} levels indicate close agreement at winds of 5–7 m s^{−1} but a systematic difference above this velocity. The short-wave roughness is higher for the smoother long-wave level, mss_{l} = 0.012. The difference holds for *U*_{10} of 7–13 m s^{−1}. Above this point, sampling at the low mss_{l} level is not available. At the higher winds, these observations can be considered to contrast ^{′}_{h}_{l} level), with the case of fully developed and/or a mix of sea and swell (the high mss_{l} level). Under this rendering one sees as much as a 60% ^{′}_{h}^{−1} this short-wave signal can vary systematically with the wave field independent of the mean wind speed. An additional check of this behavior is obtained by replacing mss_{l} with *H*_{s} in reproduction of Fig. 11. Results are not shown but they provide the same qualitative picture, lower *H*_{s} correlates with increased ^{′}_{h}

## 4. Discussion

The laser-derived slope data of section 3 provide field measurements having several key features. First, the fidelity of direct slope measurements is likely to be superior to optical glint deduction, and to the ill-suited approach of inferring higher-order wave slope statistics via temporal buoy-based acceleration data under a quasi-linear dispersion model. Moreover, these measurements do not require oil dampening to artificially impose *k*_{l}. Last, the extensive dataset provides a robust new source for intercomparison with the nine independent slick surface realizations of CM56 that have long served as a baseline for numerous wind–wave, air–sea interaction, and remote sensing modeling efforts.

In an overall sense, the accord with CM56 results seen in Fig. 5 is impressive. As one expects, the present mss_{l} data fall slightly below CM56, in line with the estimation that *k*_{l} was 20 for that study as compared with 3 here. Qualitatively, the data above *U*_{10} = 3 m s^{−1} provide further support for a logarithmic, rather than linear, mss_{l} dependence upon wind. For a quantitative fit to the data we use the saturation subrange form of Phillips (1977) where the model is scaled solely by the wind-independent coefficient in Eq. (8). Figure 5a shows that *B* = 5.7 × 10^{−3} ± 25% spans the ±*σ* for the observations. Note that the upper bound nearly matches the CM56 result where *k*_{l} is 6 times as high. This point suggests the uncertainty resident in estimating mss_{l} wind dependence based on limited field observations. As a further point, the present mean results are derived over an ensemble that includes few, if any, classical fully developed sea cases. It is more typical to find in this data a mix of over- and underdeveloped vantage points with the seas either decaying or building. The Phillips form estimates mss_{l} assuming that the total is derived only from wave scales having *k* < *k*_{p}; the peak wavenumber for a fully developed sea. Thus, at least at moderate winds, one might suppose that the central value for *B* represents the fully developed sea while the upper limit tends to a mixture of sea and swell and the lower limit toward that of young or dying seas. The supposition is more tenuous at winds above 10 m s^{−1} where *H*_{s} levels in the present dataset indicate that observations were generally for young seas. Thus a fully developed estimate likely tends toward the upper bound. These issues are addressed further in Fig. 12 to be discussed shortly. Note that use of an equilibrium model (e.g., Banner 1990) in lieu of the saturation subrange assumption leads to similar qualitative interpretation.

Regarding other gravity wave slope statistics, data here agree with the findings of CM56 regarding minimal slope skewness and an observed cross-to-alongwind slope variance ratio of about 0.85, both parameters varying little with change in the wind. Distribution peakedness levels in Fig. 5b also affirm the long-wave slope pdf nonlinearity deduced by CM56. Present observed levels are measurably higher than for CM56 and depict a wind dependence not seen in that data. These differences are likely a result of the higher fidelity in the present direct slope measurement approach. What do the differences tell us? First, that gravity wave nonlinearity is a fundamental feature in the long-wave slope pdf and is perhaps greater than supposed. Theoretically, under a narrowband dominant wave framework, kurtosis excess in the wave slope pdf is indicative of third-order nonlinear wave–wave interactions (Longuet-Higgins 1963) whereby a correlation between the dominant long-wave and short-scale gravity waves exists. This nonzero correlation is evidenced by the increased occurence of steep wavelets near the long-wave crests and flattened areas within the troughs. These interactions also act to drive directional uniformity at the short scale, consistent with the low observed difference between the along and crosswind mss_{l}. While no generalized theoretical model exists for depicting these nonlinearities over all of *S*(*k*), recent study of the non-Gaussian slope pdf (e.g., Liu et al. 1997; Chapron et al. 2000) suggests that *γ* may be empirically related to the wave spectral bandwidth parameter and to the inherent compounding of nearly Gaussian processes at multiple gravity wave scales. The wind dependence observed in Fig. 5b is consistent with a dependence on wave–wave interactions. These interactions can become predominant in the low wind, low mean slope environment where wave groupiness and multimodel wave fields can force local peakedness in the absence of wind. As wind levels increases, the data indicate *γ* tends toward a local minimum suggesting the most linearized situation comes at moderate winds, while as the wind increases past 10 m s^{−1} it may be that the weight toward young seas within this dataset provides enough active wave breaking to increase the level of wave steepening once again. Strong directionality of the intermediate-scale gravity waves may also play a role in explaining observed *γ* values, especially at light winds. In this case even weakly non-Gaussian slope distributions in the along- and crosswind directions can lead to a strongly peaked 2D distribution. While not shown, a picture similar to that of Fig. 5b emerges if one replaces *U*_{10} with mss_{l}.

Section 3c provides a view of diverse wave climates and slope pdf dynamics for various fixed wind speeds. Figure 9 illustrates the mss_{l} variability encountered as one spans inland, coastal, and open water. The obvious extreme of inland water, obtained over the shallow (depth < 5 m) Currituck Sound, provides data with no swell and limited seas, along with requisite wave dissipation. In this case, one sees mss_{l} tends to very low levels at light wind and a factor of 3 below the open ocean at the moderate speed of 7 m s^{−1}. While not shown, the levels of slope peakedness are also reduced consistent with the expected reduction in nonlinear wave–wave interactions. A more in-depth look at mss_{l} variations within the coastal zone can be found in Sun et al. (2001).

Overall, mss_{l} variability for a fixed *U*_{10} is not small and likely warrants consideration in modeling and inversion studies related to remote sensing of waves and wind near the coast. This is particularly true for sensors most sensitive to the lower wavenumbers such as the low-incidence angle radar scatterometers, altimeters, and envisioned L-band systems for sensing ocean salinity and wind.

The ad hoc wave zone partitioning used to generate Fig. 9 is one means to elicit tilt field variability. To further illustrate this variability at a fixed wind speed, results from two open-ocean aircraft data collection segments are contrasted in Fig. 12. The flight transects were 5 km long and flown in the alongwind direction in both cases. Transect wavenumber spectra (1D) and slope pdf (2D) data are provided. The pdf results for the slope magnitude are derived from the individual facet slopes as discussed in section 2. Each transect (one-dimensional) slope spectrum comes from an encounter wave height–wavenumber spectrum [*S*(*k*_{x})] computed using the along-track wave elevation spatial series data [*η*(*x*)] collected using one of the aircraft's laser altimeters. The cumulants displayed represent the alongwind variances 〈*s*^{2}_{x}

The low slope (mss_{l} = 0.014) case represents a young sea plus old swell case while the higher slope (mss_{l} = 0.024) example is one of a developed sea and swell; *H*_{s} is 0.8 m for the former and 2.1 for the latter. Clearly, for these 8 m s^{−1} wind speed cases the overall surface tilt fields differ substantially. The 2D pdf results show the substantial discrepancy between surface steepness for the two cases. Recall that these are slope measurements derived across a 1-m facet scale. The spectrally resolved data for alongwind flight legs, including cumulative alongwind mss_{l} estimates, provide a sense of the wave scales dictating mss_{l}. For example, there is measurable swell energy present for the low slope case at *k* = 0.05 rad m^{−1}, but its magnitude in the slope spectrum and upon the mss_{l} estimate is negligible. What drives the large difference between these two realizations are wave scales from *k* = 0.2 to 1 rad m^{−1}. Within the equilibrium subrange from *k* = 1–3 rad m^{−1} the differences are smaller and the contributions to mss_{l} similar.

The control of mss_{l} variability by relatively low wavenumbers suggests that operational wave buoy data may serve to examine and bound the extent of wind-independent mss_{l} variation. The nominal cutoff wavenumber for NDBC wave buoy directional wave measurements is provided on Fig. 12. This implies that a substantial fraction (∼30%–50%) of the example mss_{l} could be inferred via the measured buoy acceleration variance. Indeed, multiyear climatologies of buoy-derived mean squared acceleration for regions such as the Great Lakes, Hawaii, and the U.S. West Coast (Gourrion et al. 2002a) suggest a dynamic range in this longer wave slope variance of ±60%–80%, not unlike that observed in the mss_{l} results of Fig. 9.

Discussion of the presented radar backscatter data can be couched in terms of (i) examining multifrequency *σ*° observations, (ii) long-wave effects on altimeter backscatter, and (iii) the radar inference of surface wave information.

(i) First, regarding multifrequency observations of *σ*°, the present radar measurements were undertaken in part to reproduce past nadir-view Ka-band data (Masuko et al. 1986). While only a small sampling, those past results indicate a large difference between reported X- and Ka-band *σ*° estimates that lies far outside that predicted solely based upon the 0.6-dB difference between respective Fresnel reflection coefficients. Several studies (e.g., Jackson et al. 1992; Walsh et al. 1998) suggest those Ka-band data are much too low in an absolute sense, and inconsistent with what is known from other radar observations (e.g., Jackson et al. 1992; Chapron et al. 2000) and from optically derived slope spectral density data at high wavenumbers, while others (e.g., Apel 1994) make use of the Masuko data to infer an accordingly increased slope spectral density at these wave scales. The new data of Fig. 6a show a large 5-dB discrepancy that supports the former view. Section 3 results also show the present data to be consistent with satellite observations at Ku-band to within the error of *O*(1 dB) system calibrations. Results are also consistent with the calibrated Ku-band tower data of (Melville et al. 1991). Last, it is noted that the Masuko et al. (1986) data were obtained by averaging of log amplifier outputs. This would serve to bias both their X- and Ka-band estimates by 2–3 dB. However, this would not alter their observed 5-dB difference between frequencies. We conclude that the present Ka-band results are a preferred choice for physically based interpretations and for spaceborne sensor signal-to-noise design considerations.

The wind dependence of open-ocean aircraft Ka-band results are shown to nearly replicate the Ku-band TOPEX altimeter model function (Gourrion et al. 2002b) in Fig. 6b. This agreement between a globally derived satellite result and data from off U.S. East Coast suggests a robust mean relationship between wind and *σ*°(0°) consistent with numerous past aircraft studies (e.g., Jackson et al. 1992; Vandemark et al. 1997; Hesany et al. 2000; Chen et al. 2001). Certainly these aircraft data closely emulate Ku-band altimeter backscatter characteristics in a mean sense that suggests their utility for supporting satellite data interpretation.

The close agreement at all *U*_{10} between the Ka- and Ku-band systems is not completely in agreement with that found between C- and Ku-band *σ*° using systems aboard the TOPEX satellite (Elfouhaily et al. 1998). In that study a wind-dependent C-Ku band *σ*° difference is observed for moderate to high *U*_{10}, increasing with *U*_{10} from 0.1 to 1.0 dB. This trend is attributed to increase in short-wave spectral content following optically derived results on the increase in wave density at high wavenumbers with *U*_{10} (Hwang et al. 1996; Elfouhaily et al. 1997). This logic predicts the mean difference between Ka- and Ku-band *σ*° data should exhibit a similar wind dependence but with a slightly reduced magnitude. This study does not contain the coincident Ku- and Ka-band data needed to examine this small difference signal. The aircraft versus satellite model results of Fig. 6b show no clear wind-dependent depression of *σ*° at Ka with respect to Ku band like that seen for C and Ku band. Given the small expected signal we consider the results inconclusive. At best they suggest wind-dependent change between nadir-view *σ*° at radar *λ*_{0} of 2.1 and 0.83 cm is small. One consistency between Elfouhaily et al. (1998) and this study is seen in their Fig. 2c and the present study's Fig. 11 where both observations indicate long-wave field control on short waves as discussed below. One inconsistency is that we see no large 3-dB frequency-dependent (Ku-to-Ka) offset like that observed between C and Ku band for the TOPEX altimeter (Elfouhaily et al. 1998). It is our contention that the wind-independent 3-dB TOPEX bias is largely due to calibration error (Chapron et al. 2000). Regardless, absolute calibration for the satellite altimeter C-band systems (TOPEX and *Jason-1*) is not performed at a level sufficient to achieve scientific certainty beyond the 1–2 dB. Accurately calibrated data are needed to fully resolve this issue.

(ii) Coincident observation of *σ*^{°}_{Ka}_{l} in this dataset provides a unique perspective for assessing the impact of tilt field variability upon altimeter backscatter. It is physically expected that the total integrated slope variance, and hence altimeter backscatter, will be subject to change with *U*_{10}, but also with a changing directional long wave field that is potentially uncoupled from the wind. The quasi-optical prediction of Eq. (5) and a two-scale surface model where ^{′}_{Ka}_{l} + mss_{h} [see Eq. (7)] provide a simple framework to investigate the issue. However, obtaining comprehensive mss_{l} datasets is uncommon. Several recent satellite altimeter studies attempt to quantify such uncoupled long wave impacts using surrogates for mss_{l} such as *H*_{s}, significant slope, and wave age (Gourrion et al. 2002a,b; Hwang et al. 1998; Gommenginger et al. 2002). Present laser-derived slope measurements provide mss_{l} directly and for every data sample in the set.

Results of Fig. 9 show mss_{l} and ^{′}_{Ka}*U*_{10}. Knowing that mss_{l} variation is carried within the ^{′}_{Ka}^{′}_{Ka}*σ*°) in Fig. 9. The ^{′}_{h}

The impact of mss_{l} variability upon altimeter *σ*° is not always small. Taking for example a linear dependence between wind and mss, observed 30%–40% changes in ^{′}_{Ka}_{l} variability following work such as Hwang and Shemdin (1988).

(iii) Radar-derived slope variance estimates are explicitly linked to the model of Eq. (5) to emphasize that ^{′}_{Ka}^{′}_{h}*σ*° data under specific approximations discussed in section 2. Quantitative comparison of results in Figs. 7 and 10 to optically derived mss data remains subject to uncertainties and varied interpretation—due in part to absolute calibration issues and to the unresolved surface wave description at all scales and conditions as mentioned in section 2. There are, however, several points to be made when considering Figs. 7, 10, and 11.

First, no effective Fresnel term is invoked in deriving ^{′}_{Ka}*σ*° fall off with incidence angle shown in Eq. (5) to show that these efforts are likely underestimating the true mss because of neglect of slope pdf peakedness. This underestimation in turn leads to use or modification of the effective Fresnel adjustor. If one adjusts the Ku-band mss results of Jackson et al. (1992) or the nearly equivalent Ka-band results of Walsh et al. (1998, see their Fig. 11) by the proposed blanket factor of 1.2 then one finds their results agree very well with our ^{′}_{Ka}

Second, the variation of ^{′}_{h}^{′}_{h}^{−1}. Such a result is consistent with that inferred from space (see above, Elfouhaily et al. 1998) and in the laboratory (Donelan 1987). It is also consistent with prediction of short-wave suppression by enhanced breaking due to wind drift and wave current effects under increased mechanical wave energy (Chu et al. 1992). The present ^{′}_{h}*k* < 700 rad m^{−1}. One implication is that a multifrequency altimeter is potentially sensitive to such long-wave and short-wave dynamics. Last, note the nearly opposite behavior for ^{′}_{Ka}^{′}_{h}_{l} change at a given moderate wind speed (Figs. 9 and 11). While the total roughness ^{′}_{Ka}^{′}_{h}_{l} appears to be rising with mss_{l}, the high-frequency component is decreasing. Gourrion et al. (2002b) hypothesized this contrast to be a potential source for the ambiguity observed in the satellite altimeter wind inversion process at moderate to high wind speeds. This appears to be supported by the present results.

## 5. Summary

The field measurements provided serve to clarify and validate several ocean wave roughness observations and models commonly invoked in the study of ocean waves and radar remote sensing of ocean winds. Wave slope pdf measurements for *λ* > 2 m generally affirm the oft-cited results of Cox and Munk where oil was used to eliminate short-wave scales. Present results confirm a slope distribution peakedness that greatly exceeds that for a Gaussian and exhibits a wind dependence. Both features indicate the presence of nonlinear wave–wave interactions at the intermediate gravity wave scales and occurring at all wind speeds, particularly on the open seas.

The mean wind dependence of Ka-band radar *σ*° indicates a high level of agreement with Ku-band satellite altimeters—an expected result. Moreover, the calibrated *σ*° data also indicate that past nadir-view results at Ka band are biased low by 5 dB. The radar *σ*° data are then used to infer slope variance at higher wavenumbers, 2 cm < *λ* < 2 m, as defined in Eq. (7) via combination of laser and Ka-band nadir-view scatterometer data. A key observation is that of long-wave tilt effects upon the observed radar backscatter (and ^{′}_{Ka}_{l} over all long-wave scales, including any swell, affirms that the primary concern when addressing long wave impacts upon *σ*° or mss is divining variability within a spectrally integrated parameter, for example, mss_{l} or the mean-square acceleration, that is associated with intermediate gravity wave scales. The present study suggests this variability can be quite large, particularly near the coast, and therefore of some concern in exacting mss measurements and models. Use of terms derived from low-order wave spectral moments such as *H*_{s} or *k*_{p} are unlikely to provide equivalent sensitivity to the effect.

Last, initial open-ocean results of Fig. 10 suggest that the radar-deduced slope variance for the high-wavenumber subrange is relatively insensitive to the long waves, increasing solely with the logarithm of the wind. A closer look reveals that this is true only for light-to-moderate winds below *U*_{10} of 7 m s^{−1}. At speeds above this range, a depression in ^{′}_{h}

It is hoped that the present study serves to motivate the consideration of intermediate and long-wave scale slope measurements, and their nonlinearity, in future ocean remote sensing work. In particular, the multifrequency altimeters aboard *Jason-1* and ENVISAT, the Ku-band Tropical Rainfall Measuring Mission satellite, and developments involving GPS bistatic scattering and ocean salinity measurements. In all cases, the sensors are sensitive to these wave scales at some level. Satellite extraction of such information may prove useful to the enhanced understanding of the role that slope variance plays in gas and momentum transfer processes. Future work with this LongEZ aircraft data intends to address covariance between momentum flux variability and the wave scales resolved here.

Our thanks are given to the E. Dumas, S. Burns, D. Hines, J. French, and T. Elfouhaily for their help in system developments, calibrations, and data collection and to W. Drennan for the ASIS data. Thanks are given to the anonymous reviewers whose comments helped to improve the document considerably. This work was supported by ONR, NASA, and NOAA. Credit goes most of all to T. Crawford, the aircraft's fabricator, chief scientist, pilot, and instrument designer. His curiosity, energy, and many talents made these efforts possible and are sorely missed.

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# APPENDIX

## Antenna Beamwidth Impact on Nadir-Viewing σ°

Equation (4) represents the radar equation assumming that *σ*° is constant across the illuminated footprint area. Equation (5) indicates that the cross section falls off quite rapidly with near-nadir incidence angle. The radar antenna beamwidth used in this experiment spans 5.8° between its half-power points. The combination of these two features suggests the need to correct nadir-view *σ*° estimates to compensate for the error due to the constant *σ*° assumption. This is in contrast to the satellite altimeter where the narrow effective beamwidth is of *O*(0.1°) and the constant *σ*° assumption is certainly valid.

*σ*° measurements are derived using the power returned from a single footprint on the surface having a diameter of about 1 m. Under a composite two-scale model this measurement reflects the rough surface scattering associated with wave scales inside that

*m*-scale patch, but also the mean tilt of the underlying surface patch away from the nominal horizontal or normal reflection angle. All

*σ*° values used throughout the paper represent a 5-km along-track summation over many thousand instantaneous estimates where each estimate has a unique short scale within-footprint slope variance and a mean tilt angle (

*δ*= tan

*β*) away from normal. If one assumes that the short-scale roughness is homogeneous over the flight segment and is uncorrelated with the wave slope then one can write the segment-averaged

*σ*° asHere the antenna pattern incidence angle variation is given as

*g*(

*θ*), the slope probability density function is

*P*(

*δ*), and

*σ*

^{°}

_{GO}

*θ,*

*δ*) represents the geometrical optics cross section approximation as given by Eq. (5). An important distinction, however, is that the slope variance for this within-patch contribution is given by the short-scale component,

^{′}

_{h}

This equation will revert to the narrow antenna beam (e.g., satellite) result as the beamwidth tends to 0. In the flat surface limit where all tilt angles are zero this equation would depend only upon the convolution of the antenna pattern and cross-section functions. In this case the averaged *σ*°(*θ* = 0°) obtained using antenna beamwidths of 1–10 would fall increasingly below the true infinitely narrow beam result as the beamwidth widened.

However, the opposite effect occurs in the real-world case where the large-scale tilts become nonzero. In this case the mean reflection angle is often away from the normal specular-point direction. Now the diffuse surface scattering pattern associated with the short-scale waves is tilted away from the nominal antenna view angle *θ*_{o}. The result is that the illumination pattern is now integrated about a shifted form of the cross section [*σ*^{°}_{GO}*θ*′)] where the facet reflection angle is *θ*′ = *θ* + *δ.*

Evaluation of Eq. (A1) for varied beamwidths is performed through a Monte Carlo simulation over the tilt pdf and by assuming a Gaussian antenna pattern, shown to be in good agreement with the antennae used in the field work. Under the composite model, we assume a mean wind dependence for both short- and long-wave slopes as given by Eqs. (8) and (11). The long-wave tilt values for a given run are obtained through *N* random draws over a slightly non-Gaussian slope pdf that supports variable levels of kurtosis to evaluate results over limits consistent with those in Fig. 5b. Results are computed as the ratio [Δ*σ*°(*U*)] between *σ*° obtained for a given beamwidth antenna to that of an infinitely narrow beam system.

The cross-section correction limits mentioned above are readily obtained. For the case of no-tilt field, a maximum negative bias of −0.05 dB from the narrow-beam limit can be found for the case of moderate wind speeds and a beamwidth of 6. This decreases in magnitude as the wind (and hence ^{′}_{h}*σ*° is overestimated for all cases where the antenna beam width exceeds about 1° but errors are more pronounced as the width exceeds 3°. For the aircraft's 5.8° configuration the overestimate exceeds 1 dB at light winds and lessens to below 0.2 dB at the highest winds.

*b*

_{L}) and within-patch (

*b*

_{S}) slope variances are used to evaluate this last issue. The middle panels of Fig. A1 demonstrate that the effect of reducing short-wave variance is almost negligible as long as at least one-half of the

^{′}

_{h}

*σ*° bias results, given in decibels, and applied as

*σ*

*U*

*σ*

^{°}

_{ac}

*U*

*σ*

*U*

*σ*

^{°}

_{ac}

*U*)〉 is the flight segment averaged

*σ*° result without this correction. The correction magnitudes are small and their estimation with this model is obviously a first-order approximation. However, they are provided as a further step in providing accurate calibration of nadir-view

*σ*°.