Whitecap reflectance was measured using a PANalytical Boulder ASD FieldSpec 4 spectroradiometer with a wavelength range from 350 to 2,500 nm interpolated to a 1 nm resolution under ambient sunlight during clear sky conditions. The sensor was equipped with an 8.5° fore-optic and was optimized for the light field with dark current. The sensor was pointed to a 99% white Spectralon plaque held horizontally to the sea surface at a distance of ~2 cm from the plaque. The plaque was held in a relative azimuth orientation toward the sun to avoid user shading on the measurement. The plaque measurement provided an estimate of downwelling irradiance during the experiment and was taken periodically to normalize the measurement as described further below. After the measurement was taken over Spectralon, the sensor was extended over the target at ~0.5–1.5 m above the sea surface depending on the size of the swell and motion of the boat at an azimuth angle of ~145° from the sun to minimize sun glint. This translates to a field of view covering a 7.5–22-cm diameter circle on the sea surface.

For the natural breaking waves, measurements were made from the side of the R/V Lowell Weicker on 19 January 2016 pointing with a heading into the wind in order to maintain position with the wave field. On this day, wind speed varied from 10 to 12 m s⁻¹ measured at 3 m height at the Eastern Sound Buoy in Long Island Sound (41° 15.48′N, 72° 04.00′W) and significant wave heights estimated to be 1.5–2 m using measurements from the Central Sound Buoy (lisicos.uconn.edu). The background reflectance followed standard NASA protocols where 5-replicates were taken of a sequence of measurements from a 12% spectralon reference panel, water, and sky while maintaining a 45° zenith angle of the sensor. The sensor was positioned at an azimuth angle of ~145° from the sun to minimize sun glint. Whitecap measurements were taken in a time-series mode at 8 ms over a 20-min interval with the radiometer pointing down at a nadir angle at the sea surface for roughly 75,000 samples. Measurements were taken of the 99% spectralon reference panel at the beginning and end of sampling and after every whitecap event in an azimuthal angle facing the sun with the sensor at a nadir angle to the plaque. The time series of measurements taken over rolling breakers were considered to be spectral mixtures of whitecap, foam and undisturbed water. Four different breaking events were measured with varying intensities and contributions of breaking waves.

Measurements were also taken over manufactured breaking waves and foam from various sources including boat wakes and an outflow pipe. During these experiments, the field of view of the sensors was focused on the breaking feature and these measurements are considered to be spectral endmembers of breaking waves/foam without undisturbed water. Forward motion of the 23-m long R/V Connecticut produced a boat wake with a large breaking wave on the side of the ship on 29 June 2017 16:01–16:15 UTC at Race Point Long Island Sound (41° 14.62′N, 72° 02.52′W) with clear skies and a 20° solar zenith angle. In addition, measurements were taken of the boat wake and multiple layers of foam produced by circular motion of a small Carolina Skiff on 25 June 2015. Measurements were also taken of layers of foam produced on the sea surface by an outflow pipe from the high-flow pressurized seawater distribution system at the Rankin Seawater Laboratory, University of Connecticut 27 January 2015.

Lambertian-equivalent reflectance of the sea surface was calculated as ratio of the measurement obtained of a breaking wave and measurement of the lambertian Spectralon plaque. Normalization was conducted using a recent calibration of the white plaque which varied from 97% in ultraviolet, 99% in visible, and 93% reflective in SWIR wavelengths out to 2,500 nm. No corrections were conducted for reflection of glint or skylight from the sea surface when estimating the reflectance from background water or water enhanced by foam and whitecaps. From a satellite perspective, the removal of sun/sky glint is a separate step from removing whitecap reflectance. Hence, the methods developed here included the sun/sky glint in the background reflectance in order to differentiate the contribution from the whitecap signal from the remaining signal.

Reflectance from breaking waves, foam, and bubbles is treated with different methods in the literature and these different methods are considered in the results section. Because whitecaps are so bright, the signal is commonly thought to dwarf the contributions from water-leaving radiance and reflected skylight or glint and is treated as a “white” patch on the sea surface. For atmospheric correction approaches, the estimated whitecap reflectance is generally considered invariant of the water-leaving signal, sky conditions, and wind speed (Gordon and Wang, 1994). Whitecaps, and particularly bubble plumes, can also be treated as an augmentation above a background reflectance (Moore et al., 2000). The total reflectance (R_t) is considered to be the background water-leaving reflectance (R_b) plus some additional amount due to whitecaps (R_aug), such that:

A recent study defined an “augmented reflectance ratio” as the fractional augmentation of a whitecap above a background reflectance (Xu et al., 2015). This definition is not used in this manuscript because it is felt that an augmentation ratio does not properly reflect the physics of the system. A more nuanced approach is to treat whitecaps as a two-layered system where a semi-transparent layer of foam and bubbles overlays a background water reflectance (Frouin et al., 1996), as described further in the results. Radiative transfer modeling based on the theoretical work of Kokhanovsky (2004) and Zege et al. (2006) is presented in the Results. Radiative modeling of whitecaps bridges the measured reflectance to the optical properties of foam.

Whitecap measurements were compared to the absorption coefficient of seawater, a_w. The a_w (m⁻¹) used in this analysis were calculated for standard ranges of salinity (30–34 ppt) and temperature (0–20°C) encountered in the world ocean using the Water Optical Properties Processor (Rottgers et al., 2011). Differences in a_w with temperature and salinity are insignificant to the results presented here out to 2,500 nm, but water properties of 20°C 34 ppt are used for analyses.

Sea surface reflectance is converted to radiance and propagated through the atmosphere to estimate the radiance due to mixed pixels of whitecaps and background at the satellite. First, the reflectance is converted from water-leaving reflectance (R_w = E_u(0⁺)/E_d(0⁺)) to an estimate of the upwelling radiance leaving the sea surface in the nadir direction, L_w (note that this definition includes mixed pixels of radiance from foam and bubbles, as well as water-leaving radiance and sea surface reflected diffuse and direct irradiance). This conversion is conducted assuming the sea surface reflectance is lambertian and the downwelling irradiance reaching the sea surface can be approximated by the solar constant, F_o, adjusted by the cosine of the solar zenith angle (θ_s), the transmission of diffuse irradiance between the sun and earth, t_ds, and the mean distance between the Sun and Earth, f_s, such that for each wavelength:

Finally, this estimate of radiance at the sea surface is attenuated by the intervening atmosphere according to the diffuse transmission factor in the direction of the satellite, t_dv. For purposes of this investigation, an experimental atmosphere was used to investigate whether the water impacted by different amounts of foam and whitecap could be detected with an intervening atmosphere containing water vapor and other constituents that could obscure the signal. Hence, experimental values of at-sensor radiance due to Rayleigh scattering (L_r) and aerosols (L_a) were added to the spectra to create a typical top of the atmosphere radiance spectrum (L_TOA). An atmosphere typical to the MOBY buoy (157°11′36″W, 20°49′07″N) was used with a solar zenith angle of 42°, humidity of 72.8%, water vapor of 1.737 g cm⁻², and pressure of 1,015.19 mb. No sun glint contribution is considered at TOA because the direct and diffuse reflected skylight was included in the field measurements of water reflectance and propagated through the atmosphere with L_w. This approach neglects any atmospheric effects that may occur due to enhanced reflectance of the sea surface from whitecaps and aerosol-molecular coupling.

The observed epsilon, ε^obs, is used to assess spectral dependence of aerosols from the TOA radiance and is estimated here for water surfaces with different levels of foam. The observed aerosol reflectance for a given wavelength Raobs(λ) was estimated as the difference between the total and Rayleigh radiance components adjusted by an estimate of downwelling irradiance using F_o(λ), such that:

The ratio of this parameter with different combinations of wavebands provides the ε^obs as:

The Lw contribution is presumed to be 0 in NIR and SWIR wavelengths rather than running a bio-optical model to estimate the contribution of L_w(NIR) (Bailey et al., 2010). The addition of whitecaps, however, could impact the radiance at 670 nm and implementation of the bio-optical model. The wavelengths used to estimate this parameter are typically from the MODIS Ocean Color Sensor 748, 869, 1,240, 1,640, and 2,130 nm (Wang and Shi, 2007). Here the analysis was conducted with 753, 869, 1,253, 1,617, and 2,132 nm, similar to the bands proposed for the PACE mission.

Arithmetic means were calculated and shown with plus or minus the standard deviation. The average spectral whitecap model was developed using a dataset of measured bright white foam data. This average spectrum was used to model the fractional whitecap coverage from a different dataset of field measurements of mixed pixels of whitecap, foam, and background.

Performance of various models to retrieve the whitecap fraction of mixed pixels were considered using mean average error (MAE), mean average percent error (MAPE), coefficient of variation, bias, and R² and slope (Seegers et al., 2018). These metrics were done in logarithmic space and are reported transformed back from logarithmic space, since whitecap fraction followed a lognormal distribution. Bootstrapping was not conducted for the proposed algorithms using the mixed pixel dataset, as the data was collected over similar water types and the number of data points was considered too limited to divide into algorithm development and validation subsets.

The whitecap reflectance spectrum was fit to water absorption with nonlinear regression using least squares. A p < 0.05 is considered to be statistically significant. For the radiative transfer and fractional whitecap estimates, model parameters were fit to the data with a non-linear regression function using least squares estimation. Spectral weights were not applied, but the spectrum was constrained between 400 and 1,800 nm. For some of the spectral matching modeling, the parameter fits were constrained within a range of realistic values (e.g., 0–1 for whitecap factor). Reflectance of mixed pixels measured from 1800 to 2,500 nm had low signal to noise and were excluded in the spectral fitting analysis.

Whitecap reflectance varies depending on the layers of foam on the sea surface and the amount of submerged bubbles. Examples of reflectance measured with different manifestations of whitecap, foam and bubbles from Long Island Sound are shown in Figure 2A. These examples and their corresponding pictures illustrate how the highest reflectance across the spectrum occurs with multiple layers of foam at the sea surface from a wave breaking (Figure 2A, magenta). The color in visible wavelengths can be nearly spectrally flat or “white” with high reflectance ~0.50 for Stage A whitecaps with many layers of foam, and is considerably higher than background (black line). The spectrum decreases into the NIR and SWIR wavelengths with several apparent peaks and troughs. Gaps in the spectrum centered at 1.5 and 1.9 μm occur in regions where the atmosphere is highly absorbing and downwelling irradiance is too low for a measurable signal in the field.

High reflectance can also occur when considerable amounts of bubbles are produced at depth (Figure 2A, green lines). In comparison to breaking waves, however, the spectral shape in visible wavelengths is different likely due to multiple scattering within the water. The waters of Long Island Sound have considerable blue-absorbing colored dissolved organic matter and detritus causing waters with submerged bubbles to be less “white” in visible wavelengths and have a more green-peaked reflectance spectrum (Figure 2A, green lines). The spectra appear as amplifications of the background water reflectance (Stramski and Tegowski, 2001). However, these spectra from deep bubbles produced from a submerged engine are not necessarily representative of normal surface wave breaking and are not considered further in this analysis.

Reflectance of thin layers of foam and residual bubble plumes diminish with similar spectral shapes to the breaking waves (blue lines, Figure 2A). Of note, our field measurements of thin foam produced by whitecaps had ~18% reflectance in the visible wavelengths was nearly equivalent to 22% in visible wavelengths used as an average whitecap reflectance in current atmospheric correction algorithms. This value was estimated by Koepke (1984) who showed that reflectance varied from 20 to 55% upon initial wave breaking to 3–10% after 10 s and an time-averaged effective reflectance of whitecaps of 22% was derived. However, there is considerable uncertainty in those values because a radiometer was not used to measure reflectance but rather a measure of film density was used assuming that the maximum reflectance was 55% (Whitlock et al., 1982) and presuming background water reflectance was equal to Fresnel reflectance of sun glint. Moreover, simply averaging the reflectance over the time-course of a breaking wave may not accurately reflect the physics of wave breaking. The spectral shape of the wave can change as the wave transitions to Stage B with more bubble plumes and more of the underlying water color contributes to the color (Moore et al., 2000).

Our measurements of both natural and manufactured foam are compared to other published measurements of spectral reflectance. Past studies reveal considerable variability in whitecap reflectance varying from 75 to 40% in the visible wavelengths (Figure 2B). Many of the past measurements were conducted with multi-spectral instruments that covered discreet bands within the visible and near-infrared (NIR). Frouin et al. (1996) studied the visible and NIR of sea foam found in the turbulent surf zone. As shown in Figure 2B, sea foam reflectance was found to monotonically decrease into the near infrared wavelengths due to enhanced water absorption in these wavelengths decreasing by 40% at 850 nm, 50% at 1020 nm, and 85% at 1650 nm (Frouin et al., 1996). Measurements by Moore et al. (1998) of foam produced by the bow of a large ship have higher reflectance values (50–75%) and a more green-peaked spectrum in visible wavelengths. However, a similar decay in reflectance out to 860 nm near infrared wavelengths was found by Moore et al. (1998), as well as a decrease at 670 nm compared to blue-green wavelengths. These peaked spectra are closer to the ones produced by deep submerged bubbles in Figure 2A and this may be a result of the type of foam and deeper bubbles produced by the wake of a large ship.

Whitlock et al. (1982) was the first to measure the full spectral reflectance of different layers of foam from 350 to 2700 nm. The reflectance profiles are generally spectrally flat in visible wavelengths with a maximum of 60% and higher reflectance into the NIR and SWIR than observed in other studies (Whitlock et al., 1982). Reflectance from 500 to 800 nm shown in Figure 2B (black line) are from Whitlock et al. (1982 Figure 2A) and derived from the polynomial fit with absorption by water at wavelengths >800 nm (Whitlock et al., 1982, Equation 1). Whitlock et al. (1982) also found a strong relationship between the absorption by pure water at >800 nm and the measured whitecap spectrum and noted reflectance minima at 1,500 and 1,900 nm that correspond to absorption peaks of clean water. A fourth order polynomial model was developed to model the whitecap reflectance from absorption by water. A similar approach was used in this study, as developed below.

While the work was seminal, several problems are evident in the Whitlock et al. (1982) study. First, the values of water absorption from 400 to 800 nm, shown in Figure 2B of Whitlock et al. (1982), are roughly an order of magnitude higher than the cited values (Smith and Baker, 1981). This could explain why the polynomial model of Whitlock does not extend into visible wavelengths. Secondly, the values are quite high in NIR and SWIR compared to other datasets. A reference standard of barium sulfate (EASTMAN 6080 paint) reference surface was used and reported to have 94–99% reflectivity out to 2,000 nm with no correction. While this non-luminescent coating yields reflectance values of 95–98% over the wavelength region from 300 to 1,200 nm, it is spectrally varying from 60 to 80% out to 2,500 nm. Hence, the Whitlock et al. (1982) values are too high and these were “corrected” based on the estimated spectral reflectance of this coating (Figures 2B,C dotted line), although the amount and condition of the coating are not known. However, even with this correction, the Whitlock values are much higher in the NIR and SWIR compared to field measurements made in natural seawater, which could also be due to artifacts in the tank setup that lead to further uncertainty in the NIR and SWIR values of Whitlock et al. (1982).

Two different datasets are provided in Figure 2C and compared to the Whitlock and Frouin measurements. The blue lines in Figure 2C represent measurements of mixed pixels of naturally produced foam, bubbles and background water from rolling breakers obtained close to the water surface from a small boat. The cyan lines represent a compilation of measurements made over bright thick foam at the sea surface mostly generated from a boat wake produced by the shallow-draft 11-m long R/V Lowell Weicker. These waves have considerably larger reflectance than the 22% average used in the standard algorithm for average open ocean waves and average reflectance of ~40% in visible wavelengths. Reflectance dips are prominent particularly at 750, 980, and 1,150 nm which have enhanced liquid water absorption, a result of multiple scattering in and around the bubbles and foam. The cyan lines tend to follow the range measured by Frouin et al. (1996) despite the fact that the measurements were taken over very different types of water and generated foam (i.e., Long Island Sound vs. San Diego surf zone). A mean from our spectrum is highlighted in the thick blue line (Figure 2C) and reveals a similar decay in NIR wavelengths to Frouin et al. (1996) but with more spectral features in the NIR and SWIR.

Similar to the approach followed by Whitlock et al. (1982), we modeled the average whitecap spectrum (Figure 2C thick black line) using a polynomial fit to water absorption. The relationship between liquid water absorption and whitecap reflectance is predicted from visible to SWIR wavelengths. The high reflectance corresponds to low water absorption primarily in visible wavelengths and the low reflectance corresponds to high water absorption in NIR and SWIR. A second order polynomial fit to the data explains much of the inverse relationship (Figure 3A, R² = 0.975), but misses the complexity of the shape. A simple third order polynomial on the log-transformed water absorption is a near match to the average whitecap reflectance spectrum across the visible to SWIR wavelengths (Figure 3B):

For wavelengths < 400 nm, there is a slightly lower reflectance spectrum than that predicted by water absorption (Figure 3B). This is likely due to absorbing constituents within the water like colored dissolved organic matter and more investigation is needed to accurately explore whitecap reflectance in the ultraviolet wavelengths.

The whitecap reflectance reveals troughs that occur in local liquid absorption maxima evident in the water absorption spectrum (Figure 4). These troughs are centered around 600, 756, 970, 1,198, 1,448, and 1,932 nm and are specific to water in the liquid form, as the maxima shift when water is in solid or vapor form. For example, the snow community uses the shift from 1,030 nm of the imaginary index of refraction (i.e., absorption) of pure ice to 970 nm for liquid water in algorithms to estimate the liquid water content of melting snow (Green et al., 2002). This same absorption feature is shifted to 940 nm for water vapor and is often a channel used for estimating perceptible water vapor and other atmospheric remote sensing applications (Ningombam et al., 2016; Ibrahim et al., 2018). Hence, the liquid absorption features detectable in the whitecap reflectance measured at the sea surface may wholly or in part be obscured by water found in various forms in the atmosphere due to the close and overlapping absorption bands of water in its various forms (see analysis in Section Whitecap Modeling for the PACE sensor below).

The close relationship between whitecaps and liquid water absorption implies that these can be tied within the theoretical framework of radiative transfer modeling. Indeed whitecaps belong to a broad class of strongly multiple-scattering media where the volumetric concentration of bubbles is >70% (Kokhanovsky, 2004). Radiative transfer modeling typically applies to media with a low concentration of scatterers (<1% by volume), but progress has been made in the development of the phenomenological optics of whitecaps (Kokhanovsky, 2004; Ma et al., 2015). Zege et al. (1991) provide a radiative transfer model of closely packed optical foam. Kokhanovsky (2004) furthered this work to provide a simple model semi-infinite foam, but notes this can extend to the case of finite foamed media having a large optical thickness (which is similar to a whitecap). Whitecap reflectance has been related to the liquid bulk water absorption (a_w), the reflectance function for a single scattering albedo of 1 (i.e., a non-absorbing media) (R_o), and a spectrally neutral constant (h) through the following relationship valid for a semi-infinite foam layer (Kokhanovsky, 2004):

The values for b and Ro can be fit to experimental data or solved based on the physics using the liquid fraction (l), the average diameter of the bubbles (d), the constant B related to the real part of the index of refraction of liquid water, and Q related to the illumination conditions and observation geometry.

However, more research is needed to fully interpret these parameters in the scope of breaking waves and foam on the sea surface. Kokhanovsky (2004) was able to match the Whitlock et al. (1982) spectrum at wavelengths >800 nm only by adding a component to water absorption equivalent to 0.000270 μm⁻¹, three orders of magnitude higher than values found in the ocean (e.g., ~0.270 m⁻¹). If typical values of water absorption are used in Equation (8), then the modeled spectrum is much higher in magnitude and the water absorption features are much more dramatic than that of a typical whitecap (Figure 5, blue line). To achieve the appropriate magnitude of whitecap reflectance, the R_o parameter can also be tuned to <1.0, which may account for some loss of energy as light passes through the air-water interface (Kokhanovsky pers. comm.). If we fit our whitecap spectrum (Figure 5 black line) with Equation (8) using non-linear least squares, we arrive at R_o = 0.36 and b = 10.3 mm (Figure 5 red dotted line, R² = 0.97, RMSE = 0.0311).

This h can be deconstructed to estimate the amount of liquid water and size of bubbles using (Equation 9) above. The parameter B was assumed to ~2.3 for media similar to whitecaps following from Kokhanovsky (2004). The parameter Q can be calculated as follows (Zege et al., 1991):

Given an incidence angle of 20° equivalent to the solar zenith angle and a nadir observation angle and presuming R_o is 0.36 (modeled), Q is 4.40 for our measurement. With these assumptions, we can solve for the equivalent water thickness (dl) of 0.099 mm for our average whitecap reflectance. The question arises whether this parameter can be further decomposed into realistic liquid water content and bubble size distributions of a breaking wave.

Bubble clouds near the surface may be crudely separated into short-lived high void-fraction plumes of large bubbles close to the surface embedded in a more slowly varying low-void fraction background field of smaller bubbles extending to greater depths (Melville, 1996). Technology has allowed for better visualizations of breaking wave processes at small scales within the surface ocean (Blenkinsopp and Chaplin, 2007, 2011). As highlighted in Figure 6, modified from Blenkinsopp and Chaplin (2007, 2011), the fraction of air or “void fraction” is much larger in the above water whitecap compared to the underlying submerged bubble plumes. If we infer that the reflectance of the whitecap is largely from the above-water portion of bubbles, the void fractions range from 60 to 99%. The liquid water fraction is the non-air fraction and would then range from ~1 to 40% (Melville, 1996; Blenkinsopp and Chaplin, 2007). Using this range for l in Equation (9), the average bubble diameter, d, contributing to the measured whitecap reflectance would range from 0.16 to 1 mm.

Bubble size distributions are generally measured on the submerged plume within the water column. They generally follow a power law distribution (Blenkinsopp and Chaplin, 2010; Randolph et al., 2014; Deane et al., 2016). However, determining an average bubble diameter is dependent on the size range of bubbles under consideration and the technology used to assess their sizes. For example, acoustics tend to measure a larger size range than afforded by optical methods (Randolph et al., 2014). Most of these approaches are designed to measure bubbles plumes within the water and not surface expressions of foam. However, the larger range of diameter (1 mm) found here is consistent with measurements of bubbles sizes in plumes of breaking waves (e.g., Deane et al., 2016). More experimental research is needed to further constrain the radiative transfer model and underlying assumptions. The ultimate objective would be to use optical measurements of whitecaps from satellites to further elucidate the physics of wave breaking and gas exchange under different environmental conditions across the world ocean.

Various components of breaking waves can contribute to the reflectance (Frouin et al., 1996). The simplest formulation and the one currently employed by ocean color correction algorithms is to treat the ocean as a mixed pixel where the whitecap component has a constant reflectance, R_f, and covers a fraction of the sea surface, A, and the remaining fraction of sea surface is comprised of a constant background water-leaving reflectance, R_w, such that:

Even though area-weighted averages should be used for the whitecap and white-cap free areas in atmospheric correction routines (Gordon, 1997), the (1-A) term is not explicitly incorporated in retrievals of water-leaving reflectance. Presumably this introduces little error under most wind regimes where whitecap fraction, A, is very low and when considering standard 1-km ocean color pixels that average over a large ocean footprint. However, as satellite spatial resolution becomes smaller, the fraction of whitecap within a pixel can become higher and this error would need to be rectified. The fractional whitecap coverage in pan-sharpened Landsat or Sentinel 2 images (e.g., Figure 1A) can vary from 0 to nearly 1 within a given scene. For this modeling, A is derived as a “whitecap factor” rather than a whitecap fraction since it can be applied to an individual pixel and has no spatial dependence.

A whitecap can also be treated as semi-transparent where there is a contribution from the water layer below. Such a two-layer system can be modeled by the following equation which considers reflectance from the foam layer and an approximation of the water layer which has been attenuated by the overlying foam layer. The optical properties of diffusing materials (Duntley, 1942) can be used to consider a layered whitecap system water the water-leaving reflectance is attenuated by the overlying foam layer (Frouin et al., 1996), such that:

However, it should be noted that the contribution of the water-leaving radiance is negligible when overlain by a thick surface foam, as measured here. The utility of this formulation would only be significant when there is a thin foam and the presumed R_f is low.

Hence, another way to consider the problem is to specify separate contributions from thick foam, as well as a semitransparent thin foam/bubble layer overlying the background water, which can also contain submerged bubbles, and potentially submerged bubbles without surface foam (Zege et al., 2006; Randolph et al., 2014). This would imply a fraction covered by the opaque whitecaps (A₁) and another fraction that may be covered by semitransparent layer (A₂). If we presume that the thinner foam has the same spectral shape but reduced magnitude when compared to thick foam (e.g., Figure 2A), then the thin foam could be written as a fraction, F, of R_f and the formulation would look like:

These 3 different models (Equations 10–12) were tested using a time series of reflectance measurements made over natural mixed pixels of background, bubbles, and whitecaps (e.g., rolling breakers) (blue lines in Figure 2C). Model parameters were fit to 88 different spectra using non-linear least squares to the total reflectance measured between 400 and 1,800 nm presuming a known background reflectance of water, R_w, and the average whitecap reflectance from Equation (7) (Figure 4A). The simplest model was able to capture the spectral shape from mixed pixels with lots of foam to those just above the background, as illustrated by a range of selected spectra shown in Figure 7A. The modeled retrieval of total reflectance showed good correspondence across all wavelengths (Figure 7B) with and R² of 0.96 and a slope of 0.98. The MAPE estimate for each wavelength (Figure 7C) shows that the model achieves an average of 18.5% across the spectrum and 9.0% in visible wavelengths (400–700 nm). Application of the second model (Equation 13) which includes a term for semi-transparent whitecap provided little improvement in fit with an average MAPE of 18.69% and 8.27% in visible wavelengths. As mentioned earlier, this is likely because the whitecap reflectance is high and the contribution of the water-leaving radiance is negligible when overlain by a thick surface foam. The third model (Equation 14) allows for a thinner foam layer, but has more free parameters to fit (F, A₁, and A₂) and unique solutions were difficult to constrain. The model seemed to overfit either A1 or A2 with either fractions of 0 or 1.0. Additionally, the complexity of this model did not significantly improve the fit to the measured spectra and the MAPE was 18.20 and 8.85% in visible wavelengths. Hence, the simplest model (Equation 13) captured the large range in reflectance from mixed pixels, particularly at higher whitecap factors. This provides further evidence that the mixed pixel behaves in a linear manner and the retrieved effective whitecap factor is able to account for different thicknesses of foam on the sea surface.

The above exercise illustrates that a simple model is capable of reproducing the total reflectance from a mixed pixel of foam and background reflectance using the average whitecap reflectance spectrum (Equation 7). The retrieved whitecap factor is an “effective” whitecap coverage that incorporates different levels of foam and bubbles within the pixel varying from 0.01 to 1. With this parameter, the contribution of whitecaps can be removed from the mixed pixel in order to retrieve an estimate of the background water-leaving reflectance that is needed for implementation of ocean color products. In this section, we consider different algorithms that could be used to retrieve to the effective whitecap factor assuming that the background reflectance is not known.

Following from the previous section, an iterative procedure could be implemented to retrieve both the fractional whitecap coverage and the background reflectance with a constant whitecap reflectance by adjusting the shape of background reflectance and fractional whitecap coverage with an optimization routine. Constraints could be applied such that background reflectance is retrieved within the scope of known water-leaving reflectance shapes. Such an optimization, however, may be sensitive to other components of atmospheric correction, such as the choice of aerosol models and removal of diffuse and direct sea surface reflected solar radiance. Hence, the average whitecap reflectance (Equation 7) could be straightforwardly incorporated into existing models that solve both the water and atmospheric components simultaneously (Stamnes, 2003; Fan et al., 2017) or atmospheric correction schemes that rely on spectral matching and optimization (Steinmetz et al., 2011).

Other atmospheric correction algorithms from ocean color satellites are stepwise and incorporate an independent determination of the contribution of whitecaps to the total radiance at the top of the atmosphere (Gordon, 1997; Bailey et al., 2010; Ibrahim et al., 2018). To assess the parts of the spectrum most useful for discrimination of whitecap factor, a correlation matrix was constructed to assess how combinations of different wavelengths could be used to predict whitecap factor, A. A Normalized Difference Index (NDI) was used which is the difference between two wavelengths normalized by the sum of the wavelengths (Figure 8; Dierssen et al., 2015). The normalization constrains the index to values between −1 and 1 with 0 indicating no difference and can be used to isolate narrowband features within a hyperspectral signal. Using this index, regions of highest correlation are localized in discrete wavebands and primarily include water absorption bands at 750 nm, 900–980 nm, and 1,100–1,300 nm. Specifically, a higher effective whitecap factor resulting either from more whitecaps in a pixel or more intense breaking waves are expressed as greater reflectance troughs in the liquid water absorption bands highlighted in Figure 4 (600, 756, 980, 1,198, 1,448 nm).

The initial correlation was done on a linear scale, but a logarithmic scale is more appropriate given the distribution of whitecap factor and the radiance values (Figure 9). Areal-averaged whitecap factors can range over three orders of magnitude roughly from 0.001 to 0.1 (Brumer et al., 2017). Using a logarithmic distribution in the correlation analysis (Figure 8B), the correlation coefficients are higher and the SWIR features are more prominent compared to the linear fit. The amount of reflectance in SWIR bands (1500–1800 nm) is found to be highly correlated to the whitecap factor with R² > 0.90 for some combinations of wavebands (Figure 8B). While these SWIR bands are not specific liquid water absorption features, the enhanced reflectance in SWIR wavelengths can be important predictors of whitecap contribution and can potentially interfere with aerosol and glint atmospheric correction methods (see Section Whitecap Modeling for the PACE sensor below).

This analysis focuses on the proposed bands for incorporation in the PACE OLI sensor which include hyperspectral bands from 350 to 890 nm in 5 nm increments with additional largely heritage NIR/SWIR bands at 940, 1,038, 1,250, 1,378, 1,615, 2,130, 2,260 nm. The 1,378 nm channel is not included further in this analysis, because the atmosphere highly attenuates radiance in this band and reflectance at the sea surface is not measurable under most conditions. The PACE OLI sensors misses many of the NIR/SWIR bands related to liquid water absorption such as features around 980 and 1,200 nm. However, PACE is poised to be hyperspectral into the NIR and this analysis shows narrowband information in the 730–800 nm region that are related to the liquid water absorption features at 756 nm. In addition, the 1,038 nm band also may provide information on whitecap, although this region is likely also used for aerosols and sun glint extrapolations.

If we presume that the sea surface is a mixture of pure whitecap with whitecap-free background reflectance, then a simple linear mixing model could be developed to quantify the depth of the reflectance trough for different liquid water absorption bands. For the data collected here, the depth of the trough related to liquid water absorption is related to the derived whitecap factor for the 980 and 1,200 nm features. A baseline subtraction approach (also referred to as continuum removed) has proven to be robust for many environmental remote sensing applications (Clark, 1999; Dierssen et al., 2015; Khan et al., 2017; Garaba and Dierssen, 2018) and is explored here (Figure 10). The water absorption features at ~750 nm is not a robust indicator when the whitecap factor is low. However, the absorption features at 980 nm and 1,200 nm are well-correlated to whitecap factor across several orders of magnitude (0.02–1) and have little bias and fairly low error (Figure 10). Similar results are found for a simple band difference algorithm (Table 1). The NDI does not perform as well in predicting whitecap factor for any of these spectral regions because the normalization to the sum of the reflectance values tends to obscure the relative differences (Table 1).

While these liquid water absorption bands are not currently part of the PACE mission, this analysis suggests their addition, particularly 980 and 1,200 nm, could be valuable for predicting whitecap factor and other sea surface applications. High correlations can also be found with a multiple linear regression for select bands in the far NIR/SWIR including 880, 1,038, 1,250, and 1,615 nm (Table 1). As discussed earlier, whitecaps elevate the reflectance in the NIR/SWIR above background and the amount of signal within these bands is a good predictor of the fractional whitecap coverage. The 940 nm band was excluded from the regression analysis because this band varies with atmospheric water vapor absorption. The generality and applicability of these algorithms broadly across different oceanic regimes and atmospheric conditions remains to be tested.

Many of the features unique to whitecaps may be part or wholly obscured by the intervening atmosphere. Hence, a simple transformation was conducted to determine whether the signal observed at the sea surface could “potentially” be observed at the Top of the Atmosphere (TOA). This transformation is for a single atmospheric condition with a realistic set of aerosols and atmospheric gases and does not consider the impact of highly scattering waters on atmospheric processes (e.g., multiple scattering). It provides a glimpse of what a satellite might observe over whitecap-enhanced waters in the bands expected on PACE. The TOA radiance is nearly an order of magnitude higher in visible wavelengths, but the two datasets become closer in magnitude into the NIR and SWIR wavelengths (Figure 11A). The separate contribution of Rayleigh (L_r) and aerosols (L_a) are shown in comparison to a whitecap pixel (Lw maximum) and an unimpacted background pixel (L_w minimum) (Figure 11B). Pixels completely covered by whitecaps (A = 1) contribute more radiance than aerosols at the TOA from visible to 1,615 nm. This would not be observed on PACE, however, given the large spatial footprint of 1 km and typical values of A < 0.1. Another feature of note is that the changes in radiance are less sensitive to whitecap factor at the top of the atmosphere having steeper slopes (Figure 11C) compared to reflectance at the sea surface (Figure 11D). This could impact the ability to retrieve low whitecap factors common to open ocean conditions from TOA radiance. The impact of different atmospheric conditions will also need to be explored.

Gordon (1997) writes that “area-weighted averages” of the whitecap-covered and whitecap-free areas of the surface should be considered in the formulation for atmospheric correction (Gordon, 1997). However, current correction routines do not incorporate the (1-A) fraction for the remaining signal emanating from the sea. This may not introduce much error in the current implementation of the whitecap correction routine, which only considers low fractions of whitecaps (A < 0.02). The formulation by Gordon (1997) also presumes that reflection of the direct and diffuse skylight would be the same over whitecap-covered and background waters; however, seas with breaking waves and foam do not reflect light in the same way as flat or wind-ruffled seas. Hence, the area-weighted average of whitecap-free sea surface should potentially include the direct beam (sun glint, L_g) and diffuse skylight (L_d) in addition to the water-leaving component. A modification of the traditional algorithm for conducting atmospheric correction (Gordon, 1997) is proposed that explicitly incorporates the fraction of whitecap-covered and whitecap-free areas following:

where t is the diffuse and T is the direct transmittance of the atmosphere. This equation separates the component of the diffuse reflected skylight, L_d, from the computed Rayleigh radiance and includes it as part of the water signal. This allows for true validation of the atmospheric correction approach, whereby values of L_g, L_d, L_w, and L_f can be individually measured and compared to those derived from the atmospheric correction algorithm.

The ε^obs used to estimate the aerosol model for atmospheric correction purposes was evaluated at different combinations of NIR and SWIR wavelengths (Figure 12). As shown in Figure 12A, ε^obs is quite insensitive to whitecap factor using the NIR wavebands of 753 and 869 nm. The liquid water absorption feature at 753 nm compensates for the enhanced reflectance due to the whitecaps. However, the SWIR wavelengths at 1,253, 1,617, and 2,132 nm do not incorporate the liquid water features and ε^obs is dependent on the whitecap factor with different combinations of SWIR bands. No correlation is found between whitecap factor and ε^obs(753,869), while the correlation coefficients are 0.54 and 0.36 for ε^obs(1,253,1,617) and ε^obs(1,617,2,132), respectively. The range in ε^obs is large for Figures 12B,C and would result in different aerosol spectral models and variable amounts removed from visible wavelengths, even though the atmospheric properties and water-leaving reflectance were the same.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.