Validation of an Improved Statistical Theory for Sea Surface Whitecap Coverage Using Satellite Remote Sensing Data

The whitecap coverage at the sea surface is affected by the ratio of kinetic energy to potential energy, θ, the wave spectrum width parameter, ρ, and other factors. This paper validates an improved statistical theory for surface whitecap coverage. Based on the theoretical analysis, we find that the whitecap coverage is more sensitive to ρ than to θ, and the improved statistical theory for surface whitecap coverage is suitable in regions of rough winds and waves. The satellite-derived whitecap coverage data in the westerly wind zone is used to validate the improved theory. The comparison between the results from theory and observations displays a better performance from the improved theory relative to the other methods tested.


Introduction
Breaking wave processes are of importance to the air-sea interaction, coastal circulation, ocean remote sensing and offshore engineering. Wave breaking can be a good visual indicator of wave-current interactions [1], where the upper-ocean currents and mixing are partially driven by the wave breaking [2]. However, previous studies still have not been able to establish a comprehensive understanding of wave breaking in deep ocean. Zhang et al. [3] use a vertical distribution model of turbulent kinetic energy based on an exponential distribution method, and they demonstrate that the energy dissipation rate of breaking waves relies on wind speed and the sea state. He and Song [4] examine the onset and the strength of unforced wave breaking in a numerical wave tank, and they suggest that the application of energy growth rate can yield better results than using the initial wave steepness for estimating the fractional energy losses.
In the study of wave breaking characteristics, the whitecap is one of the most important phenomena for wave breaking. Whitecaps are the manifestation of the breaking wave on the surface. Table 1. Statistics on the relationship between whitecap coverage and wind speed, W is the whitecap coverage. U 10 is the wind speed at 10 m above the sea surface.
W increases with wave age [44] and wave height [27,45]. Cross-swell conditions, on the other hand, reduce whitecapping [43]. Considering many factors, the expression of whitecap coverage, given in Yuan et al. [30], for a sea state with infinite wind fetch and duration, the ratio of the breaking kinetic energy to potential energy is analyzed in Wang et al. [32], and the values   [44] and wave height [27,45]. Cross-swell conditions, on the other hand, reduce whitecapping [43]. Considering many factors, the expression of whitecap coverage, given in Yuan et al. [30], for a sea state with infinite wind fetch and duration, the ratio of the breaking kinetic energy to potential energy θ is analyzed in Wang et al. [32], and the values θ = 8, C en = 0.1777, n = −1.713, F T = 0.75 are obtained for a sea state with infinite wind fetch and duration; C en and n are constants. Parameter F T is a dimensionless number that represents the integral of the bubble accumulation function over the entire domain (the value of F T is within the interval 0-1) [30]. We also consider similar values to test the theoretical expressions of the whitecap coverage for the general sea state.

Theoretical Expressions of the Whitecap Coverage for the General Sea State
The expression of the whitecap coverage for the general sea state is given in Yuan et al. [30] as follows: U B ≈ 0.25 m s −1 is the minimum terminal rise speed for the bubble group concerned [30,32]. g is the acceleration due to gravity, ρ is a parameter associated with the spectrum width (more details are given in Section 3.2), α = 1 for weak nonlinear waves, π = 3.14 [30]. L is the mean wavelength, and λ = 2/3 is the coefficient derived from the Neumann spectrum, which is comparable to 0.87 as measured in the laboratory [30,32,46]. H S is the significant wave height. L = gλ T 2 z /2π, T z is the zero-crossing wave period [30,32], C D = 1.5 × 10 −3 is the drag coefficient [47].
Applying the above equations to the general sea state and substituting the values of C en , n, F T [32] into Equation (1), W can be re-written as: where the significant wave height and the zero-crossing wave period T z are derived using the third Marine Science and Numerical Modeling (MASNUM) wave model [48][49][50][51], T z is used to calculate the wavelength, L, and the whitecap coverage (Equation (3)) is directly obtained from H S and L.
The third MASNUM wave model is developed by Key Lab of Marine Science and Numerical Modeling, State Oceanic Administration, China. In the model, the wave energy spectrum balance equation and its complicated characteristic equations are derived in wave-number space. The characteristic inlaid method is applied to integrate the wave energy spectrum balance equation [48][49][50][51]. MASNUM is widely used in forecasting ocean wave movement [51]. The MASNUM wave model shows good results in the general sea state and good improvement in the high sea state, and the model is used to evaluate the wave-induced mixing in the upper ocean [48][49][50][51].

Data
The model results of the whitecap in the general wave state are compared with observed data are reported. The observed whitecap is taken from Salisbury et al. [20], which we digitalize for the month of October 2006 and then extract the data points. The method of estimating the whitecap coverage from satellite remote sensing data is described in Anguelova and Webster [19]. Salisbury et al. [20] refer to this method as the W(T B ) algorithm. The W(T B ) algorithm has been improved in several aspects [52,53], such as the use of independent input datasets in the algorithm. Salisbury et al. [20] attest the variability of the whitecap coverage extracted by the W(T B ) algorithm. The algorithm for estimating W combines the satellite T B observations with models for the rough sea surface and the foam-covered areas (whitecaps). An atmospheric model is used to obtain the changes in T B at the ocean surface. Wind speed, wind direction, the sea surface temperature, and atmospheric variables such as water vapor and cloud liquid water are necessary as inputs to the atmospheric, roughness, and foam models; for simplicity the algorithm is called the W(T B ) algorithm [20]. The datasets used are shown in Table 2. The resolution of the whitecap coverage data is 0.
We extracted the data points of the satellite-derived W estimates in Salisbury et al. [20] for the month of October 2006, to validate the improved equation for the surface whitecap coverage. We extracted 354,521 data points in the latitude range of 90 • S to 89 • N and the longitude range 180 • E to 180 • W. We then mapped these data onto 0.5 • × 0.5 • resolution grids. The wave parameters H S and T z in Equation (3) are calculated using the third MASNUM Wave Model [48][49][50][51]. QuikSCAT wind data [20] has 0.5 • × 0.5 • resolution and is applied to the third MASNUM Wave Model. 3.2. θ and ρ θ, first introduced by Yuan et al. [30], is the ratio of the breaking wave kinetic energy to the potential energy. The wave-breaking process mainly occurs near the crest of the wave front. Based on the definition by Yuan et al. [30], θ is further simplified by Wang et al. [32], and the values 8-11 lie in the middle of θ's distribution, so that in this study, θ was kept between 8 and 11 for the general state of the ocean.
ε is the spectrum width parameter, µ i is the ith-order moment of the wave frequency spectrum, and ρ is a parameter associated with the spectrum width [30]. Since the Neumann spectrum describes the well-developed wave field [46]. Yuan et al. [30] apply the spectrum to derive the practical expressions of the whitecap coverage for a sea state with infinite wind fetch and duration. The spectral moment (µ 0 , µ 2 , µ 4 ) can be obtained through the definition of the spectral moment [30], for a sea state with infinite wind fetch and duration, in Yuan et al. ρ 2 = 1/3, so here we choose 0.53-0.59 for a test, and the values of the spectral width ε are from 0.80 to 0.84, which represent the general sea state.
To explore the dependence of whitecap coverage on θ and ρ, different combinations are chosen (Table 3), where θ is kept between 8 and 11, and ρ varies from 0.53 to 0.59, with intervals of 0.6 and 0.01, respectively. The numbers in Table 3 [33,40]. The value of the whitecap coverage obtained near the equator was very small, and several large value centers were observed in the westerly zone in the Southern Hemisphere. Figure 2 (1, 15, 36) and Figure 2 (7,21,42) show that the whitecap coverage decreased with increasing θ. Figure 2 (1, 7), Figure 2 (15,21), and Figure 2 (36,42) show that the whitecap coverage decreased with increasing ρ. The effect of ρ on the whitecap coverage was greater than that of θ on the whitecap coverage, as Figure 2 shows that the change in the whitecap coverage value is larger with the variation of ρ than that of θ. To determine how θ and ρ accounts for the variability in the whitecap coverage as estimated by the model, we calculated the Pearson product-moment correlation coefficient (r) between the satellite measurements and the model results with different combinations of θ and ρ. The correlation coefficient was calculated using the following formula: where X is the model whitecap coverage, and Y is the satellite whitecap coverage. The number of the data N is 128,944. There are 128944 satellite-derived data and 128944 model data used, within a latitude range of 79 • S to 65.5 • N and a longitude range of 180 • E to 180 • W. The corresponding correlation coefficients are shown in Table 4. It was shown that the correlation between the satellite measurements and the model results decreased with increasing ρ, but the correlation coefficient varied within a small range and irregularly with a change in θ. Therefore, Figure 2 and Table 4 show the result: the whitecap coverage was more sensitive to ρ than θ.  Table 4. Correlation coefficient (r) between satellite measurements and model results. The correlation coefficient r is related to the covariance of the whitecap coverage. The first row are the values for θ while the first column are the values for ρ. The significant level is 0.05. The critical value is 0.062. According to Table 4, several combinations with higher correlation are selected as representatives to further validate the model. One can clearly see that the combinations 1, 8, 15, 29 and 36 (in Table 3) had higher correlations than other combinations. A significant occurrence of whitecap coverage was presented in the westerly region (180 • W-180 • W, 40 • S-60 • S). This area was therefore chosen for further validation of the whitecap coverage model (Figure 3). Figure 3a-f shows that the whitecap coverage decreased with increasing θ when ρ was kept constant. Several large value centers were observed in the westerly zone, with the distribution of the centers resembling that from the satellite-derived results. However, the values of the whitecap coverage obtained in the coastal areas were noted to be larger than in the satellite observations. For example, the whitecap coverage in the coastal areas around 70 • W, 70 • E and 170 • E were not well reproduced by the model. By comparing color scales, where the color represents the value of the whitecap coverage, the value of the whitecap coverage in Figure 3a-f was decreasing. The value of the whitecap coverage of Figure 3f was the closest to Figure 3g, and the distribution of the center of the whitecap coverage of Figure 3f was similar to Figure 3g. Figure 3 shows that the performance of Combination 36 was the best. θ = 11 and ρ = 0.53 for Combination 36 were thus chosen in the following validation and substituted into Equation (3).  Table 2 Table 2 for more details about the different combinations) and the satellite-derived estimates of the global whitecap coverage in October 2006. A unified scale colorbar with values from 0 to 3.5% is used.     [20]. A unified scale colorbar with values from 0 to 3.5% is used.    In the following model analysis, we analyzed the errors of the model data with the satellite data. We analyzed the distribution of the global whitecap coverage over 180 • W (E)-180 • W, 55 • S-40 • N. We then carried out a regional analysis. Figure 4 displays that both the whitecap coverage of the model and the satellite had extreme centers of high values, and the spatial distribution was very similar, but there was still some difference. The picture shows that our theoretical model underestimated the breaking processes in the equatorial area. The whitecap coverage of the theoretical model due to the weaker wind near the equator, the breaking process was not as significant as in the westerly areas. Our theoretical model thus required further improvement, especially in the areas with weak winds, like the equatorial area.  Figure 5a-d had extreme centers of high values, and the spatial distribution of (a) and (b) were similar, and the spatial distribution of (c) and (d) was also similar, but there was still some difference. The relative differences of the whitecap coverage in Figure 6 showed more information. Figure 6 shows the centers of the positive and negative anomalies. The positive anomalies corresponded to the highest whitecap coverage, which meant that the model overestimated the whitecap coverage. This may be due to the wind speed. More studies are shown in Figure 7.

Model Analysis
The maximum and mean values of the whitecap coverage over the western and eastern parts were compared when θ = 11 and ρ = 0.53. The comparison results are shown in Tables 5 and 6. For the western part, the percentage difference (|model result−satellite result|/satellite result × 100) between the model and satellite mean maximum values was 6.3%, and the percentage difference between the model and satellite mean values was 1.2%. For the eastern part, the percentage difference between the model and satellite maximum values was 11.4%, and the percentage difference between the model and the satellite mean values was 2.8%. The percentage difference of the mean value was small; the mean values of the satellite-derived and modeled whitecap coverage are similar. These results demonstrate that whitecap coverage can be well reproduced in these areas. However, there were deviations due to factors such as sea state and the atmospheric impact. Figure 7 displays the dependence of the whitecap coverage model error on wind. The Figure 7 shows that most of the error data was concentrated between −0.5%~+0.5%. When the wind speed was lower than 10.8 m/s, the deviation was negative. When the wind speed was between 10.8 and 12.9 m/s, the deviation had both negative and positive values. When the wind speed was greater than 12.9 m/s, the deviation was positive. Such variations resulted from the influence of the wind speed, wind history, and the measurement of both stage A and stage B whitecaps [16]. As wave parameters (H S ,T z ) were included in our model, the whitecap coverage of the model result was affected by the wind speed and the wave age. Nordberg et al. [56] and Ross and Cardone [57] point out that the presence of foam streaks above about 13 m/s and the ratio of the area of foam streaks to whitecaps increased with increasing wind speed, which may be a reason explaining the positive deviations for wind speeds higher than 12.9 m/s. Meanwhile, the wind speed could affect the detection of the whitecaps by the satellites.         Through the least squares method, Zhao et al. [45] make a regression analysis of the whitecap coverage as a function of wind speed, wave age, wave period, friction velocity, and breaking parameter B R respectively. It also demonstrates a strong dependence of the whitecap coverage on the wave parameters. The correlation coefficient of B R is the highest. Thus, the following Equation Through the least squares method, Zhao et al. [45] make a regression analysis of the whitecap coverage as a function of wind speed, wave age, wave period, friction velocity, and breaking parameter R B respectively. It also demonstrates a strong dependence of the whitecap coverage on the wave parameters. The correlation coefficient of R B is the highest. Thus, the following Equation (5) was chosen for testing: 2 10 , υ = 0.15 × 10 −4 m 2 /s , u * is the friction velocity, ω p is the peak angular frequency of wind-waves, T P is the wave period, and υ is the kinematic viscosity of water. Here, we chose the largest value υ = 0.15 × 10 −4 m 2 /s in Zhao et al. [45] (if it was set to be smaller, the value of the whitecap coverage would be too greatly overestimated). The result is shown in Figure 8. The plot showed that the value of the whitecap coverage was overestimated compared with the result of the satellite-derived coverage. There are very large areas, where (a) shows whitecap coverage, but none are observed in (b). Thus, the relationship between the whitecap coverage and the breaking parameter still needs further improvement. Through the least squares method, Zhao et al. [45] make a regression analysis of the whitecap coverage as a function of wind speed, wave age, wave period, friction velocity, and breaking parameter B R respectively. It also demonstrates a strong dependence of the whitecap coverage on the wave parameters. The correlation coefficient of B R is the highest. Thus, the following Equation (5)  al. [45] (if it was set to be smaller, the value of the whitecap coverage would be too greatly overestimated). The result is shown in Figure 8. The plot showed that the value of the whitecap coverage was overestimated compared with the result of the satellite-derived coverage. There are very large areas, where (a) shows whitecap coverage, but none are observed in (b). Thus, the relationship between the whitecap coverage and the breaking parameter still needs further improvement. According to the results above, our model domain was set between 40° S-60° S and 180° E-180° W. The model results were then compared with satellite measurements from Salisbury et al. [20], and with several widely accepted wind speed empirical relationships (Table 7). We used Equation (3) to According to the results above, our model domain was set between 40 • S-60 • S and 180 • E-180 • W. The model results were then compared with satellite measurements from Salisbury et al. [20], and with several widely accepted wind speed empirical relationships (Table 7). We used Equation (3)   All of the models displayed large value centers in the westerly zone. Additionally, some models showed overestimated values at and around the large value centers in Figure 9. Since there was almost no land obstruction in the Southern Hemisphere (40 • S-60 • S), the sea water moved toward the east from the west under the influence of wind where the sea area was affected by the westerly wind all year round. The winds and waves in this area were high, resulting in a generally large the whitecap coverage. Scanlon et al. [59] indicate that the dissipation source term is more closely related to the whitecap coverage. In Yuan's model [30], the whitecap coverage is not only dependent on the wind speed and the distribution of the wind, but also on the energy dissipation and the other wave parameters, especially the wave steepness. The top panel in Figure 9, which was the result from our improved model, shows that near the land, the values were overestimated, suggesting that our model was again not suitable in coastal areas. This may be due to errors in calculating wave steepness or uncertainties associated with the energy dissipation rate. The whitecap coverage of our model was based on the ratio of the breaking wave kinetic energy to the potential energy. In coastal areas, the whitecap coverage was also much influenced by the topography. However, the model performed well in the open ocean, with values ranging from 0 to 3.5%. The value of the whitecap coverage was almost similar to that of the satellite-derived result. The spatial distribution of the whitecap coverage was better than Figure 8a. Based on Monahan's empirical relationship, W = 4.5 × 10 −6 U 3.31 , the value of the region around the maximum value center was more than 3.5% in the westerly areas. The distribution was noted to have similar pattern with the wind field (QuickSCAT), suggesting a high dependence on the wind field. Figure 9d,e has the same characteristics, and the value of the whitecap coverage was larger than that in Figure 9b. Figure 9c is parameterized as a function of wind speed by fitting power laws [20]. The values also exhibited a dependence on the wind.  [20], (d) Strama's empirical relationship [40], (e) Wu's empirical relationship [37], and (f) satellite-derived result [20]. A unified scale colorbar with values from 0 to 3.5% is used. Figure 10 shows the error, which is calculated as the model results minus satellite results. The fraction of the whitecap coverage based on the wind speed empirical relation was dependent on the wind. Compared to the fraction of the whitecap coverage derived from the satellite data, a big difference in the value and spatial distribution was observed. Most in situ datasets of the whitecap coverage have been obtained in coastal, fetch-limited conditions [19]; data from the open ocean are very sparse [20]. The empirical formulas of the whitecap coverage listed in Table 1 were developed from the situ data and they were not applicable to the open ocean. The reason is that the whitecap coverage is affected by many factors: sea conditions, breaking parameters and atmospheric factors, but the above empirical formulas in Table 7 only included the wind speed (Table 7 shows the most commonly used formulas among all those listed in Table 1), and they were very dependent on the wind field. Further, Figure 10 shows that the present model (Figure 10a) well captured the fraction of the whitecap coverage, as compared to those from empirical formulas, in terms of the value and spatial distribution in the westerly zone, and the error of the present model was smaller than those from the empirical formulas. Therefore, the present model is applicable to the open ocean. It is also noted that the present model generated large errors, and the whitecap coverage is overestimated in the coastal areas. The large error may be caused by several factors, such as errors in the satellitederived whitecap coverage, errors introduced when extracting data from a digitalized image, and errors from the present model, which are based on the ratio of the breaking wave kinetic energy to  [20], (d) Strama's empirical relationship [40], (e) Wu's empirical relationship [37], and (f) satellite-derived result [20]. A unified scale colorbar with values from 0 to 3.5% is used. Figure 10 shows the error, which is calculated as the model results minus satellite results. The fraction of the whitecap coverage based on the wind speed empirical relation was dependent on the wind. Compared to the fraction of the whitecap coverage derived from the satellite data, a big difference in the value and spatial distribution was observed. Most in situ datasets of the whitecap coverage have been obtained in coastal, fetch-limited conditions [19]; data from the open ocean are very sparse [20]. The empirical formulas of the whitecap coverage listed in Table 1 were developed from the situ data and they were not applicable to the open ocean. The reason is that the whitecap coverage is affected by many factors: sea conditions, breaking parameters and atmospheric factors, but the above empirical formulas in Table 7 only included the wind speed (Table 7 shows the most commonly used formulas among all those listed in Table 1), and they were very dependent on the wind field. Further, Figure 10 shows that the present model (Figure 10a) well captured the fraction of the whitecap coverage, as compared to those from empirical formulas, in terms of the value and spatial distribution in the westerly zone, and the error of the present model was smaller than those from the empirical formulas. Therefore, the present model is applicable to the open ocean. It is also noted that the present model generated large errors, and the whitecap coverage is overestimated in the coastal areas. The large error may be caused by several factors, such as errors in the satellite-derived whitecap coverage, errors introduced when extracting data from a digitalized image, and errors from the present model, which are based on the ratio of the breaking wave kinetic energy to the potential energy, and wave breaking in the coastal areas are more complicated.  [20], (d) Strama's empirical relationship [40], and (e) Wu's empirical relationship [37]. A unified scale colorbar with values from −0.8 to 0.8% is used.

Discussion
Whitecaps are usually the "tuning knob" of a wave model [8]. The fraction of the whitecap coverage can be used to improve upper-ocean turbulence models, wave models, and can be used for research in ocean-atmosphere interactions. An accurate estimation of whitecap coverage has significant practical applications, such as in ocean observation and modeling. In this study, a more accurate surface whitecap coverage is estimated, based on the improved statistical theoretical whitecap coverage model.
Wang et al. [32] have improved the constants of the whitecap model from Yuan et al. [30]. In this paper, the improved whitecap model is used to calculate the whitecap coverage, and the results are compared to those from Salisbury et al. [20]. For the whitecap model [30], θ and ρ vary within a range of values, in order to investigate the sensitivity of whitecap coverage W to the parameters. θ and ρ, which are negatively correlated to whitecap coverage, are demonstrated to have similar effects on the value of the whitecap coverage. Table 4 shows that the correlation between the satellite and model results decrease with increasing ρ, but they cannot determine the relationship with θ. As the correlation coefficient of the whitecap coverage is more sensitive to ρ than θ, the whitecap coverage is more dependent on ρ than θ. Many factors affect the whitecap coverage, therefore requiring more improvement to the statistical theoretical whitecap model. Tables 5 and 6 show that the maximum and mean values are almost the same as the satellite data. There will be deviations due to factors such as sea state and the atmospheric impact. Figures 9 and 10 show the poor performance of the whitecap model in coastal areas, while good results are observed in the open ocean. Values near islands are noted to be overestimated by the model.
The present model produces better results than the above wind speed-related empirical formulas. Analyzing the whitecap model result, the formula for surface whitecap coverage is more  [20], (d) Strama's empirical relationship [40], and (e) Wu's empirical relationship [37]. A unified scale colorbar with values from −0.8 to 0.8% is used.

Discussion
Whitecaps are usually the "tuning knob" of a wave model [8]. The fraction of the whitecap coverage can be used to improve upper-ocean turbulence models, wave models, and can be used for research in ocean-atmosphere interactions. An accurate estimation of whitecap coverage has significant practical applications, such as in ocean observation and modeling. In this study, a more accurate surface whitecap coverage is estimated, based on the improved statistical theoretical whitecap coverage model.
Wang et al. [32] have improved the constants of the whitecap model from Yuan et al. [30]. In this paper, the improved whitecap model is used to calculate the whitecap coverage, and the results are compared to those from Salisbury et al. [20]. For the whitecap model [30], θ and ρ vary within a range of values, in order to investigate the sensitivity of whitecap coverage W to the parameters. θ and ρ, which are negatively correlated to whitecap coverage, are demonstrated to have similar effects on the value of the whitecap coverage. Table 4 shows that the correlation between the satellite and model results decrease with increasing ρ, but they cannot determine the relationship with θ. As the correlation coefficient of the whitecap coverage is more sensitive to ρ than θ, the whitecap coverage is more dependent on ρ than θ. Many factors affect the whitecap coverage, therefore requiring more improvement to the statistical theoretical whitecap model. Tables 5 and 6 show that the maximum and mean values are almost the same as the satellite data. There will be deviations due to factors such The present model produces better results than the above wind speed-related empirical formulas. Analyzing the whitecap model result, the formula for surface whitecap coverage is more suitable in high winds and waves areas, while the values of the whitecap coverage obtained near the equator and middle latitude are not well simulated (the value is underestimated). This is because the whitecap coverage of our model (Equation (3)) was developed based on the ratio of the breaking wave kinetic energy to potential energy, and it also includes the wind speed and wave parameters. It can cover the wave breaking condition in the open ocean, but it does not capture the conditions in coastal areas well, where the wave breaking conditions are more complicated, such as bottom topography effects.

Conclusions
Our whitecap coverage model still needs more improvement. The model is more suitable for the large wave areas in the open ocean, but in tropical areas and middle latitudes, the ratio of breaking wave kinetic energy to potential energy is set to be constant, which should be improved further. There are still other reasons leading to the underestimate results of the whitecap coverage. The model can further be improved, especially in the equatorial zone and in the coastal areas. Other dynamic factors also need to be considered in order to determine the exact coefficient and parameters for the statistical theoretical model. Different measurements of the degree of wave age and mean wave slope can account for 80 to 85% of the variability in whitecap coverage [20]. The minimal wind speed for wave breaking, bubble persistence time, and the wave number spectrum also influence the whitecap coverage. Further work is still required to provide a greater insight into the environmental and meteorological factors affecting whitecap formation on the sea surface.