1. INTRODUCTION

The objective of this paper is to propose an improvement on the standard protocol for above-water measurement of water-leaving radiance by using additional measurements of upwelling radiance in the solar principal plane (i.e., the vertical plane defined by Sun, zenith, and sensor) to estimate the wave slope distribution and corresponding near-surface wind speed from angular variation of sunglint.

A. Standard Protocol for Above-Water Radiometry

From the early supervised measurements in the 1970s and 1980s [1–3], above-water radiometry has become the main source of validation data for water reflectance products from “ocean color” satellite missions [4] and higher spatial resolution “land” satellite missions [5] repurposed for coastal and inland water applications. Automated pointable above-water systems, such as those of the AERONET-OC [4] and WATERHYPERNET networks [6], can be deployed for long periods with less maintenance and fouling problems compared to underwater measurement systems.

The measurement method for above-water measurement of water reflectance was studied in detail and consolidated by [7] termed hereafter M1999 and adopted as a standard method by the NASA [8] and IOCCG [9] protocol documents. The standard method and the uncertainties related to it are also discussed by [10]. Following this method, the water-leaving radiance, ${L_w}({\theta _v},\Delta \phi)$, is derived from one measurement (i.e., the average of several repeated scans) of upwelling radiance, ${L_u}({\theta _v},\Delta \phi)$, and one downwelling sky radiance measurement (i.e., the average of several repeated scans) of ${L_d}{(180^ \circ} - {\theta _v},\Delta \phi)$, where ${\theta _v}$ is the nadir-viewing angle and $\Delta \phi$ is the viewing azimuth angle relative to the Sun with $\Delta \phi ={ 0^ \circ}$ for a radiometer pointing toward the Sun and $\Delta \phi ={ 180^ \circ}$ for a radiometer pointing away from the Sun—see Fig. 1 of [10], termed hereafter R2019, for the angle convention. The water-leaving radiance is then estimated, as illustrated in Fig. 1, from the measurement equation,

 

Fig. 1. Schematic of above-water radiometry with measurement of sky radiance, ${L_d}$, and removal of the reflected skylight, ${L_r}$. Dashed arrows indicate that contributions to the skylight reflected at the air–water interface come from directions that are not directly measured by the ${L_d}$ radiance sensor, including possible contributions from the direct sunglint direction. Adapted from [10].

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Based on radiative transfer simulations for a sea surface with wave field statistics given by [11], termed hereafter as “Cox–Munk”, M1999 shows how radiance from different portions of the sky dome contributes to ${L_r}({\theta _v},\Delta \phi)$ with increasing wind speed (Fig. 2 of [11]) and recommends the viewing geometry of ${\theta _v}={ 40^ \circ}$ and $\Delta \phi ={ 135^ \circ}$ as optimal in order to minimize sensitivity of ${\rho _F}$ to wind speed while avoiding the deployment problems (shadowing) encountered at higher $\Delta \phi$. A similar conclusion was reached by [12] for above-water radiometry with a vertical polarizer, although ${\theta _v}={ 45^ \circ}$ (near the Brewster angle) is then recommended. For this viewing geometry, sunglint (i.e., direct reflected light) is almost absent except at a very high wind speed or very low Sun zenith angle. A variant on this data acquisition protocol with $\Delta \phi ={ 90^ \circ}$, and hence some sunglint at moderate wind speed and low Sun zenith angle (Fig. 9 of [7]), is often used to avoid shadow/reflections from deployment platforms [4].

The basic data acquisition protocol of measuring ${L_u}({\theta _v}={ 40^ \circ},\Delta \phi ={ 135^ \circ}{,90^ \circ})$ and ${L_d}({\theta _v}={ 140^ \circ},\Delta \phi ={ 135^ \circ},{90^ \circ})$, termed hereafter standard M1999 protocol, has become very well-established for the measurement of water-leaving radiance, and, when combined with a measurement of downwelling irradiance [13], ${E_d}$, for the measurement of remote-sensing reflectance, ${R_{\textit{rs}}} = {L_w}/{E_d}$ or water leaving radiance reflectance, ${\rho _w} = \pi {L_w}/{E_d}$. This acquisition protocol has been successfully used for many satellite validation and aquatic optics studies over the last 20 years. However, a number of shortcomings and suggestions for improvement have been raised, as reviewed in Section 4.2.1 of [13] and detailed in the references therein. One major shortcoming of this method that will be dealt with in the present study is that:

B. Estimating Wind Speed from Extraneous Sources—the Problem

To generate the lookup tables (LUTs) needed for evaluation of ${\rho _F}$ in Eq. (3), radiative transfer simulations were made by M1999 with simulated sea surfaces, where the wave slope statistics match those of the Cox–Munk model based on an input wind speed, $W$. These wave models assume a fully developed wind wave sea, where the wave field is in equilibrium with the local wind. This assumption is not valid when the wind speed is changing, is not valid for swell waves from a distant source, is not valid for an unstable atmospheric boundary layer [14,15], and, crucially for application of the standard M1999 protocol, is not valid for fetch-limited conditions typical of inland waters, estuaries, and many coastal waters. A second weakness of this approach is that the input wind speed, which may come from a local measurement for supervised shipborne measurements, or for automated networked measurements, may come from a meteorological model, may itself be subject to error, or may not represent adequately the wave field at the target location because of inadequate spatial or temporal resolution. One typical method employed to deal with fetch-limited inland waters is to limit the input wind speed to a certain threshold, e.g., $5\;{\rm m}\;{{\rm s}^{- 1}}$, or, more crudely, process all data with a single pre-imposed wind speed typically between $0\;{\rm m}\;{{\rm s}^{- 1}}$ and $4\;{\rm m}\;{{\rm s}^{- 1}}$.

In addition, specific implementations may have additional complications, and it is noted, for example by [16], that polarization effects should be included in the radiative transfer modeling. The use of Eq. (3) with a LUT, such as those of M1999, based on Cox–Munk wave statistics, assumes that the radiometer instrument is used with sufficiently long integration time or with sufficiently wide FOV so that the measurement is averaged over the full wave facet probability distribution. Faster-sampling instruments, combined with spike-filtering [17] rather than mean-averaging of replicates, effectively remove the part of the Cox–Munk wave slope distribution that gives more sunglint. Very wide FOV instruments average over a wider range of incidence angles and, hence, ${\rho _F}$—see Figure 7/C23 of [18,19]. These details are important but are implementation-specific and outside the scope of the present study, which will consider an infinitesimally small (point) FOV instrument and full mean-averaging in time over all wave slopes. Practically, the focus lies on a method for sensors mounted on autonomous pan-tilt units and deployed on fixed platforms with FOVs ranging from 1 to 7° and protocols allowing the user to average over different repetitive scans, e.g., such as the PANTHYR [6] or HYPSTAR.

 

Fig. 2. Examples of application of M1999 to a measurement made using a HYPSTAR radiometer above the Blankaart reservoir (50.988°N, 2.835°E) on (top-left) 20210305 at 12:31 UTC (with solar zenith angles, ${\theta _s}={ 57.26^ \circ}$) and (bottom-left) on 20210514 at 15:31 UTC (${\theta _s}={ 54.26^ \circ}$). Curves show upwelling reflectance, ${\rho _u} = \pi *{L_u}/{E_d}$, downwelling sky reflectance, ${\rho _d} = \pi *{L_d}/{E_d}$ multiplied by ${\rho _F}$ for a wind speed of $2\;{\rm m}\;{{\rm s}^{- 1}}$ (i.e., 0.025), and water leaving reflectance, ${\rho _w}$, after application of Eq. (1) and LUT M1999 using ${\rho _F}$ varying from 0.025 to 0.033 corresponding to wind speeds of $0\;{\rm m}\;{{\rm s}^{- 1}}$, $2\;{\rm m}\;{{\rm s}^{- 1}}$, $4\;{\rm m}\;{{\rm s}^{- 1}}$, $6\;{\rm m}\;{{\rm s}^{- 1}}$, and $10\;{\rm m}\;{{\rm s}^{- 1}}$, respectively (left). The relative difference between ${\rho _w}$ using $2\;{\rm m}\;{{\rm s}^{- 1}}$ (default value) and $0\;{\rm m}\;{{\rm s}^{- 1}}$, $4\;{\rm m}\;{{\rm s}^{- 1}}$, $6\;{\rm m}\;{{\rm s}^{- 1}}$, and $10\;{\rm m}\;{{\rm s}^{- 1}}$, respectively (right).

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C. Estimating Wind Speed from Extraneous Sources—Associated Uncertainties

Any uncertainties in the modeling of ${L_r}({\theta _v},\Delta \phi)$ are amplified to give potentially very large relative uncertainties in the desired ${L_w}$ product in situations where the reflected light, ${L_r}$, is greater than ${L_w}$ because ${L_w}$ is then the small difference between two large values via Eq. (1). As noted in Section 6.B of [13], this occurs for low reflectance waters/wavelengths or for high Sun zenith angles or for cloudy skies. It is important to note that the reflected skylight correction used in Eq. (1) leads to an absolute uncertainty (expressed in physical units) in ${L_w}$ (in contrast to the typical relative percentage uncertainties relating to factors such as instrument calibration). Subsequently, for low ${L_w}$, it will give high relative uncertainty while, for higher ${L_w}$, it will give lower relative uncertainty. This is illustrated in Fig. 2. For clear waters, when the wind speed is overestimated by only $2\;{\rm m}\;{{\rm s}^{- 1}}$, up to 75% and 50% errors are observed for ${\rho _w}$ in the blue and near-infrared (NIR) spectral range, respectively. In more turbid waters, errors are less pronounced with errors up to 25%.

D. Proposal for Estimating Effective Wind Speed by Measurements of Sunglint

As already illustrated by the photographs of Cox and Munk [11] and further discussed by Munk [20], there is a strong sensitivity of the sunglint viewing angle variability to wind speed. The peak in specular reflection decreases with wind speed, and, at higher wind speed, reflection is diffused to a wider range of angles. Based on these facts and the Cox and Munk wave statistics, Strong and Ruff [21] modeled the maximum reflection within the sunglint pattern as a function of wind speed (for wind speeds ranging from 0 to $20\;{\rm m}\;{{\rm s}^{- 1}}$). Similarly, Webber [22] related wind speed to the ratio of the maximum radiance and the radiance measured away from the sunglint pattern. Using spaceborne images, Bréon and Henriot [23] used the multidirectional observations from POLDER to retrieve wind speed and wind direction from the reflectance measurements within and around sunglint, and Harmel and Chami [24] retrieved wind speed from the multidirectional PARASOL images. Here, we propose to address the fundamental problem P1 by making additional optical measurements in the solar principal plane, thus effectively measuring the (optical effect of) the surface wave field. The extra ${L_u}$ measurements are then used to estimate the effective wind speed assuming wave slope statistics from the Cox–Munk model. This near-surface wind speed is then used as a proxy for calculating ${\rho _F}({\theta _v}={ 40^ \circ},\Delta \phi ={ 135^ \circ}{,90^ \circ})$. Here it is not essential that the estimated wind speed be correct (and for fetch-limited inland waters, it is likely to be very different from the actual wind speed). What is important is that the wave slope statistics for the ${L_u}({\theta _v}={ 40^ \circ},\Delta \phi ={ 135^ \circ}{,90^ \circ})$ viewing geometry are well estimated from the wave slope statistics deduced from the ${L_u}({\theta _v},\Delta \phi ={ 0^ \circ})$ measurements.

E. Estimation of Wind Speed—Scope and Algorithm Design Constraints

The algorithm for estimation of wind speed from sunglint measurements should obviously be sensitive to wind speed variations over a wide range of wind speeds. In this study, the range ${W} = 0 {-} 14\;{\rm m}\;{{\rm s}^{- 1}}$ is considered, although the priority is for the low and moderate wind speeds (${\lt}6\;{\rm m}\;{{\rm s}^{- 1}}$), which are optimal for satellite measurements and validation. At higher wind speeds, the standard protocol for above-water measurements encounters multiple problems, especially for the correction of the air–water interface reflectance, and the presence of wave-breaking and foam may also be problematic both for the in situ measurements and for correction of satellite measurements.

The algorithm for estimation of wind speed from sunglint measurements should also be effective for a wide range of Sun zenith angles. In this study, the range 0–80° is considered. As will be seen later, performance may be less good for very low ${\theta _s} \lt 10^ \circ$ and for very high ${\theta _s} \gt 70^ \circ$ particularly for high wind speeds. If the algorithm to estimate can also return information on performance, preferably as a measurement uncertainty, then the user can decide whether to incorporate this optical estimation of ${\rho _F}$ or alternatively ignore it and fall back on the standard meterological input for ${\rho _F}$. The algorithm for estimation of wind speed from sunglint measurements should be designed to have low sensitivity to the following unknown factors:

On many structures such as offshore platforms, it may be impossible or inadvisable to measure upwelling radiance too close to the platform, i.e., for too small of a nadir-viewing angle. Shadowing of skylight and reflections of sunlight and skylight into the target water may contaminate the upwelling radiance measurement in ways that are difficult to quantify, although some studies exist for such optical perturbations for the standard above-water viewing geometry [25,26]. Since such limitations are specific to each viewing platform and each Sun zenith/azimuth geometry, it is impossible to give a general rule here for what may be acceptable. In the present study, a pragmatic approach is adopted of allowing measurements at nadir angles of 20-80° in the principal plane. The approach could be easily adapted with more restrictive minimum nadir-viewing angles for platforms/geometries where this is necessary. Factors that are not considered explicitly in the algorithm design, but which may affect performance in specific cases, include:

For practical implementation, it will be preferable to limit the number of additional measurements that are needed to estimate $W$. While for research experiments it will be interesting to make many measurements at different nadir and azimuth angles, e.g., every 1° in nadir from 20° to 80° and every 1° in azimuth from ${-}{30^ \circ}$ to ${+}{30^ \circ}$, for operational implementation it is important to limit the time needed for these additional measurements both to minimize problems associated with natural temporal variations (including Sun zenith angle variation) and to limit power requirements at remote sites with constraints on locally generated electricity. In the present study, the algorithm will be designed with measurements at only two additional geometries and in the NIR spectral range where the water reflectance is considered as negligible.

F. Outline of the Paper

Having now explained the objective and design constraints of the new approach to replace extraneous wind speed measurements by estimating effective wind speed from optical measurements within the sunglint pattern, the radiative transfer model used for this study is described in Section 2.A. Next the angular variation of upwelling reflectances in the principal plane is investigated in Section 3.A for a wide range of wind speed and Sun zenith angle, aerosol particle types, relative humidity, and AOT. The difference in upwelling reflectance is identified as the key parameter to which effective wind speed is monotonically related. The optimal choice of the nadir-viewing angle is made and justified in Section 3.B, and algorithm performance is illustrated in Section 3.C with sensitivity to unknown aerosol properties. Nadir-pointing errors are addressed in Section 3.D. Next, to validate the performance and robustness of the method, the maximum error on the retrieved wind speed and its impact on the water leaving radiance reflectance are estimated. This twin experiment is detailed in Section 3.E. Finally the findings are summarized, and any remaining potential limitations of the approach are discussed together with future perspectives.

Table 1. Input Parameters for the OSOAA Simulations

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2. METHODS

A. Radiative Transfer Simulations

The OSOAA [27,28] vector radiative transfer code v1.6 has been used to perform simulations in the conditions summarized in Table 1. The OSOAA model simulates the light field and the polarization state of light using the plane–parallel layer assumption and successive-orders-of-scattering method in a coupled atmosphere–ocean system. To model the rough sea surface, the surface slope probability function is used [28],

Results of these simulations will be presented as (hemispherical directional) upwelling reflectances, ${\rho _u}$, defined as

 

Fig. 3. Upwelling reflectance, ${\rho _u}$, in the principal plane viewing toward the Sun as function of nadir viewing angle, ${\theta _v}$, and wind speed, $W$ (from 2 to $14\;{\rm m}\;{{\rm s}^{- 1}}$; see legend for line-style), for each Sun zenith angle, ${\theta _s}$, from 10° to 80° (vertical dashed gray line). Simulations made for 900 nm with (1) no atmosphere and a fully absorbing water body (black sea, black sky) and (2) coastal aerosols with aerosol optical thickness at 900 nm of 0.2 and a chl concentration of $1\;{\rm mg}\;{{\rm m}^{- 3}}$ (see Table 1).

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B. Twin Experiment and Metrics

To validate the method and its robustness, a twin experiment is performed. This experiment consists in comparing $\rho _w^t(\lambda ,{\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ})$ when all the environmental variables (i.e., ${W}$, AM, RH, and AOT) are perfectly known, with $\rho _w^e(\lambda ,{\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ})$, when erroneous environmental variables are used for the retrieval of $W$ (only, all other variables remain similar since on the field these variables would be measured, i.e., ${L_u}$, ${L_d}$, ${E_d}$). Quantifying the difference between $\rho _w^e(\lambda ,{\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ})$ and $\rho _w^t(\lambda ,{\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ})$ can be easily performed using OSOAA simulations since the error that will be made when estimating ${\rho _w}(\lambda ,{\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ})$ is equal to the difference between the upwelling reflectance ${\rho _u}(\lambda ,{\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ})$ retrieved with the correct wind speed and with the erroneous wind speed, respectively.

To evaluate the performance of the method, the following metrics are used:

3. RESULTS

A. Upwelling Reflectance in the Principal Plane

The variation of ${\rho _u}$ in the principal plane ($\Delta \phi ={ 180^ \circ}$) as function of ${\theta _v}$ and $W$ is shown for each ${\theta _s}$ in Fig. 3. A flat sea case, ${W} = 0\;{\rm m}\;{{\rm s}^{- 1}}$ (not shown here), acts as a simple reflective mirror of the Sun and sky showing a clear and very thin sunglint pattern. For low wind speed, ${W} = 2\;{\rm m}\;{{\rm s}^{- 1}}$, and low to moderate Sun zenith angle, ${10^ \circ} \lt {\theta _s} \lt 60^ \circ$, the nadir viewing angle with highest reflectance, ${\theta _{v0}}$, shifts slightly toward the horizon but remains close to ${\theta _s}$. For the higher Sun zenith angles, ${\theta _s} \ge {70^ \circ}$, the “mirror” analogy no longer holds, ${\rho _u}$ increases monotonically with ${\theta _v}$, and no maximum reflectance angle ${\theta _{v0}}$ can be identified. This is explained by the significant increase in Fresnel reflectance for the higher incidence angles (as clearly shown by the black sea and black sky simulations in Fig. 3 and in particular for ${\theta _s} \gt 60^ \circ$ at all wind speeds). Figure 4 compares the variation in reflectance for a black sea and black sky and for ${\theta _s}$ equaling 20° and 40°, along the principal plane, and along the azimuth for ${\theta _v} = {\theta _s}$.

 

Fig. 4. Black sea and black sky upwelling reflectance at 900 nm for (left) nadir scanning along the principal plane, and (right) azimuth scanning for ${\theta _v} = {\theta _s}$ for ${\theta _s}$ equals 20° (blue) and 40° (orange) and for different ${W}$ (see legend for line-style). The polar plot on the right shows the black sky and black sea simulations for upwelling reflectance at ${\theta _v}={ 40^ \circ}$ and ${W} = 6\;{\rm m}\;{{\rm s}^{- 1}}$. The red transect and circle indicate the nadir (principal plane) and azimuth scanning (for ${\theta _s}={ 40^ \circ}$), respectively.

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Fig. 5. Upwelling reflectance, ${\rho _u}$, in the solar principal plane viewing toward the Sun as function of nadir viewing angle, ${\theta _v}$, and wind speed, ${W}$ (see legend for line-style), for three Sun zenith angles, ${\theta _s}={ 20^ \circ}$ (left), ${\theta _s}={ 40^ \circ}$ (center), and ${\theta _s}={ 60^ \circ}$ (right). (a) ${\rm AOT} = {0.1}$ and ${\rm RH} = {98}\%$ and three different aerosol particle types, i.e., maritime, coastal and urban (see legend for color); (b) ${\rm AOT} = {0.1}$ and a coastal aerosol particle type and two different relative humidity (70% or 98%; see legend for color); (c) coastal aerosol particle type, ${\rm RH} = {98}\%$, and three different aerosol optical thicknesses (AOT) at 900 nm: a clear atmosphere (${\rm AOT} = {0.05}$, blue), a hazy atmosphere (${\rm AOT} = {0.2}$, orange), and a turbid atmosphere (${\rm AOT} = {0.5}$, green). All simulations are made at 900 nm (fully absorbing water).

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Considering the results of Figs. 3 and 4, for a wavy sea surface, it is seen that, as $W$ increases, the magnitude of the maximum reflectance reduces significantly, as does the angular variation of reflectance. For ${\theta _v} \lt {\theta _{v0}}$, the slope of the curves to the left of ${\theta _{v0}}$ decreases with wind speed. For ${\theta _v} {\gt} {\theta _{v0}}$, the slope of the curves, to the right of ${\theta _{v0}}$, is subject to more complicated processes. For instance, at ${\theta _s}={ 20^ \circ}$, the effect of waves on angular variation of ${\rho _u}$ is fairly symmetrical around ${\theta _{v0}}$ for low to moderate wind speeds (i.e., ${W} \le 10\;{\rm m}\;{{\rm s}^{- 1}}$; see Figs. 3 and 4). However, this symmetry is broken for higher wind speeds and for higher ${\theta _s}$, e.g., for ${\theta _s}={ 40^ \circ}$, the angular variation of ${\rho _u}$ around the ${\theta _{v0}}$ is notably asymmetric, even for a quite low wind speed of ${W} = 6\;{\rm m}\;{{\rm s}^{- 1}}$, because of angular variation of Fresnel reflectance and, to a lesser extent, the sky radiance (see simulations for coastal aerosols in Fig. 3).

From these results, it seems that an algorithm to retrieve $W$ should prefer the viewing range ${\theta _v} \lt{\theta _{v0}}$ over the viewing range ${\theta _v} \gt {\theta _{v0}}$, except for ${\theta _s} \lt 30^ \circ$, where practical problems of viewing at low ${\theta _v}$ (i.e., between 0 and 20°) may occur (i.e., optical perturbations from structures). Results in Figs. 3 and 4 suggest strong sensitivity of ${\rho _u}$ to $W$ and, hence, good prospects for retrieval of W from measurements of ${\rho _u}({\theta _v})$ in the principal plane, viewing toward the Sun. However, as mentioned in Section 1.E, the algorithm for retrieval of $W$ should also be designed to have low sensitivity to (1) aerosol composition, (2) viewing zenith and azimuth angle pointing errors, and (3) non-zero water-leaving radiance.

Figure 5 shows how ${\rho _u}({\theta _v})$ varies with aerosol properties, i.e., AOT, aerosol particle type, and RH. There is a small difference in ${\rho _u}({\theta _v})$ between the different aerosol particle types [see Fig. 5(a), variation with aerosol particle type] and relative humidities [see Fig. 5(b), variation with RH], but only at ${\theta _v}$ close to ${\theta _{v0}}$ and for the lowest wind speeds, ${W} \le 4\;{\rm m}\;{{\rm s}^{- 1}}$. From these results, it seems that ${\rho _u}({\theta _v})$ mainly varies with AOT [see Fig. 5(c), variation with AOT]. For the hazier atmosphere, the Sun glitter is less intense, which is intuitively reasonable. While this is the main effect, Fig. 5 also shows an increase in ${\rho _u}$ with AOT near the horizon (see, for instance, Fig. 5 for ${\theta _v} \gt 60^ \circ$ and at ${\theta _s}={ 40^ \circ}$ and ${ W} = 2\;{\rm m}\;{{\rm s}^{- 1}}$) because of increased sky radiance nearby the horizon.

B. Estimation of Wind Speed—Choice of Nadir Viewing Angles in the Principal Plane

Considering the design constraints to maximize sensitivity to wind speed (Figs. 3 and 4) and minimize sensitivity to aerosols and RH (Figs. 5), any viewing zenith angle pointing error and any non-negligible water-leaving radiance (which can be expected to be quite constant with ${\theta _v}$ because angular variation of water-leaving radiance is much weaker than angular variation of radiance reflected at the air–water interface), the following algorithm is proposed: $W$ will be estimated from $D$, the viewing zenith angle difference of ${\rho _u}$ (Fig. 6), where the latter is estimated from two measurements at viewing angles ${\theta _{v1}}$ and ${\theta _{v2}}$,

These two viewing angles are chosen to:

These considerations lead to the choice (see also Fig. 6):

Measuring $D$ from two nadir viewing angles rather than at a single angle is preferred to avoid sensitivity of the method to variations in water-leaving reflectance. Indeed, at scattering angles corresponding to the principal plane in the Sun direction (i.e., between 60° and 180°; see Fig. 3 in [31]), different water types show variations in magnitude but similar shapes in their scattering phase functions. Consequently, taking a difference in upwelling reflectance [see Eq. (8)] allows the removal of most of the light reflected by the water column such that only the reflected diffuse and direct light are considered [i.e., ${L_r}$ in Eq. (1)]. Making measurements at wavelengths where the light is almost fully absorbed by the water column (e.g., in the NIR spectral range for low to moderate turbid waters) also reduces the sensitivity of the method to variations in the water reflectance. A difference is also preferred for the computation of $D$ compared to a ratio of upwelling reflectances as the ratio is very sensitive to errors in ${\rho _u}$, in particular, when those reflectances are very low (e.g., when ${\rho _u}$ is measured away from the Sun glitter and/or at low wind speed when ${\theta _{\textit{vx}}} \ne {\theta _s}$). Hence, the suggested approach to estimate $D$ is to measure ${\rho _u}$ from the two nadir viewing angles and use this as input to a LUT defined for that ${\theta _s}$. In the most general case where there is no a priori knowledge of aerosol particle type, RH, or AOT, then the LUT is defined for the default case of coastal aerosol, ${\rm RH} = {98}\%$, and ${\rm AOT} = {0.1}$, and the sensitivity to the unknown aerosol properties will be a source of uncertainty.

C. Estimation of Wind Speed—Algorithm Sensitivity to Unknown Aerosol Properties

The desirable sensitivity of $D$ to $W$ and the undesirable sensitivity of $D$ to the aerosol model, AOT, and RH, which propagate to an uncertainty for ${ W}$ estimation, can be seen in Fig. 7. First, it is reassuring to note that for most ${\theta _s}$ (${\theta _s} \ge {10^ \circ}$ according to Fig. 7) and for all ${W} \le 14\;{\rm m}\;{{\rm s}^{- 1}}$ the dependence of $D$ on $W$ is monotonic—there will never be a non-unique solution for $W$. There may be situations in practice where there is no solution for $W$, if the measured $D$ is outside the range covered by these simulations for whatever reason (measurement problems, $W$ beyond the simulation range, cloudy skies, aerosols outside the range considered in the simulations, etc.)—these situations can be easily identified from out of range ${D}$ and flagged in an operational implementation. However, if there is a solution, it will be unique. Note, according to the OSOAA simulations (not shown here), for ${\theta _s} \le {6^ \circ}$ (only) the solution is not unique. However, ${\theta _s} \le {6^ \circ}$ is very unlikely to happen and will be considered out of scope for implementation.

 

Fig. 6. Required nadir viewing angles at both sides from the maximum reflectance along the principal plane, ${\theta _{v1}}$ and ${\theta _{v2}}$, respectively, for a given Sun zenith angle ${\theta _s}$.

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Fig. 7. Variation of ${ D}$ with windspeed, ${W}$, at 900 nm for (a) a relative humidity, RH, of 98% and different aerosol models (Coastal, Maritime and Urban; see legend for line-style) and aerosol optical thickness at 900 nm, AOT (0.05 and 0.2, see legend for color), and (b) a coastal aerosol particle type and different RH (70, 90, and 98%; see legend for line-style) and aerosol optical thickness at 900 nm, AOT (0.05 and 0.2; see legend for color).

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A steeply decreasing curve indicates high sensitivity to $W$, and closely grouped curves for the different aerosol cases indicate lower sensitivity to the aerosol properties and RH (Fig. 7). Thus, the approach is expected to work rather well for wind speeds up to about ${W} = 8\;{\rm m}\;{{\rm s}^{- 1}}$, except for ${\theta _s} \ge {60^ \circ}$. For the higher ${\theta _s}$, and especially ${\theta _s}={ 80^ \circ}$, it becomes more important to have a good estimate of aerosol properties, especially AOT. It is not surprising to see that the aerosol model does not affect the method significantly. Indeed, for a given wavelength, the sunglint, i.e., direct light, is mainly dependent on the optical thickness.

 

Fig. 8. Upwelling reflectance, ${\rho _u}$, at 900 nm along viewing zenith angles ${\theta _v}$ from 0 to 89° for two wind speeds ${W} = {2}$ and $8\;{\rm m}\;{{\rm s}^{- 1}}$ (plain and dashed line, respectively) with ${\rho _u}({\theta _{v1}})$ (black cross) and ${\rho _u}({\theta _{v2}})$ (black dot) and vertical lines for ${\theta _{v1}}$ and ${\theta _{v2}}$ (dashed and dotted line, respectively).

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Fig. 9. Variation in ${ D}$ with wind speed, ${W}$ (varying from 1 to $14\;{\rm m}\;{{\rm s}^{- 1}}$), with and without pointing errors in zenith (see legend for line-style) and azimuth (see legend for color) (upper row), and upwelling reflectance ${\rho _u}$ at 900 nm along viewing zenith angles ${\theta _v}$ from 0 to 82° and viewing azimuth angles $\Delta \phi$ from 0 to 10° for a wind speed $W$ of (a) $2\;{\rm m}\;{{\rm s}^{- 1}}$ and (b) $8\;{\rm m}\;{{\rm s}^{- 1}}$. Note the azimuthal isotropy around the solar plane.

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D. Estimation of Wind Speed—Sensitivity to Pointing Errors

Figure 8 illustrates the retrieval of the D factor along the principal plane (i.e., ${\rho _u}$ at ${\theta _{v1}}$ and ${\theta _{v2}}$) for two wind speeds (${W} = {2}$ and $8\;{\rm m}\;{{\rm s}^{- 1}}$) and ${\theta _s}$ ranging from 20 to 80°. Hence, a pointing error will shift the values for ${\rho _u}$ at ${\theta _{v1}}$ and ${\theta _{v2}}$ along ${\theta _v}$. Figure 9 shows the sensitivity of the approach to typical zenith and azimuth angle pointing errors (i.e., ${+}/ - {2^ \circ}$). Considering the different wind speed estimates that would result from such errors for a measured input of ${D}$, it can be concluded that for low ${\theta _s}$ there is some sensitivity to zenith angle error at low ${W}$ (i.e., ${\lt}2\;{\rm m}\;{{\rm s}^{- 1}}$). As shown by Fig. 8 and the 2D color plots of ${\rho _u}$ (i.e., for from 0 to 89° and from 0 to 10°), at lower wind speeds (i.e., $2\;{\rm m}\;{{\rm s}^{- 1}}$ versus $8\;{\rm m}\;{{\rm s}^{- 1}}$), the sunglint pattern has limited angular range in the zenith direction. A pointing angle error may, therefore, miss the slope of the rapid changing upwelling reflectance. For high (${ \gt} 60^ \circ$) and at low wind speed, both a zenith and azimuth pointing error will introduce significant inaccuracies in the estimation of $W$ with $D$. Indeed, for this scenario, the sunglint pattern is relatively narrow in both angular directions with high variability in upwelling reflectance (see 2D color plots legend range in Fig. 9 for ${\theta _s}={ 80^ \circ}$). Overall, the sensitivity of the method to pointing errors decreases with wind speeds as the sunglitter decreases in magnitude and its size increases along zenith and azimuth.

E. Impact of Errors in Wind Speed Estimation—Twin Experiments

Inaccuracies in the estimations of aerosols, RH, and viewing geometries will lead to erroneous estimations of the relations between $D$ and $W$. As seen in above sections, the monotonic relation between $D$ and $W$ is mostly sensitive to AOT and less to aerosol type and RH. A twin experiment is, therefore, performed with coastal aerosol particle types and for upwelling reflectance at 400, 600, and 900 nm. Viewing and illumination geometries, aerosol type, and RH are considered to be known, but an incorrect AOT is used. To ensure that the twin experiment includes the worst case scenario, wind speed is (erroneously) estimated (i.e., ${{\rm W}^e}$) from $D$ assuming an AOT of 0.2 (${{ D}_{0.2}}$) while simulations have been performed with an AOT of 0.05 (i.e., ${{D}_{0.05}}$ with true wind speed, ${{W}^t}$). Figure 10 shows ${{W}^e}$ versus ${{W}^t}$ and the difference ${{W}^e}-{{W}^t}$ for different RH and ${\theta _s}$. As shown above, the sensitivity of $D$ with $W$ decreases when wind speed and ${\theta _s}$ increase (see Figs. 7 and 9). This explains why the differences between ${{W}^e}$ and ${{W}^t}$ increase with ${\theta _s}$ and wind speed. Higher errors are also seen for very low ${\theta _s}$ and low wind speed. Due to optical perturbations from the viewing platform, ${\theta _{\textit{vx}}}$ was constrained to be ${ \gt} 20^ \circ$, which reduces the sensitivity of $D$ to wind speed (while its sensitivity to ${\theta _{\textit{vx}}} \lt 20^ \circ$ is relatively high, it is relatively low for ${20^ \circ} \lt {\theta _{\textit{vx}}}$ due to the limited size of the sunglitter). Overall, if the AOT is largely overestimated, the errors on $W$ from $D$ will range from 0 to $2\;{\rm m}\;{{\rm s}^{- 1}}$ for $1\lt {\theta _s} \le {60^ \circ}$ and up to $7.5\;{\rm m}\;{{\rm s}^{- 1}}$ for higher ${\theta _s}$.

 

Fig. 10. Comparison between the true wind speed, ${{W}^t}$, simulated with ${\rm AOT} = {0.05}$, and the erroneous wind speed, ${{W}^e}$, estimated with ${{D}_{0.05}}$ but a ${W} - {D}$ relation for ${\rm AOT} = {0.2}$ for the different ${\theta _s}$ (see legend for marker type) and RH (see legend for color, note 80% and 90% overlap) (left); barplot with mean, 25th, and 75th percentiles for the difference between ${{W}^t}-{{W}^t}$ (in ${\rm m}\;{{\rm s}^{- 1}}$) as a function of ${\theta _s}$.

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Fig. 11. Absolute percentage difference between $\rho _w^e(\lambda ,{\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ})$ (for a wind speed retrieved with ${ D} - { W}$ relation for ${\rm AOT} = {0.2}$) and $\rho _w^t(\lambda ,{\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ})$ (for a wind speed retrieved with ${D} - {W}$ relation for ${\rm AOT} = {0.05}$) as a function of $\rho _u^t(\lambda ,{\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ})$. Simulations are performed for a coastal aerosol model at 400, 600, and 900 nm, and for different RH (80 and 98%) and ${\theta _s}$.

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As stated in Section 2.B, the difference $\rho _w^e({\theta _v}={ 40^ \circ},\def\LDeqbreak{}\Delta \phi ={ 0^ \circ}) - \rho _w^t({\theta _v}={ 40^ \circ},\Delta \phi ={ 0^ \circ})$ (i.e., with an erroneous retrieval of wind speed, ${{W}^e}$, and with the correct retrieval of wind speed, ${{W}^t}$) is equal to the difference in the upwelling reflectance $\rho _u^e({\theta _v}={ 40^ \circ},\Delta \phi ={ 0^ \circ}) - \rho _u^t({\theta _v}={ 40^ \circ},\Delta \phi ={ 0^ \circ})$. Hence, comparing the upwelling reflectance retrieved with the correct AOT and with erroneous AOT allows us to estimate the impact of inaccuracies in the method on the water reflectance. Figure 11 shows the difference between $\rho _w^e$ and $\rho _w^t$ as a function of $\rho _u^t$ for different wavelengths and RH and a coastal aerosol model. The largest errors are observed when the Sun in close to zenith (${\theta _s}={ 1^ \circ}$) and for high ${\theta _s}$ (in particular for ${\theta _s}={ 80^ \circ}$). For ${\theta _s}$ between 10 and 60°, absolute errors are relatively constant with averages ranging from ${2.10^{- 4}}$ to ${1.10^5}$ for 400, 600, and 900 nm.

4. CONCLUSION

A. Summary of Findings

The standard protocol [7,10] for above-water measurement of water-leaving radiance, based on a single measurement of upwelling (water) radiance and downwelling (sky) radiance, uses extraneous wind speed input to estimate the amount of light reflected at the air–water interface, and the Cox–Munk model of wave slope statistics. This can lead to very large errors (i.e., as shown in Fig. 2, up to 75% error in ${\rho _w}$ in the blue spectral range and 50% in the NIR spectral range over clear waters and when the wind speed is overestimated by only $2\;{\rm m}\;{{\rm s}^{- 1}}$), in cases where the wind speed used, e.g., from a meteorological model or from nearby measurements, is not representative of the local wind field. In addition to the possibility of errors or spatiotemporal variability of the wind speed input itself, there are situations, such as fetch-limited coastal or inland waters, in which the Cox–Munk model does not represent well the local wave slope statistics. To overcome these problems, it is proposed here to measure optically the local wave slope statistics (using effective wind speed as a proxy variable) by measurements of upwelling radiance made in the principal plane toward the Sun. The angular variation of upwelling “sunglint” radiance in both zenith and azimuthal directions, i.e., the angular width of the “sunglitter,” is strongly related to the effective wind speed. Since the azimuthal width of the sunglitter becomes very wide for low Sun zenith angle and very narrow for high Sun zenith angle, the present study has chosen to focus on zenithal variability, where the angular width of the hotspot is closely related to the characteristic wave slope angle at least for low and moderate Sun zenith angles for which the incidence angle variability of Fresnel reflectance is low.

Vector radiative transfer simulations have been made with the OSOAA code [28], including molecular and aerosol atmospheric scattering and absorption, reflection, and transmission of light at the air–water interface (represented by wave slope statistics as function of effective wind speed), and scattering and absorption from water molecules, phytoplankton, and non-algae particles. These simulations cover a wide range of Sun zenith angle (0° to 80°) and wind speed (0 to $14\;{\rm m}\;{{\rm s}^{- 1}}$) and a variety of aerosol properties (three particle types, and 2 and 3 values for RH and optical thickness, respectively). Results have been analyzed at 900 nm since this is the longest wavelength present on typical aquatic radiometers. The method could theoretically be used at any wavelength, and it might be interesting for quality control purposes to estimate wind speed at multiple wavelengths; however, the longest wavelength is expected to give best results since this will minimize reflected diffuse light compared to the reflected direct sunlight and will also minimize any water-leaving radiance even in turbid waters.

First the zenith angle variation of upwelling reflectance is presented for all Sun zenith angles, showing a clearly defined maximum upwelling reflectance for low and moderate wind speeds and strong sensitivity to wind speed for all Sun zenith angles. An approach to estimate wind speed is defined by calculating the angular difference of upwelling reflectance between two viewing zenith angles with one near the “mirror” Sun zenith angle and the second at ${-}/ {+} {10^ \circ}$, with ${+}{10^ \circ}$ chosen for low Sun zenith angles to avoid viewing the water too close to nadir and, hence, minimize optical perturbations from the viewing platform. This angular variability in upwelling reflectance, ${\rho _u}$, is shown to be strongly and monotonically related to effective wind speed with desirable high sensitivity to wind particularly at low to moderate wind speed (${\lt}8\;{\rm m}\;{{\rm s}^{- 1}}$) for all Sun zenith angles. Results show that this angular derivative is only weakly dependent on aerosol properties and that use of a default choice of aerosol particle type, RH, and AOT does not cause large errors in wind speed estimation except for cases of (1) Sun zenith angles close to nadir, (2) wind speed close to $0\;{\rm m}\;{{\rm s}^{- 1}}$, and (3) both high Sun zenith angle cases ($ \gt = 70^ \circ$) and high wind speed (${\gt}8\;{\rm m}\;{{\rm s}^{- 1}}$). For those cases, a proper estimation of AOT using, for instance, downwelling irradiance measurements [32] or possibly the glint reflectance at several wavelengths [33] may greatly improve the wind speed retrieval via $D$. The approach of using the angular difference of upwelling reflectance is also shown to be relatively robust to typical constant zenith and azimuth pointing errors of up to 2°.

B. Limitations

The limitations of the approach are summarized here, first referring to the results obtained within the scope and design constraints of the algorithm presented in Section 3 and then referring to factors that are considered out of scope in that section. On the basis of results presented in Section 3 and discussed in Section 4.A, the main limitation of the method is expected to be for high Sun zenith angle (${ \gt} 70^ \circ$) combined with high wind conditions (${\gt}8\;{\rm m}\;{{\rm s}^{- 1}}$), where both angular variation of Fresnel reflectance and wind speed affect the sunglitter. For those cases the variation in upwelling reflectance ${\rho _u}$ with wind speed significantly decreases (see Fig. 3, ${\theta _s}={ 80^ \circ}$). In addition, at high wind speeds and high ${\theta _v}$ or very high ${\theta _s}$, simulated ${\rho _u}$ may be biased as shadowing effects by wave edges and multiple reflections of photons between wave facets are ignored in OSOAA.

One extra limitation may be for platforms in which it is not possible to make good measurements for low viewing zenith angle (e.g., $ {\lt} 30^ \circ$), as required in this method for the lowest Sun zenith angles. In such cases, it is possible to adapt the method to choose higher viewing angles to reduce optical perturbations, although this may limit accuracy of the method for low Sun zenith angles.

Azimuth-pointing errors could be significantly reduced by performing some measurements at small azimuth angles around the maximum upwelling reflectance to locate more accurately the Sun azimuth direction (i.e., the upwelling light field is expected to be symmetric in azimuth around the Sun azimuth direction but has no such symmetry in the principal plane). Azimuth-pointing errors may become most critical for high ${\theta _s}$, where the sunglint is narrow in azimuthal width.

When applying the method to real measurements, additional geometries in the solar principal plane could further improve the retrieval of $D$ (i.e., better fitting between measurements and simulations) and/or could be used as a quality check.

While the present study represents the surface wave slope statistics via a single degree of freedom, which is called here the effective wind speed, there is no need for this to correspond to the actual aerodynamic wind speed since it is only the optical impact of the wind speed, via the wave slope statistics, that will be subsequently used to correct for reflected sky radiance in the standard above-water radiometry geometry. In fact, this optical measurement of the impact of wave slope statistics in the principal plane is very closely related to the wave slope statistics in the relative azimuth directions of 90° or 135°, where it will subsequently be used. There may be some uncertainties in using this sunglint-derived wind speed in cases of highly anisotropic capillary wave fields. The method could be refined to take account of wind direction with respect to Sun; prior studies of reflected diffuse and direct light (e.g., [29]), including the Cox–Munk study itself, suggest that the optically important capillary wave fields have low variability with wind direction. Note, however, that the LUT could be further refined by adding some results with simulations including anisotropic wave fields.

The case of a highly directional wave field dominated by swell waves without associated capillary waves is not considered here and may occasionally present difficulties for the current approach of $W$ estimation (but is already problematic for the standard protocol without the current approach).

The present theoretical study does assume a small FOV measurement. Commercially available aquatic radiometers generally have FOV between 2° and 10° (FWHM). While a FOV of 2° should resolve sufficiently the angular variability, the present approach may need to be refined with an appropriate factor applied to $D$ for the larger FOV instruments. A sensor, pointing away from the nadir and with a larger FOV, e.g., 7°, will average ${\rho _u}$ over a footprint with an elliptical shape and a major-axis along the nadir viewing angles pointing toward the horizon. The larger ${\theta _s}$ (and subsequent ${\theta _{v1}}$ and ${\theta _{v2}}$), the longer the major axis is. In addition, if ${\rho _u}$ is measured close to a (maximum) turning point (i.e., at low wind speed), a larger FOV will affect the accuracy of the method (see Fig. 8). Hence, a larger FOV will lower the accuracy of the method at high ${\theta _s}$ and/or very low wind speed, and the LUT should be refined accordingly.

The present study also assumes that the upwelling radiance is averaged over time to remove any fast fluctuations from the surface wave field—this assumption is inherent to any modeling that uses wave slope statistics such as those of Cox–Munk. The exploitation of temporal fluctuations, such as sunglint flashes, could actually enhance the estimation of wind speed from upwelling reflectance measurements in the principal plane. However, this would require a fast-sampling instrument (typically not possible for a hyperspectral radiometer) and would require very careful consideration of any temporal filtering applied to data, such as spike removal, as well as of the spatial resolution/integration of the instrument.

Further uncertainties may arise because of the various radiometer instrument artifacts [34]. Since the approach is based on upwelling reflectance rather than radiance, certain instrument calibration uncertainties, e.g., relating to calibration lamp degradation and/or thermal sensitivity, are significantly reduced. Probably the most problematic instrument artifact would be imperfect cosine response of the irradiance sensor for high ${\theta _s}$ since this would affect proportionally $D$. Instrument polarization sensitivity has not been considered here but is generally small, e.g., 1% for TRIOS/RAMSES instruments [35], and so it is not expected to be a problem here for effective wind speed estimation.

The present study also assumes a clear sky. While the impact of a hazy atmosphere has been represented by simulations with higher AOT, the presence of clouds, particularly in the vicinity of the Sun, would dramatically affect results. However, such problems would probably be obvious from parallel measurements such as downwelling irradiance and/or hemispherical sky photos. Some prior filtering of clear sky conditions would be needed for operational application of the method.

Finally, a threshold needs to be defined for the time-lapse between the measurements of ${\rho _u}$ within the solar principal plane and the standard above-water radiometry acquisitions to ensure that the wave slope statistics (and at target wind speed) derived from ${L_u}({\theta _{\textit{vx}}},\Delta \phi = 0)$ are similar to the effective wave slope statics for ${L_u}({\theta _v}={ 40^ \circ},\Delta \phi ={ 90^ \circ}{,135^ \circ})$.

C. Future Perspectives

Clearly the approach should now be tested practically with real measurements at sea under different Sun zenith, sky (aerosol), wave, and water conditions. Automated pointable radiometers such as the CIMEL/SeaPRISM [4], PANTHYR/TRIOS [6], and HYPSTAR systems are ideally suited for testing and implementation of the method. For most platforms used to mount these radiometer systems, the principal plane toward Sun is usually unobstructed and possible for measurement when the standard viewing geometry of 90° or 135° azimuth relative to the Sun is usable. Viewing in the principal plane away from Sun is almost always not possible because this would involve viewing the supporting structure in all but the most slender of towers—this is clearly avoided in the current method. As mentioned above, the method is also sensitive to the the spatial resolution/integration of the instrument. Additional research will be required when applying the method to real measurements. According to the design of the radiometer, the best number of scans for averaging ${L_u}$ at ${\theta _{v1}}$ and ${\theta _{v2}}$ should be defined, and a compromise needs to be found between integration time and saturation.

Of the three problems mentioned in Section 1.A, the present study deals with only the problem of effective wind speed. The other problems are amenable to improvements from additional water and sky measurements but will be dealt with in future studies.