A model for the probability density function of downwelling irradiance under ocean waves

Meng Shen; Zao Xu; Dick K. P. Yue

doi:10.1364/OE.19.017528

1. Introduction

Understanding fluctuations of the underwater light field is important for many ocean related sciences, such as marine animal visual ecology, underwater imaging and ocean surface reflectance remote sensing [1–3]. Experimental study of downwelling light irradiance induced by the random ocean waves dates to the 1970’s [4–10]. Since irradiance signal fluctuates rapidly with both time O(ms) and space O(mm), measuring accurate irradiance signal is challenging. Specific irradiance instruments to measure these short temporal variations have been developed [11, 12], however, obtaining the distribution of extreme values is still difficult since it requires stable and long-duration measurements under varying environmental conditions such as wind, waves and inherent optical properties (IOPs) of the water. Direct numerical simulation of the underlying radiative transfer equations [11, 13–18] and hence the statistical behavior provides an alternative, but the challenges of using relatively expensive direct simulations to quantify extreme statistics remain. Theoretical predictions that might provide guidance and cross-validation of measurements and computations are much needed.

There are few theoretical results that quantify the temporal and spatial variability of underwater irradiance. Snyder and Dera [4] develop a statistic model that obtains the downwelling irradiance frequency spectral density as a linear combination of the spectra of the ocean surface elevation, slope and curvature. Veber [19] later introduces a small-angle-approximation to account for water turbidity. The limitation of all such linear theories is that the downwelling irradiance probability density function (PDF) for Guassian surface elevation (or slope) is necessarily still Gaussian, a result which turns out to obtain in experiments only in deep water. Shevernev [20] formulates a nonlinear model for the correlation and spectral density of downwelling irradiance with a truncated series expansion. However, none of them provides the explicit PDF for the downwelling irradiance.

The present analysis is motivated by Friden [21] who study the underwater irradiance by assuming that the ocean surface is composed of independent and identically distributed (IID) facets. After further assuming that the detected irradiance is a superposition of light from many facets, he claims that the downwelling irradiance conforms to a normal distribution according to central limit theorem. Friden does not provide explicit condition(s) required for the validity of the IID assumption. Furthermore, the central limit theorem may not be satisfied except in deep water.

In present paper, we start with the idea of IID facet surface (Section 2), but instead of using the central limit theorem, we show that the underwater downwelling irradiance is described by a non-homogeneous Poisson process, under a suitable condition depending on the surface slope spectrum. Under this condition, we obtain a closed-form analytic PDF of the downwelling irradiance. The effects of short surface waves (contributing to sub-facet variations), and that of volume light scattering can be included in the analytical PDF in a straightforward manner in terms of the light beam spreading associated with these effects. The explicit PDF provides an immediate asymptotic result for the probability of extreme values, as well as that of a critical depth at which surface and volume light scattering effects are in balance (and below which the latter dominates). The theoretical model is validated against existing ocean measurements and direct Monte Carlo simulations with excellent quantitative comparisons (Section 3.1). To illustrate the usefulness of the present model, we conclude with a practical problem of quantifying the effect of surface wind speed on the underlying irradiance distribution (Section 3.2)

2. Derivation of Gaussian-Poisson (GP) statistical model for downwelling irradiance PDF

2.1. Derivation of Gaussian-Poisson (GP) for flat facet surface

2.1.1. Problem description

We consider the solar light irradiance E ₀ incident normally through an one-dimensional ocean surface (Fig. 1). We choose a Cartesian coordinates x–z with z positive upwards and z=0 corresponding to the mean free surface given by z=η (x,t). Our interest is the probability distribution of the underwater light irradiance E at some receiver depth z=−D.

Fig. 1 Geometry of the method: faceted ocean surface.

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For a Gaussian random η, following Friden [21], we consider a model where the ocean surface (at any instant) is represented by a large number of independent and identically distributed (IID) flat facets. For simplicity, we assume the facet size L to be constant. Let x_i = iL, s_i be respectively the facet center position and facet slope of facet i. Consistent with a linear Gaussian surface, we assume s_i = O(ɛ) ≪ 1 is a random variable [22]. Let λ_i be the light beam width from facet i at the receiving plane and define the light beam spreading coefficient to be α_i ≡ λ_i/L.

2.1.2. Deterministic relations

At any one instant/realization, a solar light beam incident on the i^th facet is refracted according to Snell’s law [13]. At the receiver plane, the center of the beam is located at (x,z)=(ξ_i, −D), and we define ξ′_i ≡ ξ_i – x_i (Fig.1). For s_i ≪ 1, it follows that

\frac{ξ_{i} - x_{i}}{D} = \frac{m - 1}{m} s_{i} [1 + O (ɛ^{2})],

where m is the refractive index of water. Consistent with small facet slopes, we further assume unit transmission coefficient of the air-sea interface.

For flat facet case, the light beam is parallel to each other, and to leading order, we have,

λ_{i} = \bar{λ} = L [1 + O (ɛ^{2})],

α_{i} = \bar{α} = 1 + O (ɛ^{2}) .

2.1.3. Statistical model

We assume that the receiver width D_d is small relative to facet size L, and that the detected irradiance E is a collection of contributions from all possible facets in stochastic superposition process. Let N_r be the number of contributing facets to the receiver in a given realization. Since the facet slopes s_i are random variables, therefore N_r is also a random variable. Using Eq. (3) E at the receiver can be written as,

E = \sum_{i = 1}^{N_{r}} E_{0} \bar{α} = N_{r} E_{0} [1 + O (ɛ^{2})] .

We define the normalized downwelling irradiance as χ ≡ E/〈E〉, so that to leader order,

χ = \frac{E}{〈 E 〉} = \frac{N_{r}}{〈 N_{r} 〉},

where 〈·〉 represents averaging over all realizations/instants.

Our interest is the probability density function (PDF) p_χ_̄(χ) (hereafter, we represent the PDF of any random variable f by p_f̄(f)). To obtain p_χ_̄(χ), we start from p_{$\bar{ξ}$ _i}(ξ_i). From Eq. (1), this can be derived from the PDF of the ocean surface slope p_{s̄_i} (s_i). The arrival probability of the i ^th facet can then be expressed as the probability of ξ_i ∈ [− $\bar{λ}$ /2, $\bar{λ}$ /2],

P_{{\bar{ξ}}_{i}} (- \frac{\bar{λ}}{2} \leq ξ_{i} \leq \frac{\bar{λ}}{2}) = \int_{- \frac{\bar{λ}}{2}}^{\frac{\bar{λ}}{2}} p_{{\bar{ξ}}_{i}} (ξ_{i}) d ξ_{i} .

We now consider the arrival probability from all the facets in the surface. For independent and identically distributed facets, P_{$\bar{ξ}$ _i} is a function of x_i. Each facet i can be considered as an independent Bernoulli trial with different success probability P_{$\bar{ξ}$ _i}. If in addition we can assume that 〈N_r〉 is large, i.e.,

〈 N_{r} 〉 ≫ P_{{\bar{ξ}}_{i}} (- \frac{\bar{λ}}{2} \leq ξ_{i} \leq \frac{\bar{λ}}{2}),

then this process satisfies the conditions of a non-homogeneous Poisson process [23] involving the sum of large numbers of Bernoulli trials with small individual successful probabilities. Under these conditions, the PDF of N_r can be obtained as,

p_{{\bar{N}}_{r}} (N_{r}) = exp (- 〈 N_{r} 〉) \frac{{〈 N_{r} 〉}^{N_{r}}}{Γ (N_{r} + 1)},

which has variance 1/ $\bar{α}$ , where the gamma function Γ is used to approximate the continuous PDF.

To understand the condition (Eq. (7)), we use Eq. (6) to obtain

〈 N_{r} 〉 = \sum_{i = - \infty}^{\infty} P_{{\bar{ξ}}_{i}} (- \frac{\bar{λ}}{2} \leq ξ_{i} \leq \frac{\bar{λ}}{2}) = 1 + O {(\frac{L}{σ_{ξ}})}^{2},

where σ_ξ is the standard deviation of p_{$\bar{ξ}$ _i}(ξ_i). For Gaussian ocean surface [22] with surface slope standard deviation σ_s, σ_ξ can be expressed in terms of σ_s using Eq. (1),

σ_{ξ} = \frac{(m - 1) D}{m} σ_{s},

and condition (Eq. (7)) can be expressed as,

D^{*} \equiv \frac{m - 1}{m} \frac{D}{L} σ_{s} ≫ 1 .

It can be shown that L/σ_ξ ≪ 1 is automatically satisfied given condition Eq. (11) valid. Finally, in terms of the normalized irradiance, we can use Eq. (9) and Eq. (5) in Eq. (8).

p_{\bar{χ}} (χ) = \frac{exp (- 1)}{Γ (χ + 1)} .

2.2. Relationship between the facet size L in the GP model and the slope correlation length l_s of the ocean surface

The facet size L in the GP model must be chosen so that s_i satisfies the IID condition. In addition, the GP requires a sufficiently large number of contributing facets for given D. To obtain an estimate of the required L given an ocean surface, we obtain the slope correlation curve R(ζ) of the surface in terms of its moments [24],

R (ζ) = exp [- \frac{ζ^{2}}{2 {(σ_{2} / σ_{4})}^{2}}],

where, ζ is the spacial lag for the correlation function; σ ₂, σ ₄ are the ocean surface slope and curvature standard deviations respectively. We now define the slope correlation length l_s of the surface as twice the standard deviation of R(ζ) (Eq. (13)):

l_{s} = 2 \frac{σ_{2}}{σ_{4}} .

In order to satisfy IID, we require L ≥ l_s; and to maximize 〈N_r〉, we simply choose L=l_s. In terms of l_s and only physical quantities, the condition Eq. (11) for GP validity then becomes

D^{*} = \frac{m - 1}{m} \frac{D}{l_{s}} σ_{s} ≫ 1 .

2.3. Including the effect of sub-facet slope variation in the GP model

For simplicity, the facets in the original GP model are assumed to be flat. In the physical problem (of a continuously varying surface), a light beam incident over a facet is diffused by sub-facet slope variations so that the beam corresponding to a given facet is no longer parallel but spread out. This light beam spreading effect can be accounted for in the GP model in an elegant way.

In terms of the surface wave number spectrum S_η (k), the sub-facet slope variance is the components which are above the Nyquist wave number k ^* = π/l_s,

σ_{s S}^{2} = \int_{k^{*}}^{\infty} S_{η} (k) k^{2} d k .

Let h̃_i(ξ′_i) be the (time) averaged irradiance beam spreading function at z=−D for facet i for this case. Assuming that the sub-facet variations are due to σ_sS, h̃_i(ξ′_i) can be shown to be a Gaussian form. We set the half width of the beam spreading function $\tilde{ℓ}$ _i to be the standard deviation of h̃_i(ξ′_i), which can be obtained (after using Eq. (1)) as:

\tilde{ℓ} = D \frac{(m - 1)}{m} σ_{s S},

where to leading order for IID, Eq. (17) is independent of i. Physically, the original GP can be modified to account for the sub-facet spreading by an equivalent beam width and beam spreading coefficient:

λ_{i} = \tilde{λ} = 2 \tilde{ℓ} + \bar{λ} = 2 \tilde{ℓ} + l_{s},

α_{i} = \tilde{α} = 2 \tilde{ℓ} / l_{s} + \bar{α} = 2 \tilde{ℓ} / l_{s} + 1 .

We calculate the arrival probability P̃_{$\bar{ξ}$ _i} corresponding to ξ_i ∈ [− $\tilde{λ}$ /2, $\tilde{λ}$ /2] > [− $\bar{λ}$ /2, $\bar{λ}$ /2]. The conditions for non-homogeneous Poisson process still obtains, and we find,

〈 N_{r} 〉 = \sum_{i = - \infty}^{\infty} P_{{\bar{ξ}}_{i}} (- \tilde{λ} / 2 \leq ξ_{i} \leq \tilde{λ} / 2) = \tilde{α} [1 + O {(\frac{l_{s}}{σ_{ξ}})}^{2}],

p_{\bar{χ}} (χ) = exp (- \tilde{α}) \frac{{\tilde{α}}^{{\tilde{α}}_{χ + 1}}}{Γ (\tilde{α} χ + 1)} .

The variance of the normalized irradiance above is

σ_{χ}^{2} = 1 / \tilde{α}

.

The corresponding condition for the validity of Eq. (21) starting with Eq. (7) turns out to have the same expression as Eq. (11) (or Eq. (15)). This is due to the fact that the increase in the sum of the arrival probabilities 〈N_r〉 in this case is proportional to the increase in the individual arrival probability for a facet.

2.4. Including the effect of volume scattering in the GP model

An important factor that may influence the light beam width during its propagation to the detector is the hydrosol light scattering in water column. Let ĥ_i(ξ′_i) be the irradiance beam spreading function at z=−D for facet i for this case that accounts for the volume scattering function (VSF) of the water body. In terms of ĥ_i(ξ′_i),

λ_{i} = \hat{λ} = 2 \hat{ℓ} + l_{s}, α_{i} = \hat{α} = 2 \hat{ℓ} / l_{s} + 1,

where, $\hat{ℓ}$ is some characteristic spreading (half) width due to volume scattering, and again Eq. (22) is independent of i.

The length scale of ĥ(ξ′) varies with the VSF. In this paper, we choose the Henyey-Greenstein (H-G) scattering function [24]. In this case, for small angle scattering, the Fourier transform of ĥ(ξ′) can be obtained [24],

\hat{H} (k, D) = exp {- D c ω_{0} g_{0} [1 - \frac{1 - exp (- D k α_{0}}{D k α_{0}}]},

where c is the attenuation coefficient, g ₀ is the shape coefficient of H-G, ω ₀ is the single scattering albedo, and α ₀ is the characteristic scattering angle of H-G scattering function. Using an approximation for small k, the inverse Fourier transform of Eq. (23) can be obtained. Using the approximation for small ξ′, a Gaussian form is yielded for ĥ(ξ′) with standard deviation given by,

\hat{ℓ} = c g_{0} ω_{0} α_{0} D^{2} .

2.5. Summary of GP model results

In the general ocean wave case, both sub-facet variation and hydrosol scattering are present. The irradiance beam spreading function for the combined effects can be expressed as:

h (ξ') = \tilde{h} (ξ') * \hat{h} (ξ'),

where * denotes convolution operation. The combined beam spread (half) width can be expressed as,

\tilde{\hat{ℓ}} = {({\tilde{ℓ}}^{2} + {\hat{ℓ}}^{2})}^{\frac{1}{2}} .

The light beam width and spreading coefficient in the receiver plane are now (to leading order),

λ = 2 \tilde{\hat{ℓ}} + l_{s}

, and

α = 2 \tilde{\hat{ℓ}} / l_{s} + 1 .

The derivation from Eq. (20) to Eq. (21) follows, and the normalized irradiance PDF for the general case takes the form

p_{\bar{χ}} (χ) = exp (- \frac{1}{σ_{χ}^{2}}) \frac{{(\frac{1}{σ_{χ}^{2}})}^{χ / σ_{χ}^{2} + 1}}{Γ (χ / σ_{χ}^{2} + 1)},

which has a variance

σ_{χ}^{2} = \frac{1}{α} .

The condition for the validity of Eq. (28) is the same as in Eq. (15).

Note that, in the final form Eq. (28), the normalized irradiance variance $σ_{χ}^{2}$ is the only explicit parameter the PDF. Given a surface wave spectrum S_η (k), the long-wave component of the surface is captured through the choice of L in term of l_s, while the sub-facet variance part of the surface is captured through beam spreading (half) width $\tilde{ℓ}$ via Eq. (17). The IOPs of the water body is captured through $\hat{ℓ}$ by Eq. (24). Together $\tilde{ℓ}$ and $\hat{ℓ}$ determine α (Eq. (27)) which specifies $σ_{χ}^{2}$ for the PDF. For the special cases respectively of flat facet with no volume scattering, ocean wave surface with no volume scattering, and flat facet with volume scattering, the general result above reduces to the appropriate form with α= $\bar{α}$ , $\tilde{α}$ , and $\hat{α}$ .

The theoretical GP model provides an asymptotic expression for the probability of occurrence of extreme values of the normalized irradiance at a given depth. Expanding the Gamma function for $χ / σ_{χ}^{2} ≫ 1$ (keeping $σ_{χ}^{2} \sim O (1))$ ,

Γ (χ / σ_{χ}^{2} + 1) = \frac{1}{\sqrt{2 π}} {(χ / σ_{χ}^{2} = 1)}^{χ / σ_{χ}^{2} + \frac{1}{2}} e^{- (χ / σ_{χ}^{2} + 1)} [1 + O (\frac{1}{1 + χ / σ_{χ}^{2}})] .

Finally, Eq. (28) can be approximated as,

p_{\bar{χ}} (χ) \sim exp [- \frac{χ}{σ_{χ}^{2}} ln (\frac{χ}{σ_{χ}^{2}})] for χ / σ_{χ}^{2} ≫ 1 .

It is seen from Eq. (31) that the probability of extreme values of

χ / σ_{χ}^{2}

diminishes asymptotically faster than exponential but slower than Gaussian decay. The behavior of the irradiance PDF for extreme values is a significant result heretofore not obtained by direct measurements or simulations.

For increasing depth D, $\tilde{ℓ}$ ∼ D while $\hat{ℓ}$ ∼ cD ² (Eq. (17) and Eq. (24)), and hence

σ_{χ}^{2} \sim l_{s} [{(D^{2} + {(c D^{2})}^{2}]}^{- 1 / 2} .

Setting $\tilde{ℓ}$ = $\hat{ℓ}$ , we obtain a critical depth D_cr,

c ω_{0} D_{c r} = \frac{m - 1}{m} \frac{σ_{s S}}{g_{0} α_{0}} .

For D > D_cr, Eq. (28) is dominated by hydrosol light scattering over surface short wave diffusion; and vice versa for D < D_cr.

We remark finally that the results above are for the normal solar incidence case. For general (small) solar zenith angle θ_s, the derivations follow in a straightforward way provided that the linearization based on s_i = O(ɛ) ≪ 1 still obtains. For large θ_s, the IID condition for the facets may not strictly apply. In addition, multiple surface refraction-reflection can not be ruled out and the general problem may be intractable theoretically.

3. Validation and results

3.1. GP Model validation by MC simulations and experiments

The proposed theoretical GP model is validated by comparison with data from the 2008 Radiance in a Dynamic Ocean (RaDyO) Santa Barbara Channel experiment reported in [11, 12]. In addition, we perform direct Monte Carlo (MC) simulations of radiative transfer for the coupled atmosphere-ocean system including dynamic ocean surface [25]. The Monte Carlo simulation capability is optimized with state-of-art variance-reducing techniques and is fully parallelized in order to run on high performance computing platform. For the Monte Carlo simulations, we generate the linear Gaussian surface using the Walker directional spectrum [24] which is consistent with Cox-Munk measurement [22].

Figure 2 compares the GP PDF of the normalized downwelling irradiance χ Eq. (28) with that from MC simulations and experimental measurements at different (measured) depths D. Note that, in this case, θ_s = 30° is sufficient small and the irradiance statistics is indistinguishable from those for normal incidence [6]. The overall agreement among the three predictions in Fig. 2 are satisfactory, with notable deviations at larger χ for shallower depth as shown in Fig. 2(a) and Fig. 2(b). At shallower depths, the irradiance PDF is a highly skewed shape, while in deeper water shown in Fig. 2(c) and Fig. 2(d), the PDF approaches a near Gaussian distribution. The deviations for larger χ at shallower depths is more prominent for the experimental data relative to GP or MC. This is likely due to differences between the Walker spectrum used in GP and MC and the actual surface conditions in the experiment where it is known that the short wave characteristics may be easily changed by environmental factors such as wind, swell and nonlinearity [3]. For larger normalized irradiance values, experimental measurements also become increasingly difficult, requiring longer sampling duration over which the environment conditions may change. Finally, we point out that the discrepancy between GP and MC for the very shallow depth cases is due to the condition for the validity of GP model where, for the present conditions, Eq. (15) evaluates to D ^* ∼ 2.7 and 5.4 for D= 0.86m (Fig. 2(a)) and 1.7m (Fig. 2(b)).

Fig. 2 Comparison of the GP theoretical model (—) for the probability density function (PDF) of the normalized downwelling irradiance χ = E/〈E〉 with Monte Carlo (MC) simulation (···) and experimental data (- - -) at different depths below the ocean surface: D= (a) 0.86m; (b) 1.7m; (c) 2.85m; (d) 4.72m. The conditions used correspond to those in the experiment case [11] with inherent optical properties (IOPs) (attenuation coefficient) c=0.6982 m⁻¹, (absorption coefficient) a=0.0886 m⁻¹, and (scattering coefficient) b=0.6096 m⁻¹; wind speed at 10 meters above the ocean surface U ₁₀ ≈5.5 m/s; solar zenith angle θ_s = 30°; and light wavelength 532 nm.

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To compare the results over a range of depth, we plot the standard deviation of the normalized downwelling irradiance σ_χ (Eq. (29)) as function of depth in Fig. 3. The comparison between GP and measurements is excellent. For Dcω ₀ <∼ 0.5, corresponding to D ^* <∼ 2.7, there is a difference between GP and MC which, as explained earlier, marks the limit of validity of the GP model. (The critical depth in this case is D_crcω ₀ ∼ 0.37). The MC result is expected to be correct at these shallow depths, where σ_χ starts from zero value at the surface, reaches a maximum value, and then approaches the GP asymptotically as D increases. This “bump” in the shallow depth σ_χ behavior has been reported in experiments, theoretical model and numerical simulations [4, 10, 14, 24].

Fig. 3 Standard deviation of the downwelling normalized irradiance σ_χ as a function of the scattering depth Dcω ₀. Results of GP model (—) is compared with Monte Carlo (MC) simulations (···) and experimental data (⋄). The physical conditions are the same as in Fig. 2.

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3.2. Application to the effect of the wind driven wave spectrum on the irradiance PDF

Finally, to illustrate the usefulness of the theoretical model, we consider the effect of varying surface wave conditions on the underwater downwelling irradiance distribution. Unsteady wind is a difficulty encountered in experimental measurement of the irradiance distribution where it is known that the underwater light field can be rapidly affected by changes of the wind on the surface [3]. For definiteness, we use the same IOPs as before (c=0.6982 m⁻¹, a=0.0886 m⁻¹, b=0.6096 m⁻¹), normal solar incidence, and consider Walker wave spectra [24] corresponding to varying surface wind speed U ₁₀ in the range of 2∼15 m/s. The desired results obtain easily from the analytical PDF (Eq. (28)). Figure 4(a) shows the dependence of σ_χ with depth for the reference case of U ₁₀=2 m/s (with effectively no surface roughness) for the present condition. Figure 4(b) shows the wind speed dependence of σ_χ at different depths. As wind speed increases, the short-wave component σ_sS increases (while l_s decreases), and subsequently $\tilde{α}$ increases (Eqs. (17) and Eq. (19)), resulting in the monotonic decrease of the normalized irradiance standard deviation σ_χ with wind speed, a feature also observed in experiments [6]. At any given depth, the effect of further increase in U ₁₀ (beyond ∼8m/s in this case) diminishes rapidly reflecting the characteristic of the Walker spectrum. For increasing depth, σ_χ (U ₁₀) (normalized by its reference value for U ₁₀= 2m/s) rapidly approaches the deep water limit as the downwelling irradiance is dominated by volume scattering effect.

Fig. 4 Wind speed effect on σ_χ. (a) Depth dependence of σ_χ for U ₁₀=2m/s. (b) Dependence on wind speed for different depths: Dcω ₀= 0.524 (—); 1.04 (- - -); 1.74 (- · - ·); and Dcω ₀ → ∞ (···).

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4. Conclusion

Under general conditions of linear Gaussian ocean surface, we obtain the theoretical probability distribution function (Eq. (28)) for the downwelling light irradiance under water. The effects of short waves in the surface and of volume light scattering in the water are incorporated in the model in a straightforward way. It is shown that the probability of extreme values of the irradiance diminishes faster than exponential but slower than Gaussian decay (Eq. (31)). The only condition for the validity of the result is that the water depth must be sufficiently deep relative to the slope correlation length of the surface (Eq. (13)), which obtains under all but very shallow depths (for example, the theory is valid for depth greater than ∼1m for steady wind speed U ₁₀ >5m/s).

The present model is validated against existing experimental measurements and direct Monte Carlo numerical simulations obtaining quantitatively satisfactory comparisons. The theoretical model provides a powerful tool for understanding the distribution of the underwater irradiance under different environmental conditions such as surface wind speed; and is useful for the inverse problem wherein ocean surface wave conditions and/or inherent optical properties of the water column might be estimated from underwater downwelling irradiance measurements.

Acknowledgments

This study was supported by the Office of Naval Research under the Radiance in a Dynamic Ocean (RaDyO) project. Computing resources were provided by the Department of Defence High-performance Computing Modernization Program (HPCMP).

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A model for the probability density function of downwelling irradiance under ocean waves

Abstract

1. Introduction