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

doi:10.1364/OE.19.017528

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

Meng Shen, Zao Xu, and Dick K. P. Yue »View Author Affiliations

Optics Express, Vol. 19, Issue 18, pp. 17528-17538 (2011)
http://dx.doi.org/10.1364/OE.19.017528

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▸ Article Outline
– Abstract
1. Introduction
2. Derivation of Gaussian-Poisson (GP) statistical model for downwelling irradiance PDF
3. Validation and results
4. Conclusion
– References and links

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Abstract

We present a statistical model that analytically quantifies the probability density function (PDF) of the downwelling light irradiance under random ocean waves modeling the surface as independent and identically distributed flat facets. The model can incorporate the separate effects of surface short waves and volume light scattering. The theoretical model captures the characteristics of the PDF, from skewed to near-Gaussian shape as the depth increases from shallow to deep water. The model obtains a closed-form asymptotic for the probability that diminishes at a rate between exponential and Gaussian with increasing extreme values. The model is validated by comparisons with existing field measurements and Monte Carlo simulation.

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

W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish 8, 173–184 (1983). [CrossRef]

–3

J. R. V. Zaneveld and P. A. Hwang, “The influence of coherent waves on the remotely sensed reflectance,” Opt. Express 9(6), 260–266 (2001). [CrossRef] [PubMed]

]. Experimental study of downwelling light irradiance induced by the random ocean waves dates to the 1970’s [4

R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B 60(8), 1072–1083 (1970). [CrossRef]

–10

P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res. 114(15), C06025 (2009). [CrossRef]

]. 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

Y. You, D. Stramski, M. Darecki, and G. W. Kattawar, “Modeling of wave-induced irradiance fluctuations at near-surface depths in the ocean: a comparison with measurements,” Appl. Opt. 49(6), 1041–1053 (2010). [CrossRef] [PubMed]

, 12

P. Gernez, D. Stramski, and M. Darecki, “Vertical changes in the probability distribution of downwelling irradiance within the near-surface ocean under clear sky conditions,” presented at Ocean Optics XX, Anchorage, Alaska, 27 September 2010.

], 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

C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic Press, 1994).

–18

P. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. II. the hybrid matrix operator-Monte Carlo method,” Appl. Opt. 47, 1063–1071 (2008). [CrossRef] [PubMed]

] 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

R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B 60(8), 1072–1083 (1970). [CrossRef]

] 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

V. L. Veber, “On the spatial fluctuations of underwater illumination,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1982).

] 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

V. I. Shevernev, “Statistical structure of the illumination field under a wavy surface,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1973).

] 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

B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 1983). [CrossRef]

] 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.

For a Gaussian random η, following Friden [21

B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 1983). [CrossRef]

], 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

C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B 44(11), 838–850 (1954). [CrossRef]

]. 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

C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic Press, 1994).

]. 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})],

(1)

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})],

(2)

α_{i} = \bar{α} = 1 + O (ɛ^{2}) .

(3)

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})] .

(4)

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

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

(5)

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} .

(6)

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}),

(7)

then this process satisfies the conditions of a non-homogeneous Poisson process [23

R. G. Gallager, Discrete Stochastic Processes (Springer, 1996).

] 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)},

(8)

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},

(9)

where σ_ξ is the standard deviation of p_{$\bar{ξ}$
_i} (ξ_i ). For Gaussian ocean surface [22

C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B 44(11), 838–850 (1954). [CrossRef]

] 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},

(10)

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

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

(11)

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)} .

(12)

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. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).

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

(13)

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}} .

(14)

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 .

(15)

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 .

(16)

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},

(17)

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},

(18)

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

(19)

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}],

(20)

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

(21)

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,

(22)

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

R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).

]. In this case, for small angle scattering, the Fourier transform of ĥ(ξ′) can be obtained [24

R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).

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

(23)

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} .

(24)

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'),

(25)

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}} .

(26)

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 .

(27)

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)},

(28)

which has a variance

σ_{χ}^{2} = \frac{1}{α} .

(29)

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}})] .

(30)

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 .

(31)

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} .

(32)

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}} .

(33)

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

Z. Xu, “A DNS capability for obtaining underwater light field and retrieving upper ocean conditions via in-water light measurements,” thesis (MIT, 2011).

]. 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

R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).

] which is consistent with Cox-Munk measurement [22

C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B 44(11), 838–850 (1954). [CrossRef]

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

J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

]. 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

J. R. V. Zaneveld and P. A. Hwang, “The influence of coherent waves on the remotely sensed reflectance,” Opt. Express 9(6), 260–266 (2001). [CrossRef] [PubMed]

]. 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.

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

R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B 60(8), 1072–1083 (1970). [CrossRef]

, 10

P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res. 114(15), C06025 (2009). [CrossRef]

, 14

J. W. McLean and J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt. 35(18), 3261–3269 (1996). [CrossRef] [PubMed]

, 24

R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).

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.

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

J. R. V. Zaneveld and P. A. Hwang, “The influence of coherent waves on the remotely sensed reflectance,” Opt. Express 9(6), 260–266 (2001). [CrossRef] [PubMed]

]. 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

R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).

] 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

J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

]. 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 → ∞ (···).

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).

References and links

1.	W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish 8, 173–184 (1983). [CrossRef]
2.	W. C. Brown and A. K. Majumdar, “Point-spread function associated with underwater imaging through a wavy air-water interface: theory and laboratory tank experiment,” Appl. Opt. 31(36), 7650–7659 (1992). [CrossRef] [PubMed]
3.	J. R. V. Zaneveld and P. A. Hwang, “The influence of coherent waves on the remotely sensed reflectance,” Opt. Express 9(6), 260–266 (2001). [CrossRef] [PubMed]
4.	R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B 60(8), 1072–1083 (1970). [CrossRef]
5.	H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci 21, 466–470 (1971).
6.	J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).
7.	D. Stramski and L. Legendre, “Laboratory simulation of light focusing by water surface waves,” Mar. Biol 114, 341–348 (1992). [CrossRef]
8.	J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).
9.	H. W. Wijesekera, W. S. Pegau, and T. J. Boyd, “Effect of surface waves on the irradiance distribution in the upper ocean,” Opt. Express 13(23), 9257–9264 (2005). [CrossRef] [PubMed]
10.	P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res. 114(15), C06025 (2009). [CrossRef]
11.	Y. You, D. Stramski, M. Darecki, and G. W. Kattawar, “Modeling of wave-induced irradiance fluctuations at near-surface depths in the ocean: a comparison with measurements,” Appl. Opt. 49(6), 1041–1053 (2010). [CrossRef] [PubMed]
12.	P. Gernez, D. Stramski, and M. Darecki, “Vertical changes in the probability distribution of downwelling irradiance within the near-surface ocean under clear sky conditions,” presented at Ocean Optics XX, Anchorage, Alaska, 27 September 2010.
13.	C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic Press, 1994).
14.	J. W. McLean and J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt. 35(18), 3261–3269 (1996). [CrossRef] [PubMed]
15.	J. R. V. Zaneveld and E. Boss, “Influence of surface waves on measured and modeled irradiance profiles,” Appl. Opt. 40(9), 1442–1449 (2001). [CrossRef]
16.	J. Hedley, “A three-dimensional radiative transfer model for shallow water environments,” Opt. Express 16(26), 21887–21902 (2008). [CrossRef] [PubMed]
17.	P. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037–1047 (2008). [CrossRef] [PubMed]
18.	P. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. II. the hybrid matrix operator-Monte Carlo method,” Appl. Opt. 47, 1063–1071 (2008). [CrossRef] [PubMed]
19.	V. L. Veber, “On the spatial fluctuations of underwater illumination,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1982).
20.	V. I. Shevernev, “Statistical structure of the illumination field under a wavy surface,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1973).
21.	B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 1983). [CrossRef]
22.	C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B 44(11), 838–850 (1954). [CrossRef]
23.	R. G. Gallager, Discrete Stochastic Processes (Springer, 1996).
24.	R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).
25.	Z. Xu, “A DNS capability for obtaining underwater light field and retrieving upper ocean conditions via in-water light measurements,” thesis (MIT, 2011).

OCIS Codes
(000.5490) General : Probability theory, stochastic processes, and statistics
(010.4450) Atmospheric and oceanic optics : Oceanic optics
(010.5620) Atmospheric and oceanic optics : Radiative transfer

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: June 29, 2011
Revised Manuscript: August 10, 2011
Manuscript Accepted: August 12, 2011
Published: August 22, 2011

Virtual Issues
Vol. 6, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Meng Shen, Zao Xu, and Dick K. P. Yue, "A model for the probability density function of downwelling irradiance under ocean waves," Opt. Express 19, 17528-17538 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17528

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References

W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish8, 173–184 (1983). [CrossRef]
W. C. Brown and A. K. Majumdar, “Point-spread function associated with underwater imaging through a wavy air-water interface: theory and laboratory tank experiment,” Appl. Opt.31(36), 7650–7659 (1992). [CrossRef] [PubMed]
J. R. V. Zaneveld and P. A. Hwang, “The influence of coherent waves on the remotely sensed reflectance,” Opt. Express9(6), 260–266 (2001). [CrossRef] [PubMed]
R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B60(8), 1072–1083 (1970). [CrossRef]
H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci21, 466–470 (1971).
J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia23, 15–42 (1986).
D. Stramski and L. Legendre, “Laboratory simulation of light focusing by water surface waves,” Mar. Biol114, 341–348 (1992). [CrossRef]
J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia34, 13–25 (1993).
H. W. Wijesekera, W. S. Pegau, and T. J. Boyd, “Effect of surface waves on the irradiance distribution in the upper ocean,” Opt. Express13(23), 9257–9264 (2005). [CrossRef] [PubMed]
P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res.114(15), C06025 (2009). [CrossRef]
Y. You, D. Stramski, M. Darecki, and G. W. Kattawar, “Modeling of wave-induced irradiance fluctuations at near-surface depths in the ocean: a comparison with measurements,” Appl. Opt.49(6), 1041–1053 (2010). [CrossRef] [PubMed]
P. Gernez, D. Stramski, and M. Darecki, “Vertical changes in the probability distribution of downwelling irradiance within the near-surface ocean under clear sky conditions,” presented at Ocean Optics XX, Anchorage, Alaska, 27 September 2010.
C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic Press, 1994).
J. W. McLean and J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt.35(18), 3261–3269 (1996). [CrossRef] [PubMed]
J. R. V. Zaneveld and E. Boss, “Influence of surface waves on measured and modeled irradiance profiles,” Appl. Opt.40(9), 1442–1449 (2001). [CrossRef]
J. Hedley, “A three-dimensional radiative transfer model for shallow water environments,” Opt. Express16(26), 21887–21902 (2008). [CrossRef] [PubMed]
P. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt.47, 1037–1047 (2008). [CrossRef] [PubMed]
P. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. II. the hybrid matrix operator-Monte Carlo method,” Appl. Opt.47, 1063–1071 (2008). [CrossRef] [PubMed]
V. L. Veber, “On the spatial fluctuations of underwater illumination,” Izv. Atmos. Oceanic Phys.18, 735–741 (1982).
V. I. Shevernev, “Statistical structure of the illumination field under a wavy surface,” Izv. Atmos. Oceanic Phys.18, 735–741 (1973).
B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 1983). [CrossRef]
C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B44(11), 838–850 (1954). [CrossRef]
R. G. Gallager, Discrete Stochastic Processes (Springer, 1996).
R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).
Z. Xu, “A DNS capability for obtaining underwater light field and retrieving upper ocean conditions via in-water light measurements,” thesis (MIT, 2011).

Other

W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish8, 173–184 (1983). [CrossRef]
W. C. Brown and A. K. Majumdar, “Point-spread function associated with underwater imaging through a wavy air-water interface: theory and laboratory tank experiment,” Appl. Opt.31(36), 7650–7659 (1992). [CrossRef] [PubMed]
J. R. V. Zaneveld and P. A. Hwang, “The influence of coherent waves on the remotely sensed reflectance,” Opt. Express9(6), 260–266 (2001). [CrossRef] [PubMed]
R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B60(8), 1072–1083 (1970). [CrossRef]
H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci21, 466–470 (1971).
J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia23, 15–42 (1986).
D. Stramski and L. Legendre, “Laboratory simulation of light focusing by water surface waves,” Mar. Biol114, 341–348 (1992). [CrossRef]
J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia34, 13–25 (1993).
H. W. Wijesekera, W. S. Pegau, and T. J. Boyd, “Effect of surface waves on the irradiance distribution in the upper ocean,” Opt. Express13(23), 9257–9264 (2005). [CrossRef] [PubMed]
P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res.114(15), C06025 (2009). [CrossRef]
Y. You, D. Stramski, M. Darecki, and G. W. Kattawar, “Modeling of wave-induced irradiance fluctuations at near-surface depths in the ocean: a comparison with measurements,” Appl. Opt.49(6), 1041–1053 (2010). [CrossRef] [PubMed]
P. Gernez, D. Stramski, and M. Darecki, “Vertical changes in the probability distribution of downwelling irradiance within the near-surface ocean under clear sky conditions,” presented at Ocean Optics XX, Anchorage, Alaska, 27 September 2010.
C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic Press, 1994).
J. W. McLean and J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt.35(18), 3261–3269 (1996). [CrossRef] [PubMed]
J. R. V. Zaneveld and E. Boss, “Influence of surface waves on measured and modeled irradiance profiles,” Appl. Opt.40(9), 1442–1449 (2001). [CrossRef]
J. Hedley, “A three-dimensional radiative transfer model for shallow water environments,” Opt. Express16(26), 21887–21902 (2008). [CrossRef] [PubMed]
P. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt.47, 1037–1047 (2008). [CrossRef] [PubMed]
P. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. II. the hybrid matrix operator-Monte Carlo method,” Appl. Opt.47, 1063–1071 (2008). [CrossRef] [PubMed]
V. L. Veber, “On the spatial fluctuations of underwater illumination,” Izv. Atmos. Oceanic Phys.18, 735–741 (1982).
V. I. Shevernev, “Statistical structure of the illumination field under a wavy surface,” Izv. Atmos. Oceanic Phys.18, 735–741 (1973).
B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 1983). [CrossRef]
C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B44(11), 838–850 (1954). [CrossRef]
R. G. Gallager, Discrete Stochastic Processes (Springer, 1996).
R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).
Z. Xu, “A DNS capability for obtaining underwater light field and retrieving upper ocean conditions via in-water light measurements,” thesis (MIT, 2011).

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

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Abstract

1. Introduction

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

2.1.2. Deterministic relations

2.1.3. Statistical model

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

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

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

2.5. Summary of GP model results

3. Validation and results

3.1. GP Model validation by MC simulations and experiments

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

4. Conclusion

Acknowledgments

References and links

References

Other

Cited By

Figures

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