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

Optics Express, Vol. 19, Issue 18, pp. 17528-17538 (2011)

http://dx.doi.org/10.1364/OE.19.017528

Acrobat PDF (732 KB)

### 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.

© 2011 OSA

## 1. Introduction

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]

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

## 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

*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*.

*η*, following Friden [21

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

*L*to be constant. Let

*x*=

_{i}*iL*,

*s*be respectively the facet center position and facet slope of facet

_{i}*i*. Consistent with a linear Gaussian surface, we assume

*s*=

_{i}*O*(

*ɛ*) ≪ 1 is a random variable [22

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]

*λ*be the light beam width from facet

_{i}*i*at the receiving plane and define the light beam spreading coefficient to be

*α*≡

_{i}*λ*/

_{i}*L*.

#### 2.1.2. Deterministic relations

*i*facet is refracted according to Snell’s law [13]. At the receiver plane, the center of the beam is located at (

^{th}*x,z*)=(

*ξ*, −

_{i}*D*), and we define

*ξ*′

*≡*

_{i}*ξ*–

_{i}*x*(Fig.1). For

_{i}*s*≪ 1, it follows that where

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

#### 2.1.3. Statistical model

*D*is small relative to facet size

_{d}*L*, and that the detected irradiance

*E*is a collection of contributions from all possible facets in stochastic superposition process. Let

*N*be the number of contributing facets to the receiver in a given realization. Since the facet slopes

_{r}*s*are random variables, therefore

_{i}*N*is also a random variable. Using Eq. (3)

_{r}*E*at the receiver can be written as, We define the normalized downwelling irradiance as

*χ*≡

*E*/〈

*E*〉, so that to leader order, where 〈·〉 represents averaging over all realizations/instants.

*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*(

_{ξ¯i}*ξ*). From Eq. (1), this can be derived from the PDF of the ocean surface slope

_{i}*p*(

_{s̄i}*s*). The arrival probability of the

_{i}*i*

^{th}facet can then be expressed as the probability of

*ξ*∈ [−

_{i}*/2,*λ ¯

*/2],*λ ¯

*P*is a function of

_{ξ¯i}*x*. Each facet

_{i}*i*can be considered as an independent Bernoulli trial with different success probability

*P*. If in addition we can assume that 〈

_{ξ¯i}*N*〉 is large, i.e., 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

_{r}*N*can be obtained as, which has variance 1/

_{r}*, where the gamma function Γ is used to approximate the continuous PDF.*α ¯

*σ*is the standard deviation of

_{ξ}*p*(

_{ξ¯i}*ξ*). For Gaussian ocean surface [22

_{i}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]

*σ*,

_{s}*σ*can be expressed in terms of

_{ξ}*σ*using Eq. (1), and condition (Eq. (7)) can be expressed as, It can be shown that

_{s}*L*/

*σ*≪ 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).

_{ξ}### 2.2. Relationship between the facet size L in the GP model and the slope correlation length l_{s} of the ocean surface

*L*in the GP model must be chosen so that

*s*satisfies the IID condition. In addition, the GP requires a sufficiently large number of contributing facets for given

_{i}*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], 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*of the surface as twice the standard deviation of

_{s}*R*(

*ζ*) (Eq. (13)): In order to satisfy IID, we require

*L*≥

*l*; and to maximize 〈

_{s}*N*〉, we simply choose

_{r}*L*=

*l*. In terms of

_{s}*l*and only physical quantities, the condition Eq. (11) for GP validity then becomes

_{s}### 2.3. Including the effect of sub-facet slope variation in the GP model

*S*(

_{η}*k*), the sub-facet slope variance is the components which are above the Nyquist wave number

*k*

^{*}=

*π*/

*l*,

_{s}*h̃*(

_{i}*ξ′*) be the (time) averaged irradiance beam spreading function at

_{i}*z*=−

*D*for facet

*i*for this case. Assuming that the sub-facet variations are due to

*σ*,

_{sS}*h̃*(

_{i}*ξ′*) can be shown to be a Gaussian form. We set the half width of the beam spreading function

_{i}*to be the standard deviation of*ℓ ˜

_{i}*h̃*(

_{i}*ξ′*), which can be obtained (after using Eq. (1)) as: where to leading order for IID, Eq. (17) is independent of

_{i}*i*. Physically, the original GP can be modified to account for the sub-facet spreading by an equivalent beam width and beam spreading coefficient:

*P̃*corresponding to

_{ξ¯i}*ξ*∈ [−

_{i}*/2,*λ ˜

*/2] > [−*λ ˜

*/2,*λ ¯

*/2]. The conditions for non-homogeneous Poisson process still obtains, and we find, The variance of the normalized irradiance above is*λ ¯

*N*〉 in this case is proportional to the increase in the individual arrival probability for a facet.

_{r}### 2.4. Including the effect of volume scattering in the GP model

*ĥ*(

_{i}*ξ*′

*) be the irradiance beam spreading function at*

_{i}*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′*), where,

_{i}*is some characteristic spreading (half) width due to volume scattering, and again Eq. (22) is independent of*ℓ ^

*i*.

*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], 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,

### 2.5. Summary of GP model results

***denotes convolution operation. The combined beam spread (half) width can be expressed as, The light beam width and spreading coefficient in the receiver plane are now (to leading order),

*S*(

_{η}*k*), the long-wave component of the surface is captured through the choice of

*L*in term of

*l*, while the sub-facet variance part of the surface is captured through beam spreading (half) width

_{s}*via Eq. (17). The IOPs of the water body is captured through*ℓ ˜

*by Eq. (24). Together*ℓ ^

*and*ℓ ˜

*determine*ℓ ^

*α*(Eq. (27)) which specifies

*α*=

*,*α ¯

*, and*α ˜

*.*α ^

*D*,

*∼*ℓ ˜

*D*while

*∼*ℓ ^

*cD*

^{2}(Eq. (17) and Eq. (24)), and hence Setting

*=*ℓ ˜

*, we obtain a critical depth*ℓ ^

*D*, For

_{cr}*D*>

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

_{cr}*D*<

*D*.

_{cr}*θ*, the derivations follow in a straightforward way provided that the linearization based on

_{s}*s*=

_{i}*O*(

*ɛ*) ≪ 1 still obtains. For large

*θ*, 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.

_{s}## 3. Validation and results

### 3.1. GP Model validation by MC simulations and experiments

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]

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]

*χ*Eq. (28) with that from MC simulations and experimental measurements at different (measured) depths

*D*. Note that, in this case,

*θ*= 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

_{s}*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

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]

*D*

^{*}∼ 2.7 and 5.4 for

*D*= 0.86m (Fig. 2(a)) and 1.7m (Fig. 2(b)).

*σ*(Eq. (29)) as function of depth in Fig. 3. The comparison between GP and measurements is excellent. For

_{χ}*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*

_{cr}cω_{0}∼ 0.37). The MC result is expected to be correct at these shallow depths, where

*σ*starts from zero value at the surface, reaches a maximum value, and then approaches the GP asymptotically as

_{χ}*D*increases. This “bump” in the shallow depth

*σ*behavior has been reported in experiments, theoretical model and numerical simulations [4

_{χ}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]

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

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]

*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*with depth for the reference case of

_{χ}*U*

_{10}=2 m/s (with effectively no surface roughness) for the present condition. Figure 4(b) shows the wind speed dependence of

*σ*at different depths. As wind speed increases, the short-wave component

_{χ}*σ*increases (while

_{sS}*l*decreases), and subsequently

_{s}*increases (Eqs. (17) and Eq. (19)), resulting in the monotonic decrease of the normalized irradiance standard deviation*α ˜

*σ*with wind speed, a feature also observed in experiments [6]. At any given depth, the effect of further increase in

_{χ}*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.

## 4. Conclusion

*U*

_{10}>5m/s).

## Acknowledgments

## References and links

1. | W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish |

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. |

3. | J. R. V. Zaneveld and P. A. Hwang, “The influence of coherent waves on the remotely sensed reflectance,” Opt. Express |

4. | R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B |

5. | H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci |

6. | J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia |

7. | D. Stramski and L. Legendre, “Laboratory simulation of light focusing by water surface waves,” Mar. Biol |

8. | J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia |

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 |

10. | P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res. |

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. |

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, |

14. | J. W. McLean and J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” |

15. | J. R. V. Zaneveld and E. Boss, “Influence of surface waves on measured and modeled irradiance profiles,” Appl. Opt. |

16. | J. Hedley, “A three-dimensional radiative transfer model for shallow water environments,” Opt. Express |

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. |

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. |

19. | V. L. Veber, “On the spatial fluctuations of underwater illumination,” Izv. Atmos. Oceanic Phys. |

20. | V. I. Shevernev, “Statistical structure of the illumination field under a wavy surface,” Izv. Atmos. Oceanic Phys. |

21. | B. R. Frieden, |

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 |

23. | R. G. Gallager, |

24. | R. E. Walker, |

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).

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