## Monte Carlo simulation of propagation of a short light beam through turbulent oceanic flow

Optics Express, Vol. 15, Issue 21, pp. 13988-13996 (2007)

http://dx.doi.org/10.1364/OE.15.013988

Acrobat PDF (440 KB)

### Abstract

We use Monte Carlo time-dependent simulations of light pulse propagation through turbulent water laden with particles to investigate the application of Multiple Field Of View (MFOV) lidar to detect and characterize oceanic turbulence. Inhomogeneities in the refractive index induced by temperature fluctuations in turbulent ocean flows scatter light in near-forward angles, thus affecting the near-forward part of oceanic water scattering phase function. Our results show that the oceanic turbulent signal can be detected by analyzing the returns from a MFOV lidar, after re-scaling the particulate back scattering phase function.

© 2007 Optical Society of America

## 1. Introduction

01. D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. **43**, 5662–5676 (2004). [CrossRef] [PubMed]

02. D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, “Comparison of nearforward scattering on turbulence and particles,” Appl. Opt. **37**, 4669–4677 (1998). [CrossRef]

^{-7}to 10

^{-3}

*rad*demonstrate that the total scattering coefficient,

*b*, due solely to oceanic turbulence, can be on order of 10 m

^{-1}. Thus the small-angle scattering function due to turbulence is, in most cases, significantly larger than typical values due to the combined contribution of oceanic particles and the ’pure water’ scattering function, which is typically less than 0.25 m

^{-1}and 0.05 m

^{-1}in coastal and deep ocean waters respectively [1

01. D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. **43**, 5662–5676 (2004). [CrossRef] [PubMed]

03. J. S. Jaffe, “Monte-carlo modeling of underwater-image formation - validity of the linear and small-angle approximations,” Appl. Opt. **34**, 5413–5421 (1995). [CrossRef] [PubMed]

*ε*, the rate of dissipation of turbulent kinetic energy, which is inversely proportional to the size of the smallest flow structure. Typically χ ranges from 10

^{-2}°

*C*

^{2}

*s*

^{-1}a few meters below the surface [4

04. D. M. Farmer and J. R. Gemmrich, “Measurements of temperature fluctuations in breaking surface waves,” J. Phys. Oceanogr. **26**, 816–825 (1996). [CrossRef]

^{-10}°

*C*

^{2}

*s*

^{-1}in the deep ocean [5

05. T. M. Dillon, “The energetics of overturning structures: Implications for the theory of fossil turbulence,” J. Phys. Oceanogr. **14**, 541–549 (1984). [CrossRef]

*ε*attains values of 10

^{-4}

*m*

^{2}

*s*

^{-3}in the fairly energetic upper layer to 10

^{-11}

*m*

^{2}

*s*

^{-3}in the mid-water column [6

06. A. Anis and J. N. Moum, “Surface wave-turbulence interactions: scaling *ε*(*z*) near the sea surface,” J. Phys. Oceanogr. **25**, 2025–2045 (1995). [CrossRef]

*b*ranges between 10 and 30 m

_{turb}^{-1}a few meters below the surface to

*b*≃ 0.1m

_{turb}^{-1}in the deep and quiescent part water column [2

02. D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, “Comparison of nearforward scattering on turbulence and particles,” Appl. Opt. **37**, 4669–4677 (1998). [CrossRef]

*α*(in m

^{-1}) in the water column is given by the sum of absorption,

*a*, and scattering,

*b*, i.e.

*α*(

*z*) =

*a*(

*z*)+

*b*(

*z*).

*b*> 1 m

_{turb}^{-1}), implies that the photon mean path length given by

*l*≃ 1/b < 1

_{phot}*m*is short. Consequently, most remotely sensed photons emerging from any depth > 1

*m*will undergo multiple forward scattering events on turbulent flow before reaching a remote detector. Here we explore the potential optical consequences of a large scattering coefficient due to turbulence in the energetic upper ocean. Namely,

*can surface layer turbulence be detected and quantified by remote measurements?*The enhanced forward scattering on turbulence can be likened to scattering on atmospheric clouds with large water droplets. For clouds, the single-scatter lidar equation predicts a small signal variation as a function of the FOV (Field Of View) of the receiver. However, multiple scattering leads to a strong FOV dependence on signal return from inside the cloud as suggested in the pioneering work of [7] and subsequently explored by others [8

08. E. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. **37**, 2464–2472 (1998). [CrossRef]

09. L. Bissonnette, G. Roy, L. Poutier, S. Cober, and G. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte Carlo simulations and comparisons with in situ measurements,” Appl. Opt **41**, 6307–6324 (2002). [CrossRef] [PubMed]

09. L. Bissonnette, G. Roy, L. Poutier, S. Cober, and G. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte Carlo simulations and comparisons with in situ measurements,” Appl. Opt **41**, 6307–6324 (2002). [CrossRef] [PubMed]

## 2. The setup of the Monte Carlo numerical runs

11. J. Piskozub, P. Flatau, and J. Zaneveld, “Monte Carlo Study of the Scattering Error of a Quartz Reflective Absorption Tube,” J. Atmospheric and Oceanic Technol. **18**, 438–445 (2001). [CrossRef]

02. D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, “Comparison of nearforward scattering on turbulence and particles,” Appl. Opt. **37**, 4669–4677 (1998). [CrossRef]

12. V. Banakh, I. Smalikho, and C. Werner, “Numerical Simulation of the Effect of Refractive Turbulence on Coherent Lidar Return Statistics in the Atmosphere,” Appl. Opt **39**, 5403–5414 (2000). [CrossRef]

^{12}histories. Runs were executed on the USC Linux cluster, distributed among 1024 nodes. In the following figures we convert the flight-time of the returned lidar signal to an equivalent depth in 1m increments. For this work we analyzed three cases. In each of the three cases (Fig. 1) the narrow Gaussian beam, with a 10

^{-5}

*rad*divergence and wavelength

*λ*= 550

*nm*, impinges normally on a flat ocean surface. The relevant optical parameters corresponding to the medium and turbulence for each case are presented in Table 1. We additionally calculated the equivalent photon diffusive distance, defined as

*l*= 1/

_{diff}*α*· 1/(1 -

*g*) where

*g*> 0 is the average cosine of the forward scattering angle [10]. The equivalent diffusive distance represents the propagation distance after which the initially collimated photon has ’forgotten’ its original direction. The particles used in our calculations have a scattering coefficient

*b*= 0. 1

*m*

^{-1}and an absorption coefficient

*a*= 0.1/

*m*

^{-1}The underlying pure water follows the Rayleigh scattering phase function with

*b*, = 1.9 · 10

_{Rayleigh}^{-3}

*m*

^{-1}[13] and absorption

*a*, = 5.7 · 10

_{Rayleigh}^{-2}/

*m*

^{-1}. In calculations we have used a turbulent light scattering phase function corresponding to relatively large but realistic oceanic turbulence levels, with the rate of temperature variance dissipation -χ = 10

^{-3}

*deg*

*C*

^{2}

*s*

^{-1}, and a rate of dissipation of turbulent kinetic energy -

*ε*= 10

^{-6}/

*m*

^{-6}

*s*

^{-3}. The turbulent phase function was obtained experimentally [2

**37**, 4669–4677 (1998). [CrossRef]

01. D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. **43**, 5662–5676 (2004). [CrossRef] [PubMed]

*ε*and

*χ*. For computational efficiency,i.e. to shorten the computation time, we enhanced the backscattering of the particulate VSF relative to oceanic observations [14

14. M. Twardowski, E. Boss, J. Macdonald, W. Pegau, A. Barnard, and J. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Research **106**, 14,129–14,142 (2001). [CrossRef]

*°*by a water-type varying factor of 100 to 1000 - see the bottom part of Fig, 2. The larger value used for backscatter at or near 180

*°*allows more photons to return to the source thus decreasing the necessary simulation time to build the statistics of the returning photons. As it will be argued below (see Eq 1 and Eq 2) the backscattering at or near 180

*°*is a multiplicative constant of the problem considered here and the obtained results are ∝ VSF(180

*°*).

15. V. Haltrin, “One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater,” Appl. Opt. **41**, 1022–1028 (2002). [CrossRef] [PubMed]

**37**, 4669–4677 (1998). [CrossRef]

*n*= 1 .04 corresponds to waters with optical properties dominated by chlorophyll in the water column while the larger value of

*n*= 1.18 corresponds to waters dominated by inorganic sediment [14

14. M. Twardowski, E. Boss, J. Macdonald, W. Pegau, A. Barnard, and J. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Research **106**, 14,129–14,142 (2001). [CrossRef]

16. H. R. Gordon, “Sensitivity of radiative transfer to small-angle scattering in the ocean: quantitative assessment,” Appl. Opt. **32**, 7505–7511 (1993). [CrossRef] [PubMed]

*°*and 15

*°*, i.e. near-forward angles, does not significantly modify the solar irradiance observed underwater.

## 3. Theory of lidar scattering - quasi-small angle (QSA) approximation theory

09. L. Bissonnette, G. Roy, L. Poutier, S. Cober, and G. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte Carlo simulations and comparisons with in situ measurements,” Appl. Opt **41**, 6307–6324 (2002). [CrossRef] [PubMed]

*°*. This condition is valid for particles that are large compared with the lidar wavelength and requires that the lidar FOV footprint be smaller than the mean free path between scattering events. The latter condition ensures that only small angle scattering contributes to the received signal and that the range resolution based on the measured time of flight is maintained to a good approximation. In our simulation, we vary the FOV between 10

^{-5}

*rad*to 3·10

^{-2}

*rad*and the analyzed range is 20

*m*. Since 1/

*b*= 10

_{particle}*m*≫ 0.6

*m*=3·10

^{-2}·20

*m*, then the single backscattering condition is met. In addition turbulent scattering does not contribute to large angle scattering and characteristic turbulence lengthscale is much smaller than lidar FOV footprint. The QSA approximation offers a useful theorem to interpret the received lidar signal. We will apply this theorem to analyze our results obtained from the Monte Carlo simulations. Katsev et al. [17

17. I. Katsev, E. Zege, A. Prikhach, and I. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A **14**, 1338–1346 (1997). [CrossRef]

*°*(backscattering), for the radiative transfer calculations, the water column can be replaced by a fictitious water column possessing twice the extinction coefficient for the purposes of lidar calculations. Then the angular properties of the scattered beam are modeled only on the downward propagation leg. This result has been extensively verified in atmospheric lidar experiments [9

**41**, 6307–6324 (2002). [CrossRef] [PubMed]

08. E. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. **37**, 2464–2472 (1998). [CrossRef]

**41**, 6307–6324 (2002). [CrossRef] [PubMed]

*z*is the range,

*θ*is the FOV of the receiver,

*P*is the received power due to a single-scattering contribution, and

_{ss}*M*(

*z*,

*θ*) represents the multiple scattering contribution at variable FOV. The single scattering contribution

*P*(

_{ss}*z*) (by definition FOV independent) is given by:

*K*is an instrument constant,

*P*

_{0}is the laser pulse power,

*β*(

*z*,

*π*) is the particle backscatter coefficient (at 180

*°*), and

*τ*(

*z*) = ∫

^{z}_{0}

*α*(

*z*′)

*dz*′ is the integrated depth dependent optical depth or thickness. We use an approximate expression for the multiple scattering contribution

*M*(

*z*,

*θ*) developed by Eloranta [8

08. E. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. **37**, 2464–2472 (1998). [CrossRef]

*x*is defined as:

*x*=

*zθ*/[(

*z*-

*z*

_{0})

*θ*

_{0}] , the constant

*z*

_{0}is the distance between the scattering layer and the receiver,

*z*the total range,

*θ*is the FOV of the receiver, and

*θ*

_{0}is the mean-square width of the particle scattering peak, the red or the green curve, Fig. 2. The above equation, Eq. 3, is an approximate expression for double (

*k*= 1) and higher order (

*k*> 1) scattered light [8

**37**, 2464–2472 (1998). [CrossRef]

*Erf*is the error function. In our calculations the turbulent phase function has

*θ*

_{0}≃ 10

^{-4}

*rad*(Fig. 2). We can estimate Monte Carlo RT accuracy comparing the results of Case C (in the absence of turbulence) to predictions of Eq. 3. The ratio of the received lidar signal at different FOV normalized to its single scattering value is compared in Fig. 3. The blue and red lines denote the RT Monte Carlo data and the black lines the results of Eq. 3. The results of RT Monte Carlo are consistent with theoretical predictions of Eq 3. In addition, we note the effects of double and multiple scattering which are evident at FOV of 8 · 10

^{-4}rad as the returned signal approximately follows a straight line obtained from Eq 3. The effect of higher order scattering becomes pronounced for larger FOV of 3 · 10

^{-2}, where the returned signal follows a

*k*> 1 power law; the doubly scattered return obtained from Eq 3 is plotted as reference - the black line.

## 4. Results of the RT calculation for an embedded turbulent layer

17. I. Katsev, E. Zege, A. Prikhach, and I. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A **14**, 1338–1346 (1997). [CrossRef]

*M*(

*z*,

*θ*) of the FOV lidar can be expressed as:

*θ*

_{0}is the mean-square width of the highly peaked forward scattering phase function, i.e. the turbulent scattering VSF with

*θ*

_{0}≃ 10

^{-4}

*rad*(Fig. 2),

*θ*is the receiver FOV,

*α*

_{0}the constant turbulent layer extinction coefficient,

*g*- is the mean cosine, and

*z*

_{0}is the distance to the scattering turbulent layer from the receiver. In Eq 4, the initially collimated beam spread angle is given by

*θ*(

_{beam}*z*)

^{2}∝ (

*z*-

*z*

_{0}) ·

*θ*

^{2}

_{0}. This dependence of the spreading angle on the propagation distance is a hallmark of a random walk or the diffusion with distance as independent variable and the angle as a dependent variable (i.e. the diffusion in the angle space). In Eq 4, the asymptotic limit of large propagating distance is usually reached quickly. It has been shown [18

18. A. Kim and M. Moscoso, “Beam propagation in sharply peaked forward scattering media,” J. Opt. Soc. Am. A **21**(5), 797–803 (2004). [CrossRef]

*m*. To compare the formula of Eq 4 to our RT calculations we expand it for small FOV (i.e.

*θ*) as:

*z*-

*z*

_{0}) and does not depend on any turbulent parameters. This is what we observe at the smallest FOV of 8·10

_{-4}

*rad*in Fig. 4. (Note the stretch of vertical coordinates on the Fig. 4, depends on the analyzed FOV. The lidar return slopes, Fig. 4, at the onset of turbulent layer in absence of coordinate stretching are consistent with the first two terms of Eq 5). At the larger FOV (3 · 10

^{-2}

*rad*) the multiple scattering contribution,

*M*(

*z*,

*θ*), corresponding to the term containing ∝

*θ*

^{-2}

_{0}, of Eq 5 becomes dominant. In general, at larger and fixed FOV, the slope of the multiple scattering contribution

*M*(

*z*,

*θ*= FOV) depends on the width (

*θ*

^{2}

_{0}) of the turbulent scattering phase function (Fig. 2). In turn, the width of the turbulent VSF was shown [2

**37**, 4669–4677 (1998). [CrossRef]

*E*(

_{T}*k*) is the depth dependent temperature variance spectrum. The quantity 〈

*θ*

^{2}

_{0}〉 is affected by the smallest scale region of the temperature spectrum

*E*(

_{T}*k*) as discussed previously [2

**37**, 4669–4677 (1998). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

01. | D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. |

02. | D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, “Comparison of nearforward scattering on turbulence and particles,” Appl. Opt. |

03. | J. S. Jaffe, “Monte-carlo modeling of underwater-image formation - validity of the linear and small-angle approximations,” Appl. Opt. |

04. | D. M. Farmer and J. R. Gemmrich, “Measurements of temperature fluctuations in breaking surface waves,” J. Phys. Oceanogr. |

05. | T. M. Dillon, “The energetics of overturning structures: Implications for the theory of fossil turbulence,” J. Phys. Oceanogr. |

06. | A. Anis and J. N. Moum, “Surface wave-turbulence interactions: scaling |

07. | C. M. R. Platt, “Remote Sounding of High Clouds. III: Monte Carlo Calculations of Multiple-Scattered Lidar Returns,” J. Atmospheric Sciences |

08. | E. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. |

09. | L. Bissonnette, G. Roy, L. Poutier, S. Cober, and G. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte Carlo simulations and comparisons with in situ measurements,” Appl. Opt |

10. | R. E. Walker, |

11. | J. Piskozub, P. Flatau, and J. Zaneveld, “Monte Carlo Study of the Scattering Error of a Quartz Reflective Absorption Tube,” J. Atmospheric and Oceanic Technol. |

12. | V. Banakh, I. Smalikho, and C. Werner, “Numerical Simulation of the Effect of Refractive Turbulence on Coherent Lidar Return Statistics in the Atmosphere,” Appl. Opt |

13. | M. Jonasz and G. Fournier, |

14. | M. Twardowski, E. Boss, J. Macdonald, W. Pegau, A. Barnard, and J. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Research |

15. | V. Haltrin, “One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater,” Appl. Opt. |

16. | H. R. Gordon, “Sensitivity of radiative transfer to small-angle scattering in the ocean: quantitative assessment,” Appl. Opt. |

17. | I. Katsev, E. Zege, A. Prikhach, and I. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A |

18. | A. Kim and M. Moscoso, “Beam propagation in sharply peaked forward scattering media,” J. Opt. Soc. Am. A |

19. | C. F. Bohren and D. R. Huffman, |

**OCIS Codes**

(010.7060) Atmospheric and oceanic optics : Turbulence

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

**ToC Category:**

Atmospheric and oceanic optics

**History**

Original Manuscript: July 30, 2007

Revised Manuscript: September 30, 2007

Manuscript Accepted: October 6, 2007

Published: October 11, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

D. J. Bogucki, J. Piskozub, M.-E. Carr, and G. D. Spiers, "Monte Carlo simulation of propagation of a short light beam through turbulent oceanic flow," Opt. Express **15**, 13988-13996 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-21-13988

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### References

- D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, "Light scattering on oceanic turbulence," Appl. Opt. 43, 5662-5676 (2004). [CrossRef] [PubMed]
- D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, "Comparison of nearforward scattering on turbulence and particles," Appl. Opt. 37, 4669-4677 (1998). [CrossRef]
- J. S. Jaffe, "Monte-carlo modeling of underwater-image formation - validity of the linear and small-angle approximations," Appl. Opt. 34, 5413-5421 (1995). [CrossRef] [PubMed]
- D. M. Farmer and J. R. Gemmrich, "Measurements of temperature fluctuations in breaking surface waves," J. Phys. Oceanogr. 26, 816-825 (1996). [CrossRef]
- T. M. Dillon, "The energetics of overturning structures: Implications for the theory of fossil turbulence," J. Phys. Oceanogr. 14, 541-549 (1984). [CrossRef]
- A. Anis and J. N. Moum, "Surface wave-turbulence interactions: scaling ?(z) near the sea surface," J. Phys. Oceanogr. 25, 2025-2045 (1995). [CrossRef]
- C. M. R. Platt, "Remote Sounding of High Clouds. III: Monte Carlo Calculations of Multiple-Scattered Lidar Returns," J. Atmospheric Sciences 38, 156-167.
- E. Eloranta, "Practical model for the calculation of multiply scattered lidar returns," Appl. Opt. 37, 2464-2472 (1998). [CrossRef]
- L. Bissonnette, G. Roy, L. Poutier, S. Cober, and G. Isaac, "Multiple-scattering lidar retrieval method: tests on Monte Carlo simulations and comparisons with in situ measurements," Appl. Opt 41, 6307-6324 (2002). [CrossRef] [PubMed]
- R. E. Walker, Marine light field statistics, (A Wiley Interscience Publication, 1994) p. 660 .
- J. Piskozub, P. Flatau, and J. Zaneveld, "Monte Carlo Study of the Scattering Error of a Quartz Reflective Absorption Tube," J. Atmospheric and Oceanic Technol. 18, 438-445 (2001). [CrossRef]
- V. Banakh, I. Smalikho, and C. Werner, "Numerical Simulation of the Effect of Refractive Turbulence on Coherent Lidar Return Statistics in the Atmosphere," Appl. Opt 39, 5403-5414 (2000). [CrossRef]
- M. Jonasz and G. Fournier, Light Scattering by Particles in Water: Theoretical and Experimental Foundations (Academic Press, 2007).
- M. Twardowski, E. Boss, J. Macdonald, W. Pegau, A. Barnard, and J. Zaneveld, "A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters," J. Geophys. Research 106, 14,129-14,142 (2001). [CrossRef]
- V. Haltrin, "One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater," Appl. Opt. 41, 1022-1028 (2002). [CrossRef] [PubMed]
- H. R. Gordon, "Sensitivity of radiative transfer to small-angle scattering in the ocean: quantitative assessment," Appl. Opt. 32, 7505-7511 (1993). [CrossRef] [PubMed]
- I. Katsev, E. Zege, A. Prikhach, and I. Polonsky, "Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems," J. Opt. Soc. Am. A 14, 1338-1346 (1997). [CrossRef]
- A. Kim and M. Moscoso, "Beam propagation in sharply peaked forward scattering media," J. Opt. Soc. Am. A 21(5), 797-803 (2004). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, 1983).

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