## Abstract

The lidar signature from a collection of bubbles is proportional to the volume backscatter coefficient at a scattering angle of 180°. This quantity, calculated using a combination of geometric optics and diffraction, is proportional to the void fraction of the bubbles in the water for any bubble size distribution. The constant of proportionality is 233 m^{−1} sr^{−1} for clean bubbles, slightly less for bubbles coated with a thin layer of organic material, and as large as 1445 m^{−1} sr^{−1} for a thick coating of protein.

© 2010 OSA

## 1. Introduction

Bubbles in the upper ocean are important in a number of physical and chemical processes. They are a significant factor in the absorption of CO_{2} in the ocean [1,2], reducing the greenhouse effect. On the other hand, bubbles from the ocean floor may be a source of methane in the atmosphere [3,4], which would increase the greenhouse effect. Breaking bubbles produce aerosol particles in the atmosphere [5,6], which can serve as cloud-condensation nuclei. Bubbles are a source of sound in the ocean [7,8], and also affect the propagation of sound through absorption, refraction, and scattering [9,10].

Bubbles also affect the propagation of light in the ocean, primarily through scattering. For example, the color of reflected sunlight will be shifted toward the green, which can distort estimates of chlorophyll concentration based on satellite-based measurements of ocean color [11–13]. The bubbles in ship wakes have also been shown to affect both the scattered radiance and its spectrum [14].

The signal from a lidar looking down into the ocean will include a contribution from bubbles, if they are present in sufficient concentrations. Krekova et al. [15] used a Mie calculation to estimate the return expected from an airborne lidar and concluded that bubbles would represent a significant contribution at concentrations above 105 m^{−1}. Su et al. [16] measured the scattering at 180° from a propeller-generated wake in the laboratory, and concluded that wake detection in the open ocean was feasible. More recently, Li et al. [17] demonstrated the detection of bubble plumes in the laboratory using a short range lidar system. None of these works demonstrates lidar returns from bubbles in the ocean, where their size distribution and coating might be different than in the laboratory.

This paper uses a combination of geometric optics and diffraction to calculate the lidar return from a collection of bubbles.

## 2. Scattering theory

We define the lidar scattering amplitude from a bubble by

where*E*is the incident field amplitude, and

_{i}*E*is the amplitude of the field scattered directly back toward the source measured at a distance

_{s}*r*from the bubble. This distance is assumed to satisfy the condition

*r*>>

*ka*

^{2}, where

*k*is the optical wavenumber and

*a*is the bubble radius. The backscatter cross section

*σ*= │

*A*│

^{2}.

Following Arnott and Marston [18], we can approximate *A* by

*m*is the refractive index of the bubble relative to sea water,

*r*and

_{1}*r*are the Fresnel reflection coefficients for the two orthogonal linear polarizations at the specified incidence angle, and

_{2}*p*, the order of the ray path, is one less than the number of surfaces intercepted by a ray. Note that

*m*is defined such that it is equal to unity for sea water and to 0.75 for air in sea water that has a typical value for the refractive index relative to vacuum of

*n*= 1.33.

The first two terms represent the first two axial rays with *p* = 0 and *p* = 2. Higher-order axial rays are neglected. The sum over *p* in Eq. (2) represents glory rays with *p* = 3 and higher. The incidence angle of the *p*^{th} glory ray is found from [19]

*ν*is the refraction angle obtained from the incidence angle through Snell’s law:The Fresnel reflection coefficients are

_{p}The backscatter cross section obtained from this formula agrees well with a Mie calculation for large values of *ka* [19]. Because of the large phase difference between ray paths of different orders, the backscatter cross section is very sensitive to small changes in bubble size parameter [Fig. 1(a)
]. For a distribution of bubble sizes, however, the phase of different ray paths will vary randomly between bubbles, so one can approximate the backscatter cross section by the sum of the individual ray-path cross sections

*p*= 0, 2, 3, 4, 5, … with

*A*given by the individual terms in Eq. (2). Because the phases of the individual rays are ignored, the expression is greatly simplified and the underlying dependence on bubble radius becomes obvious. For an index of refraction of

_{p}*m*= 0.75 and

*a*measured in meters, the result isAveraging over bubbles of different sizes tends to this same result as the amount of averaging increases [Fig. 1(b)].

For a typical lidar wavelength of 532 nm,

and the glory ray contribution will be larger than the axial ray contribution for bubble radii greater than 7.1 μ. Note that the second term is equal to 233*V*(

*a*), where

*V*(

*a*) is the volume of a bubble of radius

*a*, and the constant 233 has units of m

^{−1}sr

^{−1}.

The volume backscattering coefficient is given by the integral

*N*(

*a*) is the size distribution of bubble density. When the axial-ray contribution can be neglected,

*F*is the total volume of air in the bubbles in a unit volume of ocean. Bubbles in the ocean are typically coated with an organic film that is collected as they rise through water containing dissolved organic materials [20–22]. These coatings may be lipids or proteins [21] with relative (to sea water) refractive indices of 1.10 and 1.20, respectively [11]. Layers can vary from a monomolecular layer 10 nm – 1 μm thick up to a thickness of 10 μm [23], or even a thickness comparable to the bubble radius [21].

_{V}The presence of an organic film increases the scattering from bubbles in the backwards direction, including the direct backscatter of interest for lidar [11,13,23]. The details of the calculation depend on the thickness of the layer, because of the interference between the reflections from the front and back. The result is a very complex calculation for layers of uniform thickness comparable to the wavelength, but is simplified for thick layers with varying thickness (>> a wavelength so the phase difference is randomly distributed between 0 and 2π). For very thin layers (<< a wavelength), the coating has not effect, and the bubble scatters light in the same way as one without a coating.

In the case where the phase difference can be considered random over a collection of bubbles, the reflection coefficients are added incoherently to get

Including the effects of coatings, we can approximate the volume backscatter coefficient $\beta \left(\pi \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}K{F}_{V}$, where *K* is provided in Table 1
for the various cases. These values clearly show that bubble coating can have a large effect on the backscatter coefficient for a given void fraction, and this dependence must be considered in lidar estimates of void fraction.

For bubbles right at the surface, the return from bubbles will be contaminated by the specular return from facets on the rough ocean surface that are normal to the lidar beam; this return also preserves the incident polarization. This glint return will produce the same signal as volume scattering with

*R*is the surface reflectivity at an incidence angle of zero,

*n*is the refractive index of sea water relative to vacuum,

*c*is the speed of light in vacuum,

*τ*is the lidar pulse length in seconds, and

*P*(

_{x,y}*x,y*) is the probability density function of surface slope components

*x*(parallel to the wind) and

*y*(perpendicular to the wind), evaluated at the lidar incidence angle, (

*x*). The approximation to

_{i},y_{i}*P*by Cox and Munk [24] can be used to estimate the glint contribution for different geometries (Fig. 2 ). The conclusion is that the best bubble data will be obtained for larger incidence angles oriented perpendicular to the wind direction.

_{x,y}The minimum void fraction that can be detected can be estimated based on the requirement that the bubble return be much greater than the glint return. For thickly coated bubbles, a glint contribution of *β* < 0.003 would produce an overestimation of void fraction by 2 – 5 × 10^{−6}, depending on the composition of the coating. Thus, a reasonable accurate estimate of void fraction should be possible down to values on the order of 10^{−5}. For comparison, void fractions from around 10^{−4} at a depth of 0.7 m [12] to over 10^{−2} at a depth of 0.3 m [25] have been reported.

## 3. Conclusions

The assumption that the phases between different ray paths for bubbles of different sizes are uncorrelated allows a very simple expression for the lidar signal from a collection of bubbles with and without organic films. That expression shows that there is a linear relationship between lidar return and bubble void fraction. While only direct backscatter was considered here, the same approach, defined by Eq. (6), could easily be used to estimate the scattering at other angles, simplifying calculations of scattered sunlight.

The results derived here are valid for either open-ocean or laboratory-generated bubbles as long as the bubbles are spherical and the number of bubbles per unit volume is low enough that multiple scattering effects can be neglected. Either of these conditions will cause a cross-polarized component in the return. The absence of a cross-polarized return in a measurement indicates that the results described here can be applied.

The simple linear relationship with void fraction also depends on the condition that the contribution from the axial rays can be neglected. This will be the case if the contribution to the void fraction by bubbles of radius less than 7 μ (for a 532 nm wavelength lidar) can be neglected. For a typical open-ocean size spectrum [26], less than 0.01% of the void fraction is from these small bubbles, so axial rays can be safely neglected.

## Acknowledgments

The work was partially supported by the U.S. Navy.

## References and links

**1. **D. K. Woolf, “Bubbles and the air-sea transfer velocity of gasses,” Atmos.-Ocean **31**, 517–540 (1993). [CrossRef]

**2. **R. S. Bortkovskii, B. N. Egorov, V. M. Kattsov, and T. V. Pavlova, “Model estimates for the mean gas exchange between the ocean and the atmosphere under the conditions of the present-day climate and its changes expected in the 21st century,” Izv., Atmos. Ocean. Phys. **43**(3), 378–383 (2007). [CrossRef]

**3. **G. K. Westbrook, K. E. Thatcher, E. J. Rohling, A. M. Piotrowski, H. Pälike, A. H. Osborne, E. G. Nisbet, T. A. Minshull, M. Lanoisellé, R. H. James, V. Hühnerbach, D. Green, R. E. Fisher, A. J. Crocker, A. Chabert, C. Bolton, A. Beszczynska-Möller, C. Berndt, and A. Aquilina, “Escape of methane gas from the seabed along the West Spitsbergen continental margin,” Geophys. Res. Lett. **36**(15), L15608 (2009), doi:. [CrossRef]

**4. **E. A. Solomon, M. Kastner, I. R. MacDonald, and I. Leifer, “Considerable methane fluxes to the atmosphere from hydrocarbon seeps in the Gulf of Mexico,” Nat. Geosci. **2**(8), 561–565 (2009). [CrossRef]

**5. **W. C. Keene, H. Maring, J. R. Maben, D. J. Kieber, A. A. P. Pszenny, E. E. Dahl, M. A. Izaguirre, A. J. Davis, M. S. Long, X. L. Zhou, L. Smoydzin, and R. Sander, “Chemical and physical characteristics of nascent aerosols produced by bursting bubbles at a model air-sea interface,” J. Geophys. Res. **112**(D21), D21202 (2007), doi:. [CrossRef]

**6. **A. Sorooshian, L. T. Padró, A. Nenes, G. Feingold, A. McComiskey, S. P. Hersey, H. Gates, H. H. Jonsson, S. D. Miller, G. L. Stephens, R. C. Flagan, and J. H. Seinfeld, “On the link between ocean biota emissions, aerosol, and maritime clouds: Airborne, ground, and satellite measurements off the coast of California,” Global Biogeochem. Cycles **23**(4), GB4007 (2009), doi:. [CrossRef]

**7. **M. R. Loewen and W. K. Melville, “An experimental investigation of the collective oscillations of bubble plumes entrained by breaking waves,” J. Acoust. Soc. Am. **95**(3), 1329–1343 (1994). [CrossRef]

**8. **J. Park, M. Garcés, D. Fee, and G. Pawlak, “Collective bubble oscillations as a component of surf infrasound,” J. Acoust. Soc. Am. **123**(5), 2506–2512 (2008). [CrossRef] [PubMed]

**9. **M. V. Hall, “A comprehensive model of wind-generated bubbles in the ocean and predictions of the effects on sound propagation at frequencies up to 40 kHz,” J. Acoust. Soc. Am. **86**(3), 1103–1117 (1989). [CrossRef]

**10. **P. A. Hwang and W. J. Teague, “Low-frequency resonant scattering of bubble clouds,” J. Atmos. Ocean. Technol. **17**(6), 847–853 (2000). [CrossRef]

**11. **X. Zhang, M. Lewis, and B. Johnson, “Influence of bubbles on scattering of light in the ocean,” Appl. Opt. **37**(27), 6525–6536 (1998). [CrossRef]

**12. **E. J. Terrill, W. K. Melville, and D. Stramski, “Bubble entrainment by breaking waves and their influence on optical scattering in the upper ocean,” J. Geophys. Res. **106**(C8), 16815–16823 (2001). [CrossRef]

**13. **X. D. Zhang, M. Lewis, M. Lee, B. Johnson, and G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. **47**, 1273–1282 (2002). [CrossRef]

**14. **X. D. Zhang, M. Lewis, W. P. Bissett, B. Johnson, and D. Kohler, “Optical influence of ship wakes,” Appl. Opt. **43**(15), 3122–3132 (2004). [CrossRef] [PubMed]

**15. **M. M. Krekova, G. M. Krekov, and V. S. Shamanaev, “Influence of air bubbles in seawater on the formation of lidar returns,” J. Atmos. Ocean. Technol. **21**(5), 819–824 (2004). [CrossRef]

**16. **L. P. Su, W. J. Zhao, X. Y. Hu, D. M. Ren, and X. Z. Liu, “Simple lidar detecting wake profiles,” J. Opt. A, Pure Appl. Opt. **9**(10), 842–847 (2007). [CrossRef]

**17. **W. Li, K. Yang, M. Xia, J. Rao, and W. Zhang, “Influence of characteristics of micro-bubble clouds on backscatter lidar signal,” Opt. Express **17**(20), 17772–17783 (2009). [CrossRef] [PubMed]

**18. **W. P. Arnott and P. L. Marston, “Optical glory of small freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A **5**(4), 496–506 (1988). [CrossRef]

**19. **P. L. Marston and D. S. Langley, “Glory- and rainbow-enhanced acoustic backscatter from fluid spheres: Models for diffracted axial focusing,” J. Acoust. Soc. Am. **73**(5), 1464–1475 (1983). [CrossRef]

**20. **B. D. Johnson and R. C. Cooke, “Generation of stabilized microbubbles in seawater,” Science **213**(4504), 209–211 (1981). [CrossRef] [PubMed]

**21. **R. E. Glazman, “Effects of adsorbed films on gas bubble radial oscillations,” J. Acoust. Soc. Am. **74**(3), 980–986 (1983). [CrossRef]

**22. **E. C. Monahan and H. G. Dam, “Bubbles: An estimate of their role in the global oceanic flux of carbon,” J. Geophys. Res. **106**(C5), 9377–9383 (2001). [CrossRef]

**23. **W. Li, K. Yang, M. Xia, D. Tan, X. Zhang, and J. Rao, “Computation for angular distribution of scattered light on a coated bubble in water,” J. Opt. A, Pure Appl. Opt. **8**(10), 926–931 (2006). [CrossRef]

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

**25. **G. B. Deane and M. D. Stokes, “Scale dependence of bubble creation mechanisms in breaking waves,” Nature **418**(6900), 839–844 (2002). [CrossRef] [PubMed]

**26. **M. V. Trevorrow, “Measurements of near-surface bubble plumes in the open ocean with implications for high-frequency sonar performance,” J. Acoust. Soc. Am. **114**(5), 2672–2684 (2003). [CrossRef] [PubMed]