Page updated: April 12, 2020
Author: Curtis Mobley
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The Quasi-Single-Scattering Approximation

For highly peaked phase functions such as those typical of ocean waters, most of the scattering is at very small scattering angles ψ. For some purposes, scattering through a small scattering angle is almost the same as no scattering at all. The quasi-single-scattering approximation (QSSA) exploits this observation by assuming that the forward-scattering part of the phase function can be represented by a Dirac delta function at scattering angle ψ = 0, with no scattering at all for 0 < ψ < 90deg. At first glance this seems like a terribly inaccurate approximation of reality for phase functions like the Petzold ”average-particle” phase function) shown in the left panel of Fig. 1. However, when plotted on linear axes as in the right panel of the figure, the approximation looks more reasonable. In practice, it can yield surprisingly accurate results for quantities that depend mostly on absorption and/or backscatter (such as Ed and Rrs).

The QSSA can be traced back to Hansen (1971), who used it in studies of reflection by planetary atmospheres. Gordon (1973) introduced it to oceanography for ocean color remote sensing of the ocean.


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Figure 1: Fig. 1. The Petzold average particle phase function plotted on log-linear and linear-linear axes.

The QSSA uses the formulas of the SSA, but treats forward scattering as no scattering at all. With this approximation, the beam attenuation coefficient c becomes

c = a + b = a + bf + bb a + bb c.

With this approximation for c, the single scattering albedo ωo and the optical depth ζ become

ωo = b c b a + bb = bb a + bb 1 B ωo,

and

ζ = cz (a + bb)z ζ.

where B is the backscatter fraction. The QSSA thus replaces c,ωo,andζ with c,ωo,andζ respectively. This is an example of a similarity transformation in which the solution of one problem is rescaled to obtain the solution to a different problem.

The QSSA was developed for reflectance calculations, so let us compute the QSSA approximation for the remote-sensing reflectance Rrs. The SSA solution for Lu(SSA),

Lu(SSA)(ζ,μ,ϕ) = ω oEd(0)β̃(μsw,ϕsw μ,ϕ) 1 μsw μeζμsw , (1)

with the approximations of the QSSA becomes

Lu(QSSA)(ζ,μ,ϕ) Ed(0) = bb a + bbβ̃(μsw,ϕsw μ,ϕ) B 1 μsw μeζμsw

When evaluated just below the sea surface at ζ = 0, this quantity is related to Rrs by

Rrs = t n2 Ed(0) Ed(inair) Lu(QSSA)(0,μ,ϕ) Ed(0) . (2)

Here t is the transmittance of radiance from water to air. For nadir-viewing radiance, μ = 1 and t 0.98. n 1.34 is the index of refraction of water, and the ratio of irradiances (close to 1 for solar zenith angles away from the horizon) converts the underwater irradiance to the above-water value used in the definition of Rrs. Thus for nadir-viewing Rrs we have

Rrs = bb a + bbβ̃(μsw,ϕsw μ = 1,ϕ = 0) B 1 μsw + 1, (3)

The factor of β̃B is determined by the shape of the total phase function, which in turn is determined by the type of particles in the ocean. The μsw factor is determined by solar angle. The remaining factor,

G = bb a + bb,

shows that, to first order, Rrs depends on the IOPs via bb(a + bb), where both a and bb are functions of depth and wavelength. G is sometimes called the Gordon parameter in recognition of his use of this quantity in numerous ocean remote-sensing studies.

Figure 2 compares the accuracy of the QSSA as a function of ωo with the SSA and with HydroLight computations that include all orders of multiple scattering. The HydroLight run used the Petzold phase function of Fig. 1, for which the backscatter fraction is B = 0.0183. The sun was placed at 30 deg in a black sky, and the sea surface was level. This figure shows that the SSA does well only for ωo < 0.3. The SSA is almost a factor of four too small at ωo = 0.7, whereas the QSSA is only 16% too small. Even at ωo = 0.85 the QSSA is only 35% too small, whereas the SSA is only one tenth of the correct value.


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Figure 2: Fig. 2. Predictions of Rrs for the SSA, QSSA, and HydroLight as a function of ωo.

The blue curve of Fig. 3 shows an Rrs spectrum computed by HydroLight for homogeneous Case 1 water with a chlorophyll concentration of 2mgm3. The phytoplankton were modeled with a Petzold average particle phase function as used above. The sun was at 30 deg in a clear sky, and the wind was 5ms1. Those sky and surface conditions violate the assumptions of collimated incident radiance and a level sea surface that underlie the SSA and QSSA. Nevertheless, the QSSA, evaluated using the IOPs generated by the HydroLight bio-optical model, gives an amazingly close prediction of the exact spectrum even though ωo was in the range of 0.80 to 0.86 between 350 and 575 nm. The QSSA of course does not capture the chlorophyll fluorescence peak near 685 nm. The SSA fails badly until near 750 nm, where ωo is less than 0.15.


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Figure 3: Fig. 3. Predictions of Rrs for the SSA, QSSA, and HydroLight as a function of wavelength for Case 1 water with a chlorophyll concentration of 2mgm3.

The excellent performance of the QSSA seems counterintuitive given the crudeness of its phase function approximation. Why, in particular, does it do better than the SSA, since the SSA uses the full phase function, which is certainly a better description of nature than a delta function for forward scattering? The answer lies in how backscattering is parameterized in the QSSA. Note from Eq. (1) that in the SSA the backscatter part of the phase function is weighted by (multiplied by) ωo. For a typical water body with a scattering coefficient four times the absorption coefficient, b = 4a, and a Petzold phase function with a backscatter fraction of B = 0.018, this gives

ωoβ̃ = b a + bβ̃ = 0.80β̃.

Equation (3) shows that the QSSA for the same water weights β̃ by

bb a + bb 1 Bβ̃ = bB a + bB 1 Bβ̃ = 3.73β̃.

Thus, for these IOPs, the QSSA weights the backscatter part of the phase function by a factor 4.66 times greater than that of the SSA. This corresponds precisely to the increase in Rrs for the QSSA compared to the SSA as seen in Fig. 2 for ωo = 0.8. In effect, the QSSA accounts for the ”missing” multiple scattering in the single-scattering formulation by increasing the amount of backscattering. In other words, the QSSA parameterizes multiple scattering within the single-scattering mathematical framework by artificially increasing the backscattering.

The main utility of the SSA and QSSA is in the insight they give to the dependence of various AOPs on the IOPs. We saw above that Rrs depends primarily on bb(a + bb). Likewise, the SSA and QSSA both show that near the sea surface Kd depends primarily on a + bb. Gordon (1977) and (1994) show comparisions between the QSSA and exact computations of other quantities such as the near-surface irradiance reflectance R and mean cosines (the inverse of the distribution functions used in those papers).

Two more comments must be made for completeness. As Gordon noted in his original paper, the desired partioning of scattered radiance is into upward vs. downward components. His development and the one above truncated the phase function out to a scattering of ψ = 90 deg. This separation of backward and forward scattering corresponds to radiance scattered into upwelling and downwelling directions, respectively, only if the sun is at the zenith. If the sun is not at the zenith, then the phase function used in the QSSA should include only the angles that contribute to upward scattering, in which case B in Eq. (3) is not exactly the backscatter fraction of the original phase function. However, as we have seen in the example calculations with the sun at a 30 deg zenith angle, the QSSA still works well for non-zenith solar angles even when evaluated with B equal to the phase function backscatter fraction. Finally, note that in the limit of no scattering, i.e., ωo 0, the QSSA reduces to the SSA.

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