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Application of the small-angle approximation (SAA) to the radiative transport equation results in a linear, integro-differential equation describing the lateral spread of a beam as it propagates through a medium (for example, Arnush D 1972, his Equation 3). That latter equation can then be solved by integral transform techniques (Arnush D 1972). The result is a closed form expression for the modulation transfer function (MTF), F(ψ, r), of the medium, where ψ is the spatial frequency in cycles per radian. The normalized point spread function, f (θ, r):
| f (θ, r) = r 2 PSF (θ, r) | (1) |
where r is the distance to the point source and PSF denotes the point spread function (PSF) of turbid medium. PSF is related to the MTF of turbid medium via the Hankel transform pair (for example, Mertens LE and Replogle 1977):
|
f (θ, r) = 2π ∫0∞J0 (2π θ ψ) F(ψ, r) ψdψ |
(2a) |
| F(ψ, r) = 2π ∫0θ max J0 (2π θ ψ) f (θ, r) θdθ | (2b) |
where J0 is the zero-th order Bessel function, ψ is the angular frequency, and θmax is the maximum angle conforming to the SAA. Wells WH 1969 suggested using θmax = 10°. In practice, θmax would be the half-angle of the field of view of the detector. The Hankel transform is obtained from the reduction of the two-dimensional Fourier transform in spherical coordinates (r, θ, φ) by invoking symmetry in the (azimuthal) φ dimension (for example, Williams CS and Becklund 2002 pp. 158-162, Ferrari JA 1995). Scattering of light by seawater is generally independent of φ, even for polarized beams, as long as the scattering angle is small.
Following the small-angle scattering approximation theory of image transfer in a turbid medium (Wells WH 1973, 1969, Dolin LS 1964 - as cited by Zege EP and Kokhanovsky 1994), the functional form of the MTF is
| F(ψ, r) = exp[-cr + bf r P(ψ)] | (3) |
where c = a + b is the attenuation coefficient, a is the absorption coefficient, and b is the scattering coefficient. The scattering coefficient in the forward direction, bf , is approximated by bf ' = ηb, where:
| η = 2π ∫0θ max p(θ ) sinθ dθ | (4) |
In the collimated illumination case, the function P(ψ) in Equation 3 is the Hankel transform of the scattering phase function (SPF), denoted by p(θ) in Equation 4. The relevant Hankel transform pair is,
|
p(θ ) = 2π ∫0∞ J0 (2π θ ψ) P(ψ) ψdψ |
(5a) |
| P(ψ) = 2π ∫0θ max J0 (2π θ ψ) p(θ ) θ dθ. | (5b) |
Note a slightly different formulation for the case of illumination by a point source (Wells WH 1969, his Equations 2 and 3). The SPF is the normalized volume scattering function (VSF). Hence, the scattering function of a turbid medium must be known to calculate the PSF of that medium, as hinted by Figure 1 in Point spread function: Definition geometry. Various analytical forms of the SPF have been used to obtain an analytical form for the PSF (see Point spread function and the small-angle approximation to the radiative transfer equation). Better results are obtained by using measured data for the SPF and performing numerical Hankel transform.
| CITATION: Swanson N. L. 2008. The small-angle approximation to the RTE with application to ocean waters (www.tpdsci.com/Tpc/RTESAASw.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 28-Oct-2008 Modified: 27-Jun-2009 Peer-reviewed: 31-Oct-2008 |
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