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Diffuse reflection coefficient or diffuse reflectance of light from water
body is an informative part of remote sensing reflectance of light from
the ocean. Diffuse reflectance contains information on content of
dissolved and suspended substances in seawater. Diffuse reflectance is
an apparent optical property that depends not only on inherent optical
properties of the seawater, but also on the parameters of illumination.
The dependence on inherent optical properties is expressed as a
dependence on a ratio of backscattering coefficient bb to absorption
coefficient a. In the open ocean under diffuse illumination of the sky
diffuse reflectance R is linearly proportional to the ratio of bb to a, i. e.
R=kbb /a, with k=0.33 according to Morel and Prieur.
The abovementioned linear equation is very good for the Type I open
ocean waters. It is also acceptable for certain types of coastal waters. In
fact, it is valid for all types of waters when the ratio of bb to a is less
than 0.1. From physical considerations R should always lie between zero
and one for any ratio bb /a between zero and infinity. The linear equation
fails to pass this criterion, i. e. it exceeds unity when bb /a becomes
greater than 1/k, or a<kbb (highly scattering water with a lot of very
small particles). It means that indiscrete use of the linear equation for
coastal waters, when parameter bb /a exceeds limitations of smallness,
can cause unacceptable errors in processing of in situ and remote
sensing optical information.
In order to estimate possible errors in determining diffuse reflectance
we used different approaches to generate diffuse reflectance as a
function of bb /a, or g=bb /(a+bb). One approach is based on numerical
calculations using Monte Carlo simulation, and other approaches were
theoretical. The input values of bb /a have been varied from very small to
very large numbers. It was found that numerically and theoretically
generated results for all varieties of input parameters satisfactory
correspond to the available experimental data.
It was found both theoretically and using Monte Carlo that diffuse
reflectance strongly depends on backscattering coefficient and has very
weak dependence on the shape of the phase function used.
Because we do not have reliable in situ measurements of diffuse
reflectances that represent the whole range of variability of inherent
optical properties, 0 < bb /(a+bb) < 1, we have to choose a dependence
which can be regarded as sufficiently “precise one” in order to be a basis
for error estimation.
Such dependence exists in literature (Haltrin, 1988) and represents an
exact solution of radiative transfer for diffuse reflection of light in a
medium with delta-hyperbolic phase function. This solution lies exactly
in the middle of two Monte Carlo and two theoretical solutions for
diffuse reflections for small values of bb /(a+bb) < 0.2, and it gives
precise and asymptotically correct values for 1 - bb /(a+bb) < < 1 (see
first two figures).
(Morel-Prieur, 1977):
The widely used linear model is very good for bb /(a+bb) < 0.1 and
very satisfactory for bb /(a+bb) ≤ 0.2, it produces wrong results for bb
/(a+bb) > 0.2. The majority of coastal water and almost all open
ocean water cases fall in the range of applicability of linear model.
But the linear model may be very inadequate in some important
and interesting coastal water conditions like hazardous blooms,
spills, etc. For the reasons to avoid possible unacceptable errors and
missing interesting optical events it is advisable to avoid using linear
model to process information related to coastal (Type II and III)
waters. All presented non-linear equations (except the KubelkaMunk equation that is not acceptable for seawater at small values of
bb /(a+bb) < 0.2, and Gordon’s equations that are not valid at
bb /(a+bb) > 0.2, and at bb /(a+bb) < 0.0001) are capable to produce
values of R that are correct for all possible values of bb /(a+bb).
Exact (Haltrin, 1988)
Self-Consistent Asymptotic
(Haltrin, 1985, 1993, 1997)
Self-Consistent Diffuse
(Haltrin, 1985)
In order to detect special optical cases the non-linear equations
should be used in automatic processing of in-situ and remotely
obtained optical information.
Two-Stream:
(Gamburtsev, 1924; Kubelka-Munk, 1931;
Sagan and Pollack, 1967)
Monte Carlo
(Gordon, Brown, and Jackobs, 1975)
Semi-Empirical
(Haltrin and Weidemann, 1996)
Direct
Diffuse
The author thanks continuing support at the Naval Research
Laboratory through the Spectral Signatures 73-5939-A1 program.
This article represents NRL contribution AB/7330-01-0168.
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