Errors generated by the use of a linear model of optical diffuse reflectance in coastal waters

1
Errors generated by the use of a linear model of
optical diffuse reflectance in coastal waters
Naval Research Laboratory, Ocean Optics Section,
Code 7333, Stennis Space Center, USA
Vladimir I. HALTRIN
e-m lthaltrin_at_nrlssc.navy.milgt
lthttp//www7333.nrlssc.navy.mil/haltringt
Introduction
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. Rk bb /a, with k0.33
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 lt k bb (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
/(abb). 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.
References
Conclusion
Linear model
(Morel-Prieur, 1977)
1. G. A. Gamburtsev, On the problem of the sea
color, Zh. RFKO, Ser. Fiz. (Journal of
Russian Physical and Chemical Society, Physics
Series), 56, 226-234 (1924). 2. P. Kubelka and
F. Munk. Ein Beitrag zur Optik der
Farbanstriche, Zeit. Techn. Phys., 12,
593-607 (1931). 3. C. Sagan and J. B. Pollack,
Anisotropic nonconservative scattering and
the clouds of Venus, J. Geophys. Res., 72,
469-477 (1967). 4. H. R. Gordon, O. B. Brown, and
M. M. Jacobs, Computed relationships between
the inherent and apparent optical properties of
a flat homogeneous ocean, Appl. Optics, 14,
417-427 (1975). 5. A. Morel, L. Prieur, Analysis
of variations in ocean color, Limnol.
Oceanogr., 22, 709-722 (1977). 6. V. I. Haltrin
(a.k.a. V. I. Khalturin), Propagation of light
in sea depth, in Remote Sensing of the Sea
and the Influence of the Atmosphere (in
Russian), V. A. Urdenko and G. Zimmermann, eds.
(Academy of Sciences of the German Democratic
Republic Institute for Space Research,
Moscow-Berlin-Sevastopol, 1985), pp. 20-62. 7. V.
I. Haltrin, Exact solution of the characteristic
equation for transfer in the anisotropically
scattering and absorbing medium, Appl. Optics,
27, 599-602 (1988). 8. V. I. Haltrin and G. W.
Kattawar Self-consistent solutions to the
equation of transfer with elastic and inelastic
scattering in oceanic optics I. Model,
Appl. Optics, 32, 5356-5367 (1993). 9. V. I.
Haltrin, and A. D. Weidemann, A Method and
Algorithm of Computing Apparent Optical
Properties of Coastal Sea Waters, in Remote
Sensing for a Sustainable Future Proceedings of
1996 International Geoscience and Remote
Sensing Symposium IGARSS96, Vol. 1,
Lincoln, Nebraska, USA, p. 305-309, 1996. 10. V.
I. Haltrin, Self-consistent approach to the
solution of the light transfer problem for
irradiances in marine waters with arbitrary
turbidity, depth and surface illumination, Appl.
Optics, 37, 3773-3784 (1998).
The widely used linear model is very good for
bb /(abb) lt 0.1 and very satisfactory for bb
/(abb) 0.2, it produces wrong results for bb
/(abb) gt 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 Kubelka-Munk equation that
is not acceptable for seawater at small values of
bb /(abb) lt 0.2, and Gordons equations that are
not valid at bb /(abb) gt 0.2, and at
bb /(abb) lt 0.0001) are capable to produce
values of R that are correct for all possible
values of bb /(abb). 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.
Non-linear solutions for Diffuse reflectance
Exact (Haltrin, 1988)
Self-Consistent Asymptotic (Haltrin, 1985, 1993,
1997)
Self-Consistent Diffuse (Haltrin, 1985)
Two-Stream (Gamburtsev, 1924 Kubelka-Munk,
1931 Sagan and Pollack, 1967)
Because we do not have reliable in situ
measurements of diffuse reflectances that
represent the whole range of variability of
inherent optical properties, 0 lt bb /(abb) lt 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 /(abb) lt 0.2,
and it gives precise and asymptotically correct
values for 1 - bb /(abb) lt lt 1 (see first two
figures).
Direct
Monte Carlo (Gordon, Brown, and Jackobs, 1975)
Diffuse
Acknowledgment
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.
Semi-Empirical (Haltrin and Weidemann, 1996)