IEEE Geoscience and Remote Sensing Magazine - December 2023 - 52

bacteria biofilms [91], [92], soil [66], [67], [68], [69], [72],
[73], plant leaves [49], [74], [75], [90], nonanimal biological
tissues (fruits and vegetables [93], [105], [106], [107], [108],
[109], [110]), and solar system objects [111], [112], [113], [114],
[115], [116], [117], [119], [120], [121].
In this section, we provide a brief description of the most
recent models developed for each of these object types, respectively:
MPBOM [92], MARMIT [72], [73], PROSPECT [49],
[75], Farrell [106], [108], and finally, Hapke [111]. The first two
models belong to the category of the one-layer models of the
Beer-Lambert law branch. PROSPECT belongs to the multilayer
category of the Beer-Lambert law branch. The two last
models belong to the RTE branch (Farrell is in the single-layer
model category for diffusion approximation, while Hapke is
in the single-layer model category for single- and multiplescattering
approximation). All these models consider an incident
incoherent and unpolarized light beam. (Note that the
Schmitt model is the only recent multilayer model for the RTE
branch, but it is outside of the scope of this article.)
ALGAE AND BACTERIA BIOFILM: MPBOM
Microphytobenthos (MPB) refers to the mixture of photosynthetic
algae and bacteria that live in the top layer of sediments
in marine environments. They are present in areas with light
penetration and are commonly found in shallow intertidal
and subtidal areas. MPB has high primary production, comparable
to phytoplankton, but it is limited to a thin biofilm layer
[123]. To recover the optical properties of MPB over the spectral
range (0.4-1
nm ), a multilayered model called MPBOM
was developed [91], [92].
MPBOM considers a homogeneous horizontal layer modeling
the compact layer of translucent microalgae cells constituted
by the MPB (medium 2), with two plane-parallel
faces adjacent to semi-infinite layers: a layer with air on the
top (medium 1) and an opaque background (medium 3) [cf.
Figure 4(a)]. The layer modeling the MPB (medium 2) is characterized
by a refractive index n and an absorption coefficient
a . The thickness L of medium 2 is considered explicitly in .a
The white light is partially transmitted from the air (medium
1) at an angle of incidence i to the algae layer (medium 2)
with a transmissivity t12
is reflected in the air with a reflectivityrt .
[cf. Figure 4(a)]. The other fraction
12 1 12=- Then, the
light accomplishes multiple internal reflections between the
MPB-background interface (of reflection coefficient
the MPB-air interface (of reflection coefficient r ).21
r )23
t21
and
After that,
the proportion of light that is not reflected is transmitted from
the algae layer to the air ().
The light traveling the homogeneous MPB layer of thickness
L follows the Beer-Lambert law. The transmission coefficient
x of the MPB layer is therefore given by
x eeLMPB
==#aa
-(1)
where
we used the fact that L is linearly dependent on the parameter
a of this model ().
aa# =
MP LB
The inherent reflectance R of the MPB and its transmittance
T are evaluated using the well-known Stokes
52
formula [65] (note that R and T are always wavelength
dependent [124])
Rr12
T=
=+
ttr x2
1-rr x2
12 21 23
21 23
tt x
12 23
1 21 23
-rr x2
12 21 23
.
12 21 23
===
(2)
(3)
The authors of the MPBOM model proposed the following
simplifications: rrr RF
where RF
and TF
=== and ttt ,TF
are the reflectance and transmittance of the
interface, respectively, given by traditional Fresnel formulae
[91]. With this simplification, the ratio of the apparent reflectance
RA
of the MPB (i.e., the reflectance of the complete structure)
to its background reflectance RB
(i.e., the reflectance of
the background when medium 2 is omitted) is linked to the
absorption coefficient a through the following formula [92]:
R
R
If measurements of RA
B
A
= e()6
-a
.
(4)
and RB are available, the formula (4) is
very easy to invert to find a . A linear relationship between the
absorption coefficient a at the chlorophyll peak absorption
wavelength (673 nm), computed thanks to MPBOM and the
MPB biomass, has been found out for different backgrounds
with a coefficient of determination R2
between 0.91 and 0.97
[92] (cf. Table 1). We recall that the coefficient of determination
R2
is a standard parameter to evaluate the pertinence of a
linear model for linking two quantities, whose definition can
be found, for example, in [125].
SOIL: MARMIT
For reflectance spectrum modeling of soil in the solar domain
(. .),
04 25 m
- n the most complete model comprises MARMIT
[72], [73]. Soil is a natural resource, a complex ecosystem that
is strongly linked to other environmental compounds (water
and air). Soil content is very important for plant growth and
for supporting many animal and human activities. Different
types of soil exist, each with distinct characteristics that provide
benefits and limitations for each specific activity.
In its original proposal [72], the MARMIT model focused
on wet soil. It represents wet soil as dry soil (medium 3) covered
with a thin layer of water (medium 2), itself covered by a
semi-infinite layer of air (medium 1) [cf. Figure 4(b)].
Similar to the single-layer MPBOM model, the round trip
of the light beam inside medium 2 is the key element to the
optical properties of this model. White light emerging from
the water to the air has undergone at least one reflection on
the soil and has therefore traveled at least twice in the water
layer. The authors of the MARMIT model take account of the
water absorption by introducing the water transmission coefficient
x
. The expression of x depends on the hypotheses
used to model the water. If pure liquid water is assumed for
the top layer (MARMIT-1), x follows the Beer-Lambert law
and thus verifies (1), where MPB
a is replaced by w
a the absorption
coefficient of pure liquid water [126], and L is the
water layer thickness.
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE DECEMBER 2023
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