Apr 1, 1986 - Steven G. Ackleson and Vytautas Klemas. A two-flow model is ...... J. E. Tyler, R. C. Smith, and W. H. Wilson, "Predicted Optical. Properties for ...
Two-flow simulation of the natural light field within a canopy of submerged aquatic plants Steven G. Ackleson and Vytautas Klemas
A two-flow model is developed to simulate a light field composed of both collimated and diffuse irradiance
within natural waters containing a canopy of bottom-adhering plants. To account for the effects of submerging a canopy, the transmittance and reflectance terms associated with each plant structure (leaves, stems, fruiting bodies, etc.) are expressed as functions of the ratio of the refractive index of the plant material to the refractive index of the surrounding media and the internal transmittance of the plant structure. Algebraic solutions to the model are shown to yield plausible physical explanations for unanticipated variations in volume reflectance spectra. The effect of bottom reflectance on the near-bottom light field is also investigated. These results indicate that within light-limited submerged aquatic plant canopies, substrate reflectance may play an important role in determining the amount of light available to the plants and, therefore, canopy productivity.
1.
Introduction
II.
Two-Flow Modeling Approach
Submerged aquatic plants are believed to play a major role in the ecosystem of coastal, estuarine, and inland waters. In Chesapeake Bay, species such as Zostera marina (eel grass) provide food, shelter, and breeding areas for waterfowl, fish, shellfish, and many other forms of aquatic life. Because these ecosystems are of enormous commercial value, there exists a need to assess periodically the abundance and health of submerged aquatic plant communities. The purpose of this work is to model the natural light field around and within a canopy of submerged aquatic plants. Such a model will have several important applications. In remote sensing, to optimize sensor design for detecting submerged vegetation and to interpret accurately aircraft or satellite imagery, a thorough understanding of the radiative transfer processes is required. The same is true for estimating productivity. Because light is essential to the growth and survival of all plants, an accurate assessment of submerged canopy productivity must include detailed knowledge of the within-canopy light field. The model development follows the well-known two-flow approach, in which the Suits modell 2 formulated for terrestrial plant canopies is combined with a model to simulate the natural light field within natural
The two-flow approach to modeling natural light fields was first reported by Schuster4 in a study of radiative transfer through a foggy atmosphere. Since then, two-flow models have been applied to a wide variety of problems pertaining to atmospheric scattering, terrestrial plant canopies, 2 5-7 and natural wa-
waters.
where
3
ters.3, 9 - 14
As the name implies, in a two-flow model the light field is conceptualized as two streams of irradiance. For natural light fields, it is common to define one stream as flowing downward through the optical media and the other as flowing upward. Where the model domain may be illuminated with direct sunlight and skylight, the downwelling stream may have both a collimated and diffuse component, while the upwardflowing stream is assumed to be diffuse. The media through which the irradiance propagates are assumed to be plane-parallel and homogeneous in optical properties. Since all irradiant flux is in the vertical direction, the problem becomes one-dimensional. The governing equations for the two-flow model having a collimated component is of the form E= UE,
(1)
[dzEdz+
dE,(z,+) The authors are with University of Delaware, College of Marine Studies, Newark, Delaware 19716. Received 25 October 1985. 0003-6935/86/071129-08$02.00/0. © 1986 Optical Society of America.
dE(z,-)
.k=dz
I E =1
dE,(z,+)
_dz
Ed(z,+)] E(
)
EC(Z,+)
_
1711
1
7=
12
L
°
'112
113
'122 '723
0
1 April 1986 / Vol. 25, No. 7 / APPLIEDOPTICS
,
33_
1129
z denotes vertical location within the optical media and is positive down, and the subscripts d and c indicate diffuse and collimated irradiances, respectively. The direction of irradiance propagation is labeled either + for downwelling or
for upwelling.
-
Ill.
in classic hydrologic optics theory 3 as 7l = -a(z)D(z,+)
The ele-
ments of the coefficient matrix n represent the various scattering and attenuation coefficients of the two-flow model, which are functions of the scattering and absorption characteristics of the optical media. Having assumed a homogeneous optical media, Eq. (1) becomes linear, and, with the assumption that absorption is greater than zero, the general solution is of the form E=t
exp(Kz),
I) = 0,
-
(
(3)
where I is a 3 X 3 identity matrix.
The above expres-
sion is identical in form to that used in eigenvector analysis. In other words, E is a solution to the twoflow model if there exists a vector of eigenvalues K and a matrix of associated eigenvectors for the coefficient
matrix
.15
Solving Eq. (3) for K and results in the general solution to the two-flow equations: 11 exp(Klz)
E = 421exp(Klz)
L
412 exp(K 2 Z)
413 exp(K 3 z)
422 exp(K 2 Z)
423 exp(K 3 z)
0
0
exp(K3Z) -
[C'
whe: re 1 = 1 + 112- '22
12
K1
2
1 + 21- '11I
+ '112 -122 K2
422 = 1 +
K1
'121- 11
'113 =
1
K3
'723('111(122 -
K3)-
K 3 )(711 1 -
13721
K3 ) -
112'21
(6d)
'122=
a(z)D(z,-) +Bb(z,-),
(6e)
'23 =
-bb(z,+) secO,
(6f)
'133 = -[a(z)
+ bf(z,+) + bb(z,+)] secO,
(6g)
where a(z) is the beam absorption coefficient, D(z,d) are radiance distribution functions, B(z,±) are the backscatter coefficients for diffuse irradiance, bf(z,+) and bb(z,+) are, respectively, the forward-scattering and backscattering coefficients for collimated irradiance, and 0 is the apparent solar zenith angle. In many historical applications of the two-flow model to natural waters, it is assumed that the downwelling radiance composing the diffuse stream is distributed similarly to the upwelling radiance. This assumption is useful because it decreases the number of coefficients required by the model, and the analytical solutions become greatly simplified. However, within natural waters, this assumption is unjustified. Within the clear waters of Lake Pend Oreille, the distribution of downwelling radiance is much different from that of upwelling radiance regardless of depth16 ; D(z,+) 1.25 and D(z,-) 2.7. Downwelling radiance intensities within optically deep water are peaked sharply within a cone of half-angle of -47° and centered about zenith, while upwelling irradiance is a minimum at nadir and increases toward the horizon. Within this work, the two-flow coefficients associated with each diffuse irradiance stream are assumed unique.
K3
('111+ 22) + [(11 - '22)2 + 4'1127211 2 ('11 + 22) - [(1112
K2=2
22)2 + 41211211
K3 = '33.
The elements of the constant vector [Cl,C2 ,C3 ]T are determined based on the boundary conditions of the particular problem. Equation (4) may be expressed in a more simplified form: E=
C.
APPLIEDOPTICS / Vol. 25, No. 7 / 1 April 1986
Optically Shallow Water
Within optically shallow water, the light field at the surface (z = 0) is measurably affected by changes in the depth and optical properties of the bottom boundary. At the surface, the quantity of downwelling irradiance, both E,(O,+) and Ed(O,+), is presumed known, and the objective here is to derive expressions for Ec(z,+), Ed(z,+), and Ed(z,-). The followingdiscussions deal primarily with a water column, but it is easy to see how each remark may be applied to either a submerged or terrestrial plant canopy. Ignoring for the present any surface effects, such as Fresnel reflectance and refraction, the initial condition just below the air-water interface is
'113
1-
'713'121- '123(7111 - K3 )
1130
(6c)
bf(z,+) secO,
K2
('122 - K3 )('111 - K3 ) - '7127721
K1 =
(6a) (6b)
'121=-Bb(Z,+),
A. = _13 '121
-Bb(Z,+)
'12 = Bb(Z,-),
(2)
where the elements of vectors and K are functions of the two-flow scattering and attenuation coefficients. Substituting Eq. (2) for E in Eq. (1) leads to
Two-Flow Model for Natural Waters
The two-flow coefficients for water are defined with-
(5)
Ed(0,+) C = E(O,-) "'b
I
(7)
LEd(O,+)J
Because the water is optically shallow, a measurable quantity of light propagates down through the water
17 TableI. Estimations of Sea SurfaceReflectance
tion within the water. Rewriting Eq. (5) as a set of simultaneous
Surface
illumination Clear sky, 0° = 60° Uniform Overcast
0.1 0.066 0.052
EJ(,+)t
13
(12)
- C 2412
411
CEd(0,-)01
- Ed(0,+)02 + E~O,+)(Q13021 - 23011)
C2=
422411-
I='dC-
Ed(d,-)
(8)
Ec(d,+)
Substituting the right side of Eq. (7) for C in Eq. (8) forms the equality Ed(d,+) -E,(d,+)
*
(9)
-E,(O,+)
The bottom boundary condition requires that upwelling irradiance leaving the bottom be equal to the portion of downwelling diffuse and collimated irradiance reflected from the bottom. At this point, it is important to make distinctions between the bottom reflectance of diffuse irradiance Pbd and that of collimated irradiance Pbc The bottom is assumed to be a Lambertian reflector, which ensures that the upwelling irradiance stream just above the bottom is diffuse. This assumption is much less critical for Ed(d,+) than for Ec(d,+). A mirror, for example, is certainly not a Lambertian reflector, yet it appears so if illuminated with totally diffuse light. In the case of collimated irradiance, not only the distribution but the quantity of reflected radiance is a function of the incident angle. The precise nature of these functions is presently unknown, and, even with the Lambertian assumption, the mo'st that one can say is that Pbd # Pbc is a possible condition. The bottom boundary condition is formulated as a linear mapping function fb so that -a, a 2 1 = a 2 Pbdal Pbca3-
(10)
-
(14)
Equation (5) now defines the subsurface light field for any location within the water column. The effect of the air/water boundary on the subsurface light field may be approximated by making the following substitutions within Eqs. (11)-(14):
Ed(O,+)
I41ol Ed(O,-)
=
(13)
41221
C3 = E(,+).
Ed(d,+)
fb
Ed(0,+)-
C
column and impinges on the bottom, z = d. Here the light field may be expressed as
Ed(d,-)
equations for z = 0 and solving for the
integration constants, C1, C2, and C3, yield
Reflectance
Ed(O,+) = Ed(O+,+)(l - Psd)'
(15)
E,(0,+) = E,(O+,+)(1-p,)
(16)
0 = arcsin(n. sin00 ),
(17)
where the parenthetical 0+ indicates an irradiance quantity measured just above the water surface, Psd and Pse are Fresnel reflectances at the water surface of diffuse and collimated irradiance, respectively, n is the refractive index of water (nw - 1.34 for visible light), and 0a is the solar zenith angle as measured above the water surface. Cox and Munk17 computed the reflectance of a smooth ocean surface under a variety of illumination conditions (Table I). Their results represent the combined reflectances of collimated sunlight and diffuse skylight. For purposes here, because the surface illumination is divided into diffuse and collimated components, Psd is probably best approximated by the overcast computation of Cox and Munk, 0.052. For the collimated component, surface reflectance may be computed using the Fresnel relationship for randomly polarized irradiance,
Lsin
05
2 (01 - 02) sin 2(01 + 02)
tan 22(01 - 02) tan (01 + 02)]
(18)
Upwelling irradiance measured just above the water surface may be defined as
_a3_
Ed(O+,-)
Applying fb to Eq. (9) and solving for Ed(,-)
(
Ed(O,-) =Ed(O,+) 22P1 \412P1
-
J
llP2
Ec(0,+) E21P2* 412P1 -
llP2
X ((Q1223- 1322)Pl + (Q1321-11023)P2 + (Q1122-
(11)
1221)P3*),
where P1
Pbd l
-
P2 = (Pbd12 -
21 22) expl(K2
-
=
Ed(O,-)(1 - Pwd)+ p dEd(°+,+) + PscE,(O+,+), (19)
yield
KI)dj,
P3 = (Pbc+ 13Pbd- 423)exp[(K3 - Kl)d].
Having defined the light field at the water surface, expressions may be derived for irradiance at any loca-
n,2
where Pwd is the internal reflectance of the water surface. Gordon1 8 reported that for a uniformly distributed light field, the internal reflectance of a smooth surface is 0.49. However, most often, the subsurface radiance distribution is not uniform but is a minimum at nadir and increased toward the horizon. Because upwelling light reaching the surface at angles greater than -47° is almost completely reflected internally, Pwd = 0.49 is probably an underestimation. Nevertheless, Gordon's value is used here for the purpose of illustration. B.
Optically Deep Water
Within optically deep water, the light field at the upper boundary is not measurably affected by changes 1 April
1986 / Vol. 25, No.7 / APPLIEDOPTICS
1131
in the depth or optical properties of the lower boundary. Expressions of irradiant intensities throughout an optically deep water coluiin may be derived by taking the limit of Eq. (11) as d -X and solving for the new integration constants: Ed(O,-) = Ed(O,+) 422+
E,(0,+) (23-
(20)
1322)
C1 = 0,
(21)
Ed(0 +) -E
(0 +)413
(2
12
(23)
Multiple Layers
In the previous sections, the water column is represented as a single homogeneous layer. Recently, how-
ever, researchers have begun to express concern for the case of vertically inhomogeneous waters. 1 9
Klemas2 0
20
Philpot
and were able to attribute variations in remotely sensed water radiance to vertical changes in the concentration of suspended and dissolved material. Within plant canopies, both terrestrial and submerged aquatic plant structures, such as leaves, stalks, and fruiting bodies, are often vertically stratified within the canopy. Within water, a canopy of bottom-adhering plants may occupy only the lower part of the water column. It would be useful, therefore, if the two-flow model could be made to simulate the case of a vertically stratified optical media. A vertically inhomogeneous water column may be simulated by dividing the column into a finite number of horizontal layers. As before, each layer is assumed to be homogeneous. The layer boundaries are transparent to streams of irradiance, both diffuse and collimated. To derive expressions of irradiance for any location within the water column, the integration constants associated with each layer must be obtained. This is accomplished by first calculating the diffuse volume reflectance of the water column at the top of each layer, starting with the bottom layer and working sequentially to the surface layer. In each case, the layer is assumed to be optically shallow. For the lower-most layer, the bottom boundary condition is simply the substrate reflectance. For all other layers, however, the bottom boundary condition becomes the volume reflectance of the water column, as measured at the top of the underlying layer. The linear mapping function associated with the lower boundary condition of layer i is Fal fbi
1a 21
a -
Two-Flow Model for a Submerged Plant Canopy
Rd..Ral
plant canopy in which the model coefficients are derived as functions of the optical properties of the plant structures and the canopy morphology. Each type of plant structure within the canopy is idealized as a suite of perfectly diffusing panels both for transmitted and reflected irradiance. Plant structures are assigned a preferred nadir orientation but are assumed distributed randomly with respect to azimuth. For example, the attenuation coefficient for diffuse irradiance is defined within the Suits model as m
U11=
ZNij cosfj(l
- rdij)
j=1
+
si
-
dij + Pdi )
-
l
(24)
where Rdi-' and Ri-i are volume reflectances of the water column measured at the top of layer i - 1 for diffuse and collimated illumination, respectively. Notice that in the case of the bottom layer, Rdi-1 = Pbd and Rci- = Pbc, and Eq. (24) is identical to Eq. (10). APPLIEDOPTICS / Vol. 25, No. 7 / 1 April 1986
(25)
where j identifies canopy structures within layer i, such as leaves, stalks, or fruiting bodies, Nij is the cumulative one-sided structure area per unit canopy volume, 4'ij is the zenith angle of a vector pointing normal to the surface of the structure, Tdij is the diffuse transmittance of the structure, and Pdij is the diffuse reflectance of the structure. Several authors have compared Suits model predictions of canopy reflectance with field measurements for canopies of cultivated wheat and cotton.6 - 8 Although the Suits model was found to overestimate consistently the measured reflectance, the discrepancies between predicted and observed values were greatest for near-IR radiation. Within the visible portion of the spectrum, the Suits model accurately predicted measured reflectance. For a submerged canopy, only the visible portion of the spectrum is relevant because water is an efficient absorber of IR radiation. The Suits model is modified to account for a submerged canopy of vegetation by adding the two-flow coefficients of the Suits model to weighted versions of the corresponding water model coefficients: v = ns + V.17.,
C3-
1132
IV.
Suitsl,2 reported a two-flow model for a terrestrial
C3 = E,(0,+).
C.
Armed with the lower boundary conditions for each layer, the only remaining task is to calculate the integration constants for each layer, starting with the surface layer and progressing sequentially to the bottom layer. In each case, the downwelling collimated and diffuse irradiance at the top of a layer constitutes the initial conditions, which, together with the lower boundary condition, define the integration constants for that layer.
(26)
where the subscript s indicates the matrix of Suits model coefficients, and the subscript w identifies the water coefficient matrix. The weighting variable V is a value between 0 and may be interpreted as the percent canopy volume occupied by water. Presumably, for most submerged canopies, the volume of water displaced by plant material is small, and, for purposes of illustration, Vw= 0.95 is used here.
25
Table 11. Beam AbsorptionCoefficientsFor Clear Natural Water
+ 0
A(nm)
a
)(nm)
a
X(nm)
a
450 460 470 480 490 500 510 520 530 540
0.0367 0.0374 0.0386 0.0397 0.0416 0.0459 0.0536 0.0579 0.0625 0.0668
550 560 570 580 590 600 610 620 630 640
0.0735 0.0806 0.0942 0.1165 0.1554 0.2242 0.2561 0.2830 0.3123 0.3418
650 660 670 680 690 700
0.3774 0.4133 0.4545 0.4909 0.5371 0.6181
where n
= nij/ na, and r* is the transmittance of the plant-air or -water interface. If the transmittance of the plant structure is Lambertian and if n > 1, Stern
reported that 440
480
520
560
600
640
680
T*
720
= 4(2n+ 1) +4n(n 2+ 2n-1) 3(n + 1)2 (n2 + 1) 2(n2 - 1)
Wavelength (nm)
Fig. 1. ESM computations of volume reflectance from a clear col-
umn of water containing a canopy of bottom-adhering plants. Values represent measurements made just above the water surface and include the reflectance of diffuse and collimated illumination from
the air-water interface. Canopy optics and morphology are held constant while the surface water layer is varied in thickness from 0 to 20 m. The 20-m case represents an optically deep surface layer.
V.
Decomposition of r 11 And pl
There is as yet no clear connection between a submerged canopy and terrestrial canopy. To investigate the effects of submerging a canopy, the plant structure transmittance and reflectance parameters, rij and Pij, must first be expressed as functions of the optical properties of the surrounding media. Allen et al.,21 in applying the work of Stern,2 2 were able to derive expressions for the transmittance and reflectance of a corn leaf as functions of the internal transmittance of the leaf Tij, the refractive index of the plant material making up the leaf nij, and the refractive index of the ambient media na: pij = 1-
* +
TU.2T*2 n4T 12:!(n22 -
'*
)
(27)
=
n4 - Tj2(n 2
T*)2'
I
Table Ill.
X(nm) 450 460 470 480 490 500 510 520 530 540
0.0437 0.0476 0.0529 0.0604 0.0626 0.0814 0.1242 0.1832 0.2449 0.2797
-
_ 1) In(n) + 2n(n2 1) ln[( 1)2 (n + ),
+
-
1).
(29)
The constraint that n > 1 requires that, for a terrestrial plant structure, nij > 1, while in the case of an aquatic plant structure, nij> 1.34. Although few measurements of nij have been reported, those that do exist within the literature 21 23 indicate that for healthy vegetation, nij has a value of between 1.41 and 1.55. Replacing the Suits parameters, rij and Pij with Eqs. (27) and (28), respectively, yield a two-flow model that may be applied to either a terrestrial or submerged canopy. It should be noticed that when V, = 0 and na = 1, as would be the case within a terrestrial canopy, the elements of X become identical to the Suits twoflow coefficients. On the other hand, when V, = 1 and Ni = 0, i.e., a water column containing no submerged plant canopy, the elements of it are identical to Preisendorfer's two-flow coefficients for water. Throughout the remainder of this work, the two-flow model for a submerged canopy is referred to as the extended Suits model or ESM. VI. Applications of the ESM in Relation to Remote Sensing
In remote sensing, the ESM may be used to identify optimum scanner specifications to test new data analysis techniques and identify possible causes for variations within imagery. As an example, the ESM is used
TO n2 *2 T-
(n2
Zostera Marina Diffuse TransmittanceAnd Reflectance
P
X(nm)
0.0036 0.0037 0.0038 0.0039 0.0041 0.0047 0.0059 0.0082 0.0102 0.0113
550 560 570 580 590 600 610 620 630 640
0.3046 0.3231 0.3266 0.3211 0.3126 0.3044 0.2941 0.2886 0.2795 0.2504
P
X(nm)
r
P
0.0120 0.0122 0.0120 0.0114 0.0108 0.0101 0.0095 0.0090 0.0084 0.0074
650 660 670 680 690 700
0.2081 0.1777 0.1555 0.1794 0.2954 0.4269
0.0065 0.0061 0.0061 0.0078 0.0121 0.0196
1April 1986 / Vol. 25, No. 7 / APPLIEDOPTICS
1133
here to calculate the volume reflectance R(0+) of an optically shallow water column containing a canopy of submerged bottom-adhering plans (Fig. 1). The calculations represent what would be measured just above the water surface and include the reflectance of diffuse and collimated illumination from the air-water interface. The water column is divided into two layers, a surface water layer and a single underlying layer containing the plant canopy. The plant canopy morphology is held constant, while the thickness of the surface water layer is varied. The two-flow scattering coefficients for water are computed from measurements of volume scattering conducted in clear ocean water2 4 and radiance distribution measured in clear lake water1 6 : Bb(+)
=
0.0059;Bb(-) = 0.047;bb(+)
=
0.0029;and
bf(+)= 0.17. The volume scattering function adopted for water is identical to Petzold's AUTEC7 measurements and is assumed to be constant between 450 and 700 nm. The beam absorption coefficient is, on the other hand, wavelength dependent and represents clear natural water (Table II).25 The morphology of the plant canopy is defined based on observations reported for Zostera marina inhabiting the southern portion of Chesapeake Bay.2 6 The canopy thickness is 1 m, and Nij = 2. Leaf orientation within a Zostera canopy is dependent on the local current regime. With large currents, the leaves become bent over in the direction of the water flow. However, when the currents are small, as assumed here, the individual leaves assume a near vertical orientation and 6 = 0. The directional transmittance and reflectance were measured from individual Zostera leaves harvested from a site in southern Chesapeake Bay adjacent to Guinea Marsh, VA (Table III). Measurements were collected using a Brice/Pheonix Scattering Photometer fitted with an Oriel 150-W tungsten lamp and a UDT Scanning Radiometer. Plant specimens were submerged within a small vessel of filtered bay water positioned on a central stage and illuminated with collimated light normal to the plant surface. Directional transmittance
was measured for look angles
ranging from 0 (normal to the specimen surface) to 45°. Diffuse transmittance was approximated as Tij(X)=L ( 0 )
where 450 nm
X
-
(30)
700 nm, LT(X,O) is the transmit-
ted radiance recorded normal to the specimen surface, Eo(X)is the irradiance illuminating the specimen, and ' is the half-angle subtended by the radiometer collector (220). The coefficient x is computed by fitting the directional transmittance to a function of the form cosx(6), where 6 is the radiometer look angle.
The problem of estimating diffuse reflectance was more difficult because directional reflectance measurements were limited to 135° (450 relative to the specimen surface). In the absence of more detailed directional reflectance data, diffuse reflectance was 1134
APPLIEDOPTICS / Vol. 25, No. 7 / 1 April 1986
estimated using the assumption that directional reflectance from the specimens was Lambertian. The reflectance of the bottom is 0.05 and is analogous to the case of Zostera colonizing a substrate of
dark mud. ESM simulations of volume reflectance spectra may be used in remote sensing to specify optimum band location, width, and radiometric sensitivity. If, in Fig. 1, the objective is to distinguish between optically deep water and plant canopies occupying shallow water, neglecting atmospheric effects, a narrowband centered around 560 nm would serve the purpose as would a
narrowband within the blue portion of the spectrum between 450 and 480 nm. The spectral region 600 nm, because of increased water absorption, may converge too rapidly to the deep water curve for deeper plant canopies to be distinguished. An interesting situation occurs between 490 and 500 nm in that the volume reflectance appears to be insensitive to changes in canopy depth. Understanding why this happens may be gained through the ESM algebraic expression of upwelling irradiance from an optically shallow water column. To avoid needless complexity, the water column is thought of as a onelayer system, and Pbd and Pbc represent the reflectance of diffuse and collimated irradiance, respectively, from the submerged plant canopy. In other words, the plant canopy, because the optical and morphological characteristics are held constant, becomes the bottom boundary. The algebraic expression is, therefore, identical to Eq. (11), where d is the variable thickness of the surface water layer. For Ed(0,-), to be unaffected by changes in the thickness of the surface water layer, the exponential depth terms within P2* and P3* must somehow be removed. Looking first at the case of diffuse surface illumination, Rd(O+)=Si (22P1* -
t2P2*) + S 2 '
(31)
where S and S 2 account for all the surface effects.
Assuming that absorption within the water is greater than zero, Eq. (31) can only be insensitive to depth if P2* = 0. Forming this equality yields Pbd=22412-
(32)
Equation (32) is identical to the expression for inwater volume reflectance of diffuse irradiance from optically deep water-included within the first term on the right-hand side of Eq. (20). In other words, the diffuse component of Ed(,-) is insensitive to the surface layer thickness when the reflectance of the lower layers, the submerged plant canopy, is equal to the deep water reflectance of the surface layer. If the surface illumination is 100% collimated and, once again, assuming that a(z) > 0, the depth-depen-
dent terms within the collimated component of may, on substituting 622/12 for Pbd, be removed if P2* =O. Forming this equality and rearranging yield
Ed(,-)
Pbc
423 -
413 12
(33)
vegetation are identical to those of the previous discus-
1.1
sions where X = 450 nm. The water depth is 1 m, the
canopy occupies the entire water column, and the substrate reflectance is varied from 0 to 1, where Pbd = Pbc The surface illumination is 90% collimated, and the solar zenith angle is 45°. As irradiance propagates down through a submerged plant canopy, the combined effects of scattering and absorption act to attenuate the intensity. Within an optically deep canopy, the light intensity will continue to decrease until no measurable quantity remains. Within an optically shallow canopy, downwelling irradiance reflected from the substrate is immediately added to the stream of upwelling diffuse irradiance. Depending on the magnitude of Pbd and Pbc, this has the effect of either decreasing or increasing the intensity of the light field near the bottom, relative to what intensities would be at a particular depth if the canopy were optically deep. If the bottom reflectance
0
+.
0.7
P
0.6 N
0.82 0.5
04
'S
0.
.
0.40~~~~~ 0
to1whr
0.2
Pd=
0.4
b
0.6
0.5
1
is zero, intensities will be lower than R(z) associated
Depth(in)
Fig. 2. ESM computations of total irradiance within a canopy of aquatic vegetation submerged within clear water. Values are expressed as a percentage of the surface illumination measured just below the air-water interface. Substrate reflectance is varied from 0 to where Pbd Pbc-
The right-hand side of Eq. (33) defines the in-water volume reflectance for optically deep water illuminated by collimated irradiance. Once again, if the underlying layers of a multiple-layer system look like an
optically deep version of the surface layer, variations in the thickness of the surface layerwill have no effect on upwelling irradiance at the surface. Vil.
Intercanopy Light Field
The intensity and spectral quality of the submarine light field play a dominant role in shaping marine productivity. Within coastal and estuarine waters, light is intercepted by benthic and planktonic algae, photosynthetic bacteria, and higher plants, and the intercepted energy is used to convert inorganic matter to organic matter-the well-known process of photosynthesis. In many cases, the intensity of the ambient light field is believed to be the single controlling factor determining the level of productivity. Van Tine and Wetzel2 7 studied the productivity of communities of bottom-adhering plants within Chesapeake Bay and concluded that "light and factors governing light energy available to submerged aquatic plants are principal controlling forces for growth and survival." The ESM may be used to simulate total irradiance E(z) at various depths within a submerged plant canopy. Total irradiance, defined as E(z) = Ed(Z,+) + Ed(z;-) + E,(z,+),
(34)
is expressed here as a percentage of the surface illumination, as measured just below the air-water interface (Fig. 2). In this form, E(Z) is bounded by 0 and 2, where the latter value is the limit as d - 0, Pbd - 1, and Pbc 1. The optical properties of the water and
with an optically deep canopy. If the bottom reflectance equals R(z) of an optically deep canopy, the nearbottom light field within an optically shallow canopy will be no different than at the same depth within an optically deep canopy. If, however, the bottom reflectance is greater than R(z) of an optically deep canopy, the intensity of the near-bottom light field will be increased. This effect of bottom reflectance on the near-bottom light field is supported by Monte Carlo simulations.2 8 In varying the bottom reflectance from 0 to 1, the data reported by Plass and Kattawar indicate that when the bottom reflectance is small relative to the deep water volume reflectance, the near-bottom light field is decreased. For relatively high-bottom reflectances, the near-bottom light field is increased. The apparent effect of bottom reflectance on the within-canopy light field has important implications for estimating the productivity of submerged aquatic vegetation. Within submerged plant canopies, a highly reflective substrate (sand) may increase the amount of available light within the canopy compared to a dark substrate (mud). Within Lower Chesapeake Bay, species such as Zostera marina are believed to be lightlimited due in part to high turbidity levels. These same canopies may be found colonizingsubstrates having a wide variety of reflectances, although the dominate component is nearly always sand. We are moved to speculate that within light-limited canopies, substrates having higher reflectances may increase the chance of plant survival. However, to date, no studies have been documented that attempt to relate substrate reflectance to plant photosynthetic rates or canopy productivity. Vil.
Conclusions
The ESM is formulated in two steps: (1) the linear combination of two-flow coefficients pertaining to natural water and to a terrestrial plant canopy and (2) the decomposition of the plant component transmittance and reflectance terms into functions of the optical 1 April 1986 / Vol. 25, No. 7 / APPLIEDOPTICS
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properties of the plant structures and the surrounding media. The two-flow coefficients for water are derived from classical hydrologic optics theory, while the coefficients pertaining to vegetation are identical to those developed by Suits.1 2 In so doing, the ESM becomes extremely versatile. Given the appropriate boundary conditions, the ESM may be applied to a submerged plant canopy, natural waters containing no vegetation, or a terrestrial plant canopy. Because the plant structure transmittance and reflectance terms are expressed as functions of the optical properties of the plant material and the surrounding media, the ESM may be used to investigate the effects of submerging a plant canopy while holding the optical properties of the plant structures constant. In problems related to remote sensing and plant canopy productivity, the ESM may be used to increase our understanding of the complicated interaction of light with submerged plant communities. The algebraic solutions to the ESM are shown here to yield plausible physical explanations for observed changes in volume reflectance spectra. In one example, volume reflectance from a water column containing a submerged plant canopy is shown to be insensitive to changes in canopy depth. On closer inspection of the algebraic solutions, it is evident that when the canopy reflectance is equal to deep water reflectance, variations in the thickness of the surface water layer have no effect on the upwelling irradiance from the water column. Bottom reflectance is shown through the ESM to effect significantly the near-bottom light field, a phenomenon which is also predicted in Monte Carlo simulations. This result has potentially far-reaching implications within the study of submerged canopy productivity. Within light-limited canopies of Lower Chesapeake Bay, it may be possible to explain variations in canopy productivity in terms of substrate reflectance. Support for this project was provided by NASA Office of Life Science through contract NAGW-374
and by NASA Goddard Space Flight Center through contract NAS5-27580 as part of the Landsat Image Data Quality Analysis Program managed by NASA Goddard Space Flight Center. We wish to thank William Philpot for his critical review of this work and helpful suggestions. References 1. G. H. Suits, "The Calculation of the Directional Reflectance of a Vegetative Canopy," Remote Sensing Environ. 2, 117 (1972). 2. G. H. Suits, "The Cause of Azimuthal Variations in Directional Reflectance of Vegetation Canopies," Remote Sensing Environ. 2, 175 (1972). 3. R. W. Preisendorfer, Hydrologic Optics (U.S. Department of Commerce, NOAA Environmental Research Laboratory, Honolulu, 1976), p. 1-6. 4. A. Schuster "Radiation Through a Foggy Atmosphere," Astrophys. J. 21, No 1, 1 (1905). 5. W. A. Allen, T. V. Gayle, and A. J. Richardson. "Plant-Canopy
6. J. E. Chance, and J. M. Cantu, "A Study of Plant Canopy
Reflectance Models," Final Faculty Research Grant Report, Pan American U. (1975). 7. J. E. Chance, and E. W. LeMaster, "Suits Reflectance Models
for Wheat and Cotton: Theoretical and Experimental Tests," Appl. Opt. 16, 407 (1977). 8. E. W. LeMaster, J. E. Chance, and C. L. Wiegand, "A Seasonal
Verification of the Suits Spectral Reflectance Model for Wheat," Photogr. Eng. Remote Sensing 46, 107 (1980). 9. S. Q. Duntley, "The Optical Properties of Diffuse Materials," J. Opt. Soc. Am. 32, No. 2, 000 (1942).
10. E. 0. Hulbert, "Propagation of Radiation in a Scattering and Absorbing Medium," J. Opt. Soc. Am. 33, 42 (1943).
11. V. J. Joseph, "Untersuchungen ber ober- und unterlichtmessungen im meere und fiber ihren zusammenhang mit durchsichtigkeits-messungen," Dtsch. Hydrogr. Z. 324 (1953). 12. S. C. Jain and J. R. Miller. "Algebraic Expression for the Diffuse Irradiance Reflectivity of Water from the Two-Flow Model," Appl. Opt. 16, 202 (1977). 13. R. Doerffer, "Applications of a Two-Flow Model for Remote Sensing of Substances in Water," Boundary layer Meteorol. 18, 221 (1980). 14. G. H. Suits, "A Versatile Directional Reflectance Model for
Natural Water Bodies, Submerged Objects, and Moist Beach Sands," Remote Sensing Environ. 16, No 12, 143 (1984). 15. W. E. Boyce, and R. C. DiPrima, Elementary Differential Equa-
tions And Boundary Value Problems (Wiley,New York, 1977). 16. J. E. Tyler, "Radiance Distribution as a Function of Depth in an Underwater Environment," Bull. Sci. Instrum. Oceanogr. 7,363 (1960). 17. C. Cox, and W. Munk, "Some Problems in Optical Oceanography," J. Mar. Res. 14, 63 (1955).
18. J. I. Gordon, Directional Radiance (Luminance) of the Sea Surface, SIO Ref. 69-20:50 pp, Visibility Laboratory, Scripps Institute of Oceanography, San Diego (1960). 19. S. Q. Duntley, R. W. Austin, W. H. Wilson, C. F. Edgerton, and S. E. Moran, Ocean Color Analysis, SIO Ref. 74-10, Visibility
Laboratory, Scripps Institute of Oceanography, San Diego (1974). 20. W. Philpot and V. Klemas, "Remote Detection of Ocean Waste," Proc. Soc. Photo-Opt. Instrum. Eng. 208, 189 (1979). 21. W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thom-
as, "Interaction of Isotropic Light with a Compact Leaf," J. Opt. Soc. Am. 59, 1376 (1969).
22. F. Stern, "Transmission of Radiation Across an Interface Between Two Dielectrics," Appl. Opt. 3, 111 (1964). 23. J. T. Woolley, "Refractive Index of Soybean Leaf Cell Walls," Plant Physiol. 55, 172 (1975).
24. T. J. Petzold, "Volume Scattering Functions for Selected Ocean Waters," SIO Ref. 72-28:79pp, Visibility Laboratory, Scripps Institute of Oceanography, San Diego (1972). 25. J. E. Tyler, R. C. Smith, and W. H. Wilson, "Predicted Optical Properties for Clear Natural Water," J. Opt. Soc. Am. 62, 83 (1972). 26. R. L. Wetzel, "Structural and Functional Aspects of the Ecology
of Submerged Aquatic Macrophyte Communities in the Lower Chesapeake Bay," Final Report to EPA, Vol. 1, Virginia Institute of Marine Science, Gloucester Point (1983). 27. R. F. Van Tine, and R. L. Wetzel, "Structural and Functional
Aspects of Submerged Aquatic Macrophyte Communities in Lower Chesapeake Bay," Final Report to EPA, Vol. 2, Virginia
Institute of Marine Science, Gloucester Point (1983). 28. G. N. Plass and G. W. Kattawar, "Monte Carlo Calculations of
Irradiance Specified by the Duntley Equations," J. Opt. Soc.
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