Two-flow simulation of the natural light field within a canopy of submerged aquatic plants

Two-flow simulation of the natural light field withina 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 irradiancewithin natural waters containing a canopy of bottom-adhering plants. To account for the effects ofsubmerging 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 materialto 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 unanticipatedvariations in volume reflectance spectra. The effect of bottom reflectance on the near-bottom light field isalso 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 plantsand, therefore, canopy productivity.

1. Introduction

Submerged aquatic plants are believed to play amajor role in the ecosystem of coastal, estuarine, andinland waters. In Chesapeake Bay, species such asZostera marina (eel grass) provide food, shelter, andbreeding areas for waterfowl, fish, shellfish, and manyother forms of aquatic life. Because these ecosystemsare of enormous commercial value, there exists a needto assess periodically the abundance and health ofsubmerged aquatic plant communities.

The purpose of this work is to model the naturallight field around and within a canopy of submergedaquatic plants. Such a model will have several impor-tant applications. In remote sensing, to optimize sen-sor design for detecting submerged vegetation and tointerpret accurately aircraft or satellite imagery, athorough understanding of the radiative transfer pro-cesses is required. The same is true for estimatingproductivity. Because light is essential to the growthand survival of all plants, an accurate assessment ofsubmerged canopy productivity must include detailedknowledge of the within-canopy light field.

The model development follows the well-knowntwo-flow approach, in which the Suits modell 2 formu-lated for terrestrial plant canopies is combined with amodel to simulate the natural light field within naturalwaters. 3

The authors are with University of Delaware, College of MarineStudies, Newark, Delaware 19716.

Received 25 October 1985.0003-6935/86/071129-08$02.00/0.© 1986 Optical Society of America.

II. Two-Flow Modeling Approach

The two-flow approach to modeling natural lightfields was first reported by Schuster4 in a study ofradiative transfer through a foggy atmosphere. Sincethen, two-flow models have been applied to a widevariety of problems pertaining to atmospheric scatter-ing, terrestrial plant canopies, 2 5-7 and natural wa-ters.3, 9- 14

As the name implies, in a two-flow model the lightfield is conceptualized as two streams of irradiance.For natural light fields, it is common to define onestream as flowing downward through the optical mediaand the other as flowing upward. Where the modeldomain may be illuminated with direct sunlight andskylight, the downwelling stream may have both acollimated and diffuse component, while the upward-flowing stream is assumed to be diffuse.

The media through which the irradiance propagatesare assumed to be plane-parallel and homogeneous inoptical properties. Since all irradiant flux is in thevertical direction, the problem becomes one-dimen-sional.

The governing equations for the two-flow modelhaving a collimated component is of the form

E= UE,

wheredE,(z,+)[dzEdz+dE(z,-) Ed(z,+)] 1711

.k= dz I E = E(1 ) 1 7= 12

dE,(z,+) EC(Z,+) ° L_dz _

(1)

'112 113'122 '723 ,

0 33_

1 April 1986 / Vol. 25, No. 7 / APPLIED OPTICS 1129

z denotes vertical location within the optical mediaand is positive down, and the subscripts d and c indi-cate diffuse and collimated irradiances, respectively.The direction of irradiance propagation is labeled ei-ther + for downwelling or - for upwelling. The ele-ments of the coefficient matrix n represent the variousscattering and attenuation coefficients of the two-flowmodel, which are functions of the scattering and ab-sorption characteristics of the optical media.

Having assumed a homogeneous optical media, Eq.(1) becomes linear, and, with the assumption that ab-sorption is greater than zero, the general solution is ofthe form

E = t exp(Kz), (2)

where the elements of vectors and K are functions ofthe two-flow scattering and attenuation coefficients.Substituting Eq. (2) for E in Eq. (1) leads to

( - I) = 0, (3)

where I is a 3 X 3 identity matrix. The above expres-sion is identical in form to that used in eigenvectoranalysis. In other words, E is a solution to the two-flow model if there exists a vector of eigenvalues K and amatrix of associated eigenvectors for the coefficientmatrix .15

Solving Eq. (3) for K and results in the generalsolution to the two-flow equations:

whe:

11 exp(Klz)

E = 421 exp(Klz)

L 0

re

1 = 1 + 112 - '22K1

2 1 + 21 - '11IK 1

412 exp(K2 Z) 413 exp(K3 z)

422 exp(K2 Z) 423 exp(K3 z)

0 exp(K3Z) - [C'

12 + '112 -122K2

'121 - 11422 = 1 +

K2

_13 = '121 '723('111 - K3) - 13721 '113

1 K3 (122 - K3 )(711 1 - K3 ) - 112'21 1- K3

'713'121 - '123(7111 - K3 )

('122 - K3)('111 - K3) - '7127721

K1 =('111 + 22) + [(11 - '22)2 + 4'1127211

2

('11 + 22) - [(111- 22)2 + 41211211K2 =2 2

K3 = '33.

The elements of the constant vector [Cl,C2 ,C3 ]T aredetermined based on the boundary conditions of theparticular problem. Equation (4) may be expressed ina more simplified form:

E = C. (5)

Ill. Two-Flow Model for Natural Waters

The two-flow coefficients for water are defined with-in classic hydrologic optics theory 3 as

7l = -a(z)D(z,+) -Bb(Z,+)

'12 = Bb(Z,-),

'113 = bf(z,+) secO,

'121 =-Bb(Z,+),

'122 = a(z)D(z,-) + Bb(z,-),

'23 = -bb(z,+) secO,

'133 = -[a(z) + bf(z,+) + bb(z,+)] secO,

(6a)

(6b)

(6c)

(6d)

(6e)

(6f)

(6g)

where a(z) is the beam absorption coefficient, D(z,d)are radiance distribution functions, B(z,±) are thebackscatter coefficients for diffuse irradiance, bf(z,+)and bb(z,+) are, respectively, the forward-scatteringand backscattering coefficients for collimated irradi-ance, and 0 is the apparent solar zenith angle.

In many historical applications of the two-flow mod-el to natural waters, it is assumed that the downwellingradiance composing the diffuse stream is distributedsimilarly to the upwelling radiance. This assumptionis useful because it decreases the number of coeffi-cients required by the model, and the analytical solu-tions become greatly simplified. However, within nat-ural waters, this assumption is unjustified. Withinthe clear waters of Lake Pend Oreille, the distributionof downwelling radiance is much different from that ofupwelling radiance regardless of depth16; D(z,+) 1.25and D(z,-) 2.7. Downwelling radiance intensitieswithin optically deep water are peaked sharply withina cone of half-angle of -47° and centered about zenith,while upwelling irradiance is a minimum at nadir andincreases toward the horizon. Within this work, thetwo-flow coefficients associated with each diffuse irra-diance stream are assumed unique.

A. Optically Shallow Water

Within optically shallow water, the light field at thesurface (z = 0) is measurably affected by changes in thedepth 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 theobjective here is to derive expressions for Ec(z,+),Ed(z,+), and Ed(z,-). The following discussions dealprimarily with a water column, but it is easy to see howeach remark may be applied to either a submerged orterrestrial plant canopy.

Ignoring for the present any surface effects, such asFresnel reflectance and refraction, the initial condi-tion just below the air-water interface is

Ed(0,+)

C = E(O,-) I "'b (7)

LEd(O,+)J

Because the water is optically shallow, a measurablequantity of light propagates down through the water

1130 APPLIED OPTICS / Vol. 25, No. 7 / 1 April 1986

Table I. Estimations of Sea Surface Reflectance17

Surfaceillumination Reflectance

Clear sky, 0° = 60° 0.1Uniform 0.066Overcast 0.052

column and impinges on the bottom, z = d. Here thelight field may be expressed as

Ed(d,+)

Ed(d,-) I= 'dC- (8)

Ec(d,+)

Substituting the right side of Eq. (7) for C in Eq. (8)forms the equality

Ed(d,+) Ed(O,+)

Ed(d,-) = I41ol Ed(O,-) * (9)-E,(d,+) -E,(O,+)

The bottom boundary condition requires that up-welling irradiance leaving the bottom be equal to theportion of downwelling diffuse and collimated irradi-ance reflected from the bottom. At this point, it isimportant to make distinctions between the bottomreflectance of diffuse irradiance Pbd and that of colli-mated irradiance Pbc The bottom is assumed to be aLambertian reflector, which ensures that the upwell-ing irradiance stream just above the bottom is diffuse.This assumption is much less critical for Ed(d,+) thanfor Ec(d,+). A mirror, for example, is certainly not aLambertian reflector, yet it appears so if illuminatedwith totally diffuse light. In the case of collimatedirradiance, not only the distribution but the quantityof reflected radiance is a function of the incident angle.The precise nature of these functions is presently un-known, and, even with the Lambertian assumption,the mo'st that one can say is that Pbd # Pbc is a possiblecondition.

The bottom boundary condition is formulated as alinear mapping function fb so that

-a,

fb a2 1 = a2 - Pbdal - Pbca3- (10)_a3_

Applying fb to Eq. (9) and solving for Ed(,-) yield

Ed(O,-) = Ed(O,+) ( 22P1 - E21P2* Ec(0,+)\412P1 - llP2 J 412P1 - llP2

X ((Q12 23- 1322)Pl + (Q1321 -11023)P2

+ (Q1122 - 1221)P3*), (11)

where

P1 Pbd l - 21

P2 = (Pbd12 - 22) expl(K2 - 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-

tion within the water. Rewriting Eq. (5) as a set ofsimultaneous equations for z = 0 and solving for theintegration constants, C1, C2, and C3, yield

C Ed(0,+) - EJ(,+)t 13 - C2412 (12)411

CEd(0,-)01 - Ed(0,+)02 + E~O,+)(Q13021 - 23011)

C2= 422411 - 41221

C3 = E(,+).

(13)

(14)

Equation (5) now defines the subsurface light field forany location within the water column.

The effect of the air/water boundary on the subsur-face light field may be approximated by making thefollowing substitutions within Eqs. (11)-(14):

Ed(O,+) = Ed(O+,+)(l - Psd)'

E,(0,+) = E,(O+,+)(1 -p,)

(15)

(16)

0 = arcsin(n. sin00 ), (17)

where the parenthetical 0+ indicates an irradiancequantity measured just above the water surface, Psdand Pse are Fresnel reflectances at the water surface ofdiffuse and collimated irradiance, respectively, n isthe refractive index of water (nw - 1.34 for visiblelight), and 0

a is the solar zenith angle as measuredabove the water surface.

Cox and Munk17 computed the reflectance of asmooth ocean surface under a variety of illuminationconditions (Table I). Their results represent the com-bined reflectances of collimated sunlight and diffuseskylight. For purposes here, because the surface illu-mination is divided into diffuse and collimated compo-nents, Psd is probably best approximated by the over-cast computation of Cox and Munk, 0.052. For thecollimated component, surface reflectance may becomputed using the Fresnel relationship for randomlypolarized irradiance,

05 Lsin2 (01 - 02) tan 2(01 - 02)sin 2(01 + 02) tan 2(01 + 02)] (18)

Upwelling irradiance measured just above the watersurface may be defined as

Ed(O+,-) = Ed(O,-)(1 - Pwd) + p dEd(°+,+) + PscE,(O+,+), (19)n,2

where Pwd is the internal reflectance of the water sur-face. Gordon18 reported that for a uniformly distrib-uted light field, the internal reflectance of a smoothsurface is 0.49. However, most often, the subsurfaceradiance distribution is not uniform but is a minimumat nadir and increased toward the horizon. Becauseupwelling light reaching the surface at angles greaterthan -47° is almost completely reflected internally,Pwd = 0.49 is probably an underestimation. Neverthe-less, Gordon's value is used here for the purpose ofillustration.

B. Optically Deep Water

Within optically deep water, the light field at theupper boundary is not measurably affected by changes

1 April 1986 / Vol. 25, No.7 / APPLIED OPTICS 1131

in the depth or optical properties of the lower bound-ary. Expressions of irradiant intensities throughoutan optically deep water coluiin may be derived bytaking the limit of Eq. (11) as d -X and solving for thenew integration constants:

Ed(O,-) = Ed(O,+) 422 + E,(0,+) (23- 1322) (20)

C1 = 0, (21)

Ed(0 +) -E (0 +)413 (2

12

C3 = E,(0,+). (23)

C. Multiple Layers

In the previous sections, the water column is repre-sented as a single homogeneous layer. Recently, how-ever, researchers have begun to express concern for thecase of vertically inhomogeneous waters.1 9 20 Philpotand Klemas20 were able to attribute variations in re-motely sensed water radiance to vertical changes in theconcentration of suspended and dissolved material.Within plant canopies, both terrestrial and submergedaquatic plant structures, such as leaves, stalks, andfruiting bodies, are often vertically stratified withinthe canopy. Within water, a canopy of bottom-adher-ing plants may occupy only the lower part of the watercolumn. It would be useful, therefore, if the two-flowmodel could be made to simulate the case of a verticallystratified optical media.

A vertically inhomogeneous water column may besimulated by dividing the column into a finite numberof horizontal layers. As before, each layer is assumedto be homogeneous. The layer boundaries are trans-parent to streams of irradiance, both diffuse and colli-mated.

To derive expressions of irradiance for any locationwithin the water column, the integration constantsassociated with each layer must be obtained. This isaccomplished by first calculating the diffuse volumereflectance of the water column at the top of each layer,starting with the bottom layer and working sequential-ly to the surface layer. In each case, the layer is as-sumed to be optically shallow. For the lower-mostlayer, the bottom boundary condition is simply thesubstrate reflectance. For all other layers, however,the bottom boundary condition becomes the volumereflectance of the water column, as measured at the topof the underlying layer. The linear mapping functionassociated with the lower boundary condition of layer iis

Fal

fbi 1a2 1 a - Rd..Ral - l (24)

C3-

where Rdi-' and Ri-i are volume reflectances of thewater column measured at the top of layer i - 1 fordiffuse and collimated illumination, respectively. No-tice that in the case of the bottom layer, Rdi-1 = Pbd andRci- = Pbc, and Eq. (24) is identical to Eq. (10).

Armed with the lower boundary conditions for eachlayer, the only remaining task is to calculate the inte-gration constants for each layer, starting with the sur-face layer and progressing sequentially to the bottomlayer. In each case, the downwelling collimated anddiffuse irradiance at the top of a layer constitutes theinitial conditions, which, together with the lowerboundary condition, define the integration constantsfor that layer.

IV. Two-Flow Model for a Submerged Plant Canopy

Suitsl,2 reported a two-flow model for a terrestrialplant canopy in which the model coefficients are de-rived as functions of the optical properties of the plantstructures and the canopy morphology. Each type ofplant structure within the canopy is idealized as a suiteof perfectly diffusing panels both for transmitted andreflected irradiance. Plant structures are assigned apreferred nadir orientation but are assumed distribut-ed randomly with respect to azimuth. For example,the attenuation coefficient for diffuse irradiance isdefined within the Suits model as

m

U11 = ZNij cosfj(l - rdij)j=1

+ si - dij + Pdi ) (25)

where j identifies canopy structures within layer i,such as leaves, stalks, or fruiting bodies, Nij is thecumulative one-sided structure area per unit canopyvolume, 4'ij is the zenith angle of a vector pointingnormal to the surface of the structure, Tdij is the diffusetransmittance of the structure, and Pdij is the diffusereflectance of the structure.

Several authors have compared Suits model predic-tions of canopy reflectance with field measurementsfor canopies of cultivated wheat and cotton.6-8 Al-though the Suits model was found to overestimateconsistently the measured reflectance, the discrepan-cies between predicted and observed values weregreatest for near-IR radiation. Within the visible por-tion of the spectrum, the Suits model accurately pre-dicted measured reflectance. For a submerged cano-py, only the visible portion of the spectrum is relevantbecause water is an efficient absorber of IR radiation.

The Suits model is modified to account for a sub-merged canopy of vegetation by adding the two-flowcoefficients of the Suits model to weighted versions ofthe corresponding water model coefficients:

v = ns + V.17., (26)

where the subscript s indicates the matrix of Suitsmodel coefficients, and the subscript w identifies thewater coefficient matrix. The weighting variable V isa value between 0 and may be interpreted as thepercent canopy volume occupied by water. Presum-ably, for most submerged canopies, the volume of wa-ter displaced by plant material is small, and, for pur-poses of illustration, Vw = 0.95 is used here.


+0

440 480 520 560 600 640 680 720

Wavelength (nm)

Fig. 1. ESM computations of volume reflectance from a clear col-umn of water containing a canopy of bottom-adhering plants. Val-ues represent measurements made just above the water surface andinclude the reflectance of diffuse and collimated illumination fromthe air-water interface. Canopy optics and morphology are heldconstant 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 r11 And pl

There is as yet no clear connection between a sub-merged canopy and terrestrial canopy. To investigatethe effects of submerging a canopy, the plant structuretransmittance and reflectance parameters, rij and Pij,must first be expressed as functions of the opticalproperties of the surrounding media.

Allen et al.,21 in applying the work of Stern,2 2 wereable to derive expressions for the transmittance andreflectance of a corn leaf as functions of the internaltransmittance of the leaf Tij, the refractive index of theplant material making up the leaf nij, and the refrac-tive index of the ambient media na:

pij = 1- * + TU.2T*2 :! 2 '*n4- T 12(n 2 - )

TO n2 *2T- = I

n4 - Tj2(n 2 T*)2'

(27)

Table 11. Beam Absorption Coefficients For Clear Natural Water2 5

A(nm) a )(nm) a X(nm) a

450 0.0367 550 0.0735 650 0.3774460 0.0374 560 0.0806 660 0.4133470 0.0386 570 0.0942 670 0.4545480 0.0397 580 0.1165 680 0.4909490 0.0416 590 0.1554 690 0.5371500 0.0459 600 0.2242 700 0.6181510 0.0536 610 0.2561520 0.0579 620 0.2830530 0.0625 630 0.3123540 0.0668 640 0.3418

where n = nij/ na, and r* is the transmittance of theplant-air or -water interface. If the transmittance ofthe plant structure is Lambertian and if n > 1, Sternreported that

T* = 4(2n + 1) + 4n(n 2+ 2n-1)3(n + 1)2 (n2 + 1)2(n2

- 1)

_ 1) In(n) + 2n(n 1) ln[( + 1). (29)(n 2 - 1)2 (n2 + ), -

The constraint that n > 1 requires that, for a terres-trial plant structure, nij > 1, while in the case of anaquatic plant structure, nij> 1.34. Although few mea-surements of nij have been reported, those that do existwithin the literature 21 23indicate that for healthy vege-tation, 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 thatmay be applied to either a terrestrial or submergedcanopy. 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 two-flow coefficients. On the other hand, when V, = 1 andNi = 0, i.e., a water column containing no submergedplant canopy, the elements of it are identical to Prei-sendorfer's two-flow coefficients for water. Through-out the remainder of this work, the two-flow model fora submerged canopy is referred to as the extendedSuits model or ESM.

VI. Applications of the ESM in Relation to RemoteSensing

In remote sensing, the ESM may be used to identifyoptimum scanner specifications to test new data analy-sis techniques and identify possible causes for varia-tions within imagery. As an example, the ESM is used

Table Ill. Zostera Marina Diffuse Transmittance And Reflectance

X(nm) P X(nm) P X(nm) r P

450 0.0437 0.0036 550 0.3046 0.0120 650 0.2081 0.0065460 0.0476 0.0037 560 0.3231 0.0122 660 0.1777 0.0061470 0.0529 0.0038 570 0.3266 0.0120 670 0.1555 0.0061480 0.0604 0.0039 580 0.3211 0.0114 680 0.1794 0.0078490 0.0626 0.0041 590 0.3126 0.0108 690 0.2954 0.0121500 0.0814 0.0047 600 0.3044 0.0101 700 0.4269 0.0196510 0.1242 0.0059 610 0.2941 0.0095520 0.1832 0.0082 620 0.2886 0.0090530 0.2449 0.0102 630 0.2795 0.0084540 0.2797 0.0113 640 0.2504 0.0074


here to calculate the volume reflectance R(0+) of anoptically shallow water column containing a canopy ofsubmerged bottom-adhering plans (Fig. 1). The cal-culations represent what would be measured justabove the water surface and include the reflectance ofdiffuse and collimated illumination from the air-waterinterface.

The water column is divided into two layers, a sur-face water layer and a single underlying layer contain-ing the plant canopy. The plant canopy morphology isheld constant, while the thickness of the surface waterlayer is varied. The two-flow scattering coefficientsfor water are computed from measurements of volumescattering conducted in clear ocean water24 and radi-ance distribution measured in clear lake water16:Bb(+) = 0.0059; Bb(-) = 0.047; bb(+) = 0.0029; andbf(+)= 0.17. The volume scattering function adoptedfor water is identical to Petzold's AUTEC7 measure-ments and is assumed to be constant between 450 and700 nm. The beam absorption coefficient is, on theother hand, wavelength dependent and representsclear natural water (Table II).25

The morphology of the plant canopy is definedbased on observations reported for Zostera marinainhabiting the southern portion of Chesapeake Bay.26

The canopy thickness is 1 m, and Nij = 2. Leaf orien-tation within a Zostera canopy is dependent on thelocal current regime. With large currents, the leavesbecome bent over in the direction of the water flow.However, when the currents are small, as assumedhere, the individual leaves assume a near vertical ori-entation and 6 = 0.

The directional transmittance and reflectance weremeasured from individual Zostera leaves harvestedfrom a site in southern Chesapeake Bay adjacent toGuinea Marsh, VA (Table III). Measurements werecollected using a Brice/Pheonix Scattering Photom-eter fitted with an Oriel 150-W tungsten lamp and aUDT Scanning Radiometer. Plant specimens weresubmerged within a small vessel of filtered bay waterpositioned on a central stage and illuminated withcollimated light normal to the plant surface. Direc-tional transmittance was measured for look anglesranging from 0 (normal to the specimen surface) to 45°.Diffuse transmittance was approximated as

Tij(X) =L ( 0 ) - (30)

where 450 nm X 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 collec-tor (220). The coefficient x is computed by fitting thedirectional transmittance to a function of the formcosx(6), where 6 is the radiometer look angle.

The problem of estimating diffuse reflectance wasmore difficult because directional reflectance mea-surements were limited to 135° (450 relative to thespecimen surface). In the absence of more detaileddirectional reflectance data, diffuse reflectance was

estimated using the assumption that directional re-flectance from the specimens was Lambertian.

The reflectance of the bottom is 0.05 and is analo-gous to the case of Zostera colonizing a substrate ofdark mud.

ESM simulations of volume reflectance spectra maybe used in remote sensing to specify optimum bandlocation, width, and radiometric sensitivity. If, in Fig.1, the objective is to distinguish between optically deepwater and plant canopies occupying shallow water,neglecting atmospheric effects, a narrowband centeredaround 560 nm would serve the purpose as would anarrowband within the blue portion of the spectrumbetween 450 and 480 nm. The spectral region 600 nm,because of increased water absorption, may convergetoo rapidly to the deep water curve for deeper plantcanopies to be distinguished.

An interesting situation occurs between 490 and 500nm in that the volume reflectance appears to be insen-sitive to changes in canopy depth. Understandingwhy this happens may be gained through the ESMalgebraic expression of upwelling irradiance from anoptically shallow water column. To avoid needlesscomplexity, the water column is thought of as a one-layer system, and Pbd and Pbc represent the reflectanceof diffuse and collimated irradiance, respectively, fromthe submerged plant canopy. In other words, theplant canopy, because the optical and morphologicalcharacteristics are held constant, becomes the bottomboundary. The algebraic expression is, therefore,identical to Eq. (11), where d is the variable thicknessof the surface water layer. For Ed(0,-), to be unaffect-ed by changes in the thickness of the surface waterlayer, the exponential depth terms within P2* and P3*must somehow be removed. Looking first at the caseof diffuse surface illumination,

Rd(O+) = Si (22P1* - t2P2*) + S2 ' (31)

where S and S2 account for all the surface effects.Assuming that absorption within the water is greaterthan zero, Eq. (31) can only be insensitive to depth ifP2* = 0. Forming this equality yields

Pbd =22412- (32)

Equation (32) is identical to the expression for in-water volume reflectance of diffuse irradiance fromoptically deep water-included within the first termon the right-hand side of Eq. (20). In other words, thediffuse component of Ed(,-) is insensitive to the sur-face layer thickness when the reflectance of the lowerlayers, the submerged plant canopy, is equal to thedeep 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 ofEd(,-) may, on substituting 622/12 for Pbd, be removedif P2* =O. Forming this equality and rearranging yield

Pbc 423 - 413 12 (33)


1.1

0

'S 0.82 04 0. .

+. 0.7 P

0~~~~~ to1whr Pd= b

0.6 N

0.5

0.4-

0 0.2 0.4 0.6 0.5 1

Depth (in)

Fig. 2. ESM computations of total irradiance within a canopy ofaquatic vegetation submerged within clear water. Values are ex-pressed as a percentage of the surface illumination measured justbelow the air-water interface. Substrate reflectance is varied from

0 to where Pbd Pbc-

The right-hand side of Eq. (33) defines the in-watervolume reflectance for optically deep water illuminat-ed by collimated irradiance. Once again, if the under-lying layers of a multiple-layer system look like anoptically deep version of the surface layer, variations inthe thickness of the surface layerwill have no effect onupwelling irradiance at the surface.

Vil. Intercanopy Light Field

The intensity and spectral quality of the submarinelight field play a dominant role in shaping marineproductivity. Within coastal and estuarine waters,light is intercepted by benthic and planktonic algae,photosynthetic bacteria, and higher plants, and theintercepted energy is used to convert inorganic matterto organic matter-the well-known process of photo-synthesis. In many cases, the intensity of the ambientlight field is believed to be the single controlling factordetermining the level of productivity. Van Tine andWetzel27 studied the productivity of communities ofbottom-adhering plants within Chesapeake Bay andconcluded that "light and factors governing light ener-gy available to submerged aquatic plants are principalcontrolling forces for growth and survival."

The ESM may be used to simulate total irradianceE(z) at various depths within a submerged plant cano-py. Total irradiance, defined as

E(z) = Ed(Z,+) + Ed(z;-) + E,(z,+), (34)

is expressed here as a percentage of the surface illumi-nation, 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, andPbc - 1. The optical properties of the water and

vegetation are identical to those of the previous discus-sions where X = 450 nm. The water depth is 1 m, thecanopy occupies the entire water column, and the sub-strate reflectance is varied from 0 to 1, where Pbd = PbcThe surface illumination is 90% collimated, and thesolar zenith angle is 45°.

As irradiance propagates down through a sub-merged plant canopy, the combined effects of scatter-ing and absorption act to attenuate the intensity.Within an optically deep canopy, the light intensitywill continue to decrease until no measurable quantityremains. Within an optically shallow canopy, down-welling irradiance reflected from the substrate is im-mediately added to the stream of upwelling diffuseirradiance. Depending on the magnitude of Pbd andPbc, this has the effect of either decreasing or increasingthe intensity of the light field near the bottom, relativeto what intensities would be at a particular depth if thecanopy were optically deep. If the bottom reflectanceis zero, intensities will be lower than R(z) associatedwith an optically deep canopy. If the bottom reflec-tance equals R(z) of an optically deep canopy, the near-bottom light field within an optically shallow canopywill be no different than at the same depth within anoptically deep canopy. If, however, the bottom reflec-tance is greater than R(z) of an optically deep canopy,the intensity of the near-bottom light field will beincreased.

This effect of bottom reflectance on the near-bottomlight field is supported by Monte Carlo simulations.28

In varying the bottom reflectance from 0 to 1, the datareported by Plass and Kattawar indicate that when thebottom reflectance is small relative to the deep watervolume reflectance, the near-bottom light field is de-creased. For relatively high-bottom reflectances, thenear-bottom light field is increased.

The apparent effect of bottom reflectance on thewithin-canopy light field has important implicationsfor estimating the productivity of submerged aquaticvegetation. Within submerged plant canopies, a high-ly reflective substrate (sand) may increase the amountof available light within the canopy compared to a darksubstrate (mud). Within Lower Chesapeake Bay, spe-cies such as Zostera marina are believed to be light-limited due in part to high turbidity levels. Thesesame canopies may be found colonizing substrates hav-ing a wide variety of reflectances, although the domi-nate component is nearly always sand. We are movedto speculate that within light-limited canopies, sub-strates having higher reflectances may increase thechance of plant survival. However, to date, no studieshave been documented that attempt to relate sub-strate reflectance to plant photosynthetic rates or can-opy productivity.

Vil. Conclusions

The ESM is formulated in two steps: (1) the linearcombination of two-flow coefficients pertaining to nat-ural water and to a terrestrial plant canopy and (2) thedecomposition of the plant component transmittanceand reflectance terms into functions of the optical


properties of the plant structures and the surroundingmedia. The two-flow coefficients for water are de-rived from classical hydrologic optics theory, while thecoefficients pertaining to vegetation are identical tothose developed by Suits.1 2 In so doing, the ESMbecomes extremely versatile. Given the appropriateboundary conditions, the ESM may be applied to asubmerged plant canopy, natural waters containing novegetation, or a terrestrial plant canopy. Because theplant structure transmittance and reflectance termsare expressed as functions of the optical properties ofthe plant material and the surrounding media, theESM may be used to investigate the effects of sub-merging a plant canopy while holding the optical prop-erties of the plant structures constant.

In problems related to remote sensing and plantcanopy productivity, the ESM may be used to increaseour understanding of the complicated interaction oflight with submerged plant communities. The alge-braic solutions to the ESM are shown here to yieldplausible physical explanations for observed changesin volume reflectance spectra. In one example, vol-ume reflectance from a water column containing asubmerged plant canopy is shown to be insensitive tochanges in canopy depth. On closer inspection of thealgebraic solutions, it is evident that when the canopyreflectance is equal to deep water reflectance, varia-tions in the thickness of the surface water layer have noeffect on the upwelling irradiance from the water col-umn.

Bottom reflectance is shown through the ESM toeffect significantly the near-bottom light field, a phe-nomenon which is also predicted in Monte Carlo simu-lations. This result has potentially far-reaching im-plications within the study of submerged canopyproductivity. Within light-limited canopies of LowerChesapeake Bay, it may be possible to explain varia-tions in canopy productivity in terms of substrate re-flectance.

Support for this project was provided by NASAOffice of Life Science through contract NAGW-374and by NASA Goddard Space Flight Center throughcontract NAS5-27580 as part of the Landsat ImageData Quality Analysis Program managed by NASAGoddard Space Flight Center. We wish to thank Wil-liam Philpot for his critical review of this work andhelpful suggestions.

References

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Documents

Two-flow simulation of the natural light field within a canopy of submerged aquatic plants