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Limnol.Oceanogr., 34(8), 1989, 1442-1452 Q 1989, by the American Society of Limnology and Oceanography, Inc. An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties J. Ronald V. Zaneveld College of Oceanography, Oregon State University, Corvallis 97331 Abstract An expression is derived for the rate at which the diffuse attenuation coefficient for vector irradiance approaches its asymptotic value in a homogeneous medium. The asymptotic approach rate is shown to be a function of boundary conditions at the surface and the asymptotic diffuse attenuation coefficient which is an inherent optical property. The asymptotic approach rate is then used to derive the vertical structure of the vector and scalar irradiances, the vector and scalar diffuse attenuation coefficients, the average cosine of the light field, and the remotely sensed reflectance at the surface, based only on the surface values of the vector and scalar irradiances and the vector and scalar diffuse attenuation coefficients. This theory is inverted and combined with previously derived radiative transfer relations to show that in principle the vertical structure of the absorption, scattering, attenuation, and backscattering coefficients can be derived from the vertical structure of the scalar and vector irradiances and the nadir radiance. An example for the western North Atlantic Ocean is provided. The vertical structure of the light in the transfer, which relates the light field and its sea is important to many disciplines. Sun- derivative to the inherent optical properties light is the energy source for the biological via the beam attenuation coefficient and the food chain, and the amount and spectrum volume scattering function. An analytical of solar energy available at a given depth solution to this equation for a homogeneous must be known if accurate productivity cal- medium was first given by Chandrasekhar culations are to be made. The oceanic biota (1950). Many other solutions have been giv- in return are a major factor in determining en (see Prieur and Morel 1973; Zaneveld the distribution of light through their ab- 1974; Jerlov 1976). Preisendorfer (1976) has sorption and scattering characteristics. So- provided an extensive treatise on the equa- lar energy that is absorbed by the ocean plays tion of radiative transfer and its applica- a role in physical oceanography, in partic- tions to optical oceanography. Tmplemen- ular in the structure of the mixed layer. Op- tation of the analytical solution to the full tical remote sensing allows us to study the equation of radiative transfer is cumber- large-scale structure of biological and op- some because it requires use of the full ra- tical parameters by measuring radiance em- diance distribution as well as the volume anating from the ocean. Remote sensing is scattering function. essential in investigating the global effects Purely numerical solutions based on of stratospheric ozone depletion and the Monte Carlo routines were developed early greenhouse effect. in the last decade (Plass and Kattawar 1972; The behavior of the light field in the sea Gordon et al. 1975) and have developed is described by the equation of radiative into powerful tools for the study of the for- ward problem, i.e. the derivation of the , structure of the underwater light field as a Acknowledgments function of the inherent optical properties. I thank James C. Kitchen for help in carrying out Another genre of numerical model, based the calculations and for preparation of the manuscript. on the deterministic solution of differential This work began during a sailing voyage in the South Pacific. The patience of the crew, Jackie, Jesse, and equations rather than on a probabilistic Eric Zaneveld, is greatly appreciated. The later stages Monte Carlo solution, is now available of this work were supported by the Office of Naval (Mobley 1989). Either type of model can Research SBIR contract NOOO14-86-C-0784.with Sea generate the equivalent of an experimental Tech, Inc., via a subcontract with Oregon State Uni- versity. The support of ONR and Sea Tech is gratefully data set. acknowledged. Approximate solutions based on the two- 1442 Asymptotic closure theory 1443 A theoretical solution of the inverse prob- Absorption coefficient, m-l lem-the derivation of the inherent optical Scattering coefficient, m-l properties from the radiance and its deriv- Backscattering coefficient, m-l ative with depth- was first given by Zane- Forward scattering coefficient, m-l veld (1974) based on the work of Zaneveld Particulate scattering coefficient, m-l an,d Pak (1972). Due to the unavailability Volume scattering function, m-l sr-1 Beam attenuation coefficient, m-l of suitable radiance and scattering function Beam attenuation coefficient of water, m-l data, the inversion has not yet been tested. Vector irradiance, W m-2 Much of the interest in two-stream so- Scalar irradiance, W m-2 lutions derives from their potential use to Downwclling scalar irradiance, W m-2 Shape factor for scattering invert the vertical structures of the up- and Shape factor for radiance downwelling irradiances to obtain the ver- Diffuse attenuation coefficient, m-l tical structure of the inherent optical prop- Vertical structure coefficient for absorption, erties, especially the backscattering. coeffi- m-l cient (Preisendorfer and Mobley 1984; Aas Attenuation coefficient for vector irradiance, m-l 1987; Stavn and Weidemann 1989). As Vertical structure coefficient for KE, m-l mentioned above, these solutions require Vertical structure coefficient for average co- very restrictive assumptions in order to ob- sine, m-* tain general solutions. As shown here, a more Attenuation coefficient for scalar irradiance, m-l general inverse solution for the vertical Attenuation coefficient for downwelling scalar structure of the backscattering coefficient can irradiance, m-l be obtained from the inversion of an ana- Asymptotic diffuse attenuation coefficient, lytic solution for the remotely sensed re- m-l Radiance, W m-2 sr-’ flectance (upwelling nadir radiance divided Wavelength, nm by the downwelling scalar irradiance) de- Average cosine rived by Zaneveld (1982). Asymptotic average cosine There continues to be a need to provide Asymptotic approach rate, m-l realistic models of the vertical structure of Remotely sensed reflectance, sr-1 Depth, m irradiance and upwelling radiance as a func- tion of the inherent optical properties. Models that allow inversion of irradiance stream model of Schuster (1905) have been and radiance to obtain the inherent optical described in great detail by Preisendorfer properties are particularly useful. In this pa- (1976). The two-stream models are quite per I develop such a model based on the restrictive in their assumptions if realistic observation that the attenuation coefficients solutions for the oceans are to be obtained for irradiance eventually become constant. (Aas 1987; Stavn and Weidemann 1989). This is the so-called asymptotic regime in The problem of inverting the vertical which the shape of the radiance distribution structure of the radiance field to obtain the is constant and the magnitude of the radi- inherent optical properties is also of con- ances and irradiances all decrease at the same siderable importance. It is not yet possible exponential rate. to measure the spectral inherent optical A semiempirical proof of the existence of properties in the ocean routinely and ac- the asymptotic regime was given by Prei- curately, although significant improvement sendorfer (1959). The final theoretical proof in this regard is expected soon (Zaneveld et was provided by Hojerslev and Zaneveld al. 1988). Even so, the determination of in- ( 1977). A review of other work regarding herent optical properties by direct instru- the asymptotic light field is given by Zane- mentation uses only small measurement veld (1974). Particularly useful is the anal- volumes while those with radiance and ir- ysis of Prieur and Morel (197 1) which gives radiance use very large volumes. Compar- the relationship between the asymptotic at- ison of the two will give considerable insight tenuation coefficient, Ko3, and the inherent into the contribution to the radiance distri- optical properties. K, is an inherent optical bution by large particles. property as it does not depend on the initial 1444 Zaneveld light field at the surface. The theory dictates where z is taken to be positive downward that the shape of the light field eventually (units given in list of symbols). Integration becomes a function of the inherent optical over 4n sr gives properties only. We thus see a gradual trans- formation of a surface-dependent light field (an apparent optical property) to one that - Wz) = a(z)Eo(z). - dz is an inherent optical property. This phe- nomenon also shows that the diffuse atten- This is Gershun’s (1939) equation, which uation coefficient K is not constant in a ho- allows us to calculate a(z) when we know mogeneous medium. The rate at which K the vertical structure of E(z) and E,(z). The is transformed from its surface value to its vector irradiance is defined as asymptotic value has not been derived pre- E(z) = viously. This is an important rate because 2s 7r the asymptotic K as well as the shape of the L(8, $, z)cos(e)sin(e) de d$, (3) asymptotic light field are inherent optical ss 0 0 properties. and the scalar irradiance is defined as In this paper advantage is taken of the 2n T knowledge that the transformation to the asymptotic state exists. The rate at which E,(z) = L(e, $, z)sin(e) de d$. ss0 0 this transformation occurs is then calculat- (4) ed and its dependence on the surface light- ing conditions and the inherent optical The inherent optical properties are de- properties is derived. When the rate of ap- fined as follows: a is the absorption coefh- proach to the asymptotic state is known, the cient; b is the volume scattering coefficient, vertical structure of the irradiance and up- where welling nadir radiance can be calculated, as K well as the remotely sensed reflectance just beneath the surface. Once the light field is asymptotic its properties are inherent and, b = 2~ S 0 P(r)Wr) dr; @a) as will be shown, can be inverted to obtain c is the beam attenuation coefficient, and the inherent optical properties, including the absorption, scattering, attenuation, and c= b -t a. Ub) backscattering coefficients. We define the diffuse attenuation coeffi- cients for vector and scalar irradiance by Relationships between the d@use attenuation coeficients and the K&z) = - $ In E(z) vertical structure coeficients for absorption and the average cosine --- 1 dJw$ @a) The equation of radiative transfer for a E(z) dz ’ medium without internal sources, or any so that cross-wavelength effects, and for which the horizontal gradients are negligible com- pared to the vertical ones is given by: E(z) = E(O)elp[-lz K&J dz] , 0) and + 7r = - cL(B,ip,z) K,(z) = - $ ln E,(z) ss 2* 0 0 P(e,s’,~,~‘)L(e’,~‘,z)sin(sl) de’ d+’ (1) _ - 1 a,(z) E,(z) dz ’ (W Asymptotic closure theory 1445 so that The average cosine used by Jerlov (1976) E,(z) = E,Oexp[ - lz KoW dz] . VW and o”er~c~~~~o~ _ ;W) . (1 4) E Substitution of Eq. 6a in 2 gives We then define the vertical structure coef- K&)JW = a(z)E,(z). (8) ficient for the average cosine by Differentiation of Eq. 8 then yields 1 &W K,(z) = - - - (15) Wz) p(z) dz ’ E(z) $ &W f KE(z) 7 Differentiation of Eq. 14 and a similar ma- *o(z) + E,(z) $ . (9) nipulation as that used to obtain Eq. 13 then = 4.4 7 yields We now divide the left-hand side by K,(z) = KE(z) - K,(z). (16) KE(z)E(z) and the right-hand side by a(z)Eo(z). These factors are equal as shown A combination of Eq. 13 and 16 gives the by Eq. 8. We then get following important result: K&z) = K,(z) - Kc(z) -- 1 CKdz) + -- 1 Wz) = KL(z) - K,(z). (17) KE(z) dz E(z) dz These new relations constitute the differ- _ 1 HO + -- 1 da(z) (10) ential form of Gershun’s equation and are . E,(z) dz a(z) dz * valid at any depth in any horizontally ho- mogeneous but vertically inhomogeneous We now define vertical structure coeffi- medium. The relationship for a homoge- cients similarly to the diffuse attenuation neous ocean, -Kdz) = KE(z) - K,(z), was coefficients. Thus the vertical structure coef- reported earlier by Hojerslev and Zaneveld ficient for the diffuse attenuation coefficient (1977). of vector irradiance is defined by An interesting result from Eq. 17 is that, in principle, we should be able to obtain the KK(z) = - - 1 -ME (114 vertical structure coefficient of the absorp- KE(z) dz ’ tion coefficient from the vertical structure so that of KE(z), K,(z) and KJz), i.e. K,(z) = W4 - K,(z) + &AZ). (18) K&z) = K,(O)exp [ - I= JG&9 dz]. (1 lb) This is different from Eq. 8 and 2 in that no intercalibration of the vector and scalar The vertical structure coefficient for the ab- irradiances is necessary. The term E/E0 does sorption coefficient is defined by not occur. Applying Eq. 12b and setting a(0) 1 da(z) = K,(O)F(O) allows us to calculate the ver- K,(z) = - - - Wd tical structure of a(z). Only p(0) needs to be a(z) dz ’ estimated. so that The forward solution in a homogeneous medium using a(z) = a(O)exp[-I= KzC4 dz]- (12b) asymptotic closure We wish to calculate the vertical structure Substitution of Eq. 6, 7, 11, and 12 into 10 of the apparent optical properties when the gives inherent optical properties are known. Hs- K&z) + K&) = K,(z) + KM. (13) jerslev and Zaneveld ( 1977) have proven 1446 Zaneveld that the diffuse attenuation coefficient KE(z) us to relate P to the boundary conditions at asymptotically reaches a constant value K, the surface. The rate at which the vector at great depth. Such asymptotic behavior irradiance approaches the asymptotic value can be modeled by thus depends on the difference of the diffuse attenuation coefficients for scalar and vector K,(z) = [K,(O) - KJexp( -Pz) irradiance at the surface, K,(O) - KE(0), and + K,. (19) the ratio of the asymptotic diffuse atten- This structure for K,(z) is supported by ob- uation coefficient and the diffuse attenua- servations (Preisendorfer 1959). That is, the tion coefficient for vector irradiance at the diffuse attenuation coefficient for vector ir- surface, Km/K,(O). Use of the boundary radiance changes rapidly at first and then conditions allows us to calculate P, and sub- asymptotically approaches Km at great depth. stitution into Eq. 19 gives the vertical struc- The rate at which KE(z) approaches the ture of KkT(z)given K, (the relationship be- asymptotic value is governed by P. In a ho- tween K, and the inherent optical properties mogeneous medium the asymptotic rate pa- is discussed later). With the vertical struc- rameter, P, is assumed to be constant. This ture of the vector irradiance attenuation allows the solution of the simplified radia- coefficient in hand, we must now derive the tive transfer equations presented here. The vertical structure of the irradiances and the term “asymptotic closure” has been adopt- remotely sensed reflectance. ed to describe the present theory. We now Using the definition of Eq. 6b, we can need to derive an expression for the asymp- integrate the expression for KE(z) in Eq. 19 totic approach rate based on the boundary and obtain conditions at the surface of the ocean. E(z) = E(O)exp x (29) Differentiation of Eq. 19 gives where d&AZ) = -P[K,(O) -- - K,]exp( -Pz). dz (20) x = -{K,z - -$exp(-Pz)] The vertical structure coefficient for the dif- fuse attenuation coefficient is then given by W&O - LII. (24) 1 dK,(z) We then calculate the scalar irradiance from Kk(z) = -- - Kd-9 dz Mz)E (4 P[ K,(O) - K,]exp( -Pz) E,(z) = a = [KE(0) - K,]exp(-Pz) + K, (2la) or eexp(--Pz) + K,}E(O)exp X. P[K,(z) - Km1 &AZ) = @lb) The average cosine is given by &(z) * Solving for P yields p(z) = a M‘(z) (22) = [KE(0) - KJe:p(-Pz) + K, ” for homogeneous water. Alternately, by (26a) substitution of Eq. 18, we get Rewriting Eq. 26a gives p = K,(z) - Wz) (23) l-- Kca - &W This equation is correct at any depth in a - 1 iw [---1 = F(O) 1 iL 1 .exp(-Pz) + 1. WW homogeneous ocean. Setting z = 0 allows Asymptotic closure theory 1447 We thus see that the asymptotic approach structure of K,(z) derived in Eq. 24. Setting rate describes the rate at which the inverse EOAZ) = E,(z) entails only a small error as of the average cosine, also known as the the reflectance is very small. An analytical distribution function, approaches its solution also exists but is somewhat cum- asymptotic value KJa. bcrsomc. It is derived next. Substitution of Zaneveld (1982) has derived a theoretical Eq. 25 into 28 gives relationship of the vertical structure of the L(a, 0) = O3 S bE(O) inherent optical properties and the remotely sensed reflectance (RSR) just beneath the surface of the ocean. He has shown that b 2?ra {[KE(0) - K&wW’z) 0 UT 0) = RSR(0) - + K,}exp y dz (30) EOd(O) where = S “fbVP&? 2n Y = 1 0 2’ -{(K, + a + bb)z - i exp(-Pz) - 1 c(z”) - f,(z”)bAz”) -L(O) - Kzol>. Let M = KE(0) - K, and N = K, + a + + KOd(zN) dz” dz’ (27) bb, so that I L(7r, 0) = where L(n, 0) is the nadir radiance at the surface, Eod(z) the downwelling scalar ir- y--$S03 E(O) bb radiance and Kod(z) the associated atten- 0 uation coefficient,fh andf, are shape factors for the scattering function and the radiance distribution, bYis the forward scattering co- efficient, and bb the backscattering coeffi- cient. Equation 27 is exact in any horizon- M + P)z + p exp(-Pz) M Ii + K,exp(-M) p exp( -Pz) dz. tally homogeneous, vertically stratified (31) ocean. It was shown that the shape functions are close to unity for all oceanic conditions. Expansion of the terms in braces into power Setting fL andf, equal to unity, and setting series and integration gives the desired re- c - bf = a + bb, it can then be shown that sult: in a homogeneous ocean the upwelling nadir UT 0) radiance at the surface, L(n, 0), is given by -= E(O) L(?r, 0) = S O3bb o z Eo(z)exp[- (a + bb)z] dz. (28) The remotely sensed reflectance can be ob- tained by substituting Eq. 7b into 28: RSR(0) = $$ z 0 = SO3bb 0% K,(z’) + a + bb dz’ dz. I (29) RSR(0) can then be obtained by numerical In order to express the remotely sensed re- integration over death. , using the vertical L flectance as the ratio of nadir radiance and 1448 Zaneveld scalar irradiance, the result must be multi- This procedure has been used frequently in plied by the average cosine: the past. A review is given by Jerlov (1976). Similar to the irradiances, the vertical UT 0) UT 0) n(()) (33) structure of the attenuation coefficient for RSR(0) = - = - E,(O) E(O) - nadir radiance can then be calculated from We have thus derived analytical expres- 1 Cu? z) sions for RSR(O), E(z), E,(z), K(z), KE(z), K(r, z) = - - - (35) and p(z) based only on boundary conditions L(x, z) dz and the inherent optical properties of the We can then use Zaneveld’s ( 1982) relation medium. These expressions form the the- oretical basis for the transformation of the UT 4 RSR(z) = - apparent optical properties at the surface, E,, (4 TWO), E(O), Eo(0), K(O), K&9, and F(O) to the inherent optical properties RSR,, Km, bb(z)/2n (36) and j& at asymptotic depths. = K(rr, z) + a(z) + bb(z) The inverse problem to calculate bb(z): Considerable effort has been expended re- cently (Preisendorfer and Mobley 1984; Aas bb(Z) = RSR(;)P%, Z) + a(Z)] . 1987; Stavn and Weidemann 1989) in trying (37) to invert two-flow models of irradiance to - - RSR(z) obtain the vertical structure of the back- 27 scattering coefficient. A problem with two- If necessary, the remotely sensed reflectance flow models is that they require severe restrictions in order to solve them. Preisen- w, 4 RSR(z) = - (38) dorfer and Mobley assumed that the back- Eodw scattering coefficients for the upwelling and downwelling streams are the same. Stavn can be approximated to within a few per- and Weidemann showed this to be incorrect cent by for many more turbid oceanic cases. They in turn assumed that the ratio of the back- J%, 4 RSR(z) = T scattering coefficients for the upwelling and 1 downwelling streams is known in order to as the reflectivity for scalar irradiance is provide a solution. This assumption implies small. considerable knowledge of the nature of the It should be noted that Eq. 34 and 37 are water being studied. not dependent on vertical homogeneity of A far less restrictive solution to the in- the water column and can be applied what- verse problem can be based on the work of ever the vertical structure of the inherent Zaneveld (1982) and the asymptotic closure optical properties. model presented here. In the inversion The asymptotic closure theory presented model presented here, it is assumed that the here allows us, in principle, to calculate the vertical structure of the vector irradiance, scattering and beam attenuation coefficients E(z), the scalar irradiance, Eo(z), and the in addition to the absorption and backscat- upwelling nadir radiance, L(n, z), are known. tering coefficients. In a homogeneous layer If E,(z) and E(z) are known, the vertical between z = z. and z = z2, Eq. 19 can be structure of K,(z) and K,(z) can then be applied in the form calculated by means of Eq. 6a and 7a. The absorption coefficient is obtained KF:(z) = [ KE(zo) - K,]exp( -Pz) from Gershun’s equation (Eq. 2): + K,. (40) E(z) The parameters P and K, apply only to the a(z) = K,(z) - (34) E,(z) ’ layer (zo, z,). In other homogeneous layers Asymptotic closure theory 1449 different parameters P and K, apply. We 0 now measure KE(zO), &(zz), and K’(zl) where z1 is halfway between z. and z2 and m then solve for K,: 20- 40- We can thus, in theory, obtain K, in a ho- mogeneous layer even though KE(z) in that 3 60- layer remains far from asymptotic. Prieur and Morel (197 1) and Timofeeva a Z (197 1) have derived relationships between & 80- K, and b/c. If we use the observation that in the ocean particulate scattering is much -8 larger than molecular scattering, the results loo- -b of Prieur and Morel’s theoretical calcula- -C tions as well as Timofeeva’s experimental results can be fit by the expression 120- - bbx10 -= b b2 Kw 1 - 0.52 ; - 0.44 F . (42) c The average cosine for the asymptotic ra- Optical Properties (l/m) diance distribution can be obtained by use Fig. 1. Inherent and apparent optical properties de- of Eq. 14 and 42: rived from the vertical structure of vector and scalar irradiance and nadir radiance. 1 - (b/c) a myriad of effects such as ship’s shadow, = (43) l- 0.52(b/c) - 0.44(b2/c2) ’ varying cloud conditions, waves, orienta- tion of the instrument platform, etc. Sepa- pi, is thus a function of b/c only. ration of the medium into homogeneous If a has been calculated from Gershun’s layers may also be difficult, although the (1939) equation, we can then solve for b and vertical structure of a(z) and bb(z) could be is c if PC0 known. derived first and used as guidance. More research is needed to extend the asymptotic b=a closure theory to inhomogeneous cases. (44) Sample analysis In principle we have thus shown that the There are few data sets which lend them- asymptotic closure theory allows us to cal- selves readily to the analysis proposed culate b and c in addition to a from Ger- above. What is needed is a numerically gen- shun’s (1939) equation and bb from Zane- erated set of E,(z), E(z), and L(w, z) along veld’s (1982) equation. The equations above with known values of a, b, bb, etc. so that can be used in a homogeneous layer even the inversion algorithms can really be tested. though KE: has not yet reached its asymp- What follows here is an analysis of Biowatt totic limit. It should be noted that there are 85 station 19-56. This data set has problems practical limitations to this approach. The in that E,(z) was measured on a different accurate measurement of apparent optical instrument platform than E(z) and L(n, z). properties is a serious problem due to their Furthermore, the depth gauges of the two very nature. They are readily influenced by instruments did not correspond. This was 1450 Zaneveld Equations 19 and 23 can be combined; we set Ko(z2) - K&2) K&2) - Kw r = K,(z,) - KE(z,) = KE(z,) - K, (45) where it is assumed that K, B [KE(z) - KJexp(--Pz). K, is then calculated from K KE(z2) - rK,(z,) = 00 (46) l-r * This equation was applied to the depth in- terval 16-36 m. K, was found to be 0.061 m-l. We then calculate j&, = a/K, = 0.68. b/c can then be calculated from Eq. 43. In this case b/c = 0.77. Use of Eq. 23 allows us to calculate P = 0.076. With this value for P Eq. 19 can be used to show that K&(z) is within 5% of its asymptotic value for z = 40 m. Therefore below 40 m we will assume that KE = K,. KE continues to change with depth, however, as a(z) changes. As long as (3660) (l/m) the absorption coefficient changes relatively Fig. 2. A comparison of observed and calculated slowly K,,(z) will remain asymptotic. There- beam attenuation coefficients. fore as long as changes in a(z) are sexp(--Pz), we can set KE(z) = K&z). We corrected by comparing beam attenuation then calculate b/c from jia = a/K, and Eq. data as taken with Sea Tech transmissome- 43. ters. During occupation of this station clear Once a(z) and b(z)lc(z) have been cal- skies prevailed, however ship drift and ori- culated, b(z) and c(z) can be obtained from entation may have biased the data. Never- Eq. 44 and 5b. We can then also calculate thelcss it seems to be the best set available. b,(z)lb(z). Figure 1 shows the calculated a, The analysis was carried out for X = 488 b, c, and bb at 488 nm. It is of interest to nm as L,(z) penetrated the deepest at that corn pare the calculated beam attenuation wavelength. Figure 1 shows the vertical coefficient at 488 nm to the one measured structure of KE(z), a(z) as calculated from with the Sea Tech transmissometer at 660 Eq. 34, and b)>(z)calculated from Eq. 37. nm. We do this by assuming the particulate These calculations are relatively straight- scattering coefficient, b,(X), to vary approx- forward. imately as X-l. We then obtain ~(660) * The absorption coefficient at 488 nm b,,(488) x 488/660 + c,,(660), where ~~(660) shows a nearly constant value to -36 m is the beam attenuation coefficient for pure depth, after which it climbs to a maximum water. Figure 2 shows the comparison be- at 46 m and then decreases to 8 1 m after tween ~(660) calculated from the apparent which it is constant. The backscattering coef- optical properties and the measured ~(660). ficient decreases steadily down to 50 m, af- ter which the instrument is not sufficiently Discussion and conclusions sensitive to detect any further changes. The introduction of the vertical structure Next, we wish to calculate b and c from coefFicients for the average cosine, the ab- the vertical structure of K. and KE. The data sorption coefbcient, and the diffuse atten- are far too noisy for direct application of uation coefficient allows us to study the dif- Eq. 4 1. However a slightly less accurate, but ference between the scalar and vector diffuse less sensitive approach can be used. attenuation coefficients, which is of interest Asymptotic closure theory 1451 because this difference is more readily mea- measured. It would be useful to increase the sured than the ratio of the vector and scalar sensitivity of radiometers so that the back- irradiances since no intercalibration of the scattering coefficient can be deduced to sensors is needed. The vertical structure greater depths. Values for b,/b determined coefficients play an important role in deri- here varied from 3.4 to 2%. The lower back- vation of the asymptotic approach rate P. scattering ratios were found in a region of P must always be positive as can be seen increased a, b, and fluorescence, indicative from the following argument. If KE(0) is <K, of simultaneous increases in phytoplankton in a homogeneous ocean, the denominator and pigment concentrations. This corre- in Eq. 23 is negative. K,(z) must then in- sponds well with Morel and Bricaud’s (198 1) crease and therefore KK(z) must be negative. observation that backscattering in pure phy- P thus is positive. Similarly when KE(0) is toplankton cultures is small. >K,, KK(z) must be positive and P again The comparison between the calculated is positive. attenuation coefficient at 660 nm and the In the theory presented here it is assumed measured one is excellent considering that that the boundary conditions K,(O) and KE(0) two different instrument platforms were used are known. For modeling purposes it will in deriving the former, as well as a X-l scat- be necessary to derive the boundary con- tering dependence. This type of analysis ditions from first principles, i.e. express them constitutes a form of optical closure in that in terms of the inherent optical properties the absorption and scattering coefficients and incoming radiance field. For inversion were calculated from the vector and scalar this is not necessary. The theory presented irradiances and the resultant beam atten- here needs to be extended to the case of a uation coefficient was compared with an en- vertically inhomogeneous ocean. The em- tirely independent measurement. The pro- phasis in this paper has been on the inver- ccdure shows that the inversion may be sion. I have shown that judicious USCof the useful to compare large-volume inherent asymptotic closure theory allows us to cal- properties derived from apparent optical culate the vertical structure of the inherent properties with small-volume inherent op- optical properties. This is certainly reason- tical properties. In that case measurements able once KE(z) is nearly asymptotic. The must bc made at the same wavelengths, and mixed layer being nearly homogeneous in the scalar and vector irradiances must be structure allowed us to calculate the inher- measured from the same platform and be ent optical properties there also. The theory properly intercalibratcd. is at present not well suited to the normally In conclusion it has been shown that the extremely noisy apparent optical properties asymptotic closure theory for irradiance is data. To obtain a reasonably accurate value useful for understanding the vertical struc- for the asymptotic approach rate and K, ture of irradiance in the sea as well as in- requires that the homogeneous layer be quite version of that structure to obtain the in- thick, which may not always be the case. herent optical properties. The measurement of inherent optical properties is not subject to the various en- References vironmental perturbations that naturally af- AAS,E. 1987. Two-streamirradiance model for deep fect the apparent properties, and so they are waters. Appl. Opt. 26: 2095-2101. potentially far less noisy. They also can be S. CHANDRASEKHAR, 1950. Radiative transfer. OX- measured at any time of the day or night. ford. As the capability of measuring the inherent GERSHUN,A. 1939. The light field. J. Math. Phys. 18: 51-151. optical properties routinely and accurately GORDON,H. R., 0. B. BROWN, AND M. M. JACOBS. develops, it is thus useful to develop theo- 1975. Computed relationships between the in- ries that readily allow the subsequent cal- herent and apparent optical properties of a flat culation of the apparent optical properties. homogeneous ocean. Appl. Opt. 14: 417-427. I have demonstrated that in addition to N. HDJERSLEV, K., AND J. R. V. ZANEVELD, 1977. A theoretical proof of the existcncc of the submarine a(z), bb(z) can routinely be determined pro- asymptotic daylight field. Univ. Copenhagen Inst. vided that the upwelling nadir radiance is Phys. Oceanogr. Rep. 34. 1452 Zaneveld JERLOY, N. G. 1976. Marine optics, 2nd ed. Elsevier. -, AND -. 1973. Apercu sur les theories du MOBLEY, C. D. 1989. A numerical model for the transfert radiatif applicables a la propagation dans computation of radiance distributions in natural la mer, p. 2.1-1 to 2.1-45. In Optics of the sea. waters with wind-roughened surfaces. Limnol. AGARD Lect. Ser. 61. Oceanogr. 34: 1473-1483. SCHUSTER, A. 1905. Radiation through a foggy at- MOREL, A., AND A. BRICAUD. 198 1. Theoretical re- mosphere. Astrophys. J. 21: l-22. sults concerning light absorption in a discrete me- STAVN, R.H., AND A.D. WEIDEMANN. 1989. Shape dium, and application to specific absorption of factors, two-flow models, and the problem of ir- phytoplankton. Deep-Sea Res. 28: 1375-l 393. radiance inversion in estimating optical parame- PLASS,G. N., AND G. W. KATTAWAR. 1972. Monte ters. Limnol. Oceanogr. 34: 1426-144 1. Carlo calculations of radiative transfer in the earth’s TIMOFEEVA, V. A. 197 1. Optical characteristics of atmosphere-ocean system. 1. Flux in the atmo- turbid media of the sea-water type. Izv. Atmos. sphere and ocean. J. Phys. Oceanogr. 2: 139-145. Ocean. Phys. 7: 863-865. PREISENDORFER, W. 1959. Theoretical proof of the R. ZANEVELD, J. R. V. 1974. New developments in the existence of characteristic diffuse light in natural theory of radiative transfer in the oceans, p. 121- waters. J. Mar. Res. 18: l-9. 133. In N. Jerlov and E. Steemann Nielsen [eds.], ‘-. 1976. Hydrologic optics. 6 V. Natl. Tech. Optical aspects of oceanography. Academic. Inform. Serv., Springfield, Va. -. 1982. Remotely sensed reflectance and its ---, AND C. D. MOBLEY. 1984. Direct and inverse dependence on vertical structure: A theoretical irradiance models in hydrologic optics. Limnol. derivation. Appl. Opt. 21: 4146-4150. Oceanogr. 29: 903-929. -, R. BARTZ, J.C. KITCHEN,AND R.W. SPINKAD. F%IEUR, L., AND A. MOREL. 197 1. Etude theoretique 1988. A reflective-tube diffuse attenuation meter du regime asymptotique: Relations entre charac- and absorption meter. Eos 69: 1124. teristiques optiques et coefficient &extinction re- -, AND H. PAK. 1972. Some aspects of the ax- latif a la penetration de la lumiere du jour. Cah. ially symmetric submarine daylight field. J. Geo- Oceanogr. 23: 35-48. phys. Res. 77: 2677-2680.

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