ZANEVELD, J. RONALD V. An asymptotic closure theory for

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ZANEVELD, J. RONALD V. An asymptotic closure theory for Powered By Docstoc
					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
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