Section 1 Theoretical aspects of

hydrologic optics
Limnol.Oceanogr., 34(8), 1989, 1389-1409
0 1989, by the American Society of Limnology and Oceanography, Inc.

Can the Lambert-Beer law be applied to the

diffuse attenuation coefficient of ocean water?
Howard R. Gordon
Department of Physics, University of Miami, Coral Gables, Florida 33 124

Abstract
Radiative transfer theory is combined with a bio-optical model of Case 1 waters and an optical
model of the atmosphere to simulate the transport of radiation in the ocean-atmosphere system.
The results are treated as experimental data to study the downwelling irradiance attenuation
coefficient. It is shown that the downwelling irradiance attenuation coefficient just beneath the
surface and the mean downwelling irradiance attenuation coefficient from the surface to the depth
where the irradiance falls to 10% of its value at the surface can be corrected for the geometric
structure of the in-water light field to yield quantities that are-to a high degree of accuracy-
inherent optical properties. For Case 1 waters these geometry-corrected attenuation coefficients
arc shown to satisfy the Lambert-Beer law with a maximum error of 5-10% depending on wave-
length. This near-validity of the Lambert-Beer law, when there are compelling reasons to believe
that it should fail, is shown to result from three independent facts: the dependence of the diffuse
attenuation coefficients on the geometric structure of the light field can be removed; pure seawater
is a much better absorber than scatterer at optical frequencies; and the phase function for particles
suspended in the ocean differs significantly from that of pure seawater. Finally, it is shown that
extrapolation of the corrected diffuse attenuation coefficients to the limit c --* c, yields quantities
that are within 2% of the corresponding quantities that would be measured for an ocean consisting
of pure seawater with the sun at zenith and the atmosphere removed.

In a series of papers, Smith and Baker phytin a, and are contained in phytoplank-
(Baker and Smith 1982, and references ton or in their detrital material. In Eq. 1,
therein) have developed a “bio-optical” the Lambert-Beer law applied to &, K,,, is
model for relating the optical properties of the contribution to Kd from the water itself,
near-surface ocean water to the content of K, is the contribution from material susi
biological material. Specifically, the atten- pended or dissolved in the water and not
uation coefficient Kd of downwelling irra- covarying with C, and Kc(C) represents the
diance Ed defined by Kd = - ( 1/Ed) dE,/dz, contribution to Kd from phytoplankton and
where z is depth, is related to the phyto- their immediate detrital material (units giv-
plankton pigment concentration C through en in list of symbols). This decomposition
of Kd is very useful for the optical analysis
Kd = K, + Kc(C) + K,, (1) of ocean water because of the relative ease
C is the concentration of Chl a and all chlo- in measuring Ed, the absence of the require-
rophyll-like pigments that absorb in the ment for absolute radiometry to determine
same spectral bands as Chl a, such as pheo- Kn, and the possibility of measuring Kd re-
motely (Austin and Petzold 198 1; Gordon
1982) and even at night (Gordon 1987). To
Acknowledgments utilize it, we plot measurements of Kd for a
This work received support from the Office of Naval given wavelength and from a variety of
Research under contract NO00 14-84-K-045 1 and grant
NO00 14-89-J- 1985, and from the National Aeronau- oceanic waters as a function of C and as-
tics and Space Administration under grant NAGW- sume the minimum envelope of the result-
273. ing curve, (Kd)min, corresponds to K, = 0.
1389
 1390 Gordon

Taking the limit of (Kd)min as C + 0 yields for a stratified ocean with sun at zenith the
K,. Then, Kc(C) is given by (Kd)mi” - K,,,. value of Kd at a given depth depended most-
(For an example of this procedure see figure ly on the properties of the medium at that
1 of Baker and Smith 1982.) If we assume depth, i.e. Kd is a local property of the me-
that Kc(C) is valid for all waters, Eq. 1 dium. These observations suggest that if Kd
can be applied to specific cases to estimate values were corrected for variations in il-
K, from Kd and C or to estimate C from Kd lumination (e.g. corrected so that they are
in waters for which Kx is known to be neg- referenced to a standard incident illumi-
ligible. These latter waters are usually re- nation) and used in Eq. 1, the error resulting
ferred to as “Case 1 waters” and are defined from the fact that Kd is not a true property
to be waters for which the optical properties of the medium would be considerably re-
are controlled by phytoplankton and their duced. But a residual error would remain
immediate detrital material (Gordon and in Eq. 1 because Kd depends on depth.
Morel 1983; Morel and Prieur 1977). Lim- In this paper the earlier computations
iting the analysis to Case 1 waters, Gordon (Gordon et al. 1975) of Kd are extended to
and Morel (1983) and Morel (1988) used cases of more realistic illumination of the
Eq. 1 to derive Kc(C) by assuming K, = 0. surface and a realistic model of the optical
Equation 1 has been criticized by Morel properties of the ocean. It begins with a re-
and Bricaud (198 1) and Stavn (1988) on the view of the equation which governs the
basis that, unlike the absorption coefficient transport of radiant energy in the ocean and
and the volume scattering function, Kd is in the atmosphere, and of the basic optical
not solely a property of the medium. This properties (e.g. absorption and scattering
is because it depends on the depth (even for coefficients, etc.) required to obtain a so-
a homogeneous ocean) and on the geometric lution. A realistic model of these optical
structure of the light field incident on the properties is then presented and the results
sea surface, as well as on the properties of of Monte Carlo simulations of the in-water
the medium. Since a given Kd is unique only light field are used to study the properties
to the particular situation in which it is mea- of Kti The analysis of Kd shows that when
sured, and there is no reason to expect that it is measured either just beneath the sur-
the three components of Kd will vary in the face, or when an average value is deter-
same manner with depth and with the struc- mined between the surface and the depth
ture of the incident light field, it is correctly where the surface irradiance is reduced to
asserted that Eq. 1 can only be an approx- 10% of its value at the surface, the resulting
imation. However, Gordon et al. (1975) Kd can be corrected to yield a quantity that
have shown with Monte Carlo simulations can be directly expressed in terms of the
of the in-water light field that for simple optical properties of the medium. Further-
modes of illumination (i.e. a sky of uniform more, the corrected Kd values are shown to
radiance or a parallel beam of irradiance depend almost linearly on the optical prop-
incident at an angle So with the vertical), erties of the medium, which in turn are lin-
the dependence of Kd on the structure of the early additive over the constituents, and so
incident light field can be removed without the corrected values will satisfy the Lam-
any knowledge of the optical properties of bert-Beer law with reasonable accuracy. Fi-
the medium. Gordon (1976) showed that nally, the computations show that the value
the correction factor required to remove the of K,,, determined by extrapolation of Kd to
light field dependence from Kd could be C = 0 in Case 1 waters is very nearly equal
computed with reasonable accuracy by to the value of Kd that would be measured
knowing only the relative amounts of sky- in an ocean consisting ofonly pure seawater.
light and direct sunlight incident on the sea
surface in the spectral band in question. Lat- The radiative transfer equation and
er, Baker and Smith (1979) directly verified optical properties
that for turbid water under clear skies, Kd Consider an ocean in which the optical
was nearly independent of 6, for b0 540”. properties and sources depend only on the
Finally, Gordon ( 1980) demonstrated that depth z. The steady state light field in the

---_l_c...?..- .._ r ~__ .- _ -” ._-- -.----_-- --.- - __

_*
 Total absorption coefficient, m-’ K KAO), m-’
Particle absorption coefficient, m-l K’ Inherent value of K<,(O),m-l
Water absorption coefficient, m-l KV Pure water component of KAO) and KAz),
Total scattering coefficient, m-l m-’
Total backscattering coefficient, m-l K, Particle component of K, m-’
Total backscattering probability KC Pigment component of KAz), m-l
Particle scattering coefficient, m-’ KC Nonpigment-nonwater component of
Water scattering coefficient, m-l KAz), m-l
Total volume scattering function, m-l (K> Mean K,Xz) from surface to zlO, m-l
sr-’ Inherent value of (K), m-’
Particle volume scattering function, m-l K Water component of (K), m-’
sr-l I Inherent value of (K),, m-l
Water volume scattering function, m-l IZMJ Particle component of (K), m-l
sr-’ W); Lambcrt-Beer value of (K) ((K), +
Specific volume scattering function (pi/ W,), m ’
C,), m2 mg-’ sr-’ (K), True value of (K), m-l
C Total attenuation coeflicient (a + b), m-l K,(z) Attenuation coelhcient for E,(z), m-l
CP Particle attenuation cocfhcient (a,, + b,), L(z; 6, sb) Radiance traveling in direction (29,V),
m-l mW (cm2 pm sr)-1
Water attenuation coefficient (a,,, + b,), x Wavelength, nm
m-l P Average cosine of L(z; 29,p)
Ci Attenuation coefficient of component i, PM, Average cosine when c, = 0
m-l 4J Average cosine when c,, = 0
C; Specific attenuation coefficient c/C,, m2 P(@) Scattering phase function (/3/b), sr-’
mg-’ ppw Particle scattering phase function (&lb,),
C Pigment concentration, mg m-3 sr-’
ci Concentration of ith constituent, mg m-3 pww Water scattering phase function (~,Jb,,J,
DO Downwelling distribution function sr-’
ho = 0) Q(z; 29,p) Intensity density of internal sources, mW
f Direct sun fraction of Ed(O) (cm3 wrn sr)-’
F Tolal forward scattering probability (1 - fJ2 Surface slope variance
bd t Fresncl transmittance of sea surface
F, Particle forward scattering probability 7 Optical depth (T = cz)
F,” Water forward scattering probability 710 Optical depth at 10% surface irradiance
FO Extraterrestrial solar irradiance, mW ekJ)
cm-2 pm-’ Aerosol optical thickness of atmosphere
Ed(z) Downwelling irradiance at z, mW cm-2 Ozone optical thickness of atmosphere
pm-l Rayleigh optical thickness of atmosphere
Edsun) Direct sun component of surface irradi- Direction of radiant energy flow
ance, mW cme2 pm-l Scattering angle
EAW Sky component of surface irradiance, Solar zenith angle
mW cm-2 pm-l Solar zenith angle below surface
E,(z) Upwelling irradiance at z, mW cm-2 Total scattering albedo (b/c)
pm-l Particle scattering albcdo (bdc,)
E,(z) Scalar irradiance at z, mW cm-2 km-’ Water scattering albedo (b,Jc,)
EoAz) Downwelling scalar irradiance at z, mW Solid angle, sr
cm-2 pm-l Depth, m
KAz) Attenuation coefficient for EAz), m-l Depth at 10% surface irradiance, m
KIxz) Inherent value of KAz) at z, m-l Surface slope components

ocean is described by the radiant power per erncd by the radiative transfer equation
unit area per unit solid angle called the ra- (RTE):
diance L(z; 0, P), where 0 and P are the cos 8 dW; 0, q)
polar and azimuth angles (in a spherical co- = -c(z)L(z; I?, P)
dz
ordinate system in which the z-axis is into
the ocean and the X- and y-axes are along + /3(z; I?‘, cp’ --) 6, P)
the ocean surface) of a vector in the direc- sw
tion the radiant energy is flowing. The dis- ,L(z; W, P’) dQ’
tribution of radiance in the ocean is gov- + Q<z;8, VP)
1392 Gordon

where c(z) is the beam attenuation coeffi- the direction with which the solar beam en-
cient, @(z; 20’, Cp’ -+ 3, p) the volume scat- ters the top of the atmosphere, and 6 the
tering function for the scattering of radiance Dirac delta function.
from a direction specified by (@, p’) to that The quantities a(z), b(z), c(z), and ,&z; 0)
specified by (ti, VP), and da’ a differential depend only on the constituents of the me-
element of solid angle around the direction dium and are known as the inherent optical
(vY, p’). The subscript s1’ on the integral in- properties (IOPs) (Preisendorfer 1976). Note
dicates that the integration is to be taken that only two of these, a and p, are required
over the entire range of dQ’ (i.e. 47r sr). The to specify the full set. They are all linearly
last term, Q(z; 6, p) is the intensity density additive over the constituents, i.e. if /3,, is
-(radiant power per unit volume per unit sol- the volume scattering function of pure sea-
id angle) of in ternal sources in the ocean water and pi that of the ith constituent,
such as fluorescence (Gordon 1979), Raman
scattering (Stavn and Weidemann 1988),
PC@) = Pw(@) + C Pi(@)-
bioluminescence (Gordon 1987), etc. i
The total scattering coefficient b(z) is re-
lated to the volume scattering function The individual IOPs are directly propor-
through tional to the concentration of the constitu-
ents. For example, pi is directly proportional
to the concentration of ith constituent c-‘i:
b(z) = p(z; W’, P’ 4 G, P) dQ’
s $1’
Pi(@) = PT Co) ci
P
2
27r p(z; 0) sin 0 de, where 0; is called the speciJic volume scat-
J0 tering function. It is convenient to define
where 0 is the angle between the directions two new IOPs: the scattering phase func-
specified by (fi’, 9’) and (fi, V). The beam tion,
attenuation coefficient is given by
@(z; d’, cp’ ---)9.9,PO)
c(z) = a(z) + b(z) P(z; I?‘, cpI -’ I?, 50) =
b(z) ’
where a(z) is the absorption coefficient of
the medium. The radiance, a, b, c, 0, and and the single scattering albedo oo( z) - b(z)/
Q all depend on the wavelength x of the c(z). The RTE then becomes
light; however, this dependence has not been
explicitly shown in these equations. It has cos 0 dW; 639 = -L(z; 9, P) + q)(z)
been shown (Case 1957) that given the ra- c(z)d z
diance incident on the sea surface [L(O; 9,
. P(z; I?‘, cp’ -+ I?, 9)
‘P) for 6 <n/2] and the sources within the s
ocean, the RTE has unique solutions if b(z)/ 4;~; W, 9’) dL?’
c(z) < 1, i.e. in an ocean that has some
absorption throughout. The RTE also ap- + Q(z; ~-8‘WC(Z),
plies to the combined ocean-atmosphere
system when z = 0 is taken to be the top of from which it is seen that there are no in-
the atmosphere; however, in this case L must ternal sources (Q = 0). When the true depth
be replaced by L mM2, where m is the index is scaled by the beam attenuation coefficient
of refraction of the medium (m M 1 for the to form the optical depth 7, where d7 = c(z)
atmosphere and M 1.33 for the ocean). Also, dz, the propagation of the radiance is de-
Snell’s law, the law of reflection, and the termined only by ~~(7) and the scattering
Frcsnel equations must be used to propagate phase function, i.e.
the radiance across the air-water interface. cos d dU7; 8, Q’>
The incident radiance on the boundary of = -L(7; 8, P) + coo(T)
d7
the ocean-atmosphere system is simply a
beam of parallel light from the sun: L(0; 6, P(7; d ‘, 9’ + 9, 9)
9) = L,6(?9 - Go)S(8 -- (PO),where L, is the
average radiance of the solar disk, (9,, Vo) (2)
 Lambert-Beer law applied to K 1393

Radiance in the ocean is difficult to mea- tially with depth, so it is useful to determine
sure because of its dependence on 9 and Cp. the exponential decay coefficient of Ed:
Thus, most measurements of the oceanic
light field are limited to integrals of the ra- ~~l&%(z)l~
diance. Three integrals particularly useful in
&(‘) = - dz
marine optics arc the downwelling Ed, up-
welling E,, and scalar E. irradiances defined or
by Km
-= - d { WU~)l >
C dT ’
Kd is the downwelling irradiance attenua-
L(z; 29,v?)cos9 sin d d9, tion coefficient. For a homogeneous ocean
(IOPs independent of z), it is a slowly vary-
ing function of depth (e.g. see Fig. 3A) and
thus for a given distribution of radiance on
the sea surface it could almost be considered
L(z; 0, P)cos 29sin d db, a property of the medium; however, it also
depends on the geometric structure of the
light field (the radiance) in the water which
and in turn depends on the distribution of ra-
diance incident on the sea surface (e.g. see
Fig. 3B). If the latter is changed, the value
of Kd will also change. Thus, in contrast to
the IOPs, Preisendorfer (1976) has called Kd
an apparent optical property (AOP) of the
The irradiances Ed and E, are the down-
medium. Although he was able to relate Kd(z)
ward and upward flux (radiant power per
to the diffuse absorption and backscattering
unit area) of radiant energy across a hori-
coefficients (which he called hybrid optical
zontal surface at depth z. The scalar irra-
properties) exactly, these coefficients are not
diance is proportional to the energy density
independent of the light field. In an analysis
of the light field at z. If the RTE (with Q =
of this relationship, Kirk (1981, 1983) has
0) is multiplied by dfi and integrated over
all Q the result is Gershun’s equation (Ger- shown that the diffuse backscattering coef-
shun 1939; Preisendorfer 196 1): ficients bear little resemblance to their in-
herent counterpart, the true backscattering
coefficient b, given by
-& EM - JW)l = -4WoW ?r
bb = 2~ /3(0)sin 0 de.
s T/2
The combination Ed - E, is called the vec-
tor irradiance and given the symbol E(z). A Thus, a direct relationship between Kd and
quantity which will be of interest later is the the IOPs has yet to be established, and this
“average cosine” fi of the light field: is one of the bases for criticism of the ap-
plication of the Lambert-Beer law to &(z).
F(z) = -E(z) Optical model of the
E”(Z) * ocean-atmosphere system
From the definitions of E and E, it is seen The present study of the efficacy of Eq. 1
that ji is the average value of cos d weighted is based on solving the RTE in the ocean-
by the radiance distribution. atmosphere to determine Ed(z) from which
Of the three irradiances defined above, &(z) is computed. To accomplish this re-
Ed has most often been measured and we quires realistic models of the IOPs of the
center our interest on it. Observations in ocean and the atmosphere. Such models are
the ocean show that Ed(z) (along with E, described below.
and E,) decreases approximately exponen- The ocean-As discussed earlier, water
1394 Gordon

and its constituents influence the in-water agation of irradiance from a point-source
light field through their effect on a and /?. embedded in the ocean. Briefly, the absorp-
For simplicity, we limit the modeling of tion coefficient a,,, has been inferred from
these properties to Case I waters. Moreover, measurements of downwelling and upwell-
WC also limit the model to those waters for ing irradiance in oligotrophic waters such
which absorbing dissolved organic mate- as the Sargasso Sea (Morel and Prieur 1977;
rials-yellow substances-are absent. The Prieur and Sathyendranath 198 1; Smith and
rationale for this is that the effect of yellow Baker 198 I), and the scattering coefficient
substance absorption on the results would b, and the volume scattering function /3,,,(O)
be identical to that of increasing the ab- have been measured directly for pure water
sorption coefficient of pure seawater by an and for saline solutions of pure water cor-
appropriate amount. Also, the medium is responding to salinities between 35 and 39o/oo
assumed to be homogeneous. Thus, the me- by Morel (1974). The resulting a,, b,, and
dium is described by an absorption coeffi- w, are given in Table 1 for the wavelengths
cient, a = a, + a,, where the subscripts w used in the present computations. Note that
and p here (and hereafter) refer to the con- these values of w, represent the upper limit
tribution from water and suspended parti- of the scattering albedo for water plus any
cles and a volume scattering function given dissolved material such as yellow sub-
by p(O) = &,(O) + p,(O). The total scattering stances, since dissolved material typically
coefficient b found by integrating ,6 over sol- found in seawater can contribute to the ab-
id angle is b = b, + bP.Likewise, the beam sorption coefficient but not to the scattering
attenuation coefficient is given by c = c, + coefficient. The scattering phase function for
G =a+b. pure seawater is from Morel (1974).
From the partial IOPs (i.e. a,,,,a,, ,6,, and The optical properties of the suspended
/3&, the parameters P(0) and w. that are re- particles for Case 1 waters can be related to
quired in Eq. 2 for the ocean can be related pigment concentration. The scattering coef-
to the similar water and particle quantities ficient of particles at 550 nm, b,(550), is
P,+,(O),P,(O), w+,,,and op through nonlinearly related to the pigment concen-
tration C through (Morel 1980).
&pkv) + WV bP = B,P.62
00 = (3) (5)
c,Ic, + 1 ’
where b,(550) is in m-l and C is in mg rnh3
(see also Gordon and Morel 1983). The con-
stant Rc- the scattering coefficient at a pig-
ment concentration of 1 mg mh3-ranges
OOP= qJyq7kJ~ + w&v (4) from 0.12 to 0.45 and has an average value
c,Ic, + 1 . of 0.30. The variation in BcVis due to the
natural variability of scattering over the
Thus, in this model, given a,,,, ‘3p, P,,,(O), and various species of phytoplankton, as well as
P,(O), representing the inherent optical variability in scattering by detrital particles
properties of the water and of the particles, associated with the phytoplankton. Simi-
w. and P(0) are specified by the ratio c,Ic,. larly the absorption coefficient of the par-
This ratio is proportional to the particle ticles has been studied as a function of c’ by
concentration. Prieur and Sathyendranath (198 l), yielding
Experimental measurements must be used for C < 10 mg m-3:
to provide a realistic parameterization of
the IOPs oW, oy, P,,,(O), and P,(O). These a,(A) = 0.06A~y(X)C0.602, (6)
IOPs are functions of the wavelength X of where a,(X) is in m-l and C is in mg me3.
the light, and therefore, in what follows, the In this equation A,(h) is the absorption coef-
wavelength will be explicitly displayed in ficient of phytoplankton normalized to 440
the equations, e.g. b,(X) refers to the scat- nm:
tering coefficient at the wavelength X. The
model used here is identical to the one I a,@)
A,(h) = -a,(440)
developed (Gordon 1987) to study the prop- *
 Lambert-Beer law applied to K 1395

Table 1. Absorption and scattering coefficients of Table 2. Model values of w,,and w,.
pure seawater.
A(nm) % WI9
440 0.86 0.253
440 0.0049 0.0145 0.253 480 0.88 0.162
480 0.0034 0.0176 0.162 550 0.93 0.029
550 0.0019 0.0638 0.029

(Bricaud et al. 1983). This depression of

The relative absorption of phytoplankton scattering would make ~~(440) and ~~(480)
A,(X) deduced by Prieur and Sathyendra- smaller than given in Table 2; however, the
nath (198 1) agrees well with absorption effect is not large, e.g. changing n from + 1
measurements made on phytoplankton cul- to - 1 only reduces ~~(440) from 0.86 to
tures by Sathyendranath (198 1). Note that 0.80. To ensure that wide departures of UV
a,(X) includes both phytoplankton and their from those used in Table 2 do not influence
detrital material and thus represents the ab- the results of this work, I have also carried
sorption of all components other than the out simulations for ~~(480) = 0.5, 0.7, and
water itself. These nonlinear relationships 0.99.
between bPand C and aPand C are believed The particle phase function is the most
to be due to a systematic variation in the difficult quantity to parameterize because it
ratio of the concentration of phytoplankton requires the individual phase functions of
to that of detrital material as a function of the plankton and the detrital material-nei-
the concentration of phytoplankton (Hob- ther of which have ever been measured in
son et al. 1973; Smith and Baker 1978a). the field. Thus, we must rely on measure-
Since b,(X) and a,(X) vary with pigment con- ments of the total particle phase function
centration in nearly the same manner, b,(h)/ (plankton plus detrital material). Measure-
a,(X) is nearly independent of the pigment ments of the volume scattering function at
concentration: 530 nm have been made for waters in sev-
eral locations with very different turbidities
(total scattering coefficients) by Petzold
(1972). When the scattering by pure sea-
water is subtracted, the resulting particle
This relation provides an estimate of o,(X) phase functions are very similar, having a
and shows that this quantity is, in the first standard deviation within about 30% of the
approximation, independent of pigment mean, over waters for which the particle
concentration. At 5 50 nm, where an average scattering coefficient varied over a factor of
Bc is known, it yields ~~(550) = 0.933, in 50. This mean particle phase function de-
good agreement with the range for those rived from Petzold’s measurements is
measured by Bricaud et al. ( 1983) for four adopted for this study and designated by the
species of cultured phytoplankton: 0.89 I symbol “M.” Also, two other particle phase
~~(550) 5 0.97. To fix reasonable values of functions are used to represent the extremes
L+,(X) at the other wavelengths of interest, of the phase functions given by Petzold’s
we require the variation of Bc with X. Fol- measurements: the mean of three phase
lowing Gordon (1987) we assume B,(X) functions measured in the turbid waters of
obeys a power law with wavelength (i.e.) San Diego Harbor and designated by “T,”
B,(X) 0~ X-” and take n = + 1. This yields and a phase function measured in the clear
B&480) ~0.34 and B&440) x 0.38. The waters of the Tongue of the Ocean, Baha-
resulting values of U,(X) and o,(X) used in mas, and designated by “C.” The three par-
the computations are provided in Table 2. ticle phase functions are shown in Fig. 1
It should be noted that the assumption B,(X) along with the phase function for scattering
0~ X-l often overestimates the dependence by the water itself [PJO)]. We see that these
of b, on X since the scattering by absorbing model phase functions differ principally in
particles (e.g. phytoplankton) tends to be their scattering at angles >25”.
depressed in the pigment absorption bands This completes the specification of the
1396 Gordon
10’4
diant energy is simulated by Monte Carlo
10’3
methods yielding the irradiance in the water.
This is a numerical solution of Eq. 2 which
lo+*
fully accounts for multiple scattering in the
IO+’
ocean and in the atmosphere, including
ocean-atmosphere coupling (i.e. light can
IO* scatter out of the ocean and then backscatter
from the atmosphere and re-enter the ocean,
10-l
etc.). In most of the simulations the sea sur-
face is flat. As in all Monte Carlo simula-
tions, the computed values of Ed(z) contain
‘OT-.-+i---++
10-1 IO4 statistical uncertainties. On the basis of the
SCATTERING ANGLE (3 (Deg )
number of photons processed in each com-
Fig. 1. Phase functions for particles and water. Phase puter run, the maximum error (SD) in Ed
functions M and C have been multiplied by 2 and 4, just beneath the surface is less than +0.3%.
respectively, to facilitate plotting. This error increases with depth, reaching
+- 1% where E, falls to 10% of its value at
quantities needed for the simulation: o,, w,, the surface and + 3% where Ed is 1% of its
P,(O), and P,(O). Varying the parameter ci/ value at the surface.
c, from 0 to 0~) results in models which To assess the errors in Eq. I, we use the
range from a particle-free ocean to an ocean simulations to provide Kd as a function of
in which the optical properties of the par- C. Specifically, the various model oceans
ticles are completely dominant. This pa- are generated by allowing the pigment con-
rameter can be related to the pigment con- centration C to vary from 0 to about 4.5 Img
centration through the bio-optical model by me3 which in turn causes c,Ic, at each wave-
noting that Cu = a, + b. and using Eq. 5 length to vary according to Eq. 7. This vari-
and 6. The result is ation in c,lc,, then induces variations in w.
and P(0) determined by Eq. 3 and 4. Figure
-c,(N = y(x)c”.6, 2, for example, shows the change in the shape
of the total phase function at 480 nm as c,l
Gm
c, is varied from 0 to 100. Note how the
where y(X) = 22.4, 18.5, and 4.9 at 440, phase function deviates strongly from that
480, and 550 nm. of pure water (Rayleigh scattering) even
The atmosphere -The atmosphere influ- when c,lc, = 1, i.e. even when the total
ences the in-water light field by distributing attenuation is shared equally between water
a portion of the near-parallel solar beam and particles. To simulate a variety of cloud-
over the entire upward hemisphere (i.e. in free situations, I have carried out the com-
producing sky light from direct sunlight). To putations for solar zenith angles of O”, 20”,
simulate the angular distribution of radia- 25”, 30”, 40”, 60”, and 80”. Also, to simulate
tion entering the ocean requires an atmo- a totally overcast sky, I have studied each
spheric model. This model atmosphere con- ocean model with the atmosphere removed
sisted of 50 layers and included the effects and a totally diffuse light field incident on
of aerosols, ozone, and Rayleigh scattering, the sea surface. Thus, only situations with
vertically distributed according to data tak- broken clouds are not considered here. A
cn from the work of Elterman (1968). The sky with broken clouds is particularly dif-
aerosol phase functions were computed by ficult to examine because the radiation field
R. Fraser (pers. comm.) from Mie theory is no longer independent of the observer’s
with the Deirmendjian (1969) Haze C size horizontal position in the medium.
distribution. This model simulates optical For the analysis, the resulting values of
properties of the cloud-free atmosphere only. E,(a) from the simulations are treated as
experimental data, albeit data collected un-
Computations and properties of I(d der carefully controlled conditions-a cloud-
With the above model for the IOPs of the free sky and a homogeneous ocean of pre-
ocean and atmosphere, the transport of ra- cisely known inherent optical properties-
 Lambert-Beer law applied to K 1397

SCATTERINGANGLE 8 (Deg.)

Fig. 2. Total phase function at 480 nm as a function

of the particle concentration. Progressing from bottom
to top on the left of the graph, c,Ic, = 0, 1, 3, 10, and
100.

from which Kd can be determined by nu-

merical differentiation and related to the
IOPs of the ocean and ultimately to the con-
stituent concentrations. Where comparison
of the resulting Kd values with computations
by Kirk (1984) is possible, i.e. Kd computed
at the midpoint of the euphotic zone using
particle phase function T with c,/c, very Kd/%
large and the atmosphere removed, the 2.6
agreement is excellent.
Figure 3A provides some samples of the
resulting profiles of the irradiance attenua-
tion coefficient. In these examples IQ/c,,, is
computed for c,,/c, = 0, 1.4, and 3.7 for
particle phase function M at 440 nm with 20 -
the sun at zenith. The deepest computed
point for each profile corresponds to r = 9.
These particular profiles represent a clear 2
ocean (i.e. C I 0.05 mg m-3) with c,/c, =
0 corresponding to pure seawater. The im- 8 A()-
mediate conclusion to be drawn from these N

simulations is, as mentioned earlier, that Kd

is dependent on depth even for a homoge- if _
neous ocean. Also, Kd near the surface in-
creases more rapidly with z as c,/c, in-
creases. In fact, from the surface to z = 100 60-
m, Kd increases by 2.5, 10, and 20% for cJ
CW= 0, 1.4, and 3.7, respectively. Figure 3B
provides &/c,,, at 440 nm as a function of
depth with c,,/c, = 5.6 (C = 0.1 mg m-3) for
three solar zenith angles (o. = O”, 20”, and 808
cases from left to right are for c,/c, = 0, 1.4, and 3.7.
B. Computed dependence of J&/c,,,on depth at 440 nm
for particle phase function M with q/c, = 5.6. The
Fig. 3. A. Computed dependence of &/cW on depth three casesfrom left to right correspond to 19~= O”,20”,
at 440 nm for particle phase function M. The three 40”.
1398 Gordon

0
0 0.2 0.4

1-w
0.6 0.8
-k-h-------1
'-@-+I
0.8

Fig. 5. As Fig. 4, but for (K)/c.

Fig. 4. K/c as a function of 1 - q,. The symbol
code is given on Fig. 1. Note, 1 - wO= a/c.
ficult to measure because of wave-induced
40”). It shows that I& in the upper 70 m is light-field fluctuations, K is valuable for
also strongly dependent on the incident ra- showing the way toward a useful represen-
diance distribution as well as on depth. tation of Kd in terms of the IOPs. In con-
These computations confirm the argu- trast, (K) is relatively simple to measure.
ment that Kd cannot be considered an in- On the basis of the number of photons con-
herent optical property because it depends tributing to K and (K), the statistical un-
on depth and on the incident radiance dis- certainty 6K/c in K/c is x +O.O 12, while the
tribution. [An exception to this of course is relative error 6(K) in (K) is about t-0.006,
the asymptotic light field (z -+ 00) for which i.e. 6(K)/(K) x kO.006.
it has been shown (Preisendorfer 19 59) that Figures 4 and 5 provide the computations
Kd becomes independent of depth and in- of K/c and (K)lc as a function of o. for 334
dependent of the incident radiance distri- simulations comprising various values of 1-9,
bution. The asymptotic values of I&/c, for and C for each particle phase function and
the examples in Fig. 3 are 1.OO, 1.38, 1.95, wavelength. Note that 1 - o. = a/c, so these
and 2.42 for c,,Ic, = 0, 1.4, 3.7, and 5.6.1 figures relate K and (K) to the absorption
Thus, if we attempt to use Kd as an inherent coefficient a. Although a strong trend of in-
optical property, it is necessary to specify creasing K/c and (K)lc with an increasing
in some manner the depth at which the mea- absorption component in the total atten-
sured value applies and to remove the de- uation is observed, it is clear (as expected)
pendence on the incident radiance distri- that the variation in K and (K) cannot be
bution. Here we focus on the irradiance explained solely on the basis of the total
attenuation coefficient (K) just beneath the absorption and scattering coefficients of the
surface, i.e. medium alone.
To proceed further it is useful to review
K = lim KCI(7), the results of earlier investigations. Gordon
T >o et al. (1975) found that the dependence of
Kd(r) on the scattering phase function could
and on the average diffuse attenuation coef- be approximately removed by expressing
ficient ((K)) over the upper half of the eu- KLt(7)as a function of ooF, where F is the
photic zone, forward scattering probability (F = 1 -- bb,
EL- 1nb% IOY&KOl where bb = b,lb), rather than o. alone. Also,
c 710
they Found that the effect of the nature of
the illumination of the ocean on Kd(7) could
where 710 is the optical depth for which Ed be understood by examining Kd(7)/Do(7),
falls to 10% of its value just beneath the where Do(~) was the downwelling distribu-
surface [EJT~~)/E~(O) = 0.11. Although dif- tion function (Preisendorfer 196 1) for a to-
 Lambert-Beer law applied to K 1399

Table 3. D,, just beneath the sea surface.

$0 440 nm 480 nm 550 nm 1/cos o,,
0" 1.034 1.027 1.019 1.ooo
20” 1.074 1.065 1.055 1.035
25” 1.088 1.077 1.067 1.054
30” 1.105 1.100 1.093 1.079
40” 1.158 1.154 1.149 1.142
60” 1.286 1.293 1.299 1.315
80” ’ 1.284 1.311 1.346 1.484
Di ffisc 1.197 1.197 1,197

OK I I I I I I I I I
0 0.2 0.4 0.6 0.8
tally absorbing ocean with the same surface l-u.@

illumination, i.e. Fig. 6. K/cDo as a function of 1 - w$. The symbol

code is given on Fig. 1.

decrease in Do, Also, for b. 160” the dif-

ference between Do(O) and l/cos b,, is usu-
with o. = 0, where EOd is the downwelling ally 5 3%. Do(~) also depends on r; however,
scalar irradiancedefined by this dependence is of little interest here.
Applying the observations from previous
studies to the computations in Figs. 4 and
5, w. is replaced by woF and K/c and ( K)/c
are replaced by K/CD, and (K)/cDo, where
In the revised suggested notation (Morel Do is the value of Do(O) taken from Table
and Smith 1982) for optical oceanography 3. This new scaling of the computations is
Do(T) = l/jid for &+, = 0, where & is the presented in Figs. 6 and 7. We see that when
“average cosine” of the downwelling light the computations are presented in this man-
field evaluated just beneath the surface. In ner, K/cDo and (K)/cDo fall on what appear
the Gordon et al. (1975) study there was no to be universal curves. The curves on the
atmosphere over the ocean, and in that case figures are least-squares fits of the points to
Do@) = l/cos tiow, where bow is the solar
zenith angle measured beneath the sea sur-
5 = 5 k,(l - C&-J)n, (8)
face. In the present simulations this is no 0 n=l
longer valid because of the presence of the
atmosphere, and Do is also dependent on
wavelength because the amount of skylight and
produced by scattering in the atmosphere is
a function of wavelength. Therefore, Do has g = i (k),(l - OOfln, (9)
been computed at each wavelength and for 0 n=l
each solar zenith angle by directly solving
with k, = 1.0617, k, = -0.0370, (k), =
the transfer equation for the given X and 6,
with o. = 0. The results of this computation
1.3197, (k)2 = -0.7559, and (k), = 0.4655.
The average error in the least-squares fit to
for Do just beneath the surface (7 = 0) are
Eq. 8 and 9, is 1.8 and 2.2%. [Replacing
given in Table 3. The statistical errors in
Ed(O) and EOd(0) are t-0.3%, so the error in
Do(O) by Do(~lo) in Eq. 9 provides no sig-
nificant increase in the quality of the ex-
Do(O) will normally be 5 +0.4%. Note that,
pansion.] Also, a linear fit of K/CD0 to (1 -
as expected, Do usually increases with in-
o,F) is almost as good as Eq. 8, i.e.
creasing 6,; however, for 440 nm the con-
tribution from the increasing amount of
skylight compared to direct sunlight from K
- = 1.0395( 1 - CL@),
b. = 60” to b, = 80” actually causes a small CD,
1400 Gordon

o.o(w 0.2 0.4

I-q,F
0.6 0.8

Fig. 8. K/CD, for a particle free ocean (Rayleigh

scattering). K/CD,-0; (K)lcD,-0.

which implies that

always be the same at a given point in the
K ocean). A particular setting, wherein the (fiat)
- = l.O395(a + bJ,
DO sea surface is illuminated by the sun at ze-
nith with the atmosphere absent, is unique
with an average error of 2.5%. The points as far as Kd(z) is concerned. For a given z
on Figs. 6 and 7 with 1 - ooF > 0.85 do the value of K,(z) in this setting is a mini-
not fit Eq. 8 and 9 quite as well as the rest mum over all possible modes of illumina-
because these points correspond to pure sea- tion. Thus, it is reasonable to refer to Kd(z)
water-the phase function of which differs in this situation as the inherent irradiance
considerably from an ocean containing par- attenuation coefficient and give it the special
ticles (see Fig. 2). The K/cDo - (1 - o,fl symbol K$(z). Likewise, Kr and (K)’ are the
relationship computed for an ocean free of inherent values of K and (K) (i.e. the values
particles is presented in Fig. 8 (for So = 0 that would be measured in an imaginary
and no atmosphere); it differs considerably ocean-atmosphere system above). The
from that in Figs. 6 and 7. Since the mini- quantities K/Do and ( K)lDo represent. ex-
mum value of (1 - w,JJ is 0.85 (near 400 cellent approximations to Kz and (K)‘, i.e.
nm), and over the range 0.85 5 (1 - wz,,,) the results of measurements in real situa-
5 1 the K/cDo - (1 - w,F) relationship for tions can be transformed to this ideal setting
water and for the strongly forward scattering through simple division by Do.
particles is very similar, the computations To consider applying this result to a real
for the model ocean all fall very near the ocean, we must examine the effect of surface
universal curves even though there is a large roughness on this simple observation. To
variation in the shape of the scattering phase include surface waves in the radiative trans-
function. fer code requires a statistical model of the
The above analysis shows that K/Do and waves. For simplicity, we assume that the
(K)lDo can be written as explicit algebraic surface roughness has no preferred direction
functions of the inherent optical properties (i.e. the structure of the surface is indepen-
c, wo, and F (independent of the geometry dent of wind direction). Then with the mea-
of the incident light field) with an accuracy surements of Cox and Munk (1954) the
that is likely better than the accuracy with probability density that the sea surface at a
which K or (K) can be measured. Therefore given point has slope components z, and z,,
we are justified in regarding the quantities in the x and y directions is given approxr-
K/Do and (K)/Do as inherent optical prop- mately by
erties. Note that if the mode of illumination
of the ocean never varied, the distinction
1 z;+z;
between the IOPs and the AOPs would blur
(i.e. for a given set of IOPs the AOPs would
PC%,zy) = 27rcr exp - - CT2
( )
 Lambert-Beer law applied to K 1401

where c2, the slope variance, is related to Table 4. Computed D, and diffuse attenuation coef-
the wind speed V (in m s-l) through ficients at 480 nm and tiO = 60” as a function of the
surface roughness parameter c.
rT2= 0.003 + 0.005 12 K
(r DO K/c K/cDo (WC (WcDo
The rough surface described by p(z,, z,,) 0.0 1.293 0.2624 0.2029 0.2914 0.2254
is incorporated into the Monte Carlo radia- 0.1 1.306 0.2632 0.2015 0.2924 0.226 1
tive transfer code used in this work in a 0.2 1.333 0.2733 0.2050 0.2954 0.2285
manner similar to that described by Plass 0.3 1.373 0.2833 0.2063 0.2999 0.23 19
et al. (1975). A complete examination of the
effect of surface roughness on K and (K)
requires a significant computational effort; where t(i) is the irradiance transmittance for
however, only a few computations are re- light from source i and D,(i) the value of
quired to show that the basic result above- Do that would result from source i acting
division by DO renders K and (K) inherent alone. For a cloud-free atmosphere the only
optical properties - is still valid for an ocean sources are the sun and the sky so this equa-
with waves. A sample of the computations tion reduces to
carried out is presented in Table 4 which
provides computations of DO, K, and (K) Do = fl>o(sun) + (1 - f-Po(W ( 11)
as a function of the surface roughness at 480 where f is the fraction of direct sunlight in
nm for 6, = 60” and c,lc, = 12.3 (C z 0.5 the incident irradiance transmitted through
mg m-3). Note the slow increase in D, with the interface, i.e.
c indicating an increasingly diffuse incident
light field beneath the surface as the rough- t(sun)E,(sun)
ness increases. This increases K with in- f=
t(sun)Ed(sun) + t(sky)EJsky) ’
creasing roughness; however, division of K
by DO provides a quantity that is nearly in-
dependent of surface roughness. Interest- If skylight is assumed to have a uniform
ingly, the effect of surface roughness on both radiance distribution [i.e. radiance (bright-
(K)/c and (K)/cD, is small (~3%) up to ness) independent of direction of viewing],
wind speeds of 17 m s-l. These computa- Eq. 11 simplifies to
tions suggest that K/D, and (K)/D, remain
inherent optical properties even in the pres- Do = -cosf d + 1.197(1 -j-). (12)
ence of surface waves; however, the value ow
of D, used to form these ratios must be that
Given TJ,, the only unknown isJ: It can be
which is valid in the presence of the rough
estimated by placing an irradiance meter
surface.
above the surface, measuring the total in-
cident irradiance E,(sun) + E,(sky), and
Experimental estimation of D, then measuring the sky irradiance E,(sky)
Determination of D,, from field measure- by casting a shadow over the opal diffuser
ments requires the radiance distribution in- of the instrument.
cident on the sea surface. This can be quan- The efficacy of Eq. 12 is tested with the
titatively determined using a camera Monte Carlo simulations, where E,(sun) is
equipped with a fisheye lens (Smith 1974; computed from
Smith et al. 1970); however, analysis of the
resulting sky photographs is not simple. E,(sun) = cos 19,Fo
Earlier (Gordon 1976) I proposed a simple ’ exp[-(rA + TR + %d
scheme for estimating D,. Briefly, if Ed(i) is + cos ?Jo] (13
the irradiance incident on the sea surface with F. the extraterrestrial solar irradiance
from source i (e.g. direct sunlight, skylight, and r,& TR, and 702 the contributions to the
clouds, etc.), then it is easy to show that optical thickness of the atmosphere from
aerosol scattering, molecular (Rayleigh)
scattering, and ozone absorption. E,(sky) is
then determined by subtraction from the
1402 Gordon

total irradiance falling on the sea surface. between the estimate obtained with Eq. 13
Even though Eq. 13 is exact, for our pur- and 14.
poses it underestimates E,(sun) because all In the presence of surface waves, com-
photons scattered by the aerosol are as- putation of the correct value of Do is facil-
sumed to be uniformly distributed over the itated by the empirical observation that Do
sky; in reality a significant fraction of the increases approximately in proportion to a2
aerosol scattering is through small angles for wind speeds up to M 20 m s-l. This is
and these scattered photons are still trav- demonstrated in Fig. 9 for an overcast sky
eling in nearly the same direction as the and for solar illumination (no atmosphere)
unscattered photons. To compensate for this with 19, = 60”, 70”, and 80”. The dots on
effect, we can obtain an upper limit on Fig. 9 are the computed values of Do and
E,(sun) by ignoring the aerosol scattering the lines are least-squares fits to Do = cl +
entirely, i.e. by computing E,(sun) accord- c2c2,where c1 and c2 are constants. The lcast-
ing to squares lines allow estimation of Do with
an error of 5 2%. For b. 5 50” the variation
E,(sun) = cos fioFo in Do for 0 I (r 5 0.3 is ~2%. Thus, for 6,
’ exP[- @R + TOz) I 50”, Do can be computed by assuming that
+ cos So], (14 the sea surface is flat; for larger values of
6, (or for an overcast sky) the flat-surface
which clearly overestimates E,(sun) since values of D,(sun) and D,(sky) for use in Eq.
aerosol scattering does make some contri- 11 must be increased in accordance with
bution to E,(sky). Thus, for our purposes Fig. 9.
Eq. 13 and 14 provide lower- and upper- Finally, in the atmospheric model used
bound estimates of E,(sun) and therefore of here rA at 550 nm was taken to be 0.25. This
J: Comparison between Do computed from is very conservative, since it would corre-
E<q. 12 using Eq. 13 for Ed(sun) and the spond to a coastal atmosphere (it is typical
“exact” values (Table 3) shows that for 0 5 of a continental aerosol) and is a factor of
Go I 60” the error is < +. 3%, and for Go = 2-3 higher than would be expected for a
80” Eq. 12 yields a value for Do that is 5- “clear” marine atmosphere. Therefore an
8% too low. The corresponding computa- estimate off based on Eq. 14 alone (i.e.
tions with Eq. 14 for Eti(sun) show that for without any measurements above the sur-
0 5 Go 2 60” the error is < i2%, and for face) should provide excellent estimates of
19~= 80” the computed value is 0.5-4% too Do in clear marine atmospheres.
high.
We can apply this computation to the The Lambert-Beer law applied to &
“shadow” method suggested above for es- Having established that measurements of
timating J: Assume that the object used to K and (K) can be transformed into inherent
cast the shadow of the sun is a circular disk optical properties in a variety of realistic
of diameter somewhat larger than the col- situations, we now turn to the main ques-
lecting face of the irradiance meter. Then, tion of this paper: the extent to which Kd
if the disk is relatively close to the irradiance satisfies the Lambert-Beer law. Consider an
meter, a portion of the sky in the vicinity ocean consisting of m components, one of
of the sun is also obscured. This would ap- which is pure seawater. Let CTbe the specific
proximately correspond to estimatingfwith attenuation coefficient ofconstituent i. Then,
Eq. 14, i.e. photons scattered at small angles ci = C~Cj, and the total attenuation coeffi-
from the sun would be included in E,(sun). cient can bc written
Conversely, if the disk were at a great dis-
tance from the instrument only the solar
disk itself would be obscured, and photons C = Cw + 5 Cj
i-l
scattered at small angles from the sun be- m
come part of Ed(sky)- approximately cor- = Cw -t 2 CrCi
responding to using Eq. 13 to estimate x r=l
Thus we conclude that the shadow method
of determining f should yield values of Do where Cj is the concentration of the ith con-
 Lambert-Beer law applied to K 1403

as functions of the inherent optical prop-

erties and because measurements made un-
der a variety of environmental conditions
(i.e. a variety of D,s) are often combined
for statistical analysis (Baker and Smith
1982; Morel 1988).
Clearly, if Eq. 10 is used for K/Do, its
0 = linear dependence on the inherent optical
1.24, :
properties means that the error in Eq. 15 is
no more than the error in Eq. 10, i.e.
K
- = l.O395(a + bb)
DO
Fig. 9. Do as a function of a2. The curves from
bottom to top correspond to a completely overcast sky rm m 1
and to solar illumination with &, = 60”, 70”, and 80”. = 1.0395 z ai + 2 (bb)i
1i=l i=l 1

stitucnt. These relationships comprise the

Lambert-Beer law, i.e. the individual atten-
= 5 l.O395[ai + (bb)j]*
i=l
uation coefficients are proportional to the
individual concentrations and the total at- = 4
z
tenuation coefficient is a linear sum of the i=l D,’
individual or partial attenuation coeffi-
cients. [On the surface, Eq. 5 and 6 seem to Thus, any error in the Lambert-Beer law
suggest that cl, is not proportional to the over and above the error due to the fact that
concentration of phytoplankton, but rather division of K and (K) by Do does not re-
on the concentration to the 0.6 power. How- move all of the geometric effects, i.e. the
ever, this is an artifact because cp in the scatter of the points about the smooth curve
present bio-optical model includes not only in Figs. 6 and 7 is due to nonlinearities in
the contribution of phytoplankton but also the dependence of K and (K)/D, on the
the contribution from detrital material- the inherent optical properties. To understand
relative concentration of which varies with the magnitude of this nonlinear contribu-
the concentration of phytoplankton (Hob- tion to the error we consider a hypothetical
son et al. 1973). In reality the attenuation model. Assume that Eq. 9 for (K)/cD,-the
coefficient of particles in Case 1 waters more nonlinear of the two relationships-
should be written cr, = &Cph + ciCd, where is exact and that the ocean consists only of
the subscripts ph and d refer to phytoplank- water and plankton. We use w, and wy from
ton and detritus.] Since K/Do and (K)/D, Table 2 with F, = 0.50. Particle phase func-
are inherent optical properties, the relevant tions C, M, and T have FP = 0.98 19,0.9856,
question concerns the validity of the expres- and 0.9880, so FP is chosen to be 0.985. The
sions relative concentration of particles, as mea-
sured by c/c, is then varied from 0 to 1.
K
-= m Kj The true value of ( K)/Do, (K) JDo, is com-
c (15)
DO 1=1D, puted from Eq. 9 using the value of woF for
the mixture, i.e. with
and

o,F = qF&I> + %Fwcw

(K) - m (K)j c,+c, -
DO
z: Do -
(16)
i=l
The Lambert-Beer law value, ( K),/Do, is
For an individual observation, the D,s can- computed from
cel from these equations; however, we will
keep Do on both sides of these equations (WB _
-
(Ww +
-
(K),
because Eq. 8-10 express K/D, and (K)/D, Do Do Do
1404 Gordon

in (1 - uoF) to better fit the computed (K)/

Do near 1 - w,F = 1, a different error dis-
tribution with c,/c would result. To assess
the true error .the reader should sketch what
he or she believes to be the “best” smooth
curve (or segmented curve) through the
points in Fig. 7 and then use the graphical
method above to assess the error in the
Lambert-Beer law, including the scatter of
the points about the “best” line. When I do
this for the model values of c+,, Fp, ww, and
F,, I find the maximum error to be about
ParticleFraction ( c,,I c ) 5% for 440 nm and about 10% for 550 nm
Fig. 10. Relative error (%) in (KjB as a function of for Case 1 waters.
the relative concentration of particles. The optical model developed for this work
is strictly applicable to Case 1 waters only;
however, it is of interest to consider the
with (K) wlDo and (K),lD, individually de- possible extension of these results to Case
termined by Eq. 9 using w,,,.Fwand opFp, re- 2 waters (i.e. Case 1 waters subjected to high
spectively. This procedure can be visualized concentration of suspended sediment and/
graphically by considering Fig. 7. The value or absorbing yellow substances). The basic
of (K),/cD, falls on a straight line between relationships between K/Do and ( K)/Do and
two points on the least-squares curve lo- the IOPs provided in Figs. 6 and 7 are ex-
cated at woF = owFwand woF = a$“, while pected to be valid in Case 2 waters since
(K),/cD, is on the curve itself. The relative one of the particle phase functions used in
error in (K),lD, is then computed by means the simulations was measured in Case 2
of waters and additional values of op (0.50,
0.70, and 0.99) over and above those in
(& - (WT Table 2 were used in the computations and
have been included in the analysis.
The Case 1 example concerned adding
This error is shown in Fig. 10 as a function strong scatterers (plankton and their detri-
of (c,/c). We see that the maximum error tus) to a strongly absorbing medium (sea-
at 440 nm is ~3%, and the maximum error water). If very weak scatterers (wp < I), or
at 550 nm is ~66%. Had w,(440) = 0.80 been nonscatterers such as yellow substances (wp
used in this example the error at 440 nm = 0), are added to pure seawater, the error
would have been <2%. Since ~~(440) = 0.86 in the Lambert-Beer law can be assessed by
is near the upper limit for phytoplankton examination of the region of Fig. 7 near 1
[e.g. Bricaud et al. 1983 measured 0,(440) -ypz 1. Since water has o. 50.30, the
= 0.88 for the coccolithophore Emiliana error in the Lambert-Beer law is essentially
huxleyi which is known to be a very strong the error in the “best” fit to the simulations
scatterer], it is believed that the error at 440 in this region, i.e. about +2%. If particles
nm will typically be < 3% and often -K2%. that scatter more weakly than phytoplank-
With Eq. 7, c,/c can be related to the pig- ton, i.e. wp ~0.8, are added to water, the
ment concentration for this two-component Lambert-Beer law should be even better sat-
example. The maximum error in the Lam- isfied than for Case 1 waters at 440 nm.
bert-Beer law at 440 nm occurs when C M Only the extreme case of adding nonab-
0.09 mg m-3, while at 550 nm it occurs sorbing particles wp = 1 to pure seawater
when C z 1.70 mg m-3. These errors are remains to be considered. For this the Lam-
to a certain extent dependent on the use of bert-Beer law always predicts a (K) value
Eq. 9. Had a different fit to the simulations that is too small, and the error can become
been used, e.g. a higher order polynomial excessive (-30%) in the region 1 - ooF s
 Lambert-Beer law applied to K 1405

0.3. In fact, the error is ~20% when oOF 5 scatterer at optical frequencies (1 - FWo,
0.8 at 440 nm and when wOF 5 0.7 at 550 2 0.85); and the phase functions for par-
nm. Thus, the Lambert-Beer law will fail in ticles suspended in the ocean differs signif-
Case 2 waters dominated by high concen- icantly from that of pure seawater (Fig. 1).
trations of nonabsorbing suspended mate- Finally it is of interest to determine the
rials; however, if high concentrations of yel- accuracy with which one can estimate the
low substances occur simultaneously with diffuse attenuation coefficient of an ocean
the nonabsorbing particles, wOFfor the mix- consisting solely of pure seawater through
ture may be sufficiently low so the error in extrapolation of Kd values measured in a
the Lambert-Beer law is not excessive. Un- real ocean to the limit of zero particle con-
fortunately, the Lambert-Beer law will also centration. As mentioned earlier, this is the
fail in certain coccolithophore blooms in scheme that Smith and Baker and others
Case 1 waters as well. For example, the op- have used to estimate Kd for pure water
tical properties of blooms of E. huxleyi have (Baker and Smith 1982; Smith and Baker
been observed (Holligan et al. 1983) to be 1978a,b, 198 1). For this purpose we have
dominated at times by nonabsorbing de- computed (K) as a function of c at 480 nm
tached coccoliths (uP = 1). The same diffi- by letting C in Eq. 13 range from 0 to 4.5
culties with the Lambert-Beer law that oc- mg m-3. Figure 11 shows the results for Go
cur in sediment-dominated Case 2 waters = O”, 60°, and for overcast skies. The lines
w’ill also apply to such blooms even though on the graph correspond to linear least-
they satisfy the definition of Case 1 waters, squares fits to the computed points with C
i.e. the optical properties are determined by >O (c, > 0 or c > c,), i.e. the point on each
water and by phytoplankton and their im- line corresponding to pure water was left
mediate detrital material. out of the fit. The least-squares line was then
It is important to understand that the near- extrapolated to cP = 0 to determine (K) in
validity of the Lamb&-Beer law rests the absence of particles, which corresponds
squarely on the near-linearity of the rela- to extrapolating C to zero pigment concen-
tionships shown in Figs. 6 and 7, i.e. that tration. As seen from the figure, the extrap-
the quantities involved must bc inherent olated line falls very close to the computed
optical properties is a necessary but not suf- values of (K) for pure seawater. In fact, the
ficient condition for the validity of the law. difference between the computed and ex-
For example, if all particles in Case 1 waters trapolated values of (K), are, respectively,
were sufficiently small to scatter light with 3.8, 1.9, and 1.5%. In this example, the in-
the same phase function as pure seawater, cident illumination is the same for each val-
the dependence of K/CD, and (K)/cD, on 1 ue of c along the individual least-squares
- q,F would be given by Fig. 8. In such a lines, and the linearity of the (K) - c re-
case, if phytoplankton and detritus were lationship again verifies that the Lambert-
mixed with water at 550 nm, the Lambert- Beer law is valid for (K) in Case 1 waters
Beer law value of K for the resulting mixture when the mode of illumination is held fixed.
would fall along a straight lint from 1 - In practice it would be impossible to ar-
oOF x 0.54 to 1 - wOFx 1, while the actual range this experimentally. In the field, each
K values would fall along the curve. Clearly, data point would likely correspond to a dif-
very large departures from Lambert-Beer ferent incident light field. However, we have
law would be seen for all values of c,/c in seen that dividing (K) by D,, removes most
such an ocean. Thus, the near-validity of of the effects of the geometric structure of
the Lambert-Beer law in the case of a re- the light field. To assess the efficacy of de-
alistic ocean is seen to result from the in- termining (K), from extrapolation to cP =
terplay of three independent facts: the de- 0 in more realistic situations, I applied the
pendence of the diffuse attenuation above extrapolation procedure to (K)lD,,
coefficients on the geometric structure of the obtained from all of the simulations (i.e. all
light field can be removed (division by DO); illumination conditions were treated equal-
pure seawater is a much better absorber than ly and included in the analysis). Figure 12
1406 Gordon

---,---i---I-‘-’ Table 5. The inherent (K)!, and the extrapolated

- 1=480nm
value of (K),JD, in m-l for the three wavelengths.
440 nm 480 nm 550 nm

(KkL 0.0182 0.0202 0.0652

UOw’Do 0.0178 0.0202 0.0667

mental determination of (K)‘, must be car-

ried out by excluding turbid waters from the
0.W Wd
analysis.
0.1 0.2
c (me’)
0.3 0.4
From the extrapolated values of K, and
(K),, it is natural to try to estimate a,,,. In
Fig. 11. (K) at 480 nm as a function of c for particle
phase function M. The lower and upper lines are for fact, the values of a, used here were com-
6, = 0” and 60”, and the center line is for an overcast puted by Smith and Baker (198 1) from their
sky. estimated values of K,,, using
K, = aM,+ (b&,, = a, + 0.5b,
shows the results of the extrapolation at 480 where (bJw is the backscattering coefficient
nm, and Table 5 compares the extrapolated of pure seawater, along with Morel’s labo-
value of (K) JDO with the inherent value of ratory measurements of b, (Morel 1974).
(K), (i.e. (K);,), the value of (K) for an However, Fig. 8 shows that, for w,F near 1,
ocean composed of pure seawater computed better approximations are
with the atmosphere removed and with the
sun at zenith. Table 5 suggests that the ex- K’, = a, + 0.62b,
trapolation procedure can yield (K)‘, to and
within -2%. Note, however, that Fig. 12
shows that large errors in (K)‘, are possible (K)‘, = a, + 0.72b,,
if it is determined from a small amount of leading to values of a,,,that are slightly lower
data with cP % c,. For example, if the high- than Smith and Baker’s in the spectral re-
est value of (K)lD,, at c E 0.27 m-’ and the gion below 550 nm. By looking at the vari-
lowest value at c e 0.42 m-l were used, the ation of Kd with a and 19in the asymptotic
extrapolated value of ( K),lDo would be light field for isotropic scattering, Bohren
M 0.038 m-l -an error of nearly a factor of (1984) suggested that the Smith and Baker
2. Thus, as we would expect, the experi- a, values were too low and proposed larger
values based on using K, = a, + b,. My
computations here support Bohren’s con-
tention; however, the true correction is less
than half as large at the surface as he pro-
posed.
My conclusions concerning the validity
of the Lambert-Beer law are in conflict with
those of Stavn (1988). He suggested that
there is a systematic error in the Lambert-
Beer law and that this error can lead to large
errors in partitioning absorption between
water and plankton. Briefly, Stavn based his
0.00 I 1
0.1
’
0.2
1 ’
0.3
1 ’ 0.4
’ ’ objections on the exact relationship
c (m-9

Fig. 12. (IQ/Do as a function c at 480 nm. The Kd = ; (1 - R) + RK,

points are Monte Carlo simulations for various values
of lYo;the line is a least-squares fit to the points with
c, > c,. where R = E,,IEd and Ku = -(l/E,) clE,/
 Lumber-t-Beer law applied to K 1407

dz, which can be derived in a straightfor- the sea surface and independent of depth.
ward manner from Gershun’s law and is When the revised values of a, determined
valid for all z. If R K 1 and Ku x Kd, which from KL are used along with the actual val-
are usually excellent approximations in Case ues of K, near the surface (K, = D,K’,) and
1 waters or yellow substance-dominated the small increase of KI, with depth is con-
Case 2 waters, then Kd M al& The identity sidered, the errors on which Stavn based his
a = a, + aP can be rewritten objection to the Lambert-Beer law vanish
(within his error limits) in the upper 20 m
of the water column in three of the four cases
he examined. Note that this occurs even
without considering the rather large uncer-
tainties Smith and Baker give for their es-
timated K, values. Below this surface layer,
where &, and j&, are the values of p in an it appears that Stavn’s conclusion becomes
ocean with the same radiance distribution valid, i.e. the Lambert-Beer law may lead
incident on the sea surface and the same to significant systematic errors when ap-
surface roughness, but for which cp = 0 and plied to Kd(z) at a given depth z for z ?zlo.
Determination of these errors, a question
CW= 0, respectively. Note that the values of
p, and &, at a given depth are fixed (i.e. they which is very important for the interpre-
do not depend on c,Ic,J. Using Kd e a/j& tation of measurements from moorings, will
require further study both through mea-
Kv x a,+,/&,,etc., we have
surement with modern instrumentation and
through simulation. Only the application of
K/(z)= the Lambert-Beer law to K and (K) has been
addressed here.

[-x41&J(z).
+ i&(z)
Concluding remarks
By simulating the transport of radiation
This equation shows that the Lambert-Beer in a realistic ocean-atmosphere system and
law applied to Kd(z) is approximately valid treating the results as experimental data ob-
only when p(z) M i&,,(z) z i&(z), which oc- tained under carefully controlled condi-
curs near the surface where the value of p tions, it has been shown that K and (K),
is determined mostly by the radiance dis- when modified through division by Do are,
tribution incident on the sea surface and on to a high degree of accuracy, inherent optical
surface roughness. In fact, near the surface properties. A simple scheme for estimating
i&3 = iL(z> = i&m x l/D,. As we move Do for individual experimental situations is
deeper into the water column, F deviates provided. Furthermore it is shown that for
systematically from fi,,, and & with &(z) < Case 1 waters K/Do and (K)/Do satisfy the
p(z) < J&,(Z). On this basis, we expect that Lambert-Beer law to a reasonable degree of
the Lambert-Beer law should work reason- accuracy (maximum error z 5-lo%, de-
ably well near the surface but should lead pending on wavelength). The errors are not
to systematic errors as depth is increased. significantly increased for Case 2 waters as
The present paper quantitatively assesses long as waters for which the IOPs are dom-
these errors, showing that the law applied inated by nonabsorbing suspended particles
to K and (K) works well for z sz,,. Stavn are avoided. The near-validity of the Lam-
on the other hand suggested that large errors bert-Beer law in this situation, where there
are possible even near z = 0. His analysis, are compelling reasons to believe that it
however, was based on the Smith and Baker should fail, is traced to three independent
(198 1) estimations of K, and a,, which we facts: the dependence of the diffuse atten-
have seen are inconsistent with one another. uation coefficients on the geometric struc-
Stavn also assumed that K,,, is independent ture of the light field can be removed; pure
of the distribution of radiance incident on seawater is a much better absorber than
1408 Gordon

scatterer at optical frequencies; and the phase plankton and detritus, yellow substances,
functions for particles suspended in the etc.) and separation of the components can
ocean differs significantly from those of pure only be carried out in a statistical sense.
seawater. If any of these facts were false the Application of these results to field ex-
Lambert-Beer law would fail. Finally, it is periments presents several difficulties. The
shown that extrapolation of K/Do and (K)l first stems from the fact that K/Do, which
Do to the limit c -+ c, yields quantities that satisfies the Lambert-Beer law better than
are within 2% of KL and (K)i, i.e. the value ( K)lDo, is very difficult to measure in prac-
of K and (K) that would be measured for tice due to the strongly fluctuating irradi-
an ocean consisting of pure seawater with ance at the surface resulting from the pres-
the sun at zenith and the atmosphere re- ence of surface capillary waves and the
moved. difficulty of accurately determining the depth
When Do is left out of the analysis, the of the instrument near the surface due to
result is a rather large additional error in the presence of surface gravity waves. Thus,
the Lambert-Beer law (e.g. see Figs. 4 and measurement of (K)/Do, which is signifi-
5). However, Table 3 shows that when mea- cantly less influenced by the surface effects,
surements are restricted to situations where is preferred from an experimental point of
0 I Go 5 40” the total variation in Do is view; however, in the case of oceanic water,
only from 1 to about 1.16, a + 8% variation the mixed layer must be sufficiently deep so
around 1.08. An analysis of (K) so restrict- that .zlo = ~~~/c is within the mixed layer
ed shows that in Case 1 waters the error and the water can be treated as homoge-
doubles over that when Do is included. Thus, neous. For the limiting case of an ocean .free
when measurements are restricted to clear of particles, this would require a mixed layer
skies near noon, the Lambert-Beer law ap- of = 125, 115, and 35 m at 440, 480, and
plied to (K) itself should be in error by no 550 nm. A second difficulty concerns the
more than lo-20%. This may account for determination of Do in the presence of bro-
the success of the law for in situ observation ken clouds. In this case Eq. 12 will not apply
and analysis of phytoplankton absorption and the only viable method is to photograph
(i.e. investigators may have restricted the the sky with a fisheye lens. Finally,
analysis to data taken under “ideal” con- the presence of whitecaps on the sea surface
ditions similar to these). will further modify the internal geometry of
The analysis of oceanic properties with the light field and influence DO. Their effect
Kd is useful because of the relative simplic-. cannot be discussed further without knowl-
ity of the instrumentation required for its edge of their optical properties.
measurement. The near-validity of the References
Lambert-Beer law for all but the most
strongly scattering of natural waters allows AUSTIN, R. W., AND T. J. PETZOLD. 1981. Remote
sensing of the diffuse attenuation coefficient of sea
the partial diffuse attenuation coefficients, water using the Coastal Zone Color Scanner, p.
in the sense of Eq. 1, 15, and 16, to be 239-256. Zn J. R. F. Gower [cd.], Oceanography
determined. Since the partial, as well as the from space. Plenum.
total, Kd functions (K and (K)) are in the BAKER, K. S., AND R. C. SMITH. 1979. Quasi-inherent
characteristics ofthe diffuse attenuation coefficient
first approximation proportional to a + bb for irradiance, p. 60-63. Zn Ocean Optics 6, Proc.
for those species for which a B bb (e.g. SPIE 208.
phytoplankton and dissolved organic ma- -,AND-. 1982. Bio-optical classification
terial), measurement of K or (K) provides and model of natural waters. 2. Limnol. Oceanogr.
a direct estimate of a from KilDo or (K)il 27: 500-509.
BOHREN, C. F. 1984. Absorption of pure water: New
Do. Thus, until the development of an in upper bounds between 400 and 580 nm. Appl.
situ spectral absorption meter, measure- Opt. 23: 2868.
ment of Kd appears to be the only available BRICAUD, A., A. MOREL, AND L. PRIEUR. 1983. Op-
tical efficiency factors of some phytoplankters.
in situ means of estimating a(X). Note, how- Limnol. Oceanogr. 28: 8 16-832.
ever, that in general the medium will con- CASE, K. M. 1957. Transfer problems and the reci-
tain more than two components (e.g. water, procity principle. Rev. Mod. Phys. 29: 651-663.
 Lambert-Beer law applied to K 1409

Cox, C., AND W. MUNK. 1954. Measurements of the in relation to its biogenous matter content (Case
roughness of the sea surface from photographs of I waters). J. Geophys. Res. 93: 10,749-10,768.
the sun’s glitter. J. Opt. Sot. Am. 44: 838-850. -, AND A. BRICAUD. 1981. Theoretical results
DEIRMENDJIAN, D. 1969. Electromagnetic scattering - concerning light absorption in a discrete medium,
on spherical polydispersions. Elsevier. and application to specific absorption of phyto-
ELTERMAN, L. 1968. UV, visible, and IR attenuation plankton. Deep-Sea Res. 28: 1375-l 393.
for altitudes to 50 km. AFCRL Rep. 68-0153. AND L. PRIEUR. 1977. Analysis of variations
AFCRL, Bedford, Mass. in’ocean color. Limnol. Oceanogr. 22: 709-722.
GERSHUN, A. 1939. The light field. J. Math. Phys. -, AND R. C. SMITH. 1982. Terminology and
18: 51-151. units in optical oceanography. Mar. Geod. 5: 335-
GORDON, H. R. 1976. Radiative transfer in the ocean: 349.
A method for determination of absorption and PETZOLD, T. J. 1972. Volume scattering functions for
scattering properties. Appl. Opt. 15: 26 1l-26 13. selected natural waters. Scripps Inst. Oceanogr.
-. 1979. The diffuse reflectance of the ocean: Visibility Lab. SIO Ref. 72-78.
The theory of its augmentation by chlorophyll a PLASS, G. N., G. w. KATTAWAR, AND J. A. GUINN.
fluorescence at 685 nm. Appl. Opt. 18: 116l-l 166. 1975. Radiative transfer in the earth’s atmo-
-. 1980. Irradiance attenuation coefficient in a sphere: Influence of ocean waves. Appl. Opt. 14:
stratified ocean: A local property of the medium. 1924-l 936.
Appl. Opt. 19: 2092-2094. PREISENDORFER,R. W. 1959. Theoretical proof of the
- 1982. Interpretation of airborne oceanic Li- existence of characteristic diffuse light in natural
da;: Effects of multiple scattering. Appl. Opt. 21: waters. J. Mar. Res. 18: l-9.
2996-300 1. -. 196 1. Application ofradiative transfer theory
-. 1987. A bio-optical model describing the dis- to light measurements in the sea. Int. Union Geod.
tribution of irradiance at the sea surface resulting Geophys. Monogr. 10, p. 1l-30.
from a point source embedded in the ocean. Appl. -. 1976. Hydrologic optics. V. 1. NTIS PB
Opt. 26: 4133-4148. 259793/8ST. Natl. Tech. Inform. Serv., Spring-
-, 0. B. BROWN, AND M. M. JACOBS. 1975. field, Va.
Computed relationships between the inherent and PRIEUR, L., ANDS.SATHYENDRANATH. 1981. Anop-
apparent optical properties of a flat homogeneous tical classification of coastal and oceanic waters
ocean. Appl. Opt. 14: 417-427. based on the specific absorption curves of phy-
-, AND A. Y. MOREL. 1983. Remote assessment toplankton pigments, dissolved organic matter, and
of ocean color for interpretation of satellite visible other particulate materials. Limnol. Oceanogr. 26:
imagery: A review. Springer. 67 l-689.
HOBSON, L. A., D. W. MENZEL, AND R. T. BARBER. SATHYENDRANATH, S. 198 1. Influence des substances
1973. Primary productivity and the sizes of pools en solution et en suspension dans les eaux de mer
of organic carbon in the mixed layer of the ocean. sur l’absorption er la reflectance. Modclisation et
Mar. Biol. 19: 298-306. applications a’la teledetection. Ph.D. thesis, 3rd
HOLLIGAN, P. M., M. VIOLLIER, D. S. HARBOUR, P. cycle, Univ. Pierre et Marie Curie, Paris. 123 p.
CAMUS, AND M. CHAMPAGNE-PHILIPPE. 1983. SMITH, R. C. 1974. Structure of solar radiation in the
Satellite and ship studies of coccolithophore pro- upper layers of the sea, p. 95-l 19. Zn N. G. Jerlov
duction along the continental shelf edge. Nature and E. Steemann Nielsen [eds.], Optical aspects of
304: 339-342. oceanography. Academic.
KIRK, J. T. 0. 198 1. Monte Carlo study of the nature - R. W. AUSTIN, AND J. E. TYLER. 1970. An
of the underwater light field in, and the relation- oceanographic radiance distribution camera sys-
ships between optical properties of, turbid yellow tem. Appl. Opt. 9: 20 15-2022.
waters. Aust. J. Mar. Freshwater Rcs. 32: 5 17- - AND K. S. BAKER. 1978a. The bio-optical
532. state of ocean waters and remote sensing. Limnol.
-. 1983. Light and photosynthesis in aquatic Oceanogr. 23: 247-259.
ccosystcms. Cambridge. -,AND- 1978b. Optical classification of
-. 1984. Dependence of relationship between natural waters. ‘Limnol. Oceanogr. 23: 260-267.
inherent and apparent optical properties of water -, AND -. 198 1. Optical properties of the
on solar altitude. Limnol. Oceanogr. 29: 350-356. clearest natural waters (200-800 nm). Appl. Opt.
MOREL, A. 1974. Optical properties of pure water 20: 177-l 84.
and pure sea water, p. l-24. Zn N. G. Jerlov and STAVN, R. H. 1988. Lambcrt-Beer law in ocean waters:
E. Stccmann Nielsen [eds.], Optical aspects of Optical properties of water and of dissolved/sus-
oceanography. Academic. pended material, optical energy budgets. Appl. Opt.
- 1980. In-water and remote measurement of 27: 222-23 1.
ocean color. Boundary-Layer Meteorol. 18: 177- - AND A. D. WEIDEMANN. 1988. Opticalmod-
201. cling of clear ocean light fields: Raman scattering
- . 1988. Optical modeling of the upper ocean effects. Appl. Opt. 27: 4002-4011.