KØBENHAVNS UNIVERSITET '

INSTITUT F O R FYSISK OCEANOGRAFI

F U N D A M E N T A L S O F THE SOLAR RADIATION F I E L D

IN AIR AND U N D E R W A T E R

by

N i e l s K. H ø j e r s l e v

R E P O R T N o . 47 C O P E N H A G E N JANUARY 1-.9&4.

KØBENHAVNS UNIVERSITET

INSTITUT FOR FYSISK OCEANOGRAFI

FUNDAMENTALS OF THE SOLAR RADIATION FIELD

IN AIR AND UNDERWATER

by

Niels K. Højers lev

All r ights r e s e r v e d No par t of this work may be reproduced in any form whatsoever without the permiss ion of the

Institute of Phys ica l Oceanography University of Copenhagen

REPORT No. 47 COPENHAGEN JANUARY 1984

HC. a-tryk København

FUNDAMENTALS OF THE SOLAR RADIATION FIELD

IN AIR AND UNDERWATER - A REVIEW

by

N. K. Højerslev

Institute of Physical Oceanography

Haraldsgade 6, 2200 N, Copenhagen, Denmark.

General Introduction:

The fundamental and currently applied optical quantities in me­

teorology, oceanography and limnology are defined.

The nomenclature and units used throughout the text follow the

recommendations by the IAPSO Working Group on Symbols, Units and

Nomenclature in Physical Oceanography. The classical American

nomenclature and the nomenclature recommended by the IAMAP Ra­

diation Commission in Atmospheric Physics are presented for

reasons of comparison.

The equation of radiative transfer and related equations are

derived following classical schemes.

Finally, the measuring principles for determining the optical

quantities given in the text are outlined.

J_. Definitions, Units and Nomenclature.

The analysis of the daylight radiance distribution and the op­

tical quantities derived thereof leads to the following consi­

derations :

3

The amount of radiant energy, d Q(X)dA, in a specified wave­

length interval, (X,X+dX), which is transported across an ele­

ment of area, dA, and in directions confined to an element of

solid angle d_ü> during a time, _d_t_, is expressed in terms of ra­

diance ("specific intensity" or "power") J , by the expression:

d Q d X = L cos a dX dA dw dt (J) (1)

in which dX , dA, dai and dt are all infinitesimal quantities and

ot_ is the angle which the direction considered makes with the

outward normal to dA. It is implicitely assumed that the spec­

tral distribution of the radiant energy, d3Q dX, is piece-wise

continuous.

h

QtMJJ-m"1

X m

Fig. 1.

rtidA

Q(X,tîd\

Fig. 2.

The radiance, L, is a function of, i) three space coordinates,

ii) one angle which can be given in terms of a zenith angle and

an azimuth angle, iii) wavelength and, iv) time i.e.

L = L (r, a, X, t).

In hydrosol optics the three space coordinates are normally gi­

ven in a three dimensional Cartesian coordinate-system as a

frame of reference. The X-coordinate will then be the latitude,

the Y-coordinate the longitude and the Z-coordinate (oriented

downwards) the water depth. The zenith angle (the angle between

zenith and the incident light confined to the element of solid

angle, dw) is termed 0 and the azimuth angle is termed cp. The

vertical plane determined by the position of the sun and the

point of observation defines a> = 0. The above conventions imply

that dA in Fig. 2S is oriented horizontally at a smooth water

surface. Consequently, the unit vector n and the Z-coordinate

will both have the same orientation.

5

From Equation (1) a number of optical quantities can be defined

(see also Table 1):

Radiant flux, $ :

The radiant flux is considered to be a flux of radiant energy

i.e. radiant energy per unit time. Applying this concept to

Equation (1) the definition of $ is straightforward:

^-rr- dX = d2 $ d X = L cos 0 dX dA dw (W) dt

(2)

Radiant intensity, I :

A given light source emits radiant fluxes in various directions

at certain radiant intensities C"intensity"). If the radiant

flux from such a light source becomes increasingly more colli-

mated in one direction, the radiant intensity is said to in­

crease when observing the light source towards this very same

direction. Improved collimation for say a light beam is obtai­

ned by diminishing its divergence i.e. by diminishing do>. It is

therefore natural to define radiant intensity as radiant flux

per unit solid angle of emittance (or acceptance). Using Equa­

tion (2) the definition of I becomes :

d 2 $ d3Q dt d üj dA = d tu

dX = L cos 0 dX dA (W sr"1 ) (3)

Note that a perfect parallel light beam is non-existing accor­

ding to this definition.

Irradiance (in general), E :

The irradiance ("illumination") of a certain infinitesimal ob­

ject is defined as the radiant flux per unit area. Using Equa­

tion (2 ) the definition of E becomes :

d 2 $ dA

d X = L cos Ö d X dw (W.m 2 ) (4)

Several irradiance parameters as well as parameters derived there­

of are applied in hydrosol optics. Irradiance parameters widely

used are defined and listed below :

Scalar irradiance, E • o —

The integral of a radiance distribution at a point over all di­

rections 6, cp about the point i.e. :

E dX = dX /. L dw ' (5) o 4'ir

in which /, stands for inteqration over all solid angles in the 4 TT ^

upper and the lower hemisphere and d to = sin © d © dcp.

Spherical irradiance, E __ :

Limit of the ratio of radiant flux onto a spherical surface to

the area of the surface, as the ratio of the sphere tends to­

ward zero with its centre fixed i.e. :

E dx = dx * / 4-rr: (6)

where * is the radiant flux onto the sphere of radius r. r

The followinq relation between E and E holds : y o s

E dX = A E dX o s (7)

Downward i r r a d i a n c e , E , :

The r a d i a n t f l u x on an i n f i n i t e s i m a l e l e m e n t o f t h e u p p e r f a c e

(0 £ 0 £ 9 0 , 0 £ cp ^ 360 ) o f a h o r i z o n t a l s u r f a c e c o n t a i ­

n i n g t h e p o i n t b e i n g c o n s i d e r e d , d i v i d e d by t h e a r e a o f t h a t

e l e m e n t i . e . :

E , d X = dX J* _ L cos 0 dw d 2TT ( 8 )

in which /,, stands for inteqration over all solid angles in

the upper hemisphere.

Upward irradiance. E :

The radiant flux on an infinitesimal element of the lower face

(90° <, 0 ^180°, 0° ,< cp £360°) of a horizontal surface contai­

ning the point being considered, divided by the area of that

element i.e. :

E d X = d X / „ L l cos 0 u - 2 7T dw (9)

in which /« stands for integration over all solid angles in

lower hemisphere. Since E > 0 the integrand reads L I cos © I u ^

i n s t e a d o f L cos 0.

V e c t o r i r r a d i a n c e , E :

The v e c t o r ( n e t ) i r r a d i a n c e i s s i m p l y d e f i n e d as :

E dX = (E . - E v dX = dX I , L cos 0 dw d u ; 4TT ( 1 0 )

Downward scalar irradiance, E , : i o c | _

The downward scalar irradiance is defined as :

E . dX = dX /„ L dw (11 ) od 2 TT

Upward scalar irradiance, E : —*- ï ou— The upward scalar irradiance is defined as :

E dX = dA J _ L du (12) OU -2 TT

The following relation is valid :

E dX = (E + E ) dX (13)

o ou ou

Note the difference in signs in Equations (10) and (13), re­

spectively .

Quanta irradiance, q :

The photosynthetic available daylight (PAR) in natural waters

is confined to the approximate spectral range 350 nm X 700 nm -9

(nm = nanometers = 10 m ) .

The quanta irradiance is normally measured as the number of pho­

tons coming from the upper hemisphere and impinging on a hori­

zontal unit area per unit time. The number of photons N (Xo) dXo

per unit area and unit time in the spectral range dX0can be ex­

pressed in terms of the downward irradiance E , dX :. d «

N (X ) dX = E, (X ) X h~1 v ~1 dX (14) o o a o a o o

-34 -1 in which h = 6.626 • 10 J-s is Planck's constant, Q _ -I

v = 3 * 1 0 m • s~ is the propagation velocity of lightand X

is the wavelength - both in vacuo. This velocity holds practically

speaking the same value in air but becomes reduced in water accor­

ding to the relation :

v r v / n (m . s"1) (15) o

in which n_ is the refraction index equal to 4/3 for essentially

all natural waters and wavelengths. For the change in wavelength

between j_ri vacuo (or air) and water, respectively, there exists

a relation analogous to Equation (15) :

X = Xo / n (m) (16)

because the frequency for the radiant energy is invariable to

changes in refractive indices. Accordingly, the number of pho­

tons will not change discontinuously at any optical interface

assuming no reflection. Applying Equations (8), (14) and (15),

the quanta irradiance in water is then defined as :

q = / E (x) X h"' v dX =/ N (X) dX 3 5 0 ^ 350

(17)

In photosynthetic studies it is however preferable to deal with

the number of photons coming from all directions rather than

from only the upper hemisphere. Consequently, photosynthetic

available daylight should preferably be defined through Equa­

tion ( 5 ) :

; 700

E (X) A h 1 v dX (18) 350

As a first order approximation the following correction is re­

commended (Højerslev, 1978) :

q = 1.3 . q (19)

2_. Quantities related to radiance and irradiance

The optical properties of natural waters (or of any medium for

that matter) can be divided into two mutually exclusive classes

consisting of inherent and apparent properties. An inherent

property is one that is a physical constant always having the

same magnitude for a given water mass with a definite composi­

tion of dissolved and suspended matter i.e. variations in the

inherent properties are solely caused by concentration changes

of dissolved and suspended matter in the water mass. An apparent

property is one for which this is not the case.

Often this definition is given in terms of the environmental

radiance distribution : An inherent property is one that is

independent of changes in the radiance distribution ; an appa­

rent property is one for which this is not the case (Preisen-

dorfer, 1961). Quantities related to irradiance quantities are

clearly all apparent properties although they demonstrate quite

commonly quasi-inherent behaviour.

Apparent properties :

The vertical attenuation coefficient, K. , for the radiance, L,

in any direction, (0,cp), at a certain depth z, is the most fun­

damental apparent property in hydrosol optics although not wide­

ly used. The definition of K. reads : L

a L a z

(m-1) (20)

Note that K. depends on the concentration and nature of the sus­

pended and dissolved matter,, the depth, the position, the orientation

(Q,co), the solar elevation, the cloud cover, the sea state, the depth to the

bottom and the albedo of the bottom. K. can be positive, nega­

tive and consequently zero at certain depths and surface light

conditions without violating the principle of conservation of

energy. However, at. great depths in homogeneous waters, K. be­

comes a constant for all 0 ,cp - the so-called asymptotic state

(Højerslev and Zaneveld, 1977).

Ver t ical Cldi ffuse")at tenuat ion coefficients for the irradiances

defined in Equations (5) - (12) and (17) can be defined in the

same manner as K. in Equation (20). The most relevant defini­

tions are listed below :

K = _ _1_ JLL. (m-1) (21)

E 3 z o

in which K is the vertical attenuation coefficient for the sea-

lar irradiance.

d

in which K , is the vertical attenuation coefficient for the d downward irradiance.

K = - 1 |_&l Cm"1) (23) u E a z

u

in which K is the vertical attenuation coefficient for the up­

ward irradiance.

Kr = - _L_ JLL (m~1) (24) E E 3z

in which Kf is the vertical attenuation coefficient for the vec­

tor (net) irradiance.

K = - - L . l i (m~1) (25) q q 9 z •

in which K_ is the vertical attenuation coefficient for the H

downward quanta irradiance between 350 nm and 700 nm.

The depth of the euphotic zone is often defined through the two

relations below :

q [Z ~- °> „ = 100 (26) q ( 2 = z ( 1 % ) ) or

z (1?o) = l n 1 0° a 4- 6

( 2 7 )

< K > < K > q q

10

in which z (1 %) is the depth of the euphotic zone i. e. the depth

at which the just below surface quanta irradiance, q(0), is re­

duced by a factor of 100 (see Equation (26)) and :

1 ,z(n) < Kn > = -, 1 »\ Sn K o <z'> d z ' ( 2 8 )

q z M /o ; o q

All the vertical attenuation coefficients defined in Equations

(21) - (28) attain always positive values in "lossy" and passive

(no internal light sources) natural waters. In some rare cases

with high concentrations of plankton pigments, the water cannot

considered to be passive around 685 nm due to intense fluor-

escense. So in this spectral region even negative K-values can

be observed (see Equation (79) and Højerslev, 1975). In homo-qeneous natural waters K , K., K (and consequently Kr) are y o d u ^ J E quite independent on the surface light conditions and the depth

i.e. they display quasi-inherent features. This is however not

the case for K except for turbid waters (say z ( 1 ?o ) < 10 m) . q = For inhomoqeneous natural waters K , K ., K and K vary only

^ o d u q

slightly with respect to the solar elevation (e.g. Højerslev,

1974 a,1974 b and 1982). This is the rationale for decomposing

any of the above K-functions into forms like :

K = K + K + K . + K (29)

w p ph y

in which the four vertical attenuation coefficients originates

from the vertical attenuation due to pure seawater (index w ) ,

suspended inorganic matter and detritus (index p ) , living phy-

toplankton (index ph) and dissolved organic matter like yellow

substance (index y). Such decompositions are not permissible in a strict sense (see the definitions of an apparent property) but has proven their usefulness at a number of occasions.

Irradiance ratio, R :

The irradiance ratio is defined as the ratio of the upward to

the downward irradiance at a depth in the sea i.e. :

E dX R = — (30)

r dX d

Even in stratified waters R is slowly varying with depth since

K - K . = -1 15. (31 ) u d R 9z J

and K , s K . Changes i n the s o l a r e l e v a t i o n s has a minor e f fec t d u on R being of the order of "\ü% or l e s s .

11

Average cosine, \i :

The average cosine is defined as the ratio of the vector irra-

diance to the scalar irradiance at a depth in the sea i.e. :

5 = LI* = d X J4TT L C O S 0 d0i = < cos 0 >, (32) E dA dA /. L dw

o 4TT

according to Equations (5) and (10). The average cosine is of

fundamental importance in photosynthetic studies. It is normally

equal to 0.75 with an overall variation of the order of ±0.15

for all wavelengths in the visible part of the spectrum and a

very large group of water types - if not them all.

The average cosine varies only slightly with depth or surface

light conditions but may change to some extent with respect to

wavelength (e.g. Højerslev, 1977 and 1978).

Distribution factors, D, and D :

— • ! d u —

The downward distribution factor D . is defined in a reciprocal

but similar way as the average cosine since : Enrl dA dX / 9 L du) -1

D . = -Q£1 -~ *E = < c o s 0 > (33) d E, dX dX /_ t cos 6 du> 2TT

d - 2ir The upward distribution factor D is defined as :

r u E0()dÀ dA / L du

D = -°Ü— = =^21 (34) u Eu dX dA J_2Tr L | cos 9 I dco in which D - 2.5 ± 0.3 for most cases. If the radiance in the u

lower hemisphere is assumed isotropic i.e. L = constant for

90° S Q Â 1 8 0° a n d °° S V i 3 6 0° t h e n D = TT.

_3. Inherent properties :

Absorption coefficient, .a :

The internal absorptance of an infinitesimally thin layer of

the medium (hydrosol) normal to the beam, divided by the thick­

ness, A r , of the layer. Since the absorptance, A, is defined

as the ratio of the radiant energy flux lost from the beam by

means of absorption, (transformation of radiant energy into

sensible heat, chemical energy, fluorescent energy etc.) to the in­

cident flux, the definition for a becomes :

12

a = - M = Ar

Å 3> 0 Ar

Cm - 1) (35)

(Total) Scattering coefficient, _b :

The internal scatterance of an infinitesimally thin layer of the

medium (hydrosol) normal to the beam, divided by the thickness,

Ar, of the layer.

Since the scatterance, B, is defined as the ratio of the radiant

flux scattered from the beam, (causing no loss in radiant ener­

gy but solely changes in direction of photon pathways) to the

incident flux, the definition for b becomes :

b = - M = Ar

A $ $ Ar

(m 1) (36)

Beam Attenuation coefficient, .c_ :

The internal attenuance of an infinitesimally thin layer of the

medium (hydrosol) normal to the beam, divided by the thickness,

Ar, of the layer.

Since the attenuance, C, is defined as the ratio of the radiant

flux lost from the beam by means of absorption and scattering,

to the incident flux, the definition for j: becomes:

c = -AC _ A $ (rrT1) (37) Ar $ Ar

For a (non-existing) parallel and monochromatic beam of infinitesi­

mal width the following relation is valid :

a + b = c (38 )

If the beam is polychromatic, part of the radiant flux being ab­

sorbed at some wavelengths might be re-emitted as a fluorescent

radiant flux at higher wavelengths. For this case a + b # c,

when measured by using a polychromatic emitting light source

combined with a monochromatic receiving light sensor.

Volume scattering function, ß(0) :

The radiant intensity (from an infinitesimal volume element, dV,

in a given direction forming the angle B to the incident beam)

per unit of irradiance on the volume and per unit volume i.e. :

13

dl(0)

Fig. 3

ß (0) = dl O ) E dV

or alternatively

3 (ø) = L (0) _ L

(m sr ) (39)

(40) L (0) dl dw

in which L (0) is the radiance at the volume surface perpendicu­

lar to the incident beam and L (ø) the outgoing radiance from dV

in the direction 0. Note that L (0) and L (0) are radiances at

the volume element. In the real experimental situation when

measuring 8(0) in situ (see fig. 4) dV is surrounded by the h y -

drosol and Equation (40) shall read :

ijht sensor, 6*6

light source

ight sensor, 0*0

Fig . 4

ß (0) - L (6) 9e|)-Vrl

L (0) dl dw (41)

in which c is the attenuation coefficient of the surrounding

hydrosol and dw the solid angle of acceptance for the light sen­

sor .

Forward and backward scattering coefficients, bc and b, :

tering coe

8(0,9) dw

The forward scattering coefficient, b f, is simply defined as :

; 2* ; „/2 X o o (nT1) (42)

14

and t h e b a c k w a r d s c a t t e r i n g c o e f f i c i e n t , b . , as :

b b = ; 2 7 T S^/2 ß(0,cp) dw Cm"1 ) ( 4 3 )

S i n c e b = b f + b. : f b

b = !, B ( 0 , cp ) dio (m~ 1 ) ( 4 4 )

I t i s o f t e n assumed a p r i o r i t h a t ß i s independent o f cp i . e • :

b = 2iT / * ß ( 6 ) s i n 0 d 0 ( m - 1 ) ( 4 5 )

The scattering properties of a hydrosol is not completely de­

scribed by the volume scattering function. This is only the

case if the radiance field is described in terms of streams of

photons (a scalar formulation). If polarization is considered

for the radiance field (a vector formulation) the scattering

properties of a hydrosol is completely described through a so-

called Mueller scattering matrix consisting of 16 (4 x 4) ele­

ments. Assumptions concerning the nature of the suspended mat­

ter reduce the number of elements in a deducable manner. How­

ever, throughout the text only the scalar formulation for radia­

tive transfer in hydrosols is applied. The complete theory of

radiative transfer can be found in various textbooks and review

articles (e.g. Chandrasekhar, 1950, Van de Hülst, 1957, and Mo­

rel, 1973).

it* Equation of radiative transfer :

In order to derive the classical equation for radiative transfer

in a medium like a hydrosel in a fairly general and simple way

some introductory assumptions are necessary :

1) The radiant flux sources are all external and the boundaries

of the hydrosol extend towards infinity. (Later, the case of

internal flux sources is treated as well - see Equation (69)).

2) Only a scalar theory for the radiative transfer is considered

i. e.. the light field is unpolarized.

Note : In the following the symbols will be used as defined ear­

lier without further explanations.

First consider : L = L (x, y, z, 0, cp, X, t).

There are practically speaking two classes of time dependences

- or two typical time domains :

1) The time dependence connected with the propagation velocity

15

of light in the hydrosol i.e. v = v/n, and

2) variations in the atmospheric light conditions due to changes

in solar elevation, cloud cover, etc.

Concerning the first class it can be said that light is not di­

stributed instantaneously from a given light source. There will

be a certain time retardation effect, j5_t, determined by y_ and

the travelled distance j5r_ during the period 61. Accordingly,

or = v6t.

The variation of L along the path, 6r, becomes thus due to the

above effect £L = _L ÈL.. The latter is the change of L during the $7 v 3t

time Ot at a volume dV around the point (x, y, z ) . In hydrosol

optics this term can be safely neglected because typical values — f\

for or and __y_ lead to 6t - 10 s, which cannot be recorded by

conventional meters.

The second class of time domains has then to be included for

an appropriate time scale range between say one second and 12

hours. In practice, these time variations are corrected for by

using deck photometers or similar devices. So, for most measu­

ring situations it can simply be assumed that the radiance field

is quasi-stationary, i.e. L = L (x, y, z, 0, <D , À ) .

UÔ,¥,r ) (•\*',r)

Fig. 5

A simple derivation of the classical equation of radiative

transfer is given below. It is assumed that the light field is

monochromatic for reasons of simplicity at this point.

The radiant flux on dA (see Fig. 5) is :

d2 $ (r) = L CO, cp, r) dA dœ (46)

The radiant intensity scattered the angle y away from the volume

element dV = dA or amounts :

16

d2 I (y) = P(Y) dE dV = 3(Y) d* * ( r ) dA 6r = 8(Y) d2 * (r) 6r (47)

dA

The total radiant flux scattered away from dV is obtained by

integration over all directions i.e. :

-/ 4 TT * (Y) dtü = -d2 * (r) 6r /. 6(Y) du = 4 TT

(48) -L (0, cp, r) dA 6r dw • b

by applying Equations (44), (46) and (47).

The volume element absorbs in addition part of the radiant flux

in a manner given by Lambert-Beers law or by Equation (35) :

6 (d2 0 (r)) = - a d2 $ (r) 6r = - L ( 0, cp, r) dA ôr d to • a (49)

The total change of the radiant flux along _r and due to dV be­

comes :

- (a + b) L (0, cp, r) dA 6r dw (50)

in which a + b = c.

The radiance field surrounding dV is exemplified by L (©', cp ', r)

in Fig. 5. Part of this radiance field becomes scattered into

the directions of the considered pathway r_. The radiant intensity

scattered the angle o_ away (i.e. scattered from the 0' , cp' -

direction into the 0, cp - direction) amounts :

d2 I (0, cp) = 3(a) dE dV = $(a) d* ? ( r ) dV = ds

[3 (ot) L (0'7 cp', r) dm' ds d V ds

(51)

in which d_s_ is a cross sectional area to the 0' , cp' - direction.

The total radiant flux scattered int.o the 0, cp - direction by

dV is likewise before obtained by integration over all direc­

tions i.e. :

4TT I (0, cp) = dl (0,cp) = dV /. ß(a) L (0', cp', r) du' (52)

4TT

It can easily be shown that

cos a = cos ©cos 0' + sin 0 sin 0' cos (cp-cp1) (53)

by forming the scalar product of vectors of unit length sepa­

rated by the angle a and given in spherical coordinates. Since

dl (0, cp) • du)=d 2 $ (r), Equation (52) can be written as

$ = dco dV J ß(a) L (0', cp' , r) dca* 4TT

(54)

which expresses the change in the radiant flux due to the ra­

diance field surrounding dV. Consequently, the overall change

17

in the radiant flux is obtained by adding Equations (50) and (54).

The overall change in the radiant flux along ^r_ equals also :

d 2 $ (r + 6r) - d 2 $ (r) = <S ( d* $ (r>> <5r (55) ôr

C o m b i n a t i o n s of Equa ' t i ons ( 4 6 ) , ( 5 0 ) , ( 5 4 ) and ( 5 5 ) r e s u l t i n t h e

c l a s s i c a l t i m e - i n d e p e n d e n t e q u a t i o n f o r r a d i a t i v e t r a n s f e r t

5L (0 <p, r ) = _ c ( r ) L (0 , tp, r ) + j ^ L ( 0 " , cp ' , r ) 3 (a ) d w ' ( 5 6 )

i n w h i c h r = r ( x , y , z ) and dw' = s i n 8 ' d o ' d cp'.

Now , -*±- - —..iL., g r a d L = 9_L s i r i 0 cos cp + 2L s in 0 s i n ro + _ cos 0 so or | r I 6 x 6y ' ôz

t h e r a d i a t i v e t r a n s f e r i n t h e d e f i n e d c o o r d i n a t e - s y s t e m i s now

d e s c r i b e d by :

6L ( x , y , z , e , < p ) s i n 0 c o s + 6L ( x , y , z ,6 , ep) s i n 0 s i n rø +

öx oy

6 L i x ^ z r e 7 ^ cos 0 = - c ( x , y , z ) L ( x , y , z , 0 , c o ) +

/ 4 7 r L ( x , y , z , 0' , cp') ß ( x , y , z , 0, cp, 0» , cp» ) s i n 0' d 0* d cp' ( 5 7 )

In order to deal with the radiance - or other related optical

quantities in a manageable way - it is often assumed that the

hydrosol is horizontally stratified leading to -^ = -•?-=- = 0. o x <5y

For t h e m a r i n e e n v i r o n m e n t , t h e r a t i o e s •— s i n 0 cos cp /ÈJL cos 0 ox / o z

and —k sin 0 cos cp / — cos 0 come close to zero for all (x.y.z. tfy / 6z

0, cp) because the tilt of the j3-isosurfaces and the a^-isosurfaces

with the horizontal is small - even in oceanic frontal zones.

T— = -r - = 0 is therefore an acceptable assumption for these re-ox oy gions.

In lakes, however, it is not quite so. If the tilt of the B_ -

isosurfaces and the ja-isosurfaces with the horizontal is of the

order of 45 - which can be the case - the two beforementioned

ratioes may be of the order of unity for all ( x , y , z , 0, cp) .

Finally, the shadowing effect from the measuring platform or

bottom albedoes might result in appreciable horizontal grad,, L

contributions,

Equation (57) illustrates clearly the difficulties in measuring

L. It has to be measured in at least 2 vertical points, 4 hori-

18

zontal points and in a number of G, f0 - directions in order to

make proper use of Equation (57). This is the rationale for

measuring other optical parameters derived from the equation of

radiative transfer.

Integration of Equation (57) over all solid angles., / . doo ,

gives straightforward :

x A r T— f, L s i n 0 c o s cp do) + — : - a E ( 5 8 ) o x 4ÏÏ ô z o

by a p p l y i n g É q u a t i o n s ( 1 0 ) , ( 5 ) , ( 4 4 ) and ( 3 8 ) i n t h e o r d e r men­

t i o n e d . Note t h a t

- — - / . L s i n 6 s i n c p d w = TT— J \ . L s i n 2 6 s i n c ? d O d r o - 0

6 y 4ÏÏ Sy 4 T

due to the definition of cp, L (cp) ^ L (-(p) and sin cp = - sin (-co),

respectively. (L is normally symmetrical around the vertical plane through

the sun and the point of observation and the orientation of this plane cor­

responds to © = 0). The (p - dependence of L can be approximated by an ex­

pression of the form : L M = A £ + ' (e > 1) (59)

G - COS CO

for any point in the hydrosol. A is a function of the L and 7 r J — max

L . for any fixed G (Lundgren and Høierslev, 1971) and e is a min 7 y j » __ E + I constant. The expression • is the parametric represen-e - cos cp

tation of an ellipse.

The integral /, L sin 0 cos cp dco is clearly much less than

S, L do) = E because of the changss in sign for cos cp due to

the 0 - 2TT integration and since sin 0 ^ 1 for all Gin the 0-TT

range. Consequently, conventional scale analysis gives, when

ignoring signs :

/ 6E / ~~ J. L s i n 0 cos cp d w / a E << - * - £ /aE ~ 1 ( 6 0 ) 6 x 4TT / o ox / o

From t h i s c o n d i t i o n :

<5E <• a ox or K - a - - a cot 9 (61) E o o z

o o o

Since K ~ a, 0 ~- 45 i.e. for a tilt even equal to 45 of the o . H

_a-isosur f aces with the horizontal, Equation (60) leads to :

aE >> ~- / . L s i n B cos cp dw ( 6 2 ) 0 O X H-TT

resulting in the now justified and highly useful equation :

Sz - ' o (63)

19

valid in all natural waters displaying even the highest inhomo-geneities with respect to suspended and dissolved organic mat­ter. Note, that E, E and a depends on the position (x,y,z) for

' ' o —

this considered case and that the wavelength dependence has been

ignored until now.

It is sometimes convenient . to consider another form of Equation

(63) :

a = K . r I - R + _!_ ÉK ] £ d (64) d L K . dz J E

d o which is simply another formulation of Equation (63) obtained by using Equations (10), (22), (30), and (63). Since ^ = R I K, - K l << R • b and R < 0.1 (normally being less

dz ' d u ' in the oceans and many lakes) it can safely be assumed for most

natural waters that ~ ~ 0. Accordingly, nz

a * K. (1 - R) Li (65) d E

o

Equation (63) can finally be written as

a = y • KE (66)

applying Equations (24), (32) and (63).

In the marine environment p is a slowly varying function with

respect to surface light conditions and depth (e.g. Højerslev,

1977, 1978) i.e. the vertical variation of §_ is directly reflec­

ted in the vertical variation of Kp (or K . ) . This empirical

observation is reflected in the most simple oceanic model for K . which states : d

K . ~ a + b, ~ a + 0.02 b (67) d b

A- The time-independent equation of polychromatic radiative transfer

for a horizontally stratified medium.

The results obtained in Chapter 4 - especially Equation (57) -

are directly applicable for this Chapter in which a multi-spec­

tral light source like the sun will be considered (in Chapter 4

only a monochromatic light source was assumed).

The conditions stated in the head-lines of this Chapter give

strictly :

20

iL - iL - iL - o (68) 8t 3x ~ 3y

A multispectral light field might give rise to fluorescence in

some spectral regions. This effect, F^ has to be taken into

account together with the findings in Equation (57) which for

the above case then becomes (Højerslev, 1975) :

cos 0 3 L ( z ' f> C-p> XoA = - c (z, A ) I (z, 0, cp, X )

+ L„ (z, 0, (p, X ) (z, X, X ) (69)

where L„ is the path function

L^ = J ß(z, 0, cp , 0' , ©' , XQ) L(z, 0', <p', XQ) sin 0* dø' dtp' (70)

and F# is the source function of fluorescent light at X

F„ = / E (z, X) f(z, TT/2, A, A ) dX. "*" o o o

(71)

Equation (69) can be proven from the following considerations :

The volume scattering

fined by the relation

The volume scattering function ß(z, a, A ) is for this case de-3 O

dl . (z, OL x ) dX B(z, a, X ) = scat ! —° °

° E (z, XQ) dXQ dV (72)

dl , (z, rv, X ) d X is the radiant intensity of 'the scat ' ' o o 7 where

light scattered in the direction a from the incident beam, and

E (z, X ) dX is the irradiance at wavelength A on the volume 7 o o o

element dV at depth z. The scattering coefficient JD is found

by integration of ß ( z , et, X ) over all solid angles : b(z, X ) = J, ß(z, a, X ) dw

O 47T O (73)

In case of isotropic scattering, Equations (73) and (70) give :

î ( z' a' V = "4? b ( z ' Xo } = 3(Z»'ÏÏ/2, X0) (74)

and

U ( z , 0, cp, A ) = E (z, A ) ß(z, TT/2, A J . (75) O O ' O ' " " ' ' "O

Since daylight acts like a broad spectral light source, emis­

sion of fluorescent radiant energy takes place in the sea when

excited by the shortwave part of the daylight.

A volume scattering function of fluorescent radiant intensity

f(z, ot, X, A ) is introduced by the relation :

f(z, a, A, Xo).jdA0 d2I fluor (z, a, X, X ) dX

E ( z, A) dA . dV (76)

21

where d 2I r, (z, a, X, X ) dX is the radiant intensity of flu-fluor ' ' ' o o 7

orescent light at wavelength X excited by daylight at wave­

length X and emitted in the direction oc from the incident beam,

E(z, X) dX is the daylight radiant flux at wavelength X im­

pinging upon the volume element dV at the depth z. The total

scattered radiant intensity of fluorescent light at wavelength

X in the direction ot from the incident beam is obtained from o integration of Equation (76) over all wavelengths

(77)

Obviously, fluorescent radiant intensities are scattered iso-

tropically which means that

f(z, Ci, X, X ) = f(z, TT/2, X, X ). (78)

o o

The source function of fluorescent light F+ can now be derived

analogous to the derivation of the pathfunction L^ in the case

of isotropic scattering - see Equation (72) and (75) :

dl_ (z, a, X ) = / E (z, X) f(z, a, X, X ) dX dV. fluor o o ' o

F„ = J E n (z, X)- f (Z,TT/2, X, X j dX (79)

The scalar irradiance E (z, X) can be measured by means of a o • J

light absorption meter (Højerslev, 1975) and the volume scat­

tering function of fluorescent light f(z, TT/2 , X, X ) by aid of o

a conventional fluorimeter (Parker 1968) measuring through 90° the excitation spectrum at the emission wavelenqth X of a wa-

3 o

ter sample taken at depth z. In practice it suffices to inte­

grate the expression in Equation (79) between 295 nm (lower li­

mit of incoming daylight radiant energy) and X .

An optically homogeneous hydrosol acts like a monochromator at

great depths. In the clearest ( oligotrophic) waters the wave­

length of maximum transmission is around 460 nm so K , (X = 460 nm) d

attains its minimum among all the K.-functions . Applying this

to Equation (22) it can be seen directly that at great depths in

these parts of the ocean only the blue (460 nm) daylight compo­

nent will be present.

In turbid (not necessarily eutrophic) waters the wavelength of

maximum transmission can even be found at wavelengths where the

pure water itself absorbs strongly i.e. above 600 nm. But again,

these water masses act as monochromators at great optical depths

= / Kd (z1) dz'. Note that the water depth, _z, can be small,

say less than 10 m. All together, at great depths the time-in­

dependent equation 6f radiative transfer - Equation 69) reduces

to :

22

cos ø 3 L (z, 9, <p, X Q I = _ c ( x ) L ( ø x ) + • ( ø {p x ) (80)

gz o o * o

in which the wavelength dependence can be disregarded. Assuming

either, i) a and R are constants, or, ii) -§• is a constant, Hø­

jerslev and Zaneveld (1977) have proven theoretically - without

using further assumptions than those contained in Equation (80)

- the existence of the so-called asymptotic regime (z -* °°) for

which :

| ^ = 0 and - ± |1 = constant = k 3cp L 3z

(81 )

so in the asymptotic regime, the radiance distribution is sym­

metrical around the vertical and the form of this distribution I 3 L is invariant to changes with depth. The finding -_ -L = k implies L 3L

that the K-functions in Equations (21) - (24) also attain the value k and that ^ = = ^T^ = r^- = 0> exactly. — dz dz dz dz

The existence of the asymptotic regime finally implies that the

average of all possible K. -functions - and thereby of all the K-

functions - (calculated from the surface or from any.-finite

depth z

lim JL / z o

Z •+ °°

lim

and down to the asymptotic regime-) equals j<, i.e. :

K. (z' , 9, cp) dz • = k or L

z-z0

SZ K (z1 , 0, ©) dz ZQ L

If the form of the L - d i s t r i b u t i o n and the first d e r i v a t i v e of L 31 with respect to depth, 5—, are given in the asymptotic regime,

then the inherent and apparent optical properties of this regime

can be derived (Zaneveld, 1974) :

f 4^ P' (cos 0 ) cos 0 sin 0 dø

c = -3z TT

(82)

L P (cosG) sin8 d0 n

ß(Y) = I 2n +

n = o

P (cosY) n

TT 6L •To ^ p

n (cos e ) c o s e sinØdø" S J1 L P (cos 0) sin 0 dG

n

(83)

in which P is the Legendre polynomial of the order n. It is possible to n • • *,

integrate Equation (83) over all solid angles in order to ob­

tain b and then make the subtraction c - b = a in order to get a.

However, observing that Equation (66) is strictly valid, K^ = ^ = - _L JLL and applying E q u a t i o n (32) it is s t r a i g h t f o r w a r d to d e -

L a Z

rive a

23

- Jn^Lcos Q s i n Q dQ 9t 71

(84)

6,. Measurements of basic optical quantities and quantities

derived thereof.

General :

Fig. 6

In this chapter the définit ions for $, I, L and E, respecti­

vely, have to be used along with either Equations (57) or

(80).

Assume two infinitesimal elements of area ds1 and ds ? separated

by the distance r. The orientations of ds1 and ds„ are given by —• —•

the two unit vectors n^ and n„, respectively - see Fig. 6. ds,,

is supposed to have a uniform radiance L, because ds1 is small.

Now, the radiant intensity dl in the direction 91 is defined by

(avoiding the wavelength dependence) :

d2 * dl =

dw (85)

in which the solid angle dco,. is given by :

dw. ds? cos 92 (86)

The irradiance dE at ds? must be :

dE - rf2 ^ = dl cos 02

ds " T1 (87)

This formula embodies the well-known inverse-square law

and the cosine-law of irradiance.

2k

The radiance L for the outgoing field is by definition (2):

d2 $ dl L = ds4 cos 0. dw as. cos 0 i l l 1 1

or by using Equation (86) :

d2 « r2

L = ds. ds„ cos G1 cos 0„

(88)

(89)

The same expression is found for the incoming field. The symme­

try of the equation :

d2 $ - L ds^ ds2 cos 0*j cos 09 (90)

requires that the radiance for the outgoing field be equivalent

to that for the incoming field. From Equations (87) and (89)

the irradiance at ds9 may be written in the form :

dE = äl_i = L cos 0O dœ0 (91) ds 2 — 2

Radiance :

As has been seen, the full knowledge of the radiance field in a

medium with known boundary conditions permits derivation of all

the optical quantities of interest. Equations (85) and (88)

tell that a radiance measurement can be performed by having a

radiant flux receiving device with a narrow solid angle of ac­

ceptance - a so-called Gershun tube which can be oriented in

all 0, cp - directions.

Irradiance and apparent properties :

An E .-irradiance meter will, turned 180° downwards become an E -d ' u

irradiance meter. Such two lowerings at the same position de­

termine the following optical quantities :

E . , E„ , E , K, , Ki( , Kr and R (92) d ' "u ' w ' "d ' "u ' "E

An E ,-irradiance meter is designed to collect all the radiant

flux from the upper hemisphere impinging on the collector area

in a manner given by Equation (91). If the collector transmis-

25

sion of photons to the light sensor changes according to a cos 0-

law ( 0 = 0 corresponding to zenith) Equation (91) demonstrates

that such a meter integrates the upper radiance field in the de­

sired manner :

!?-u d E /„ L cosø dw Z7T

(93)

Similar arguments apply for the E -irradiance meter.

An E ,-irradiance meter will, turned 180° downwards, become an od ' >

E -irradiance meter. Such two lowerinqs at the same position ou y r determines the following optical quantities :

E , , E , E , K . , K and K (94)

o d ' o u ' o ' od ou o

Combining these quantities with those in Equation (93) more op­

tical quantities can be calculated : Dd , Du, ii and a (95)

In order to measure E , (or E ) the radiance field has to be od ou

integrated over all solid angles in the upper hemisphere. This

can be done by applying a thin-walled small spherical collector

(having cos 0 -collecting and cos 0 -emitting properties) below

which an infinitesimal light sensor is placed. Moreover, this

meter shall have a shading opaque device extending towards in­

finity in the horizontal direction without causing changes in

the original light field. It is of course an impossibility but

for all practical purposes it can be shown that the shading o-

paque device has only to have a radius being four times greater

than the radius of the sphere. E , (and E ) is not commonly od ou

measured, however. Normally, either the quantities f (E + E) and k (E - E) or E alone can be measured routinely and without

* o o • / •

difficulty. All details are given by Højerslev (1975).

Inherent properties :

It is known that :

a + b = a + J ß dw = c (96)

E q u a t i o n ( 9 6 ) shows t h a t t h e two i n d e p e n d e n t and most f u n d a m e n ­

t a l i n h e r e n t p r o p e r t i e s are _a and _ß from which _b arid £ can be d e r i v e d .

However, l i k e i n the case of rad iance, _g_ i s cumbersome and d i f f i c u l t to

26

measure. Moreover, j is difficult to measure directly in situ,

but examples of direct a_-measurements in-vi tro have been repor­

ted (e.g. Tarn and Patel, 1979).

Determinations of a (z) are based on measurements of either, i)

E , , E and E or, ii) k ( E + E) and I (E - E ) from which d u o o o

]J . Kp = a can be calculated (e.g. Højerslev 1973, 1975). It is

therefore natural to focuse on c, b and _ß in the order mentioned.

The measuring principle in the conventional c.-meter is based on

the equation of radiative transfer written in the form :

|^ = - cL + L* (97) 3r *

In order to minimize L^ a thin near-parallel and fairly intense

light beam and a small solid angle of acceptance for the recei­

ving light sensor are essential. Mechanical shading devices in

order to minimize the part of L^ which is due to the daylight

seem to be impractical for a variety of reasons.

With a properly designed £-;fiieter L^/cL ~ 0 i.e.- :

|^ = - cL (98) or

or upon integration with respect to j? and rearranging :

c = J- £n f-4H (m"1) <"> r L ( r )

i n which r_ i s the p-a th , length i n w a t e r , L (o) and L ( r ) p r o p o r ­

t i o n a l t o the s i g n a l r ead ings i n a i r and w a t e r , r e s p e c t i v e l y .

However, i n p r a c t i c e Egua t ion (99) i s not v a l i d because,

i ) the reflectance in the Mwindow"-air system i s greater than in the "win-

dow"-water system and, i i ) the divergence (which a l l physical l i g h t beams

posses) d i m i n i s e s by a f a c t o r o f n 2 ~ 1™. when go ing from the a i r

i n t o the w a t e r . The proper equa t i on f o r c_ shou ld read i n s t e a d :

c = k (A) 1 In M ° > (m~1) (100) r L ( r )

in which k (X) is a constant depending solely on the geometry of

the £-meter and on the wavelength. k (X) has to be determined

by a calibration procedure using a water sample of known c (A).

Normally k (\) -0.80 (e.g. Bartz et al., 1978).

Using the equation of radiative transfer in the form :

cos 0 17 = " cL + L* (101)

27

it is directly seen that _c can be derived from any horizontal

(cos 0 = 0 ) measurement of L and L„ since :

(7) - L« (90, cp, z) V , ~ L (90, cp, z)

(102)

L (90, cp , z) is measured by means of a Gershun tube as earlier

described. L# (90, cp, z) is also measured by such a tube but

the direct radiance coming from, the direction 90, cp is blocked

by a black plate at some distance. The method is highly imprac­

tical in oligotrophic waters since the mechanical set-up must

have a length of the order 20 meters. In turbid lake waters

the conditions leading to Equation (101) might not be fulfilled-

see earlier in the text.

The measuring principle in a £_-meter is depicted in Fig. 4 and

described through Equation (41) for the general case i. e. a

measurement over all directions. In practice, the optimum J3-

measurements are performed in the interval 0.1 £ 8 ^ 170 . It

shall be emphasized that the scattering volume changes in size

with 0 according to : dV (0) = dV (0 = 90 ) / sinØ, so correc­

tions due to this effect must be carried out. Having determined

3 (0) from a measurement in one plane, _b is obtained upon inte­

gration :

0-170 2 Tf

e > 0.1 8 (0) sin© d0 (103)

If e = 3 , say, the relative error ^ü calculated from Mie theo-Ab b

ry is expected to be found in the range 0.02 ,< j b <^ o. 93 under

extreme conditions : 0.01 ^ H _< 1 (Højerslev, in press).

Finally, the measuring principle for the _b-meter is depicted in

Fig. 7 and explained below :

Lambert emitter (L ê

senior \ , ' ^

— T —- " " » dm.

1 Fig. 7

28

The b-meter consist of a Lambert emitter having the radiance L . — ^ o

The radiant intensity emitted then varies with the angle 0 ac­

cording to

I (0) r I (0 = 90°) sin 0 (104)

The light sensor is a Gershun tube having a small solid angle of

acceptance, dw , so that no direct light from the emitter to­

wards the Tight sensor is recorded.

Only light scattered in one direction by the cone volume defined

by dw is measured. 7 o

The intensity at the scattering volume dV amounts :

I (90°) sin 0 e -(h/sin 0 ) c (105)

by using geometrical considerations and Equation (104). The ra­

diant flux through dV i.e. through the cross sectional area dA

can be written as :

I (x) dw E (x) dA (106)

in which x is the distance from the light sensor to the scatte­

ring volume dV and

dw dA (107) (h/sin 0 ) 2

Applying the above equations the irradiance E (x) at dV becomes :

I (90a)sin3e - (h/sin 0) c E yx) (108)

The radiant intensity scattered by dV into the light sensor is

found by Equation (39) :

dl = E (x) R (0) x2 dx dw (109)

o

so the radiance emitted from dV to the light sensor amounts by

definitions :

dl_ dl - ex x dw

(110)

Geometrical considerations give : x = r - h cot 0 and

dx = (h/sin20) d0 so combinations of Equations (103) - (105)

result in

dL = 1 < 9 Q S (0) sin 0 e- c!> + h (iîïTë - c o t G ) ] h

(111 )

29

The function e s i n 9 is close to unity for fall

scattering angles where the light scattering is appreciable.

Accordingly, the total radiance recorded by the light sensor is

obtained by integrating Equation (111) :

L = I (90°) e- or ; % ( e ) s i n 0 d 0 s I (90°) . e- cr . fa ( m )

h e 2TT h

in which e, 6 , h and r are fixed through the geometry of the b

-meter and I (90 ) a physical constant to be determined through _ c r

a calibration. The term e has to be measured independently.

Z* Acknowledgements

This paper was presented upon invitation in Woods Hole, Massa­

chusetts, USA from September 12 - 16, 1983. The Special Program

panel on Marine Sciences of the NATO Science Committee supported

an Advanced Research Institute Workshop on "Photochemistry of

Natural Waters" to which I address my sincere thanks.

Due thanks are also given to the local organizers Dr. 0. Zafi-

riou and Ms. S. Kadar for their most competent and agreeable

way of handling the meeting.

30

TABLE 1

QUANTITY [ U n i

Rad ian t Energy

Radiant F lux

Radiant In tens i ty

Radiance

Irradiance

Radiant ex i tance

Downward i r rad iance

Upward i r rad iance

Net i r rad iance

Scalar i r rad iance

Downward scalar i r rad iance

Upward scalar i r rad iance

Spherical i r rad iance

Irradiance ra t i o

D is t r ibu t ion fac to rs

Wavelength

Absorpt ion c o e f f i c i e n t

Volume sca t te r ing f u n c t i o n

(Total) Sca t te r ing c o e f f i c i e n t

(Beam) A t t e n u a t i o n c o e f f i c i e n t

Forward sca t te r ing c o e f f i c i e n t

Backward sca t te r ing c o e f f i c i e n t

t s ]

[ J ]

[ W ]

[W s r " 1 ]

i [W- m sr ]

[ W - m ~ 2 ]

[ W T T T 2 ]

[W - m " 2 ]

[ W T T T 2 ] ;

[ W - m " 2 ]

[ W - m - 2 ]

[ W T T T 2 ]

[ W - m - 2 ]

[ W - m - 2 ]

[ m ]

[ m " 1 ]

r -1 - 1 , Lm sr j

Cm"1 ]

[ r r f 1 ]

[ m " 1 ]

' [ r rT 1 ]

IAPSO IAMAP AMERICAN

RECOMMENDED RECOMMENDED CLASSIC SYMBOL

Q

$

I

L

E

M

E d

E u

E

E 0

E od

. E ou

E s

R =

y =

D d =

D = u

X

a

6(0)

b

c

b f

K

E / E . u d

E/E 0

. Eod

E d

F

E u

SYMBOL

Q

$

I

L

E

M

E <-),

E (+),

-

-

-

-

-

-

-

-

X

a a

Ys

a s

a e

-

_

SYMBOL

U

P

J

N

H

W

E + H (-)

E t H (+)

H "

h

h (-)

h (+)

-

R = H(+)/H(-

-

D - = h - / H -

D+ = h+ /H+

X

a

ofe)

s

a

f

b

8. References

31

BARTZ, R., 2ANEVELD, J.R.V. and PAK, H., 1978. A transmission

meter for profiling and observations in water. SP IE

Vol. 160. Ocean Optics V 1978 : p 102 - 108.

CHANDRASEKHAR, S., 1950. Radiative transfer. Oxford Univ.Press,

London : 393 pp

HØJERSLEV, N.K., 1973. Inherent and apparent optical properties

of the Western Mediterranean and the Hardangerfjord. Rep.

Inst. Phys. Oceanography, Univ. Copenhagen, 21 : 70 pp

HØJERSLEV, N.K., 1974a. Inherent and apparent optical proper­

ties of the Baltic. Rep. Inst. Phys. Oceanography, Univ.

Copenhagen, 23 : 89 pp

HØJERSLEV, N.K., 1974b. Daylight measurements for photosynthe-

tic studies in the Western Mediterranean. Rep. Inst.Phys.

Oceanogr., Univ. Copenhagen. 26 : 38 pp

HØJERSLEV: N.K., 1975. A spectral light absorption meter for

measurements in the sea. Limn. Oceanpgr., 20 (6) :

p 1024 - 1034

HØJERSLEV, N.K., 1977. Inherent and apparent optical properties

of the North Sea - Fladen Ground Experiment - FLEX 75.

Rep. Inst. Phys. Oceanography, Univ. Copenhagen, 32 : 97 pp

HØJERSLEV: N.K., and ZANEVELD, J.R.V., 1977. A theoretical proof

of the existence of the submarine asymptotic daylight field.

Rep. Inst. Phys. Oceanography, Univ. Copenhagen, 34 : 16 pp

HØJERSLEV, N.K., 1978. Daylight measurements appropriate for

photosynthetic studies in natural sea water. J. Cons. int.

Explor. Mer, 38 : p 131 - 146.

HØJERSLEV, N.K., 1982. Bio-optical properties of the Fladen

Ground: "Meteor"-FLEX-75 and FLEX-76. J. Cons. int. Explor.

Mer, 40 : p 272 - 290

HØJERSLEV, N.K., in press. Bio-optical measurements in the

Southwest Florida Shelf Ecosystem. J. Cons. int. Explor.

Mer.

LUNDGREN: B. and HØJERSLEV, N.K., 1971. Daylight measurements

in the Sargasso Sea. Results from the "DANA" expedition

January - April 1966. Rep. Inst. Phys. Oceanography, Univ.

Copenhagen, M : 44 pp

MOREL, A,, 1973. Diffusion de la lumière par les eaux de mer.

Résultats expérimentaux et approche théorique. In : Optics

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