Notes

702

Submitted: 4 August 1993 Accepted: 24 November 1993 Amended: 3 January 1994

STEEMANN NIELSEN, E. 197 5. Marine photosynthesis with special emphasis on the ecological aspects. Elsevier. TYLER, J. E. [ED.]. 1966. Report on the second meeting of the joint group experts on photosynthetic radiant energy. UNESCO Tech. Pap. Mar. Sci. 2.

Limnol. Oceanogr., 39(3), 1994, 702-706 0 1994, by the American Society of Limnology

and Oceanography,

Inc.

Characteristics of the light field in highly turbid waters: A Monte Carlo study Abstract-The nature of the light field in waters with very high ratios of scattering coefficient to absorption coefficient (b : a) has been explored by Monte Carlo modeling. The progressive change in the angular structure of the field at a specified optical depth (z,) as b : a increases up to an extreme value of 200 is illustrated in terms of the changes in average cosines &[z,~], i&,[zIF1],~[z,,]) and irradiance reflectance (R[z,,]). The dependence of the vertical attenuation coefficient for downward irradiance on absorption and scattering is satisfactorily represented by

Kd(z,,) = (u2 + 0.245~6)“. A nomogram is presented, with the help of which the values of the absorption and scattering coefficients for highly turbid water bodies can be estimated from underwater irradiance data.

There are many interesting and important aquatic ecosystems in which the turbidity of the water is exceptionally high. Examples are areas of the open ocean where coccolithophores are blooming, many estuaries especially in the region of the turbidity maximum, shallow lakes with unconsolidated sediments, lakes and rivers receiving glacial meltwater, rivers in eroding catchments, and ljords. I describe here an investigation, by Monte Carlo modeling, of the nature of the light field that is established within such water bodies. In a previous investigation (Kirk 198 1a, b) waters with scattering : absorption (b : a) ratios up to 30 and irradiance reflectance (E,/E,) values up to 24% were studied, and relationships between reflectance, b: a, and average cosine (p) were elucidated, partly because of their inAcknowledgments

This work received support from the Ocean Optics Program of the Office of Naval Research under grant NO00 1491-7-1366.

trinsic interest, but also because they provide a simple method for estimating absorption and scattering coefficients from underwater irradiance measurements. It was thought at the time that this range of b : a and R values would cover the entire range of optical water types of significance. It has since become apparent, however, that waters with even higher reflectance values and scattering : absorption ratios, while not common, do occur. For example, in the turbid Rhode River estuary (Maryland), Gallegos et al. (1990) found in a series of 15 measurements (different sites, different dates) that in two cases b: a exceeded 30, being in one case as high as 79. In the Darling River, southeastern Australia, Oliver (1990) measured a b : a ratio of 3 3 on one occasion. In the glacier-fed Lake Tekapo, New Zealand, Vant and Davies-Colley (1984) observed an irradiance reflectance of 32%. In a coccolithophore bloom in the Gulf of Maine, Balch et al. (1991) observed reflectance values of 2739% in the blue-green part of the spectrum. So that the relationships between R, ji, and b : a could be understood and methods for estimating a and b from irradiance data be devised, even for exceptionally turbid water bodies such as these, additional Monte Carlo simulations of the light field have now been carried out for a range of optical water types up to a scattering : absorption ratio of 200. Calculation of the light field established within water bodies having various values of the ratio of scattering to absorption coefficient was carried out by the Monte Carlo procedure as described previously (Kirk 198 1a, b, 1984). As before, the water was assumed to have a scattering phase function identical to that measured by Petzold ( 1972) in San Diego harborthe most turbid of the waters he studied. This

703

Notes phase function should be applicable to most natural waters above a certain minimum turbidity. Each simulation run involved modeling the fates of lo6 photons for the specified values of a and b on a MicroVax computer. The photons were assumed to consist of a parallel stream incident on a flat water surface. The previous calculations had covered waters with b: a values up to 30. In the present study the light field was calculated for 19 different waters with b : a values from 35 to 200, and these data are combined with those obtained previously. The results, as before, are presented in terms of the values of the various properties of the light field at one particular optical depth, namely that depth (z,) at which downward irradiance (Ed) is reduced to 10% of the value just below the surface. Defining optical depth, 5; by { = &z, and assuming Ed(z) is attenuated in accordance with exp( -&z), then z, corresponds to < = 2.3. The earlier work (Kirk 198 1b) had shown that for waters with b : a > 20 the angular structure of the light field at z, was independent of the solar altitude. Accordingly, the present study was carried out only for vertically incident light. As the ratio of scattering to absorption of the water changes, so the angular structure of the light field at a given optical depth must change, and this is conveniently expressed in terms of the average cosines (fcr downwelling, upwelling, and total light) and irradiance reflectance. The three average cosines (j&, j&, and fi) are the average values in an infinitesimally small volume element at a point in the field of the cosine of the zenith (b, &) or nadir (i&J angle of the downwelling, the upwelling, or all the photons, respectively. Further discussion of these parameters is given by Kirk (1983, 1991). Figure 1 shows how the average cosines and the reflectance at depth z, vary as the scattering : absorption ratio increases, up to a value of 200. As expected, and as observed in the earlier study, the angular distribution of the field at the 10% irradiance depth becomes progressively broader, less vertical, as scattering increases, as indicated by the progressive fall in I and j&(z,). What is of significance here are the actual numerical values toward which they appear to be heading at extreme b : a ratios. If the ratio of scattering to absorption were to increase indefinitely, the angular

,0.6

I

p(zmj---------------

60

80

100

-----a-___, 120

140

160

180

- 0.1

200

b:a

Fig. 1. Average cosines and irradiance z, as a function of b : a.

reflectance

at

structure of the light field would approach the limiting situation of an isotropic radiance distribution. In such a case the downwelling and upwelling flux would each have an average cosine of 0.5, and the total field would have an average cosine equal to zero. How quickly an underwater light field will approach angular isotropy as b : a increases will be a function of the shape of the scattering phase function. The wider the average angle of scattering or, putting it another way, the lower the value of b,, the average cosine of scattering (Kirk 199 l), the nearer the light field will be to isotropy for any given b :a. Like phase functions for all natural waters, the one used in the present calculations has most of its scattering concentrated at small forward angles, and ii, is high (0.922). Consequently, even at the very high scattering : absorption ratio of 200 : 1, the underwater light field at z, retains some directional character, although it is highly diffuse: &(z,) at 0.538 and p(z,) at 0.146 are 92 and 85%, respectively, toward complete isotropy. The upwelling flux, originating as it does predominantly in backscattering, is always highly diffuse even at low b: a values. Nevertheless, F,(z,) does show some increase with the scattering : absorption ratio, rising from -0.38 at low scattering to 0.466 (i.e. 93% of its maximum theoretical value) at b: a =

200. It is in the ratio of the upwelling to the downwelling flux that the underwater light field most slowly approaches isotropy with increasing scattering. Even at b: a = 200, R(z,) has climbed only to 56%; in a completely diffuse field, reflectance would be 100%. Figure 2 shows the radiance distribution at z, in the

704

Notes sequently &, the vertical attenuation coefficient for downward irradiance, is a function of b as well as a. Previous Monte Carlo studies (Kirk 1981a, 1984, 1991) have shown that, over the range b : a = O-20, the dependence of Kd on a and b closely conforms to the relationship

Kd = i [a2 + G(po)ab]1’2.

(1)

p. is the cosine of the refracted solar beam beneath the surface. G(& is a function that specifies the relative contribution of scattering to vertical attenuation and whose value is determined by the shape of the scattering phase function as well as by po. When, as in the present case, we are confining our attention to vertically incident light (pO = 1.O), Eq. 1 simplifies to

Kd = [a2 + G( l.O)ab]“,

(2)

which for some purposes is more usefully expressed as the ratio of Kd to the absorption coefficient l/2

1 Fig. 2. Radiance distribution at z,, plotted as a polar diagram for waters with b : a values of 8, 40, and 200. The length of the arrow from the origin at angle 6’corresponds to the value of the radiance at zenith angle 0 at depth z, in a water body with the specified b:a ratio. Each curve represents the locus of the tip of the arrow over all values of 0 for that particular water type.

form of a polar diagram for waters with b :a values of 8,40, and 200. The progressive transfer of radiance from the downwelling to the upwelling hemisphere with increased relative scattering is clearly evident, as is the fact that even at b : a = 200 downwelling radiance still predominates. Although it is only absorption that actually extinguishes photons in the water, scattering does intensify attenuation of irradiance with depth partly by increasing the average pathlength of the photons per unit vertical distance traversed (thus enhancing absorption losses) and partly by increasing the upward scattering of photons from the downwelling stream. Con-

.

(3)

The coefficient G has somewhat different numerical values in accordance with whether we are dealing with K,(avg), the average value of Kd through the euphotic zone, or Kd(z,), the localized value of Kd at depth z,; here we deal with the latter. A question we can now address is: does the relationship between Kd, a, and b, expressed in Eq. 2 or 3, still apply to waters in which the scattering : absorption ratio is many-fold higher than those for which the relationship was first observed? To answer this we first derive our best estimate of G(l.O) from the data set itself, using a rearranged form of Eq. 3: G(l.O) = $+I

- l}

(4)

with the value of Kd(z,) obtained in the Monte Carlo calculation at each specified b : a value. In the range b: a 2-200, G( 1.O) varied only from 0.233 to 0.264, and for 27 individual values of b : a distributed across the range, had an average value of 0.245. For comparison

Notes

0.01 0

’

20

’

40

1

60

’

80

’

100

’

120

’

140

1

160

1

180

200

Fig. 3. Ratio of vertical attenuation coefficient to absorption coefficient as a function of b : a. K,(z,)la determined from Monte Carlo calculation (-) or from Eq. 5 (---).

purposes it is noted that in a previous study, a value of 0.256 was arrived at for G(l .O) in waters with b : a up to 30. The continuous line in Fig. 3 shows Kdz,,)/a determined from the Monte Carlo-calculated light field for b : a values up to 200; the dotted line shows the values calculated from b : a, usw

F

= [l

+ 0.245ir2.

(5)

It can be seen that the agreement is very good, the two lines being virtually indistinguishable for b : a values up to -70; even at higher scattering : absorption ratios, the discrepancy is slight. When b: a = 200, K,(z,)la calculated from Eq. 5 differs from that arising out of the Monte Carlo calculation by only 2.5%. Thus, Eq. 2 and 3 should satisfactorily describe the dependence of Kd on a and b even in waters of extremely high turbidity. In highly turbid waters the angular structure of the light field at z, is virtually independent of the angle of the surface-incident light. It is in fact entirely determined by the ratio of scattering to absorption and the scattering phase function. Given that most natural turbid waters will have a very similar phase function, it follows that it is b: a which determines the angular structure. Functions of the angular structure, such as R(z,) and fi(z,), are thus rigidly linked, not only to each other but also to the value of b : a in the manner indicated in

705 200

1.0

180

0.9

160

0.8

140

0.7

120

0.6

100

0.5

80

0.4

60

0.3

40

0.2

20

0.1

0 3.0

0.1

0.2 0.3 0.4

0.0 0.5 0.6

w&J Fig. 4. Relationship at the 10% subsurface irradiance depth (z,J between irradiance reflectance [R(z,,)], average cosine [&z,,)], and ratio of scattering to absorption (b : a).

Fig. 1. Because of this linkage, if we measure R(z,) for a turbid water body, then we should be able to arrive at the corresponding values of b : a and I. To simplify this task, I used the data for R(z,), I, and b : a to prepare a nomogram (Fig. 4). With the measured value of irradiance reflectance at z, for a given water body, the corresponding values of b : a and fi(z,) can be read off. From the same irradiance data set the vertical attenuation coefficient, KE(z,), for the net downward irradiance (Ed - E,) at z, can be obtained (e.g. from measurements just above and below z,). Alternatively, if a good value of KE(z,) is not available, then Kd(z,) can be used as an estimate of KE since in practice Kd and KE are very close in value. From KE(zm) and p(z,) the absorption coefficient can be determined with the Gershun relationship (Jerlov 1976)

a = iiKE; the value of 5 : a obtained turild waters nrevious one

b can then be determined

(6)

using from the nomogram. For highly this nomogram thus replaces the (Kirk 198 1b), which was devised

706

Notes

for, and has been applied by various workers to, waters with b: a values up to 30 and reflectances up to 24%. The range of scattering : absorption ratios examined in this study, extending as it does up to b : a = 200, should encompass all turbid natural waters of any importance. With the help of the nomogram in Fig. 4, comparatively straightforward irradiance measurements on the underwater light field in turbid water bodies can be used to derive realistic values of the absorption and scattering coefficients-properties that are otherwise quite difficult to determine in such waters. It is of particular interest that the relationship between Kd and the absorption and scattering coefficients expressed in Eq. 2 or 3, originally obtained for waters with moderate ratios of b to a, applies equally well (Fig. 3) when this ratio is increased to extreme values. This observation does suggest that the relationship, rather than being an accidental one which fortuitously (albeit usefully) happens to apply to Kd, a, and b values just over a certain range, is in fact of some fundamental significance. A characteristic feature of turbid waters which is confirmed by the present series of calculations is that the light field is well on the way toward its asymptotic angular distribution even at quite modest optical depths. It was noted earlier (Kirk 198 1a) that in water with b : a = 5, there was little further change in the angular distribution of the, field beyond the optical depth (z,,, { = 4.6) at which downward irradiance was reduced to 1% of the subsurface value. The optical depths (0 beyond which there is no significant further change in angular distribution (as judged by the values of average cosines and reflectance) fall to - 1.5 at b : a = 20 and -1.0 at b:a = 60.

John T. 0. Kirk CSIRO Division of Plant Industry G.P.O. Box 1600 Canberra, A.C.T. 260 1, Australia

References BALCH, W.M.,P.M. HOLLIGAN,S.G. ACKLESON,AND K. J. Voss. 199 1. Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine. Limnol. Oceanogr. 36: 629-643. GALLEGOS,~. L.,D.L. CORRELL,AND J. W. PIERCE. 1990. Modeling spectral diffuse attenuation, absorption, and scattering coefficients in a turbid estuary. Limnol. Oceanogr. 35: 1486-l 502. JERLOV, N. G. 1976. Marine optics. Elsevier. KIRK, J. T. 0. 198 la. Monte Carlo study of the nature of the underwater light field in, and the relationships between optical properties of, turbid yellow waters. Aust. J. Mar. Freshwater Res. 32: 5 17-532. -. 198 1b. Estimation of the scattering coefficient of natural waters using underwater irradiance measurements. Aust. J. Mar. Freshwater Res. 32: 533539. -. 1983. Light and photosynthesis in aquatic ecosystems. Cambridge. -. 1984. Dependence of relationship between inherent and apparent optical properties of water on solar altitude. Limnol. Oceanogr. 29: 350-356. -, 1991. Volume scattering function, average cosines, and the underwater light field. Limnol. Oceanogr. 36: 455467. OLIVER, R. L. 1990. Optical properties of waters in the Murray-Darling basin, southeastern Australia. Aust. J. Mar. Freshwater Res. 41: 581-601. PETZOLD, T. J. 1972. Volume scattering functions for selected ocean waters. Scripps Inst. Oceanogr. Publ. 72-78. 79 p. VANT, W. N., AND R. J. DAVIES-C• LLEY. 1984. Factors affecting clarity of New Zealand lakes. New Zealand J. Mar. Freshwater Res. 18: 367-377.

Submitted: 23 July 1993 Accepted: 11 October 1993 Amended: 10 December 1993