In the absence of scattering (b = 0), Kd, the vertical attenuation coefficient for downward irradiance, is determined only by the absorption
Optical depth, Z
Fig. 6.17 Vertical attenuation coefficient for downward irradiance as a function of optical depth for b/a = 5, and a = 1.0 m"1. Data obtained by Monte Carlo calculation for vertically incident light.702
coefficient and the zenith angle, 0, of the light beam within the water, in accordance with Kd = a/cos 0.
In a scattering medium, Kd is increased, partly because of the altered angular distribution of the downwelling light, and partly because of upward scattering of the downwelling stream. Concurrently with the progressive change in angular distribution with depth, so the value of Kd increases with depth, levelling off at that optical depth at which the asymptotic radiance distribution becomes established. Figure 6.17 shows Kd as a function of optical depth for vertically incident light in a medium for which b/a = 5.0.
While Kd always increases with depth when the incident light is parallel, it is possible that the converse might be true if the incident light were diffuse, e.g. from an overcast sky. If, as could be the case, especially in water with a low value of b/a, the final asymptotic radiance distribution were more vertical than the initial radiance distribution, then Kd would decrease with depth.
The relation between Kd and the absorption and scattering coefficients may, for a given incident light field and a given /(d), be expressed with complete generality by expressing Kd/a as a function of b/a. The absolute values of a, b and Kd do not have to be specified: it is the ratios that are
important. A certain ratio of b to a gives rise to a specific ratio of Kd to a, regardless of the actual value of any of the coefficients. The calculated value of Kd/a at zm for vertically incident light is plotted against b/a in Fig. 6.18. The value of Kd starts off equal to a when b = 0, and rises progressively as b increases, in a linear manner to begin with, but curving over at high ratios of b to a. The values of Kd (zm)/a over the whole range up to b/a = 30 were found702 to fit very closely to a curve specified by the equation
G is a coefficient that can be regarded as determining the relative contribution of scattering to vertical attenuation of irradiance, and its value is determined by the shape of the scattering phase function. For a water with the San Diego harbour ¡3(9), as in Fig. 6.18, G has the value 0.256.
Similar equations to 6.22 and 6.23 can be written in which Kd(zm) is replaced by Kd (av), the average value of Kd through the zone within which downward irradiance is reduced to 1% of that penetrating the surface: in this case the coefficient, G, has a slightly different value (0.231). Although Kd(zm) is a more precisely defined, and theoretically satisfactory, form of Kd, we shall nevertheless from here on direct our attention mainly to Kd(av), since this is the Kd most commonly measured in the field. Kd(zm) and Kd(av) are in fact normally close to each other in value, and show the same kinds of dependence on the inherent optical properties of the medium.
The way in which Kd varies with solar altitude is conveniently expressed in terms of its dependence on m0, the cosine of the refracted solar beam just beneath the surface. This dependence is due not only to the anticipated change, with angle, in the pathlength of the photons per vertical metre traversed, which gives rise to a dependence of Kd on (1/m0), but in addition to the fact that the coefficient G also varies with solar angle in accordance with
where g1 and g2 are constants for a particular scattering phase func-tion.706,712 Thus we can write
a M0
or the corresponding forms, such as
in which we substitute for G(m0). For water bodies with the San Diego phase function, in the version of eqn 6.27 applicable to Kd(av), the constants have the values g1 = 0.425 and g2 = 0.19. In the version for Kd(zm), g1 = 0.473 and g2 = 0.2l8. Using these values of g1 and g2, eqn 6.27 can be used to calculate Kd from m0, a and b for most of the waters that limnologists and coastal oceanographers deal with, and is of considerable predictive value in relation to the optical water quality of these
709 711
aquatic ecosystems. ,
Fig. 6.19 Variation of G(1.0) with the average cosine of scattering of the
Fig. 6.19 Variation of G(1.0) with the average cosine of scattering of the
In the various regions of the open ocean, the scattering phase function can differ significantly in shape from the phase function of coastal and inland waters. The relationships embodied in eqns 6.22 to 6.27 nevertheless still apply.712 It is the value of the coefficient G(m0) that varies with the shape of the phase function. A useful measure of the shape of the scattering phase function is the average cosine of scattering (see §1.4), and we can use this to illustrate the nature of the dependence of G(m0) on the characteristics of the phase function. It is helpful to bear in mind that = 1.0 corresponds to all the photons being scattered forward without any change in direction, and that a progressive decrease in from 1.0 towards zero corresponds to photons being scattered through wider and wider angles. To simplify matters we can confine our attention initially to G(1.0), the value of G for vertically incident light (m0 = 1.0). Monte Carlo calculations of the light field in a wide range of optical water types712 show (Fig. 6.19), as we might expect, that the dependence of vertical attenuation on scattering, expressed through G(1.0), diminishes steeply as the scattering becomes increasingly concentrated within narrow forward angles (ps increases towards 1.0), and disappears altogether when = 1.0. Linear regression gives water.
With r2 = 0.997. To a reasonable approximation this can alternatively be expressed as
The relationships embodied in eqns 6.22-6.27 for Kd as a function of a and b are empirical in nature, derived as they are from analysis of the light fields generated by realistic Monte Carlo modelling of the fate of solar radiation in idealized water bodies having a wide range of optical properties. Maffione (1998) has shown that, at least for highly turbid media, a relationship equivalent to that in eqn 6.23 can be derived on purely theoretical grounds. This equation was found to be useful851 for interpreting the propagation of light through sea ice, a highly scattering optical medium.
The dependence of the apparent optical property, Kd, on the inherent optical properties of the water, a and b, as expressed in eqn 6.26 or 6.27, can be put to the test in waters where apparent optical properties (AOP) and inherent optical properties (IOP) data are both available. The equation has been found to work well for Kd (PAR) in a number of inland waters in Australia,706,433,723 the USA348,1446 and Japan;95 also for Kd(1) in several monochromatic wavebands in the visible region in the Clyde Sea (Scotland).148,888
As an indication of the extent to which scattering intensifies vertical attenuation of light it may be noted from Fig. 6.18 that b/a ratios of about 5 and 12 increase Kd by 50 and 100%, respectively. Comparable effects can be shown for real water bodies. In Table 6.3, data that show the effect of scattering on attenuation of the whole photosynthetic waveband in four Australian inland waters of varying turbidity are presented. The table compares the values of Kd for PAR calculated from the total absorption spectrum of the water, assuming no scattering, with those actually measured within the water with an irradiance (PAR) meter. The ratio of the observed to the calculated value of Kd is a measure of the extent to which scattering intensifies light attenuation. The increase in Kd(PAR) due to scattering ranged from 16% in Burrinjuck Dam, which was rather clear at the time, to more than three-fold in the very turbid water of Lake George.
In our consideration of the ways in which scattering affects attenuation we can now make use of eqn 1.63, derived from radiative transfer theory by Preisendorfer (1961), which expresses the vertical attenuation coefficient for downward irradiance as a function of the diffuse absorption and backscattering coefficients.
Table 6.3 Effects of light scattering on vertical attenuation coefficient (Kd) for irradiance of PAR (400-700 nm), in four water bodies of differing turbidity, in southeast Australia. Kd values calculated from the measured absorption spectra are compared with those obtained in situ with a quanta meter.
Table 6.3 Effects of light scattering on vertical attenuation coefficient (Kd) for irradiance of PAR (400-700 nm), in four water bodies of differing turbidity, in southeast Australia. Kd values calculated from the measured absorption spectra are compared with those obtained in situ with a quanta meter.
Water body |
Turbidity (NTU) |
Kd (calculated) (m-1) |
Kd (observed) (m-1) |
Effect of scattering |
Burrinjuck |
1.8 |
0.775 |
0.90 |
1.16 |
Dam | ||||
L. Ginninderra |
4.6 |
0.547 |
0.90 |
1.65 |
L. Burley |
17.4 |
1.333 |
2.43 |
1.82 |
Griffin | ||||
L. George |
49.0 |
1.742 |
5.67 |
3.25 |
Although this relation is not, in view of the difficulty of measuring the diffuse coefficients, of everyday practical use, it is conceptually valuable in helping us to understand the nature and relative importance of the radiation transfer mechanisms underlying the attenuation process. It tells us that attenuation of downward irradiance is the result of three different processes, represented by the three different terms on the right-hand side of the equation. Absorption from the downwelling stream is accounted for by the diffuse absorption coefficient ad(z), which is equal to a/pd (z), and so must increase as scattering causes pd to decrease. Light is also removed from the downwelling stream by upward scattering of the photons: this process is represented by the diffuse backscattering coefficient for the downwelling stream, bbd(z). Opposing these two processes is the downward scattering of photons from the upwelling stream, which acts to increase the downward flux: this is represented by the term, - bbu(z)R(z), where bbu(z) is the diffuse backscattering coefficient for the upwelling stream and R(z) is irradi-ance reflectance.
For any given values of a and b, and having specified ¡3(6) and the incident light field, then if the radiance distribution at depth z m is calculated by a Monte Carlo, or other, procedure, it is possible to calculate the diffuse optical properties at that depth.700 Thus all the terms on the right-hand side of eqn 1.63 may be determined for a series of values of b and a, and so the relative importance of the three different processes can be evaluated for different types of water.
eqn 1.63 are plotted alongside the corresponding values of Kd at zm (data from Kirk, 1981a).
Figure 6.20 shows the change in value of each of these three terms, compared to the value of Kd, at zm, as the ratio of scattering to absorption coefficients increases from 0.0 to 30.0. At low values of scattering, up to b/a ratios of about three, attenuation is almost entirely due to absorption, and furthermore the scattering-induced increase in attenuation is mainly a consequence of the increase in absorption resulting from the changed angular distribution of the downwelling flux. Upward scattering of the b/a downwelling stream contributes only slightly to attenuation.
When b/a has risen to about seven, upward scattering of the down-welling stream accounts for about 25% of all the attenuation and contributes as much to the increase in attenuation as does the increase in absorption of the downwelling light. At b/a values of about 11 and 20, upward scattering of the downwelling light accounts for 0.5 and 1.0 times eqn 1.63 are plotted alongside the corresponding values of Kd at zm (data from Kirk, 1981a).
Figure 6.20 shows the change in value of each of these three terms, compared to the value of Kd, at zm, as the ratio of scattering to absorption coefficients increases from 0.0 to 30.0. At low values of scattering, up to b/a ratios of about three, attenuation is almost entirely due to absorption, and furthermore the scattering-induced increase in attenuation is mainly a consequence of the increase in absorption resulting from the changed angular distribution of the downwelling flux. Upward scattering of the b/a downwelling stream contributes only slightly to attenuation.
When b/a has risen to about seven, upward scattering of the down-welling stream accounts for about 25% of all the attenuation and contributes as much to the increase in attenuation as does the increase in absorption of the downwelling light. At b/a values of about 11 and 20, upward scattering of the downwelling light accounts for 0.5 and 1.0 times as much attenuation, respectively, as does absorption, and at higher b/a values it becomes the major mechanism for attenuation. The downward scattering of the upwelling stream, which acts to diminish attenuation, has little effect over the lower part of the range of b/a values studied, but becomes significant from about b/a = 6 onwards, and counteracts a large part of the attenuation due to the other two processes at b/a values in the range 12 to 30.
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