Angular distribution of the underwater light field

As sunlight enters a water body, immediately it penetrates the surface its angular distribution begins to change – to become less directional, more diffuse – as a result of scattering of the photons. The greater the depth, 6.6 Angular distribution of the underwater light field 181

the greater the proportion of photons that have been scattered at least once. The angular distribution produced is not, however, a function of scattering alone: the less vertically a photon is travelling, the greater its pathlength in traversing a given depth, and the greater the probability of its being absorbed within that depth. Thus the more obliquely travelling photons are more intensely removed by absorption and this effect pre-vents the establishment of a completely isotropic field. The resultant angular distribution is determined by this interaction between the absorp-tion and the scattering processes. Eventually the angular distribuabsorp-tion of light intensity takes on a fixed form – referred to as the asymptotic radiance distribution – which is symmetrical about the vertical axis and whose shape is determined only by the values of the absorption coeffi-cient, the scattering coefficient and the volume scattering func-tion.338,636,1076,1092,1368

That such an equilibrium radiance distribution would be established at great depth was predicted on theoretical grounds independently by Whitney (1941) in the USA and Poole (1945) in Ireland, and mathematical proofs of its existence were given by Preisendorfer (1959) and Højerslev and Zaneveld (1977). Early measurements by Jerlov and Liljequist (1938) in the Baltic Sea showed some movement of the radiance distribution towards a symmetrical state with increasing depth.

The particularly accurate measurements of Tyler (1960) down to 66 m in Lake Pend Oreille, USA, demonstrated the very close approach of the radiance distribution to the predicted symmetrical state at this near asymptotic depth.

The progression of the angular structure of the underwater light field towards the asymptotic state can be seen from the change in the radiance distribution with increasing depth. This is illustrated in Fig. 6.11, which is based on the measurements of Tyler (1960) in Lake Pend Oreille.Figure 6.11a is for radiance at different vertical angles in the plane of the Sun. Near the surface the light field is highly direc-tional with most of the flux coming from the approximate direction of the Sun. With increasing depth the peak in the radiance distribution becomes broader as its centre shifts towardsy ¼0. The final, asymp-totic, radiance distribution would be symmetrical about the zenith. It is noticeable that in the plane of the Sun this final state had not quite been reached even at 66 m. Figure 6.11b shows that in a plane almost at right angles to the Sun, the radiance distribution is nearly symmet-rical about the vertical even near the surface. At intermediate azimuth angles, radiance distributions intermediate between those in Fig. 6.11a and b exist.

Figure 6.12 shows the near-asymptotic radiance distribution in Lake Pend Oreille in the form of a polar diagram. If in a natural water the ratio of scattering to absorption increases to high levels, then the shape of such a polar diagram of the asymptotic radiance distribution tends towards a circle (i.e. the light field approaches the completely diffuse state). If, however, scattering decreases to low values relative to absorption, then the polar diagram takes on the form of a narrow, downward pencil.¹⁰⁷⁵

Aas and Højerslev (1999) analysed a large data set of underwater angular radiance distributions, 70 from the western Mediterranean and 12 from Lake Pend Oreille, Idaho (USA). They were able to find two simple functions that between them can be used to characterize the angular radiance distribution. The Efunction represents the eccentricity of the ellipse that approximates the azimuthal radiance distribution in

10⁷

Away from Sun Towards Sun Away from Sun Towards Sun ZENITH ANGLE

0° 60° 120° 180° 120° 60° 0° 60° 120° 180°

Fig. 6.11 Radiance distribution of underwater light field at different depths.

Plotted from measurements at 480 nm made by Tyler (1960) in Lake Pend Oreille, USA, with solar altitude 56.6, scattering coefficient 0.285 m¹and absorption coefficient 0.117 m¹. (a) Radiance distribution in the plane of the Sun. (b) Radiance distribution in a plane nearly at right angles to the plane of the Sun.

6.6 Angular distribution of the underwater light field 183

polar coordinates: it provides the azimuthal mean of downward radiance at a given depth with an average error 7% and of upward radiance with an error of1%. The parameter,a, plays a similar role for the azimuthal mean of upward radiance. It describes the zenith angle dependence of the azimuthal mean of upward radiance with an average error of 7% in clear ocean water, increasing to 20% in turbid lake water. Determination ofErequires measurement of two azimuthal values of radiance:ais obtained from measurements of any two of the quantities,L(180),EuandE0u.

In the context of photosynthesis, what is significant about a radiance.

distribution is its relevance to the rapidity of attenuation of light intensity Fig. 6.12 Near-asymptotic radiance in Lake Pend Oreille, USA, plotted as a polar diagram. The data are the same as those in the lowest curve of Fig. 6.11b.

with depth. The azimuthal distribution of radiance at each vertical angle has no bearing on this. It is therefore legitimate, and in the interests of simplicity advantageous, to express the angular distribution of the light field at any depth in terms of the radiance averaged over all azimuth angles at each vertical angle.Figure 6.13shows a series of such radiance distributions at increasing depth, calculated by the Monte Carlo procedure, for PAR in an Australian lake. The much more rapid approach to a final symmetrical distribution in this case is partly the result of averaging over all azimuth angles, and partly because of the increased scattering relative to absorption in this lake relative to Lake Pend Oreille.

An even simpler way of expressing the angular distribution of the light field is in terms of the three average cosines (see }1.3) and the irradiance reflectance. In Fig. 6.14 the values of these parameters, obtained by Monte Carlo calculation,^702,704are plotted against the optical depth (z¼Kdz) in water in whichb/a¼5.0. The average cosine and the average downward cosine initially both diminish sharply in value but then begin to level off as the angular distribution of the light approaches the asymptotic radiance distribution. There is no significant further change in the angular distribution beyond the depth (zeu,z¼4.6) at which irradi-ance has been reduced to 1% of the subsurface value, and indeed most of

Fig. 6.13 Radiance distribution of underwater light field averaged over all azimuth angles in a turbid lake. Radiance values for total PAR in Lake Burley Griffin, Australia, were obtained by Monte Carlo calculation using the method of Kirk (l98la, c) on the basis of the measured absorption and scattering properties (absorption spectrum,Fig. 3.9b; scattering coefficient, 15.0 m¹; solar elevation, 32).

6.6 Angular distribution of the underwater light field 185

the change takes place before zm (z¼2.3) is reached. The decrease in with depth towards its asymptotic value is approximately exponential.¹⁰⁶ It will be noted that for vertically incident light_dstarts (atz¼0) from a value of slightly less than 1.0. This is because the downwelling light just below the surface includes not only those photons that have just passed down through the surface (and these do have d ¼1.0), but also those photons that were part of the upwelling stream and that have just been reflected downwards from the surface. These latter photons have a _d much less than 1.0 and so, although they constitute only a small propor-tion of the total, they bring the average value ofdsignificantly below 1.0.

For light incident on the surface at angles other than the vertical, the behaviour is much the same as for perpendicularly incident light except that and _d just below the surface have lower values, and the light

1.0

0.8

0.6

0.4

0.2

0.0

0.0 1.0 2.0 3.0

Optical depth, z

4.0 5.0 6.0

0.00 0.02 0.04 0.06

Average cosine Reflectance

R mu

md

m

Fig. 6.14 Variation of average cosines for downwellingðdÞupwellingðuÞ and totalðÞ flux, and irradiance reflectance (R) with optical depth (z¼Kdz) in water with b/a ¼ 5. Data obtained by Monte Carlo calculation.⁷⁰² (Vertically incident light __________. Light incident at 45---.)