## 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, 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 prevents the establishment of a completely isotropic field. The resultant angular distribution is determined by this interaction between the absorption and the scattering processes. Eventually the angular distribution 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 coefficient, 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 Hojerslev 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 directional 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 towards 0 = 0°. The final, asymptotic, 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 symmetrical about the vertical even near the surface. At intermediate azimuth angles, radiance distributions intermediate between those in Fig. 6.11a and b exist.

### ZENITH ANGLE

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 m_1 and absorption coefficient 0.117m-1. (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.

### ZENITH ANGLE

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 m_1 and absorption coefficient 0.117m-1. (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.

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 Hojerslev (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 e function represents the eccentricity of the ellipse that approximates the azimuthal radiance distribution in

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 of ~1%. 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 of e requires measurement of two azimuthal values of radiance: a is obtained from measurements of any two of the quantities, L(180°), Eu and E0u.

In the context of photosynthesis, what is significant about a radiance. distribution is its relevance to the rapidity of attenuation of light intensity

Zifnith anjii

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, 15.0m—1; solar elevation, 32°).

OZ&rn fl.60n,

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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, 15.0m—1; solar elevation, 32°).

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 6.13 shows 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,702,704 are plotted against the optical depth (Z = Kdz) in water in which b/a = 5.0. 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, Z = 4.6) at which irradiance has been reduced to 1% of the subsurface value, and indeed most of

Optical depth, Z

Fig. 6.14 Variation of average cosines for downwelling (fid) upwelling (fiu) and total (fi) flux, and irradiance reflectance (R) with optical depth (Z = Kdz) in water with b/a = 5. Data obtained by Monte Carlo calculation.702 (Vertically incident light_. Light incident at 45° )

Optical depth, Z

Fig. 6.14 Variation of average cosines for downwelling (fid) upwelling (fiu) and total (fi) flux, and irradiance reflectance (R) with optical depth (Z = Kdz) in water with b/a = 5. Data obtained by Monte Carlo calculation.702 (Vertically incident light_. Light incident at 45° )

the change takes place before zm (Z = 2.3) is reached. The decrease in fi with depth towards its asymptotic value is approximately exponential.106 It will be noted that for vertically incident light fid starts (at z = 0) 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 fid = 1.0), 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 fid much less than 1.0 and so, although they constitute only a small proportion of the total, they bring the average value of fid significantly 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 fi and fid just below the surface have lower values, and the light

Fig. 6.15 Vector diagram of radiance distribution of upwelling light at 4.24m depth in Lake Pend Oreille, USA. Radiance measurements of Tyler (1960) averaged over all azimuth angles. Radiance vectors are at nadir angle intervals of 10°.

Fig. 6.15 Vector diagram of radiance distribution of upwelling light at 4.24m depth in Lake Pend Oreille, USA. Radiance measurements of Tyler (1960) averaged over all azimuth angles. Radiance vectors are at nadir angle intervals of 10°.

field reaches its asymptotic state at shallower depths: compare the curves for 0 = 0° and 0 = 45°. In the case of a non-perpendicular incident beam it should be realized that although the angular distribution averaged over all azimuths has settled down to almost its final form at zeu, the downward radiance distribution in the vertical plane of the Sun is still significantly different from the asymptotic distribution: this can be seen in Fig. 6.11a in the curves for 29 m (zeu « 28 m).

The angular distribution of the upwelling stream acquires its final form a very short distance below the surface. In Lake Pend Oreille, USA, varied only between 0.37 and 0.34 at all depths between 4.2 and i^o ^ TOO

66.1 m. Monte Carlo calculations indicate that for waters with b/a values ranging from 0.1 to 20, is typically in the range 0.35 to 0.42 throughout the euphotic zone. That the upwelling stream should have these characteristics is not surprising since it consists predominantly of back-scattered light: backscattering does not vary strongly with angle and so rather similar upward radiance distributions are produced whatever the downward radiance distribution.

The radiance distribution of the upwelling light stream near the surface in Lake Pend Oreille is shown in the form of a polar diagram in Fig. 6.15. If, as is sometimes assumed (see §6.4), the upwelling flux has the same radiance distribution as that above a Lambertian reflector (same radiance at all angles), all the radiance vectors in Fig. 6.15 would be the same length. It is clear from the data that the upward radiance distribution is far from Lambertian.

As might be expected from the changes in angular distribution with depth, irradiance reflectance increases with depth but levels off at a final value concurrently with the settling down of ft and ftd to their final values (Fig. 6.14). Thus, in natural water bodies the depth at which the asymptotic radiance distribution (averaged over all azimuth angles) is established can be found by determining at what depth the value of irradiance reflectance ceases to increase.

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