## Indirect estimation of scattering properties

While few aquatic laboratories carry out measurements of the fundamental scattering properties of natural waters, many routinely measure underwater irradiance (§5.1). Since irradiance at any depth is in part determined by the scattering properties of the water, there is the possibility that information on the scattering properties might be derived from the measured irradiance values. For water with a specified volume scattering function and with incident light at a given angle, then at any given optical depth (see §1.6 for definition), the irradiance reflectance, R, and the average cosine, p, of the light are functions only of the ratio, b/a, of scattering coefficient to absorption coefficient. Conversely, if the value of reflectance at a certain optical depth is given, then the values of p and b/a are fixed, and in principle determinable. Kirk (1981a, b) used Monte Carlo numerical modelling (§§5.5 and 6.7) to determine the relations between b/a for the medium and R and p at a fixed optical depth, for water with a normalized volume scattering function identical to that determined by Petzold (1972) for the turbid water of San Diego harbour. It was considered that these relations would be approximately valid for most natural waters of moderate to high turbidity. Using the computer-derived curves it is possible, given a measured value of irradiance reflectance at a specified optical depth, such as Z = 2.3 (irradiance reduced to 10% of the subsurface value), to read off the corresponding values of p and b/a. The irradiance values are used to estimate KE, the vertical attenuation coefficient for net downward irradiance (§1.3), and the value of a is calculated from the relation a = p KE (§1.7). Knowing a and b/a, the value of b may then be obtained.

As a test of the validity of this procedure, Weidemann and Bannister (1986) compared, for Irondequoit Bay, L. Ontario, estimates of a derived from irradiance measurements as indicated above, with estimates obtained by summing the measured absorption coefficients due to gilvin, particulate matter and water: agreement was good. Furthermore, estimates of p read off from the p = f(R) curve agreed with those obtained from the measured ratio of net downward, to scalar, irradiance (eqn 1.15): a similar finding was made by Oliver (1990) for a range of river and lake waters in the Murray Darling basin, Australia.

Values of scattering coefficient obtained in this way by Kirk (1981b) for various bodies in Southeastern Australia were found to correlate very closely with the values of nephelometric turbidity, (Tn), a parameter that (see above) we might reasonably expect to be linearly related to the scattering coefficient: the average ratio of b to Tn was 0.92m~lNTU~l. Using literature data for Lake Pend Oreille in Idaho, USA,1076 the method was found to give a value of b differing by only 5% from that derived from the beam attenuation and absorption coefficients.

With the help of a separately derived expression (eqn 6.11) for K as a function of a, b and m0 (the cosine of the refracted solar beam beneath the surface), and another for b/a as a function of R at the 10% irradiance depth (zm), it is possible to replace the graphical procedure described above with an explicit expression by means of which a can be calculated from the measured irradiance profile and the solar altitude

where GE(m0) = 0.473 m0 - 0.22 (§6.7). To obtain b, the value of a is then multiplied by b/a.

Di Toro (1978), using radiation transfer theory with a number of simplifying assumptions, arrived at an approximate relation between the scattering coefficient and a certain function of the reflectance and vertical attenuation coefficient for irradiance. For the turbid waters of San Francisco Bay, he observed a good linear relation between the values of b calculated in this way and the nephelometric turbidity: the actual ratio of scattering coefficient to turbidity was 1.1 m _1FTU-1, which is in reasonable agreement with the value of 0.92 m ^NTU-1 (NTU and FTU being equivalent) obtained by Kirk (1981b). Combining these results with similar findings by numerous other workers282,433,1014,1401,1402,1445,1446 we arrive at the useful and convenient conclusion that, at least for waters of moderate to high turbidity, the scattering coefficient (m-1) has approximately the same numerical value as the nephelometric turbidity (NTU),