As an alternative to measuring the whole volume scattering function in order to determine b, b(0) can be measured at one convenient fixed angle, and by making reasonable assumptions about the likely shape of the volume scattering function in the type of water under study (see below), an approximate value of b can be estimated by proportion. In marine waters, for example, the ratio of the volume scattering function at 45 ° to the total scattering coefficient (b(45 °)/b) is commonly in the range 0.021 to 0.035 sr_1.636 From an analysis of their own measurements of b(0) and b in the Pacific and Indian Oceans and in the Black Sea, Kopelevich and Burenkov (1971) concluded that the error in estimating b from singleangle measurements of b($) is lower for angles less than 15°:4° was considered suitable. A linear regression of the type log b = cilog ß{d)+c2

where c1 and c2 are constants, was found to give more accurate values for b than a simple proportionality relation such as b = constant x b(0). On the basis both of Mie scattering calculations, and analysis of literature data on volume scattering functions for ocean waters, Oishi (1990) concluded that there is an approximately constant ratio between the back-scattering coefficient and the volume scattering function at 120 so that bb can be calculated from a scattering measurement at 120 using the relationship

As a function of the volume scattering function, the backscattering coefficient can be expressed (§1.4) in the form where w(0) is a dimensionless constant, sometimes referred to as the conversion factor, which makes this equality true at angle 0.849 Equation 4.5 can be regarded as a more general form of eqn 4.4, and so for backscattering meters using the Oishi principle (which in fact works quite well over a wide angular range) but operating at an angle other than 120 w(0) is determined empirically. Equation 4.4 corresponds to w(120 °)« 1.1. Single-angle estimates of bb are often carried out at 140 For coastal shelf waters off New Jersey (USA), Boss and Pegau (2001) found an average value of 1.18 for w(140 °); for Black Sea coastal waters Chami et al. (2006a) obtained an average value of 1.21.

As can readily be verified, eqn 4.4 works very well for the two volume scattering functions listed in Table 4.2. On the basis of Mie theory calculations, Oishi concluded that this relationship should be unaffected by wavelength. Field measurements of b(0) by Chami et al. (2006a) confirm that this is the case in coastal waters of the Black Sea.

The WET Labs ECO BB scattering meter makes use of this principle to measure bb. The light from an LED, modulated at 1 khz, is emitted into the water and scattered light is received by a detector positioned where the acceptance angle forms a 117 ° intersection with the source beam.1179 Versions of the instrument operating at 470, 532 or 660 nm bb « 7 £(120°)

Jk/2

which for any given value of 0 can be replaced with bb = 2k X(0)ß(0)

are available; another version incorporates all three wavelengths. The HOBI Labs HydroScat-2 Backscattering Sensor849,605 also uses a modulated LED as a light source and measures light back-scattered at an angle centred on 140 It operates at two wavelengths, 420 and 700 nm: versions with other pairs of wavelengths are available. The HydroScat-4 measures backscattering at four wavelengths.604 These instruments also measure chlorophyll fluorescence.

If, for a given water, the beam attenuation coefficient and the absorption coefficient can both be measured, then the scattering coefficient can be obtained by difference (b = c - a). This indirect approach is in principle less accurate since the estimate of b must combine the separate errors in the determination of c and a. To obtain b by a direct measurement of scattered light is preferable. Nevertheless, when both c and a data are available from, for example, an instrument such as the ac-9, then it is worth while collecting values of b as well.

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