The Secchi Disk
The previous pages give us the background needed to derive the maximum depth at which a Secchi disk can be seen.
Consider only the case of looking straight down, and drop the direction arguments in luminances and contrasts, e.g. . The underlying idea is that a disk at some depth is illuminated by the downwelling plane illuminance . The luminance reﬂected by the disk then propagates upward to the observer as a narrow beam of luminance. The development then proceeds as follows.
The downwelling plane illuminance at depth is given by
where denotes the average over 0 to .
The target is assumed to be a Lambertian reﬂector with an illuminance reﬂectance of . The luminance reﬂected by the target is then
The backgound water is also assumed to be a Lambertian reﬂector, so that
The luminance of the background water is then
The inherent contrast at depth is
The apparent contrast of the Secchi disk as seen from just below the sea surface is
(Note that in this development the argument 0 refers to depth , not to the distance from the target, which is .)
The luminance diﬀerence law
allows the apparent contrast to be written as
by (??) into (6).
This equation gives the apparent contrast of the Secchi disk as seen from just below the water surface. For viewing from above the surface, we must account for loss of contrast caused by the water surface. This loss is due both to refraction by waves and to surface-reﬂected sky light. Thus
where denotes the transmission of contrast, not of luminance or illuminance.
The Secchi depth is the depth at which the apparent contrast in air falls below a threshhold contrast . Solving for when gives
Studies with human observers show that depends on the angular subtense of the disk and on the ambient luminance (e.g., Table 1 of Preisendorfer (1986)). The values of vary from about 6 to 9 for a disk with , depending on the water reﬂectance (which is 0.015 to 0.1; Table 2 of Preisendorfer (1986)). The HydroLight code uses as its default.
Note that Eq. (12) must be solved interatively because and are averages over the (unknown) Secchi depth . This is easily done after solution of the radiative transfer equation to some depth greater than over the visible wavelengths. The photopic and can then be computed from and . The values of and just below the water surface (at depth 0) are then used to get an initial estimate of , which is then used to compute an improved estimate of the depth-averaged and , and so on. Convergence is obtained within a few interations.
Lee et al. (2015) re-examined the classic theory of the Secchi disk. They assumed that
- The disk needs not be angularly small and can perturb the ambient light ﬁeld seen near the edge of the disk.
- Visibility is not based on target vs background luminance diﬀerences at the sharp edge of the disk, but on on diﬀerences in target and background reﬂectances.
- Visibility is determined by the wavelength where the disk is most visible (which can change with depth and between water bodies), rather than on broadband photopic variables.
They argue that the classic analysis should
- Replace the photopic with , where is the wavelength at which is a minimum; and
- Replace the photopic with .
One end result of their analysis is a formula of the form (Eq. 28 of their paper)
where depends on a diﬀerence in reﬂectances, rather than on contrasts as seen in Eq. (11). This formula has the great virtue that can be estimated from multi- or hyperspectral satellite imagery.
Comparison of measured and computed by Eq. (13) gives reasonable agreement (see Fig. 6 of their paper). However, comparison of Lee et al. predictions with those of the classic theory have not been made.