Page updated: April 16, 2020
Author: Curtis Mobley
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The Secchi Disk

The previous pages give us the background needed to derive the maximum depth at which a Secchi disk can be seen.

0.1 The Classical Secchi Depth Model

Consider only the case of looking straight down, and drop the direction arguments in luminances and contrasts, e.g. LvB(z,ξ̂) = LvB(z). The underlying idea is that a disk at some depth z is illuminated by the downwelling plane illuminance Edv(z). 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 z is given by

Edv(z) = Edv(0)exp[Kdvzz], (1)

where z denotes the average over 0 to z.

The target is assumed to be a Lambertian reflector with an illuminance reflectance of RvT . The luminance reflected by the target is then

LvT (z) = Edv(z)RvT π. (2)

The backgound water is also assumed to be a Lambertian reflector, so that

RvB(z) = Euv(z) Edv(z). (3)

The luminance of the background water is then

LvB(z) = Edv(z)RvB(z)π. (4)

The inherent contrast at depth z is

Cin(z) = LvT (z) LvB(z) LvB(z) = RvT RvB(z) RvB(z) (5)

where the last equation follows from (2) and (4) into (5).

The apparent contrast of the Secchi disk as seen from just below the sea surface is

Cap(0) = LvT (0) LvB(0) LvB(0) (6)

(Note that in this development the argument 0 refers to depth z, not to the distance from the target, which is z.)

The luminance difference law

LvT (0) LBv(0) = [LvT (z) Lv(z)]exp[cvzz] (7)

allows the apparent contrast to be written as

Cap(0) = [LvT (z) LvB(z)] LvB(0) exp[cvzz] (8)

by (??) into (6).

Inserting (2) and (4) into (8) then gives

Cap(0) = RvT RvB(z) RvB(0) Edv(z) Edv(0)exp[cvzz] (9)

Assuming that RvB(0) = RvB(z) and using (1) and (5) gives

Cap(0) = Cin(z)exp[(Kdvz + cvz)z] (10)

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

Cap(air) = 𝒯 Cap(0) = 𝒯 Cin(z)exp[(Kdvz + cvz)z]

where 𝒯 denotes the transmission of contrast, not of luminance or illuminance.

The Secchi depth zSD is the depth at which the apparent contrast in air falls below a threshhold contrast CT . Solving for zSD when Cap(air) = CT gives

zSD = ln 𝒯 Cin(z) CT KdvzSD + cvzSD (11) Γ KdvzSD + cvzSD. (12)

Studies with human observers show that CT depends on the angular subtense of the disk and on the ambient luminance (e.g., Table 1 of Preisendorfer (1986)). The values of Γ vary from about 6 to 9 for a disk with RvT = 0.85, depending on the water reflectance RvB (which is 0.015 to 0.1; Table 2 of Preisendorfer (1986)). The HydroLight code uses Γ = 8 as its default.

Note that Eq. (12) must be solved interatively because KdvzSD and cvzSD are averages over the (unknown) Secchi depth zSD. This is easily done after solution of the radiative transfer equation to some depth greater than zSD over the visible wavelengths. The photopic Kdv(z) and cv(z) can then be computed from Ed(z,λ) and c(z,λ). The values of Kdv and cv just below the water surface (at depth 0) are then used to get an initial estimate of zSD, which is then used to compute an improved estimate of the depth-averaged Kdv and cv, and so on. Convergence is obtained within a few interations.

0.2 The Secchi Depth Model of Lee et al.

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 Kdv(z) with Kd(z,λo), where λo is the wavelength at which Kd(z,λ) is a minimum; and
  • Replace the photopic cv(z) with 1.5Kd(z,λo).

One end result of their analysis is a formula of the form (Eq. 28 of their paper)

zSD = γ 2.5Kd(z,λo), (13)

where γ depends on a difference in reflectances, rather than on contrasts as seen in Eq. (11). This formula has the great virtue that Kd(z,λo) can be estimated from multi- or hyperspectral satellite imagery.

Comparison of zSD measured and computed by Eq. (13) gives reasonable agreement (see Fig. 6 of their paper). However, comparison of Lee et al. zSD predictions with those of the classic theory have not been made.

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