**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. ${L}_{vB}\left(z,\widehat{\xi}\right)={L}_{vB}\left(z\right)$. The underlying idea is that a disk at some depth $z$ is illuminated by the downwelling plane illuminance ${E}_{dv}\left(z\right)$. 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

$${E}_{dv}\left(z\right)={E}_{dv}\left(0\right)exp\left[-{\u27e8{K}_{dv}\u27e9}_{z}z\right]\phantom{\rule{0.3em}{0ex}},$$ | (1) |

where ${\u27e8\cdots \phantom{\rule{0.3em}{0ex}}\u27e9}_{z}$ denotes the average over 0 to $z$.

The target is assumed to be a Lambertian reﬂector with an illuminance reﬂectance of ${R}_{vT}$. The luminance reﬂected by the target is then

$${L}_{vT}\left(z\right)={E}_{dv}\left(z\right)\phantom{\rule{0.3em}{0ex}}{R}_{vT}\u2215\pi \phantom{\rule{0.3em}{0ex}}.$$ | (2) |

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

$${R}_{vB}\left(z\right)=\frac{{E}_{uv}\left(z\right)}{{E}_{dv}\left(z\right)}\phantom{\rule{0.3em}{0ex}}.$$ | (3) |

The luminance of the background water is then

$${L}_{vB}\left(z\right)={E}_{dv}\left(z\right)\phantom{\rule{0.3em}{0ex}}{R}_{vB}\left(z\right)\u2215\pi \phantom{\rule{0.3em}{0ex}}.$$ | (4) |

The inherent contrast at depth $z$ is

$$\begin{array}{llll}\hfill {C}_{in}\left(z\right)=& \phantom{\rule{1em}{0ex}}\frac{{L}_{vT}\left(z\right)-{L}_{vB}\left(z\right)}{{L}_{vB}\left(z\right)}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill =& \phantom{\rule{1em}{0ex}}\frac{{R}_{vT}-{R}_{vB}\left(z\right)}{{R}_{vB}\left(z\right)}\phantom{\rule{2em}{0ex}}& \hfill \text{(5)}\end{array}$$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

$${C}_{ap}\left(0\right)=\frac{{L}_{vT}\left(0\right)-{L}_{vB}\left(0\right)}{{L}_{vB}\left(0\right)}$$ | (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

$${L}_{vT}\left(0\right)-{L}_{Bv}\left(0\right)=\left[{L}_{vT}\left(z\right)-{L}_{v}\left(z\right)\right]exp\left[-{\u27e8{c}_{v}\u27e9}_{z}\phantom{\rule{0.3em}{0ex}}z\right]$$ | (7) |

allows the apparent contrast to be written as

$${C}_{ap}\left(0\right)=\frac{\left[{L}_{vT}\left(z\right)-{L}_{vB}\left(z\right)\right]}{{L}_{vB}\left(0\right)}exp\left[-{\u27e8{c}_{v}\u27e9}_{z}\phantom{\rule{0.3em}{0ex}}z\right]$$ | (8) |

by (??) into (6).

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

Assuming that ${R}_{vB}\left(0\right)={R}_{vB}\left(z\right)$ and using (1) and (5) gives

$${C}_{ap}\left(0\right)={C}_{in}\left(z\right)exp\left[-\left({\u27e8{K}_{dv}\u27e9}_{z}+{\u27e8{c}_{v}\u27e9}_{z}\right)\phantom{\rule{0.3em}{0ex}}z\right]$$ | (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

$${C}_{ap}\left(air\right)=\mathcal{\mathcal{T}}{C}_{ap}\left(0\right)=\mathcal{\mathcal{T}}{C}_{in}\left(z\right)exp\left[-\left({\u27e8{K}_{dv}\u27e9}_{z}+{\u27e8{c}_{v}\u27e9}_{z}\right)\phantom{\rule{0.3em}{0ex}}z\right]$$ |

where $\mathcal{\mathcal{T}}$ denotes the transmission of contrast, not of luminance or illuminance.

The Secchi depth ${z}_{SD}$ is the depth at which the apparent contrast in air falls below a threshhold contrast ${C}_{T}$. Solving for ${z}_{SD}$ when ${C}_{ap}\left(air\right)={C}_{T}$ gives

$$\begin{array}{lll}\hfill {z}_{SD}=& \phantom{\rule{1em}{0ex}}\frac{ln\left[\frac{\mathcal{\mathcal{T}}{C}_{in}\left(z\right)}{{C}_{T}}\right]}{{\u27e8{K}_{dv}\u27e9}_{{z}_{SD}}+{\u27e8{c}_{v}\u27e9}_{{z}_{SD}}}\phantom{\rule{2em}{0ex}}& \hfill \text{(11)}\\ \hfill \equiv & \phantom{\rule{1em}{0ex}}\frac{\Gamma}{{\u27e8{K}_{dv}\u27e9}_{{z}_{SD}}+{\u27e8{c}_{v}\u27e9}_{{z}_{SD}}}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(12)}\end{array}$$Studies with human observers show that ${C}_{T}$ depends on the angular subtense of the disk and on the ambient luminance (e.g., Table 1 of Preisendorfer (1986)). The values of $\Gamma $ vary from about 6 to 9 for a disk with ${R}_{vT}=0.85$, depending on the water reﬂectance ${R}_{vB}$ (which is 0.015 to 0.1; Table 2 of Preisendorfer (1986)). The HydroLight code uses $\Gamma =8$ as its default.

Note that Eq. (12) must be solved interatively because ${\u27e8{K}_{dv}\u27e9}_{{z}_{SD}}$ and ${\u27e8{c}_{v}\u27e9}_{{z}_{SD}}$ are averages over the (unknown) Secchi depth ${z}_{SD}$. This is easily done after solution of the radiative transfer equation to some depth greater than ${z}_{SD}$ over the visible wavelengths. The photopic ${K}_{dv}\left(z\right)$ and ${c}_{v}\left(z\right)$ can then be computed from ${E}_{d}\left(z,\lambda \right)$ and $c\left(z,\lambda \right)$. The values of ${K}_{dv}$ and ${c}_{v}$ just below the water surface (at depth 0) are then used to get an initial estimate of ${z}_{SD}$, which is then used to compute an improved estimate of the depth-averaged ${K}_{dv}$ and ${c}_{v}$, 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 ${K}_{dv}\left(z\right)$ with ${K}_{d}\left(z,{\lambda}_{o}\right)$, where ${\lambda}_{o}$ is the wavelength at which ${K}_{d}\left(z,\lambda \right)$ is a minimum; and
- Replace the photopic ${c}_{v}\left(z\right)$ with $1.5{K}_{d}\left(z,{\lambda}_{o}\right)$.

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

$${z}_{SD}=\frac{\gamma}{2.5{K}_{d}\left(z,{\lambda}_{o}\right)}\phantom{\rule{0.3em}{0ex}},$$ | (13) |

where $\gamma $ depends on a diﬀerence in reﬂectances, rather than on contrasts as seen in Eq. (11). This formula has the great virtue that ${K}_{d}\left(z,{\lambda}_{o}\right)$ can be estimated from multi- or hyperspectral satellite imagery.

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