# A New Algorithm to Estimate Diffuse Attenuation Coefficient from Secchi Disk Depth

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## Abstract

**:**

^{2}= 0.965, MAE = 0.083, RMSD = 0.239, BIAS = 0.01, and MPI = 0.854). Previous models considering a single optical depth figure at which the $SD$ disappears did not capture the marine optical complexity. Our classification of 113 stations with spectral absorption data into Jerlov water types indicated that no unique correspondence existed between estimated ${K}_{d}$ (PAR) and water type, making it ambiguous to associate compatible inherent optical properties and chlorophyll with ${Z}_{SD}$. Although obtaining ${K}_{d}$ ($\mathrm{PAR}$) from ${Z}_{SD}$ is simple/low-cost, care should be taken in the methodology used to measure ${Z}_{SD}$ to ensure consistent results across different optical marine conditions.

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

^{−1}. and 0.3 to 50 m, respectively.

^{2}= 0.957), which is lower than $O{D}_{SD}$ values reported in the literature. Typically, studies performed in the 20th century considered that low $O{D}_{SD}$ values resulted from increased turbidity [13,45]. However, the authors of [18] mentioned that in addition to the above, a factor to consider is that when ${K}_{d}\text{}\left(\mathrm{PAR}\right)$ is estimated using shallow depths, this tends to be higher than values estimated for deeper layers. This trend is most evident in oceanic stations with deeper ${Z}_{eu}$ relative to stations where light penetration is lower. In other words, lower $O{D}_{SD}$ values may be obtained when the calculation of ${K}_{d}\text{}\left(\mathrm{PAR}\right)$ considers the light profile from the surface to a depth close to ${Z}_{eu}$, or at least to ${Z}_{SD}$ [18]. In determining $O{D}_{SD}$, ${{K}_{d}}_{in\text{}situ}$ was calculated using light profiles close to ${Z}_{eu}$ or ${Z}_{SD}$.

^{2}= 0.957). These coefficients are lower relative to those reported by Montes-Hugo and Álvarez-Borrego [17] (a = 1.45 and b = 1.10), likely because of the narrower sampling interval (1–12 m) used by these authors.

^{2}= 0.965). These ${Z}_{SD}$ ranges represent two contrasting conditions: turbid water with ${Z}_{SD}$ < 2.20 m (Equation (15)) and clear water with ${Z}_{SD}$ ≥ 5.37 m (Equation (17)); plus a transition zone with 2.20 m ≤ ${Z}_{SD}$ < 5.37 m (Equation (16)).

^{−1}(clear waters). In more optically complex waters (${K}_{d}$ > 2 m

^{−1}), the difference between in-situ and modelled data increases. The model of Lee et al. [18] tends to fit a positive exponential (J-shaped) function departing from the 1:1 line, while model c is a closer fit to the 1:1 line. This is evident for the full database (679 observations) and for two independent cruises with different optical conditions (Cal9709: oceanic/coastal conditions; Ties9802: estuarine conditions).

^{−1}(blue circles); the turbid water case, i.e., ${Z}_{SD}$ < 2.20 m, to ${{K}_{d}}_{insitu}$ values above 0.4 m

^{−1}(brown circles); and the transition zone, to intermediate ${{K}_{d}}_{insitu}$ values. In general, ${K}_{d}\left(PAR\right)$ is inversely related to ${Z}_{SD}$ [1,9], but with a variable dependence according to the ${Z}_{SD}$ range, a behavior due to the differential influence of the components that contribute to light attenuation as ${Z}_{SD}$ change [21].

^{−1}. The turbid-water case (${Z}_{SD}$ < 2.20 m) included the most turbid coastal waters (5–9), with ${{K}_{d}}_{insitu}$ values above 0.4 m

^{−1}. Additionally, the transition zone shows the interaction between groups 4, 5, 6, and 7, comprising stations ranging from clear (4) to turbid (5–7) waters, where some stations (for example, group 6) show turbid waters while other stations in the same group belong to the transition zone. Figure 4b also shows an overlap of water types for a given ${Z}_{SD}$ in the group of oceanic waters. This also occurs for the group of clear coastal waters that includes types 1, 2, and 4, with an overlap between the few type 2 and type 1 cases. In addition, if data representing type 3 were available, these would presumably overlap with type 4. Finally, and consistent with the above, the group of turbid coastal waters (types 5–9) exhibits an overlap between optical types. These results show that the relationship between ${Z}_{SD}(\mathrm{or}\text{}\mathrm{deduced}{K}_{d}\left(PAR\right))$ and water type, as determined from a (λ), is not unique (i.e., a given ${Z}_{SD}$ or range of ${Z}_{SD}$ values may be associated to different water types), thus limiting the ability to derive specific inherent optical properties from ${Z}_{SD}$ as the only variable, even in statistic terms. This is hardly surprising, since ${K}_{d}\left(PAR\right)$ is non-spectral and depends not only on a (λ) but also on spectral particulate backscattering [23]. Note that the model proposed by Solonenko and Mobley [30] to associate the inherent optical properties with Jerlov water type does not consider the contribution of non-phytoplankton particulate material, which comprises phytoplankton detritus plus other organic and mineral particles. They argue that it is sufficient to model absorption as a function of chlorophyll and CDOM in both Case-1 and Case-2 waters, and even for the most turbid waters. In areas highly influenced by non-phytoplankton particulate matter (e.g., mineral sources), however, the absorption of these components likely influences the association of the inherent optical properties with water type. Therefore, Jerlov’s classification needs to be adapted to account for such scenarios in turbid coastal environments, including coastal lagoons, estuaries, and river mouths.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparative analysis between the attenuation coefficient estimated in situ (${{K}_{d}}_{in\text{}situ}$) and the attenuation coefficients modeled. (

**a**) ${{K}_{d}}_{model\text{}a}$, (

**b**) ${{K}_{d}}_{model\text{}b}$, and (

**c**) ${{K}_{d}}_{model\text{}c}$.

**Figure 3.**Comparison of ${{K}_{d}}_{in\text{}situ}$ vs. ${{K}_{d}}_{\text{}model}$ calculated with the model of Lee et al. [18] (circles) and with model c (crosses), for the full database (black) and for two cruises with different optical conditions: Cal9709 (yellow and blue); Ties9802 (red and green).

**Figure 4.**(

**a**) Relationship between ${Z}_{SD}$ and ${{K}_{d}}_{insitu}$ according to three ${Z}_{SD}$ ranges (679 stations); (

**b**) relationship between ${Z}_{SD}$ and ${{K}_{d}}_{insitu}$ according to Jerlov water type (113 stations with absorption data).

**Figure 5.**Intermediate water types (dashed lines), added to the original Jerlov coastal water types.

**Figure 6.**Least-squares fitting method for four selected stations. The black line indicates $a{\left(\lambda \right)}_{model}$ values for the different water types; the dotted line, $a{\left(\lambda \right)}_{in\text{}situ}$ for stations. ${\chi}_{Crit}^{2}$ marks the maximum allowable tolerance for the fit. (

**a**) Example of the fit for type II, where station D05 gave the best fit; (

**b**) example of the fit for type 4, where station B06 gave the best fit; (

**c**) example of the fit for type 6, where station 247 gave the best fit; (

**d**) example of the fit for type 8, where station 071 gave the best fit.

Reference | Equation | ${\mathit{Z}}_{\mathit{S}\mathit{D}}\text{}\mathbf{Intervals}\text{}\left(\mathbf{m}\right)$ | MAE | RMSD | BIAS | MPI |
---|---|---|---|---|---|---|

Poole and Atkins [9] | ${K}_{d}=1.7/{Z}_{SD}$ | 1.9–35 | 0.182 | 0.285 | 0.217 | 0.104 |

Poole and Atkins [9] Holmes [15] | ${K}_{d}=1.7/{Z}_{SD}$ ${K}_{d}=1.4/{Z}_{SD}$ | 1.9–35 2–12 | 0.125 | 0.273 | 0.120 | 0.416 |

Megard and Berman [16] | ${K}_{d}=1.54/{Z}_{SD}$ | 6–46 | 0.142 | 0.285 | 0.118 | 0.229 |

Lee et al. [18] | ${K}_{d}=1.48/{Z}_{SD}$ | All intervals | 0.134 | 0.285 | 0.078 | 0.354 |

Montes-Hugo and Álvarez-Borrego [17] | ${K}_{d}=1.45/{({Z}_{SD})}^{1.10}$ | 1–12 | 0.141 | 0.359 | −0.013 | 0.250 |

Model a | ${K}_{d}=1.37/{Z}_{SD}$ | All intervals | 0.118 | 0.285 | 0.003 | 0.583 |

Model b | ${K}_{d}=1.18/{({Z}_{SD})}^{0.92}$ | All intervals | 0.097 | 0.265 | 0.002 | 0.708 |

Model c | Equations (15)–(17) | <2.20 Transition zone ≥5.37 | 0.083 | 0.239 | 0.001 | 0.854 |

Reference | Equation | ${\mathit{Z}}_{\mathit{S}\mathit{D}}\text{}\mathbf{Intervals}\text{}\left(\mathbf{m}\right)$ | MAE | RMSD | BIAS | MPI |
---|---|---|---|---|---|---|

Poole and Atkins [9] | ${K}_{d}=1.7/{Z}_{SD}$ | 1.9–35 | 0.041 | 0.073 | −0.028 | 0.074 |

Poole and Atkins [9] Holmes [15] | ${K}_{d}=1.7/{Z}_{SD}$ ${K}_{d}=1.4/{Z}_{SD}$ | 1.9–35 2–12 | 0.034 | 0.063 | −0.006 | 0.460 |

Megard and Berman [16] | ${K}_{d}=1.54/{Z}_{SD}$ | 6–46 | 0.032 | 0.063 | −0.011 | 0.425 |

Lee et al. [18] | ${K}_{d}=1.48/{Z}_{SD}$ | All intervals | 0.031 | 0.062 | −0.005 | 0.740 |

Montes-Hugo and Álvarez-Borrego [17] | ${K}_{d}=1.45/{({Z}_{SD})}^{1.10}$ | 1–12 | 0.034 | 0.003 | 0.021 | 0.425 |

Model a | ${K}_{d}=1.37/{Z}_{SD}$ | All intervals | 0.029 | 0.063 | 0.007 | 0.592 |

Model b | ${K}_{d}=1.18/{({Z}_{SD})}^{0.92}$ | All intervals | 0.032 | 0.072 | 0.009 | 0.388 |

Model c | Equations (15)–(17) | <2.20 | 0.026 | 0.062 | 0.005 | 0.814 |

${\mathit{K}}_{\mathit{d}}$_{satellite model} (490) | Standard SeaDAS product | Transition zone ≥5.37 | 0.079 | 0.187 | 0.015 | 0.074 |

**Table 3.**Number of observations (N) classified into a Jerlov water type for the 113 stations with a.

Descriptors | Oceanic | Coastal | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Water Type | I | IA | IB | II | III | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

N | 2 | 4 | 4 | 8 | 26 | 21 | 2 | - | 6 | 5 | 7 | 14 | 6 | 8 |

**Table 4.**Evaluation of the ${{K}_{d}}_{in\text{}situ}$ versus a specific model by water type and model c. The bold show the best results.

Water Type | Model | Equation | MAE | RMSD | BIAS |
---|---|---|---|---|---|

Oceanic group | Model c Model oceanic | Equations (15)–(17) ${K}_{d}=0.089/{({Z}_{SD})}^{0.518}$ | 0.0160.550 | 0.0180.020 | 0.0160.055 |

Coastal group | Model c Model coastal | Equations (15)–(17) ${K}_{d}=1.79/{({Z}_{SD})}^{0.978}$ | 0.1490.346 | 0.2600.469 | 0.037−0.343 |

III | Model c Model III | Equations (15)–(17) ${K}_{d}=0.37/{({Z}_{SD})}^{0.673}$ | 0.0110.027 | 0.0090.010 | 0.0110.027 |

1 | Model c Model 1 | Equations (15)–(17) ${K}_{d}=0.66/{({Z}_{SD})}^{0.784}$ | 0.0210.027 | 0.0270.027 | 0.0160.025 |

7 | Model c Model 7 | Equations (15)–(17) ${K}_{d}=0.95/{({Z}_{SD})}^{0.667}$ | 0.0510.105 | 0.0530.074 | −0.0320.105 |

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Castillo-Ramírez, A.; Santamaría-del-Ángel, E.; González-Silvera, A.; Frouin, R.; Sebastiá-Frasquet, M.-T.; Tan, J.; Lopez-Calderon, J.; Sánchez-Velasco, L.; Enríquez-Paredes, L.
A New Algorithm to Estimate Diffuse Attenuation Coefficient from Secchi Disk Depth. *J. Mar. Sci. Eng.* **2020**, *8*, 558.
https://doi.org/10.3390/jmse8080558

**AMA Style**

Castillo-Ramírez A, Santamaría-del-Ángel E, González-Silvera A, Frouin R, Sebastiá-Frasquet M-T, Tan J, Lopez-Calderon J, Sánchez-Velasco L, Enríquez-Paredes L.
A New Algorithm to Estimate Diffuse Attenuation Coefficient from Secchi Disk Depth. *Journal of Marine Science and Engineering*. 2020; 8(8):558.
https://doi.org/10.3390/jmse8080558

**Chicago/Turabian Style**

Castillo-Ramírez, Alejandra, Eduardo Santamaría-del-Ángel, Adriana González-Silvera, Robert Frouin, María-Teresa Sebastiá-Frasquet, Jing Tan, Jorge Lopez-Calderon, Laura Sánchez-Velasco, and Luis Enríquez-Paredes.
2020. "A New Algorithm to Estimate Diffuse Attenuation Coefficient from Secchi Disk Depth" *Journal of Marine Science and Engineering* 8, no. 8: 558.
https://doi.org/10.3390/jmse8080558