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

The vertical diffuse attenuation coefficient Kd (PAR) is used for calculating the euphotic zone, the first optical depth that is important for primary productivity models. Currently, Kd (PAR) can be estimated using an irradiometer or a Secchi disk (SD). The main objective of this work is to define a model that can be applied to a wide range of optical marine conditions to estimate Kd (PAR) by SD. We used irradiance profiles and SD depth (ZSD) from 679 stations in various marine regions. Three parametric models were developed, and their statistical performance was evaluated in view of previous approaches reported and remote sensing data. The best results were obtained with an adaptive model representing three cases: clear-water, turbid-water, and a transition zone (R2 = 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 Kd (PAR) and water type, making it ambiguous to associate compatible inherent optical properties and chlorophyll with ZSD. Although obtaining Kd (PAR) from ZSD is simple/low-cost, care should be taken in the methodology used to measure ZSD to ensure consistent results across different optical marine conditions.


Introduction
Sunlight in the electromagnetic spectrum region between 400 and 700 nm, known as photosynthetically active radiation (PAR) [1], plays an important role in the physics, biology, and chemistry of the oceans [2]. PAR is directly linked to biological processes such as photosynthesis, which constitutes the base of the food chain in marine ecosystems [1,[3][4][5]. In other words, primary productivity largely depends on the sunlight field in the water column, which decreases exponentially with depth.
Another approach used to estimate K d (PAR) through SD without involving OD SD involves the development of empirical models based on a nonlinear relationship between K d (PAR) and 1/Z SD : where a and b are the empirical coefficients of the model, which vary according to the study area.
The majority of the work on models given by Equation (2) has been conducted for inland water bodies [26][27][28]. For marine environments, Montes-Hugo and Alvarez-Borrego [17] proposed values of 1.45 and 1.10 for a and b, respectively, for coastal waters in the northeast of Baja California and southern California, an area with Z SD from 2 to 12 m. Jerlov [29] classified oceanic water into five types (I, IA, IB, II, and III) and coastal water into five types (1, 3, 5, 7, and 9), based on hyperspectral K d (K d (λ)) for each wavelength), where a larger number indicates higher turbidity. Each of the 10 Jerlov water types has a typical light absorption coefficient associated (a(λ) model ) [30]. This absorption coefficient takes into account pure sea water absorption, a w (λ), chlorophyll-based absorption, a phy (λ), and CDOM absorption, a CDOM (λ) [30] (Equation (3)): a(λ) model = a w (λ) + a phy (λ) + a CDOM (λ) (3) where λ is wavelength. Based on the above, estimating K d (PAR) from Z SD measurements has proven to be a useful, cost-effective methodology to detect changes in marine ecosystems. However, it is essential that the K d (PAR) estimates be robust and objective, since a miscalculation of K d (PAR) would lead to errors in estimates of euphotic zone and first optical depths. The main objective of this work is to establish an empirical relationship that could be applied to a wide range of optical conditions occurring in marine environments. A second objective is to examine whether the K d (PAR) derived from Z SD could be related to a specific optical water type (in Jerlov's sense), which would broaden the application of the SD approach and enhance its usefulness as a descriptor of water optical properties. Likewise, and considering that the current development of spectroradiometer-type remote sensors allows producing synoptic measurements of K d at 490 nm (K d (490)), this study considered data from these remote sensors as one of the models to evaluate.

Materials and Methods
In this study, we compiled 679 pairs of K d (PAR) and Z SD data recorded under well-illuminated (sunny) conditions, with an irradiance meter and SD, respectively. These encompass conditions from clear oceanic to turbid coastal waters (Figure 1a,b). The source of data was as follows: 212 data pairs from the Tara Ocean Consortium [31] (Figure 1a); 195 from the NASA SeaBass archive (Chesapeake bay and CALCOFI zone) [32]; 103 from Gallegos et al. [21]; 74 from the Phytoplankton Ecology Group (POPEYE) database at Universidad Autónoma de Baja California; 71 from Montes-Hugo and Alvarez-Borrego [17]; 24 from Megard and Berman [16]. Hereafter, K d (PAR) recorded from irradiance-meter data will be named K din situ .  [31] and Megard and Berman [16]. (b) Stations of the NASA SeaBass file [32], POPEYE database, Gallegos et al. [21], and Montes-Hugo and Alvarez-Borrego [17].
for NASA SeaBass [32] and POPEYE data was estimated for each station based on PAR measurements, in accordance with the criteria of Kirk [1], expressed in Equation (4): where is the slope of a linear model and the dependent variable is the natural logarithm of irradiance as a function of depth ( ). For the Tara Ocean Consortium data [31], was estimated from using the following Equation [1]: The data of Megard and Berman [16], Montes-Hugo and Alvarez-Borrego [17], and Gallegos et al. [21] were taken from those reported in the respective publications.
Based on Steyerberg [33] and the IOCCG report [34], the database was randomly split into two groups. Group 1 was used for the development of models; it comprises 478 data pairs (70% of total data). Group 2 was used for validation purposes; it comprises 201 data pairs (30% of total data). To reduce the random error in the selection of the two datasets and to test the robustness of models, 10 replicates were obtained by randomly selecting 10 different datasets for modeling, along with the respective validation datasets.
Three parametric models were tested to predict ( ) based on ; the model with the best fit was selected as per the criteria of Xu et al. [35] for least-squares analysis. The first approximation (model a) was based on Equation (1) (2)); and the third (model c) followed the concepts of Equation (2); in the latter  [31] and Megard and Berman [16]. (b) Stations of the NASA SeaBass file [32], POPEYE database, Gallegos et al. [21], and Montes-Hugo and Alvarez-Borrego [17].
K din situ for NASA SeaBass [32] and POPEYE data was estimated for each station based on PAR measurements, in accordance with the criteria of Kirk [1], expressed in Equation (4): where K din situ is the slope of a linear model and the dependent variable is the natural logarithm of irradiance as a function of depth (Z). For the Tara Ocean Consortium data [31], K din situ was estimated from Z eu using the following equation [1]: The K din situ data of Megard and Berman [16], Montes-Hugo and Alvarez-Borrego [17], and Gallegos et al. [21] were taken from those reported in the respective publications.
Based on Steyerberg [33] and the IOCCG report [34], the database was randomly split into two groups. Group 1 was used for the development of models; it comprises 478 data pairs (70% of total data). Group 2 was used for validation purposes; it comprises 201 data pairs (30% of total data). To reduce the random error in the selection of the two datasets and to test the robustness of models, 10 replicates were obtained by randomly selecting 10 different datasets for modeling, along with the respective validation datasets.
Three parametric models were tested to predict K d (PAR) based on Z SD ; the model with the best fit was selected as per the criteria of Xu et al. [35] for least-squares analysis. The first approximation (model a) was based on Equation (1), where K d (PAR) is calculated using fixed (or single) estimates of OD SD ; the second (model b) was based on the nonlinear empirical relationship between K din situ and 1 Z SD (Equation (2)); and the third (model c) followed the concepts of Equation (2); in the latter model, Z SD was split into three zones, namely two extreme cases (clear and turbid waters) and a transition zone, as follows: In the transition zone, where p3 ≤ log 10 (Z SD ) < p4, log 10 (K d ) is modeled as a linear combination of the two models: The model parameters p1, p2, p3, p4, p5, and p6 are retrieved through the least-squares fitting. . Sensor data to construct these images were downloaded from [36] to 1B level and processed at levels 2 and 3 using SeaDAS 7.5.1. (National Aeronautics and Space Administration) The multi-sensor composites were performed following the criteria by [37,38].
where n is the total number of data included in this analysis, K din situ − K dmodel is residual observations, and (K din situ − K dmodel ) is the absolute value of residuals.
where BIAS is the residual mean. According to MAE and RMSD, lower values represent better results, whereas BIAS values closer to zero mean better results. To determine which is the best model, a model performance index (MPI) was estimated (Equation (12)) based on the three statistic descriptors mentioned above: where R MAE is the MAE rank, R RMSD is the RMSD rank, R |BIAS| is the rank of absolute BIAS values, and p is the total number of models to be compared. Ranks and their respective matches were calculated following the criteria of [39][40][41]. MPI ranges from 0 to 1, where values closer to 1 mean a better model. To accomplish the objective of examine whether the K d (PAR) derived from Z SD could be related to a specific optical water type (in Jerlov's sense), we compiled in-situ absorption coefficient data (a(λ) in situ ). These were available for 113 stations only. The data for the POPEYE database and the CALCOFI zone were processed following the protocol of Mitchell et al. [42], whereas the data for the Chesapeake Bay area were analyzed following the protocol of Mueller and Austin [43]. Pope and Fry [12]) values were used for a w (λ). a(λ) in situ spectra were classified into the different Jerlov water types. The classification was obtained by comparing a(λ) in situ with a(λ) model using a least-squares fitting method based on Xu et al. [35]. The statistical significance of this fit was determined through a goodness-of-fit test following Zar [44]. POPEYE absorption data ranged from 400 to 700 nm, and SeaBaSS data from 400 to 650 mm. Thus, the goodness-of-fit test was carried out with different degrees of freedom, according to each dataset (POPEYE: α = 0.05, χ 2 Crit = 21.02, df = 12; SeaBASS: α = 0.05, χ 2 Crit = 18.30, df = 10).

Results and Discussion
Our database includes both oceanic and coastal stations (Figure 1), so the models presented in this work were developed and evaluated considering a wide range of optical conditions. The three models proposed in this work represent K din situ and Z SD values ranging from 0.030 to 3.217 m −1 . and 0.3 to 50 m, respectively.
Model a based on Equation (1) and that gives rise to Equation (13) yielded an OD SD of 1.37 (R 2 = 0.957), which is lower than OD SD values reported in the literature. Typically, studies performed in the 20th century considered that low OD 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 (PAR) 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 OD SD values may be obtained when the calculation of K d (PAR) considers the light profile from the surface to a depth close to Z eu , or at least to Z SD [18]. In determining OD SD , K din situ was calculated using light profiles close to Z eu or Z SD .
Model b was based on a nonlinear fit between K din situ and Z SD , (Equation (14)), assuming that this represents the continuous function of the empirical relationship between K d (PAR) and Z SD . This model resulted in regression coefficients a = 1.18 and b = 0.92 (R 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.
Model c is adaptive and followed the same approach as model b, i.e., it comprises three equations that consider three Z SD ranges (Equations (15)-(17)) (R 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)).
The equation for the transition zone (Equation (16)) is more complex since attempting to discern stations located between these two extremes (turbid-to-clear vs. clear-to-turbid stations), while keeping continuity in Z SD over the entire Z SD range. This partitioning into Z SD ranges in the model c allowed us to reduce variability in the data, i.e., achieve a better fit for each case, therefore providing better performance overall.
For evaluating models a, b, and c, K din situ (Equations (4) and (5)) was compared versus K dmodel a (Equation (13)), K dmodel b (Equation (14)), and K dmodel c (Equations (15)-(17)) ( Figure 2), as well as versus K d estimated from the different models for marine waters reported in the literature (Table 1). These comparisons were evaluated based on MAE (Equation (9)), RMSD (Equation (10)), BIAS (Equation (11)), and MPI (Equation (12)). The above showed that model c (Figure 2c c allowed us to reduce variability in the data, i.e., achieve a better fit for each case, therefore providing better performance overall. For evaluating models a, b, and c, (Equations (4) and (5)) was compared versus (Equation (13)), (Equation (14)), and (Equations (15)-(17)) ( Figure 2), as well as versus estimated from the different models for marine waters reported in the literature (Table 1). These comparisons were evaluated based on MAE (Equation (9)), RMSD (Equation (10)), BIAS (Equation (11)), and MPI (Equation (12)). The above showed that model c (Figure 2c) yielded the best fit (lowest MAE, RMSD, BIAS close to zero, and MPI closest to one), MAE of 0.083, RMSD of 0.239, BIAS of 0.01, and MPI of 0.854.  Table 1 represents an improvement versus previous methods, showing that model c yields the best performance considering the statistical descriptors and the index mentioned above. In particular, with respect to the most recent model [18], model c yields a MAE reduction of 0.051, an RMSD reduction of 0.046, and a MPI increase of 0.5. If models developed for particular or regional marine conditions are considered, such as the model of Lugo-Fernández et al. [46], which divide their conditions for use into summer and nonsummer for the north Gulf of Mexico an area influenced by the Mississippi river mouth, highly dispersed results may be obtained as this area is affected by high inland CDOM inputs. Additionally, we consider that the conditions in this model depend on river flows rather than seasonality. Given that our database includes stations from both northern and southern hemispheres, the criteria for   Table 1 represents an improvement versus previous methods, showing that model c yields the best performance considering the statistical descriptors and the index mentioned above. In particular, with respect to the most recent model [18], model c yields a MAE reduction of 0.051, an RMSD reduction of 0.046, and a MPI increase of 0.5.
If models developed for particular or regional marine conditions are considered, such as the model of Lugo-Fernández et al. [46], which divide their conditions for use into summer and non-summer for the north Gulf of Mexico an area influenced by the Mississippi river mouth, highly dispersed results may be obtained as this area is affected by high inland CDOM inputs. Additionally, we consider that the conditions in this model depend on river flows rather than seasonality. Given that our database includes stations from both northern and southern hemispheres, the criteria for applying this model become complex. As an example, we assessed the model of Montes-Hugo and Álvarez-Borrego [17], which was developed only for coastal waters in northeast Baja California and southern California; we observed that the application of this model to different marine conditions yields suboptimal results, thus highlighting the constraints in generalizing empirical models that were originally established from a limited range of conditions.
The evaluation of models a and b revealed that the use of a single OD SD value [9,15,16,18] for different optical conditions of water resulted in higher RMSD (Table 1), relative to using two values, i.e., the 1.70 figure proposed by Poole and Atkins [9] for clear waters and the 1.44 figure estimated by Holmes [15] for turbid waters. If we consider the broad range of optical conditions that occur in the marine environment, the above confirms that a single value is certainly not representative of the range of optical conditions. In addition, it was noted that model b yields lower RMSD values relative to OD SD .
The three equations adequately represented the range of optical diversity in marine environments (from clear ocean waters to very turbid coastal waters) in a simplified way, and the model can be used even under conditions of phytoplankton blooms such as those described in Santamaria-del-Angel et al. [47] and Aguilar-Maldonado et al. [48].
In order to estimate K d (490) values from remote sensors for comparison versus K din situ and K d figures estimated with the models in Table 1, 1 km-resolution multi-sensor images were constructed. From the original 679 data, only 383 were used for this comparison. Table 2 evidences that model c is the best model, yielding an MPI of 0.814, whereas the models with the worst fit to K din situ were the model of Pooole and Atkins [9] and the satellite model, with an MPI of 0.074. To note, the model of [9] was derived from only 14 data that are not representative of the range of optical conditions in the ocean. For its part, the satellite model represents K d (490), which is a spectral portion of the K d (PAR) measured in situ. Austin and Petzoid [49] proposed the first approach of K d (490) using the ratio between leaving-water radiances (Lwr) between 443 and 550 nm (Lwr443/Lwr550), as well as the K d (490) for optically pure seawater and some linear approximation coefficients. A number of approaches have been developed from there, such as those of [32], which essentially have yielded improved estimates of the coefficients used, being specific to each sensor.
Additionally, several authors have estimated Z SD from satellite images, mostly focusing on inland water bodies [50][51][52][53][54]. In the marine environment, Kim et al. [55] estimated Z SD from K d (490) based on concepts derived from [56][57][58]. All these point to the need to measure other intrinsic optical properties, such as backscattering at least at 490 nm. As this work did not have in-situ backscattering observations available, we might think that these could be estimated through standard models such as the Generalized Inherent Optical Property (GIOP) model; however, Betancur-Turizo et al. [59] warn that care should be taken with these models because, by being too simplified, they do not reflect the variability of the parameter in a reliable way. This cautionary statement has been addressed recently. The works by Jiang et al. [60] and Liu et al. [61] have aimed to improve these estimates to obtain more accurate satellite Z SD values for various types of water. It is clear that this is a dynamic and constantly evolving line of research.
Although data derived from satellite observations are promising and this subject of research needs to be further refined in the future, we recommend further monitoring of marine optical properties while observing the data quality, and complementing in-situ reflectance measurements.
Based on the above and following the criteria established by [62], the use of multi-sensor data will allow us, in the long term, an improved integration of in-situ measurements and satellite images. This will result in time series that will be suitable for use in marine monitoring programs, defining weekly, monthly, seasonal, and annual variability scales, as well as their trends. Figure 3 compares K din situ and K dmodel calculated using model c and the model of Lee et al. [18]. The results show that both models performed similarly for K din situ values lower than 0.1 m −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).
observations available, we might think that these could be estimated through standard models su as the Generalized Inherent Optical Property (GIOP) model; however, Betancur-Turizo et al. [ warn that care should be taken with these models because, by being too simplified, they do not refl the variability of the parameter in a reliable way. This cautionary statement has been address recently. The works by Jiang et al. [60] and Liu et al. [61] have aimed to improve these estimates obtain more accurate satellite values for various types of water. It is clear that this is a dynam and constantly evolving line of research.
Although data derived from satellite observations are promising and this subject of resea needs to be further refined in the future, we recommend further monitoring of marine opti properties while observing the data quality, and complementing in-situ reflectance measurement Based on the above and following the criteria established by [62], the use of multi-sensor d will allow us, in the long term, an improved integration of in-situ measurements and satellite imag This will result in time series that will be suitable for use in marine monitoring programs, defin weekly, monthly, seasonal, and annual variability scales, as well as their trends. Figure 3 compares and calculated using model c and the model of Lee et [18]. The results show that both models performed similarly for values lower than 0.1 (clear waters). In more optically complex waters ( > 2 m −1 ), the difference between in-situ a modelled data increases. The model of Lee et al. [18] tends to fit a positive exponential (J-shap function departing from the 1:1 line, while model c is a closer fit to the 1:1 line. This is evident for full database (679 observations) and for two independent cruises with different optical conditio (Cal9709: oceanic/coastal conditions; Ties9802: estuarine conditions). The primary objective of this work is to define a model applicable to a wide range of opti conditions in marine environments. Under this premise, model c in an adaptive model that yie better predictions relative to models that consider an value only. In order to use this mode field cruises, a table was added as supplementary material for (Table S1), derived fr this model and based on . The most accurate estimate of with model c allows obtain better results in the calculation of primary productivity, which is a major variable that can be used studies ranging from understanding the global carbon cycle to fisheries aspects [63]. A prop The primary objective of this work is to define a model applicable to a wide range of optical conditions in marine environments. Under this premise, model c in an adaptive model that yields better predictions relative to models that consider an OD SD value only. In order to use this model in field cruises, a table was added as supplementary material for K d (PAR) (Table S1), derived from this model and based on Z SD . The most accurate estimate of K d (PAR) with model c allows obtaining better results in the calculation of primary productivity, which is a major variable that can be used in studies ranging from understanding the global carbon cycle to fisheries aspects [63]. A proper understanding of primary organic production will not only allow estimating the total organic matter produced in the oceans, but also supports the comparison of ocean regions as potential food sources [14]. Figure 4a shows K d values associated with the three Z SD ranges. The clear water case, i.e., Z SD ≥ 5.37 m, corresponds to K din situ values of less than 0.2 m −1 (blue circles); the turbid water case, i.e., Z SD < 2.20 m, to K din situ values above 0.4 m −1 (brown circles); and the transition zone, to intermediate K din situ values. In general, K d (PAR) 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]. Figure 4a shows values associated with the three ranges. The clear water case, i.e., ≥ 5.37 m, corresponds to values of less than 0.2 m −1 (blue circles); the turbid water case, i.e., < 2.20 m, to values above 0.4 m −1 (brown circles); and the transition zone, to intermediate values. In general, ( ) is inversely related to [1,9], but with a variable dependence according to the range, a behavior due to the differential influence of the components that contribute to light attenuation as change [21]. The analysis of ( ) resulted in 92 stations classified into a Jerlov water type; the remaining 21 stations failed to meet the criteria for inclusion into a given water type. Intermediate water types were therefore added, i.e., 2, 4, 6, and 8 ( Figure 5), supporting the allocation of a water type to the 113 ( ) spectra. The analysis of a(λ) in situ resulted in 92 stations classified into a Jerlov water type; the remaining 21 stations failed to meet the criteria for inclusion into a given water type. Intermediate water types were therefore added, i.e., 2, 4, 6, and 8 ( Figure 5), supporting the allocation of a water type to the 113 a(λ) in situ spectra.   Figure 6 illustrates the classification procedure into Jerlov water types. Four stations were selected as examples, which were contrasted according to a goodness-of-fit test based on a leastsquares fitting method [35]. In each panel of Figure 6, ( ) of the selected stations and ( ) of different types of water (II, 4, 6, and 8) are displayed with the respective values that represent the integration of the square distances between ( ) and ( ) . The water type for which is minimum was selected (if the value was zero, this would imply a perfect fit, i.e., ( ) is equal to ( ) ). In Figure 6a, station D05 presented a = 1.46, this being the smallest value relative to all other stations, indicating that it must be classified as a type-II water. In Figure 6b, the lowest corresponded to station B06, so it was classified as type 4. For water type 6, which is exemplified in Figure 6c, station 247 showed the lowest . When stations were contrasted with water type 8 (Figure 6d), station 071 had the lowest .  Figure 6 illustrates the classification procedure into Jerlov water types. Four stations were selected as examples, which were contrasted according to a goodness-of-fit test based on a least-squares fitting method [35]. In each panel of Figure 6, a(λ) in situ of the selected stations and a(λ) model of different types of water (II, 4, 6, and 8) are displayed with the respective χ 2 Cal values that represent the integration of the square distances between a(λ) in situ and a(λ) model . The water type for which χ 2 Cal is minimum was selected (if the value was zero, this would imply a perfect fit, i.e., a(λ) in situ is equal to a(λ) model ). In Figure 6a, station D05 presented a χ 2 Cal = 1.46, this being the smallest value relative to all other stations, indicating that it must be classified as a type-II water. In Figure 6b, the lowest χ 2 Cal corresponded to station B06, so it was classified as type 4. For water type 6, which is exemplified in Figure 6c, station 247 showed the lowest χ 2 Cal . When stations were contrasted with water type 8 (Figure 6d) The classification into Jerlov water types yielded the results shown in Table 3, with 48 oceanic stations (i.e., types I-III) and 65 coastal stations (i.e., types 1-9). No water type 3 was found in the reduced data set. Figure 4b shows that the clear water case ( ≥ 5.37 m.) comprises all the ocean groups (I, IA, IB, II, III) and the most transparent coastal groups (1)(2)(3)(4), with values of less than 0.2 m −1 . The turbid-water case ( < 2.20 m) included the most turbid coastal waters (5-9), with 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 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 (or deduced ( ) ) and water type, as determined from The classification into Jerlov water types yielded the results shown in Table 3, with 48 oceanic stations (i.e., types I-III) and 65 coastal stations (i.e., types 1-9). No water type 3 was found in the reduced data set. Figure 4b shows that the clear water case (Z SD ≥ 5.37 m.) comprises all the ocean groups (I, IA, IB, II, III) and the most transparent coastal groups (1)(2)(3)(4), with K din situ values of less than 0.2 m −1 . The turbid-water case (Z SD < 2.20 m) included the most turbid coastal waters (5-9), with K din situ 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 (or deduced K d (PAR)) 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 (PAR) 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. Considering the 113 data that were optically classified into the different Jerlov water types (Table 3), a model might theoretically be derived for each type of water; however, the number of observations for some optical types is small. It was therefore decided to split the 113 data into two groups, oceanic and coastal, and work only with those types of water including more than 10 observations, aiming of derive a specific model for each. For estimating the models, observations for each type of water (oceanic, coastal, III, 1, and 7) were randomly sorted into two groups (50% for modeling and 50% for validation purposes). Once the models were obtained, these were applied to the dataset to validate them, and contrasted against model c, which is the best model from the previous analyses. These comparisons are shown in Table 4. In this case, MPI (Equation (12)) was not suitable due to the small number of observations; thus, the best model was the one showing the lowest MAE (Equation (9)) and RMSD (Equation (10)) values, and the BIAS (Equation (11)) value closest to zero.  Model c is the best fit in all cases because this model contemplates three intervals of Z SD ; also, in each interval, this model makes an approximation based on a nonlinear model (Equation (2)). Consequently, this is an adaptive model that estimates the best combination regardless of the optical type of water being analyzed. In this regard, we consider that the development of model c simplifies of the estimate K d based on Z SD regardless of the type of water. Model c can be applied to different marine optical conditions; however, it is advisable to continue obtaining measurements of Jerlov's water types to determine the contribution of each individual component present in seawater to the calculation of K d .
Continuing with the simultaneous measurement of Z SD and K d with an irradiance-meter will facilitate obtaining better coefficients in each equation of the adaptive model. Additional measurements should be performed, especially in turbid coastal waters, including coastal lagoons, estuaries, and river mouths.
There is the need to implement marine monitoring systems to detect changes in ecosystems. Water turbidity may provide information on environmental alterations, and the monitoring thereof based on estimations of K d (PAR) from Z SD using empirical models such as model c is sufficiently accurate and cost-effective. The potential sources of error in these estimates, which were analyzed by Preisendorfer [10], should however be kept in mind to ensure that the results obtained reflect the true variability across water parcels.

Conclusions
K d is a valuable tool for monitoring both spatio-temporal changes and long-term trends in the water column. The estimation of K d based on Z SD measurements is a low-cost methodology that can be used systematically in oceanographic campaigns due to its easy application/implementation. However, to ensure consistent results, care must be taken regarding the methodology to measure Z SD . The classical approaches for calculating K d from Z SD , based on a single OD SD value, do not accurately represent the broad range of optical conditions that occur in the marine environment. The adaptive model proposed in this work represents the optical diversity under various conditions, ranging from clear ocean waters to highly turbid coastal waters. It provides continuity across Z SD and, importantly, improves K d retrieval under clear, turbid, and transition scenarios. This model calculates K d considering three ranges of Z SD values, as described by the following equations: The marine optical water classification is a potential approach in studies addressing the relationship between K d (PAR) and optical characteristics. However, our classification of 113 stations with absorption data into Jerlov water types indicated that no unique water type could be related to a given Z SD or K d (PAR), thus making it difficult to use Z SD measurements to infer the optical properties and chlorophyll concentrations of particular water types. This is likely due to the fact that K d (PAR) is the overall result of the variability in both the quantity and type of material (dissolved or particulate matter) present in the water column. In addition, it should be noted that different optical types may occur at the same time in the water column, so that K d (PAR) may result from the combination of different water types. Given this complexity, data derived from satellite observations are promising and this field of research should be further refined in the future.
Supplementary Materials: The following are available online at http://www.mdpi.com/2077-1312/8/8/558/s1. Table S1. Derived K_d from model c using Z_SD ranging from 0.5 to 50 m. Funding: This research was funded by the projects "Influencia de remolinos de mesoescala sobre hábitats de larvas de peces (con énfasis en especies de importancia comercial) en la zona de mínimo de oxígeno del océano pacífico frente a México: océano abierto y efecto de islas" ("Influence of Mesoscale Eddies on the Habitats of Fish Larvae (with Emphasis on Species of Commercial Importance) in the Oxygen Minimum Layer of the Pacific Ocean off Mexico: Open Ocean and Effect of Islands") (SEP-CONACyT 236864), "Fronteras de la Ciencia: Probando paradigmas sobre la expansión de la zona del mínimo de oxígeno: reducción del hábitat vertical del zooplancton y su efecto en el ecosistema pelágico mediante métodos de muestreo autónomos" ("