The vertical spectral diffuse attenuation coefficient of Kd is an important optical property related to the penetration and availability of light underwater, which is of fundamental interest in studies of ocean physics and biology. Models developed in the recent decades were mainly based on theoretical analyses and numerical (radiative transfer) simulations to estimate this property in optically deep waters, thus leaving inadequate knowledge of its variability at multiple depths and wavelengths, covering a wide range of solar incident geometry, in turbid coastal waters. In the present study, a new model is developed to quantify the vertical, spatial and temporal variability of Kd at multiple wavelengths and to quantify its dependence with respect to solar incident geometry under differing sky conditions. Thus, the new model is derived as a function of inherent optical properties (IOPs – absorption a and backscattering bb), solar zenith angle and depth parameters. The model results are rigorously evaluated using time-series and discrete in situ data from clear and turbid coastal waters. The Kd values derived from the new model are found to agree with measured data within the mean relative error 0.02~6.24% and R2 0.94~0.99. By contrast, the existing models have large errors when applied to the same data sets. Statistical results of the new model for the vertical spectral distribution of Kd in clear oceanic waters (for different solar zenith and in-water conditions) are also good when compared to those of the existing models. These results suggest that the new model can provide an improved interpretation about the variation of the vertical spectral diffuse attenuation coefficient of downwelling irradiance, which will have important implications for ocean physics, biogeochemical cycles and underwater applications in both relatively clear and turbid coastal waters.
© 2013 Optical Society of America
Diffuse attenuation coefficient of downwelling irradiance Kd is an important optical property often used in determination of turbidity and classification water types [1, 2], photosynthetic and biological processes in the water column [3, 4] and heat transfer in the upper ocean [5–7]. It is a very useful quantity for many optical analyses of ocean water because of the relative ease in measuring downwelling irradiance Ed and the possibility of measuring Kd remotely [8–10]. It is defined as11]. Thus, Kd has been realized as a descriptor of the underwater light field variation with depth, solar altitude, and time [12, 13].
Kd is an apparent optical property (AOP) that depends on both the medium and geometric structure of the ambient light field. However, as shown in many observations, it is often insensitive to environmental effects except for extreme conditions; therefore it is often regarded as a quasi-inherent optical property depending strongly on changes in IOPs of the water body . This benign behavior of Kd was exploited by Jerlov  to develop a classification scheme for oceanic waters. An approximate formula was devised to describe the relationship between Kd and IOPs.15].
Though Kd is a relatively simple quantity to measure in situ, its relationship to a, scattering coefficient b, and backscattering bb is more complicated because Kd is also influenced by the angular structure of the underwater light field and IOPs of the water column [2, 16, 17]. The simplest form, provided by Gordon , though valid only for the limited sky and clear oceanic conditions, is given below,2, 16]. Note that this relationship is more applicable for clear oceanic waters only, where IOPs are small and vertically homogenous. In turbid coastal waters, scattering typically causes Kd to increase asymptotically and this behavior needs to be accounted for in the model. To understand the nature of Kd, efforts have been made to link Kd with water’s IOPs through Monte Carlo simulations [2, 18] leading to a more complicated form of equation [Eq. (4)],18]. VSF is a property that is seldom measured in the field and cannot be analytically derived especially for turbid waters. Thus, constraining the latter term is more complicated as to make G extremely difficult to parameterize, particularly when the sun is not directly over-head . Thus, Kirk’s approximations of G in calculating the average Kd from the surface to the depth (z1-z2) are not applicable for turbid coastal waters.
Later, Mobley  developed a two flow model providing a simple relationship between Kd and the IOPs,Eq. (1)], Lee et al.  derived a quasi-analytical model based on RTE to estimate Kd as follows,8, 19, 20], which may not account for all water conditions. Since R is typically small in optically deep water, the second term on the right side of Eq. (5) is often ignored leading to the common expression:
More recently, Pan and Zimmerman  developed a model based on the Hydrolight numerical RT simulations to estimate Kd. The calculations were verified with Hydrolight simulations and validated against in situ observations from clear oceanic waters. In order to run the Hydrolight model, one must define the boundary conditions at the surface, the IOPs (i.e., absorption, scattering and scattering phase function) of the water body, and bottom boundary conditions. However, a recent study by Sundarabalan et al.  found that such numerical simulations based on the Hydrolight simulations introduce large errors in the underwater light fields for turbid coastal waters, because of the assumption of constant phase function along the depth and simplified boundary conditions (flat surface and bottom). However, it should be noted that the use of homogeneous water and level sea surfaces are not a limitation of Hydrolight itself, which can handle depth-dependent IOPs and wind-blown sea surfaces as well. To overcome some the above problems, it is necessary to develop a new model to obtain approximately realistic values of the vertical spectral diffuse attenuation coefficient in turbid coastal waters for varying solar zenith angles and in-water optical conditions.
In this study, a new model is developed as a function of angular distribution of incident light and water’s absorption and backscattering coefficients along the depth in relatively clear and turbid waters. The model performance is verified with time series in situ data as well as the in situ data obtained at discrete stations off Point Calimere and Chennai, southeast part of India. These in situ data were collected for different solar zenith angles, sky and water conditions before and during the southwest monsoon. To further demonstrate the efficiency of the new model, its results are also compared with those of the existing models.
2. Data and methods
2.1 In situ data
Several optical parameters were measured in relatively clear and highly turbid waters off Point Calimere and Chennai (13 08.534 N; 80 21.009 E) before and during the southwest monsoon (13–21 May 2012, 15-18 August 2012, and 31 August 2013). Figure 1(a) shows the location map and sampling stations off Point Calimere; this region is shallow (maximum depth 18m) and always dominated by high levels of suspended sediments (resulting from bottom resuspension) caused by tides, alongshore currents and winds [see Figs. 1(c)–1(e)]. In contrast, seawaters off Chennai are relatively clear with very low turbidity [Fig. 1(b)]. The time-series and discrete measurements of various IOPs and AOPs made from several stations around these regions were used to evaluate the performance of the model for predicting the vertical, spatial and temporal variability of Kd for different wavelengths, solar zenith angles and in-water optical conditions. The environmental/ other parameters such as the surface wind speed, cloud condition, and water depth were recorded for each station where the in situ profiling measurements were obtained (Tables 1 and 2).
2.2 Measurement of inherent optical properties
The In situ profiles (vertical) of the inherent optical properties were obtained by several sensors (procured from the WETLAB Inc.), namely AC-S, BB9, FLNTU, and CTD (DH4 used for data collection). These sensors mounted on an underwater frame were lowered with help of a winch system in the ship and a single cable was used for transmitting power to the instruments and data to a rugged laptop computer on the deck. The AC-S instrument was used to measure the absorption coefficient and the beam attenuation coefficient in the entire visible wavelengths (400 to 700). Temperature  and salinity corrections  were applied to measured absorption and attenuation coefficients, and then scattering correction  was applied to the salinity-temperature corrected absorption data. As a result, and were obtained. Here, and denote the total and pure water components of the absorption and attenuation coefficients. Pure water coefficients of absorption and attenuation were added to the resultant data in order to obtain the total absorption and total attenuation coefficients. The total scattering was obtained by subtracting the total attenuation and absorption coefficients. The BB9 sensor was used to measure backscattering at nine wavelengths 412, 440, 488, 510, 532, 595, 650, 676 and 715. The FLNTU sensors were used to measure chlorophyll and turbidity, while the SBE SeaBird CTD sensors were used to obtain the conductivity-temperature-depth profile data.
2.3 Radiometric measurements
The underwater radiometric profiling measurements were made using three RAMSES (Trios) hyperspectral radiometers; i.e., one ARC and two ACC radiometers used to measure upwelling radiance, upwelling irradiance and downwelling irradiance respectively. The irradiance sensor has a built-in pressure sensor which provides the corresponding depth in the water column. All these radiometric quantities were measured in the visible and near-infrared (350–950 nm) wavelengths with spectral accuracy of 3.3nm. These sensors mounted on an underwater frame were deployed in the same way as the WETLAB sensors. The MSDA_XE software was used to record the radiance and irradiance data and export them from the database to a PC on the deck for further processing. Since the radiance sensor was immersed in water, the immersion factors (wavelength-dependent correction factors) from Ohde et al.  were applied to the measured radiance signal (not used in this study). Data from the radiometers and WETLAB sensors were then interpolated to common depth and wavelength for further analysis.
3. Model description
In Eq. (1), Ed (z) needs to be measured within an infinitesimal range of z to know the vertical diffuse attenuation coefficient. It is nearly impossible to accurately determine Kd (z) in the field due to wave introduced fluctuations in the subsurface light field . This obstacle is overcome by calculating the diffuse attenuation coefficient between the irradiances measured over distant depths as described below. The variables z1 and z2 are set far apart to ensure reliable measurements of Ed change. In situ measurements for Kd determination can be carried out with at least two depth values, and hence the average is defined byEq. (8)] in the upper layers of the water column where angular distribution of incident light has its effects .
In the past decades, many remote sensing and in situ methods have been used to determine diffuse attenuation coefficient [9, 18, 26–31]. Efforts have been made to relate Kd with water’s IOPs such as the absorption, scattering and backscattering coefficients. Thus, Kd is simplified as ,33, 34]. As noted earlier, these simple approximations work fairly well for clear oceanic waters but are progressively limited in turbid coastal waters. Thus, it is important to develop a new model of Kd using a, bb and θ for estimating its vertical spectral distributions in relatively clear and turbid coastal waters.
Building on the ideas of Gordon , Kirk [11, 18], Mobley  and Lee et al.  gives rise to the development of a new model for calculating the vertical spectral diffuse attenuation coefficient (Kd) (wavelengths 400-700 nm) for the near-surface, intermediate and bottom layers and for various solar zenith angles and sky conditions. Thus, Kd (λ, θ, z) is modeled based on the concept that AOPs of the ocean are determined by the IOPs of seawater and solar zenith angle (θ) through the radiative transfer equation . If the values of a, bb, θ and sea state conditions are known, Kd (λ, θ, z) values can be calculated by the new model. The model input parameters are derived from in situ measurements, wherein preprocessing steps are used to interpolate and categorize these measured data for our analysis. It includes the computation of solar zenith angles and selection of data sets for different depth intervals. The solar zenith angle for each station is calculated based on the recorded information (year, date, time, longitude, and latitude). The average Kd values between depth layers are then calculated as follows,36] coupled with empirical proportionality constants A1, A2 and A to extend the range of applicability from clear waters to coastal waters [Eq. (13)].Eqs. (14)–(19). The dependency of the newly defined parameters A1 on θ, and A2 on bb/a are shown in Fig. 2 (top panels) A1 decreases gradually with increasing solar zenith angle, while A2 shows a steep decrease with increasing values of bb/a. Dependency of the derived coefficients C1, C2 and C3 on solar zenith angle and depth are shown in Fig. 2 (bottom panels). C1 increases with solar zenith angle and also increases linearly with increasing depth, whereas C2 and C3 almost remain constant at lower solar zenith angles and show an increasing trend at higher solar zenith angles. C2 and C3 decrease with increasing depth. These parameterizations are necessary to take into account some of the complex issues associated with internal sources (i.e., Raman scattering and chlorophyll fluorescence which are difficult to distinguish with radiometric measurements), vertical inhomogeneities and solar incident radiation. Rearranging the above equations [Eqs. (13)–(19)] gives rise to the following model for predicting Kd,Table 2]. The new model takes into account the vertical inhomogeneities in the absorption and backscattering coefficients which usually pose special problems for detailed modeling of the underwater light fields in turbid coastal waters .
4. Model performance assessment
The performance of the new model for deriving the vertical spectral diffuse attenuation coefficient is evaluated using independent in situ data from three cruises (13–21 May 2012, 15-18 August 2012, and 31 August 2013 before and during the southwest monsoon) in highly turbid coastal waters off Point Calimere and relatively clear waters off Chennai. Three time series in situ data and those obtained at discrete stations are used for evaluation of the new model and existing models. The detailed information of these data sets (e.g., a(490), bb(490), [Chl] and turbidity) is given in Tables 1 and 2. To gain insights into the model performance, scatter plots of the modeled Kd values versus in situ Kd values and errors between these data are examined. Two basic statistical measures used in this study are the root mean square error (RMSE) and mean relative error (MRE), which are defined as follows:
5.1 In situ IOPs data
The spectral absorption and backscattering coefficients (a and bb) are the important inherent optical properties often used in the radiative transfer simulations and bio-optical modeling studies to determine the underwater light fields [9, 17, 21, 38]. The magnitude and spectral dependence of these coefficients are not only determined by seawater alone but also by other optically important constituents including phytoplankton, colored dissolved organic matter (CDOM) and non-algal particles. The phytoplankton, non-algal particulates and seawater absorb and scatter light, whereas CDOM is a dominant absorber of light but contributes negligibly to scattering .
5.2 In situ time series IOP profiles along the depth
To demonstrate the performance of the new model and existing models, we use the IOP profile data obtained from a wide range of waters in the coastal ocean as determined by vertical inhomogeneity and stratified conditions. Figure 3 shows the depth-averaged vertical profiles of IOPs (a and bb) for a time series coastal station off Point Calimere before and during the southwest monsoon (for brevity, these profiles are shown only for some key wavelengths 412, 443, 490, 510 and 551nm). In May 2012, a and bb values are significantly high at solar zenith angles −43° ~12° although showing less variation along the depth. When the solar zenith angle ~51°, the absorption coefficients are stable from the top layer to the middle layer and decrease slightly with depth at the bottom layer. The backscattering coefficients show a similar trend at the surface and middle layers, but with an abrupt increase at the bottom layer. The high a and bb values and an increasing trend of bb could be due to benthic resuspension and advection of the organic and inorganic particulate materials caused by the alongshore currents and tides. The time series measurements of a and bb during the southwest monsoon (August 2012) show relatively low a values and very low bb values compared to those of the pre-southwest monsoon. Note that the absorption coefficients are slightly elevated at the surface and bottom layers around −45° and −23°, being reduced noticeably at the intermediate layer. On the other hand, the absorption coefficients decrease with increasing depth at 3° and 42°. The bb profiles show a slight variation (i.e., a decreasing or increasing trend) at −45° and 42° but are more stable along the depth at −23° and 3°.
The vertical profiles of the depth-averaged IOPs from discrete stations obtained before the southwest monsoon (May 2012), display a and bb values increasing steadily with depth [Fig. 4] due to advection of the resuspended materials from coastal waters to offshore waters (as captured at these stations). The increase in bb is due to the increase in the amount of inorganic particles, which scatter light at larger angles than organic particles . During the southwest monsoon, the a and bb values are reduced although fluctuating slightly along the depth. The bb values are significantly lower than those observed before the commencement of southwest monsoon, which may be indicative of relatively clear waters encountered at these stations during this period.
For clear oceanic waters off Chennai, both the a and bb coefficients are inevitably low as compared to those observed in relatively clear and turbid coastal waters [Fig. 5] [Table 2]. These coefficients are mainly dominated by the seawater itself when Chl and turbidity levels are in the range of 0.11~3.69 µg L−1 and 0.01~2.91NTU respectively (slightly high values observed only at the deep layer). Nevertheless, the vertical profiles of these IOPs show significant fluctuations at 4° and 21° or vertically homogenous patterns at 43°. Thus, these IOP profile data can provide a better evaluation of the Kd models for different wavelengths, in-water optical conditions and solar zenith angles.
5.3 Model implementation
The spectra of Kd generated from the new model and existing models were compared with the corresponding data from field measurements collected during three cruises in both clear oceanic waters and highly turbid waters. First, we perform these comparisons for three time-series Kd data (two from highly turbid waters off Point Calimere and one from clear waters off Chennai) for the entire visible wavelengths (400-700nm), three depth layers and different zenith angles [Fig. 6 (where a-c correspond to the surface, intermediate and bottom layers respectively)]. Note that the modeled and in situ spectra Kd(λ,z,θ) match very well in both magnitude and spectral shape in coastal waters with a wide range of turbidity conditions off Point Calimere during July and August 2012. Because of high turbidity [see Table 2] encountered before the commencement of southwest monsoon, the values of Kd(λ) are increasingly high in the short wavelengths (blue), decreasing steadily from the blue to the green, and then increasing gradually from the green to the NIR wavelengths. The down-slope (from the blue to the green wavelength) is generally steeper than the up-slope (from the red to the NIR wavelengths) for these conditions. The increase in Kd(λ) at the blue wavelengths are likely due to the increased attenuation of light by particulate matter (mostly sediments and perhaps dissolved substances) in the water column . There is a slight deviation of the modeled Kd(λ) (overestimation in the blue and / or red/NIR wavelengths) in the intermediate and bottom layers at −43° and surface and bottom layers at 12°, which may be caused by the radiometric/photometric measurements including errors associated with the effects of sea surface waves, tides and ship conditions, and by time and space variations in environmental variables. As a result, these deviations are also likely due to the enhanced contribution of resuspended sediments to the backscattering coefficients at this station. Examination of the time-series data of the southwest monsoon from the same station shows good agreement between the modeled Kd(λ) and in situ Kd(λ) for all depth layers and solar zenith conditions [Fig. 6] (bottom panels). During this period, turbidity level is significantly reduced (0.44~1.94 NTU) but the chlorophyll level is elevated up to 12.6 µg L−1 compared to those obtained before the southwest monsoon (thus reduced bb and nearly the same a values in these moderately turbid waters). Consequently, the diffuse attenuation coefficient is almost equal at the blue and NIR wavelengths, with a pronounced trough at the green wavelengths. This trend of the decreased and increased Kd values in the short wavelengths and NIR wavelengths is attributable to the reduced particle concentrations and increased water content (its absorption thus enhanced the attenuation coefficient at the NIR wavelengths). It is evident that these in situ Kd(λ) spectra for optically/vertically homogenous and inhomogeneous conditions are well reproduced (often virtually indistinguishable from the measured spectra) by the new model. Application of the other two models (Model 1  and Model 2 ) to these time series in situ data from highly turbid waters indicates that their Kd(λ) spectra are very much biased with large errors across the wavelengths for different depth layers. As a result, significant deviations with respect to the measured Kd(λ) spectra are expected with these models.
To see how the models work for different waters, Fig. 7 shows the comparison of modeled Kd versus in situ Kd values (for three different depth layers) from four discrete stations covering relatively clear and moderately turbid waters off Point Calimere during July 2012 and August 2012. In relatively clear waters, the Kd(λ) values are small at the blue and green wavelengths but increasing towards the longer wavelengths as a result of the increased water absorption (e.g., St-1 and St-4). In moderately turbid waters, the Kd(λ) values show a reverse trend as already reported for highly turbid waters at the time series station. For these stations, the new model Kd(λ) values are in remarkable agreement (in spectral shape and magnitude) with the measured Kd(λ) values. Small differences are observed at the longer wavelengths (especially at the intermediate and bottom layers at St-1 and surface and bottom layers at St-4), possibly due to the effect of solar zenith angles (for the near-surface layers) and/ or changes in the water column optical conditions.
It is important to note that these deviations in modeled Kd(λ) values above 600nm for the intermediate and bottom layers may be caused by the existence of internal light sources, the most important of which are Raman scattering and chlorophyll fluorescence whose contributions to Kd variability are found to be minimal in turbid waters [14, 41]. However, the Kd(λ) values produced by the existing models are very much overestimated or underestimated when compared to the measured Kd(λ) spectra in relatively clear and moderately turbid waters off Point Calimere. These results (for the times series and discrete stations) suggest the existing models appear to be insensitive to changes in the angular distribution of light reaching the sea surface and water column optical conditions (including vertical/horizontal variations).
The performance of these models is also evaluated using another set of the time series in situ data obtained in clear oceanic waters off Chennai on 31 August 2013. These waters are different from the previous cases, exhibiting very low turbidity 0.01-2.91 and chlorophyll 0.11~3.69µg L−1 which significantly reduced both the absorption and backscattering coefficients. Figure 8 shows the comparison of modeled Kd(λ,z,θ) and in situ Kd(λ,z,θ) for this time series data set (for limited coverage). These plots depict an increasing dependence of Kd on the solar zenith angle and its resonance structure is typical of the absorption by water molecules from the blue to the red wavelengths [15, 18].
Physically, Kd corresponds to the attenuation of irradiance and is proportional to the path length of photon in water. For pure water, Kd is expected to be inversely proportional to µ [14, 18]. However, in naturally occurring water bodies, scattering is caused by water molecules and suspended particulate matters which cause the underwater light field to become more isotropic and its dependence on illumination geometry is limited. Thus, the correlation between Kd and incident solar zenith angle is less significant within the short wavelength domain (λ<570 nm) [Fig. 8]. It also shows that the Kd values are distinctly low at λ<570 nm and remarkably high in the longer wavelengths (λ>570 nm). An increasing trend in Kd values is due to the absorption effect of seawater. The solar zenith angle has a profound effect on Kd at λ > 600nm, although this correlation between Kd and solar zenith angle diminishes with increasing depth due to more isotropic light field at greater depth. Monte Carlo simulations in such clear oceanic waters showed that Raman scattering produces strong vertical gradients in the average cosine and the diffuse attenuation coefficient of irradiance, especially at longer wavelengths and for low chlorophyll waters . Since many of these varying factors are taken into account with the newly defined calibration coefficients, the new model produces more accurate Kd(λ,z,θ) values than those of the existing models for different wavelengths, solar zenith angles and in-water optical conditions [Fig. 8].
To further illustrate the differences between model and in situ Kd values over a range of solar zenith angles, these two data sets (only for some key wavelengths) are plotted as a function of solar zenith angles for the time series measurements in coastal waters off Point Calimere (before and during the southwest monsoon, May 2012 and August 2012 respectively) [Fig. 9]. These plots not only show good agreement between modeled (new model) and in situ Kd values but also depict their variation for different solar zenith angles and varying in-water optical conditions. Angular dependencies of Kd at seven wavelengths are clearly evident in these plots; i.e., the Kd values are consistently high in the upper layers (May 2012) for medium and high solar positions (e.g., θ = −30~0°), while these features are suppressed or becoming independent of solar zenith angles at the bottom layers. These high Kd values apparently coincide with high levels of suspended sediments that were seen advected by the alongshore currents and tidal waters in May 2012 [see Figs. 1(d) and 1(e)]. This means that Kd tends to be less dependent on the solar zenith angles owing to the increasing of scattering effects of these suspended particulate (inorganic) materials in the water column (dominated by the advection process). To our knowledge, this is the first time to show the dependency of Kd on solar zenith angles and in-water (highly turbid) optical conditions with experiments and modeling results.
On the other hand, the new model reproduces measured Kd for relatively clear and moderately turbid waters sampled during August 2012. Note that the Kd values show similar angular patterns in the near-surface layer but are reversed in the intermediate and bottom layers, thus not following the simple Kd ∝ 1/μ relationship. Kd tends to become low (with increasing wavelengths λ<555nm) at the intermediate and bottom layers when the solar zenith angle is within −30°~30°, which may be caused by the prevailing low particle conditions during these time periods. However, the Kd(670) values are slightly elevated at the intermediate and bottom layers likely due to the presence of more detrital/inorganic particles. These variations in Kd for different solar and in-water optical conditions confirm the previous hypothesis that this property is a quasi-continuous function of the solar zenith angle and water’s IOPs in relatively clear and moderately turbid oceanic waters . Unfortunately, other two models are unable to provide accurate Kd values that exhibit nearly same patterns (variations) regardless of the solar zenith angles and in-water optical conditions.
Figures 10 and 11 show the variations of the model and in situ Kd for different wavelengths and depth layers at the time-series station [Fig. 10] and discrete stations [Fig. 11] off Point Calimere before and during the southwest monsoon (May 2012 and August 2012). At many stations, the Kd values derived from the new model are consistent with the in situ Kd values, and both these Kd profiles tend to increase with increasing depth and become more prominent at the bottom layers due to the contribution of scattering. A slight deviation of the modeled Kd at this layer may be attributed to inelastic scattering which is not explicitly determined and included in our study (because the internal sources caused by Raman scattering and chlorophyll fluorescence cannot be discriminated from each other by radiometric measurements). In fact, a fraction of this deviation may also be caused by the additional unqualified error caused by small-scale vertical and horizontal variability in the water column optical properties (considering the timing of the deployment of WETLAB sensors and Trios sensors, with a deployment internal of 10 minutes apart) in the dynamic coastal ocean.
Nevertheless, the observed variability in Kd is associated with the IOP variability attributed to tides, alongshore currents and winds and changes in incident solar radiation. Under these prevailing conditions, the Kd profiles obtained with the existing models are not generally agreeable with the in situ Kd profiles from both relatively clear and highly turbid coastal waters (For brevity the profile comparisons not shown for clear waters off Chennai).
To better illustrate the performance of the new model and existing models, Fig. 12 shows the modeled and in situ Kd data at some key wavelengths from three cruises (including data from three depth layers and for different solar zenith conditions). The new model has nearly identical results compared to the in situ data. Further, the error plots confirm that the important feature of the new model is its stability whatever the wavelengths being considered [Fig. 13]. In particular, the RMSE, MRE, Bias and Intercept values associated with the new model are apparently closer to zero and slope and R2 values are closer to unity. These low errors and high slope and R2 values indicate that there is good agreement between the model and measurement data. Not surprisingly, the existing models produce inaccurate Kd data that are highly scattered (overestimation or underestimation) above and below the 1:1 line with respect to the measured data. As a result, their errors are not only very high but fluctuating across the wavelengths (indicating poor performance of these models). These results suggest that the new model is very promising in reproducing measured Kd values across the entire visible wavelengths and for different solar zenith and in-water optical conditions in both clear and highly turbid coastal waters.
6. Relationship between Kd(360) and Kd(490) for ocean optics and remote sensing
Many studies have been carried out in the past decades to acquire knowledge of ultra-violet (UV) radiation penetration which is important for the study phytoplankton dynamics [43, 44], quantifying the photoreaction rates of DOM in the water column , understanding coral beaching in shallow waters. Past studies derived simple relationships by relating CDOM absorption or DOC (dissolved organic carbon) to Kd, which take into account the contributions of the absorption coefficient to Kd but eventually ignore the contributions of the backscattering coefficient. Experimental and modeling studies have shown that the magnitude of backscattering for relatively clear and turbid coastal waters (particle-rich waters) could be much larger in the short wavelength region .
To quantify and understand the penetration of the light, the depth (Z(360)10%) corresponding to 10% of its surface values  is calculated from Kd(360) (i.e., ). If one calculates Kd(360) analytically, the a and bb coefficients (at 360nm) should be derived from Rrs-IOPs inversion methods, where Rrs(360-400) is usually measured in situ since the present satellite sensors do not provide this quantity directly. To overcome this limitation, Kd(360) is related to Kd(490) by the following relationship derived from Fig. 14.10]. Note that we use the wavelength 490 for the estimation of Kd(360), since our experimental and modeling study shows that the impact of solar zenith and water’s IOPs on estimating Kd(490) is minimum and its estimation accuracy is significantly higher at this wavelength . Lee et al.  provided a similar relationship for a lower (restricted) range of Kd(360) values (0.02~0.08 m−1) that is applicable only for clear oceanic waters. The proposed method [Eq. (23)] would be useful for satellite application studies as well, to know the UV radiation availability in the water column and their spatial and temporal distributions in global coastal oceanic waters.
7. Discussion and conclusions
Several empirical and semianalytical models are often used to derive Kd based on satellite measurements and IOPs measured in situ or stimulated [17, 48–53]. Kd is a significant parameter for precise assessment of the light intensity at depth  and classifying water types . For the vast ocean, satellite remote sensing plays an important role to obtain repetitive and fine-scale measurements of Kd. However, many algorithms which depend on the blue-green ratio of radiances or reflectances were found to overestimate or underestimate Kd(490) values in both clear and turbid waters where the backscattering caused by suspended matters and the absorption by dissolved organic matters increase light attenuation in the water column. Many empirical relationships are used to estimate values of Kd from remotely sensed data by relating Kd and the spectral ratio of water leaving-radiance at two wavelengths [26, 27]. However, these algorithms suffer from large uncertainties in the derived products , and are insufficient to enhance our present understanding of the vertical spectral distributions of Kd in turbid coastal waters [17, 50]. Many of these models are generally applicable for waters with low particle concentrations. Recently, Wang et al.  developed a new Kd model by relating the backscattering coefficient at the wavelength 490nm to the irradiance reflectance just beneath the surface at the red wavelength. Though this Kd model is applicable for turbid waters, it is merged with the standard models so as to find its applicability in both clear and turbid waters.
Many semianalytical approaches based on radiative transfer simulations have also been developed that provide the relationship between Kd(z) and IOPs [2, 8, 9, 15–18, 40, 55]. While the empirical algorithms suffer from large uncertainty and are insufficient to provide a better understanding of the spatial and temporal variability of Kd, these semianalytical approaches overcome such limitations by improving the accuracy of Kd but are valid only for limited sky and oceanic conditions and are generally applicable to clear and moderately turbid waters. Lee et al.  developed a Kd model for vertically homogenous water based on the radiative transfer modeling results, where the model parameters are derived from Hydrolight simulations using the particle phase function under clear sky conditions and wind speed of 5 m s−1. It means that further refinement is required for cloudy conditions and high wind speed conditions. Parameterizations of the modeled values are obtained from Look-up tables, which are not appropriate for all water conditions. However, challenges remain to extend the models to vertically stratified waters. On the other hand, the formula [Eq. (4)] proposed by Kirk  underestimates Kd(z) and finds difficulty to parameterize the appropriate coefficients associated with the scattering phase function. As the field data used were collected from waters with medium particles (Chl <3mg m−3) and medium solar position (θ <60°), approximation of parameter values was done by calculating average Kd from surface to depth receiving 1% and 10% of surface irradiance. Such approximations fail especially in turbid waters.
Based on the asymptotic closure theory [29, 56, 57], Kd at relatively high solar position is derived. These models may become rapidly invalid when solar elevation is low. However, a more complicated asymptotic closure model was developed by McCormick , but deriving the parameters from IOPs and solar elevation remains extremely difficult. Although the Eq. (7) has been widely accepted and applied to ocean optics, it fails (underestimates Kd) in particle rich waters because of the dominance of scattering effect. It works better for upper layer than for lower layer. Though several models are available to estimate Kd, none of these models provide Kd values over the entire visible spectral bands. Estimation of Kd over the entire wavelength is necessary for many applications – including classification of waters, classification of algal blooms, estimation of light intensity at depth, the study of heat budget [6, 7, 59, 60] and photosynthesis [3, 34, 61].
To overcome many of these limitations, a new diffuse attenuation coefficient model Kd (z,λ,θ) has been developed and systematically evaluated using the in situ data obtained from turbid waters off Point Calimere (where the water properties are highly dominated by high levels of suspended sediments, detrital particles and phytoplankton) and relatively clear waters off Chennai. The new model has been derived based on the absorption and backscattering coefficients for the entire visible wavelength range (400-700nm), different depth layers (surface, intermediate and bottom layers) and varying zenith angles. On the basis of extensive analysis of field data, the values of the model parameters have been derived by fitting the data with the model. The model parameters vary with solar zenith angle, depth and bb/a. The results of the new model have been compared with the in situ Kd profiles for different depths, solar zenith angles and wavelengths. It is found that the new Kd model is valid for both relatively clear and turbid coastal waters. Its applicability has also been assessed in waters with low, medium and particle concentrations (where Chl ranges from ~0.1 to 13 µg L−1) and for varying solar zenith angles (morning to evening). The new model is found to be valid in these waters characterized by the overall range of a (0.12m−1 ~1.118m−1), bb (0.001 m−1 ~0.3 m−1) and turbidity (~0.44-16 NTU). Compared to the existing models whose results are highly fluctuating (overestimation or underestimation) for different wavelengths, depth layers, water types and solar zenith conditions, the model has definitely improved the accuracy of Kd in both relatively clear and turbid coastal waters (percent MRE between the model and in situ measurements being less than 13.34%, other errors being very small and R2 always >0.94). Since the new Kd model yields accurate Kd values for the entire visible spectrum (400-700 nm) and different depth layers in both clear and turbid waters, it can be applied to a wide range of coastal oceanic waters.
A thorough evaluation of the results demonstrates that the new model has higher capability in both relatively clear and highly turbid waters. It provides a rapid method to obtain the vertical spectral distributions of Kd from IOPs measurements with significantly higher accuracy, thereby offering improvement over other models in both relatively clear and highly turbid waters within the coastal oceanic domain. It may be extended for satellite applications, but its accuracy depends significantly on the inverse retrievals of IOPs from Rrs by empirical or semianalytical algorithms. The new model will have important implications for understanding ocean physics, biogeochemical cycles, and underwater applications in highly dynamic coastal oceanic waters.
This work was supported by grants from the NRB (Project Number OEC1112106NRBXPSHA). This work was also supported by IIT Madras, Chennai-600036. We would like to thank D. Rajasekhar, The Head, Vessel Management Cell (VMC), and Director of National Institute of Ocean Technology (NIOT) for providing the Sagar Purvi and Sagar Pachimi (Coastal Research Vessels) to Indian Institute of Technology (IIT) Madras, Chennai, India for making various bio-optical and underwater light field measurements. We thank Scientists N. Ravi and K Shashikumar for arranging the vessels on time and other members of VMC for their help during the in situ measurements. We are thankful to the anonymous reviewers for the valuable comments, which helped to improve the quality of the manuscript.
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