1. Introduction

The propagation of light in natural waters is governed by their inherent optical properties (IOPs). Two basic IOPs that characterize the attenuation of a light beam through absorption and scattering processes at a particular light wavelength, λ (which is generally reported in vacuum in units of nm), are the total spectral absorption coefficient of seawater, a(λ) (m⁻¹), and the spectral volume scattering function, β(ψ, λ) (m⁻¹ sr⁻¹) where ψ is the scattering angle [1]. The total absorption coefficient a(λ) is typically divided into three or four additive components, i.e., absorption by pure water, a_w(λ), absorption by chromophoric dissolved organic matter (CDOM), a_g(λ), and absorption by suspended particles, a_p(λ), in which the latter coefficient can be further partitioned into contributions by phytoplankton, a_ph(λ), and non-algal particles (NAP), a_d(λ). Given the methodology of measuring a_d(λ) following extraction and removal from the sample of pigments that are contained primarily in phytoplankton [2], it is worthwhile to recall that the NAP component can be thought of as representing the absorption by depigmented particulate matter. Although the values of a_w(λ) are quite well quantified from the near-ultraviolet (near-UV) through the visible (VIS) and into the infrared (IR) region of the electromagnetic spectrum [3,4], the particulate and dissolved components of seawater are highly variable and drive the large variability of a(λ) in the ocean. The two absorption components, a_g(λ) and a_d(λ), exhibit similarities in the spectral shapes with a general tendency to increase with decreasing wavelength, and are often summed together as a_dg(λ) = a_g(λ) + a_d(λ). All three of these absorption coefficients have been often approximated as an exponentially decaying function of light wavelength from the UV through the VIS as defined by:

A result of the nearly exponential increase of the CDOM and NAP absorption coefficients with decreasing wavelength is that they contribute significantly to the non-water component of absorption, a_nw(λ) ≡ a_pg(λ) = a(λ) – a_w(λ), in the UV and blue spectral regions [6,9,11]. The spectral characteristics of these absorption coefficients are often employed to understand physical and biogeochemical processes in the ocean. Specifically, CDOM absorption properties in the UV and blue spectral regions can be utilized to trace the transport of dissolved organic carbon [12,13], as well as monitor changes and identify the evolution of water masses in aquatic environments [14–18]. Furthermore, information about the characteristics of non-algal particulate matter, including measurements of a_d(λ), has provided substantial insights about processes such as the remineralization of carbon and the vertical flux of particulate matter [19–22]. Another consequence directly resulting from the spectral characteristics of CDOM and NAP absorption is that variations in their properties, such as their concentration and chemical composition, have the potential to dramatically impact UV and VIS radiation fields in marine environments. Furthermore, exposure of CDOM itself to UV leads to its photodegradation over time and further alters the light environment [23–26]. Variations in the UV radiation field can impact various photochemical reactions in water [25,27,28], and an increase in exposure of marine organisms to UV radiation is potentially harmful [29–32]. As a result, changes in CDOM or NAP concentrations that alter the UV radiation environment directly impact the structure of vertical distributions and migration patterns of organisms in marine ecosystems [33].

A specific objective of ocean color satellite missions is the derivation of absorption coefficients of seawater constituents in support of characterization of the concentration and composition of particulate and dissolved matter as well as light propagation in the upper ocean layer. A variety of inverse ocean color models relying on measurements of ocean remote-sensing reflectance R_rs(λ) in the VIS spectral bands have been developed over recent decades to aid in pursuing this objective [34]. One important class of semi-analytical inverse models aims at deriving the total IOPs of seawater from ocean reflectance measurements [35–42]. These models can be applied in conjunction with stand-alone independently-developed models for partitioning the total absorption coefficient, a(λ), or its non-water component, a_nw(λ), into the constituent absorption coefficients. Thus, in this approach the derivation of constituent absorption coefficients is based on the sequential application of multiple algorithms operating in a stepwise fashion. There are four main types of absorption partitioning models: first, models that partition a_nw(λ) into the three constituent absorption coefficients, a_ph(λ), a_d(λ), and a_g(λ) [43–46], second, models that partition a_nw(λ) into a_ph(λ) and a_dg(λ) [47,48], third, models that partition a_dg(λ) into a_g(λ) and a_d(λ) [46,49], and finally, models that partition a_p(λ) into a_ph(λ) and a_d(λ) [50–54]. Most partitioning models utilize assumptions about model outputs, most notably the spectral shapes of constituent absorption coefficients, which can substantially limit the model’s applicability and performance across diverse aquatic environments. To alleviate these limitations some absorption partitioning models have been designed with a purpose to avoid or include only weakly restrictive assumptions about the spectral shapes of output absorption coefficients [46,47,49,54]. With regard to the spectral domain of applicability, the existing absorption partitioning models have been, however, limited to the VIS range.

Recently, studies have also been conducted with the aim to estimate a_g(λ) in the UV either directly from measurements of R_rs(λ) in the VIS using a multivariate statistical approach [55], or from a single-wavelength estimate of a_dg(443) derived from a semi-analytical inverse reflectance model [35,38], which also operates in the VIS range [56]. The estimation of a_dg(λ) and a_ph(λ) in the near-UV at a single wavelength of 380 nm was also explored with a modified semi-analytical inverse reflectance model [37], which requires the input of R_rs(λ) at 380 nm rather than R_rs(λ) just from the VIS range [57]. While these studies reflect a growing interest to estimate the constituent absorption coefficients from optical measurements in the VIS range, they also reveal a need for further enhancement of these capabilities. In particular, there is a need to address the estimation of both non-phytoplankton components of absorption, i.e., the a_d(λ) and a_g(λ) coefficients, in the UV from measurements in the VIS.

Given the demonstrated potential of absorption partitioning models to separate a_g(λ) and a_d(λ) components from a_dg(λ) in the VIS [46,49], the goal of this study is to determine the optimal extrapolation method to extend these spectra from the VIS to the near-UV. The portion of near-UV that is considered in this study covers the wavelength range of 350 to 400 nm. Our focus is on this near-UV range rather than a wider range including shorter UV wavelengths primarily because of limitations and increased uncertainties with decreasing UV wavelength in the current-state-of-the-art methodology for measuring the spectral particulate absorption coefficient a_p(λ) including its a_d(λ) and a_ph(λ) components [58,59,60]. In addition, the new Ocean Color Instrument (OCI) to be deployed on the upcoming NASA Plankton, Aerosol, Cloud, ocean Ecosystem (PACE) satellite mission will have the capabilities to make measurements of R_rs(λ) in this near-UV portion of the spectrum [61]. At present, however, the capabilities and performance of inverse optical models that utilize near-UV measurements, in particular the inverse reflectance and absorption partitioning models, are not yet proven. Therefore, while the extrapolation model presented in this study is expected to have an immediate applicability to measurements and optical models operating in the VIS, it can also provide a valuable tool for comparison and assessment of forthcoming optical models that utilize near-UV measurements.

For formulation of the extrapolation model, we assembled a development dataset of field measurements of a_g(λ) and a_d(λ), and hence their sum a_dg(λ), from a diverse range of marine optical environments. We also assembled an independent dataset of field measurements of a_g(λ) and a_d(λ) for the purpose of model validation. The absorption coefficients in both these datasets were collected over the spectral range from the UV through near-IR with high spectral sampling interval and they satisfied a set of method-related and data quality criteria. One of the key method-related criteria used in this study is the inclusion of a_d(λ) data that were obtained solely from spectrophotometric measurements with the inside integrating-sphere filter-pad technique, which reduces the risk of biased measurements, including the biased spectral shape, compared to more traditional transmittance filter-pad technique [59,60]. Several approaches to extrapolation into the near-UV based on absorption data in the VIS were tested, and the formulation, performance, and validation of the optimal model are described.

2. Methods

2.1 Assembly of the development dataset of absorption coefficients

A development dataset of field measurements containing spectra of constituent absorption coefficients was assembled for the formulation and evaluation of different extrapolation methods to extend a_g(λ), a_d(λ), and a_dg(λ) from the VIS into the near-UV. The outcomes of this study aim to extend existing models to the UV, which commonly work to derive absorption coefficients from satellite data, so the assembled dataset includes only near-surface water samples. Surface data are defined as a sampling depth ≤ 5 m if the water depth is < 200 m or a sampling depth ≤ 15 m if the water depth is ≥ 200 m.

Multiple data sources were initially considered, but only those which contain data that pass basic methodology-related inclusion criteria were used to create a preliminary dataset for further quality control and processing. All measurements of constituent absorption coefficients considered in this study were made on discrete water samples using state-of-the-art techniques providing data at high spectral sampling intervals over a wide spectral region. First, measurements of spectral absorption coefficients a_g(λ) and a_p(λ) including the non-algal particulate component a_d(λ) were required to span the UV-VIS range, at a minimum from 350 to 700 nm for a_g(λ) and 350 to 800 nm for a_p(λ) and a_d(λ), with a 1 nm sampling interval. Second, the spectra of a_g(λ) were measured either with the spectrophotometric method using a 10-cm path length cuvette or a long path length liquid waveguide capillary cell following standard sampling and measurement protocols [62,63]. Finally, a methodological prerequisite for inclusion of a_p(λ) and a_d(λ) data was that they were obtained with the spectrophotometric filter-pad method using the inside integrating-sphere configuration of measurement, which provides high-quality results owing largely to negligible or very small artifacts caused by light scattering during measurements of both sample filters and blank filters [59,60,64,65]. The a_d(λ) spectra were measured on the same sample filters as the a_p(λ) spectra after subjecting the filters to methanol treatment which extracts pigments present mostly in phytoplankton [2]. It must be noted that most historical data of a_p(λ) and a_d(λ) were measured with the filter-pad method using a transmittance configuration of measurement, which can result in significant error in both the magnitude and spectral shape of measured absorption [59,60]. Thus, measurements with the transmittance method were not included in this study.

The preliminary dataset of absorption coefficients that passed these basic methodology-related inclusion criteria consisted of 1610 unique water samples from 18 oceanographic experiments including 42 distinct cruises that collected 1493 spectra of a_g(λ) and 722 spectra of a_p(λ) and a_d(λ). We note that this dataset also includes the spectra of a_ph(λ) derived as a difference between the measured a_p(λ) and a_d(λ), however, our interest in this study is focused on the a_d(λ) component of particulate absorption. Additional processing was applied to the a_g(λ) spectra within the preliminary dataset. Most a_g(λ) spectra within the preliminary dataset had low signal-to-noise ratios in the green-to-red spectral region and were corrected by extrapolating the measurements with higher signal in the short-wavelength portion of the VIS spectrum to longer wavelengths. The extrapolation to the long-wavelength portion of the spectrum utilized a linear fit to log_e-transformed a_g(λ) values versus light wavelength within the 400 nm to λ_cutoff range where λ_cutoff is the cutoff wavelength at which a_g(λ_cutoff) drops to instrumental noise level. This level was assumed to be 0.03 m⁻¹ for the 10-cm cuvette spectrophotometric method and 0.005 m⁻¹ for the liquid waveguide capillary cell and is consistent with previously reported values [56,63]. The long-wavelength extrapolations were generally applied at wavelengths greater than 500 nm. If the noise level for a given sampling method occurred below 440 nm the measured spectrum of a_g(λ) was excluded from our final dataset.

Non-algal particulate absorption spectra available in the preliminary dataset were also subjected to additional processing. To remove instrumental noise, all a_d(λ) spectra were smoothed across the entire spectral range with a 7 nm moving median window and then three consecutive 7 nm moving mean windows. This method of smoothing was found adequate to smooth out the potential instrumental noise present in the measured spectra without affecting the shape of the spectra which generally does not exhibit any sharp spectral features. Additionally, it was assumed the phytoplankton contribution to a_p(λ) is zero in the near-IR region. According to this assumption, a spectrally constant offset was applied to measured a_d(λ) to ensure that the resultant a_d(λ) is equal to measured a_p(λ) in the near-IR. Specifically, a_p(λ) measurements were first smoothed from 720 nm to the long-wavelength end of the spectrum by applying the same smoothing method as applied to a_d(λ) data. The difference between the average a_p(λ) and average a_d(λ) within the 780–820 nm range, or from 780 nm to the last available wavelength if data were not available up to 820 nm, was determined and added back to the entire a_d(λ) spectrum to correct for any offset between the original measurements of a_p(λ) and a_d(λ) in the near-IR. After processing a_d(λ), it was evident that some spectra contained absorption features in spectral regions corresponding to phytoplankton pigments resulting from an incomplete pigment extraction process. To address this potential issue, all a_d(λ) spectra were analyzed to quantify the height of the residual absorption peak in the spectral region between 650 and 700 nm by applying an exponential fit to a_d(λ) values at two discrete wavelengths of 650 and 700 nm. This curve was treated as the expected portion of the a_d(λ) spectrum in the 650–700 nm range which is unaffected by potentially incomplete extraction of phytoplankton pigments. If the maximum value of measured a_d(λ) between 650 and 700 nm was greater than 8% above its expected fit value, the a_d(λ) spectrum was removed from further analysis and excluded from the final dataset. It is described below that a necessary condition of our extrapolation model is the input of an a_dg(λ) spectrum, so from the final dataset we also omit a_d(λ) without a matching concurrent measurement of a_g(λ).

A final quality control was then applied to the remaining a_g(λ) and a_d(λ) by inspecting each individual spectrum visually to identify spectra that were characterized by the presence of clearly erroneous features in the UV-VIS range. Typical visual manifestation of erroneous features included an unrealistic spectral shape and a low signal-to-noise ratio. The spectra exhibiting such erroneous features were not included in the final dataset. The entire data quality control process, which consisted of the application of initial basic methodology-related criteria and several additional inclusion and exclusion criteria, allowed us to greatly minimize the risk of including measurements of a_g(λ) and a_d(λ) that are subject to potentially significant errors in the final dataset.

The final model development dataset of a_g(λ), a_d(λ), and a_dg(λ) consists of 1294 unique a_g(λ) and 409 a_d(λ) spectra with concurrent a_g(λ) measurements as described in Table 1. The subset of data with concurrent measurements of a_d(λ) and a_g(λ) also includes the a_dg(λ) spectra obtained as a sum of a_d(λ) and a_g(λ). Table 1 also provides a list of field experiments and cruises where these data were collected.

Table 1. Final Model Development Dataset of Absorption Coefficients, a_g(λ) and a_d(λ).^a

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Figure 1 depicts the geographic locations of all oceanographic stations where samples of absorption measurements included within the final development dataset were collected. This final dataset represents a diverse set of marine environments ranging from ultraoligotrophic waters found across the subtropical gyre within the southeastern Pacific collected on the BIOSOPE cruise to turbid waters in coastal and Arctic regions. Most samples with concurrent measurements of a_g(λ) and a_d(λ) come from polar and coastal regions. The only cruise in open ocean waters where concurrent measurements of a_g(λ) and a_d(λ) were made and met all inclusion criteria of our final dataset is the CLIVAR P16S cruise in the southern Pacific.

 

Fig. 1. Geographic locations of oceanographic stations where near-surface measurements of the spectral absorption coefficients were collected for generation of the final model development dataset utilized in this study. Colored markers represent measurements of a_g(λ) (number of measurements N = 1294) with the marker color corresponding to ocean basins as indicated in the legend, with the Arctic Ocean defined in this study as north of 60°N and the Southern Ocean defined as south of 60°S. Black dots in the center of markers indicate samples with concurrent measurements of a_g(λ) and a_d(λ) (N = 409).

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The final absorption spectra of a_g(λ), a_d(λ), and a_dg(λ) as well as their corresponding histograms at λ = 443 nm are displayed in Fig. 2. The shapes of a_g(λ) spectra are generally close to linear behavior when depicted on the semi-logarithmic scale (Fig. 2(a)) which is consistent with the nearly exponential shape. The average exponential slope parameter describing the decrease of a_g(λ) with increasing wavelength calculated over the spectral range 350–500 nm for the entire final dataset is 0.0157 nm⁻¹ (standard deviation SD = 0.0023 nm⁻¹) which is in general agreement with the reported values in the literature [9]. Compared to a_g(λ), the spectra of a_d(λ) (Fig. 2(b)), and also a_dg(λ) (Fig. 2(c)), exhibit less linear behavior on the semi-logarithmic scale as their spectral slopes tend to decrease with increasing wavelength. The average spectral slope of a_d(λ) spectra for the entire final dataset is 0.0076 nm⁻¹ (SD = 0.0013 nm⁻¹), which was calculated over the spectral range 380–730 nm, excluding 400–480 nm and 620–710 nm ranges to remove the potential methodological artifact associated with any remaining residual absorption by phytoplankton pigments and to maintain methodological consistency with previous work [66]. The average slope of a_d(λ) is smaller than values reported in other studies, for example 0.011 nm⁻¹ [67,68], 0.0123 nm⁻¹ [9], and 0.0094 nm⁻¹ [66]. Most previous measurements of a_p(λ) and a_d(λ) utilized the spectrophotometric filter-pad method in the transmittance configuration of measurement [66–68] and some measurements were also made in the transmittance-reflectance configuration [9]. The transmittance method typically utilizes a null-point correction which assumes no particulate absorption in the near-IR. Following this assumption, the null-point correction typically consists of subtracting the average value of measured a_d(λ) within some portion of the near-IR from all spectral values of a_d(λ). We note that similar null-point correction is applied to measurements of a_p(λ) although this coefficient is not of direct interest to the present study. Importantly, it can be shown that the wavelength-independent null-point correction leads to an artificially increased spectral slope of a_p(λ) and a_d(λ) compared to the a_p(λ) and a_d(λ) that exhibit some real (non-zero) absorption in the near-IR. The inside integrating-sphere spectrophotometric configuration of filter-pad method allows to measure and retain the non-zero values of a_p(λ) and a_d(λ) in the near-IR if the examined sample exhibits such absorption feature, and the null-point correction is not applied. Because the particulate absorption measurements in our dataset were all made with the superior methodology of the inside integrating-sphere spectrophotometric configuration, the differences in the average spectral slope of a_d(λ) between our dataset and previous studies appear to be related to an overestimation of this parameter due to null-point correction that was commonly used in the previous studies.

 

Fig. 2. Spectral absorption coefficients (a) a_g(λ), (b) a_dg(λ), and (c) a_d(λ) available in the final model development dataset. Blue lines are data from the Pacific Ocean, green lines are data from the Atlantic Ocean, red lines are data from the Indian Ocean, yellow lines are data from the Arctic Ocean (defined in this study as north of 60°N), and lavender lines are data from the Southern Ocean (defined in this study as south of 60°S). (d)–(f) Histograms of each absorption coefficient at λ = 443 nm are color coded in the same manner as (a)–(c).

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The a_g(λ) spectra in the final dataset span approximately three orders of magnitude at 350 nm and this dynamic range remains relatively the same across the spectrum to the wavelength of 700 nm. The a_g(443) histograms (Fig. 2(d)) display a bimodal distribution that is associated with the large number of samples collected in either the Pacific or Atlantic Ocean basins. The samples collected from the Pacific Ocean typically have smaller magnitudes of a_g(443), whereas a_g(443) data from the Atlantic Ocean more often originate from coastal waters and have larger a_g(443) values. Samples of a_d(λ) spectra span approximately four orders of magnitude at 350 nm and this dynamic range remains consistent across the spectrum. In contrast to a_g(443), the histograms for both a_d(443) and a_dg(443) do not exhibit a clear bimodal shape (Fig. 2(e),(f)), and the majority of data come from experiments conducted in coastal Atlantic waters.

The formulation of the extrapolation model presented hereafter implements the complete development dataset of available near-surface water samples. This dataset was not partitioned into two independent subsets of data for separate purposes of model formulation and validation because we did not want to compromise on the representativeness of the model development dataset across the entire dynamic ranges of a_g(λ), a_d(λ), and a_dg(λ). In particular, partitioning of the development dataset into two subsets of data would result not only in reduction of total amount of data in the development dataset but, more importantly, would reduce the amount of data in portions of dynamic range where there are relatively few data, most notably in the ranges of relatively low and high values of a_d(λ) and a_dg(λ) (Fig. 2(e),(f)). This could adversely impact the formulation of the extrapolation model and its performance over the entire investigated dynamic range of absorption coefficients. Therefore, for the purpose of validation of the extrapolation model with independent data we compiled a separate dataset as described below.

2.2 Assembly of an independent validation dataset of absorption coefficients

An independent dataset of measured absorption spectra was compiled to validate the extrapolation model. The same sources of data used to create the model development dataset were employed in the assembly of the independent validation dataset. The validation dataset includes sub-surface measurements on samples collected between depths of 15 and 50 m, which satisfied the same method-related and data quality criteria as the model development dataset described in Section 2.1. This approach to validation with sub-surface (but not very deep) data is reasonable and appears as the only approach available at the present time which satisfies the method-related inclusion criteria used in the development dataset, especially in view of the overall limited availability of particulate absorption measurements with the inside integrating-sphere filter-pad technique. In the future, as more data with prerequisite methodology are collected, the validation analysis can be extended by including near-surface data. The model formulation itself can also be refined in the future with more data, if deemed appropriate.

The assembled validation dataset consists of 149 concurrent measurements of a_g(λ) and a_d(λ) from a variety of marine environments. We inspected all spectra individually within the validation dataset and found the general behavior and shapes of the spectra are consistent with those in the development dataset. However, we also observed that the validation dataset exhibits a narrower dynamic range of magnitudes of all three absorption coefficients compared to the development dataset. Nevertheless, given that the dynamic ranges overlap significantly between the validation and development datasets including the range of relatively low absorption magnitudes, the validation dataset provides a suitable basis for assessing the performance of the extrapolation model with independent data.

2.3 Methods for extrapolating absorption coefficients from the VIS into the UV

Although the spectral shape of non-phytoplankton constituent absorption coefficients, a_g(λ), a_d(λ), and a_dg(λ), have often been approximated with an exponential function of light wavelength, there is experimental evidence that the spectral shapes generally do not maintain a fixed exponential slope across the broad range of wavelengths from the UV through the VIS spectral region [9,69–73]. To address this challenge, functions other than a single exponential function of wavelength were examined in the past as potential descriptors of a_g(λ) and a_dg(λ) spectra, such as power function, double exponential function, and stretched exponential function [69,73].

In this study, five different extrapolation functions were investigated to extend a_g(λ), a_d(λ), and a_dg(λ) spectra from the VIS into the near-UV region between 350 and 400 nm: (i) an exponential function calculated by linear regression applied to log_e-transformed absorption data vs. wavelength (ELR), (ii) an exponential function calculated by non-linear regression to absorption data vs. wavelength (ENLR), (iii) a stretched exponential function calculated by non-linear regression to absorption data vs. wavelength (SENLR), (iv) a power function calculated by linear regression to log₁₀-transformed absorption data vs. log₁₀-transformed wavelength (PLR), and (v) a power function calculated by non-linear regression to absorption data vs. wavelength (PNLR). Table 2 summarizes the different regression models.

Table 2. Regression Models Evaluated for Extrapolation^a

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We examined the blue spectral range from 400 to 500 nm as a basis for fitting the regression models to absorption measurements in the VIS region. This choice appears justifiable as the spectral slope of non-phytoplankton absorption coefficients in the near-UV is most similar to the spectral slope in the blue spectral region, for example for a_g(λ) [72]. Multiple spectral windows with variable width from within the 400–500 nm range were tested. For each tested spectral window, we fixed the shortest (start) wavelength at 400 nm to ensure that measured data nearest the extrapolation UV region were always included in the formulation of each regression. The longest (stop) wavelength, λ_stop, of the spectral windows was allowed to vary in 1 nm increments from 405 nm (i.e., the narrowest window of 400–405 nm) to 500 nm (i.e., the widest window of 400–500 nm). We did not examine the regression windows that extend beyond 500 nm because such broader spectral windows are expected to have limited capability to predict the spectral shape in the UV. In addition, as the magnitudes of a_g(λ), a_d(λ), and a_dg(λ) decrease with wavelength beyond 500 nm, the measurements at longer wavelengths are at increased risk of being affected by low signal-to-noise ratio or dropping to the instrumentation sensitivity limit, in particular for low CDOM absorption in clear waters. To maintain spectral continuity across the UV-VIS boundary, the absorption values calculated from the regression were adjusted such that the predicted value at 400 nm is equal to the measured value at 400 nm and the spectral slope obtained from the regression is preserved. Standard MATLAB fitting routines were applied to determine the regression parameters for all linear and non-linear fits, where non-linear regressions were determined using the Levenberg-Marquardt algorithm [74,75].

It is important to note that differences between the ELR and ENLR or between PLR and PNLR regression models arise from different weighting of each measurement during the determination of each regression fit. Whereas linear regressions involving the use of log_e-transformed or log₁₀-transformed data weigh each measurement within the regression spectral window equally, nonlinear regressions give more weight to higher-signal measurements typically found at shorter wavelengths within the regression window, so in our case the wavelengths nearest 400 nm. A linear regression model with an equal weighting of measurements can have drawbacks if applied over a broad spectral range extending to long-wavelength portion of the VIS because it would include data with low signal-to-noise ratio, and would be subject to higher uncertainties [8,14,69]. In our analysis, however, this potential disadvantage is circumvented by using a regression window extending to at most 500 nm, which generally ensures that spectral measurements of constituent absorption coefficients maintain a high signal-to-noise ratio. Figure 3 compares two example measurements of a_g(λ) from differing marine optical environments with results from all five regression models calculated using a regression window of 400–500 nm. These examples demonstrate the differences between the fitted spectra by each type of regression model and also highlight a tendency for increased discrepancy between the fitted and measured spectral values as the wavelength decreases in the near-UV range of 350 to 400 nm, which is of particular interest to this study in the context of determining the optimal extrapolation model from the VIS into the UV.

 

Fig. 3. Comparison of measured spectra of a_g(λ) with fitted curves from exponential linear regression (ELR), exponential nonlinear regression (ENLR), stretched exponential nonlinear regression (SENLR), power linear regression (PLR), and power nonlinear regression (PNLR) using a regression spectral window of 400–500 nm. (a) Example data from coastal waters collected during the MR17-05C cruise in the Chukchi Sea, and (b) example data from open ocean waters collected during the CLIVAR I8SI9N cruise in the Pacific Ocean.

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The analysis aiming at determination of the optimal regression model and spectral window for the purpose of extrapolation from the blue to the near-UV spectral region was focused on data of a_g(λ) and a_dg(λ). This approach is justified by existing experimental evidence that the absorption spectra of CDOM, a_g(λ), are less likely to exhibit significant departures from a single-slope exponential-like behavior than the spectra of non-algal particles, a_d(λ), especially in the blue spectral region [9]. The analysis of our dataset supported this conjecture. Thus, it can be expected that fitting the a_g(λ) measurements with a regression model in the blue spectral region for the purpose of subsequent extrapolation to the near-UV region will generally lead to better results compared to a similar approach applied to a_d(λ). In addition, it is important to note that in the UV and blue spectral regions the magnitude of a_g(λ) is often larger than the magnitude of a_d(λ) across different marine environments, which implies that a_dg(λ) is often dominated by the contribution of CDOM [9,11]. This tendency is also observed in our dataset. Specifically, for the dataset of 409 concurrent measurements of a_g(λ) and a_d(λ), the average fractional contribution of a_g(λ) to a_dg(λ) at the example wavelength of 400 nm is 0.778 (SD = 0.165). These results suggest that, after a_g(λ), the spectral measurements of a_dg(λ) appear to be the second-best candidate for the application of a regression model in the blue spectral region with a purpose of subsequent extrapolation to near-UV. Thus, the third absorption coefficient associated with non-algal particles, a_d(λ), does not require fitting a regression model to measured data in the blue and extrapolation to the near-UV because it can be derived as a difference between a_dg(λ) and a_g(λ). In summary, for the formulation of the optimal extrapolation model, we evaluated different extrapolation methods (i.e., different regression models and spectral windows in the blue spectral region) by assessing the degree of agreement between extrapolated and measured values of a_g(λ) and a_dg(λ) in the near-UV. For a_d(λ), analogous agreement in the near-UV was evaluated using the model-derived values of a_d(λ) obtained as a difference between the extrapolated a_dg(λ) and a_g(λ).

2.4 Statistical evaluation of extrapolation models

A suite of statistical metrics between predicted values from the model, P_i, and observed values, O_i (where the integer index i varies from 1 to N and N is the number of data points), for a given spectral value of absorption coefficient was used to assess the performance of each extrapolation method within the 350–400 nm spectral region. This analysis was made with the model development dataset. The statistical metrics include the nondimensional median ratio, MdR, defined as the median of the ratio between P_i and O_i, the median bias, MB, defined as the median difference between P_i and O_i, the median absolute percent difference, MdAPD = median(|P_i − O_i | / O_i) × 100%, and the root-mean-square deviation, RMSD = (N⁻¹ Σ(P_i − O_i)²)^0.5. These metrics provide information on bias and random error statistics. Additionally, Model II linear regression (a reduced major axis method) was applied to scatterplots of log₁₀-transformed predicted versus measured values of absorption coefficients in the UV range to further evaluate each extrapolation method. We report on the values of two regression parameters, A which is the slope, and B which is equal to 10^I where I is the y-intercept of linear regression of log₁₀-transformed variables. The scenario with both A and B equal to 1 is most desirable in a sense that it is indicative of no biasing effects across the dynamic range of the examined absorption coefficient. The Pearson correlation coefficient, r, between the log₁₀-transformed predicted and measured values was also calculated. Hereafter, we typically illustrate the statistical metrics of MdR and MdAPD spectrally across the near-UV spectral region to allow for a comparison between each of these statistics when evaluating the output of different extrapolation models. Ultimately, the selection of the optimal extrapolation model was based on comparisons of multiple statistical metrics to establish the best regression model and best regression spectral window in the blue spectral region.

3. Results and discussion

3.1 Determination and assessment of the optimal extrapolation method

Figure 4 depicts the aggregate error statistics MdR and MdAPD for a_g(λ) and a_dg(λ) at the example wavelength of 350 nm obtained from the extrapolation of data in the blue spectral region using five regression models. These error statistics were calculated using the model development dataset and are reported at 350 nm where the largest uncertainties are expected because this wavelength is furthest away from the regression region, which has been confirmed by our analysis shown below. These results are plotted as a function of λ_stop. We recall that λ_stop defines the regression spectral window which is set to start at 400 nm and end at λ_stop that was allowed to vary between 405 and 500 nm in these calculations. Thus, Fig. 4 summarizes the consequences associated with the choice of both the type of regression model and the regression spectral window on the quality of the extrapolation of a_g(λ) and a_dg(λ) from the blue to the near-UV.

 

Fig. 4. Median ratio, MdR, and median absolute percent difference, MdAPD, vs. stop wavelength (λ_stop) of the spectral regression window in the VIS range for (a, b) extrapolated a_g(350) (N = 1294) and (c, d) extrapolated a_dg(350) (N = 409). The regression window always starts at 400 nm and ends at varying λ_stop from 405 to 500 nm in 1 nm increments. The statistical parameters of a_g(350) and a_dg(350) are shown for five regression models as indicated in panel (a) (see text for more details).

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The spectral error statistics in Fig. 4 depict the characteristics in the uncertainty of UV absorption estimates obtained from exponential and power function regressions. The three exponential regression models, ELR, ENLR, and SENLR, have similar performance as indicated by similar curves of MdR and MdAPD across the range of λ_stop. The same is true for the power function regressions, PLR and PNLR, whose spectral error statistics are very similar. However, Fig. 4 also shows there are substantial differences in the magnitude and pattern of error statistics between the two groups of regression functions. Other statistical metrics also exhibited differences (not shown) which further supports the conjecture that the optimal regression window (i.e., the optimal λ_stop) is different for each group of functions. Specifically, the patterns of error statistics in Fig. 4 suggest that for all three exponential functions the 400–450 nm range can be used as the optimal regression spectral window for the UV determinations of both a_g(λ) and a_dg(λ). This is demonstrated by relatively good balance in the optimization of both statistical parameters displayed in Fig. 4, which is achieved in the vicinity of λ_stop = 450 nm in the case of estimation of both a_g(350) and a_dg(350). Specifically, if λ_stop = 450 nm MdR for estimates of a_g(350) and a_dg(350) assumes the value closest or nearly closest to 1 (Fig. 4(a),(c)) and MdAPD for estimates of a_g(350) and a_dg(350) assumes the lowest or nearly lowest value across the range of λ_stop (Fig. 4(b),(d)). These MdR values are about 0.98 and the MdAPD values are about 4.5% to 6%. We note that although the MdR and MdAPD statistics in Fig. 4 can maintain a relatively good level at other stop wavelengths, especially for λ_stop > 450 nm, the selection of λ_stop = 450 nm has been made by consideration of the combination of patterns of the statistical parameters of estimates of both a_g(λ) and a_dg(λ) not only at 350 nm but also at other near-UV wavelengths, which is addressed in more detail below. In addition, the selection of λ_stop = 450 nm also included the consideration of statistics not shown in Fig. 4, such as the linear regression of predicted vs. measured absorption coefficients at near-UV wavelengths and the correlation between the spectral slopes of measured absorption coefficients in the near-UV and the spectral slopes calculated over different spectral subranges from the 400–500 nm range.

The results in Fig. 4 also indicate that for the power functions, PLR and PNLR, the optimal regression window is much narrower compared to the optimal regression window for the exponential functions. Specifically, the optimal combination of MdR (Fig. 4(a),(c)) and MdAPD (Fig. 4(b),(d)) values of estimates of a_g(350) and a_dg(350) is observed when the PLR or PNLR regression is calculated over the spectral window from 400 nm to λ_stop ≅ 425 nm. In addition, the MdR and MdAPD values exhibit significant sensitivity to the choice of λ_stop, including the wavelengths in the vicinity of λ_stop = 425 nm. We disregarded the power function regression models in further analysis because of such sensitivity and the relatively narrow optimal regression window of only 25 nm which implies a relatively small number of absorption data within this window. These features of the power function models can be disadvantageous for the model’s predictive accuracy. Therefore, we chose to consider the exponential regression model as a means to estimate a_g(λ) and a_dg(λ) in the near-UV from data in the blue spectral region.

As illustrated in Fig. 4, the estimation of a_g(λ) or a_dg(λ) in the near-UV is similar regardless of which of the three exponential regression models, ELR, ENLR, or SENLR, is used. In particular, this result holds precisely for the selected optimal regression window of 400–450 nm, i.e., when λ_stop = 450 nm. Therefore, for further analysis we chose the simplest version of the exponential regression model, ELR (Table 2). We recall that the ELR involves the use of ordinary linear regression with log_e-transformed values of spectral absorption coefficients, so each spectral value within the regression window of 400–450 nm has the same statistical weight in the calculation of the fitted function.

Figure 5 provides additional support for selection of the 400–450 nm spectral range as the optimal regression window. This figure depicts the error statistics, MdR and MdAPD, plotted as a function of λ_stop for estimates of a_g(λ) and a_dg(λ) at five example near-UV wavelengths ranging from 350 to 390 nm in 10 nm increments. These absorption estimates were obtained by extrapolation of the exponential regression model, ELR, that was calculated using different regression windows as indicated by varying λ_stop. When λ_stop is around 450 nm, the MdR values are close to 1 for all near-UV wavelengths and assume values either slightly below 1 (i.e., slightly negative bias) for wavelengths closer to 350 nm or above 1 (slightly positive bias) for wavelengths closer to 400 nm (Fig. 5(a),(c)). When λ_stop decreases from 450 nm towards 400 nm there is a tendency for negative bias in the a_g(λ) and a_dg(λ) estimates at all near-UV wavelengths. When λ_stop increases from the 450–460 nm range towards 500 nm, there is also a tendency for negative bias in the a_dg(λ) estimates at most near-UV wavelengths (Fig. 5(c)), although no such pattern is observed for a_g(λ) (Fig. 5(a)). Thus, λ_stop = 450 nm appears to represent good choice from the standpoint of MdR statistic for both the a_g(λ) and a_dg(λ) estimates within the entire near-UV region. In addition, when λ_stop = 450 nm the MdAPD statistic remains low (below 6%) for both a_g(λ) and a_dg(λ) estimates at all near-UV wavelengths, although slightly lower values of MdAPD are observed at shorter λ_stop (Fig. 5(b),(d)). On the other hand, MdAPD tends to increase when λ_stop continues to increase beyond 450 nm. The tradeoffs between the error statistics highlighted in Fig. 5 justify the selection of λ_stop = 450 nm, and the complete results presented in both Fig. 4 and Fig. 5 support the conjecture that the optimal extrapolation method to estimate a_g(λ) and a_dg(λ) in the near-UV from the VIS can be based on the use of the ELR model within the spectral regression window of 400–450 nm.

 

Fig. 5. Median ratio, MdR, and median absolute percent difference, MdAPD, vs. stop wavelength (λ_stop) of (a, b) extrapolated a_g (N = 1294) and (c, d) extrapolated a_dg (N = 409) at five near-UV wavelengths (350, 360, 370, 380, and 390 nm) as obtained from the ELR regression model applied to the spectral regression window starting at 400 nm and ending at varying λ_stop with ELR.

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The assessment of the optimal extrapolation method for estimating a_g(λ) and a_dg(λ) in the near-UV is provided in Fig. 6 which depicts the correlation plots between the extrapolated and measured values of the absorption coefficients at 350 and 380 nm, as obtained from the analysis of the model development dataset. Similar correlation plots are also illustrated for the a_d(λ) coefficient where the predicted values of a_d(λ) were calculated as a difference between the extrapolated values of a_dg(λ) and a_g(λ). The correlation plots generally demonstrate a very good agreement between extrapolated and measured values of a_g(λ) and a_dg(λ) as the data align well with the 1:1 line and show little scatter about this line (Fig. 6(a),(b),(d),(e)). This is also supported by the closeness of the 1:1 line and the best-fit lines from the Model II linear regression applied to log₁₀-transformed extrapolated vs. measured data of a_g(λ) or a_dg(λ). However, despite such overall good agreement, the extrapolation method tends to underestimate a_g(λ) and a_dg(λ) at 350 nm when the magnitude of these coefficients is low (Fig. 6(a),(b)). This tendency is not clearly observed at 380 nm (Fig. 6(d),(e)) indicating the potential effects of underestimation decrease with increasing wavelength between 350 nm and 400 nm. With regard to a_d(λ), the correlation plots suggest that while the extrapolation method performs generally well, there is a persistent underestimation of a_d(λ) at 350 nm across the entire range of a_d(λ) (Fig. 6(c)). Although the tendency for underestimation still appears to be present in the data for 380 nm, it is greatly reduced (Fig. 6(f)). In summary, the results presented in Fig. 6 suggest the need to develop a correction to minimize or eliminate the potential presence of bias in the predictions of absorption coefficients at near-UV wavelengths.

 

Fig. 6. Scatter plots comparing extrapolated (a_g and a_dg) or calculated (a_d) vs. measured absorption coefficients at 350 nm (a–c) and 380 nm (d–f). The statistical parameters including Pearson correlation coefficient, r, between log₁₀-transformed predicted and measured values, as well as slope, A, and coefficient B, where B = 10^I and I is the y-intercept obtained from the Model-II linear regression of log₁₀-transformed predicted vs. measured values are depicted in each panel. The 1:1 line and best-fit line derived from the Model II linear regression to log₁₀-transformed data are represented by the solid gray and dashed black lines, respectively.

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3.2 Development of correction functions and assessment of the final extrapolation model

We evaluated several options to correct the predicted absorption coefficients in the near-UV and determined that the most effective approach involves the use of two separate correction functions: one for correcting the extrapolated a_g(λ) and the other for correcting a_d(λ) that is initially calculated as a difference between the extrapolated a_dg(λ) and a_g(λ). Note that in this correction approach the extrapolated a_dg(λ) is not subject to any direct correction. Instead, the final a_dg(λ) is obtained as a sum of corrected a_g(λ) and corrected a_d(λ) which ensures the closure between the three absorption coefficients. Details of the correction functions are described below.

The development of the correction of a_g(λ) was based on the analysis of the relationship between the ratio of measured to extrapolated a_g(λ) as a function of the magnitude of extrapolated a_g(λ) at each wavelength within the near-UV region, i.e., from 350 to 399 nm in 1 nm increments. This analysis provided a means to quantify the potential bias in extrapolated a_g(λ) as a function of a_g(λ) across the near-UV spectral region. Specifically, for any given near-UV wavelength, if the ratio of measured to extrapolated a_g(λ) shows a clear tendency to be greater than 1 for a given value of extrapolated a_g(λ), it means that the extrapolated a_g(λ) tends to be negatively biased. On the other hand, if the ratio tends to be smaller than 1 the extrapolated a_g(λ) is positively biased. Values of the ratio that are evenly distributed around 1 indicate no bias. Examples of this analysis for two wavelengths, 350 nm and 380 nm, are presented in Fig. 7. The results for 350 nm are selected in this example illustration because at this wavelength we identified the presence of largest negative bias in extrapolated a_g(λ) at low values of a_g(λ) (Fig. 7(a)). Figure 7(a) displays all 1294 data points from our model development dataset as well as a trend in these data represented by the median values of the ratio of measured to extrapolated a_g(350) (red crosses). These median values were calculated within 19 successive bins of extrapolated a_g(350) data across the entire range of a_g(350). As seen, the median values increase significantly above 1 when the extrapolated a_g(350) has low magnitude, specifically when a_g(350) is less than about 0.1 m⁻¹. Thus, in this range the extrapolated a_g(350) tends to be underestimated. The other important result in Fig. 7(a) indicates that as the magnitude of extrapolated a_g(350) increases beyond approximately 0.2 m⁻¹, the bias becomes very small or negligible because the median ratio of measured to extrapolated a_g(350) remains very close to 1. Similar patterns were observed at wavelengths longer than 350 nm, however, the extent of negative bias at low values of extrapolated a_g(λ) was found to decrease gradually with increasing wavelength. Eventually, at a wavelength of about 380 nm and beyond towards 400 nm, no significant bias was identified in extrapolated a_g(λ) across the entire dynamic range of a_g(λ). This result is illustrated in Fig. 7(b) for λ = 380 nm where the median ratio of measured to extrapolated a_g(380) remains close to 1 across the range of a_g(380). Therefore, we formulated a correction function that applies the correction to extrapolated a_g(λ) in the wavelength range between 350 nm and 380 nm, and no correction is applied at wavelengths ≥ 380 nm.

 

Fig. 7. The relationship between the ratio of measured to extrapolated a_g(λ) vs. extrapolated a_g(λ) at 350 nm (a) and 380 nm (b) obtained from the optimal (ELR) extrapolation method (see text for more details). The red line and crosses depict the median of each bin between every x-tick mark, except above 1 m⁻¹ where the median of all points is calculated and plotted at 5 m⁻¹. The blue dashed line denotes the non-linear fit to the function E_g(λ), where the steepness parameter, α(λ), and shift parameter, β(λ), derived from the fit are displayed in each panel.

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To establish the correction function for a_g(λ), several formulas were tested as potential candidates to provide the best fit to the relationship between the individual points of the measured to extrapolated a_g(λ) as a function of extrapolated a_g(λ). As a result of this analysis, the final best-fit function for any given wavelength λ is defined as:

To account for the weakening of the dependence of E_g(λ) on the magnitude of extrapolated a_g(λ) as the near-UV wavelength increases from 350 to 380 nm, the weighting factors, w₁(λ) and w₂(λ), are applied to obtain the final correction function, CF_g(λ), as follows:

Table 3. Spectral Values of the Parameters of the E_g(λ) Function^a

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The correction for bias in a_d(λ) estimates is based on the analysis of data such as shown in Fig. 6(c),(f) but for all near-UV wavelengths. This analysis demonstrated that the predicted a_d(λ), which is calculated as a difference between the extrapolated a_dg(λ) and a_g(λ), tends to be biased low with the extent of bias decreasing as the wavelength increases from 350 to 400 nm. In addition, it was determined that for any given λ there was no need to introduce a correction for bias that is dependent on the magnitude of a_d(λ) because the underestimation of a_d(λ) persisted quite consistently across the entire range of a_d(λ), which is particularly evident in the data at 350 nm (Fig. 6(c)). The formulation of the correction for a_d(λ) is therefore simpler than that for a_g(λ). Specifically, this correction is based on the spectral values of MdR of predicted to measured a_d(λ) (Fig. 8) and the final corrected a_d(λ) at any given wavelength λ is obtained by dividing the uncorrected a_d(λ) by the value of MdR at a given λ. The values of MdR from 350 to 399 nm in 1 nm increments are provided in Table 4. The value of MdR is the smallest and differs the most from 1 at 350 nm, so the correction for underestimation is largest at this wavelength. The MdR values increase as a function of wavelength and approach the value of 1 near 390 nm and beyond towards 400 nm (Fig. 8). Thus, in this long-wavelength portion of the near-UV region the correction of a_d(λ) is minimal. It is also to be noted that as a result of such correction based on the spectral values of MdR, the median aggregate bias of corrected a_d(λ) is forced to zero at all near-UV wavelengths.

 

Fig. 8. Median ratio, MdR, of modeled to measured a_d(λ) in the near-UV spectral region. Modeled values of a_d(λ) were calculated from the difference a_dg(λ) − a_g(λ), where values of a_dg(λ) and a_g(λ) were determined with the optimal (ELR) extrapolation method. The numerical values of MdR are reported in Table 4 and utilized to correct the initial estimates of a_d(λ) (see text for more details).

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Table 4. Spectral Values of MdR for a_d(λ)^a

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A diagram that depicts the sequence of steps for the complete extrapolation model is presented in Fig. 9. In summary, in the first step the input data of spectral a_g(λ) and a_dg(λ) in the VIS are assembled. If the input data are based on field measurements of a_g(λ) and a_d(λ), as in the present study, the values of a_dg(λ) are calculated as a sum of a_g(λ) and a_d(λ). However, it is to be noted that a_g(λ) and a_dg(λ) can also originate from models such as absorption partitioning models where prior knowledge of a_d(λ) is not necessarily needed. In the second step, the input data of a_g(λ) and a_dg(λ) are subject to the ELR regression which provides a basis for extrapolation from the VIS into the near-UV. This is then followed by calculation of a_d(λ) in the near-UV as a difference between the extrapolated a_dg(λ) and a_g(λ). In the third step, two separate corrections are applied to minimize or eliminate bias in the near-UV estimates of a_g(λ) and a_d(λ) obtained in the second step. In the final step, the corrected values of a_g(λ) and a_d(λ) are summed to obtain a_dg(λ) in the near-UV region. This fourth step completes the extrapolation model by providing the final estimates of a_g(λ), a_d(λ), and a_dg(λ) in the near-UV, which satisfy the closure equation a_dg(λ) = a_g(λ) + a_d(λ).

 

Fig. 9. Schematic describing the final extrapolation model for extending the spectral absorption coefficients a_g(λ), a_d(λ), and a_dg(λ) from the VIS to the near-UV spectral region. Step 1 depicts the input absorption data in the VIS. The extrapolation subsection (Step 2) outlines the initial optimal (ELR) extrapolation method to determine the three absorption coefficients in the near-UV. The correction subsection (Step 3) describes the two independent corrections applied to a_g(λ) and a_d(λ) (for model parameters see Tables 3,4) to obtain the final output of each absorption coefficient in the near-UV (Step 4).

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An assessment of the final extrapolation model to obtain a_g(λ), a_d(λ), and a_dg(λ) in the near-UV spectral region is presented in Table 5 and Fig. 10. These results were obtained from the analysis of the model development dataset. The aggregate error statistics of estimates of the three absorption coefficients indicate an overall good performance of the model (Table 5). The aggregate bias of a_g(λ) and a_dg(λ) estimates is very small or negligible across the near-UV region as indicated by MdB which is close to 0 m⁻¹ and MdR which is very near 1 to within about 1%. Given the formulation of the correction applied to a_d(λ), the bias in this absorption coefficient was completely removed at all near-UV wavelengths, so MdB and MdR values are exactly equal to 0 m⁻¹ and 1, respectively. The MdAPD and RMSD statistics indicate that the largest uncertainty in estimates of a_g(λ), a_d(λ), and a_dg(λ) occurs at 350 nm. For example, the MdAPD values for the estimates of a_g(350), a_d(350), and a_dg(350) are 5.14%, 10.45%, and 4.54%, respectively (Table 5). This is expected as 350 nm is the furthest away from 400 nm where the transition occurs between the near-UV extrapolation range and the VIS range containing the input data used in the extrapolation model. Importantly, the MdAPD values are small across the near-UV region, typically below 5%, which further supports the overall good performance of the extrapolation model. In addition, it is notable that the error statistics of final corrected absorption coefficients obtained in step 4 of the model are improved compared to the statistics of initial estimates of absorption coefficients obtained in step 2 of the model. Consistent with the correction design and purposes, these improvements are most significant within the short-wavelength portion of the near-UV region. It is also notable that in the process of estimating a_g(λ), the correction procedure in step 4 of the model brings in especially significant improvements at low values of a_g(λ). This can be demonstrated by comparing the slope (A) and intercept-related (B) parameters of Model II linear regression of (log₁₀-transformed) predicted vs. measured values of a_g(350) where the predicted values are obtained in step 2 (i.e., before correction) or step 4 (i.e., after correction) of the extrapolation model and the measured values are restricted to the low-magnitude range less than 0.1 m⁻¹. In this range we have 536 data points available for this regression analysis. In the case of a_g(350) predicted in step 2 of the model, the A and B values are 1.128 and 1.350, respectively. When the corrected values of a_g(350) from step 4 of the model are used in this analysis, the A and B parameters are closer to the most desirable value of 1, specifically 0.959 and 0.886, respectively.

 

Fig. 10. The performance of the final extrapolation model applied to the development dataset and illustrated with scatter plots of the model-derived vs. measured absorption coefficients at 350 nm (a–c) and 380 nm (d–f). The statistical parameters including Pearson correlation coefficient, r, between log₁₀-transformed predicted and measured values, as well as slope, A, and coefficient B, where B = 10^I and I is the y-intercept obtained from the Model-II linear regression of log₁₀-transformed predicted vs. measured values are depicted in each panel. The 1:1 line and best-fit line derived from the Model II linear regression to log₁₀-transformed data are represented by the solid gray and dashed black lines, respectively.

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Table 5. Error Statistics of the Final Extrapolation Model^a

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Figure 10 is analogous to Fig. 6 but shows the correlation plots of the final corrected estimates of a_g(λ), a_dg(λ), and a_d(λ) versus measured values of these coefficients at 350 nm and 380 nm. Compared to results in Fig. 6, Fig. 10(a),(b) demonstrates improved estimates of a_g(350) and a_dg(350), especially for low-magnitude data points which are now distributed nearer and more evenly around the 1:1 line. Significant improvement is also seen for a_d(350) in Fig. 10(c) where the biasing tendency observed in Fig. 6 has been essentially removed. The differences between the correlation plots presented in Fig. 10 and Fig. 6 for λ = 380 nm are naturally small because the correction at this wavelength either vanishes (for a_g) or becomes small (for a_d). In summary, the aggregate error statistics (Table 5) and correlation plots (Fig. 10) demonstrate the final extrapolation model involving the correction functions (Fig. 9) provides a means to estimate with satisfactory accuracy the three non-phytoplankton components of absorption, a_g(λ), a_d(λ), and a_dg(λ) in the near-UV (350 to 400 nm) spectral region.

3.3 Application of the final extrapolation model using spectrally subsampled data in the VIS

Although the extrapolation model described in previous sections was developed and assessed with hyperspectral absorption measurements available at 1 nm intervals, it is also of interest to evaluate the performance of this model using the input absorption data at lower spectral sampling intervals within the VIS region. Testing such scenarios can be relevant in the context of absorption coefficients derived from inverse optical models at certain wavelengths in the VIS which are available on satellite ocean color sensors. The past and current satellite sensors, such as SeaWiFS, MODIS, VIIRS, MERIS, or OLCI, are multispectral and have only a few spectral bands in the blue spectral region which is utilized as a basis in our extrapolation model. In contrast, the Ocean Color Instrument (OCI) to be launched in 2024 on NASA’s Plankton, Aerosol, Cloud, ocean Ecosystem (PACE) satellite mission will provide a capability for high spectral sampling-interval measurements at 5 nm intervals from the near-UV (350 nm) through the VIS into the near-IR (890 nm).

We have evaluated the performance of our extrapolation model for a few specific spectral scenarios in which the 1 nm hyperspectral absorption measurements in our model development dataset were subsampled in the VIS spectral range to provide input data to the ELR regression component of the extrapolation model with lower spectral sampling interval. Below we present results for three scenarios of subsampled spectral data of a_g(λ) and a_dg(λ) in the VIS range which serve as input to the model. The first scenario includes the input data in the 400–450 nm range at 5 nm intervals, which is consistent with the spectral characteristics of the PACE-OCI and represents rather minor degradation of spectral resolution compared to the original 1 vnm data used in the formulation of the extrapolation model. The second scenario utilizes the input data only at two wavelengths, 412 and 443 nm, which correspond to spectral bands available on MODIS. Finally, the third scenario makes use of input data at three wavelengths from the blue spectral region which are available on MODIS, 412, 443, and 488 nm. Note, however, that in this case the wavelength of 488 nm is outside the 400–450 nm range which was used in the extrapolation model based on original 1 nm data. We also note that we tested the two remaining combinations of the blue spectral bands corresponding to those available with the MODIS sensor, but the absorption estimates in the near-UV were not superior to those obtained with the aforementioned two- or three-spectral band scenarios.

Figure 11 compares the error statistics for two scenarios of estimation of the three absorption coefficients in the near-UV region: one scenario utilized the input data with the original 1 nm interval in the ELR regression component of the extrapolation model, and the other scenario utilized the input data with a 5 nm interval. In both cases the absorption coefficients were estimated at 1 nm intervals within the near-UV region, so the values of error parameters in Fig. 11 are also plotted at 1 nm intervals. These results demonstrate that compared to the use of 1 nm input data, the use of 5 nm data within the 400–450 nm regression window yields very similar or nearly identical values of MdR and MdAPD across the near-UV region for all three absorption coefficients. This is indicative of essentially no degradation in performance of the extrapolation model when the spectral sampling interval of input data within the 400–450 nm regression window is increased from the 1 nm to 5 nm interval.

 

Fig. 11. Median ratio, MdR, (a-c) and median absolute percent difference, MdAPD, (d-f) in the near-UV spectral region for estimates of a_g(λ), a_dg(λ), and a_d(λ) obtained from the final extrapolation model. The results are calculated with input data from the VIS spectral region with the original 1 nm spectral sampling interval (black lines) and by subsampling the input data into a 5 nm spectral interval (gray lines).

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Fig. 12. Same statistics as presented in Fig. 11, but results determined with input data with the original 1 nm spectral sampling interval (black lines) are compared with results obtained by subsampling the input data into three MODIS bands (gold lines) and two MODIS bands (light blue lines) from the blue portion of the VIS spectral region.

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Figure 12 depicts analogous results to Fig. 11, but the scenario of 1 nm input data is compared with two MODIS-like scenarios in which only two or three wavelengths from the blue spectral region are used as input to the ELR regression analysis. These results show that the exponential fit to only two or three data points in the blue region results in a decline in performance of the extrapolation model. The MdR and MdAPD statistics for estimates of the three absorption coefficients in the near-UV region are clearly inferior when the input data are only at 412 and 443 nm or 412, 443, and 488 nm, if compared to the 1 nm input data within the 400–450 nm regression window. The most significant worsening of error statistics is observed for the estimates of a_d(λ) when the input data at the three wavelengths, 412, 443, and 488 nm, are used. If the two MODIS-like scenarios are compared to each other, the use of two wavelengths of 412 nm and 443 nm tends to provide superior estimation of near-UV absorption coefficients from the extrapolation model than the use of three wavelengths which additionally include 488 nm. It is also notable that the error statistics for the two-wavelength scenario are quite satisfactory for all three absorption coefficients across the near-UV region as the MdR values remain within the range between about 1.02 and less than 1.06 and the MdAPD values remain below 10% with the exception of a_d(λ) estimation in the short-wavelength end of near-UV region where MdAPD reaches about 24%. Thus, for the potential application of the extrapolation model when the input absorption data in the VIS are available only at limited number of spectral bands such as in multispectral satellite ocean color sensors, our results suggest the use of two bands from the 400–450 nm range, such as 412 nm and 443 nm available on MODIS, rather than the use of additional absorption data at longer wavelengths from the blue and/or green spectral region.

3.4 Validation of the extrapolation model with an independent dataset

The performance of the final extrapolation model for estimating a_g(λ), a_dg(λ), and a_d(λ) in the near-UV when applied to the validation dataset (N = 149) is provided in the correlation plots between extrapolated and measured values at 350 and 380 nm (Fig. 13). We observe a generally good overall agreement between modeled and measured values of absorption coefficients. The most notable differences between measured and modeled data are observed as underestimations of a_d(350) at low magnitudes and more scatter in data of a_d(350) about the 1:1 line compared to a_g(350) and a_dg(350). Specifically, for a_d(350) the MdR is 0.975 and MdAPD is 11%. These statistics are comparable to the model performance found in Section 3.2 when applied to the development dataset. We also examined the validation dataset using the input data to the extrapolation model that were subsampled at 5 nm intervals in the VIS spectral region (not shown), and we obtained similar results to those with 1 nm input data. The validation analysis with the independent field dataset supports the good performance of the final extrapolation model and its capability to extend a_g(λ), a_dg(λ), and a_d(λ) from the VIS to the near-UV.

 

Fig. 13. The performance of the final extrapolation model applied to the validation dataset (N = 149) as scatter plots of model-derived versus measured absorption coefficients at 350 nm (a–c) and 380 nm (d–f). The statistical parameters of the Pearson correlation coefficient, r, between log₁₀-transformed predicted and measured values, slope, A, coefficient B, where B = 10^I and I is the y-intercept obtained from the Model-II linear regression of log₁₀-transformed predicted vs. measured values, median ratio, MdR, median absolute percent difference, MdAPD, and root-mean-square deviation, RMSD, are depicted in each panel. The 1:1 line and best-fit line derived from the Model II linear regression to log₁₀-transformed data are represented by the solid gray and dashed black lines, respectively.

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4. Summary

In this study we developed and assessed a method for extrapolating data of constituent absorption coefficients, a_g(λ), a_d(λ), and a_dg(λ), from the VIS spectral region, specifically its blue portion, into the near-UV (350 to 400 nm) spectral region. The extrapolation model was developed for the purpose of applicability to absorption data obtained either from measurements or models in the VIS range when the data in the UV are lacking or are deemed to be subject to significantly larger uncertainty than the available data in the VIS. One example scenario of applicability of the extrapolation model is the availability of data of constituent absorption coefficients derived from absorption partitioning models that were developed specifically for the VIS spectral region. Another example is the availability of constituent absorption data at VIS spectral bands as derived from inverse reflectance models, such as those designed for operation with satellite multispectral measurements of ocean color in the VIS. In both example cases, it is desirable to have an extrapolation model providing a reliable capability to extend data of a_g(λ), a_d(λ), and a_dg(λ) from the VIS into the UV with satisfactory accuracy for a wide range of oceanic conditions.

We formulated the extrapolation model using an assembled development dataset that includes field measurements of 1294 a_g(λ) spectra and 409 concurrent spectra of a_d(λ) and a_g(λ), and hence also 409 spectra of a_dg(λ). These measurements are available at 1 nm intervals over a broad spectral range from the UV to near-IR, and were collected within the near-surface ocean layer in diverse environments which made this dataset well suited for the purpose of our analysis. Importantly, the model development dataset includes data that satisfy several method-related and data quality criteria, most notably the accepted hyperspectral a_d(λ) data were obtained solely from spectrophotometric measurements using the inside integrating-sphere filter-pad technique. We determined that the exponential function fitted to the 1 nm data of a_g(λ) and a_dg(λ) within the 400–450 nm spectral window provides an adequate first step of the extrapolation model. The initial estimates of a_d(λ) in the near-UV region are calculated as a difference between the extrapolated values of a_dg(λ) and a_g(λ). Then, to further improve the overall performance of the extrapolation model, we formulated the correction functions to minimize or eliminate the bias in the initial estimates of a_g(λ) and a_d(λ) in the near-UV. The significance of these corrections is, however, restricted mostly to the short-wavelength portion of the near-UV region, and in the case of a_g(λ) to low magnitudes of this coefficient. The final products of the extrapolation model include the corrected values of a_g(λ) and a_d(λ) as well as the final a_dg(λ) calculated as a sum of corrected a_g(λ) and a_d(λ).

The comparative analysis of model-derived estimates and measured values of the absorption coefficients using the model development dataset indicates a very good performance of the extrapolation model across the near-UV region. For example, the spectral values of aggregate bias at different wavelengths in the 350–400 range are generally less than 1% and the uncertainty in terms of median absolute percent difference (MdAPD) is typically less than 5% with the exception of about 10% for a_d(λ) estimates at or near 350 nm. The extrapolation model provides equally satisfactory performance if the exponential fit in the 400–450 nm spectral window is applied to subsampled absorption data at 5 nm intervals. We also determined that if absorption data are available only at relatively few wavelengths in the VIS including its blue spectral region, the extrapolation model still performs reasonably well if the exponential function is fitted to data at just two wavelengths in the 400–450 nm range, such as 412 nm and 443 nm available on MODIS satellite ocean color sensor. In this case, when compared to the 1 or 5 nm input data in the blue, the error statistics for near-UV absorption estimates deteriorates the most for a_d(λ) with MdAPD values reaching 20–25% at the short-wavelength end of the near-UV region. The extrapolation model was also validated with an independent dataset of subsurface (depths of 15 to 50 m) measurements which were collected at the same oceanographic stations and processed using the same protocols as our model development dataset. This validation analysis supports the good performance of the extrapolation model and its capability to extend the spectral absorption coefficients a_g(λ), a_dg(λ), and a_d(λ) from the VIS to the near-UV.

Whereas, the present study utilized the field datasets of absorption measurements to formulate and assess the performance of the extrapolation model, it will be desirable to further test this model in conjunction with input absorption data in the VIS derived from absorption partitioning models. For example, the partitioning models that were developed for the VIS spectral region and would be suitable for such analysis are described in Zheng and Stramski (2013) [47], Zheng et al. (2015) [46], and Stramski et al. (2019) [49]. In addition, in the future it will be desirable to test the extrapolation model with additional independent field datasets. A prerequisite for such analysis is the acquisition of more high-quality hyperspectral data of constituent absorption coefficients which satisfy the methodology-related and data quality criteria, especially the measurements of a_p(λ) and a_d(λ) with the spectrophotometric filter-pad technique using the inside integrating-sphere configuration or another absorption technique providing equivalent or better quality of particulate absorption data, such as PSICAM technique [65].

In summary, the significance of the proposed extrapolation model stems primarily from its applicability to and enhancement of existing and potential future absorption partitioning models that are formulated for the VIS spectral region. However, because the absorption partitioning models can be used in conjunction with inverse reflectance models with an ultimate goal of estimating the constituent absorption coefficients from ocean color data, our extrapolation model can also be relevant as a potential component of a suite of inverse optical models that are applied to satellite ocean color observations. At present, to our knowledge, no validated or sufficiently mature capabilities to estimate the constituent absorption coefficients in the UV from inverse reflectance and absorption partitioning models that utilize UV measurements are available, so our extrapolation model provides a tool for immediate applicability to existing models that operate in the VIS. Our present extrapolation model can also provide a valuable tool for comparison and assessment of future optical models that utilize UV measurements.