Radiometric products determined from fixed-depth and continuous in-water profile data collected at a coastal site characterized by moderately complex waters were compared to investigate differences and limitations between the two measurement methods. The analysis focused on measurements performed with the same radiometer system sequentially deployed at discrete depths (i.e., 1 and 3 m) and successively used to profile the water column. Within the 412–683-nm spectral interval, comparisons show uncertainties of 2%–4%, 3%–5%, and 2% for the subsurface values of upwelling radiance, Lun, upward irradiance, Eun, and downward irradiance, Edn, all normalized with respect to the above-water downward irradiance. The related spectral biases vary from −2% to 1% for Lun, are in the range of 2%–3% for Eun, and are lower than 0.5% for Edn. Derived products like the irradiance reflectance, R, Q factor at nadir, Q, and normalized water leaving radiance, LWN, exhibit spectral uncertainties of 4%–6%, 2%–3%, and 2%–4%. The related spectral biases vary from 1% to 3%, 2% to 3%, and −2% to 1%, respectively. An analysis of these results indicates a general diminishing of uncertainties and biases with a decrease of the diffuse attenuation coefficient, Kd, determined at 490 nm, for most of the quantities investigated. Exceptions are Edn and Kd because an increase of Kd reduces the perturbations due to wave effects on downward irradiance measurements. An evaluation of the perturbing effects due to the presence of optical stratifications, which lead to a nonlinear decrease with depth of log-transformed radiometric measurements, shows an expected increase in uncertainty and bias specifically evident for Ku, Eun, Kl, and Lun, and derived quantities like R, Q, and LWN. Overall results, supported by a t-test analysis, indicate the possibility of using moorings in moderately complex coastal waters to determine LWN with a slightly higher uncertainty with respect to that achievable with continuous profiling systems.
Highly accurate in situ radiometric data are a requirement for ocean color activities like bio-optical modeling, vicarious calibration of sensors in space, and validation of remote sensing products (McClain et al. 2004). The specific radiometric quantity relevant to ocean color applications is the normalized water-leaving radiance determined from the subsurface upwelling radiance. The latter can be derived from in-water radiometric profiles of continuous or multiple fixed-depth measurements, and its determination relies on the assumption of linear decrease with depth of log-transformed upwelling radiance (Smith and Baker 1984). This fundamental assumption is supported by the exponential decrease with depth of light intensity in a homogeneous water column (Kirk 1983).
While continuous profile data provide the capability of choosing on a case-by-case basis the optimum extrapolation interval satisfying the linearity condition, profile data obtained from measurements performed at a few discrete depths (hereafter simply called fixed-depth profile data) must necessarily depend on the assumption that the linearity condition is satisfied from the surface down to the maximum discrete measurement depth defining the bottom boundary of the extrapolation interval. Because of this, systems like bio-optical buoys relying on radiometers operated at fixed depths are generally deployed in oceanic waters, where the linear decrease of log-transformed data is easily satisfied at near-surface depths not appreciably affected by Raman scattering (Waters 1995; Zheng et al. 2002). Data from optically complex coastal waters, for which the linearity condition is not always fulfilled, are however increasingly required for a more comprehensive development and validation of remote sensing products (Chang et al. 2006; Mélin et al. 2007). Despite the measurement difficulties, which include large biofouling perturbations on in-water optics (Chavez et al. 2000), coastal bio-optical buoys might become a permanent source of data in addition to (i) above-water radiometer systems continuously operated from suitable deployment platforms (Zibordi et al. 2006b), and (ii) above- or in-water measurement systems operated during regular or occasional oceanographic campaigns (Darecki et al. 2003).
Best practice would always suggest comparisons between radiometric products determined through different measurement methods to verify their equivalence. This need is strengthened at coastal sites where the complexity of seawater optical properties might challenge measurements and their successive analysis. In agreement with this need, a major objective of the current work is the quantification of differences between radiometric products determined from fixed-depth and continuous profile data collected in moderately complex coastal waters at a site already extensively used for the validation of satellite products (Mélin et al. 2007; Zibordi et al. 2006a).
Measurement methods currently adopted by the ocean color community for in-water radiometry mostly rely on that documented by Smith and Baker (1984, 1986). This combines achievements from former experimental studies [see the historical overviews in Jerlov (1968) and Tyler and Smith (1969)] and basically requires that profiles of radiometric data are collected in the water column in combination with above-water downward irradiance measurements. While the in-water data are used to determine the subsurface values of radiance and irradiance by simply extrapolating their values just below the water surface (i.e., at depth 0−), the above-water downward irradiance data are used to minimize the effects of illumination changes on in-water measurements during data collection.
In-water profiles of radiometric quantities generally result from measurements performed with radiometers operated on moorings, or on winched and free-fall systems. In the first case, measurements are a function of the discrete depths selected for the radiometers and of the acquisition rate and logging interval. In the second case measurements are a function of the sampling depth interval and of the depth resolution as defined by the system acquisition rate and deployment speed.
The use of winched systems, allowing for the production of continuous profiles, had extensive application over decades and largely supported the study of apparent optical properties within the water column (e.g., Smith et al. 1984). Since the late 1980s, the design of free-fall systems, characterized by high buoyancy and providing the possibility of performing measurements at some distance from the deployment structure (Lewis et al. 1986; Waters et al. 1990), has allowed for collecting continuous profiles ideally not affected by perturbations due to ship shading and ship roll. Potential for radiometric profiling in coastal waters at high frequency over long duration is now offered by autonomous moored profilers (Dunne et al. 2002).
The concept of multiple radiometers deployed at different fixed depths was introduced by Dera et al. (1972), who envisaged its applicability to buoy systems. Actually, since late 1990s, the methodology based on fixed-depth radiometers has become the basis for measurements performed through bio-optical buoys to support satellite ocean color applications (Clark et al. 1997, 2001; Kishino et al. 1997; Pinkerton et al. 2003; Antoine et al. 2008; Kuwahara et al. 2008).
Surface effects like wave focusing and defocusing, perturbation of depth measurements by waves, nonhomogeneities in the vertical distribution of bio-optical properties typical of coastal waters, and inelastic processes like chlorophyll fluorescence or Raman scattering, might affect the radiance distribution in the water column and consequently the linear decrease of log-transformed data with depth. This may lead to uncertainties differently affecting the extrapolation of subsurface values from fixed-depth and continuous profile data. As a consequence, the selection of the sampling depths for fixed-depth measurements in the near-surface layer depends on balancing the perturbations due to (i) wave effects (decreased by selecting relatively large extrapolation intervals; e.g., 10 m), and (ii) gradients in the vertical distribution of the seawater optically significant constituents (minimized by picking relatively small extrapolation intervals close to the sea surface; i.e., within 5 m or less).
Recent studies on the equivalence between radiometric products determined from fixed-depth and continuous profile data presented comparisons of subsurface values derived from one fixed-depth radiometer and a profiling system (Hooker et al. 2002; Chang et al. 2003), as well as from two fixed-depth radiometers and a profiling system (Antoine et al. 2008). These studies mostly showed differences between products derived from various measurement methods using independent measuring systems, without explicitly quantifying uncertainties related to the measuring systems (e.g., absolute calibration, immersion factor, self-shading), and the measurement methods (i.e., fixed-depth and continuous profiling). The rationale of the present work is then to advance results presented in the former valuable studies by specifically investigating the equivalence between methods when made independent from the applied instruments and processing schemes.
3. Data and methods
The methodology here applied to compare radiometric products determined from fixed-depth and continuous profile data relies on the use of a single package of radiometers deployed through a winch from an oceanographic tower. Additionally, the processing of data related to the two methods is performed with the same code. These solutions completely remove any uncertainty due to differences in measuring systems and processing codes from the comparison of the two methods.
The data used in this study were collected at the Acqua Alta Oceanographic Tower (AAOT) positioned eight nautical miles offshore of the Venice lagoon (45.314°N, 12.508°E) in the northern Adriatic Sea (see Fig. 1). Since 1995 this platform has been used to support ocean color validation activities through a comprehensive data collection within the framework of the Coastal Atmosphere and Sea Time Series (CoASTS) program (Zibordi et al. 2002). The measurement site, located in a frontal region characterized by a succession of case-1 and case-2 waters, is considered well representative of coastal zones (regions permanently or occasionally affected by bottom resuspension, coastal erosion, river inputs, or relevant anthropogenic impact) with an occurrence of roughly 65% case-1 waters (Berthon et al. 2002; D’Alimonte et al. 2007).
The data relevant to the current study are in-water radiometric profiles taken with the Wire-Stabilized Profiling Environmental Radiometer (WiSPER): a winched system equipped with Satlantic (Halifax, Canada) OCR-200 and OCI-200 radiometers installed on a custom rig. Specifically, WiSPER provides simultaneous measurements of upwelling radiance, Lu(z, λ), upward irradiance, Eu(z, λ), downward irradiance, Ed(z, λ), as function of depth z, and above-water downward irradiance, Ed(0+, λ), all at the 412-, 443-, 490-, 510-, 555-, 665-, and 683-nm center wavelengths λ, with a 6-Hz acquisition rate (Zibordi et al. 2004). The rigidity and stability of the measuring system are maintained through two taut wires anchored between the tower and a weight on the sea bottom that prevent the movement of the rig out of the vertical plane of the wires. In addition, different from winched or crane systems operated from ships subject to roll, the immovability of the AAOT does not produce mechanical perturbations adding to wave effects.
Measurements for the comparison of quantities determined from fixed-depth and continuous down- and upcast profiles were performed taking (i) fixed-depth data by sequentially positioning WiSPER at 1- and at 3-m nominal depths for 3 min each, to mimic the collection of optical buoy data; and (ii) continuous profile data with WiSPER deployed at a speed of 0.1 m s−1. In both cases reference Ed(0+, λ) data were measured with a radiometer operated at the top of the AAOT. The wire stabilization and the relatively low deployment speed ensured the collection of continuous profile data with a high depth resolution leading to a detailed characterization of the vertical radiometric features within the measurement layer. The selection of the two fixed depths was made by observing that the near-surface homogenous layer at the site generally goes from the surface down to approximately 2–5-m depth (Zibordi et al. 2004). The choice of the fixed nominal depths of 1 and 3 m was specifically made to produce a large fraction of measurements in the homogeneous near-surface layer (i.e., diminishing the probability of getting data affected by the presence of gradients in the vertical distribution of seawater optically significant components). It is reported that similar depths (i.e., 0.9 and 2.9 m) were actually applied to a bio-optical buoy deployed in nearly coastal waters (Pinkerton and Aiken 1999). Slightly larger depths (i.e., 2, 4, and additionally 10 m) were applied to a different buoy system deployed in turbid waters (Chang et al. 2006).
The WiSPER measurements have been processed using a code developed for the analysis of CoASTS data (D’Alimonte et al. 2002). The applied processing steps are here summarized for completeness. In-water radiometric quantities (in physical units) are normalized with respect to the above-water downward irradiance Ed(0+, λ, t), with t explicitly expressing dependence on time, according to
where n (z, λ, t) identifies the normalized radiometric quantities [i.e., Lun (z, λ, t), Eun (z, λ, t), and Edn (z, λ, t)] obtained from (z, λ, t) [i.e., Lu(z, λ, t), Eu(z, λ, t), and Ed(z, λ, t)]. Absolute radiometric quantities, 0(z, λ, t0), are determined as if they were all taken at depths z at the same time t0 according to
where Ed(0+, λ, t0) indicates the above-water downward irradiance at time t0 (which is generally chosen to coincide with the start of the cast). Omitting the variable t, the subsurface quantities n(0−, λ) [i.e., Lun(0−, λ), Eun(0−, λ), and Edn(0−, λ)] or 0(0−, λ) [i.e., Lu(0−, λ), Eu(0−, λ), and Ed(0−, λ)] are then determined as the exponential of the intercept resulting from the least squares linear regressions of ln n(z, λ) or ln 0(z, λ), versus z within the extrapolation interval identified by z0 < z < z1. The extrapolation limits applied in this study are z0 = 0.3 m and 2 < z1 < 5 m for continuous profile data, and z0 = 0 m and z1 = 5 m for fixed-depth profile data. Under the premise that computations are performed using actual depth measurements, the boundaries selected for the extrapolation interval of fixed-depth profile data (i.e., 0 and 5 m) allow for processing measurements performed at the nominal deployment depths (i.e., 1 and 3 m) regardless of the depth fluctuations due to wave perturbations.
The negative values of the slopes of the regression fits are the so-called diffuse attenuation coefficients (λ) [i.e., Kl(λ), Ku(λ), and Kd(λ)] for the selected extrapolation intervals. The appropriateness of the extrapolation interval satisfying the requirement of linear decay of lnn(z, λ) [or equally of ln0(z, λ)] for continuous profiles is evaluated on a case-by-case basis by successive trials.
In addition to the basic products 0(0−, λ) and (λ), derived quantities like the subsurface Q factor at nadir, Q(λ), the subsurface irradiance reflectance, R(λ), and the normalized water-leaving radiance, LWN(λ), are computed according to
In agreement with the scheme outlined by Zibordi et al. (2004), the normalized water-leaving radiance is also computed using above-water downward irradiance values derived from Ed(0−, λ) and it is hereafter identified as WN(λ).
c. Self-shading, superstructure, and bottom perturbations
WiSPER data collected at the AAOT are affected by instrument self-shading due to the size (3.5-in. diameter) of the OCI-200 and OCR-200 radiometers, tower shading due to the close deployment distance (7 m) from the main superstructure, and bottom perturbations due to the shallow water depth (17 m). Despite applying the same measuring system and deployment platform, perturbations affecting continuous and fixed-depth profile data recorded at different times and depths are likely to slightly differ. However, it has been considered more appropriate to not apply corrections for these perturbations to data products. This choice is mostly justified by the impossibility of determining independent corrections for each data profile through the application of the schemes developed within the framework of the CoASTS program and operationally applied to the WiSPER data collected at the AAOT (Zibordi et al. 2002). In particular, referring to the self- and tower shading correction schemes (Zibordi and Ferrari 1995; Doyle and Zibordi 2002), the optical and geometric parameters defining the actual measurement conditions required for determining the corrections are gathered once for each sequence of measurements and not on a profile-by-profile basis. Consequently, corrections would not account for variability due to short-term environmental changes. Additionally, the correction schemes for self- and tower-shading perturbations were implemented and validated for subsurface radiometric measurements only, and not for data at various depths. As a consequence, the differences between the correction factors determined for fixed-depth and continuous profile data products would only depend on changes in sun position during the collection of individual matchups (i.e., less than 15 min). These differences have been quantified through a sensitivity analysis performed for self- and tower-shading correction factors determined under the assumption of negligible changes in the seawater inherent optical properties. Results from replicates of continuous profile data collected within 15 min from each other have shown that differences solely due to changes in the sun position exhibit spectrally averaged biases of less than 0.05% for tower-shading corrected Lun, Eun, and Edn and of less than 0.1% for self-shading corrected Lun and Eun, all with respect to uncorrected values.
The CoASTS correction scheme applied for bottom perturbations (Zibordi et al. 2002) relies on parameters determined from the radiometric profile itself (i.e., the bottom reflectance and the average diffuse attenuation coefficient for the whole water column). Thus its application to both fixed-depth and continuous profile data would lead to differences that are more dependent on the uncertainty of the input parameters (which are specific to each method) rather than the actual effects of the bottom reflectance. Similar to self- and tower-shading corrections, the nonapplication of corrections for bottom effects is, however, expected to be of no impact to the quality of the study. In fact, absolute corrections determined at the AAOT for Lun (0−) and Eun (0−) generally exhibit average values of 1%–2% at 555 nm (i.e., at the most affected wavelength). Consequently, actual average differences between corrections for bottom perturbations differently affecting fixed-depth and continuous profile data are expected to be much lower than those values and mostly constrained to the green center wavelengths.
4. Data analysis
The comparison has been focused on close-in-time (so-called matchups) radiometric products from fixed-depth and continuous profile data. The dataset analyzed includes 105 matchups collected during clear sky conditions (i.e., with clouds at the horizon and coverage lower than 1/8) within the framework of 10 different CoASTS campaigns in the period July 2003–July 2006. The average values of quantities identifying the measurement conditions for the dataset are listed in Table 1. Specific quantities, all determined following the protocols described in Zibordi et al. 2002, are the absorption of colored dissolved organic matter (CDOM), ay, at 412 nm; the total chlorophyll a, Chla (i.e., chlorophyll a plus chlorophyllide a); and the total suspended matter, TSM. Additional data describing the measurement conditions are the wind speed, Ws, the sun zenith, θ0, and the in-water diffuse attenuation coefficients. These latter are summarized in Table 2 and were determined from continuous profiles of Lun, Eun, and Edn.
Illustrative examples of fixed-depth and continuous profile data are presented in Figs. 2 and 3. Specifically, Fig. 2 represents a condition exhibiting an almost linear decrease with depth of log-transformed radiometric data and large wave effects. The latter are very pronounced in Edn and extend quite deep in the water column because of the low diffuse attenuation coefficient [i.e., Kd(490) = 0.07 m−1]. In contrast, Fig. 3 shows a condition characterized by relatively small wave perturbations and a nonlinear decrease with depth of log-transformed radiometric data. The fixed-depth measurements are presented through red circles and appear as clusters around the 1- and 3-m nominal depths. They overlay the continuous profile data displayed through black circles. Aside from focusing and defocusing effects by waves, which produce random changes in radiometric values (i.e., stochastic deviations from the mean at each fixed depth), it is also possible to observe depth variations around each nominal fixed depth due to changes in wave height during data collection.
The data analysis is presented through the following basic radiometric products (i.e., those directly computed from profile data): the subsurface normalized values, Lun (0−, λ), Eun (0−, λ), and Edn (0−, λ), and the associated diffuse attenuation coefficients within the extrapolation interval, Kl(λ), Ku(λ), and Kd(λ). The choice of presenting results for the normalized quantities, Lun (0−, λ), Eun (0−, λ), and Edn (0−, λ), instead of those for the absolute quantities, Lu(0−, λ), Eu(0−, λ), and Ed(0−, λ), is justified by the need to minimize the effects of differences in Ed(0+, λ) between fixed-depth and continuous profiles collected at slightly different times (i.e., within 15 min). Additional radiometric products included in the analysis are R(λ), Q(λ), LWN (λ), and WN(λ).
A sample of these data products, including Lun, Kd, and R, is presented in Fig. 4. Notable is the large variability displayed by all products. In particular, different R(λ) spectra show maxima at 490, 510, or 555 nm, which indicate waters dominated by various fractions of pigmented and nonpigmented particles in addition to colored dissolved organic matter.
The comparison of products obtained from fixed-depth and continuous profile data is presented and summarized through average percent differences, ψ, and average absolute percent differences, |ψ|, of N matchup profiles. While ψ highlights the existence of a bias between the compared quantities, |ψ| quantifies the average uncertainty.
The values of ψ are computed through
where i is the profile index and ψi is given by
with superscript F indicating the subsurface quantities computed from fixed-depth profiles and superscript C indicating the reference quantities computed from continuous profiles. The absolute values of ψi, |ψi|, are used to compute the average absolute percent differences |ψ| according to
Results from the data analysis are summarized in Table 3 at the various center wavelengths for basic and derived radiometric products. Specific comparisons between Lun, Eun, Edn, and Kd determined from the two methods are additionally displayed in Fig. 5 through scatterplots of data at the center wavelengths of 443, 555, and 665 nm assumed representative of the values for the 412–683 spectral region.
Results in Table 3 generally exhibit different uncertainties and biases for data in the 412–555-nm spectral range with respect to data at 665 and 683 nm. Specifically, Lun exhibits uncertainties slightly varying from 2.9% at 412 nm to 1.9% at 510 nm, increasing above 4% at 665 and 683 nm. The related biases spectrally increase from −0.5% to 1.1% in the 412–555-nm range, while they show values of 0.1% and −2.1% at 665 and 683 nm, respectively. The upward irradiance Eun exhibits uncertainties of the order of 3% between 412 and 490 nm, increasing then to 5.0% and 4.4% at 665 and 683 nm, respectively. The related biases increase from 2.0% at 412 nm to approximately 3% at 555 and 665 nm. A decrease to 0.8% is then observed at 683 nm. The downward irradiance Edn exhibits uncertainties decreasing from 2.4% at 412 nm to 1.5% at 555 nm, increasing then to approximately 2% at 665 and 683 nm. The related biases are below 0.4%.
As expected, the values and spectral variations of uncertainties and biases observed for Lun, Eun, and Edn are related to the respective diffuse attenuation coefficients Kl, Ku, and Kd. Specifically, Kl exhibits uncertainties increasing from 5.8% at 412 nm to 11.2% at 555 nm, and decreasing to 5.8% and 7.1% at 665 and 683 nm. The related biases increase from 2.0% at 412 nm to 4.3% at 555, and drop to −0.6% and −4.4% at 665 and 683 nm, respectively. Uncertainties in Ku show a spectral increase from 9.1% at 412 nm to 20.3% at 555 nm, and fall to 6.7% at 665 and 683 nm. The related biases increase from 7.6% at 412 nm to 16.9% at 555 nm, and drop to 2.2% and 0% at 665 and 683 nm, respectively. Among the diffuse attenuation coefficients, the uncertainties and biases are the lowest for Kd. In particular, uncertainties exhibit values increasing from 6.0% at 412 nm to 9.1% at 490 nm, and again decreasing to 6.9%, 2.6%, and 2.3% at 555, 665, and 683 nm, respectively. The related biases are generally within ±0.7%, with the exception of 2.6% at 555 nm.
The spectral dependence of uncertainties observed for Kl and Ku is mostly explained by the increase of the spectral effects due to optical stratifications and wave perturbations, with a decrease of the attenuation coefficient (cf. Kl and Ku values in Table 2 with the related uncertainties in Table 3). Instead, uncertainties exhibit a less pronounced spectral dependence for Kd. This is explained by an expected nonappreciable dependence of Kd on optical stratification effects and a higher sensitivity to wave perturbations within the near-surface homogeneous layer utilized for its determination.
The small biases (within ±0.4%) observed for Edn suggest the equivalence of the fixed-depth and continuous profile methods for the determination of this specific quantity. The appreciable biases determined for Lun, showing values of −2.1% at 683 nm and of 1.1% at 555 nm, and those even more pronounced for Eun at all center wavelengths, exhibiting a minimum of 0.8% at 683 nm and a maximum of 3.6% at 555 nm, indicate the presence of perturbing elements that might affect the linearity with depth of the log-transformed radiometric data. Possible explanations are (i) surface waves causing focusing and defocusing of sunlight and thus leading to large fluctuations in the field of view of sensors at depths that also depend on wave height (Zaneveld et al. 2001); (ii) inelastic processes due to Raman scattering, and, CDOM and chlorophyll a fluorescence; and (iii) the presence of vertical gradients in the concentration of optically significant materials in the upper sea layer.
Former analysis on the effects of waves in the determination of subsurface radiometric quantities performed with continuous profiles characterized by different depth resolutions, and thus implicitly sensitive to wave effects, did not highlight any appreciable bias in the computed subsurface values of Eu and Lu (Zibordi et al. 2004). This suggests a negligible contribution of wave perturbations to the bias observed for Eun and Lun. A theoretical evaluation of the effects of inelastic processes was performed applying the HYDROLIGHT code (Sequoia, Redmond) to simulate radiometric processes in a homogeneous water column at fixed depths using input parameters reflecting the average measurement conditions characterizing the present dataset (see Table 1). Including or excluding inelastic processes, results have shown differences generally well within 0.2% for subsurface values extrapolated from modeled data at 1 and 3 m in the 412–555-nm spectral range. However at 670 nm, close to the 665- and 683-nm center wavelength included in the experimental dataset, the observed differences are lower than 0.2% for Ed and approximately 3% for both Lu and Eu. These results implicitly suggest that gradients in the vertical distribution of optically significant constituents, with the additional contribution of fluorescence effects by chlorophyll a at 665 and 683 nm, might be the major source for the appreciable biases observed for Eun at all center wavelengths and for Lun at 555 and 683 nm.
When compared to Edn, the higher sensitivity of Eun and (to a lesser extent) of Lun to optical stratifications in the water column is explained by the different physical processes leading to the measurement of these quantities. This can be illustrated considering the case of a homogeneous subsurface layer on top of a second layer exhibiting different optical properties (which is a frequent situation in coastal regions). When data profiling is restricted to the first homogeneous layer, Edn is mostly determined from radiance contributions due to forward transmission of sunlight within this layer. Differently, Eun and Lun are determined from radiance contributions due to backward transmission of light within both layers (where the relative contribution from each layer varies with depth). Consequently, contrary to Lun and Eun, Edn measurements performed within the homogeneous deployment layer are almost unaffected by in-water optical stratifications occurring at deeper layers.
As expected, uncertainties and biases of derived data products result from those characterizing the quantities used for their determination. An example is LWN showing uncertainties and biases identical to those of Lun from which it is directly computed. Differently, the uncertainties for WN result from the statistical composition of the independent uncertainties of Lun and Edn. Specifically, WN exhibits uncertainties generally slightly higher than those obtained for Lun (and consequently of LWN), with values close to 3% in the 412–555-nm spectral region and close to 5% at 665 and 683 nm. The related biases are close to 1% in the 412–555 spectral region and become slightly more negative than those observed for LWN at 665 and 683 nm with values of −0.3% and −2.3%, respectively (these latter differences are probably explained by chlorophyll a fluorescence).
The uncertainties determined for R result from the statistical composition of the independent uncertainties of Eun and Edn, and exhibit values varying from 3.5% to 4.0% in the 412–555-nm spectral region and values of 5.6% and 4.9% at 665–683 nm, respectively. The related biases vary from 2.4% to 2.9%, with the exception of 1.1% at 683 nm.
The uncertainties determined for Q depend on those of Lun and Eun. But different from WN and R, the uncertainties of the basic radiometric products used to determine Q exhibit some degree of correlation and thus do not simply add statistically. As a consequence, Q exhibits uncertainties that are generally analogous to those of Lun with values varying from 2.9% down to 2.3% in the 412–555 nm spectral region and values of 3.3% at 665 and 683 nm. The related biases show values spectrally decreasing from 2.7% to 2.1% in the 412–555 nm spectral region and values close to 3% at 665 and 683 nm.
Results presented in Table 3 are hereafter discussed as a function of the diffuse attenuation coefficient and of the in-water optical stratification within the extrapolation interval, which might affect the determination of radiometric products. In addition, uncertainties in band ratios of radiometric products from continuous and fixed-depth profile data are also investigated. Supplementary analyses are finally proposed to quantify uncertainties due to environmental perturbations for each method using replicate measurements, and to investigate the statistical significance of the observed differences between methods.
a. Sensitivity to Kd and to in-water optical stratification
Tables 4 and 5 show results obtained from the comparison of radiometric products determined from fixed-depth and continuous profile data at 443, 555, and 665 nm as a function of Kd and of an empirical stratification index.
The analysis as a function of the diffuse attenuation coefficient has been carried out considering two classes of Kd values. Specifically, matchup data have been partitioned using the threshold Kd(490) = 0.14 m−1. This has been set consistently with a former study focused on the analysis of wave perturbations on radiometric products determined from in-water continuous profiles (Zibordi et al. 2004). Data in Table 4 indicate that a decrease of Kd leads to a general reduction in the uncertainty and bias values for most of the quantities investigated, with the exceptions of Edn and Kd. This is explained by a large increase of perturbations due to wave effects on downward irradiance measurements as Kd decreases.
The analysis based on the stratification index has relied on a flag empirically set using the extrapolation interval chosen for the continuous profile data: radiometric products derived from fixed-depth profile data are assumed affected by optical stratifications when the nominal discrete deployment depths (i.e., 1 and 3 m) are not fully included in the extrapolation interval selected for the processing of continuous profile data (it is recalled that during data processing, the extrapolation interval of each continuous profile is iteratively chosen to satisfy the requirement of linear decrease with depth of log-transformed radiometric data). Results in Table 5 indicate an appreciable increase in uncertainty and bias for Ku, Eun, Kl, and Lun and the derived quantities for cases showing an optical stratification within the extrapolation interval (e.g., uncertainty and bias of 24.6% and 23.8%, respectively, for Ku). Differently, a slight decrease is observed for Kd and Edn. This is again explained by the fact that these quantities are more affected by wave perturbations than by gradients in the vertical distribution of optically significant constituents. In fact, the large occurrence of cases characterized by relatively high Kd in the subset identifying stratified conditions (which implies less pronounced wave effects) leads to smaller uncertainties and biases in Kd and Edn with respect to those resulting from cases related to homogeneous conditions characterized by relatively low Kd values.
b. Band ratios
The rationale for comparing band ratios of radiometric products determined from the different methods is provided by their frequent application in empirical modeling to derive seawater inherent optical properties or the concentration of biogeochemical constituents. As expected, the use of band ratios leads to a decrease in uncertainties. Specifically, results presented in Table 6 show biases generally well below 1% for all the basic quantities (exceptions are data at 665 nm). In addition, the uncertainties for R, LWN, and WN exhibit values generally in the range of 1%–2%, except values of 3%–4% for the 665/555 ratio. Uncertainties lower than 1% are generally observed for Q; an exception is the value of 2% determined for the 665/555 ratio.
c. Environmental perturbations
A comprehensive appreciation of the two methods applied in coastal regions requires a quantification of the effects of environmental perturbations in the determination of the various radiometric products. By assuming 100% repeatability of measurements in identical environmental conditions for each method, an estimate of environmental perturbations has been made by comparing results from successive (both continuous and fixed depth) profile data typically collected within 15 min. For each method, results are presented in Table 7 on the basis of 36 cases representative of the different measurement conditions. These results are complemented by the data in Fig. 6 illustrating values and short-term variability of the beam attenuation coefficient at 412 nm, c(412), and the backscattering coefficient at 443 nm, bb(443), determined with an AC-9 (Wet Laboratories, Philomath, Oregon) and a Hydroscat-6 (HOBI Laboratories, Tucson, Arizona), respectively (Berthon et al. 2007). Graphs show values of c(412) in the range of 0.1–3.6 m−1 with short-term average variations of 1.8% as determined from absolute differences between replicate measurements. Similarly, bb(443) exhibits values in the range of 0.001–0.028 m−1 with average short-term variations of 2.3%.
As expected the uncertainties due to environmental perturbations for most of the quantities determined from each method are lower than those resulting from the comparison of data from fixed-depth and continuous profiles (see Tables 3, 7).
When restricting the analysis to continuous profile data at 555 nm, the largest differences among the basic radiometric quantities are observed for the diffuse attenuation coefficients (9.7%, 5.1%, and 3.9% for Kd, Kl, and Ku, respectively) while the lowest values are observed for Lun, Eun, and Edn (1.3%, 1.5%, and 2.1%, respectively). Among derived quantities, the largest values are observed for WN and R (2.7%), while the lowest values are observed for Q and LWN (1.0% and 1.3%, respectively).
For basic radiometric products derived from fixed-depth profile data at 555 nm, in agreement with results from continuous profiles, the largest perturbations due to environmental effects are also observed for the diffuse attenuation coefficients (10.4%, 9.1%, and 3.8% for Ku, Kl, and Kd, respectively) while the lowest values are observed for Edn, Lun, and Eun (0.8%, 1.8%, and 2.3%, respectively). Similarly to continuous profiles, the largest values among derived quantities are observed for R and WN (3.1% and 2.2%, respectively), while the lowest values are observed for LWN and Q (1.8% and 1.9%, respectively).
Results from fixed-depth data, when compared to those obtained from continuous profile data, indicate a slight reduction of environmental effects in products derived from Ed measurements and a slight general increase for the other products. This finding is again explained by perturbations due to wave effects, which are minimized by increasing the number of data used to compute the radiometric products. In fact, the low sensitivity to near-surface optical stratifications of Ed measurements, and the large number of individual data taken at each fixed depth (more than 1000, compared to the approximately 400 cumulative measurements, including both the down- and upcasts, available on average for the extrapolation process with continuous profiles), fully support the reduction of environmental uncertainties (mostly due to wave perturbations) in Edn and Kd. On the contrary, for the quantities derived from Lu and Eu measurements the increased number of observations available for fixed depth with respect to continuous profile data does not appear to produce any benefit. The increase in environmental uncertainties observed for these latter products, which translates into an increased difficulty of capturing the actual slope defining the linear decrease with depth of log-transformed data, is likely due to the effects of nonhomogeneities in the near-surface seawater optical properties. These effects appear mitigated by the extrapolation process performed on continuous profile data, probably due to the iterative selection of extrapolation intervals.
d. Significance of observed differences between methods
The differences between methods have been investigated through an additional analysis relying on a hypothesis testing procedure based on the t test (e.g., DeGroot and Schervish 2002). It is briefly recalled that hypothesis testing is a probabilistic scheme to decide if observations provide enough statistical evidence to reject the hypothesis initially believed correct, which is the null hypothesis. In this study, the null hypothesis is that the methods are equivalent (i.e., differences are only due to random noise in measurements). The statistical evidence is given by the P value, which is the probability that two methods equivalent by definition would produce differences equal or larger than those actually observed between data products determined from fixed-depth and continuous profile data. It is pointed out that the null hypothesis is increasingly satisfied with an increase of the P value. The confidence threshold, which infers the validity of the null hypothesis, is defined through the level of significance. This is typically set to 0.05, indicating that when the P value is lower than a confidence threshold of 5%, the null hypothesis is rejected and consequently the two methods cannot be considered statistically equivalent.
As detailed in the appendix, the P value is determined from the logarithm of the differences, di, between products derived from fixed-depth, nF(i), and continuous nC(i), profile data:
The logarithm transforms a multiplicative factor into an additive contribution, and hence the uncertainties of di have the same statistical weight for all observations regardless of the specific values of nF(i) and nC(i). The requirement of normality of the di distributions for the considered radiometric products has been verified with the Kolgomorov–Smirnov test (DeGroot and Schervish 2002; Press et al. 1992).
The results shown in Table 8 indicate that there is no significant difference between the two methods for Lun, Edn, Kl, Kd, and LWN. Instead, the difference is generally highly significant for Eun, Ku, R, and Q. Moreover, P values close to the confidence threshold are observed for WN (665 nm excluded). In agreement with the results presented in Tables 4 and 5, the t test indicates that differences between the two methods are less significant for data collected when Kd(490) < 0.14 m−1 and for homogenous conditions (explicit results are not presented). In the presence of optically stratified conditions (results not presented), biases indicating systematic significant differences between methods are also observed for Eu and related parameters (Ku, R, and Q). Finally, the analysis shows that there is no statistically significant difference between methods for LWN, the primary product for remote sensing applications. With the exception of results at 683 (likely affected by chlorophyll a fluorescence) and 555 nm (likely due to the high penetration depth of light making radiometric products more dependent on in-water optical stratifications), the P values are significantly higher than 5%.
Additional and practical insight to interpret the Student’s t test analysis is offered by the absolute value of the ratio, ρ, between the bias, ψ, and the scattering, |ψ|, representing the average symmetry of the data products distribution along the ideal one-to-one line (see Fig. 3). This additional statistical quantity, also presented in Table 8, is used to question whether the average uncertainty (i.e., the scattering between data products derived with the two methods, quantified as |ψ| and depending on many factors including the environmental variability) justifies the corresponding average systematic difference (i.e., quantified as ψ and more strictly related to methods, assuming that the environmental variability can be neglected on average). The symmetry is complete for ρ = 0. Instead, when ρ = 1, data products from the fixed-depth profile method are systematically overestimated or underestimated with respect to those obtained by applying the continuous profile method. As an example (see Table 8), the scattering and the bias between Eun data products at 412 nm are 3.1% and 2.0%, which result in ρ = 0.7 (i.e., a significant asymmetry). In this case the P value is 0 (i.e., P < 5%) and indicates that the bias has a systematic component. Also, the Kl data products exhibit a bias of 2.0%, associated with a scattering of 5.8%. Now the corresponding P value is 34% (i.e., P > 5%), thus suggesting that there are no systematic differences between methods, in agreement with the smaller asymmetry factor of 0.3.
As a rule of thumb, large P values correspond to differences between methods likely depending on environmental factors. By the same token, it is important to observe that symmetry values higher than 0.5 correspond to P values systematically less or equal to the confidence threshold of 5%. Regrettably, a straight relation between ρ and P value cannot be envisaged because of the dependence of the P value on the number of observations. Furthermore, it is stressed that ρ and the P value are defined on the basis of different quantities (the first is derived from the difference between the logarithm of data products, while the latter is the absolute value of the ratio of the average data bias and scattering).
It is finally emphasized that the t test cannot (and has not been used to) state if the scattering and/or the bias satisfy operational requirements for applying the investigated methods to specific operational tasks (e.g., calibration and validation activities). In fact, a small bias may be statistically significant (when compared to the scattering) but negligible from an applicative perspective.
6. Summary and conclusions
The equivalence of methods based on fixed-depth and continuous profile data for the determination of subsurface radiometric products has been investigated at a coastal site in the northern Adriatic Sea characterized by moderately complex waters. Fixed-depth and continuous profile data have been produced with the same measuring system with a 6-Hz acquisition rate. Fixed-depth profile data have been collected recording measurements at 1- and 3-m nominal depths for 3 min each. Continuous profile data have been collected with a deployment speed of approximately 0.1 m s−1. The comparison of results from 105 matchups spanning over different measurement conditions and evaluated through data in the 412–683-nm spectral range indicates uncertainties of 2%–4%, 3%–5%, and 2% for Lun, Eun, and Edn, respectively. The related biases vary from −2% to 1% for Lun, are in the range of 2%–3% for Eun, and are lower than 0.5% for Edn. The uncertainties and biases for Kl, Ku, and Kd exhibit much higher values when compared to Lun, Eun, and Edn. Specifically, uncertainties exhibit values of 6%–11%, 7%–20%, and 2%–9% while the related biases vary from −4% to 4%, 0% to 17%, and −1% to 3% for Kl, Ku, and Kd, respectively. The derived products R, Q, LWN, and WN exhibit uncertainties of 4%–6%, 2%–3%, 2%–4%, and 3%–5%. The related biases exhibit values varying between 1% and 3%, 2% and 3%, −2% and 1%, and −2% and 1%, respectively. All the former uncertainties and biases vary spectrally, and are appreciably related to the seawater attenuation coefficient and to nonhomogeneities in the near-surface vertical distribution of seawater optically significant constituents.
An evaluation of results obtained by partitioning the dataset with a threshold applied to Kd indicates a general diminishing with a Kd(490) decrease of the uncertainty and bias for most of the quantities investigated. Exceptions are Edn and Kd because a decrease of Kd produces an increase of the perturbations due to wave effects on downward irradiance measurements. An evaluation of the perturbing effects of optical stratifications in the extrapolation interval applied for fixed-depth profile data, which leads to a nonlinear decrease with depth of log-transformed radiometric measurements, indicates an appreciable increase in uncertainty and bias for Ku, Eun, Kl, Lun, and the related derived quantities. Only a slight worsening of results is observed for Kd and Edn. This is explained by the fact that these latter quantities are more affected by wave perturbations than by slight gradients in the vertical distribution of optically significant constituents.
The effects of environmental perturbations in the determination of radiometric products from continuous and fixed-depth profiles have been evaluated through the analysis of replicate profiles. As expected, results exhibit uncertainties for each individual method that are generally appreciably lower than those resulting from the comparison of radiometric products from fixed-depth and continuous profile data. In addition, the environmental uncertainties estimated for products derived from Ed are lower for fixed-depth than for continuous profile data. On the contrary, an increase in environmental perturbations is observed on all other derived products from fixed-depth measurements.
Statistical differences between the two methods have also been investigated through a hypothesis testing procedure based on the t test. Results confirmed that systematic differences between methods tend to decrease with low attenuation coefficients and vertical homogeneity of the optical properties of seawater. Results also indicate that fixed-depth profile data, when compared to continuous profile data, do not add systematic differences to LWN at most of the sampling center wavelengths. This finding is fully supported by the uncertainties determined for LWN from comparisons based on percent differences (see Table 3). In fact, those uncertainties exhibit values (i.e., 2.8%, 2.3%, and 4.2% at 443, 555, and 665 nm, respectively) close to the corresponding quadrature sum (i.e., 2.5%, 2.2%, and 4.7%) of the environmental perturbations given in Table 7 for fixed-depth and continuous radiometric profiles, respectively.
In conclusion, as long as the discrete deployment depths are properly selected, overall results indicate statistical equivalence of fixed-depth and continuous profile data products in supporting satellite ocean color applications in moderately complex coastal waters. For the case addressed in this study, the analysis of LWN data products has shown a slightly higher uncertainty for values determined from discrete depths with respect to continuous profiles with an increase of approximately 1% (mostly due to a higher contribution of environmental perturbations to fixed depth, with respect to continuous profile radiometric data products), and with a bias generally within ±1% (exception is −2% at 683 nm, likely due to inelastic processes).
Hypothesis testing is a probabilistic framework to decide if observations provide enough statistical evidence to reject the hypothesis initially believed to be correct. The statistical test is checked with (i) the level of significance, α, which defines the confidence threshold; and (ii) the P value (the larger the P value, the better the observations satisfy the null hypothesis). In this study, the t test (e.g., DeGroot and Schervish 2002) is applied to verify the statistical differences between the considered methods. The prerequisite for the applicability of the t test is the normality of data distributions.
If a variable Z follows a standard normal distribution and Y indicates the sum of the square of N independent standard normal variables, the Student’s ts random variable is given by the ratio
This variable can be derived from N data points (di, …, dN) sampled from a normal distribution with mean μ and variance σ2. By indicating with d the sample mean, and defining S2 = ∑i = 1N (di − d)2, Z = N1/2 (d − μ) / σ, and Y = S2 / σ2, Z consequently follows a standard normal distribution and
is the t-test variable with N − 1 degrees of freedom.
In the current analysis, the t test is used to verify that the di values are sampled from a normal population with an expected mean μ = 0. If ts indicates the value derived through Eq. (A2), the resulting P value is
Corresponding author address: Dr. Giuseppe Zibordi, Via E. Fermi 1, Ispra 21027, Italy. Email: email@example.com