1. Introduction

The collection of field measurements with in-water optical systems underpins a number of ocean color applications including the assessment and vicarious calibration of space-born radiometric data [1–3], or the development of inversion schemes to obtain maps of derived products [4, 5]. Following early developments [6], in-water radiometers can be roughly separated into fixed-depth and profiling systems [7]. Fixed-depth radiometers are generally operated on bio-optical buoys [8,9]. In-water radiometric profiling is instead performed through winched or free-fall systems. Winched systems were extensively used for many years and are still applied on specific deployment platforms [10, 11]. Free-fall systems introduced during the late 1980s, have been increasingly used to perform in-situ measurements from ships during oceanographic campaigns [7, 11, 12].

Uncertainty requirements for insitu radiometric measurements are strictly connected to the specific application [7]. For instance, the validation of satellite radiometric products such as the water-leaving radiance determined from at-the-satellite radiance corrected for atmospheric perturbations, implies the use of insitu data affected by uncertainties generally below 5% across the visible spectrum. These uncertainties should be even lower for insitu data applied for the indirect calibration of space sensors (i.e., system vicarious calibration).

A comprehensive quantification of the uncertainties affecting insitu measurements implies accounting for contributions due to calibration, sensors performance (e.g., linearity and temperature responses), effects of environmental variability (e.g., light focusing and defocusing by wave facets), data reduction algorithms (e.g., extrapolation of sub-surface values), and perturbations by instrument housing and deployment platform (e.g., shading). It is also important that each of these contributions is kept to within small values, ideally not exceeding 1–2% to avoid that combined uncertainties may exceed the target thresholds.

Within such a general context, the scope of the present study is to investigate the effects of integration time on optical data from in-water profiling systems. Analyzed quantities are the downward irradiance E_d, the upward irradiance E_u and the upwelling radiance L_u, henceforth generically denoted ℜ. Considered data products are the subsurface value ℜ₀ and the diffuse attenuation coefficient K_ℜ, derived from the regression of radiometric measurements acquired at different depths and hereafter indicated as regression products. The sampling setting is defined by a constant deployment speed v_prf and a dynamical integration time t that increases at the lessening of the radiometric signal. Specifically, the in-water i-th optical measurement is obtained through the accumulation of photons collected within a depth interval Δ_z =[z_i, z_i + v_prf · t]. The recorded value denoted with the tilde notation is

The effects of integration time can be individually studied in the ideal condition of a still sea. In real cases, environmental factors and data reduction solutions combine with the uncertainties affecting ℜ₀ and K_ℜ. For instance, wind-driven waves at the air-water interface, besides focusing and defocusing the light [13–18], can influence the response of the pressure gauge at the depth of the optical system. Linear and non-linear data reduction schemes can also lead to slightly different results [19]. Overall, the effects of integration time, which convolve with uncertainties from environmental perturbations and processing solutions, differently affect the various radiometric quantities.

All these elements are considered in the present study to stress measurement and processing solutions in view of increasing the quality of insitu radiometric data products from hyperspectral in-water free-fall profiles. The study is executed using the MOX Monte Carlo code to simulate radiometric fields for different measurement conditions by applying high-performance computing solutions [14, 19–24]. The response of in-water optical sensors is investigated based on a two-dimensional representation of the underwater light distribution.

Study results are obtained assuming an ideal hyperspectral radiometer with negligible delays (i.e., null latency time) between the end of a measurement and the start of the successive one, signal digitization through a 24-bit analog-to-digital converter (ADC) and an overall uncertainty of 10 counts. Actual measurements are then discussed by: 1) addressing wavelength-specific integration times; 2) accounting for 16-bit digitization and non-negligible latency time; and 3) evaluating the effects of an increase of the radiometric measurements per unit depth adopting the multicasting scheme.

2. Data and methods

This section presents the methods applied to: 1) model the sky-radiance distribution; 2) represent the air-sea interface as a function of the wind speed v_wnd; 3) generate the in-water radiometric fields; 4) produce virtual radiometric profiles accounting for the effects of integration time; and 5) compute regression products using linear LN and non-linear NL reduction schemes.

The radiance distribution of a clear sky is largely determined by the sun zenith and the optical properties of the atmospheric constituents (mostly aerosols and air molecules). Similarly, the in-water light distribution depends on the seawater Inherent Optical Properties (IOPs) and, on a lesser extent, on sea-surface waves. To minimize computing time, in this work the sky- and in-water radiance distribution are modeled separately. This allows for applying the same sky dome to initialize the photon tracing in a number of in-water radiometric simulations with different IOPs and/or sea-surface configurations.

2.1. Sky-radiance distribution

In addition to the overcast illumination case, the diffuse sky radiance is defined for a sun zenith θ_sun of 30° [see Fig. 1 and Table 1] in plane-parallel, homogeneous and cloud-free conditions [25] with ideal Lambertian bottom. The photons tracing is at first performed in the atmospheric domain. The cumulative distribution function (CDF) for initializing the photon direction above the sea-surface is then derived from the solar plane of the full sky dome. The diffuse-to-total ratio ρ_d between diffuse and total irradiance is also evaluated (Table 2 [23]).

 

Fig. 1 Panel (a) shows an example of the distribution of the diffuse sky-radiance (i.e., without the solar disk) for θ_sun = 30° and in units of W m⁻² nm⁻¹. The probability density function (PDF) and the cumulative distribution function (CDF) of incoming photon direction θ in the solar plane are presented in Panels (b) and (c), respectively.

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Table 1. Parameters for Simulating the Sky-radiance Distribution at λ =490nm.

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Table 2. Parameters Determining the Clear Sky and Overcast Illumination Conditions.

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Atmospheric optical properties are specified at λ =490 nm chosen as representative wavelength for this analysis. The scattering optical thickness of the atmosphere is ${\tau }_b={\tau }_b^{\mathrm{M}}+{\tau }_b^{\mathrm{A}}$, where ${\tau }_b^{\mathrm{M}}$ and ${\tau }_b^{\mathrm{A}}$ denote the molecules M and aerosol A contributions, respectively. The absorption optical thickness is instead ${\tau }_a={\tau }_a^{\mathrm{O}}+{\tau }_a^{\mathrm{G}}$, with ${\tau }_a^{\mathrm{O}}$ and ${\tau }_a^{\mathrm{G}}$ accounting for weak contributions of ozone O and permanent gases G, respectively. The absorption a_atm and scattering b_atm coefficients are defined scaling the optical thickness by the atmosphere height, and assuming that the scattering rather than absorption determines the aerosol contribution.

The Monte Carlo path length l of the photon trajectory in the atmosphere is

The photon weight is updated as

Photons ending their trajectory in the atmosphere undergo Rayleigh scattering of air-molecules if u < µ, where $u\in \mathcal{U}(0,1)$ and $\mu ={\tau }_b^{\mathrm{M}}/{\tau }_b$ [26, 27]. The Mie theory defines the aerosol scattering when u ≥ µ [27–31].

2.2. Sea surface modeling

The sea surface is defined over a domain of 40m length discretized by N_s =32768 points (2¹⁵), which leads to a resolution of Δ_s = L_x/N_s = 1.2mm and spatial coordinate x(s) = s · Δ_s with s = 1, …, N_s. The sea-surface elevation z(s) is expressed under the assumption of linear wave theory applying the Inverse Fast Fourier Transform (IFFT, [32])

A fully developed sea is modeled in this study by: 1) computing the two-sided discrete values of the elevation variance; 2) sampling the amplitudes of the harmonic components with normal distributions; and 3) defining the IFFT Hermitian coefficients to obtain a real-valued z(r). The spectral density function $\mathcal{S}(k)$ is iteratively adjusted until converging to the target value [23]. Statistical figures of wind-induced surface perturbations are summarized in Table 3 for v_wnd values of 4 and 8ms⁻¹.

Table 3. Examples of Sea-surface Statistical Figures.

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2.3. In-water radiometric simulations

Ray tracing starts at a random point above the sea surface determined by the photon cosine directors µ_x and µ_z, as well as the photon weight w. The photon source is either the sun (direct light) or the sky (diffuse light) quantified in Table 2 as a function of the diffuse-to-total ratio ρ_d of the downward irradiance. If the photon comes from the sun, the initial travel direction is given by the sun zenith angle θ_sun and its weight is the unity. Otherwise, the cumulative distribution function derived from the sky radiance is used to initialize the photon direction and weight [see Fig. 1(c)]. The Snell and Fresnel equations redefine the photon properties across the sea-surface.

In-water radiometric fields are simulated in a domain with horizontal Δx = 0.01 m and vertical Δz =0.001m resolution (Fig. 2 and Table 4). The in-water free path length traveled by a photon before absorption or scattering follows the exponential probability density function p(τ)=exp(−τ), where τ =c · r is the optical distance, c is the attenuation of seawater [m⁻¹], and r [m] is the geometric distance. The Monte Carlo photon tracing is executed solving P τ =u for τ, where P is the cumulative distribution function of the optical distance and $u\in \mathcal{U}(0,1)$.

 

Fig. 2 Schematics of photon tracing. The radiometric sensor of the virtual profiler is represented by the “ray collecting bins” located at the nodes of the simulation grid.

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Table 4. Settings of In-water Radiometric Simulations.

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The photon weight is scaled at the end of each free path by the single scattering albedo ω = (c − a)/c, where a is the absorption coefficient of seawater [m⁻¹]. The new direction is obtained from the volume scattering function β(θ) [m⁻¹ sr⁻¹]. Namely, the scattering coefficient b = c − a and scattering phase function $\tilde{\beta }=\beta /b$ define the probability density function $p(\theta )=2\pi \tilde{\beta }(\theta )\sin (\theta )$. Each zenith angle of scattering θ is sampled by solving P(θ) =u for θ, where P is the cumulative distribution function of the scattering angle θ and $u\in \mathcal{U}(0,1)$. The azimuth scattering angle is randomly chosen between 0 and 2π. If a photon hits the left (right) side of the domain, its trajectory is continued from the right (left) side at the same depth thanks to the periodicity of sea-surface waves.

A photon that reaches the lower boundary of the simulation domain is upward scattered with its weight scaled by the irradiance reflectance factor R_b = E_u/E_d. The bottom depths z_b and the related reflectances reported in Table 5 have been pre-determined for each water type by performing a preliminary set of low-resolution simulations in a water column of 100m neglecting the bottom effects (details not presented). This solution, which attempts to approximate an infinitely deep sea, has been implemented to meet computational constraints.

Table 5. Inherent Optical Properties (Absorption a, Attenuation c and Single Scattering Albedo ω) as well as Properties of the Virtual Bottom (Depth z_b and Reflectance Value R_b) Adopted to Perform Monte Carlo Simulations for Different Marine Optical Cases. The Coefficients of the Fournier-Forand Volume Scattering Function [34,35] are: Slope of the Junge Distribution m =3.5835 and Refraction Index n =1.34.

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The new cosine directors of photons scattered by the virtual bottom are ${\mu }_{\mathrm{z}}=\sqrt{u}$ and ${\mu }_{\mathrm{x}}=\sqrt{1-u_{\mathrm{z}}^2}$ sign(q − 0.5), where q and u are independent random numbers in $\mathcal{U}(0,1)$ and sign(x) = x/|x|. The photon tracing ends when its weight becomes lower than the threshold ζ_w =10⁻⁶.

Each time a photon intersects a ray-collecting-bin representing the detector [see Fig. 2], its weight is added to the corresponding entry of matrices expressing the radiometric fields according to the following rules: 1) µ_z < 0 for E_d; 2) µ_z > 0 for E_u; and 3) µ_z > cos(FAFOV/2) for L_u, where the full-angle field of view (FAFOV) for radiance measurements is set to 20°. Photons are traced within the entire water column, but their contribution to radiometric fields (tracking of the ray trajectory) is taken into account only between 0 and −10m to meet computing time and memory requirements.

The response of in-water optical sensors is analyzed in the following cases:

 

Fig. 3 Example of an E_d radiometric field simulated for highly absorbing marine waters (CDOM-dominated), v_wnd =4ms⁻¹ and θ_sun =30°.

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2.3.1. Virtual profiling

In the reality, waves travel at the surface while the radiometric sensors descend vertically to generate the optical profile. To optimize simulation efficiency, the present study considers a diagonal virtual profile below a static sea surface as illustrated in Fig. 4 [14, 19, 36]. Differences between the phase velocities of sea-surface waves are neglected assuming that all translate at the same v_wav speed

 

Fig. 4 Example of diagonal virtual trajectory below a fixed sea surface applied to compute the optical profile (see text for details).

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2.3.2. Pressure gauge correction

The still level z₀ =0, at which radiometric data products are ideally referred to, can differ from the depth sensed by a pressure gauge under a wavy sea-surface z_s(x). This effect is considered applying a pressure transfer model derived from the linear wave theory [37–39]. The adjusted depth z_g(x) of a photon collecting bin located at the grid-depth z (i.e., the actual distance from z₀) is computed as

 

Fig. 5 (a) Example of iso-depth lines accounting for the pressure gauge correction due to surface waves generated with v_wnd = 8ms⁻¹. (b) Standard deviation of the difference between subsequent iso-depth lines.

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2.3.3. Integrated radiometric values and measurement depth definition

Insitu radiometric measurements are simulated by relying on the schematic of Fig. 6, and the integration times with the related radiometric intervals listed in Table 6. The Min and Max radiometric values were determined accounting for 15% and 80% of the digitization range, respectively. Noise was added to the digitized signal by sampling from a normal distribution with standard deviation of 10 counts regardless of the integration time.

 

Fig. 6 Schematic of the process determining the integration times applied for measuring E_d, E_u and L_u values. The radiometric boundaries are specified in Table 6.

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Table 6. Values of the Integration Time for E_d and L_u [see also Fig. 6]. The Intervals for E_u are the Same as for E_d. These Intervals were Determined to Mimic those Actually Implemented in HyperOCR Radiometers Manufactured by Satlantic (Halifax, NS, Canada).

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Alternative measurement depths are considered for each radiometric value resulting from the accumulation of photons (integration) over time intervals between 8 and 2048ms. These include:

The Wgt method assumes an exponential decay of the radiometric quantity $\Re (z)={\Re }_0\cdot {\mathrm{e}}^{K_{\Re }\cdot z}$, where K_ℜ > 0, z is negative downward and ℜ₀ is set to 1 without loss of generality (Fig. 7). The integrated value ${\tilde{\Re }}_{z_i}^{z_i+{\mathrm{\Delta }}_z}$ between z_i and z_i + Δ_z (for Δ_z <0) is

 

Fig. 7 Schematics of the weighted (Wgt) measurement depth determined as a function of the diffuse attenuation coefficient.

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In summary, the Wgt method associates the depth z^∗ to the integrated radiometric value ${\tilde{\Re }}_{z_i}^{z_i+{\mathrm{\Delta }}_z}$.

2.3.4. Data reduction schemes

Linear and non-linear regression schemes [19] are both considered for the determination of radiometric regression products. The rationale is to determine differences between the two methods accounting for the convoluted effects of the integration time with other perturbing factors such as the light focusing and the pressure gauge response below a wavy sea surface.

The linear regression

The classical solution to determine ℜ₀ and K_ℜ is based on the linear regression (LN) as a function of z of the log-transformed ℜ(z) values [7, 40, 41]. The underlying assumption is the exponential decay

The non-linear regression

The average of log-transformed values is equal or lower than the logarithm of their averages because of the inequality between the arithmetic and the geometric mean [19, 42]. This can bias linear regression results. Offset corrections are difficult since perturbations due to light focusing and defocusing largely vary in the water column [11, 15, 16, 36, 43–46]. The nonlinear NL approach [47] has been proposed to compute ℜ₀ and K_ℜ by minimizing the SSE^NL error function

3. Results

3.1. Statistical figures

The depth separation between the data points of the Full resolution virtual profile is about the same as the vertical grid spacing of the simulation domain (i.e., Δz = 0.001m; Table 4). The corresponding NL data products are denoted as ${\left\{{\left[\mathfrak{J}\right]}_{\text{FULL}}^{\text{NL}}\right\}}^{\mathrm{S}}$, where ℑ indicates the ℜ₀ or K_ℜ regression products, and S denotes the environmental case (i.e., water type, illumination condition, wind speed) and the measurement setting (i.e., sensor deployment speed). The ${\left\{{\left[\mathfrak{J}\right]}_{\text{FULL}}^{\text{NL}}\right\}}^{\mathrm{S}}$ values, not affected by the integration time nor biased by the log-transformation (Sec. 2.3.4), represent the reference quantities for this study. Alternative regression results compared with ${\left\{{\left[\mathfrak{J}\right]}_{\text{FULL}}^{\text{NL}}\right\}}^{\mathrm{S}}$, instead, depend on the sampling depth definition (Sec. 2.3.3) and the data reduction scheme (LN or NL, Sec. 2.3.4). As an example, ${\left\{{\left[\mathfrak{J}\right]}_{\text{FST}}^{\text{LN}}\right\}}^{\mathrm{S}}$ indicates the LN regression of integrated radiometric data adopting the Fst depth definition. The percent difference ${\left\{{\left[{\delta }_{\mathfrak{J}}\right]}_{\text{FST}}^{\text{LN}}\right\}}^{\mathrm{S}}$ between ${\left\{{\left[\mathfrak{J}\right]}_{\text{FST}}^{\text{LN}}\right\}}^{\mathrm{S}}$ and ${\left\{{\left[\mathfrak{J}\right]}_{\text{FULL}}^{\text{NL}}\right\}}^{\mathrm{S}}$ is computed as

Data regressions are performed in a layer determined by the depths −0.2 and −4m. E_d, E_u and L_u data products are computed for a number of different settings, including: 1) deployment speeds of the optical system; 2) environmental conditions—i.e., sky-radiance distribution as detailed Table 2, sea-surface statistics as detailed in Table 3, and marine water type as detailed in Table 5; 3) LN and NL regression methods; and 4) measurement depths as defined in Sec. 2.3.3. Specific analyses are presented to illustrate the independent effects of integration time (Sec. 3.2) and the additional influence of environmental perturbations (Sec. 3.3). A list of symbols for the identification of data products analyzed in this study is presented in Table 7.

Table 7. List of Symbols of the Data Products Analyzed in this Study.

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3.2. The effects of integration time for a still sea and overcast illumination

A still sea and overcast illumination are considered in this section to exclude perturbations due to light focusing and the influence on the pressure gauge by the surface waves. On this basis, Fig. 8 documents the data density in the water column (i.e., the number of radiometric measurements per unit depth) when the optical system is deployed at v_prf =0.2ms⁻¹. The inset of each panel shows the integration time as a function of depth [see also Table 6]. The Case-1, CDOM-dominated and NAP-dominated water cases are presented in the row panels from top to bottom, while results for E_d, E_u and L_u are from left to right. The sampling density varies from a maximum of about 40m⁻¹ for E_d close to the sea-surface in oligotrophic Case-1 waters, to a very few records per meter at 4 m depth for E_u and L_u in the CDOM-dominated case. Results document how the integration time increases with depth due to the lessening of the radiometric signal. A particular case is the E_d optical profile in Case-1 waters presented in Fig. 8(a), where the integration time remains constant because of the low attenuation coefficient. It is also noted that the integration time of E_u quickly settles to its maximum value for all the considered water types, as shown in Figs. 8(b), 8(e) and 8(h). This highlights the expected limitations in using the same radiometric thresholds to define the integration time for both E_d and E_u values, as specified in Table 6.

 

Fig. 8 Number of radiometric measurements per unit depth (i.e., density of data) for overcast illumination and still sea surface. The deployment speed of the optical system is v_prf = 0.2ms⁻¹. The Case-1, CDOM-dominated and NAP-dominated water cases are ordered from top to bottom. Results for E_d, E_u and L_u are reported from the left to the right column panels. The integration time as a function of depth is displayed in each panel inset.

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The percent differences affecting data products as determined through Eq. (15), are presented in Fig. 9. The first and the second column panels refer to E_d0 and $K_{E_{\mathrm{d}}}$, respectively. Equivalent statistical figures for E_u and L_u regression products are documented in the subsequent column pairs. The effects of integration time on the reduction of optical profiles for Case-1 and NAP-dominated waters show: 1) a low influence on E_d data products; and 2) larger percent biases for $K_{E_{\mathrm{u}}}$ and $K_{L_{\mathrm{u}}}$ than for E_u0 and L_u0. Figs. 9(d) and 9(j) provide an example of how the effects of the integration time on $K_{E_{\mathrm{u}}}$ increase with the deployment speed of the optical system. It is noted that the Fst and the Lst depth definitions lead to an overestimate and underestimate of $K_{E_{\mathrm{u}}}$ values, respectively, as a consequence of the high integration time illustrated in Fig. 8 for E_u data. A similar tendency, but with smaller values, can be observed for $K_{L_{\mathrm{u}}}$ in Figs. 9(f) and 9(l). The effects of the integration time are slightly higher for NAP-dominated than for Case-1 waters. Figs. 9(i) and 9(k) indicate that the biases affecting E_u0 and L_u0 tend to be opposite to those affecting the attenuation coefficients (e.g., ${\left[K_{L_{\mathrm{u}}}\right]}_{\text{FST}}^{\text{NL}}<0$ corresponds to ${\left[K_{L0}\right]}_{\text{FST}}^{\text{NL}}<0$).

 

Fig. 9 Statistical figures from the comparison of regression products determined with Eq. (15) for a still sea and overcast sky.

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Figs. 9(m)–9(r) display an increase of the effects of the integration time for CDOM-dominated waters due to the hight absorption coefficient [see Table 5]. The overall trends of the Fst and Lst depths to respectively overestimate and underestimate the diffuse attenuation coefficient is here confirmed, as well as the opposite sign of the offsets affecting the attenuation coefficients and subsurface radiometric values. Some heterogeneity clearly affects the tendencies of the offset values as a function of the deployment speed. It is then noted that the change of integration time with depth can itself be a perturbing component. This can be explained considering the separation of a generic pool of samples into disjoint sets. If all partitions have the same size, then the average of the mean values of individual subsets (group means) is equal to the mean value of the entire dataset (grand mean) [49]. In the case of subsets with different sizes, instead, the average of the group means can under- or over-estimate the dataset grand mean depending on how the subsets were constructed. A similar scenario can influence the reduction of optical profile data and slightly change the offsets with respect to the case of a constant integration time (details not presented).

3.3. The convoluted effects of integration time with environmental factors

A wavy sea surface (v_wnd = 4 ms⁻¹) and clear sky (θ_sun = 30°) are now accounted to define complementary conditions with respect to the still sea and overcast sky of the previous section. The aim is verifying how the effects of integration time on the reduction of optical profile data combine with environmental factors in a sampling scenario of practical interest for ocean color applications. As expected, the number of integrated radiometric values as a function of depth shown in Fig. 10 for clear sky conditions indicates an increase in data density due to the higher total irradiance with respect to the overcast case [see Fig. 8 and Table 2].

 

Fig. 10 As in Fig. 8, but considering a clear sky (θ_sun = 30°) and a wavy sea-surface (v_wnd =4ms⁻¹).

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Following the same process applied for a steady sea surface and overcast sky, results obtained from simulated profiles for a wavy surface and clear sky are summarized in Fig. 11. Specifically, Fig. 11(a) indicates that the linear regression scheme produces ${\left[E_{\mathrm{d}0}\right]}_{\text{FULL}}^{\text{LN}}$ values 10% lower than the reference ones ${\left[E_{\mathrm{d}0}\right]}_{\text{FULL}}^{\text{NL}}$. This underestimate depends on the log-transformation of data affected by light focusing and defocusing effects as detailed in Section 2.3.4 and several published studies [14, 19, 23]. The manifold biases appearing in Fig. 11(b), instead, partially depend on the very low $K_{E_{\mathrm{d}}}$ values of the considered Case-1 water, which implies computing relative difference of quantities close to zero. This clearly confirms the difficulty to accurately determine the diffuse attenuation coefficient in oligotrophic waters. The example also shows that ${\left[E_{\mathrm{d}0}\right]}_{\text{DEF}}^{\text{LN}}~{\left[E_{\mathrm{d}0}\right]}_{\text{FULL}}^{\text{NL}}$, where the acronym Def denotes any of the considered depths. As already observed for the overcast sky condition, the lower integration time in proximity of the sea surface has a larger influence on $K_{E_{\mathrm{d}}}$ than E_d0 [see Table 8]. Figs. 11(c) and 11(e) document a negligible effect of the integration time on E_u0 and L_u0. Instead Fig. 11(d) and partially Fig. 11(f) indicate that the Fst and Lst depth definitions induce a positive and a negative bias in both $K_{E_{\mathrm{u}}}$ and $K_{L_{\mathrm{u}}}$ values, respectively. The biases affecting the results for NAP-dominated waters presented in Figs. 11(g)–11(l), follow trends similar to those reported for Case-1 waters in Figs. 11(a)–11(f). The main difference is the lessening of the $K_{E_{\mathrm{d}}}$ biases—cf., Fig. 11(b) and Fig. 11(h).

 

Fig. 11 As in Fig. 9, but in the presence of a clear sky (θ_sun =30°) and a wind-driven sea surface (v_wnd =4ms⁻¹).

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Table 8. Values of Differences δ (Indicating both ${\delta }_{{\Re }_0}$ or ${\delta }_{K_{\Re }}$) Determined with Eq. (15) and Values of the Related Standard Deviations σ (i.e., σ_ℜ or σ_K), from Fig. 11 for the Avg and Wgt Depths. Values are Provided for the Different Water Types, Deployment Speeds and Regression Methods (i.e., NL and LN). The Number of Measurements per Unit Depth (i.e., Meas. Density) Refers to the Extrapolation Interval.

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The CDOM-dominated case of Figs. 11(m)–11(r) attests the under- and the over-estimation of E_d0, E_u0 and L_u0 values when considering the Fst and the Lst measurement depths, respectively. It is also verified that this tendency mostly appears opposite to the bias reported for $K_{E_{\mathrm{d}}}$, $K_{E_{\mathrm{u}}}$ and $K_{L_{\mathrm{u}}}$, as observed for the overcast and still sea simulation case [see Fig. 9]. Namely, an underestimate of the subsurface value due to the Fst depths corresponds to an overestimate of the diffuse attenuation coefficient. Opposite results can be observed with the Lst depths. It is finally noted that ${\left[{\delta }_{\mathfrak{J}}\right]}_{\text{WGT}}^{\text{NL}}$ as well as ${\left[{\delta }_{\mathfrak{J}}\right]}_{\text{AVG}}^{\text{NL}}$ show the best data reduction performances among the various depth definitions. This highlights the importance to adopt a proper determination of measurement depth to limit the impact of perturbations due to integration time.

In view of supporting a more comprehensive analysis of results, the mean values (generically indicated as δ) of computed ${\delta }_{{\Re }_0}$ and ${\delta }_{K_{\Re }}$ values displayed in Fig. 11, are also summarized in Table 8 together with their standard deviations σ for the Avg and Wgt depths exhibiting best regression performances. Values of δ or σ outside the ideal uncertainty interval of ±1% are highlighted in shade of gray. By remarking that these results refer to simulated measurements from 24-bit radiometers, values of δ or σ outside the selected interval are observed for both NL and LN regression products and in particular: 1) for E_d E_u and L_u in CDOM-dominated waters for most of the deployment speeds; and 2) for E_u in NAP-dominated waters with deployment speed of 1ms⁻¹. It is also observed that δ and σ tend to be more pronounced with a decrease of the density of simulated measurements in the regression layer, which is naturally associated to an increase of the deployment speed. When specifically looking at results for K_ℜ, in addition to the cases already identified for E_d, E_u and L_u affected by a relatively low density of simulated measurements, both δ and σ determined for $K_{E_{\mathrm{d}}}$ largely exceed the ±1% limit in clear waters as a result of wave perturbations.

4. Discussion

Building on previous analyses, this section progresses the discussion towards elements of practical relevance for actual field measurements. In particular, it addresses: 1) the impact of the spectral dependence of radiometric quantities on regression products; 2) the performance of hyperspectral radiometers commonly used in the field; and finally 3) the application of the multicasting technique to enhance precision of regression products.

4.1. Spectral dynamics and integration time

Radiance and irradiance from natural waters vary spectrally. An ideal measurement solution would require that the integration time is optimized for each wavelength, but technological constraints impose limitations. The integration time is generally determined using the maximum radiometric value across the measured spectrum in agreement with the scheme detailed in Fig. 6. Nevertheless, the sensor spectral response is low in the blue and high in the red, as well as the spectral distribution of light from natural waters is low in the red and highly variable in the blue. Because of this, the portions of measured spectrum away from the wavelength at which the integration time is determined, are characterized by a signal-to-noise ratio (SNR) decreasing with the signal amplitude and consequently may exhibit increasing uncertainties.

A case study equivalent to that considered in Fig. 11 has then been performed for a signal 10 times lower than that previously applied. These additional results are presented in Fig. 12.

 

Fig. 12 As in Fig. 11, but for a signal 10 times lower than that applied for the determination of the integration time.

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In other words, in view of investigating the impact of signal dynamics, the two analyses of Fig. 11 and Fig. 12 refer to the same integration times, but the latter is based on a lower signal. Results, which do not exhibit major differences with respect to the ideal case presented in Fig. 11, indicate that even a non favorable integration time, such as that applied to data away from maxima, does not appreciably affect the radiometric products when measurements benefit of a high SNR. It is in fact recalled that both analyses rely on the assumption of a 24-bit digitization.

4.2. Performance of commonly used in-water hyperspectral field radiometers

So far this study has neglected the latency time, which implies the assumption of a null delay between the end of a measurement and the start of the successive one. This delay is due to a number of actions comprising the transfer of the charge accumulated in the elements of the photo-detector array, its conversion to a digital value and the handling of this latter. The overall delay is not related to the collection of photons and can be significant in current field radiometers when compared to the integration time. Latency times of 200–300ms, varying from unit to unit, were observed for HyperOCRs. These latency times are responsible for a reduction of data points in radiometric profiles that may lower the accuracy of regression results. The number of data in the radiometric profiles can further decrease due to the use of an internal shutter to determine the dark-signal. In particular, HyperOCRs are often programmed to collect 1 dark-signal after 5 sequential radiometric measurements.

An additional source of uncertainty is a SNR lower than that insofar considered in this study. Specifically, HyperOCRs digitize the radiometric signal with a 16-bit ADC. With this respect, Table 9 summarizes the SNR applied for sub-surface data simulated as a function of water type for the 16- and 24-bits systems. Excluding any source of uncertainty other than the digitization one, a specific analysis is presented to document the ability to actually mimic HyperOCRs profile data considering 16-bit ADC and fully accounting for both a latency time of 250 ms and the capability of regularly measuring the dark-signal with the internal shutter. Equivalent to the 24-bit analysis presented in Sec. 4.1, a noise sampled from a normal distribution with standard deviation of 10 digital counts has been added to the digitized radiometric values. This standard deviation was actually determined from measurements of the dark signal performed with HyperOCR radiometers operated in thermally stable conditions.

Table 9. Values of the Signal-to-noise-ratio (SNR) Characterizing the Sub-surface Radiometric Quantities Simulated for Different Water Types and for 16- and 24-bit Systems.

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Fig. 13 displays the density of L_u and E_d radiometric data obtained from actual HyperOCR radiometers and the corresponding simulated values. Notably, experimental and simulated results exhibit very close densities of about 10 measurements per meter.

 

Fig. 13 Number of simulated data per unit depth obtained during clear sky with latency time of 250ms and 1 dark-signal recording after every 5 successive measurements. E_d and L_u results are presented in the top and bottom row panels, respectively. Experimental and simulated values (referring to v_prf =0.2ms⁻¹, v_wnd =4ms⁻¹, θ_sun =30°, a=0.15m⁻¹ and b=0.05m⁻¹) are displayed in the left and right column, respectively.

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Fig. 14 shows the equivalent analyses presented in Fig. 11. Results indicate the impact of an expected lowering in performance of radiometric measurements due to a decrease of the SNR implicit of the use of a 16-bit digitization alternative to the 24-bit one. The decreased measurement quality is is more evident across the various depth definitions for E_u and L_u in CDOM-dominated waters in agreement with the lower SNR values shown in Table 9. In particular, the quality of the data indicates the difficulty to determine radiometric products for E_u in highly absorbing waters as a result of the assumption of equal sensitivity for both E_d and E_u. This is clearly illustrated by the sample virtual profiles displayed in Fig. 15. A higher sensitivity would definitively increase the measurement accuracy of both L_u and E_u beyond that documented in Fig. 14.

 

Fig. 14 As in Fig. 11, but considering 16-bit ADC and accounting for both the latency time of 250ms and dark-signal recording.

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Fig. 15 Virtual optical profiles for Case-1, NAP and CDOM-dominated waters, considering 16-bit ADC, latency time of 250ms and dark-signal recording, and assuming a clear sky and a wind-driven sea surface. The deployment speed is v_prf =0.2ms⁻¹. E_d, E_u and L_u are in the row panels from top to bottom. Full resolution points are denoted by the + symbol in shade of gray. The integrated values are labeled as Δ, ∇ and ○ for the Fst, Lst, and Wgt depths, respectively.

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4.3. Multicast profiles

Enhancing the density of optical profile data improves the precision of regression results in the presence of perturbations that do not induce a systematic under or overestimation. An example is the use of the multicasting scheme that combines multiple casts into a single profile. This was shown to lessen the sea-surface focusing and defocusing effects in regression products from measurements performed at fixed acquisition rate [11, 19]. Here, by considering profiles performed at variable acquisition rate implicit of the application of a variable integration time, a new analysis is presented for 16-bit systems to illustrate the impact of data density on regression products.

Examples of single-cast and multicast profiles are displayed in Fig. 15 and Fig. 16, respectively. In both cases, notable is the impact of integration time: integrated values exhibit a lower dependence to wave perturbations with respect to full resolution data of Fig. 15(a) and Fig. 16(a). While statistical regression results for the single-cast case have been already presented in Fig. 14, those related to the multicast case are summarized in Fig. 17. The comparison between the two analyses confirms that increasing the number of data records per unit depth through the multicasting scheme, improves the precision of regression products by minimizing the effects of environmental perturbations.

 

Fig. 16 As in Fig. 15, but adopting the multi-cast scheme with 10 independent casts combined into a single profile.

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Fig. 17 As in Fig. 14, but adopting the multi-cast scheme with 10 optical profiles.

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In view of providing guidelines for profiling with hyperspectral systems, Tables 10, 11 and 12 summarize statistical analysis for E_d, E_u and L_u regression products, respectively, for the single-cast and multicasting schemes, as well as a for different water types and various deployment speeds. This analysis is restricted to Wgt depths and to the realistic case featuring a 16-bit ADC, latency time of 250ms and dark-signal recording. It is specified that: 1) results are always obtained from a pool of 10 independent samples; and 2) each sample of the pool is an individual cast in the single-cast scheme, while alternatively it refers to the combination of 5 or 10 casts in the multicasting one.

Table 10. E_d Regression Determined Products Accounting for 16-bit ADC, Latency Time of 250ms and Dark-signal Recording. Both the Single-cast and the Multicast Profiles (the Latter Built with 5 or alternatively 10 casts) are Considered. Statistical Figures are Provided for the Sole Wgt Depths, both the NL and the LN Regression Methods, the Various Water Types and Deployment Speeds of 0.2, 0.6 and 1.0 ms⁻¹. Symbols µ, CV, δ and σ Refer to the Mean of Regression Products, their Coefficient of Variation, the Percent Differences with Respect to Reference Values Determined with Eq. 15 and the Related Standard Deviations, respectively. It is further Mentioned that Results Refer to a Pool of 10 Samples for each Case. The Mean Density of Measurements is also Reported to Identify cases Leading to a Better Determination of Regression Products.

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Table 11. As in Fig. 12, but for E_u Data Regression Products.

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Table 12. As in Fig. 10, but for L_u Data Regression Products.

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Statistical figures are indicated as follows: 1) µ is the average of the data regression products; 2) CV is the variation coefficient of µ; 3) δ is the average of percent differences determined with Eq. 15, between the investigated regression products and the corresponding values obtained with the NL regression of full-resolution radiometric values—i.e., the reference quantity in this work; and finally 4) σ is the standard deviation of the δ values. Table entries in shade of gray highlight values outside and uncertainty target interval of ±2% for δ or σ values (slightly increased with respect to that applied to discuss results in Table 8 for 24-bit simulated data).

Table 10 indicates that the CV values of E_d0 are in most of the cases within 2% for a deployment speed of 0.2ms⁻¹, when the multicast scheme benefits of the combination of 10 independent casts. The values of CV tend to increase at higher deployment speeds and for a lower number of casts. This further confirms the need for increasing the measurements density to enhance the precision of regression results. It is noted the similarity between the values of CV and σ values, which is explained by their dependence on the closeness between the average regression products (i.e., µ) and the reference quantity (i.e., NL full-resolution regression products).

In agreement with the analysis presented in Sec. 3.3 for an ideal instrument, the present evaluation of the multicasting scheme for hyperspectral measurements simulated for radiometers commonly used in the field, further remarks the difficulty to accurately retrieve $K_{E_{\mathrm{d}}}$ in Case-1 waters due to the low attenuation coefficient and the presence of wave perturbations. Results in Table 11 document the quality of E_u regression products. Namely, all δ and σ values for E_u0 are within the ±2% uncertainty interval in Case-1 and NAP-dominated waters. A performance degradation is nevertheless observed for $K_{E_{\mathrm{u}}}$ determined in the Case-1 waters when the data density is less than 20 measurements per unit depth. The low SNR characterizing E_u in CDOM-dominated waters is again the limiting factor for an accurate determination of both E_u0 and $K_{E_{\mathrm{u}}}$.

Finally, Table 12 shows an overall higher accuracy of L_u results with respect to the E_d and E_u regression products discussed above. Typically, a measurement density of at least 10m⁻¹ leads to values of δ and σ for L_u0 within the ±2% target interval considered in this latter analysis regardless of the water type. As already noted for $K_{E_{\mathrm{d}}}$ and $K_{E_{\mathrm{u}}}$, also the determination of $K_{L_{\mathrm{u}}}$ in Case-1 waters is subject to a decrease in performance with respect to L_u0. It is finally observed that the multicasting technique remains of strategic importance in CDOM-dominated waters.

5. Summary and conclusions

Ocean color applications for environmental monitoring and climate change studies rely on highly-accurate insitu validation measurements. However, the determination of field radiometric data products is sometimes based on assumptions whose validity is restricted to ideal cases. Matter of this study has been evaluating the effects of integration time on in-water radiometric data collected with hyperspectral profiling systems. The analysis has been executed simulating the radiative transfer processes with the MOX Monte Carlo code to represent realistic measurement conditions. Simulation features include the capability of discretizing the sea-surface with a resolution of 0.0012m, and the in-water fields with 0.01 and 0.001m horizontal and vertical resolutions, respectively.

Virtual E_d, E_u and L_u radiometric measurements were computed from the accumulation of photons over time intervals ranging from 8 to 2048ms. Optical profiles were then constructed to evaluate subsurface radiometric values and diffuse attenuation coefficients assuming a 24-bit ADC. Both linear and non-linear regression methods have been applied to obtain data products from alternative determinations of measurement depths. The study has so documented the best performance of the Wgt depth definition relying on depth values weighted by the diffuse attenuation coefficient. However, close statistical figures (in most cases within 1%) have been obtained with the Avg depths relying on the mean depth within the integration interval. Instead, both the Fst and Lst depth schemes (i.e., corresponding to the beginning or the end, respectively, of each integration interval) should not be adopted to avoid biased estimates of diffuse attenuation coefficients and sub-surface values, mostly for E_u and L_u data collected in waters exhibiting a high absorption coefficient.

The analysis of simulated optical profiles has provided additional manifold insights. Largely biased K_Ed values have been observed in Case-1 waters due to the combined effects of integration time and light focusing. A main finding is however that the integration time can partially compensate the underestimates produced by the linear fit of log-transformed data. The complementary use of both linear and non-linear schemes for the regression of optical profile measurements is then devised as a solution to evaluate the effects of light focusing and defocusing in an operational framework addressed to account for the accuracy of data products.

The study has also investigated the capability of simulating profile data obtained from HyperOCR in-water hyperspectral systems fully accounting for their features (i.e., 16-bit digitization with 10-count noise, 250ms latency time and 1 dark-signal recording after every 5 successive measurements). As expected, the decrease in the density of profile data and the lessening of SNR with respect to the ideal case (i.e., 24-bit digitization with 10-count noise and negligible latency time) significantly affects the radiometric accuracy challenging measurements in highly absorbing waters.

Overall, the study indicates that an inappropriate definition of the measurement depths likely added to an unfavorable SNR, may become a source of large uncertainties for data products determined from in water hyperspectral profile data. It is thus essential that, in view of meeting requirements for satellite ocean color applications, the suggested Wgt or alternatively the Avg schemes are applied. The analysis also indicates that an increase of the number of measurements per unit depth obtained through the multicasting technique leads to an increase of the precision of regression products, also for systems characterized by variable acquisition rates.

Finally, the work further reinforces the relevance of numerical simulations of photons transport as a mean to advance the understanding of uncertainty budgets affecting the collection of insitu radiometric data and to investigate the related measurement protocols.