Effects of integration time on in-water radiometric profiles

Abstract: This work investigates the effects of integration time on in-water downward irradiance Ed, upward irradiance Eu and upwelling radiance Lu profile data acquired with free-fall hyperspectral systems. Analyzed quantities are the subsurface value and the diffuse attenuation coefficient derived by applying linear and non-linear regression schemes. Case studies include oligotrophic waters (Case-1), as well as waters dominated by colored dissolved organic matter (CDOM) and non-algal particles (NAP). Assuming a 24-bit digitization, measurements resulting from the accumulation of photons over integration times varying between 8 and 2048ms are evaluated at depths corresponding to: 1) the beginning of each integration interval (Fst); 2) the end of each integration interval (Lst); 3) the averages of Fst and Lst values (Avg); and finally 4) the values weighted accounting for the diffuse attenuation coefficient of water (Wgt). Statistical figures show that the effects of integration time can bias results well above 5% as a function of the depth definition. Results indicate the validity of the Wgt depth definition and the fair applicability of the Avg one. Instead, both the Fst and Lst depths should not be adopted since they may introduce pronounced biases in Eu and Lu regression products for highly absorbing waters. Finally, the study reconfirms the relevance of combining multiple radiometric casts into a single profile to increase precision of regression products.


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][2][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 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 pass 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 . Considered data products are the subsurface value 0 and the diffuse attenuation coefficient K , derived from the regression of radiometric measurements acquired at different depths (negative downwards in this study) 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 increment ∆ z = −v prf · t. The recorded value denoted with the tilde notation is where the sampling depth z * i ∈ [z i , z i + ∆ z ] has to be determined so that a difference between the measured˜ (z * i ) and the actual (z * i ) radiometric value is minimized and consequently does not affect the determination of 0 and K . This requirement is addressed in the present study accounting that is a non-linear function of z.
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 0 and K . For instance, wind-driven waves at the air-water interface, besides focusing and defocusing the light [13][14][15][16][17][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, 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][20][21][22][23][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.

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 field; 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 distributions 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.  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.

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]). Atmospheric optical properties are specified at λ = 490 nm chosen as representative wavelength for this analysis. The scattering optical thickness of the atmosphere is τ and τ A b denote the molecules M and aerosol A contributions, respectively. The absorption optical thickness is instead τ a = τ O a + τ G a , with τ O a and τ G a 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 where c atm = a atm + b atm is the attenuation coefficient and u is a random number sampled from a uniform distribution between 0 and 1; i.e., u ∈ U(0, 1). The photon weight is updated as where ω atm = b atm /c atm is the single scattering albedo.  Photons ending their trajectory in the atmosphere undergo Rayleigh scattering by air-molecules if u < µ, where u ∈ U(0, 1) and 26,27]. The Mie theory defines the aerosol scattering when u ≥ µ [27-31].

Sea surface modeling
The sea surface is defined over a domain of 40m length discretized by N s = 32768 points (2 15 ), which leads to a resolution of ∆ s = L x /N s = 1.2 mm and spatial coordinate x(s) = s · ∆ 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]) where the harmonic componentsẑ(u) are indexed in the wavenumber k domain. Theẑ(u) values are set based on the Elfouhaily omnidirectional spectral density S(k) [32, 33], which determines the variance of the sea-surface elevation as a function of the wind speed v wnd and additionally of the wave age. 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(s). The spectral density function 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 .

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   In-water radiometric fields are simulated in a domain with horizontal 0.01 m and vertical 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 coefficient 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 ∈ U(0, 1).
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 functionβ = β/b define the probability density function p(θ) = 2πβ(θ) sin(θ). 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 ∈ 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.
The new cosine directors of photons scattered by the virtual bottom are µ z = √ u and µ x = 1 − µ 2 z · sign(q − 0.5), where q and u are independent random numbers in 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 The response of in-water optical sensors is analyzed in the following cases: 1. oligotrophic waters (Case-1).

waters characterized by high concentration of Colored Dissolved Organic Matter (CDOMdominated).
3. waters characterized by high concentration of Non-Algal Particles (NAP-dominated). Table 5 reports the corresponding IOP values whereas Fig. 3 shows an example of E d simulations.

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 v wav = g · l wav 2π , where l wav = 20m is the longest wave that the IFFT can represent in the simulation domain (Eq. (4); [23, 32]) and g is the acceleration due to gravity. The angle of the virtual profile θ prf with respect to zenith is then with v prf values between 0.2 and 1.0ms −1 .

Pressure gauge correction
The still level z 0 = 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 z 0 ) is computed as where k wav = 2π/l wav denotes the wave number, l wav is set to 20m as in Eq. (5), and the water column is assumed infinitely deep. Fig. 5 shows how the effect of the pressure gauge correction varies with depth.  Table 6.

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.
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: 1) the depths corresponding to the beginning of each integration interval (F ); 2) those corresponding to the end of each integration interval (L ); 3) the averages of F and L values (A ); and finally 4) the values weighted accounting for the diffuse attenuation coefficient of water determined for each radiometric quantity (W ).
The W method assumes an exponential decay of the radiometric quantity (z) = 0 · e K ·z , where K > 0, z is negative downward and 0 is set to 1 without loss of generality (Fig. 7). The integrated value˜ and there is a z * i depth between z i and z i + ∆ z such that In summary, the W method associates the depth z * i to the integrated radiometric value˜

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 0 and K is based on the linear regression (LN) as a function of z of the log-transformed (z) values [7,40,41]. The underlying assumption is the exponential decay in a water layer where K is constant (it is remarked that different from common conventions, depth values are negative in this work). In the logarithmic form, Eq. (11) becomes The LN method utilizes Eq. (12) to determine regression parameters by minimizing the sum-ofsquares SSE error which is solved as a linear system of equations.

The non-linear regression
without taking the logarithm of˜ (z * i ) values. The partial derivatives of Eq. (14) however lead to a non-linear system and the Trust-Region algorithm [47, 48] is applied in this work to minimize SSE NL . A set of trials based on the re-initialization of the Trust-Region scheme are also executed to verify the convergence of the NL solution.

Statistical figures
The depth separation between the data points of the F resolution virtual profile is about the same as the vertical grid spacing of the simulation domain (i.e.,0.001m; see Table 4). The corresponding NL data products are denoted as {[ ] NL F } S , where indicates the 0 or K 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

Topic Symbol Description
Radiometry where n is the index of the radiometric profile in a pool of N = 10 independent ones that differ from each other for the random initialization of the deployment point of the virtual optical sensor at the surface. Data regressions are performed in a layer determined by the depths −0.2 and −4 m. 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.

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 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.
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 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 NAPdominated waters show: 1) a low influence on E d data products; and 2) larger percent biases for K E u and K L u than for E u0 and L u0 . Figs. 9(d) and 9(j) provide an example of how the effects of integration time on K E u increase with the deployment speed of the optical system. It is noted that the F and the L depth definitions lead to an overestimate and underestimate of K E u values, respectively, as a consequence of the long integration time illustrated in Fig. 8 for E u data. A    similar tendency, but with smaller values, can be observed for K L 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., . Figures 9(m)-9(r) display an increase of the effects of the integration time for CDOM-dominated waters due to the high absorption coefficient [see Table 5]. The overall trends of the F and L depths to respectively overestimate and underestimate the diffuse attenuation coefficient are 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).

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 for defining 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].
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) [14,19,23]. The manifold biases appearing in Fig. 11(b), instead, partially depend on the very low K E 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 with measurements performed in the subsurface layer. The example also shows that , where the acronym D 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 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 F and L depth definitions induce a positive and a negative bias in both K E u and K L 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 d biases-cf., Fig. 11(b) and Fig. 11(h).
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 F and the L measurement depths, respectively. It is also verified that this tendency mostly appears opposite to the bias reported for K E d , K E u and K L u , as observed for the overcast and still sea simulation case [see Fig. 9]. Namely, an underestimate of the subsurface value due to the F depths corresponds to an overestimate of   the diffuse attenuation coefficient. Opposite results can be observed with the L depths. It is finally noted that [δ ] NL W as well as [δ ] NL A 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 δ 0 and δ K values displayed in Fig. 11, are also summarized in Table 8 together with their standard deviations σ for the A and W depths exhibiting best regression performances. Values of δ or σ outside the ideal target 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 1 ms −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 , 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 d largely exceed the ±1% limit in clear waters as a result of wave perturbations.

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.

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.
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.

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   Fig. 11, but for a signal 10 times lower than that applied for the determination of the integration time.
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. 16 Bits  Figure 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. Figure 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 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.

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.
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 singlecast and multicasting schemes, as well as for different water types and various deployment speeds. This analysis is restricted to W depths and to the realistic case featuring a 16-bit ADC, latency time of 250 ms 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. (i) Fig. 16. As in Fig. 15, but adopting the multicasting scheme with 10 independent casts combined into a single profile.    Table  11. As in Fig. 12, but for E u Regression Products. Depl.
Meas.   Table  12. As in Fig. 10, but for L u Regression Products.

Depl.
Meas.  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 a target interval of ±2% for CV, 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 cases within 2% for a deployment speed of 0.2 ms −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 values of CV and σ values, which is explained by their dependence on the closeness between the average regression products (i.e., µ) and 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 d in Case-1 waters in the subsurface layer 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 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 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 d and K E u , also the determination of K L 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.

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 0.0012 m, and the in-water fields with 0.01 and 0.001 m 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 documented the best performance of the W 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 A depths relying on the mean depth within the integration interval. Instead, both the F and L depths (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 E d 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, 250 ms 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 affect the radiometric accuracy, challenging regression products 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 W or alternatively the A 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.