1. INTRODUCTION

Hyperspectral measurements of the water-leaving radiance ($L_w$) or of derived quantities (e.g., the remote-sensing reflectance $R_{\mathrm{RS}}$) have become increasingly popular due to the consolidation of technology and of measurement protocols. These hyperspectral measurements, different from multispectral data, allow for a more comprehensive identification and quantification of the optically significant constituents suspended and dissolved in the water column, and thus a better characterization of the different water masses. Generic requirements on the accuracy of in situ radiometric measurements in support of satellite ocean color applications indicate the need to restrict uncertainties below 5% in the visible spectral region [1]. Such a target implies the adoption of strict measurement protocols and an accurate characterization and calibration of instruments [1]. Stray light, which is an unwanted distortion (i.e., redistribution of the energy) of the measured spectrum due to internal reflections, imperfections of the grating, and the presence of scratches or dust on the optical components, may become a significant source of measurement uncertainty for hyperspectral radiometers. In the specific case of in situ ocean color measurements, this unwanted light perturbation is often the source of significant errors in measured radiances and irradiances, especially in spectral regions where the sensitivity of the instrument or the signal-to-noise ratio is low as in the ultraviolet and near-infrared.

A number of studies have already focused on the correction of stray light [2 –4]. Specifically, a commercial CCD-array spectrograph was analyzed and the corresponding stray light correction matrix was assessed and tested using both broadband and narrowband sources [2]. Results indicated the possibility of reducing the impact of stray light by 1–2 orders of magnitude as well as the need to redetermine the correction matrix only when the instrument undergoes modifications of the optical or detection components. The same methodology was further investigated in successive studies focusing on marine applications. In particular, the correction was validated using various colored sources, and its performance was assessed using Monte Carlo simulations [3]. Results confirmed the validity of the methodology, stressing the importance of correcting for stray light especially when the spectral distribution of the measurement target differs significantly from the spectral distribution of the calibration source. In a different study, a correction was applied to CCD- and CMOS-array-based spectroradiometers used for underwater measurements [4]. In this case, the effects of stray light perturbations exhibited values ranging from 2% to 15%.

The objective of this work is to investigate stray light effects in above-water radiometric measurements performed with commercial radiometers (i.e., RAMSES) manufactured by TriOS Mess- und Datentechnik GmbH (Rastede, Germany) and widely applied by the ocean color community. Quantification of stray light effects in actual data products was performed with the aid of in situ measurements collected in the Mediterranean Sea in waters exhibiting different optical features.

The radiometers evaluated in this study are described in Section 2. The methodology applied to generate the stray light correction is outlined in Section 3. The validation of the correction scheme is presented in Section 4. The in situ data are presented in Section 5, and the results are summarized in Section 6. A discussion on inter-instrument stray light variability and on the processing algorithm is undertaken in Section 7. Finally, the main conclusions are drawn in Section 8.

2. DESCRIPTION OF THE INSTRUMENTS

TriOS RAMSES-ARC and RAMSES-ACC are stand-alone highly integrated hyperspectral radiometers designed to measure radiance and irradiance, respectively, from the ultraviolet to the visible spectral range in the 320–950 nm spectral interval. Both TriOS RAMSES-ARC and RAMSES-ACC are equipped with ZEISS (Oberkochen, Germany) Monolithic Miniature Spectrometers (MMS-1).

These spectrometer modules have a 256-channel silicon photodiode array characterized by an average spectral sampling of 3.3 nm per element of the detector array, and a spectral accuracy of 0.3 nm. Data are rendered with a spectral resolution of approximately 10 nm. Measurements are acquired with an integration time varying between 4 and 8192 ms depending on the illumination conditions. Three different instruments have been analyzed for this study: two RAMSES-ARC radiance sensors with a full-angle field of view (FAFOV) of 7° (with serial number SAM-8313 and SAM-8346) and one RAMSES-ACC irradiance sensor (with serial number SAM-835C). TriOS RAMSES-ARC and RAMSES-ACC are shown in Fig. 1.

 

Fig. 1. TRIOS RAMSES-ARC radiance (upper panel) and RAMSES-ACC irradiance (lower panel) sensors (courtesy of TriOS).

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3. STRAY LIGHT GENERATION AND CORRECTION

Stray lights are generated inside sensor optics by a number of sources such as light reflected or scattered by fore optics, diffraction gratings, or filters [5]. The stray light generation/correction model adopted in this study is summarized in Fig. 2. The input signal enters into the instrument where the signal measured by the detector results from the sum of the incoming signal, stray light, and the noise of the detector itself (i.e., the thermal background and additional electronic offset). The overall process has been investigated using a model allowing 1) subtraction of the noise from the measured signal, 2) removal of the stray light perturbations by applying the instrument-specific stray light correction matrix determined using the laboratory characterization of the sensor response, and 3) readdition of the noise. Adding the noise back to the corrected measurement permits assessing the effects of the correction at different levels of processing using the same tools for both the original and corrected measurements.

 

Fig. 2. Diagram illustrating the stray light generation and correction model.

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The core components of the stray light correction scheme summarized in Fig. 2, i.e., the stray light correction matrices, were determined using the following steps.

A. Determination of the Spectral Line Spread Function

A tunable monochromatic source with high spectral purity was used to determine the line spread function (LSF) of the hyperspectral radiometers. Specifically, a double grating monochromator Bentham DTMS300 was used to select 3 nm wide spectral bands from the flux of a 450 W xenon arc lamp. The monochromator output was then applied to fill the radiance or irradiance collector of the hyperspectral radiometer under characterization.

Effective central wavelengths for each detector element were determined by gradual adjustments of the monochromator wavelength to reach the central maximum in the output signal of the radiometer. For each detector element of the radiometer, 24 spectra were acquired. Each spectrum was processed as described in Section 3.B and then combined together (see Section 3.C) to represent a single line in the LSF matrix.

It is reported that the average differences between the monochromator and radiometer center wavelengths were smaller than 0.5 nm. A verification of the wavelength calibration of the spectrometers confirmed uncertainties lower than 1 nm.

Finally, tests were also performed on one RAMSES-ARC radiance sensor to evaluate potential stray light sources due to light contributions from outside the spectral range of the detector (320–950 nm). These tests were made by placing a bandpass filter centered at 248 nm with 20 nm bandwidth, or alternatively a long-pass filter with 900 nm cut-off wavelength, in front of the radiance collector. The response to out-of-range contributions from ultraviolet and near-infrared was determined as the ratio between the spectral radiances measured with and without the filter using Xenon 1000 W and FEL 1000 W sources, respectively. In the first case results showed values below 0.01% with respect to the input radiance across the response range of the sensor. In the second case results exhibited values varying between 0.05% and 0.1%. The previous results show negligible stray light from contributions outside the range of response of the sensor in the near-infrared, and additionally some indication of nonrelevant contributions from the ultraviolet spectral regions (even though limited to a narrow band away from the lower limit of the detector spectral response).

These data have confirmed confidence in the 300–950 nm spectral characterization interval applied for the determination of LSFs for RAMSES sensors.

In view of maximizing the input flux at the entrance optics of radiometers during stray light characterization, the RAMSES sensors were coupled to the monochromator with a polytetrafluoroethylene (PTFE) diffusing film of 4 mm thickness. This solution, alternative to the coupling of the radiometers to an integrating sphere illuminated by the monochromator, allowed 1) overfilling the entrance optics as required to reproduce the actual sensor performance during field measurements and 2) lessening the instrument noise by minimizing the integration time. In fact a too long integration time may appreciably affect the accuracy of data through an increase of the uncertainty in sensor noise.

B. Noise Removal

Both the thermal and electric noises affecting the signal detected by each element of the detector array were quantified and removed by following the manufacturer instructions [6].

It is mentioned that the blackened detector elements correspond to the last 18 elements of the array, and are, obviously, do not contribute to the 320–950 nm interval.

The noise correction applied for the thermal contribution is the result of a characterization performed by the manufacturer for each sensor. It is thus likely that residual noise may still affect measurements after correction. This residual noise is of the order of a few digital counts and can be safely neglected in most of cases (i.e., when compared to the $\sim {10}^3$ and $\sim {10}^5$ digital counts characterizing thermal noise and generic measurements, respectively). Nevertheless, due to the very stringent signal-to-noise requirements set to achieve an accurate characterization of the LSFs, the residual thermal noise has been estimated for each measurement by analyzing the histogram of each measurement spectrum. Under the hypothesis that stray light generated by a monochromatic signal only affects a limited number of detector elements, the residual thermal noise has been quantified as the mode of each measurement histogram and then removed from the LSF estimate.

C. Sensitivity Enhancement

In view of enhancing the sensitivity of the LSF, two sets of noise-free measurements were performed with two different integration times: a first series of 12 raw spectra, followed by a second series of 12 overexposed spectra acquired with an integration time 16 times larger than that characterizing the first series. To avoid the possible inclusion in the LSF of nonlinearity due to the long integration time of the overexposed spectra, only detector elements showing values in the lower 85% of the sensor dynamic range (0-55000, over 65535) were included in the calculation (corresponding to approximately 98% of the elements of each spectrum). Oversaturated measurements were used to improve the signal-to-noise ratio outside the main diagonal of the LSF matrix. This induced a reduction of the noise in the stray light correction of the original signal largely appreciable at the extremes of the corrected spectra.

D. Normalization

The next step in the data analysis was the normalization of the LSFs. Specifically, each LSF was normalized to the integral of the signal inside the bandpass of each detector element, the so-called in-band (IB) response. The widths of IBs were determined from the second derivative of each LSF with the limits of IBs defined by the two wavelengths corresponding to zero crossings (which indicate changes in convexity of the LSF). Results show a mean IB of approximately 23 nm (i.e., seven detector elements) for the considered sensors.

A set of selected LSFs, normalized to their maximum value, is presented in Fig. 3: the IB regions are shown in white and bounded by the segments over-plotted in black. The portions of the spectra outside the IB regions are shaded. Consistently with previous investigations [7], IB limits approximately correspond to values exhibiting 2%–3% of the peak value (1% level is highlighted by the dashed black line). The 50% level is displayed as well through the dashed red line. This level typically identifies bands extending over three to five detector elements loosely corresponding to the expected 10-nm bandwidth.

 

Fig. 3. Selected examples of in-band regions determined at sample center wavelengths. Spectra are normalized to their maximum value; IB regions are shown in white and bounded by the segments over-plotted in black, while the portions of the spectra outside the IB regions are in gray. The dashed red and black lines indicate the 50% and 1% peak levels, respectively.

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E. IB Set to Zero

In agreement with consolidated methods [2,3], the stray light distribution function (SDF) $g$, for the center wavelength of each detector element, was constructed by preserving the values of the LSF normalized with respect to the in-band area (i.e., $\mathrm{LSF}/{\mathrm{\Sigma }}_{\mathrm{IB}}(\mathrm{LSF})$) outside each IB and setting to zero the values inside

F. Matrix Inversion

By arranging all the SDFs into one matrix $G=[g_{i,j}]$ and expressing the spectral stray light contribution as $G{Y}_{\mathrm{IB}}$ with the true signal at the considered center wavelength as $Y_{\mathrm{IB}}$, the measured signal $Y_m$ is given by

The spectral stray light correction matrix $H$ was then determined by inverting $A$, i.e., $H=A^{-1}$.

The procedure is summarized in Fig. 4.

 

Fig. 4. Schematic of the algorithm applied for the determination of the inverse stray light distribution function (SDF) matrix using $N$ independent (here $N=12$) line spread functions (LSFs) with characterized in-band (IB) responses.

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The SDF matrix $(A=[I+G])$ for SAM-8346 (radiance sensor) is shown in Fig. 5. Measurement and excitation wavelengths are on the $x$ and $y$ axes, respectively. The values of the matrix are displayed in log-scale.

 

Fig. 5. SDF matrix for SAM-8346. The measurement and excitation wavelengths are on the $x$ and $y$ axes in units of nanometers, respectively. The values of the matrix coefficients, in normalized raw counts, are displayed in log-scale. The green and red ellipses highlight spectrally extended stray light perturbations. The black straight lines near the diagonal identify the IB regions.

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As expected, Fig. 5 shows that most of the signal falls inside the IB regions while a high percentage of the residual signal is distributed on the detector elements close to the diagonal as indicated by the black straight lines. The largest dispersion is observed below 550 nm due to the input signal between 320 and 400 nm (see the asymmetry of signals around the $1:1$ line in Fig. 5). Additionally, two spectrally extended stray light perturbations can be noticed. The first (see the green ellipse in Fig. 5) affects data around 630–700 nm and exhibits a signal approximately two orders of magnitude lower than that of the input. This perturbation is generated by the input signal in the spectral region located below 340 nm. The signal related to the second spectrally extended perturbation (see the red ellipse in Fig. 5) is approximately three orders of magnitude lower than the input signal. This affects almost the entire spectrum and is generated by the input signal in the spectral region between 500 and 900 nm.

Despite exhibiting some differences, the SDF matrices of the considered sensors show a high correlation. The average absolute difference ${\mathrm{\Delta }}_A$ between the SDF matrix of SAM-8346 (radiance sensor, $A_{8346}$) and those of SAM-8313 (radiance sensor, $A_{8313}$) and SAM-835C (irradiance sensor, $A_{835C}$) has been quantified as

The resulting matrix is displayed in Fig. 6 with the same color bar of Fig. 5.

 

Fig. 6. Average of absolute differences ${\mathrm{\Delta }}_A$ between the SDF matrix of SAM-8346 and those of SAM-8313 and SAM-835C, in normalized raw counts. The measurement and excitation wavelengths are displayed on the $x$ and $y$ axes in units of nanometers, respectively. The values of the matrix coefficients are displayed in log-scale.

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It is noted that the zones exhibiting the largest difference correspond to those characterized by the highest stray light. The maximum difference varies between ${10}^{-3}$ and ${10}^{-2}$, and is observed close to the diagonal and in correspondence to the green ellipse displayed in Fig. 5. Excluding the previous spectral regions, the differences are always lower than ${10}^{-4}$. This result supports the assumption that there may be similar stray light perturbations in all the considered radiometers, regardless of the fore optics differentiating radiance and irradiance sensors.

4. CONSISTENCY AND SENSITIVITY TESTS

The determination of the SDF matrix (A) is an ill-conditioned problem. In the considered case, the condition numbers determined from the characterization of the three sensors are 1.08 for SAM-8346 (radiance sensor), 1.05 for SAM-8313 (radiance sensor), and 1.08 for SAM-835C (irradiance sensor), consistent with values produced in previous investigation for different spectrometers (i.e., 1.07 and 1.86) [2,5].

The goodness of the inversion of the SDF matrix has been evaluated through the following analysis.

First, in agreement with an independent study [2], a set of monochromatic signals (randomly selected among the LSFs applied to estimate the SDF matrix) has been analyzed using the corresponding stray light correction matrix. The effects of the corrections are illustrated in Fig. 7 at the excitation wavelength of 576 nm applying the SDF matrix corresponding to SAM-8346 (radiance sensor). The input noise-free signal is shown in blue, while the stray light corrected signal is in red. Both signals are plotted in log-scale as a function of wavelength, with the 1-count level indicated by the dashed green line. It is mentioned that for convenience all data are handled as floating numbers and values smaller than ${10}^{-2}$ are forced to ${10}^{-2}$.

 

Fig. 7. Application of the stray light correction matrix corresponding to SAM-8346 to a monochromatic signal at 576 nm. The measured and the corrected signals are shown in blue and red, respectively. The 1-count level is indicated by the dashed line.

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The presence of stray light in the measured signal is evident: the excitation at 576 nm generates strong tails in the closest detector elements, more pronounced at shorter wavelengths. In addition, consistent with the SDF matrix in Fig. 5, a weak spurious signal around 430 nm can be seen. After applying the stray light correction, the signal appears almost unchanged in the IB region, but it is sensibly reduced outside with values generally lower than one to two raw counts.

As an additional test, the sensitivity of the corrected signal to variations in the measured signal has been evaluated. Consistently with analysis performed in different investigations [5], two noise contributions have been added to the measured signal before its correction—namely sinusoidal and random signals, both with maximum amplitude of 0.5% of the input signal applied for the determination of the SDF (approximately half of the maximum signal outside the IB). Results of the correction are shown in Figs. 8(a) and 8(b), where the input signals are in blue, the retrieved ones are in red, and their differences are over-plotted in black for the sinusoidal [Fig. 8(a)] and random [Fig. 8(b)] noise contributions.

 

Fig. 8. Results from the sensitivity analysis performed with (a) sinusoidal and (b) random noise contributions. The input noise contributions are in blue, the retrieved signal is in red, and their difference is in black; all are expressed by the percentage of the signal applied for the determination of the SDF.

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According to the results presented in Figs. 8(a) and 8(b), both the sinusoidal and random signals are retrieved with an uncertainty in the range of 0.05%–0.1% of the input signal, with values more pronounced at shorter wavelengths. This unequivocally indicates that the inverse SDF matrix introduces only slight distortions in the retrieved signal, thus confirming the validity of the applied method.

5. FIELD MEASUREMENTS

The SDF matrices, obtained as described in Section 3, have been used to process above-water measurements performed in agreement with current protocols [1]. Briefly, each measurement sequence comprises the collection of successive values—in this case 18 during a 3 min interval—of total radiance from the sea surface $L_T(\theta ,\mathrm{\Delta }\varphi ,\lambda )$, sky radiance $L_i({\theta }^{\prime },\mathrm{\Delta }\varphi ,\lambda )$, and downward irradiance $E_d(0^{+},\lambda )$, simultaneously acquired with viewing angles $\theta$ and ${\theta }^{\prime }$, and relative azimuth with respect to the sun $\mathrm{\Delta }\varphi$. More specifically, measurements were performed with $\theta =40°$, ${\theta }^{\prime }=140°$, and $\mathrm{\Delta }\varphi =90°$. Out of the successive measurements collected for each quantity $L_T$, $L_i$, and $E_d$, the averages of $L_i$ and $E_d$ values are used in the following calculations. Conversely, $L_T$ measurements were sorted according to the average radiance between 450 and 650 nm, and only the lowest 20% of the measurements are considered in the calculation of the average $L_T$ in view of minimizing the contributions of sun and sky glint. The spectral water-leaving radiance $L_w(\theta ,\mathrm{\Delta }\varphi ,\lambda )$ is then calculated as

6. RESULTS FROM THE STRAY LIGHT CORRECTION

A schematic view of the data processing applied to quantify the effects of stray light corrections is shown in Fig. 9. The analysis is performed for the three different levels of processing indicated by the black arrows in Fig. 9:

 

Fig. 9. Schematic of the processing chain applied to quantify the impact of stray light corrections. Black arrows indicate the levels of processing (i.e., L0, L1, and L2) at which the comparisons are performed.

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In detail, two separate processors have been implemented—namely, the Level 1 processor to calibrate and preprocess (i.e., quality control) and the Level 2 processor computing higher-level quantities (e.g., $R_{\mathrm{RS}}$) based on the corresponding Level 1 data and ancillary information such as the parameters defining measurement geometry, sea state, and cloud cover.

As shown in Fig. 9, the measured and stray light corrected raw data undergo the same steps:

Results from the statistical analysis are presented and discussed relying on actual field measurements collected in ideal illumination conditions (i.e., clear sky).

Stray light analyses performed using specific correction matrices for each sensor are presented in this section. The implications of permuting the three matrices as well as those of using a common stray light correction matrix for all the sensors, and the impact of modifying the sensor calibration model, are addressed later in Section 7.

SDF matrices corresponding to SAM-8346 (radiance sensor), SAM-8313 (radiance sensor), and SAM-835C (irradiance sensor) have been used to correct measurements of total radiance from the sea surface ($L_T$), sky radiance ($L_i$), and downward irradiance ($E_d$), respectively.

The stray light correction has been applied to 18 measurement stations carried out with clear sky during two oceanographic campaigns performed in the Mediterranean Sea.

To evaluate the effects of the stray light correction, the percent correction $\epsilon$ at each center wavelength is computed as

 

Fig. 10. Mean stray light correction values $\mu$ and related standard deviations $\sigma$ (indicated as error bars) determined from the $\epsilon$ values of 18 measurement stations for data at (a) Level 0, (b) Level 1, and (c) Level 2.

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Statistical values are given in Table 1 for the root mean square (rms) determined for spectral bands 10 nm wide at center wavelengths of particular interest. The values in brackets indicate the number of available data at Levels 0, 1, and 2, respectively, applied in the computations.

Table 1. Root Mean Square (rms) of Percent Differences between Noncorrected and Corrected Measurements at Different Levels of Processing: Level 0 (L0), Level 1 (L1), and Level 2 (L2)^a

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According to Fig. 10 and data in Table 1, the mean correction values at Level 0 vary between approximately 1.5% and 7.7%. These mean correction values are more pronounced at longer wavelengths, and especially around 600 nm, where they also exhibit larger differences (2%–5%) among $L_T$, $L_i$, and $E_d$. At Level 1, the mean correction ranges from approximately $-1\%$ to 1%, except for $L_T$, whose correction deviates from the other curves in the interval of 570–670 nm, reaching its maximum (i.e., less than 4%) around 600 nm. Except for the aforementioned interval, the differences among the correction applied to different radiometric quantities are mitigated by the calibration process and become generally smaller than 0.5%. Finally, at Level 2, the mean correction is within $-0.5\%$ and 0.5% below 570 nm, it increases up to approximately 7% around 600 nm, and it decreases back to 0.5% at 670 nm. Concerning the standard deviation $\sigma$, its values exhibit a general increase in the red spectral region. This is particularly pronounced for L2 data products (i.e., $R_{\mathrm{RS}}$) in agreement with the very low values characterizing that part of the spectrum. Specifically, for $R_{\mathrm{RS}}$ the values of $\sigma$ increase from approximately 0.2% in the blue–green spectral region to more than 2% in the red. Finally, the mean rms values closely follow the mean correction values.

As expected, the largest standard deviation is observed for $L_T$ because the correction is a function of the input signal that varies with the water type. To quantify the dependence of the stray light correction on water type, two cases, characterized by different chlorophyll-a concentrations (Chla), have been selected and analyzed in more detail. Results are presented in the following subsection.

A. Sensitivity to the Water Type

Two sample measurement stations have been selected from the dataset. The first sample was acquired in the southern Balearic Sea characterized by very oligotrophic waters with $\mathrm{Chla}=0.1\mathrm{\mu g}{\mathrm{l}}^{-1}$, while the second sample was collected in northern Adriatic Sea affected by suspended matter from rivers with $\mathrm{Chla}=1.9\mathrm{\mu g}{\mathrm{l}}^{-1}$. The effects of stray light corrections are analyzed in the next paragraphs for $L_T$ at Level 0 and Level 1, and for $R_{\mathrm{RS}}$ at Level 2.

1. Level 0

Spectra of $L_T$ for the two sample measurement stations are displayed in Fig. 11(a): the blue spectra indicate a southern Balearic Sea station (SB), while the red spectra indicate a northern Adriatic Sea station (NA). For comparison, the spectrum corresponding to the standard source applied for calibration (CAL) is shown as well with a solid black line. The percent corrections $\epsilon$ calculated from Eq. (12) are displayed in Fig. 11(b) following the same color code applied in Fig. 11(a).

 

Fig. 11. (a) Spectra in raw counts of $L_T$ from selected stations SB (blue) and NA (red), as well as the spectrum from the calibration source (CAL, black) after stray light correction. (b) Percent correction applied to the three raw-data spectra.

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The derived corrections for SB and NA vary from 1.5% to 7.5% of the input signals, with minima around 420 nm and relative maxima near 610 and 520 nm. The shape of the two correction curves determined for SB and NA can differ by up to 2% close to 610 nm due to the different shapes and amplitudes of the corresponding input signals. The corrections determined for NA are almost constant between 410 and 550 nm with values of approximately 2%–3%. Differently, the corrections for SB exhibit a more marked spectral dependence with values increasing from 2% at 410 nm to approximately 3% at 570 nm, with a local maximum of 3% at 520 nm.

The corrections applied to the spectrum of the standard source used for calibration (CAL) are similar in shape to those determined for NA, even if they are higher below 590 nm and lower at longer wavelengths. This is consistent with the patterns of the two input signals that are very similar at shorter wavelengths with NA higher than CAL up to 590 nm.

2. Level 1

The same comparison has been performed for $L_T$ using data in physical units. It is mentioned that signals have been calibrated (i.e., converted from raw counts into physical units) using a different series of calibration factors. In fact, in the case of uncorrected field measurements the calibration factors have been calculated using measurement of the standard source uncorrected for stray lights. Contrarily, when performing stray light correction on the field measurements, the same correction has been applied to the measured values of the standard source. Results are shown in Fig. 12.

 

Fig. 12. (a) Calibrated spectra of $L_T$ from stations SB (blue) and NA (red), as well as a spectrum from standard source applied for calibrations (black) after the stray light correction. (b) Percent correction $\epsilon$ applied to calibrated spectra.

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Concerning the corrections [see Fig. 12(b)], their shapes are strongly influenced by the calibration process and by their spectral values. The stray light correction values vary from approximately $-1.5\%$ at 400 nm to $+1\%$ in the red spectral region, with fluctuations generally smaller than 1%. An exception is NA near 610 nm exhibiting corrections reaching 2%. The 0% line is crossed around 510 nm for SB, and near 580 nm for NA. As observed for the raw counts, the corrections applied to SB and NA spectra are very close, with mean differences of 0.5%.

3. Level 2

The same analysis presented for Level 0 and Level 1 data has also been performed for data at Level 2, with $R_{\mathrm{RS}}$ determined with and without stray light corrections. Remote-sensing reflectances and the relative corrections for SB and NA station data are shown in Figs. 13(a) and 13(b), respectively. Corrections for SB and NA have different shapes. NA is characterized by a nearly flat profile centered at 0.5%, except for the 2% peak at 610 nm. Conversely, the percent correction applied to SB is between $-0.5\%$ and 1% below 580 nm, increasing to approximately 9% between 600 and 650 nm, and then decreasing with high fluctuations to an average value of 4% above 650 nm. The high percentage correction determined for SB and its fluctuations above 600 nm can be explained mainly by the very low values of $R_{\mathrm{RS}}$ in the considered spectral region.

 

Fig. 13. (a) $R_{\mathrm{RS}}$ from stations SB (blue) and NA (red) after stray light correction. (b) Percent correction $\epsilon$ applied.

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7. DISCUSSION

The instrument-specific SDF matrices have been applied in the previous analysis. Now, considering the very small differences among the SDFs determined for the considered sensors (see Fig. 6), the possibility of applying different correction matrices, or a single correction matrix for all sensors from the same class of instruments, is investigated. Finally, the effects of applying calibration functions, instead of calibration coefficients, are also discussed.

A. Dependence of the Stray Light Correction on Different Matrix Permutations

In this subsection, the effect of applying different permutations of the three available SDF matrices on the stray light correction is assessed. The six possible combinations are listed in Table 2, where correspondence to the reference configuration analyzed in Section 6 is indicated as Configuration 1.

Table 2. Possible Combinations of SDF Matrices Versus Sensors

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Relative statistics determined comparing the results obtained using Configurations 2–6 with respect to those obtained using Configuration 1 are presented in Table 3 in terms of the rms of percent differences $\epsilon =100·(X_{C1}-X_{Cy})/X_{C1}$, where $X_{C1}$ and $X_{Cy}$ indicate quantities determined applying Configuration 1 and Configurations $y$ (i.e., 2–6).

Table 3. Root Mean Square (rms) of Percent Differences between Data Corrected for Stray Light Perturbations Using Configurations 2–6 with respect to Configuration 1 at Different Levels of Processing: Level 0 (L0), Level 1 (L1), and Level 2 (L2)

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As shown in Table 3, relative differences are very low at all processing levels. At Level 0, the rms is generally lower than 0.5% below 510 nm, but can slightly exceed 1% at 665 nm. At Level 1, the rms is always lower than 0.3%, with the highest values at 412 and 665 nm. At Level 2, the rms is generally lower than 1%. However, configurations C3 and C6 exhibit higher rms with values up to 1.36% at 665 nm.

B. Feasibility of a Common SDF Matrix for the Whole Class of Radiometers

Stray light characterization of a hyperspectral radiometer is a demanding task in terms of expertise, instrumentation, and time. This can make the full characterization of a set of instruments unfeasible for many of the users. In this context, the possibility of using one single matrix for an entire class of instruments should be assessed.

In the following analysis, a common matrix is used to correct all the collected measurements, regardless of the sensor. Statistical results obtained from the comparison of corrections determined applying the same matrix with respect to the three different sensors are summarized in Table 4. The values of rms for Levels 0 and 1 are already included in Table 3 and are not repeated. Considering Level 2, they are computed as $\epsilon =100·(R_{{\mathrm{RS}}_{3M}}-R_{{\mathrm{RS}}_{1M}})/R_{{\mathrm{RS}}_{3M}}$, where $R_{{\mathrm{RS}}_{3M}}$ and $R_{{\mathrm{RS}}_{1M}}$ indicate the $R_{\mathrm{RS}}$ determined applying three independent matrices and one single matrix, respectively.

Table 4. Root Mean Square (rms) of the Percent Differences between Level 2 (L2) Data Corrected for Stray Light Perturbations Using One Common Matrix Versus Three Different Matrices at Different Levels of Processing

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The values in Table 4 are very close to those shown in Table 3. The rms of the relative percentage differences is generally lower than 0.7%, except at 665 nm, where it can slightly exceed 1% with the correction matrices determined for SAM-8313 and SAM-835C. Considering the very low values of the remote-sensing reflectance above 650 nm, these values support the possibility of applying a single SDF matrix to correct for stray light above-water measurements from the same class of radiometers at the expense of a relatively small increase in uncertainty.

C. Application of a Calibration Function

In the previous tests, the calibration of the radiometers has been simplified by applying a calibration factor to each detector element, as indicated in Eq. (11). With this solution, the spectral responsivity function of each element of the detector array is assumed to be rectangular and the calibration matrix is reduced to its diagonal [11]. To investigate the effects of substituting the calibration factor with a calibration function, a triangle-shaped function of 10 nm full width at half-maximum (FWHM) has been chosen to describe the spectral responsivity function of each element of the detector array.

This more complex spectral responsivity function entails the necessity of introducing a deconvolution/convolution block in the stray light generation and correction models illustrated in Fig. 2 (now revisited in Fig. 14). The deconvolution of the measured signal from the sensor spectral responsivity function is performed iteratively, as described in a previous work [12]. Applying the stray light generation and correction model including the deconvolution/convolution process, new SDF matrices have been computed and all the measurements have been reprocessed. Results from the reprocessing are summarized in Table 5. In this case, the percent difference has been calculated between the corrected measurements with and without including the deconvolution/convolution step in the processing chain, as $\epsilon =100·(X_{n\mathrm{DC}}-X_{\mathrm{DC}})/X_{n\mathrm{DC}}$, where $X_{n\mathrm{DC}}$ and $X_{\mathrm{DC}}$ indicate quantities determined without and with application of the deconvolution/convolution step, respectively.

Table 5. Root Mean Square (rms) of Percent Differences $\epsilon$ Induced by the Introduction of the Deconvolution/Convolution Step in the Processing Chain

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Fig. 14. Stray light generation and correction model including the deconvolution/convolution process.

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The introduction of the deconvolution/convolution step produces slight differences in the results, but always lower than 0.7%.

8. SUMMARY AND CONCLUSIONS

Stray light is an appreciable source of uncertainty for radiometric measurements performed with hyperspectral sensors. Thus the characterization of stray light perturbations can be of utmost importance when the application of data products requires high accuracy (e.g., satellite validation) and in the spectral regions where the sensitivity of the instrument or the signal-to-noise ratio is particularly low (i.e., toward the ultraviolet and in the near-infrared). As a novelty, in this study stray light correction has been implemented and tested on above-water measurements.

Results based on above-water in situ measurements performed with RAMSES hyperspectral radiometers in the Mediterranean Sea confirm the importance of an accurate characterization of stray light. At Level 0 (i.e., raw data) stray light contributes to 2%–3% of the measured signal, but it can exceed 7% in certain regions of the spectrum where the input signal is low. Stray light is partially corrected by the calibration of the sensor; in fact at Level 1 (i.e., radiance or irradiance) it is generally well within $\pm 1\%$ of the measured signal with higher values near 610 nm only. At Level 2 (i.e., $R_{\mathrm{RS}}$), stray light contribution is partially compensated by the composition of the different spectral quantities (i.e., total radiance from the sea $L_T$, sky radiance $L_i$, and downward irradiance $E_d$). The correction is, on average, low in the blue and green spectral regions (typically lower than $\pm 1\%$) and larger in the red spectral region (with peaks of up to 9% in eutrophic waters).

Permutation of the stray light matrices across the various sensors applied for $L_T$, $L_i$, and $E_d$ measurements has shown that the corrections only slightly vary, with rms generally lower than 1%, except for some spectral interval in the red where they approach 1.4%.

The effect of using a single stray light correction matrix for all the sensors involved in measurements of the quantities needed for the determination of the $R_{\mathrm{RS}}$ (i.e., $L_T$, $L_i$, and $E_d$) has been investigated for L2 data. Results indicate rms differences lower than 0.7% for $R_{\mathrm{RS}}$ over most of the spectrum, increasing to 1.1% at 665 nm, consistent with the results obtained from the permutation of the correction matrices.

Finally, the effect of substituting calibration factors with calibration functions in the stray light correction process has also been evaluated. Results show differences lower than 0.7% between the two calibration schemes at all levels of processing.

In summary, still considering that the study relies on the characterization of three RAMSES radiometers only, overall results lead to the following general conclusion: even though the stray light contribution is significant at Level 0, it is partially corrected by the calibration process of the sensor at Level 1, and by the composition of the different radiometric signals at Level 2. Thus, correcting above-water radiometric measurements for stray light definitively increases the accuracy of radiance and irradiance measurements. Still, when considering above-water applications and specifically the uncertainty budget affecting the determination of $R_{\mathrm{RS}}$ [1], the comprehensive characterization of each RAMSES radiometer applied for the determination of in situ radiometric quantities (i.e., $L_T$, $L_i$, and $E_d$) appears unnecessary, and one common stray light correction matrix could be considered for the processing of data for all the sensors from the same class of instruments based on spectrographs from the same production batch.