Polarimetric characteristics of a class of hyperspectral radiometers

Marco Talone; Giuseppe Zibordi

doi:10.1364/AO.55.010092

1. INTRODUCTION

The systematic and global monitoring of the sea through satellite ocean color is of major importance for climate and environmental activities. In fact, it allows for the quantification of optically significant constituents like pigment concentrations used as a proxy for phytoplankton biomass. Because of this, ocean color has been included among essential climate variables [1,2] of the global climate observing system (GCOS): this implies uninterrupted observations as well as continuous development of methods to minimize uncertainties in data products.

Current requirements for satellite ocean color radiometry indicate uncertainties lower than 5% in the blue–green spectral regions and radiometric stability of 0.5% per decade [2,3]. These ambitious goals entail continuous efforts in the calibration of space sensors and in the validation of data products. It is therefore essential that in situ measurements directed to support calibration and validation activities are performed with state-of-the-art technology and by applying consolidated measurement methods.

Optical radiometers, commonly used to produce in situ reference measurements in support of indirect calibration (i.e., vicarious calibration) or validation activities are often assumed to be insensitive to the polarization of light. This hypothesis, however, does not always reflect actual performance for sensors utilizing gratings, slits, beam splitters, or mirrors [4].

Within such a general context, the present study focuses on the analysis of the polarimetric characteristics of a class of hyperspectral radiometers widely used by the ocean color community. Specifically, a set of 11 commercial radiometers (i.e., RAMSES) manufactured by TriOS Mess-und Datentechnik GmbH (Rastede, Germany) were analyzed and their polarimetric characteristics quantified in view of determining the uncertainties affecting in situ data products from above-water radiometry when polarization sensitivity is neglected.

2. BACKGROUND ON THE STOKES VECTOR AND THE MUELLER MATRIX

The polarimetric characteristics of a light source can be conveniently described through the Stokes vector and Mueller matrix representations. In particular, by omitting the explicit dependence on wavelength, a source $(S)$ (either radiance or irradiance) can be expressed by the following four-element Stokes vector [5]:

S = [\begin{matrix} S_{0} \\ S_{1} \\ S_{2} \\ S_{3} \end{matrix}] = S_{0} [\begin{matrix} 1 \\ s_{1} \\ s_{2} \\ s_{3} \end{matrix}],

where elements

S_{0}, S_{1}, S_{2}

, and

S_{3}

, alternatively indicated as

I, Q, U

, and

V

, designate the intensity of the radiance or irradiance (i.e.,

S_{0}

), the difference between horizontal and vertical components (i.e.,

S_{1} = S_{0 °} - S_{90 °}

), the difference between the two linearly polarized components at 45° and 135° (i.e.,

S_{2} = S_{45 °} - S_{135 °}

), and the difference between the two circularly polarized right and left components (i.e.,

S_{3} = S_{RHC} - S_{LHC}

), respectively. It is noted that the formalism used in the right term of Eq. (1) makes explicit the parameters defining the polarimetric characteristics of the light source:

s_{1} = S_{1} / S_{0}

,

s_{2} = S_{2} / S_{0}

, and

s_{3} = S_{3} / S_{0}

.

Consistently with the Stokes vector representation, any optical system can be described by the transmittance of its components through the $4 \times 4$ Mueller matrix $(T)$ as [3]

T = [\begin{matrix} T_{00} & T_{01} & T_{02} & T_{03} \\ T_{10} & T_{11} & T_{12} & T_{13} \\ T_{20} & T_{21} & T_{22} & T_{23} \\ T_{30} & T_{31} & T_{32} & T_{33} \end{matrix}],

where

T_{i j}

indicates the fraction of the

j

th component of the input signal transmitted to the

i

th component of the output signal.

The result from the interaction of a light source defined by the Stokes vector $S$ with an optical system determined by the Mueller matrix $T$ is indicated as $S_{T}$ and is given by the matrix multiplication $S_{T} = T \cdot S$ :

S_{T} = [\begin{matrix} T_{00} \cdot S_{0} + T_{01} \cdot S_{1} + T_{02} \cdot S_{2} + T_{03} \cdot S_{3} \\ T_{10} \cdot S_{0} + T_{11} \cdot S_{1} + T_{12} \cdot S_{2} + T_{13} \cdot S_{3} \\ T_{20} \cdot S_{0} + T_{21} \cdot S_{1} + T_{22} \cdot S_{2} + T_{23} \cdot S_{3} \\ T_{30} \cdot S_{0} + T_{31} \cdot S_{1} + T_{32} \cdot S_{2} + T_{33} \cdot S_{3} \end{matrix}] .

When considering the specific case of a radiometer, it can be ideally modeled as two separate components (see Fig. 1): the first representing the optics of the sensor, solely accounting for its polarimetric characteristics; and the second representing the detector, only sensitive to the intensity of the optics output.

Fig. 1. Block representation of an optical radiometer.

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Defining the optics and the detector through their respective Mueller matrices $r$ and $R$ , the measured value (DN) is given by

D N = R \cdot r \cdot S = S_{0} [\begin{matrix} R & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \cdot [\begin{matrix} R_{00} & R_{01} & R_{02} & R_{03} \\ R_{10} & R_{11} & R_{12} & R_{13} \\ R_{20} & R_{21} & R_{22} & R_{23} \\ R_{30} & R_{31} & R_{32} & R_{33} \end{matrix}] \cdot [\begin{matrix} 1 \\ s_{1} \\ s_{2} \\ s_{3} \end{matrix}],

where the single element composing the matrix

R

,

R

, is the detector responsivity defined by the variation of the radiometer output per unit of incident radiance or irradiance. Underlying assumptions are the spectral invariance of

R

in each radiometer spectral band and the homogeneity of the incident flux in the radiometer field of view.

From Eq. (4), $D N$ is determined by

D N = S_{0} \cdot (R_{00} + s_{1} \cdot R_{01} + s_{2} \cdot R_{02} + s_{3} \cdot R_{03}) \cdot R = S_{0} \cdot R_{00} \cdot (1 + s_{1} \cdot r_{1} + s_{2} \cdot r_{2} + s_{3} \cdot r_{3}) \cdot R,

where

R_{00}

indicates the responsivity of the radiometer to unpolarized light, and parameters

r_{1} = R_{01} / R_{00}

,

r_{2} = R_{02} / R_{00}

, and

r_{3} = R_{03} / R_{00}

define its polarimetric characteristics.

3. INSTRUMENTS, METHODS, AND DATA

The determination of the polarimetric characteristics of radiometers is performed assuming that all the optical components involved exhibit almost-ideal performance (i.e., nonidealities are limited to a few percent [6]). This assumption has been verified within the limits of the measurement setup. Assessments are presented in the Appendix for both radiance and irradiance sources, polarizers, and depolarizers. The description of the radiometers and the methodology applied for their characterization are presented in the next section. In addition, possible uncertainties induced by the measurement setup are also addressed and discussed.

A. Radiometers

Eleven TriOS RAMSES hyperspectral radiometers have been characterized: eight RAMSES-ARC radiance sensors (with serial numbers SAM-82CD, SAM-82CF, SAM-8313, SAM-8346, SAM-84C2, SAM-82C3, SAM-8508, and SAM-850D) and three RAMSES-ACC irradiance sensors (with serial numbers SAM-835C, SAM-82C1, and SAM-84C0).

These radiometers rely on ZEISS (Oberkochen, Germany) monolithic miniature spectrometers (MMS-1). This spectrometer, which builds on a 256 channel silicon photodiode array coupled to a grating, has a spectral resolution better than 10 nm in the 320–950 nm spectral interval with an average spectral sampling of 3.3 nm per element of the detector array and a spectral accuracy of 0.3 nm. Specific to MMS-1 spectrometers is the fiber optics connector for light supply, the entrance slit, the concave image grating, and the photodiode array, which are all permanently glued to each other to increase ruggedness and ensure permanent alignment. Additionally, light sensitivity is optimized through a fiber bundle cross-section converter with single fibers arranged in a linear configuration to form the entry slit (http://www.zeiss.com/spectroscopy/products/spectrometer-modules/monolithic-miniature-spectrometer-mms.html).

B. Polarimetric Characterization

A FEL-1000 W lamp has been used as a point source for the execution of measurements. The determination of the polarimetric characteristics of irradiance sensors has been obtained by pointing the radiometers normally to the lamp. Differently, the characterization of radiance sensors has been performed by pointing the radiometers at 45 deg incidence angle to a 99% reflectance plaque illuminated by a FEL-1000 W lamp. The stability of the source has been monitored by measuring the current flowing through the lamp using a shunt in series with the lamp. In particular, the typical maximum change observed in the 8 A current applied has been 0.3 mA. This implies variations of 0.03% in the lamp flux at 400 nm, decreasing to less than 0.02% at 700 nm [7]. All characterizations have been restricted to the 400–750 nm spectral region, which is of major interest for ocean color applications but also imposed by the spectral features of the depolarizers applied in the study.

The experimental methodology applied for the polarimetric characterization of radiometers, which implies the determination of parameters $r_{1}$ , $r_{2}$ , and $r_{3}$ , is comprehensively described in [5,6,8] and summarized hereafter. In general terms, the radiometer to be characterized needs to be pointed toward a stable source with a linear polarizer placed in between.

P = [\begin{matrix} s & d \cdot \cos 2 ψ & d \cdot \sin 2 ψ & 0 \\ d \cdot \cos 2 ψ & s \cdot \cos^{2} 2 ψ + p \cdot \sin^{2} 2 ψ & (s - p) \cdot \sin 2 ψ \cdot \cos 2 ψ & - q \cdot \sin 2 ψ \\ d \cdot \sin 2 ψ & (s - p) \cdot \sin 2 ψ \cdot \cos 2 ψ & s \cdot \sin^{2} 2 ψ + p \cdot \cos^{2} 2 ψ & q \cdot \cos 2 ψ \\ 0 & q \cdot \sin 2 ψ & - q \cdot \cos 2 ψ & p \end{matrix}] .

Specifically, the configurations used for radiance and irradiance sensors are displayed in the top and bottom panels of Fig. 2, respectively.

Fig. 2. Measurement configurations applied for the determination the polarimetric characteristics of (a) radiance and (b) irradiance sensors.

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The orientation of the polarizer is kept constant with respect to the source, whereas the sensor is rotated around its main axis to perform successive measurements at different rotating angles. This solution allows the sensor to look at a linearly polarized light realized by coupling a source and a polarizer.

Following the flow diagram illustrated in Fig. 3, the signal $S_{P}$ resulting from the interaction of the source with the polarizer is given by the product of the respective Mueller matrices $S$ and $P$ . The generic Mueller matrix $P$ for a polarizer is defined by Eq. (6), where $ψ$ indicates the rotation angle between the source and the polarizer, and parameters $s$ , $d$ , $p$ , and $q$ are functions of the maximum and minimum attenuations ( $k_{1}$ and $k_{2}$ , respectively) and of the retardance $(δ)$ of the polarizer. Specifically, where:

s = \frac{1}{2} (k_{1} + k_{2}),

d = \frac{1}{2} (k_{1} - k_{2}),

p = \sqrt{k_{1} \cdot k_{2}} \cdot \cos δ,

and

q = \sqrt{k_{1} \cdot k_{2}} \cdot \sin δ .

Fig. 3. Flow diagram for the determination of the polarimetric characteristics of a radiometer.

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Assuming the polarizer is oriented with respect to $ψ = 0$ , the matrix $P$ reduces to

P = [\begin{matrix} s & d & 0 & 0 \\ d & s & 0 & 0 \\ 0 & 0 & p & q \\ 0 & 0 & - q & p \end{matrix}]

and

S_{P} = P \cdot S = S_{0} [\begin{matrix} s + s_{1} \cdot d \\ d + s_{1} \cdot s \\ s_{2} \cdot p + s_{3} \cdot q \\ - s_{2} \cdot q \cdot + s_{3} \cdot p \end{matrix}] .

When

S_{P}

is rotated by an angle

φ

by applying the Mueller matrix

M

M = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos 2 φ & \sin 2 φ & 0 \\ 0 & - \sin 2 φ & \cos 2 φ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}],

the resulting matrix

S_{MP}

is

S_{MP} = M \cdot S_{P} = S_{0} [\begin{matrix} s + s_{1} \cdot d \\ (d + s_{1} \cdot s) \cdot \cos 2 φ + (s_{2} \cdot p + s_{3} \cdot q) \cdot \sin 2 φ \\ - (d + s_{1} \cdot s) \cdot \sin 2 φ + (s_{2} \cdot p + s_{3} \cdot q) \cdot \cos 2 φ \\ - s_{2} \cdot q + s_{3} \cdot p \end{matrix}] .

The value

D N

measured by the radiometer is expressed by the product

r \cdot S_{MP} \cdot R

:

D N = S_{0} \cdot R_{00} \cdot (s + s_{1} \cdot d + r_{1} \cdot d \cdot \cos 2 φ + r_{1} \cdot s_{1} \cdot s \cdot \cos 2 φ + r_{1} \cdot s_{2} \cdot p \cdot \sin 2 φ + r_{1} \cdot s_{3} \cdot q \cdot \sin 2 φ - r_{2} \cdot d \cdot \sin 2 φ - r_{2} \cdot s_{1} \cdot s \cdot \sin 2 φ + r_{2} \cdot s_{2} \cdot p \cdot \cos 2 φ + r_{2} \cdot s_{3} \cdot q \cdot \cos 2 φ - r_{3} \cdot s_{2} \cdot q + r_{3} \cdot s_{3} \cdot p) \cdot R .

Assuming the polarimetric parameters of both the radiometer (

r_{1}

,

r_{2}

, and

r_{3}

) and the source (

s_{1}

,

s_{2}

, and

s_{3}

) are of the order of a few percent, thus allowing to assume that any second-order addend can be safely neglected being below the instrument noise (i.e., tentatively below

10^{- 4}

of the maximum output value of RAMSES radiometers), Eq. (15) reduces to

D N ≅ S_{0} \cdot R_{00} \cdot (s + s_{1} \cdot d + r_{1} \cdot d \cdot \cos 2 φ - r_{2} \cdot d \cdot \sin 2 φ) \cdot R .

The parameters

r_{1}

and

r_{2}

can be determined through optimal fitting (in this study, it has been performed by using the Powell function described in Section 10.5 of [9]). Parameter

r_{3}

is neglected under the assumption that the sensitivity to circular polarization is weak or that its contribution is commonly negligible for sky radiance and relatively small for sea radiance [10–12].

C. Inaccuracy of the Measurement Setup

All measurements were performed in a temperature/humidity-controlled environment where average excursions of temperature and relative humidity have been within 2°C and 20%, respectively. Sensors as well as all the other optical components have been installed on an optical bench screened with curtains and baffles to minimize ambient stray light.

Inaccuracies of the measurement setup, which may lead to measurement uncertainties, have been quantified by clearing the optical path between the sensor and the source, through the process illustrated by the flow diagram in Fig. 4.

Fig. 4. Flow diagram for the assessment performed to quantify the inaccuracy of the measurement setup.

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As illustrated in Fig. 4, the signal related to the source $S$ is virtually rotated through the matrix $M$ and transmitted to the radiometer $r$ , and the measured value $D N$ is

D N = r \cdot S_{M} \cdot R = S_{0} \cdot R_{00} \cdot (1 + r_{1} \cdot s_{1} \cdot \cos 2 φ + r_{1} \cdot s_{2} \cdot \sin 2 φ - r_{2} \cdot s_{1} \cdot \sin 2 φ + r_{2} \cdot s_{2} \cdot \cos 2 φ + r_{3} \cdot s_{3}) \cdot R .

By neglecting second-order addends, Eq. (17) reduces to

D N = S_{0} \cdot R_{00} \cdot R .

Any residual signal could be due to sensor–source misalignment, spatial inhomogeneity or temporal instability of the source, or inaccuracy of the derived $r$ . Two examples of residual signal are displayed in Fig. 5, as determined with SAM-84C3 but with independent installations of the radiometer (i.e., using different 99% Spectralon reflectance plaques together with a complete reinstallation of the measurement setup). In Fig. 5, stars represent the actual measurements while curves are the fitted values. Measurements and fits are only displayed for a selected set of wavelengths, namely, 412, 443, 490, 555, 670, and 750 nm. Equivalent results have been obtained for other RAMSES-ARC radiance and also RAMSES-ACC irradiance sensors.

Fig. 5. Background signal due to measurement inaccuracy as determined with SAM-84C3 using two independent installations (i.e., different 99% Spectralon reflectance plaques and by reinstalling the measurement setup).

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Results from the analysis of the measurement inaccuracy, which are systematically applied for the correction of background values from measurement sequences during data processing, indicate values of the order of 0.1%–0.2% without any marked spectral feature. This latter finding may indicate a geometric dependence of the residual signal.

It is mentioned that in addition to the correction applied to remove the residual signal, before and after each characterization, multiple ambient measurements have been performed and their average has also been removed from each measurement in view of correcting for laboratory stray light.

4. RESULTS AND DISCUSSION

Main results from the characterization of the various radiometers are presented and discussed hereafter together with an evaluation of the effects of polarization sensitivity on above-water measurements. It is mentioned that handling of RAMSES radiometer data is comprehensively described in Talone et al. [13].

A. Polarization Sensitivity of the Considered Class of Radiometers

Polarimetric characteristics of the eleven RAMSES radiometers have been determined as described in Section 1. Outcomes are separately presented for radiance and irradiance sensors.

1. RAMSES-ARC Radiance Sensors

Results for RAMSES-ARC radiance sensors are displayed in Fig. 6 as a function of wavelength. Radiometers are grouped two-by-two according to the level of field exploitation: SAM-8508 and SAM-850D manufactured in 2016, never used; SAM-84C2 and SAM-84C3 manufactured in 2015, only once used in the field; SAM-82CD and SAM-82CF manufactured in 2010, and SAM-8313 and SAM-8346 manufactured in 2012, all intensively used during several field campaigns. It is specified that the custom radiometers SAM-82CD and SAM-82CF have full-angle field of view of 3 deg different from the 7 deg of any other RAMSES-ARC sensor.

Fig. 6. Parameter $r_{1}$ obtained for radiance sensors SAM-8346 and SAM-8313 (blue), SAM-82CD and SAM-82CF (red), SAM-84C2 and SAM-84C3 (light blue), and SAM-850D and SAM-8508 (green).

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Radiometers have been reciprocally aligned to maximize parameter $r_{1}$ (at the expense of $r_{2}$ ) by commonly adjusting source-to-sensor orientation. For this reason, results are presented in terms of $r_{1}$ only. Equivalent formulations can be obtained including both $r_{1}$ and $r_{2}$ by changing the relative source and sensor orientations.

As shown in Fig. 6, the difference between horizontal and vertical polarization sensitivities increases with wavelength. Additionally, assuming the radiometers were all manufactured with identical optical components, the polarization sensitivity appears to increase with the radiometer age and use. In particular, $r_{1}$ exhibits a maximum value lower than 0.4% at 750 nm for new radiance sensors but reaches 1.5% for the most used units. However, such a finding does not appear supported by additional investigations performed on radiometers SAM-8346 and SAM-84C2 after recent field deployments (i.e., pre- and post-field characterizations did not show any appreciable change in polarization sensitivities).

Applying Eq. (16) and assuming $r_{2} = 0$ for the specific measurement geometry, the percent change $Δ$ of the measured signal as a function of $φ$ with respect to the mean value $\bar{D N}$ can be approximated as

Δ (φ) = 100 \cdot \frac{(D N (φ) - \bar{D N})}{\bar{D N}} ≅ \frac{r_{1} \cdot d \cdot \cos 2 φ}{s + s_{1} \cdot d} .

The percent change $Δ (φ)$ is shown in Fig. 7 for the particular case of RAMSES SAM-8346.

Fig. 7. Spectral percent change $Δ$ of the measured signal as a function of the rotation angle $φ$ for SAM-8346 due to the polarization sensitivity of the radiance sensor.

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Figure 7 indicates that a better optimal fitting of measurements is achieved at the longest wavelengths. Differently, poorer results are obtained below 500 nm, which are likely due to the low values of $Δ (φ)$ (i.e., lower than 0.1%) competing with the accuracy of polarimetric characterizations.

2. RAMSES-ACC Irradiance Sensors

Parameters $r_{1}$ determined for the three RAMSES-ACC irradiance sensors are displayed in Fig. 8 as a function of wavelength.

Fig. 8. Parameter $r_{1}$ obtained for the irradiance sensors SAM-835C (blue), SAM-82C1 (red), and SAM-84C0 (green).

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Data indicate that the difference between horizontal and vertical polarization sensitivity for the considered irradiance sensors is lower than approximately 0.6%, with the highest values at the longest wavelengths. Values obtained for SAM-835C and SAM-82C1 are always lower that 0.3% and are very similar to each other. SAM-84C0 of more recent production when compared to the previous sensors, instead presents higher values comprised between 0.3% and 0.6%. The angular percent change $Δ (φ)$ is illustrated in Fig. 9 for SAM-82C1.

Fig. 9. Spectral percent difference $Δ$ of the measured signal as a function of the rotation angle $φ$ for SAM-82C1 due to the polarization sensitivity of the irradiance sensor.

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Similar to the case of radiance sensors, a better optimal fitting of measurements is obtained at the longer wavelengths exhibiting higher values of $Δ (φ)$ .

Previous results confirm the depolarizing effect of the collector at the entrance of the radiometer as opposite to radiance sensors.

B. Effects on Field Measurements

RAMSES radiometers are commonly applied to perform above-water radiometric measurements in support of satellite ocean color validation activities. By neglecting the nonistrophic distribution of the in-water light field and also high glint perturbations, the remote sensing reflectance $R_{RS}$ is determined from

R_{RS} = \frac{L_{w}}{E_{d}} = \frac{1}{E_{d}} (L_{T} - ρ L_{i}),

where

E_{d}

is the downward irradiance at the sea surface,

L_{w}

is the water-leaving radiance computed from

L_{T}

(total radiance from the sea) and

L_{i}

(sky radiance), both measured with appropriate geometries [14], and finally

ρ

is the reflectance of the surface ideally determined accounting for sky radiance polarization and sea surface effects [15,16].

When neglecting the polarization sensitivity and consequently the sensor orientation with respect to the preferred polarization plane, polarization perturbations in $L_{T}$ , $L_{i}$ , and $E_{d}$ measurements likely affect computed $R_{RS}$ . This source of uncertainty in $R_{RS}$ has been investigated through the application of parameter $r_{1}$ and additionally accounting for the degree of linear polarization (DoLP) of light [17]:

DoLP = \frac{\sqrt{S_{1}^{2} + S_{2}^{2}}}{S_{0}} .

Ignoring the polarization perturbations in

E_{d}

measurements because of the relatively small values of

r_{1}

determined for RAMSES irradiance sensors, the perturbations in

L_{T}

and

L_{i}

have been estimated through additive contributions

Δ L_{T} = r_{1} \cdot {DoLP}_{L_{T}} \cdot L_{T}

and

Δ L_{i} = r_{1} \cdot {DoLP}_{L_{i}} \cdot L_{i}

to

L_{T}

and

L_{i}

, respectively. In agreement with a viewing geometry commonly applied for

L_{i}

and

L_{T}

measurements (i.e., relative azimuth of 90° with respect to the Sun, and viewing angles of 40° and 140° for

L_{i}

and

L_{T}

, respectively [14]), representative DoLP values of 0.25, 0.50, and 0.75 for

L_{T}

(i.e.,

{DoLP}_{L_{T}}

) and of 0.40 for

L_{i}

(i.e.,

{DoLP}_{L_{i}}

) have been considered [15–18]. The sensitivity analysis has been performed using field measurements collected with clear-sky conditions in oligotrophic waters in the Western Mediterranean Sea and in sediment-dominated waters in the northern Adriatic Sea. Measured

L_{T}

and

L_{i}

, and computed

R_{RS}

spectra, are displayed in Figs. 10 and 11.

Fig. 10. Spectra of (a) $L_{T}$ and $L_{i}$ and (b) $R_{RS}$ from the Western Mediterranean sea oligotrophic/mesotrophic waters. Measurements were performed with radiometers SAM-8346 $(L_{T})$ , SAM-8313 $(L_{i})$ , and SAM-835C $(E_{d})$ on 14 April 2014 at 12:17 GMT.

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Fig. 11. Spectra of (a) $L_{T}$ and $L_{i}$ and (b) $R_{RS}$ from the northern Adriatic Sea sediment-dominated waters. Measurements were performed with radiometers SAM-8346 $(L_{T})$ , SAM-8313 $(L_{i})$ , and SAM-835C $(E_{d})$ on 4 April 2015 at 8:11 GMT.

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In view of focusing the analysis on those values maximizing perturbations due to polarization sensitivity, results are only presented for the largest spectral values of $r_{1}$ displayed in Fig. 6. The percent differences with respect to $L_{w}$ of both $Δ L_{T}$ and $ρ Δ L_{i}$ are displayed as a function of wavelength in Figs. 12 and 13. A tentative value of $ρ$ equal to 0.028 has been applied for both case studies.

Fig. 12. Effect of the radiometer’s polarization sensitivity on (a) $L_{T}$ and (b) $L_{i}$ measurements from the Western Mediterranean Sea oligotrophic waters, expressed in percent of $L_{w}$ .

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Fig. 13. Effect of radiometers polarization sensitivity on (a) $L_{T}$ and (b) $L_{i}$ measurements from the northern Adriatic Sea sediment-dominated waters, expressed in percent of $L_{w}$ . To facilitate direct comparisons, the y scale is deliberately set identical to that of Fig. 12.

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Results from the analysis of perturbations induced by neglecting polarization sensitivity are displayed in Figs. 14 and 15 through percent differences $ε$ of the $R_{RS}$ values displayed in Figs. 10 and 11 with respect to those determined accounting for the additive contributions $Δ L_{T}$ shown by the solid lines and $\pm ρ Δ L_{i}$ represented by the error bars.

Fig. 14. Effects of polarization sensitivity on $R_{RS}$ from the Western Mediterranean Sea oligotrophic waters. Results have been determined with ${DoLP}_{L_{T}}$ equal to 25%, 50%, and 75% (displayed in black, blue, and red, respectively). In view of reducing the complexity of the figure, data are only presented for positive values of $Δ L_{T}$ , which maximize perturbations due to polarization sensitivity. Error bars refer to values of $\pm Δ L_{i}$ determined with ${DoLP}_{L_{i}}$ equal to 40%.

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Fig. 15. Effect of polarization sensitivity on $R_{RS}$ spectral values from the northern Adriatic Sea sediment-dominated waters. Results have been determined with ${DoLP}_{L_{T}}$ equal to 25%, 50%, and 75% (displayed in black, blue, and red, respectively). In view of reducing the complexity of the figure, data are only presented for positive values of $Δ L_{T}$ , which maximize perturbations due to polarization sensitivity. Error bars refer to values of $\pm Δ L_{i}$ determined with ${DoLP}_{L_{i}}$ equal to 40%. To facilitate direct comparisons, the y scale is deliberately set identical to that of Fig. 14.

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Results, still limited to two case studies, consistently indicate that spectral differences $ε$ in $R_{RS}$ are limited to few tenths of a percent below 570 nm. Diversely, they may more importantly impact $R_{RS}$ at longer wavelengths as a function of the water type. Specifically, as shown in Figs. 14 and 15, $ε$ increases with wavelength up to 5% between 600 and 650 nm and then exceeds 10% above 700 nm in case of oligotrophic waters (see Fig. 14), while it never exceeds 1.4% in the case of sediment-dominated waters (see Fig. 15). In view of better evaluating results, the individual $L_{i}$ and $L_{T}$ contributions are separately investigated. When solely considering $L_{i}$ , polarization effects on $R_{RS}$ are always lower than 0.1% for sediment-dominated waters (i.e., comparable to the uncertainties driving the determination of the related polarimetric characteristics). However, in case of oligotrophic waters, the contribution of $L_{i}$ to the polarization effects on $R_{RS}$ increases above 570 nm, exceeding 5% beyond 730 nm. Higher effects are observed for $L_{T}$ whose contribution to $R_{RS}$ can be up to several times larger than that quantified for $L_{i}$ . Thus, when analyzing $R_{RS}$ , differences in $ε$ due to polarization sensitivity are mostly driven by $L_{T}$ below 570 nm. At longer wavelengths, the $L_{i}$ contribution is only negligible in the case of sediment-dominated waters, while it may become a large fraction of $ε$ for oligotrophic waters.

When considering uncertainty requirements for satellite ocean color radiometry [2], the previous results indicate that the polarization sensitivity in the blue–green spectral region of the considered class of radiometers can be disregarded at the expense of a moderate spectral increase in the measurement uncertainty as a function of illumination conditions and water type. Definitively, polarimetric sensitivity should be considered when highly accurate measurements above approximately 570 nm are required, especially in oligotrophic and mesotrophic waters.

5. CONCLUSIONS

The objective of this work was the determination of the polarimetric characteristics of a class of hyperspectral radiometers in view of quantifying their effects on in situ $R_{RS}$ from above-water measurements of $E_{d}$ , $L_{i}$ , and $L_{T}$ . The characteristics of eleven hyperspectral radiometers were investigated (i.e., three RAMSES-ACC irradiance and eight RAMSES-ARC radiance units).

Results indicate that the polarimetric characteristics may appreciably vary from unit to unit. For all instruments, the polarization sensitivity exhibits an increase with wavelength. In the case of radiance sensors, the difference at 400 nm exhibits values varying from below 0.1% to 0.3%, while at 750 nm differences vary from approximately 0.4% up to 1.3%. In the case of irradiance sensors, the largest differences between horizontal and vertical polarization sensitivities vary from 0.3% at 400 nm to approximately 0.6% at 750 nm.

Concerning in situ above-water radiometric measurements, sensor orientation with respect to its preferred polarization plane can be reasonably neglected for $E_{d}$ measurements. Differently, in the case of $L_{T}$ and $L_{i}$ , it can lead to appreciable polarization effects affecting the determination of $R_{RS}$ as a function of water type and the state of polarization of sky and in-water light fields. An attempt to quantify such perturbations indicates differences in $R_{RS}$ computed accounting and alternatively neglecting polarization sensitivity always lower than approximately 0.7% up to 570 nm (0.3% for sediment-dominated waters). Above 570 nm, these differences slightly increase with the wavelength, reaching the maximum value of 1.4% at 750 nm in case of sediment-dominated waters, while they can exceed 10% beyond 730 nm for oligotrophic waters.

The previous results, which provide valuable uncertainty estimates, do not necessarily indicate the need for the application of measurement and data reduction methods accounting for polarization sensitivity of RAMSES hyperspectral radiometers in the blue–green spectral region. Nonetheless, this should be envisaged when highly accurate measurements are required beyond approximately 570 nm, especially in oligotrophic and mesotrophic waters.

APPENDIX A: CHARACTERIZATION OF THE OTHER OPTICAL COMPONENTS

A. Light Sources

The polarimetric characterization of the irradiance (i.e., a FEL-1000 W) and radiance (i.e., a FEL-1000 W combined with a 99% reflectance plaque) sources have been performed by fixing a polarizer on the radiometer in front of the fore optics (see Fig. 16). The radiometer was then rotated around its axis and successive measurements were performed at different angles.

Fig. 16. Measurement configuration for the determination of the polarimetric characteristics of the radiance (a) and irradiance (b) sources.

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Figure 17 illustrates the flow diagram of the characterization process. In this case, the applied solution allows us to look at the source by selecting different linear polarization states as a function of the rotation angle $φ$ .

Fig. 17. Flow diagram for the determination of the polarimetric characteristics of a source.

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Equivalent to the determination of the polarimetric characteristics of a radiometer, by following the scheme in Fig. 17, we find

S_{M} = S_{0} [\begin{matrix} 1 \\ s_{1} \cdot \cos 2 φ + s_{2} \cdot \sin 2 φ \\ - s_{1} \cdot \sin 2 φ + s_{2} \cdot \cos 2 φ \\ s_{3} \end{matrix}]

and

S_{PM} = P \cdot S_{M} = S_{0} [\begin{matrix} s + (s_{1} \cdot \cos 2 φ + s_{2} \cdot \sin 2 φ) \cdot d \\ d + (s_{1} \cdot \cos 2 φ + s_{2} \cdot \sin 2 φ) \cdot s \\ (- s_{1} \cdot \sin 2 φ + s_{2} \cdot \cos 2 φ) \cdot p + s_{3} \cdot q \\ - (- s_{1} \cdot \sin 2 φ + s_{2} \cdot \cos 2 φ) \cdot q + s_{3} \cdot p \end{matrix}] .

Consequently, the measured value

D N

is

D N = r \cdot S_{P M} \cdot R = S_{0} \cdot R_{00} \cdot (s + s_{1} \cdot d \cdot \cos 2 φ + s_{2} \cdot d \cdot \sin 2 φ + r_{1} \cdot d + r_{1} \cdot s_{1} \cdot s \cdot \cos 2 φ + r_{1} \cdot s_{2} \cdot s \cdot \sin 2 φ - r_{2} \cdot s_{1} \cdot p \cdot \sin 2 φ + r_{2} \cdot s_{2} \cdot p \cdot \cos 2 φ + r_{2} \cdot s_{3} \cdot q - r_{3} \cdot s_{1} \cdot q \cdot \sin 2 φ - r_{3} \cdot s_{2} \cdot q \cdot \cos 2 φ + r_{3} \cdot s_{3} \cdot p) \cdot R .

By neglecting the second-order addends, Eq. (A3) reduces to

D N = S_{0} \cdot R_{00} \cdot (s + r_{1} \cdot d + s_{1} \cdot d \cdot \cos 2 φ + s_{2} \cdot d \cdot \sin 2 φ) \cdot R .

Parameters

s_{1}

and

s_{2}

for the source can be obtained by optimal fitting, while parameter

s_{3}

is neglected in agreement with assumptions already applied for

r_{3}

.

Results from the polarimetric characterization of a FEL-1000 W lamp, as obtained from independent measurements performed with three irradiance sensors, are shown in Fig. 18 as a function of wavelength, in terms of mean value and standard deviation of parameter $s_{1}$ . Similar to the radiometer characterization, parameter $s_{2}$ is set to zero by properly adjusting the measurement geometry.

Fig. 18. Parameters $s_{1}$ obtained for a FEL-1000 W lamp (i.e., H-96531). The error bars on the y axis indicate the standard deviation determined from the polarimetric characterization of the lamp through measurements performed with three different RAMSES-ACC irradiance sensors.

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In agreement with previous studies [5,19], results indicate a difference between horizontal and vertical polarizations (i.e., $s_{1}$ ) varying between 2.5% and 2.7%, slightly increasing with wavelength.

As mentioned, the determination of $s_{1}$ is obtained by optimal fitting of measurements acquired under different rotation angles $φ$ (i.e., 21 with 18 deg steps). The percent change $Δ$ of the measured signal as a function of $φ$ , as determined with respect to the mean value $\bar{D N}$ , is shown in Fig. 19 for the particular case of characterizations performed with the RAMSES SAM-82C1 irradiance sensor.

Fig. 19. Spectral percent change $Δ$ of the measured signal as a function of the rotation angle $φ$ determined with RAMSES SAM-82C1 pointing at the FEL-1000 W lamp.

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Spectralon reflectance plaques from Labsphere (North Sutton, New Hampshire, USA) are commonly used to produce radiance fields from irradiance sources obtained with FEL-1000 W lamps [14]. This solution leads to less pronounced polarimetric characteristics of the radiance source when compared to those of the irradiance. This is highlighted in Fig. 20 through the spectral mean values and standard deviations of parameter $s_{1}$ from measurements performed with eight RAMSES-ARC radiance sensors. The average value of $s_{1}$ increases with wavelength from 0.15% to 0.25%, confirming the results obtained in specific investigations on polarization sensitivity of Spectralon plaques [20,21]. Specifically, the expected DOLP is approximately 3%–4% of that characterizing the light incident at the plaque [20,21]. This suggests expected values of approximately 0.1% in the specific case of this study. Differences between expected (i.e., 0.1%) and quantified (i.e., 0.15%–0.25%) values are likely explained by the uncertainties intrinsic of the measurement setup.

Fig. 20. Parameter $s_{1}$ determined for the radiance source obtained with the plaque illuminated by a FEL-1000 W lamp. The error bars on the y axis indicate the standard deviation obtained from the polarimetric characterization of the plaque as determined using measurements performed with eight different RAMSES-ARC radiance sensors.

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As with the characterization of the lamp, the percent change $Δ$ as a function of the rotation angle $φ$ is shown in Fig. 21 for the case of characterizations performed with the SAM-8346 radiance sensor.

Fig. 21. Spectral percent change $Δ$ of the measured signal as a function of the rotation angle $φ$ determined with RAMSES SAM-8346 pointing at the radiance source.

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B. Polarizers

Three linear polarizers have been used in this study. A 100 mm diameter dichroic sheet polarizer $(F)$ from Melles Griot (Rochester, New York, USA) was applied for the characterization of radiometers, and two 50 mm diameter dichroich glass polarizers $(G)$ and $(H)$ from Edmund Optics (Barrington, New Jersey, USA) were used for the characterization of sources. The characteristics of these polarizers have been investigated based on the method proposed by Nicodemus [6]. Specifically, the determination of parameters $p$ and $q$ is avoided because they do not affect the characterization of both sensors and sources [see Eqs. (15) and (A3)].

Consistent with Section 1, Mueller matrices of the three polarizers are denoted as $F (ψ_{f})$ , $G (ψ_{g})$ , and $H (ψ_{h})$ and their respective parameters as $(s_{f}, d_{f}, p_{f}, q_{f})$ , $(s_{g}, d_{g}, p_{g}, q_{g})$ , and $(s_{h}, d_{h}, p_{h}, q_{h})$ .

The measurement setup is illustrated in Fig. 22 considering the specific case of polarizers $F$ and $G$ . It is composed of an ideally unpolarized light source obtained by illuminating a reflectance plaque with the FEL-1000 W lamp, a radiometer assumed insensitive to polarization [in this case, realized by coupling a depolarizer to the fore optics of a RAMSES-ARC radiance sensor (i.e., SAM-8346)], and the two polarizers placed in the optical path between the radiometer and the source. It is anticipated that the polarimetric characteristics of the radiance source (i.e., the lamp flux reflected by the plaque) are considered negligible when compared to those of the polarizer.

Fig. 22. Measurement configuration for the determination of parameters $s$ and $d$ of polarizers $F$ and $G$ .

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Each measurement sequence has been performed according to the following steps:

i. First, parameters $s$ are determined by placing one polarizer at a time between the source and the sensor and measuring the radiance with and without the polarizer. If the source were unpolarized, the measurements should be independent of the rotation angle. To minimize this effect, $M$ measurements are performed with different rotation angles $ψ$ of the polarizer and then averaged. Parameters $s$ are calculated as $s = 1 / M \cdot \sum_{M} [D N_{p} (ψ) / D N_{u} (ψ)]$ , where $D N_{p} (ψ)$ and $D N_{u} (ψ)$ are the radiances measured with and without the polarizer, respectively.
ii. Parameters $d$ are retrieved using two coupled polarizers. While the orientation of the first polarizer is kept constant with respect to the sensor, the other polarizer is rotated. In this case, by solely considering polarizers $F$ and $G$ , the radiance measured by the sensor is $D N_{F G} (ψ_{f g}) = D N \cdot (s_{f} s_{g} + d_{f} d_{g} \cos (2 ψ_{f g}))$ , where $ψ_{f g}$ is the angle between the two polarizers. The product $d_{f} d_{g}$ is determined from measurements performed at different angles $ψ_{f g}$ and from optimal fitting of $D N_{F G} (ψ_{f g})$ . The parameters $d$ for each polarizer are calculated by combining the products obtained with the different configurations $d_{f} d_{g}$ , $d_{f} d_{h}$ , and $d_{g} d_{h}$ .

The spectral values of parameters $s$ determined for polarizers $F$ , $G$ , and $H$ applying the previous scheme are shown in Fig. 23(a). Extinction ratios defined as $(s^{2} - d^{2}) / (s^{2} + d^{2})$ are displayed in Fig. 23(b) for the same polarizers. These latter values roughly range between $10^{- 2}$ and $10^{- 3}$ , while specifications indicate values of $10^{- 4}$ . Differences are explained by uncertainties of approximately 1% affecting the experimental estimates of extinction ratios as a result of uncertainties of 0.1% impacting the individual determinations of $s$ and $d$ .

Fig. 23. Parameters $s$ (a) and extinction ratios (b) determined for polarizers $F$ , $G$ , and $H$ , displayed in blue, red, and green, respectively.

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C. Depolarizer

A quartz-wedge achromatic depolarizer from Thorlabs (Newton, New Jersey, USA) has been used to realize the unpolarized sensor required for the characterization of polarizers. Its performance has been verified by determining the polarimetric sensitivity of the same radiometers (both radiance and irradiance sensors) with and without coupling the depolarizer to their fore optics. Results are summarized in Fig. 24 through parameters $r_{1}$ determined from radiometers without the depolarizer (denoted by $r_{1}$ ) and alternatively with the depolarizer (denoted by $r_{1}^{D}$ ).

Fig. 24. Values of parameters $r_{1}$ obtained by characterizing the radiance sensor SAM-8346 (blue) and the irradiance sensor SAM-82C1 (red) without the depolarizer (denoted as $r_{1}$ ) and with the depolarizer (denoted as $r_{1}^{D}$ ).

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The same test has been performed using a quartz Lyot depolarizer from Edmund Optics. It is noted that these depolarizers are only applicable over wide spectral bands, tentatively larger than 50 nm. Results (not shown) indicate that after spectral averaging, differences between values of $r_{1}^{D}$ obtained with the different depolarizers are generally lower than 0.1%.

Funding

Joint Research Centre (JRC).

Acknowledgment

The authors would like to thank the anonymous reviewers whose comments and suggestions largely improved the quality of this work. Davide D’Alimonte is duly acknowledged for his willingness to make available his RAMSES radiometers for this analysis.

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Polarimetric characteristics of a class of hyperspectral radiometers

Abstract

1. INTRODUCTION

2. BACKGROUND ON THE STOKES VECTOR AND THE MUELLER MATRIX

3. INSTRUMENTS, METHODS, AND DATA

A. Radiometers

B. Polarimetric Characterization

C. Inaccuracy of the Measurement Setup

4. RESULTS AND DISCUSSION

A. Polarization Sensitivity of the Considered Class of Radiometers

1. RAMSES-ARC Radiance Sensors

2. RAMSES-ACC Irradiance Sensors

B. Effects on Field Measurements

5. CONCLUSIONS

APPENDIX A: CHARACTERIZATION OF THE OTHER OPTICAL COMPONENTS

A. Light Sources

B. Polarizers

C. Depolarizer

Funding

Acknowledgment

REFERENCES

Cited By

Figures (24)

Equations (25)

Applied Optics