# Measuring and Modeling the Polarized Upwelling Radiance Distribution in Clear and Coastal Waters
## 2 Methods
### Measurements of the Polarized Upwelling Radiance Distribution  
In Hawaii, measurements of the polarized upwelling radiance distribution were accomplished using the PolRADS camera system [15]. PolRADS is based on the NuRADS camera system [12], which is a compact (30 cm diameter, 30 cm length), multispectral camera that images the upwelling radiance distribution in six narrow (\(\approx\)10 nm full-width at half-maximum, FWHM) spectral bands centered at 412, 436, 486, 526, 548, and 615 nm. In the PolRADS instrument, three synchronized NuRADS cameras are used, each with a linear polarizer, to simultaneously acquire images. Combining the images from three NuRADS cameras, when in PolRADS configuration, allows for retrieval of three elements of the Stokes vector (\(U\), \(Q\), and \(U\), but not \(V\)), as well as _DOLP_. In the Ligurian Sea and New York Bight, measurements of the polarized upwelling radiance distribution were accomplished with the DPOL camera system [16]. DPOL is similar to PolRADS in the sense that simultaneous measurements are made using multiple fisheye lenses with different polarizing filters in the optical path. DPOL differs from PolRADS by using optical-fiber bundles to project images from all lenses through a single spectral filter and onto a single CCD array. Eliminating the redundant filters and cameras makes DPOL much smaller than PolRADS, thereby reducing the instrument shadow. Furthermore, DPOL has four lenses, rather than the three on PolRADS, enabling retrieval of all four Stokes vector elements. The spectral filters used on DPOL were also \(\approx\)10 nm FWHM and were centered at 442, 488, 520, 550, 589, and 650 nm. For both PolRADS and DPOL, the channels above 600 nm will not be used because of instrument self-shading. In addition, because of its absence on DPOL, the 412 nm channel on PolRads will also not be discussed. Data acquisition for both the PolRADS and DPOL systems used filter changers to rotate interference filters and thus sequentially acquire images in each wavelength band. Typical exposure times were less than one second. However, acquiring a set of images from all wavelengths took about two minutes due to the time required to read and store the data from the CCD. With both systems, typical deployments lasted from one to several hours, enabling multiple acquisitions over a range of solar zenith angles. Reduction of the raw images consisted of applying calibration factors [15; 16] and averaging images in both space and time to reduce environmental noise. After calibration, but before averaging, every image was inspected to find the anti-solar point, to correct the geometry of the image, and to check for obstructions in the field of view such as fish, the power/data cable, the side of the ship, or other instruments. Calibrated images of the Stokes vectors were then averaged in 10-min bins, excluding those that had been flagged as unacceptable in the inspection stage. The symmetry of the images about the principal plane was exploited to further average both halves of each image. In addition, spatial binning of 3 \(\times\) 3-pixel windows was performed to produce final average images at a 1\({}^{\circ}\times\) 1\({}^{\circ}\) resolution. Each pixel in the final, reduced image, therefore, could have been an average of up to 90 raw pixels (5 images \(\times\) 2 image halves \(\times\) 9-pixel window). The mean and coefficient of variation (CV = standard deviation divided by the mean) of the Stokes vector components were computed for each pixel in the reduced image using the up-to-90 raw pixels in the original images. The degree of linear polarization _DOLP_ was then computed for each pixel from the mean \(I\), \(Q\), and \(U\) Stokes vector components (Equation (3)). Because \(V\) was not available from the PolRADS data, _DOLP_ was calculated rather than the degree of polarization (_DOP_); however, in the upwelling lightfield, they are equivalent because the magnitude of \(V\) is negligible [23].