I(psi) = ( I' + Q'cos(2psi) + U'sin(2psi) )/2
where I', Q', and U' represent the first 3 Stokes parameters.
The calibrated reflectance factor data in the Sunflower Polarization data base actually represent a ratio of two intensity values that fall on the detector - the response to the sunflower canopy divided by the response to the Barium Sulfate reflectance standard. Therefore the reflectance factors, RF, in the data base can be modeled as:
( Is' + Qs'cos(2psi) + Us'sin(2psi) )/2
RF = ---------------------------------------
( Ir' + Qr'cos(2psi) + Ur'sin(2psi) )/2
where:
Is', Qs', Us' represent Stokes parameters for the Sunflowers
Ir', Qr', Ur' represent Stokes parameters for the Barium Sulfate panel
If one assumes that the barium sulfate reference panel exhibits very little polarization affects, then Qr' and Ur' are 0. (Previous work indicates that barium sulfate reference panels do exhibit some polarization affects - less than 1% for zenith angles less than 50 degrees and up to 2% for view zenith angles of 80 degrees.) Assuming that Qr' and Ur' are 0, the refectance factor can then be modeled as:
( Is' + Qs'cos(2psi) + Us'sin(2psi) )/2
RF = ---------------------------------------
Ir'/2
or
RF = Is'/Ir' + Qs'cos(2psi)/Ir' + Us'sin(2psi)/Ir'
The measurements for the three polarizer angles allow one to determine the three unknowns - Is'/Ir', Qs'/Ir' and Us'/Ir'. These parameters from now on will be labeled as I, Q, and U. The matrix setup for the three reflectance factor measurements at three polarizing angles (psi1, psi2 & psi3) is:
- - - - - -
|RF1| | 1 cos(2*psi1) sin(2*psi1) | | I |
|RF2| = | 1 cos(2*psi2) sin(2*psi2) | * | Q |
|RF3| | 1 cos(2*psi3) sin(2*psi3) | | U |
- - - - - -
The matrix setup for computing I, Q and U is:
- - - - - -
| I | | 1 cos(2*psi1) sin(2*psi1) | | RF1 |
| Q | = INV | 1 cos(2*psi2) sin(2*psi2) | * | RF2 |
| U | | 1 cos(2*psi3) sin(2*psi3) | | RF3 |
- - - - - -
Also these three parameter allow one to compute three other parameters:
The algorithm for averaging two sets of reflectance factor triplets and then computing I, Q and U is:
- - - -
| I | | (RF1a + RF1b)/2 |
| Q | = INVM * | (RF2a + RF2b)/2 |
| U | | (RF3a + RF3b)/2 |
- - - -
where INVM is the inverse matrix from above.
The algorithm for averaging after computing I, Q and U is:
- - - - - - - -
| I | | | RF1a | | RF1a | | 1
| Q | = | INVM * | RF2a | + INVM * | RF2a | | * -
| U | | | RF3a | | RF3a | | 2
- - - - - - - -
Note the above two equations are the same. Therefore it does not make any difference whether one averages the data before or after computing the I, Q and U parameters for a linear combination type of algorithm which the box car average is.
Another question does arise though concerning how to compute the percent polarization and chi angle of polarization for the simulated 1 degree by 12 degree field of view (FOV).
During the process of averaging the data into 2 degree bins, there were some bins which were empty. I decided to fill these bins with interpolated data from that available in the bins before and after. I interpolated the original reflectance factor triplets. I noticed that in some cases, the resulting calculated values for percent polarization and chi angle of polarization from the interpolated reflectance factor triplets were not the within the range of the respective values for the bins on either side.
This does make sense to me after thinking about it because the percent polarization (PP) and chi angle (X) of polarization are not a linear combinations of I, Q, and U. I plotted a series of measured data and interpolated data (mean of each view zenith pair for a view azimuth direction) which indicates that the computed PP and X parameters based on interpolated values can be quite different from that calculated from two measured values used for the interpolation. The affect is very apparent in the 1 degree field of view data and not so apparent in the 12 degree field of view data. This makes sense because the 1 degree field of view data are almost completely independent. The areas observed by a 12 degee FOV for a sequence of two samples separated by two degree is very nearly the same.
I decided that the proper procedure to simulate a 1 by 12 degree FOV is to average the original reflectance data (or I-Q-U parameters) and then compute PP and X. This process better estimates the polarization properties of the 1 by 12 degree FOV to compare to the Cimel 12 by 12 degree FOV.
Saying it in another way, averaging after computing the PP and X parameters will create a smoothed version of these 1 degree FOV parameters which will not necessarily be representative of larger FOV data. The percent polarization properties of smaller FOV data appear to always be larger than that for larger FOV data.
See figures comparing the two averaging techniques for computing 1x12 degree percent polarization data.
A data file was created that contains the obove six parameters for each observation. This data file, labeled as 'Single Observation Data', is available; see below.
A second data file was also created. A 'scan' of data for an azimuth direction consists of view zenith observations approximately every two degrees from 75 degrees to -75 degrees and back to 75 degrees. These data were averaged into two degree view zenith angle bins. Also for this data file, the Barnes data were averaged to simulate a 12 degree by 1 degree field of view that is sampled at every 2 degrees using a straight box car average to reduce the variation in the data. This data file, labeled as '2 Degree Bin Data', is also available; see below.
NOTE: Before downloading be sure to set your World Wide Web viewer so that these files will be downloaded to disk. (Otherwise the file, which is very large, will be displayed on the screen).
The data in the files are in a tab delimited format. The first row of data contains the titles for the columns. All view azimuth angles refer to the direction as measured from the target to the sensor. Relative azimuth refers to the above view azimuth angle minus the solar azimuth angle.
Changes (1/19/95):
Single Observation Data: Corrections were made in some Sky data view azimuth angle errors.
2 Degree Bin Data: Changes were made in some Sky view azimuth angles, view azimuth angles for all data and the procedure for the box car average. I found that the view azimuth angles in the '2 Degree Bin Data' generated in late November reflected the angle from the sensor to the scene not the scene to the sensor. This caused the relative azimuth angle to be off by 180 degrees.
Changes (1/26/95):
The data for July 11, 12, 15 and 16 were added.
Also some errors were found in the angles used for the polarizers for the Cimel Green, Red and IR bands in the July 25, 27 and 29 data. (The angles used were those for the Cimel radiometers with the 1 degree FOV's not the 12 degree FOV's as they should have been.) Also the reference polarizer angle is now taken into account for the polarizer angle 'triplets'. This affects the three Cimel bands and the MMR Blue and Red bands. The corrections cause changes in the July 25, 27 and 29 data for the Chi Angle of Polarization for the above bands and in the Percent of Polarization for only the Cimel Red band.
Polarizer Angles Used in Computations
The Matlab M-files used to create the single observation data and 2-degree bin data are available.
The PP, X, Rp data represent the calculated Percent Polarization, Chi angle of plane of polarization and the Reflectance factor due to polarization for the green, red and infrared bands of the Cimel and MMR multiband radiometers.
The MMR Band 7 data plots represent the reflectance factor for MMR channel 7. There was one MMR channel 7 in each radiometer that did not have a polarizer. The field of view of this channel was 1 degree. These measurements can be used to compare the measurement from the 3 different instruments which were viewing slightly different scenes.
Some other pieces of information that are useful when analyzing the data are the relative sizes and locations of the field of views as a function of view zenith angle and the view zenith angle as a function of the plant row. This information is probably most useful for the 1-degree FOV data since this data represents only portions of a row width for a broad range of view zenith angles. The 12-degree FOV data represents the average of several row widths even for a view zenith angle of zero degrees.
The evaluation of the single observation reflectance factor data and I-Q-U processed data for the sky and the sunflowers indicate that the variation in the 1-degree data does represent 'real signal'. There is still some concern that some of the variation in the percent polarization and chi angle of polarization values for the 1-degree data may be due to the fact that the 3 radiometer channels (with 3 different polarizer angles) used to measure the polarization are not perfectly alligned. See the section on the evaluation of the single observation data for more information.
The Matlab M-files use to create the above plots are available.