what does it mean to store cross products in SAR product data file?

Question

I'm reading the UAVSAR data format
For definition of two types of file we have:

MLC files (.mlc): calibrated multi-looked cross products, floating point format, three files8 bytes per pixel, three files 4 bytes per pixel, little endian.

Ground projected files (.grd): calibrated complex cross products projected to the ground in simple geographic coordinates (latitude, longitude). There is a fixed number of looks for each pixel. Floating point, little endian, 8 or 4 bytes per pixel.

I know that the schematic form of SLC file is like this:

where I and Q are successively the real and imaginary part of the complex signal for each pixel after sampling and analog to digital conversion but if I take cross product as explained here, how can I imagine MLC or GRD format schematically?
I mean which components in the SLC file are multiplied by each other to form the cross product of each pixel?
What does cross product of the pixel mean in a physical sense?

Answer 1

If we study UAVSAR data format completely, it is said that:

• MLC data:

  The MLC is a pure binary file with no header bytes. Three of the files are complex floating point, 8 byte per pixel. These complex products are derived from the average (usually 3 pixels in range, and 12 pixels in azimuth, given precisely by " Number of Range Looks in MLC " and "Number of Azimuth Looks in MLC" in the annotation file) of the product of each SLC pixel and correspond to:

                                        ShhShv*
  
                                        ShhSvv*  
  
                                        ShvSvv*  
  

  Three of the files are real floating point, 4 bytes per pixel. These real powers are derived from the average (usually 3 pixels in range, and 12 pixels in azimuth, given precsely by " Number of Range Looks in MLC " and "Number of Azimuth Looks in MLC" in the annotation file) of the product of each SLC pixel and correspond to:

                                        ShhShh*   
  
                                        ShvShv*   
  
                                        SvvSvv*
  

and

• GRD data:

  The grd files consists of three real floating point, 4 bytes per pixel, and three complex floating point, 8 bytes per pixel files. The number of lines and samples may be found in the annotation file and are given by grd_mag.set_rows and grd_mag.set_cols respectively. The grd files contain projected multi-looked data for crossproducts HHHH, HVHV, VVVV, HHHV, HVHV, and HVVV:

                                       ShhShv*  
  
                                       ShhSvv*  
  
                                       ShvSvv*  
  
                                       ShhShh*   
  
                                       ShvShv*  
  
                                       SvvSvv*
  

From the four SLC files that is stored in each of them and for the monostatic backscattering case we can form the scattering matrix as follows:

for each pixel. If we take the Lexicographic matrix basis set as follows:

We can make the 3-D Lexicographic feature vector:

now we can acquire the covariance matrix for each pixel from the ensemble average of the cross product of 3-D Lexicographic feature vector (hear cross product is the same as tensor product or outer product)

which is a hermitian matrix. So we have 6 files, 3 are complex floating point and the off-diagonal terms of the covariance matrix (without the constant coefficient) is stored in them and the other 3 are floating point and the diagonal terms of the covariance matrix is stored in them.
As said ensemble averaging is done by 3 pixels in range, and 12 pixels in azimuth