ANSI-C program: m-chi.c
NAME
m-chi - Calculate m-chi decomposition using information derived
from the Stokes parameters
SYNOPSIS
m-chi <s0> <m> <s2chi> <S_par>
<c1> <c2 <c3>
| <s0> | Stokes parameter for the total power in the H and V polarizations <|E_h|**2 + |E_v|**2> |
| <m> | degree of polarization: sqrt(s1**2 +
s2**2 + s3**2)/s0
(float) |
| <s2chi> | (input) sin(2*chi) = -s3/(m*s0),
2*chi: latitude of
the Stokes vector on the Poincare sphere: sin(2*chi) > 0 measured field is LCP sin(2*chi) < 0 measured field is RCP |
| <S_par> | (input) MLI image parameter file associated with the Stokes parameter data files |
| <c1> | (output) s0*m*(1 + sin(2*chi))/2 (float) |
| <c2> | (output) s0*(1 - m) depolarized component (float) |
| <c3> | (output) s0*m*(1 - sin(2*chi))/2 (float) |
EXAMPLE
m-chi 20080409_RCP.m 20080409_RCP.s2chi
20080409_RCP.mli.par 20080409_RCP.c1 20080409_RCP.c2
20080409_RCP.c3
calculates the three components of the m-chi decomposition that are proportional to the single-bounce, random, and double-bounce components of the backscatter
DESCRIPTION
The m-chi decomposition was developed for use with the Stokes parameters from the field scattered when the illuminating field is circularly polarized. Circular polarization has the property that it is rotationally invariant, and that the sense of the circularity flips when the number of bounces is odd and remains unchanged for an even number of bounces. Differentiation of single or double-bounce scattering is considered an important parameter that can be derived using polarimetry. Urban regions are often characterized by a large fraction of double-bounce reflections, while open fields or water exhibit single-bounce scattering.
Raney [1] [2] has proposed the use of the m-chi decomposition for use with compact polarization (CP) radars. In a SAR instrument using CP, a single polarization is transmitted and 2 orthogonal channels are received. Selection of circular polarization for the transmitted signal has the advantage of rotational invariance. In this case polarization signatures are robust with respect to Faraday rotation caused by the ionosphere and the rotational orientation of the target. The Indian RISAT-1 satellite as well as the planned Radarsat constellation use circular polarized transmit and horizontal (H) and vertical (V) polarized receive channels. It is also possible to synthesize compact polarization data (circular transmit, H,V linear receive) from quad-polarized (HH, HV, VH, VV) data with the Gamma software, see quad2cp.
The m-chi decomposition uses the Stokes parameters of the scattered field to evaluate the fraction of power that is randomly scattered, and the relative ratio of the single and double-bounce components of the coherent field. The scattered field is modeled as consisting of the sum of a randomly polarized component and a coherently polarized component.
The Stokes parameters (see the documentation on stokes) are derived from local second-order statistics of the SLC images of the two receive channels. The s0 Stokes parameter is the total energy of the scattered field and is the sum of the polarized and unpolarized field components. The Stokes parameters s1, s2, and s3 represent a vector in 3 dimensions on the Poincaré sphere, see http://en.wikipedia.org/wiki/Polarization_(waves) and http://spie.org/x32375.xml. Each point on the sphere characterizes a particular polarization state. The Stokes parameter s3 describes the circularity of the scattered field and is assigned to the z-axis coordinate, while the s1 and s2 parameters represent components of the field that are linear (H,V) or linear (45, 135) degrees. The north pole of the sphere represents a Left Circularly Polarized (LCP) wave, while the south pole represents Right Circularly Polarized (RCP) wave.
Specifically, when circular polarization is used to illuminate the scene, then the the s3 Stokes parameter combined with the degree of polarization m and the total power s0 can be used to differentiate if the backscatter is predominantly single or double-bounce. The parameter -s3/(m*s0) is proportional to sin(2*chi) where chi is the ellipticity angle of the polarization ellipse and 2*chi is the latitude of the Stokes vector on the Poincaré sphere. The degree of polarization m is the ratio of the polarized component of the field to the total scattered power.
m = (s12 + s22 + s32)1/2 / s0
Hence m*s0 is the coherent scattered field component power. The m-chi decomposition considers 3 components:
c1 = s0 * m * (1 + sin(2*chi))/2
c2 = s0 * (1.0 - m) depolarized component
c3 = s0 * m * (1 - sin(2*chi))/2
Since sin(2*chi) is positive for LCP waves, if the illumination is RCP then c1 will be larger than 1 for single-bounce, and less than 1 for double-bounce reflections. Conversely, if the illumination is LCP, c1 will be larger than 1 for double-bounce and less than 1 for single-bounce reflections. The quantities m, and sin(2*chi) are calculated from the Stokes parameters using the program stokes_qm.
For display purposes using an RGB or HSI composite image the usual color assignments are that double-bounce is the red channel, single-bounce is the blue channel, and random scattering to the green channel. Generating the 3-channel composite image can either be done using the program ras3pwr or generating 3 dB gray-scale images using ras_dB from the 3 components c1, c2, and c3 and combining them using the programs ras_to_hsi or ras_to_rgb.
[1] Raney, R. K. et al., "The m-chi decomposition of hybrid dual-polarimetric radar data with application to lunar craters," Journal of Geophysical Research, vol 117, doi: 10.1029/2011JE003986, May, 2012.
[2] Raney, R. K. et al. (2011), "The lunar Mini-RF Radars: Hybrid polarimetric architecture and initial results," Proc. IEEE, 99(5), 808-823, doi: 10.1109/JPROC.2010.2084970.
SEE ALSO
typedef_ISP.h, ras_to_hsi,
ras_to_rgb,
ras_dB,
stokes_qm