Gamma LAT: Reference Manual

m-alpha

ANSI-C program: m-alpha.c

NAME
m-alpha - Calculate m-alpha polarimetric decomposition using information derived from the Stokes parameters. alpha is a parameter in the H/anisotropy/alpha decomposition 

SYNOPSIS
m-alpha <s0> <m> <alpha> <S_par> <c1> <c2 <c3>

EXAMPLE
m-alpha 20080409_RCP.m 20080409_RCP.alpha 20080409_RCP.mli.par 20080409_RCP.c1 20080409_RCP.c2 20080409_RCP.c3

calculates the three components of the m-alpha decomposition that are proportional to the single-bounce, random, and double-bounce components of the backscatter

DESCRIPTION

The m-alpha 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.

Cloude has extended the use of the H/anisotropy/alpha filter to compact polarization [3] and proposed use of alpha parameter for representation of the single or double-bounce scattering. Based on specific simplifying assumptions about the scattering mechanism the coherency matrix [T] can be constructed from information contained in the Stokes parameters.

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.

The alpha parameter of the H/anisotropy/alpha decomposition is given by Cloude as:

alpha = atan(sqrt(s₁**2 + s₂**2)/s₃))/2

note that s₁ and s₂ are related to the linearly polarized components of the field and s₃ is related to the circularly polarized components of the field

Specifically, when circular polarization is used to illuminate the scene, then 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 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-alpha decomposition considers 3 components:

c₁ = s₀*m*cos (1 + cos(2*alpha))/2

c₂ = s₀ *(1 - m) depolarized component 

c3 = s₀*m*(1 - cos(2*alpha))/2

Since cos(2*alpha) 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 alpha 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, Maym 2012 doi: 10.1029/2011JE003986.

[2] Raney, R. K. et al. (2011), "The lunar Mini-RF Radars: Hybrid polarimetric architecture and initial results," Proc. IEEE, 99(5), 808-823, 2011, doi: 10.1109/JPROC.2010.2084970.

[3] Cloude, S.R., "Compact Decomposition Theory," IEEE Geoscience and Remote Sensing Letters, vol. 9, no. 1, Jan 2012, pp. 28-32, doi: 10.1109/LGRS.2011.2158983.

SEE ALSO
typedef_ISP.h, ras_to_hsi,  ras_to_rgb, ras_dB, stokes_qm