THE M−CHI DECOMPOSITION OF HYBRID DUAL-POLARIMETRIC RADAR DATA

R. Keith Raney, Joshua T.S. Cahill, G. Wesley Patterson, and D. Benjamin J. Bussey

The Johns Hopkins University/APL, 11100 Johns Hopkins Road, Laurel, MD, 20723-6099 USA

Keith Raney (k.raney@ieee.org) (410) 849-3611

CONTEXT

Compact polarization modes currently are being

implemented for Earth-orbiting synthetic aperture

radars (SAR) by four national programs (India,

Japan, Argentina, and Canada), the first of which is

scheduled to be launched in January 2012. The radar

data user community is relatively unfamiliar with

analysis methods for data from this class of radars.

This paper describes an applicable analysis

methodology that has proven to be effective when

applied to data from two radars in orbit about the

Moon. Mini-RF aboard NASA’s Lunar

Reconnaissance Orbiter (2009-) [1], together with its

precursor, Mini-SAR on India’s lunar Chandrayaan-

1 satellite [2] (2008-9), were the first space-based

compact polarimetric SARs. These radars are hybrid

dual-polarimetric, receiving orthogonal linear

polarizations, while transmitting circular

polarization (CL-pol) [3]. The precedent for this

architecture may be found in radars used for

meteorological measurements [4], and radar

astronomy [5]. The Mini-RF and Mini-SAR radars

offer the same suite of polarimetric information from

lunar orbit as Earth-based radar astronomical

observations of the Moon, since both types measure

the 2x2 covariance matrix of the backscattered field.

These data are represented in convenient form

through the classical Stokes parameters.

In the established practice of radar astronomy,

the four Stokes parameters (S1, S2, S3, S4), lead to

child products that are used individually, of which

CPR [6] and the degree of linear polarization [7] are

well known examples. The Stokes parameters also

support matrix decomposition techniques that to date

are relatively unknown in radar astronomy, although

they are well established analysis tools in Earth-

observing SAR data. Techniques such as the

“entropy-alpha” method [8], developed for quad-pol

3x3 data matrices, are not directly applicable to the

simpler CL-pol architecture whose data are only 2x2

matrices. However, the grounding principle of

decomposition—using two or more child parameters

jointly to distinguish between classes of radar

backscatter—applies directly to compact-

polarimetric data in general, and to CL-pol data in

particular.

METHODOLOGY

The degree of polarization, m, has long been

recognized as the single most important parameter

characteristic of a partially-polarized EM field, and

is defined by

m = (S22 + S3

2 + S42)½ / S1 (1)

The close relationship between entropy and degree

of depolarization (1-m) has been verified

experimentally [9]. The degree of depolarization (1-

5093978-1-4673-1159-5/12/$31.00 ©2012 IEEE IGARSS 2012

m) is indicative of randomly-polarized backscatter,

typically arising from radar-quasi-transparent

volumetric materials, such as lunar regolith or forest

canopy. The degree of polarization m is a natural

choice fore the first decomposition variable for

hybrid dual-polarimetric data.

The Poincaré ellipticity parameter � is an

obvious and the most robust choice for the second

decomposition variable. It is one of the three

classical principal components (m, �, �) that are

necessary and sufficient to describe the polarized

portion of a partially-polarized quasi-monochromatic

EM field of average strength S1. Further, the sign of

� is an unambiguous indicator of even versus odd

bounce backscatter, even when the radiated EM field

is not perfectly circularly polarized.

The m−chi decomposition methodology has

proven to be an excellent analysis tool for Mini-RF

hybrid-polarimetric data. In this formulation the key

inputs are m, and the degree of circularity

sin2� = − S4/mS1 (2)

The m−chi decomposition may be expressed

through a color-coded image, where

B = [mS1(1 – sin2�)/2]1/2

R = [mS1(1 + sin2�)/2]1/2 (3)

G = [S1(1 – m)]1/2

In this formulation, Blue indicates single-bounce

(and Bragg) backscattering, Red corresponds to

double-bounce, and Green represents the randomly

polarized constituent.

In the special case of forestry, it has been shown

that the entropy-alpha decomposition derived from

the 3x3 matrix typical of an Earth-observing

quadrature-polarimetric SAR, following application

of the “random volume over ground” model, reduces

to an expression for data from a hybrid dual-

polarimetric radar that is equivalent to the m−chi

decomposition of Eqn 3 [10]. Our results suggest

that this also should be generalizable to other

applications.

An alternative to � could be the relative phase �

between the received linearly polarized components

[3]. Like �, � has the advantage that it is sensitive to

the even versus odd bounce characteristics of the

backscatter. However, � also is dependent on �, the

orientation of the polarization ellipse of the

backscattered EM field. Thus, if there is a significant

linearly polarized component in the transmitted field

(as is the case for the imperfect circularly polarized

field of the Mini-RF radar [11]), then a change in the

angular orientation of any dihedral structure in the

scene could cause the sign of � to reverse polarity.

A LUNAR DOUBLE-BOUNCE EXAMPLE

One situation in which there often is clear

double-bounce geometry at the lunar surface is the

backscatter from an impact crater’s floor and far

wall, which together form a large natural dihedral.

The surrounding imagery arises from features that

are typical of the surface, and these reflections are

mapped at their appropriate distance (range) from

the radar. In contrast, backscatter that corresponds to

forward scatter from the floor of the crater to the far

wall, and then back to the radar, travels an extra

distance. These double-bounce reflections will

appear at greater range in the radar image, hence

appearing as if they come from an area that lies

beyond the far crater rim. Such

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Figure 1. An m-delta decomposition (a) and an m-chi decomposition (b) of floor-wall double

bounce from the crater Kies C. (26°S, 26°W), observed by Mini-RF at S-band zoom [11].

double-bounce signatures will be strongest when the

crater walls are terraced or relatively steep, where in

this context terracing or steepness depends on the

age, materials, and perhaps layering exposed by the

generating impact. Figure 1 shows two

interpretations of such an observation. The double-

bounce return is indicated unambiguously through

the �m-chi decomposition (Fig 1b) by the red “halo”

at the far range side of the crater. The range extent of

the double-bounce halo is proportional to the depth

of the crater floor below the rim.

It is instructive to look at these same data

through an m−delta decomposition, as in Fig 1a. In

this case, the halo is split into two colors, red and

blue. This indicates that the sign of � has been

reversed from one side to the other, which is caused

by the orientation � of the axis of the floor-wall

dihedral feature relative to the linear component of

the radar’s incoming illumination. This effect

suggests that a three-component decomposition (m,

�, �) could offer further insight into the detailed

structure of the crater wall than is available from an

m-chi decomposition, thus taking advantage of the

known ellipticity of the transmitted field.

CONCLUSIONS

The Mini-RF and Mini-SAR instruments are the

first compact polarimetric space-based imaging

radars. Their architecture is hybrid-polarimetric,

transmitting (quasi-) circular polarization, and

receiving orthogonal linear polarizations and their

relative phase. The four Stokes parameters that are

necessary and sufficient to fully characterize the

observed backscattered EM field are calculated from

the received linearly polarized data. The Stokes

parameters can be used to formulate an m-chi

(a)

Radar look direction

m-delta decomposition

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decomposition of the scene, which is a new

technique. This method facilitates unambiguous

interpretation of surface features according to single

(odd) or double (even) bounce signatures in the

polarized portion of the reflections, and

characterization of the randomly polarized

constituents. The m-chi decomposition has proven to

be robust in the event that the transmitted field is not

perfectly circularly polarized. Analysis of lunar data

suggests that an m-chi-psi three-component

decomposition strategy should provide additional

backscatter classification finesse. These methods are

directly applicable to data anticipated from Earth-

observing compact-polarimetric radars.

The authors acknowledge with gratitude the

many essential contributions from the Mini-RF team.

The project was supported through contracts with

NASA.

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