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2 BASIC SCATTERING MECHANISMS
Since positions of scatterers are randomly dis-
tributed, radar measurements for N independent scat-
tering mechanisms can be added incoherently (Van de
Hulst, 1957). That is, the total Mueller matrix F for
N mechanisms is simply the sum of the N individual
Mueller matrices F;,
N
P= F (1)
j=
F can be constructed, therefore, if all F; are known.
Here only four basic scattering mechanisms are con-
sidered.
2.1 Double Bounce Scattering
This mechanism typically models the scattering from
the dihedral-corner-reflector-like structures such as
the trunk-ground structure in forested areas and the
wall-ground structure in urban areas. It has been
shown through detailed studies using Stratton-Chu in-
tegral techniques, that the scattering matrix S for the
double bounce scattering can be written as (Dong and
Richards, 1995a):
Shh Suh 1 0 :
Si ES | E g = 0 1 eis $1 (2)
hv uu Va
where Si = San 1s a function of several parameters
such as the incidence angle, dimensions of trunks and
walls, and the dielectric properties of trunks, walls
and ground surfaces. a and ó are often referred to as
the PI (polarisation index, defined as the ratio of HH
to VV polarisation responses) and PD (polarisation
phase difference, defined as the phase difference be-
tween the HH and the VV backscattered fields) values.
In the case of perfectly conducting material, à — 1
and ó — 180? regardless of incidence angle. Usually,
a is about 4 — 6 and ó about 140 — 160? for the
trunk-ground structure, (Dong and Richards, 1995b).
The Mueller matrix for the double bounce scattering
is (the relationship between the scattering matrix and
the symmetrical Mueller matrix is given by Dubois
and Norikane (1987), and JPL (1995)),
ed itor f 0
el cH 0 0
Rl 3 1. ul E PINE
0 0 yg cos OH Ta sind
0 0 — 7g sind — 730086
2.2 Bragg Scattering
Small perturbation techniques have been success-
fully used to obtain the polarisation dependence from
slightly rough surfaces such as sea surfaces (Rice, 1950,
Valenzuela, 1967, and Elachi, 1987). The Mueller ma-
trix of the first-order Bragg model is,
Bil -i:5:0 0
SE 9B
e 9 0
El 1-00 155 (4)
0 V5 2
0 0 5 ++
where /, the mean PI value, is in general less than
one. Mathematically,
8 = |ann/œvol” (5)
e—l
(cos + Ve = sin?0)”
e(sin?0 -- 1) — sin?0
(ecos0 4- Vc — sin38)
0 and « are the incidence angle and the dielectric con-
stant of the surface, respectively. The mean PD value
is considered as zero (the Bragg scattering undergoes
a single bounce). ;
Œhh =
ayy = (6 — 1)
(7)
2.3 Single Bounce Scattering
This mechanism typically models the direct specular
reflections from facets of the ground and/or building
roofs perpendicular to the incident direction, and large
branches whose axes are perpendicular to the inci-
dent direction, and so on. The Mueller matrix for
this mechanism is.
100 0
0308 9d:
Ps = dua: Din eh (8)
du =;
The co-polarised response from forest crown volume
backscattering can be included in this mechanism. If
the orientations of leaves, twigs and small branches are
assumed to be uniformly distributed, the backscatter-
ing response will be independent of polarisation, giv-
ing the same HH and VV response. The backscatter-
ing from the trihedral-corner-reflector-like, wall-wall-
ground structure can also be classified into this mech-
anism, since the scattering undergoes odd bounces.
2.4 Cross Scattering
The polarimetric response of a point or distributed tar-
get in general can be decomposed into co- and cross-
polarised responses. If we are only interested in the to-
tal cross-polarised component, we can assume theoret-
ically that the total cross-polarised response is caused
by a hypothetical cross scattering mechanism whose
scattering matrix 1s,
Sm E: (9)
197
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996
