Theory of GNSS Reflectometry - GNSS Remote Sensing: Theory, Methods and Applications - page 195

where R k and R ? are the Fresnel coefficients; v and h are unit polarization vectors
for the incident vertical and incident horizontal waves respectively;

h s the

v s and

same for the reflected waves; and M is

ik s e ik s R 0
4R 0

q j q z j
k i q z

1

M D

C . k i v s / 2 I

(8.34)

. k i h s / 2

Z e i. k s k i / d 2 r 0

ik s e ik s R 0
4R 0

q j q z j
k i q z

1

D (8.35)

. k i h s / 2 C . k i v s / 2

Finally, Ulaby et al. ( 1982 ) also develop the set of dot-products

v s k i

D sin cos s cos. s / C cos sin s (8.36)

v k s D cos sin s cos. s / C sin cos s (8.37)

h s k i

D sin sin. s / (8.38)

h k s

D sin s sin. s / (8.39)

where and s are the incident and scattered angle, while and s are the azimuthal
incidence and scattered angles respectively.
Because GNSS signals are transmitted at circular polarizations, a change of
polarization base is required. The fields in Eqs. 8.30 - 8.33 must be introduced in

1
2 Œ.E vv E hh / C i.E v h C E h v /

E crosspol D (8.40)

1
2 Œ.E vv C E hh / C i.E v h E h v /

E copol D (8.41)

8.6.1.2 KA in Physical Optics Approximation (KPO)

Unlike the Geometrical Optics solution, the Physical Optics Approximation of
Eq. 8.25 accounts for contributions of the scattered field over the entire rough
surface, not only well-oriented facets . However, this analysis is limited to surfaces
with small slopes. As given in Ulaby et al. ( 1982 , Sect. 12-4.5), the scattered fields
are

E 0 Z

ik s e ik s R 0
4R 0

U pq e i. k s k i / r 0 d 2 r 0

E pq D

(8.42)

where U pq , for linear polarizations, are detailed in Ulaby et al. ( 1982 ,
Appendix 12C). In order to transform into circular polarization basis, Eqs. 8.40
and 8.41 need to be applied.