Cryospheric Sensing Using GNSS-R - GNSS Remote Sensing: Theory, Methods and Applications - page 250

The following equations describe the equations of the amplitude U i and the delay
i with which a ray that incises into the snow, propagates down to the i -layer,
rebounds, and propagates back to the snow-air interface, finally reaches the receiver.
Their derivations can be found in Cardellach et al. ( 2012 ).

D R i;iC1 … kDi

1 T k1k T kk1 e 2˛ k d k

U i (11.4)

k D

being

H k
cos. k /

d k D

And the phase

k

X

D i k

X

D i

H k
cos . k / .

i D 0 C 2n k D k / sin . 0 / (11.5)

k D 1 k D 1

being

D k D 2H k tan . k / (11.6)

where (see Fig. 11.1 for some definitions):

￿ R ij and T ij stand for Fresnel cross-polar reflection and co-polar transmission

coefficients in the interface between the i -th and j -th layers;

j Ima f p k gj ;

2

￿ ˛ k is the attenuation coefficient in the k-th layer: ˛ k D

￿ H k and k are the layer's thickness an d inc idence angle ( sk etched in Fig. 11.1 ),

and k follow Snell's refraction law: p k1 sin k1 D p k sin k ;
￿ 0 is the differential delay between the snow-air specular reflection and the direct
radio-link: 0 D 2H 0 cos 0 ;
￿Andn k is the refractive index of the k-th layer: n k

D p k .

The Fresnel coefficients, in circular basis are

1
2 .T k C T ? /

T D T co D (11.7)

1
2 .R k R ? /

R D R cross D (11.8)

with

2 cos k1
cos k1 C

T ? D q k

k1

(11.9)

sin 2 k1