Cryospheric Sensing Using GNSS-R - GNSS Remote Sensing: Theory, Methods and Applications - page 252

The received complex waveform is not simply the complex sum of all these
i -fields, but the modulation of the GNSS signals must also be taken into account.
The i -layer contribution can be obtained as the GPS C/A auto-correlation function
(triangle function) shifted by the delay at which the i -layer reflected signal arrives
(with respect to the direct signal), i (Eq. 11.5 ), and multiplied by the amplitude
corresponding to the i -layer, U i (Eq. 11.4 ). Moreover, the i-layer contribution,
once at the receiver coordinates, has a different phase than the phase of the direct
ray contribution, that is: there is a rotation in complex space between the two
contributions. The phase is given by

D 2 i i (11.13)

The total received waveform will thus be:

N layers
X

j i j
L C=A

Y./ D Y dir . / C /U i e j i

Œ.1:0 (11.14)

i D0

where is the delay-variable along the waveform; Y dir is the GPS C/A auto-
correlation function (triangle function) centered at the nominal zero-delay, and it
represents the contamination by the direct signal being captured by the horizon-
looking antenna. It should not be a big contribution, since this antenna is cross-polar
(LHCP) whereas direct signals are essentially co-polar (RHCP); L C= A is the length

of the GPS C/A triangle function, and j is the complex unity, j D p 1.
Equation 11.14 is thus the MRSR model, where U i and i are given by Eqs. 11.4
and 11.5 respectively.
From the model it is also possible to obtain the theoretical value of the
interference frequency between the ray reflected in the i -th layer and the direct
radio-link:

1 d i

dt

f i ŒH z D (11.15)

In particular, the interferences between the direct signal and the external layer
specular reflection is

2H 0

cos e de
dt

f surf ŒH z D (11.16)

(e for elevation angle). Interferences between the direct radio-link and rays reflected
off layers beneath the surface have frequencies more negative than Eq. 11.16 .

11.1.2 Dry Snow Observable: Lag-Hologram

With radio-holographic techniques it is possible to obtain the maximum spatial
compression of the main ray separately from that of the other rays' trajectories.
This makes it possible to evaluate the intensity of radio waves at each ray trajectory