Theory of GNSS Reflectometry - GNSS Remote Sensing: Theory, Methods and Applications - page 200

correlation frequency f c . This approach to the GNSS-R modelling is appropriate to
study any issue for which the complex nature of the field is relevant (for instance,
coherence of the signals). However, it involves the implementation of realistic Earth
surface topography realizations at high resolution grids (x and y < ), for

proper integration of R S d 2 r 0 . It is computationally expensive, and studies based on
this approach need to implement large sets of surface realizations for each surface
conditions for the conclusions to be statistically meaningful.

8.6.4 The Bi-static Radar Equation for GNSS

Modulated Signals

In most of the GNSS-R observational scenarios the time during which the signals
can be coherently integrated is relatively small because the carrier phase of individ-
ual reflection points sum together in unpredictable ways at the receiver, changing
along the dynamic conditions, and thus introducing random phase behavior. As
it will be explained in Sect. 8.7 , this phase behavior is a limiting factor for the
coherent integration time. Moreover, this random-like behavior provokes speckle:
fluctuations in the total received power level due to constructive and destructive
interferences between individual reflections. Speckle (also called fading) is a
major source of noise. For these reasons, the integration strategy tends to chose
a short coherent integration time ( than signal coherence time) and relatively
long non-coherent averaging (to reduce the effects of the speckle noise). The
phase information of the signal is lost during the non-coherent integration process,
resulting in amplitude < j Y j > or power < j Y j

2 > waveforms.
Power waveforms are typically modelled using the radar equation, bi-static for
the GNSS-R case. The main reference for GNSS-R bi-static radar equation was
given in Zavorotny and Voronovich ( 2000 ), publication where it was comprehen-
sively deduced from expressions similar to Eq. 8.48 . That derivation considered
Gaussian surface statistics and assumed KGO scattering. The formulation below
follows ( Cardellach 2002 ):

2 >

< j Y pq .; f c / j

4 i Z

G T . r 0 /G R . r 0 / pq . r 0 /ƒ 2 . . r 0 // j S.f c f D . r 0 // j

2

2

P T
.4/ 2

d 2 r 0

D

R 2 . T ; r 0 /R 2 . r 0 ; R /

S

(8.49)

where

￿ P T , G T , and G R are the transmitted power, transmitter's antenna gain, and

receiver's antenna gain respectively;

￿ is the electromagnetic wavelength;