IEEE Geoscience and Remote Sensing Magazine - December 2014 - 20

Al
C

Dop
ppler
Doppler

y-Coordinate

Cl
Dl

Bl
B
x-Coordinate

Delay
(a)

5 m/s
10 m/s
15 m/s
20 m/s
25 m/s

Correlation Power, dB

0
-5
-10
-15
-20
-25
-30
-1

2
3
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5
Time Delay, Chip

(b)
Figure 8. (a) A concept of delay-Doppler mapping; (b) An exam-

ple of the 1-D delay waveforms normalized by its peak values for
various winds.
correlation function [85]. Such a correlation function is usually associated with "multiscale," or "fractal" surfaces with
a wide range of roughness length scales with the smallest
ones comparable to, or shorter than, the electromagnetic
wavelength. The MSS for such surfaces are often infinite.
To regularize this divergence, usually an additional "cutoff
wavenumber" parameter is introduced which truncates the
surface spectrum at high-frequencies beyond which the
contribution to the MSS is neglected [109], [110].
Besides surface roughness, another important parameter enters into analytical scattering models. It is the dielectric permittivity of the lower media such as ocean water,
soil, snowpack, or ice. For example, for the KA-GO the dielectric permittivity defines the value of the local Fresnel
reflection coefficient, which controls how much energy is
reflected back to the upper hemisphere, and how much is
transmitted downward into the lower medium. If the lower
medium is absorbing (has losses due to conduction and is
associated with the imaginary part of the dielectric permittivity), this would limit penetration of the radiation into
the lower medium. The penetration depth is an important
parameter in soil-surface scattering studies.
There are semi-empirical models for dielectric permittivity for sea water, soil, snow, and sea ice which relate
the dielectric permittivity to the temperature and salinity
20

in the case of sea water [111], or to water content and the
chemical and physical composition of the medium [112]-
[114]. In some cases, the internal structure (layering, or
depth profile) within the penetration depth might play an
important role in GNSS signal reflection from soil [41] and
from snowpack [115].
E. ThE DElay-DopplEr Map
From the bistatic radar equation (11), it is seen that the
delay-Doppler map emerges as a convolution of the WAF
with the BRCS function within the antenna footprint described by its gain pattern. In a sense, delay-Doppler mapping creates an image of the scattering coefficient in the
delay-Doppler domain. The WAF is close to unity within
an area formed by the annulus zone and the Doppler zone,
and tends to zero outside this area. Physically, it means
that there are certain contours on the surface for which the
scattered signal has the same traveling path length upon
arrival to the receiver. Similarly, there are certain surface
contours for which the scattered signal acquires the same
Doppler frequency shift. Geometrically, the boundaries of
annulus zones are formed by the intersection of the equaldelay ellipsoid, formed by GNSS signals, with the Earth's
surface. These equi-range lines can be regarded as ellipses if
we locally approximate Earth's surface by a plane. The equiDoppler lines imposed by sinc-function S ^ f h can be found
from the equation
1
v ^ vr h - Vv RX $ nv ^ vr h]A /m,
Df / fc ! 2T = 7Vv TX $ m
i

(15)

where Vv TX and Vv RX are, accordingly, velocities of the transmitter and the receiver with respect to the Earth's surface.
They can be regarded as hyperbolae on the Earth's surface.
At this point, one can notice that the delay-Doppler map
created by bistatic GNSS radar has much in common with
the delay-Doppler map of unfocused synthetic aperture
radar (SAR) [77]. Indeed, in our case, we have the same
Doppler frequency/time delay format and pixels formed
by the intersection of equi-Doppler and equi-range lines.
The differences are that the geometry of these lines is more
complex than in the case of SAR due to the bistatic configuration, and that here we have forward bistatic scattering
instead of backscattering for SAR.
Fig. 8a depicts an idealized case of the satellite receiver
flying at 600-km altitude, in the same plane as the GNSS
transmitter, so the equi-Doppler lines (black lines on the
left panel) are perpendicular to the AB line, the intersection
of the incidence plane with the Earth's surface. The green
ellipses are equi-range lines having their common focus at
point O. The right panel presents the corresponding DDM
which has a characteristic horseshoe shape. Pixels in the
surface coordinate domain formed by intersecting equirange and equi-Doppler lines (on the left) and pixels in the
delay-Doppler domain of the DDM (on the right) are connected to each other. The zero-level intensity of the DDM
(dark blue area) corresponds to non-intersecting equi-range
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