This article explains how to derive the Field of View
(footprint) for a simple satellite-mounted antenna, for
typical Low-Earth Orbiting (LEO) satellites for two cases:
•
Nadir pointing - that is when the antenna’s beam
centre (called the “boresight”) is pointing towards the
geometric centre of Earth, and for a simple antenna,
results in a nearly circular coverage area projected
onto the Earth.
•
Off-Nadir pointing - that is when the antenna’s
boresight is offset from the Nadir direction either in 2
dimensions or 3 dimensions. We shall only consider
the simple case of 2-dimensions. Off-Nadir pointing
results in an Elliptical coverage area projected onto
the Earth.
Another term often used is “illumination” meaning the
Field of View (FoV) projected onto the Earth’s surface. The
antenna’s beam can be thought of as being similar to a
torch beam. As with a torch pointing directly at the ground,
we observe a circular illumination. When the torch is tilted
at an angle such that its beam intersects with the ground,
we observe elliptical illumination.
The geometric FoV (described in this article) and the actual
FoV may be significantly different due to effects such as
refraction and reflection e.g. from the ionosphere, or even
the Earth’s surface itself. But leaving aside such effects, the
geometric analysis is the same for a wide range of
applications such as radio links (both reception and
transmission), radiometric sensors and of course cameras.
For the purposes of this article we shall consider the
sensor to be a narrow-beam radio antenna.
Let the satellite’s direction of motion be +x (its velocity
vector), Nadir is the -z unit vector, and the +y unit vector
pointing out of the page as shown below. The satellite’s
orbital height is h and the Earth’s radius (approximately
6372 km) is RE. In the figure below, the antenna is Nadir
pointing. The point on the Earth’s surface directly below
the satellite is called the Sub-Satellite Point (SSP).
When the satellite’s sensor (e.g. an antenna) is positioned
with an off-Nadir angle, the antenna is slewed by an angle s
relative to Nadir, in the x-z plane as shown below.
The slewing of the antenna may be achieved by means of a
fixed or movable orientation of the antenna itself relative
to the spacecraft or, (especially in the case of very small
satellites), by moving the entire satellite so that it’s antenna
is orientated in the desired direction.
Antenna Beam Definition
First, we need to define the antenna beam. Antennas used
for command, control, telemetry or as a payload sensor
are usually directional. An antenna’s gain depends upon its
directivity and its efficiency. The polar plot, like that shown
below is used to plot the antenna’s gain with respect to
angle from the “boresight”, usually taken to be 0 degrees,
and often (but by no means always) the point of highest
gain. The gain is expressed in dB relative to an isotropic
antenna (dBi). The gain plot is usually normalised with
respect to the maximum gain. The half-power beam width
(HPBW) is an important measure for an antenna and
fundamental to the calculation of coverage on the Earth.
As shown above, the HPBW is the angle from where the gain,
relative to maximum drops by 3dB, either side of the boresight. In
the above example, the HPBW is 35 degrees. The antenna half-
angle, a is also defined. This is a = HPBW/2 i.e. 17.5 degrees in the
figure above.
Satellite Field of View
Nadir-pointing Field of View Geometry
First we consider the simple case of a narrow-beam
antenna pointing directly at the Earth’s centre at a height h
above the Earth’s surfacae i.e. the antenna’s boresight is
aligned with the satellite’s Nadir axis and angle s=0
degrees. The figure below illustrates the geometry and is
not drawn to scale.
The satellite is orbiting at a height h above the Earth’s
surface and is therefore a total distance of RE+h from the
Earth’s centre, O. For the Nadir-pointing case, a triangle
OGS is formed with angles of antenna half-angle a, the
enclosed angle f and the Geocentric semi-angle q. G is a
point on the ground. We want to calculate the diameter of
the Field of View (FoV), D for a given height h and antenna
half-angle, a. We can write,
therefore,
Since,
,
substituting for f,
then to derive the geocentric semi-angle q, we can write,
Now, (working in radians of course) the diameter of the
FoV, D is simply given by,
For example, if a = 17.5 degrees, h = 550km and we take RE
to be 6372km, q = 0.0273 radians, and D = 348km. This
represents the maximum coverage diameter where the
ground point G is at the edge of the FoV. The angle of
elevation e at G is derived as follows:
Taking the elevation e as the angle between the tangent at
G (i.e. a line at a right-angle to OG) and the satellite,
Substituting for f,
therefore,
As the point G is moved closer towards the sub-satellite
point, SSP at the centre of the FoV, the angle of elevation e
will increase until the point where the satellite is directly
overhead G and e becomes 90 degrees. Note, e produced
above is the minimum usable elevation, e
min
.
Minimum Usable Elevation
Consider a groundstation at G’ that is outside the FoV of
the satellite, as shown below.
The satellite is visible with a line-of-sight path from G’ with
an elevation angle e but the point G’ lies outside the FoV. As
the satellite moves towards G’, e increases. Once e reaches
the minimum usable elevation e
min
, the grounstation falls
within the FoV. The relationship between minimum
elevation and FoV is especially important for space-ground
communications (e.g. Telemetry and Telecommand)
because the grounstation will have a minimum usable
elevation for the site which is determined by factors such
as terrain, atmospheric conditions and local noise and
interference. The contact (or pass) time that the ground
staiton can offer depends upon the minimum usable
elevation of the satellite’s communications antenna and the
minimum useable elevation for the groundstation. The
groundstation and satellite beams must intersect for there
to be communication. The nature of the orbit has a
significant effect on the pass time. If the satellite does not
fly over the groundstation, but just comes over the horizon
at a low angle of elevation then dips below the horizon
again, the pass may not be useable unless the
groundstation antenna has a very low minimum usable
elevation.
It is obvious that wide beam-width antennas are better
suited to communication with a ground-station because its
emin is lower. Where the groundstation offers a low
minimum angle of elevation e.g. 5-10 degrees and the
satellite’s communication antenna’s half-power beam width
is also wide, the pass time can be increased significantly.
For payload sensors making observations of the Earth e.g.
a camera, Synthetic Aperture Radar (SAR), ship detection by
AIS, or other similar applications, the beam width may be
narrow, or even steerable. A Nadir-pointing sensor will
have a FoV centred on the Sub-satellite point (SSP) and may
detect features both in front of the satellite and to the rear
as it passes over the region of interest.
Off-Nadir Pointing Field of View Geometry
If the antenna is offset from the Nadir-pointing vector by
an angle s in the X-Z plane, so that the boresight of the
antenna points either forwards or backwards instead of
vertically towards the Earth’s centre, it is said to be off-
nadir pointing as shown below. The figure shows the side-
view (X-Z plane) and the vertical profile viewed from the
direction of motion (i.e. the satellite is travelling out of the
page towards us) in the Y-Z plane.
The FoV now projected onto the Earth’s surface is an
ellipse rather than a circle as is the case for the Nadir-
pointing beam. Such a FoV may be referred to as a Swath,
a term usually reserved for Earth Observation sensors e.g.
SAR. The analysis we present here is restricted to tilt in the
X-Z plane but many missions may rotate the beam in the
X-Y plane (i.e. to “look” to the side from the satellite’s
perspective) so that the FoV is projected to the side of the
satellite’s ground track.
The ellipse has a long dimension, the Semi-major axis and
a short dimension, the Semi-minor axis. If the antenna is
tilted in the X-Z plane as shown, the geometry of the Semi-
minor axis in the Y-Z plane is the same as that presented
for the Nadir-pointing FoV. We make the assumption that
in three dimensions the antenna beam pattern is a volume
of revolution so that its half-power beam-width in the X-Y
plane is the same as that in the X-Z plane.
The Field of View analysis for the off-Nadir pointing case is
similar to that for the Nadir-pointing case. The elliptical
FoV now has two limbs: the near limb, closest to the sub-
satellite point and the far limb, furthest away from the
sub-satellite point. The antenna boresight is offset from
the Nadir axis by an angle s. Using the trigonometry
presented for the Nadir-pointing analysis, we derive two
geocentric semi-angles, q
N
and q
F
enabling the near and
far distances D
NEAR
and D
FAR
respectively
to be derived. The
Semi-major axis of the ellipse, D
SEMI-MAJOR
is then given by
D
FAR
- D
NEAR
.
By substituting,
for the near limb,
and
for the far limb,
Note that the antenna half-angle a is offset by angle s.
Likewise the elevation of the satellite at the near ground
point G
NEAR
and the far ground point G
FAR
is given by,
and,
Summary
We have presented a simple geometric analysis to derive
the Nadir-pointing Field of View for a beam antenna on a
satellite, and using a similar analysis, the Off-Nadir Field of
View. The off-Nadir analysis considered two limbs - the
near limb, and the far limb. For each limb the Nadir-
pointing analysis was applied to derive an expression for
the Semi-major axis of the elliptical FoV.
In most cases, the Nadir-pointing FoV is circular, whereas
the off-Nadir FoV is an ellipse. This 2-dimensional analysis
considered tilt in the X-Z plane (X is direction of satellite
motion, Z is the Nadir-Zenith axis). However, the analysis
can equally be applied to a “side-looking” FoV in the Y-Z
plane. A 3-dimensional analysis is of course significantly
more complex compare to 2-dimensions, but not
inherently difficult using spherical geometry instead of
plane geometry.
The minimum usable elevation was derived, which is
important in analysing communications coverage and
therefore pass duration for a space-ground
communications link. It was shown that although a
satellite may be visible from the ground, the
communications path may not be open due to the
beamwidth of the communications antennas.