2.6.1.1.5.2 Image Pixel Geolocation
2.6.1.1.5.2.1 Physical Justification
2.6.1.1.5.2.1.1 The Image
Coordinates used for AATSR
The scanning geometry of
AATSR is such that measured
pixels lie on a sequence of
curved instrument scans. In
order to present
undistorted images, and to
ensure collocation of the
forward and nadir views, the
measured instrument pixels
must be mapped onto a
uniform grid. Instrument
pixels are mapped onto a
rectangular xy
grid aligned to the
instrument swath.
This process requires that
the relationship between the
map coordinates of the
pixel and its latitude and
longitude be defined. If the
coordinates of the pixel
with respect to rectangular
axes are x and
y, the relationship
of x and y
to the pixel latitude
and longitude defines the
projection in use. In theory
one could adopt any map
projection one chose. In
practice a projection is
defined for AATSR that is
related to the AATSR swath.
Over a small area we may
regard the surface of the
Earth as a plane, and can
define Cartesian axes
x and y in
the surface so that the
y axis is tangent
to the satellite ground
track. The x
coordinate of a pixel
within the area is then
given by its normal distance
from the y axis
(the ground track), while
the y coordinate
is measured along the ground
track from the origin to the
foot of the normal from the pixel.
Over the wider area of the
AATSR swath we cannot ignore
the curvature of the Earth.
However the same principle
is adopted to define
pixel x and
y coordinates. We
can still define a curve of
constant y
coordinate, that intersects
the ground track at
right angles, through any pixel.
For any pixel P that is
sufficiently close
^{1} to the
satellite ground track there
will be a unique point Q on
the ground track such
that the vertical plane
through Q whose intersection
on the ellipsoid crosses the
ground track at right angles
also passes through the
pixel. By vertical we mean
that the plane includes the
normal to the ellipsoid at
the intersection point Q.
The plane thus defines a
normal section (in the
sense to be defined below)
at Q, and it intersects the
ellipsoid in a curve that
both passes through the
pixel and crosses the
ground track at right
angles. The azimuth of the
curve where it crosses the
ground track is the normal
section azimuth of the pixel
measured at Q.
(^{1} If the point P
is sufficiently far from the
ground track on the
'concave' side,
then Q is not unique.)
This curve is defined as the
curve of constant y
coordinate through the
pixel. The x
coordinate of the pixel is
then the distance of the
pixel from the ground track
measured along this curve,
while the y
coordinate is given by the
distance of the
intersection point Q from
the origin, measured along
the ground track. The origin
of the y
coordinate is defined to be
the ascending node of
the orbit on the equator. In
this way a coordinate system
is defined that is centred
on the satellite ground track.
The resultant projection does
not correspond to one of the
standard projections of
cartography, and it is not
orthomorphic (that is to
say, away from the ground
track the scales in the x
and y directions are not
exactly the same). Moreover
the scale in the y
direction, which varies with
distance from the ground
track, does not do so in a
way that is precisely
symmetrical about the ground
track (because the
ground track is curved, in
the sense that it is not a
geodesic on the surface).
Nevertheless these effects
are quantitatively
small, while the mapping is
conveniently matched to the
instrument swath.
Furthermore the mapping is
unambiguously defined, and
this is sufficient to
ensure proper collocation of
the forward and nadir views.
2.6.1.1.5.2.1.2 Geometrical
Relationships on the Ellipsoid
The system of coordinates
must take account of true
distances on the ellipsoid,
so that the map scale (the
relationship between the
nominal and true distance
between the image samples)
is uniform across the Earth,
instead of showing a
dependence on latitude.
This section gives for
reference a number of
standard geometrical
relationships relating to
the ellipsoid. Most are
given without proof, but
can be found in standard
texts such as Bomford, G.
Geodesy (4th edition),
Clarendon Press, Oxford.
2.6.1.1.5.2.1.2.1 The Reference Ellipsoid
The reference ellipsoid
is the ellipsoid of
revolution formed by
rotating an ellipse
about its semiminor
axis (which will
coincide with the
axis of rotation of the
Earth). The
crosssection of the
surface in any plane
containing the axis of
rotation will be an
ellipse. If we
define axes x and y to
coincide with,
respectively, the
semimajor and
semiminor axes of this
ellipse, we can write
its equation in the
standard form
  eq 2.71 
where a is the
semimajor axis and
b is the
semiminor axis. The
ratio (a  b)/a
is the flattening,
f. It is
related to the
eccentricity e by
  eq 2.72 

Figure 2.6 
Figure2.6
shows a point P
(x',
y') on the
surface. The angle ´ is
the geocentric latitude
of P given by
  eq 2.73 
The line PQ is the normal
to the surface at P. The
angle that
this makes with the
x axis (or the
equatorial plane) is the
geodetic latitude of P;
it is related to the
coordinates by
  eq 2.74 
The radius of curvature
of the surface in the
meridional plane at
point P is given by
  eq 2.75 
The radius of curvature
in the vertical plane
orthogonal to the
meridional plane (in figure2.6 ,
the plane containing
PQ and normal to the
plane of the diagram) is
  eq 2.76 
This is sometimes termed
the radius of curvature
in prime vertical. The
point Q is the common
intersection of the
surface normals at
points on the parallel
of latitude through P,
and the length PQ can be
shown to equal the
normal radius of
curvature N.
Thus the centre of
curvature in prime
vertical lies on the
minor axis of the
ellipse. Note that the
alternative notation
, is often
used in the
literature for the radii
of curvature N,
R respectively,
and we shall use this
notation below.
The equation of the line
PQ is
  eq 2.77 
and its intercept Q at
x = 0 is
  eq 2.78 
Thus the projection of PQ
onto the y axis is
  eq 2.79 
and finally we have an
expression for the
coordinates of P in
terms of N and
,
  eq 2.80 
Define a righthanded set
of cartesian axes
X, Y,
Z with origin
at the centre of the
Earth, with Z
directed northwards
along the axis of
rotation, and with
X in the plane
of the prime meridian.
The XY plane is
therefore the plane
of the equator. If figure2.6
represents the
meridional plane at
longitude , the
X, Y,
Z coordinates
of a point (x,
y) in the plane are
  eq 2.81 
and so the geocentric
Cartesian coordinates
(X, Y,
Z) of the point
P are
2.6.1.1.5.2.1.2.2 Length and
azimuth on the ellipsoid
We also require formulae
for the length and
azimuth of the curves
joining two points 1 and
2 on the ellipsoid.
Firstly it is necessary
to be clear as to what
is meant, since three
different curves are in
question. The geodesic
is the unique curve of
shortest length between
the two points 1 and 2;
however, many of the
standard formulae refer
not to the geodesic but
to one of the two normal
sections between the points.
The geodesic does not lie
in a plane, nor does its
azimuth
coincide with the
bearing of point 2 that
would be measured with a
levelled theodolite at
point 1, although the
deviations will be very
small for short lines.
There are an infinity of
plane sections through
the ellipsoid including
points 1 and 2. In
particular we can
consider two; the
intersection of the
ellipsoid with the plane
containing point 2 and
the normal to the
surface at point 1,
which we may call
S_{1}, and the
intersection with the
plane containing point 1
and the normal to the
surface at point 2,
which we may call
S_{2}. These are
the normal sections, and
their azimuths, measured
at the points at which
the plane section
contains the normal to
the surface, are the
normal section azimuths.
Thus the angle between
the curve S_{1}
and the meridian plane
through point 1 is the
normal section azimuth
of point 2 measured from
point 1, and this angle
is the bearing of point
2 that would be measured
with a levelled
theodolite at point 1.
The geodesic S lies
between these two arcs
S_{1} and
S_{2}, somewhat
as shown in figure2.7 .
The angles between
these curves are very small.

Figure 2.7 
It is possible to give an
exact expression for the
normal section azimuth
if the latitude and
longitude of the two
points are known. Let
the two points be P and
Q, and let their
geodetic latitude and
longitude be (
_{1},
_{1}) and (
_{2},
_{2})
respectively. Consider a
sphere with its centre
at O', the
intersection of the
normal through P with
the axis of rotation.
This sphere is tangent
to the ellipsoid at the
parallel of latitude
through P, but does not
otherwise coincide with
it. Consider the
spherical triangle
NPQ' drawn on this
sphere (Figure2.9 ),
where N is the
intersection of the axis
with the sphere, and
Q' is the
intersection of the line
O'Q with the
sphere. Then the plane
O'PQ', which
also contains point Q,
is the normal section at
P containing Q, and the
normal section azimuth
of Q measured from P is
the angle at P
(NPQ'). In the
triangle, the angle N is
known, since it is given
by the difference in the
longitudes of Q and P.
The sides NP and NQ
are also known, so that
the triangle can be
solved for angle P. NP
is the geodetic
colatitude of point P.
The calculation of
the arc NQ is more complicated.

Figure 2.8 
Suppose O is the centre
of the Earth. We know
that the z coordinate of
O' is
=
e^{2}
N_{1} sin
_{1}
At the same time the z
coordinate of Q is (Equation
eq.
2.82 12)
N_{2} (1 
e^{2}) sin
_{2}
Thus in figure2.8 ,
which shows the meridian
plane through Q, the
length O'Z (where Z
is the projection of Q
on the polar axis) is
N_{2} (1 
e^{2}) sin
_{2} +
e^{2}
N_{1} sin
_{1}
(Note the subscripts 1
and 2 refer to the
points P and Q
respectively.) The
length ZQ is
N_{2} cos
_{2}
and so we have
tan
= cos
_{2} / ((1 
e^{2}) sin
_{2} +
e^{2}(N_{1}/N_{2})
sin
_{1})   eq 2.84 
from which the angle can be determined.
Now consider the
spherical triangle
PNQ' (Figure2.9 ),
in which the angle
A is the
azimuth of the normal
section PQ at P. From
the standard formulae of
spherical trigonometry
we can obtain
explicit expressions for
sin A and cos
A in terms of
known quantities and of
the unknown angle .

Figure 2.9 
From the sine rule we get
sin A =
sin
sin
cosec
  eq 2.85 
Two applications of the
cosine rule give
cos A = [cos
cos
_{1}  sin
_{1}sin cos
] cosec
Elimination of
cot
A = cosec
[cot
cos
_{1}  sin
_{1} cos
]   eq 2.86 
= sin
_{1} cosec
[cot / tan
_{1}  cos
]
where
  eq 2.87 
and
  eq 2.88 
from Equation
eq.
2.84 .
This is
Cunningham's
Azimuth formula (Bomford).
An exact form for the
length of the normal
section is given by
Bomford. Various
approximate forms are
also available, and will
be discussed below.
2.6.1.1.5.2.1.2.3 Arc length on the ellipsoid
To define the geolocation
grid (to determine the
latitude and longitude
of a general grid
point), we first defined
the latitude and
longitudes of a series
of points on the ground
track. Then we
determined the
coordinates of the
other grid points on the
assumption that each
lay on a line through a
ground track point, at a
given azimuth and at a
given distance (the grid
x coordinate) from the point.
This is an example of one
of the general problems
that arises in surveying
and geodesy, to find the
coordinates of a point
given that it is at
a known distance, along
a line at a known
azimuth, from a point
whose coordinates are
known. Suppose that the
two points are
P_{1} and
P_{2}, and that
their geodetic latitude
and longitude are
respectively (
_{1},
_{1}), (
_{2},
_{2}). The
problem is to find (
_{2},
_{2}) given the
distance L
between P_{1}
and P_{2}, and
the azimuth measured
from North at
P_{1},
together with the
latitude and longitude
of P_{1}. A
formula that gives (
_{2},
_{2}) in terms
of the known
quantities is a direct formula.
To determine the
x and
y coordinates
of a point given its
latitude and longitude,
we require the inverse
relationship, that gives
the length and azimuth
of the line between two
points on the ellipsoid,
as a function of the
coordinates
(latitude and longitude)
of the two points. An
inverse formula for the
distance (though not the
azimuth) is also needed
to generate the
table of ground track points.
A variety of different
geodetic formulae exist,
of varying degrees of
complexity and accuracy.
The present discussion
is based on the
formulae of Clarke and
Robbins; Robbins's
formula is essentially
the inverse of that of
Clarke, and we consider
it first.
Suppose that the
coordinates (latitude
and longitude) of two
points P and Q are
known. Given these
quantities, the normal
section azimuth of Q
measured at P can be
computed using
Cunningham's
azimuth formula (above).
In the same spherical
triangle (See figure2.9 )
the angular distance
(arc PQ
measured at the centre
of curvature in
prime vertical at P, or
O' in figure2.8
can also be derived from
equation
eq.
2.85 ,
equation
eq.
2.86
and equation
eq.
2.87 .
The corresponding
length on the auxiliary
sphere centred at
O' is (
_{1}
). For small
separations PQ it is
clear that this
length is very close to
the desired length of
the normal section PQ,
so that the latter can
be derived by the
application of a small
correction to . In effect
this is what
Robbins's formula does.
The radii of curvature in
prime vertical at the
points P, Q are as follows:
_{1} =
a/
  eq 2.89 
_{2} =
a/
  eq 2.90 
where a is the
semimajor axis (the
equatorial radius) of
the Earth; cf. equation
eq.
2.76 .
We can then
calculate the angle as in equation
eq.
2.84 :
  eq 2.91 
  eq 2.92 
= arc
cot
  eq 2.93 
Calculate the geodetic
correction coefficients
  eq 2.94 
  eq 2.95 
where
  eq 2.96 
The line segment between
points P and Q is then
given as a function of
the angle :
  eq 2.97 
plus terms in higher
powers of .
Clarke's direct
formula, which we quote
without proof, is then
given as follows.
Suppose we are given the
latitude and longitude
of point P
(represented by
subscript 1), and the
length L of the
arc PQ and the normal
section azimuth
A of point Q;
Calculate
  eq 2.98 
  eq 2.99 
Calculate the radius in
prime vertical at
latitude
_{1},
_{1} =
a/
  eq 2.100 
Then
  eq 2.101 
Refer again to figure2.9 .
The quantity just
calculated is the
angular distance
PQ' ( in the
diagram). Once this is
known, the triangle can
be solved by standard
techniques for the
longitude difference
and for the
angle .
Finally the latitude of
Q can be determined from
the latter as follows.
  eq 2.102 
  eq 2.103 
  eq 2.104 
  eq 2.105 
Equation
eq.
2.102
to equation
eq.
2.105
are essentially a
variant of
Cunningham's
formula, but adapted to
the context that
_{2}, the radius
of curvature at point Q,
is not known.
How many terms is it
necessary to take in the
series (enclosed in
square brackets) in Robbins's
formula
eq.
2.97 to ensure a
given degree of
accuracy? Table 2.9
below gives an upper
limit on the magnitude
of each term, including
some we have not quoted
explicitly, for three
different line lengths;
25 km (corresponding
to the interval between
alongtrack points used
in computing the tables
of along track distance
), 32 km (ditto for a
larger granule size)
and 275 km,
corresponding to the
maximum acrosstrack
distance computed during
the generation of the
reference grid.

Table 2.9 Magnitude of terms in Robbins's Formula. (Terms in mm.)

Term

L
= 25 km

32 km

275 km

^{2}

0.427 mm 
0.895 mm 
568.0 mm 
^{3}

1.25 e3 
3.37 e3 
18.37 
^{4}

6.57 e7 
2.25 e6 
0.106 
^{5}

3.21 e9 
1.41 e8 
5.69 e3 
Clearly it is sufficient
for our purposes to take
only the first two
terms. Table 2.10
shows the similar
terms in Clarke's
formula
eq.
2.101 .

Table 2.10 Magnitude of terms in Clarke's Formula. ( Terms in mm.)

Term

L
= 25 km

32 km

275 km

(L/
_{1})^{2}

0.427 mm 
0.895 mm 
568.0 mm 
(L/
_{1})^{3}

1.25 e3 
3.37 e3 
18.37 
2.6.1.1.5.2.1.2.4 Solution of
Ellipsoidal triangles
Finally, we require a
method for the solution
of a triangle on the
ellipsoid. There are no
exact formulae for
treating this case, and
so the solution
adopted must be based on
an approximation. The
approximation used here
is based on
Legendre's theorem,
or more strictly on
its extension to the ellipsoid.
Legendre's Theorem
proper is a result
relating to spherical
triangles. It is well
known, and is not
difficult to prove, that
the area of a
spherical triangle is
proportional to the
spherical excess of the
triangle; this is the
amount by which the sum
of its three angles
exceeds radians
(180°). Thus,
suppose ABC is a
triangle drawn on a
sphere of radius
R. The sum of
the angles A, B and C
will exceed by the amount
E = (A + B
+ C) 
Then the area of the
triangle ABC is
ER2, where
E is in
radians. This result is exact.
Let the sides of the
triangle ABC be
a, b,
c and consider
the auxiliary plane
triangle
A'B'C'
that has sides of the
same length as ABC. The
angles of
A'B'C'
will of course add up to
;
A' +
B' +
C' =
Legendre's theorem
states that to a good
approximation for
sufficiently small
triangles the area of
the auxiliary plane
triangle
A'B'C' is
the same as that of the
spherical triangle ABC,
and that each of its
angles can be derived by
reducing the
corresponding angle of
the spherical triangle
by onethird of the
spherical excess. Thus
A' =
A  E/3
and similarly
B' =
B  E/3
C' =
C  E/3.
It follows that any
problem in spherical
trigonometry can be
reduced to an equivalent
problem in plane
trigonometry by reducing
the angles according
to the above equations.
It can further be shown
(Bomford) that
Legendre's theorem
can be extended to
triangles on the
ellipsoid. Thus the
statements above are
true (to an appropriate
approximation) if the
triangle ABC is drawn on
the ellipsoid, and the
sides of the triangle
are defined as the
geodesics joining the
vertices A, B, C. In
this case the area of
the triangle is given by
the expression ER
^{2} as above
provided a suitable
value for the radius is
given. Bomford gives the
mean radius
  eq 2.111 
where the subscripts 1, 2
and 3 refer to the three
vertices A, B and C, and
gives higher order terms
for this case. Thus:
 Any trigonometric
calculation on the
ellipsoid may be
reduced to a plane
calculation if
desired, by
correcting the
angles by
E/3 as above.
 Because the
relationship of the
angles of the
spheroidal triangle
to those of the
auxiliary plane
triangle are the
same as those of
the angles of a
spherical triangle
having the same
sides on a sphere of
radius R,
then to the extent
that the higher
order terms
are negligible, we
may use the formulae
of spherical
trigonometry to
solve the spheroidal
triangle, if we work
with the corrected angles.
The algorithm is based on
these statements.
2.6.1.1.5.2.1.3 Calculation of Image
coordinates of Scan Pixel
Both the calculation of the
image coordinates of a scan
pixel and the image pixel
geolocation (next section)
make use of a
precalculated table of the
positions of a series of
points along the ground
track, spaced by 1 image
granule (32 image rows,
corresponding to
approximately 32 km). Thus
the table consists of the
geodetic latitude and
longitude of a series of
points i. The table
is derived with the aid
of the orbit propagator.
Consider a pixel P. If a
geodesic is drawn through P
to meet the ground track at
a right angle in point X,
the point X will lie between
the tabular points Q[i] and
Q[i+1], as shown
schematically in figure 1.
The triangle PQ[i]Q[i+1] is
completely determined if the
latitudes and longitudes
of its vertices are known,
so the angle at Q[i] and
Q[i+1] can be determined and
the lengths PX and XQ[i] can
be determined.
The x coordinate of the
pixel is then taken to be
the distance PX of the pixel
from the groundtrack. The y
coordinate is determined
from the distance, measured
along the groundtrack,
between the i'th
tabular point and the point
X, which represents the
difference between the y
coordinate of X (and hence
of P) and that of the
i'th tabular point.
Thus the distance Q[i]X is
added to the Y
coordinate of Q[i] to give
the y coordinate of P.

Figure 2.10 
For the purposes of this
calculation, the ground
track between points Q[i]
and Q[i+1] is regarded as
approximated by the normal
section between them.
This is justified by the
short length (32 km) of the
segment; the ground track is
not a geodesic, and cannot
be approximated by a
normal section over long
distances. The definition of
the table of ground track
points is essentially a
means to by which the ground
track is approximated by a
piecewise continuous curve
made up of 32 km normal sections.
2.6.1.1.5.2.1.4 Image Pixel geolocation
The generation of the
latitude and longitude as a
function of x and
y makes use of the
table of ground track points
defined above. Each ground
track point represents the
centre of an image row.
Consider a point Qi,
corresponding to alongtrack
coordinate y. The
azimuth A of the
ground track at this point
can be determined, and
the azimuth of the image row
through Qi is therefore
A + 90°. A
direct geodetic formula can
therefore be used to
determine the latitude and
longitude of any image pixel
at a distance x
along the image row. Details
are given in the
Algorithm Description (following).
2.6.1.1.5.2.2 Algorithm Description
2.6.1.1.5.2.2.1 Summary
2.6.1.1.5.2.2.1.1 Generate
Geolocation Arrays
An initial module
generates lookup tables
for use in the
subsequent geolocation
and image pixel
coordinate
determination modules.
The tables comprise:
 Tables of the
latitude, longitude,
and y coordinate of
a series of
subsatellite
points; these points
are equally spaced
in time, at an
interval
corresponding to 1
product granule (32
scan lines), and
extend for
sufficient time to
cover the whole of
the data to be
processed. They
define the satellite
ground track, for
use in the
calculation of the
pixel x and y
coordinates and
start sufficiently
far south of the
ascending node to
permit the
regridding of scans
that are north of
the equator in the
forward view
although the
subsatellite point
is south of the
equator. This table
is derived with the
aid of the ENVISAT
orbit propagation subroutine.
 Tables of the
latitudes and
longitudes of a
rectangular grid of
points covering the
satellite swath.
These points are
spaced by 25 km
acrosstrack, and at
the granule interval
defined above in the
alongtrack
direction. Thus the
points in the centre
of the column
coincide with those
described at (1)
above, while there
are 23 points in the
acrosstrack
direction. Thus in
the acrosstrack
direction the
points extend 275 km
on either side of
the ground track.
The coordinates of
the image pixels for
land flagging are
derived by linear
interpolation in
this table, and the
table forms the
basis of the grid
pixel latitude and
longitude ADS. The
grid begins at the
start of the
first granule of the
product (y
coordinate 0) and
ends sufficiently
far beyond the end
of data to ensure
that the last
instrument
scan can be fully geolocated.
2.6.1.1.5.2.2.1.2 Calculate Pixel x
and y coordinates
The xy (acrosstrack and
alongtrack) coordinates
of each tie point pixel
are derived from the
instrument pixel
latitude and longitude.
The x and y coordinates
may in principle be
calculated for each scan
pixel, but in practice
they are only directly
calculated for a
series of tie points.
The coordinates of the
remaining points are
then determined by
linear interpolation
between these tie points
in a later module
(Interpolate Pixel Positions).
The calculation makes use
of a precalculated
table giving the
latitudes and longitudes
of a series of points
along the ground track,
equally spaced in time
at an interval
corresponding to 1
granule. This table is
derived with the aid of
the orbit propagator in
an earlier module.
2.6.1.1.5.2.2.1.3 Image Pixel Positions
Linear interpolation is
performed to determine
the latitude and
longitude coordinates of
the grid pixels for use
in the subsequent
stages. These
interpolated
coordinates are used
for internally by the
processor but are not
transmitted to the
output product, and so
this interpolation
will not be described in
detail here.
2.6.1.1.5.2.2.2 Image Pixel Geolocation
2.6.1.1.5.2.2.2.1 Alongtrack
lookup tables
The orbit propagator is
used to generate the
lookup table that
define a series of
points on the ground
track separated by a
fixed time interval
T. The time
step
T is chosen to
correspond to one
product granule; that
is, to 32 image
rows. The image rows are
spaced in time by the
duration of 1 instrument
scan, which is 0.15 s,
so that
T = 4.8 s.
Thus the orbit propagator
is used to determine the
latitude and longitude
of the subsatellite
point at a sequence of times
t(k) =
T0 + (k  K)T, k
{0, 1,
2, 3, ... }   eq 2.112 
where the time origin T0
is defined by the
product limits. (It will
usually correspond to
the ascending node of
the orbit). K is a
constant offset to
ensure that the table
extends far enough south
of the equator to permit
the full geolocation of
forward view scans.
The ground trace velocity
components are also
determined at each
tabular point, from
which the azimuth of the
ground track can be determined.
Given the latitude and
longitude of each ground
track point k,
the interval
ds(k) between
points k,
k + 1 is
calculated by an
application of Robbins
formula (See 2.6.1.1.5.2.1.2.2. )
The alongtrack
coordinate of each
tabular point is
determined from
s(k) =
s(k1) + ds(k), k
{1,
2, 3, ... }   eq 2.114 
y(k) =
s(k)  s(K), k {1, 2,
3, ... }   eq 2.115 
The final step merely
redefines the origin of
the y coordinate to be
the tabular point
corresponding to T0.
(The image scan y
coordinate field in
each product ADS records
is derived from the
quantity y(k).)
2.6.1.1.5.2.2.2.2 Reference Grid of
Image Pixel Coordinates
The latitude and
longitude are calculated
for a series of tie
points in the image
plane. The principle of
the calculation is that
for any point Q on
the ground track, we can
construct the normal
section through Q that
intersects the ground
track at right angles.
All points on the
normal section have the
same y coordinate,
which is the y
coordinate of Q. If G
is any point on the
normal section, the
distance QG is equal
by definition to the
image x coordinate of
G, and if the geodetic
latitude and longitude
of Q are known, the
geodetic latitude
and longitude of G can
be derived from the
known x coordinate and
azimuth using
Clarke's formula.
Each tabular point on the
satellite ground track
correspond to the centre
of an image row. Image
pixel coordinates are
now computed for a
series of 23 tie points
on the image row
(including the centre
point). The tie points
are 25 km apart, and
range from x = 275 km
to x = 275 km. Let
the tie points be
indexed by j, j
{0, 1,
2, ... 22}. Then the
ground track point
itself corresponds to
j = 11.
For each k we then have
grid_lat(k
 K, 11) = track_lat[k]   eq 2.116 
grid_long(k
 K, 11) = track_long[k]   eq 2.117 
The azimuth A of
the image row is
determined from the
ground trace velocities.
The azimuth of the ground
track at Q is '
(measured clockwise
from the meridian; note
the sign convention here
is opposite to that we
have used elsewhere).
the latitudes and
longitudes of 11
points to the left and
right of the
subsatellite point are
calculated, again using
an interval of exactly
25.0 km between the
points. These points
are in the acrosstrack
direction, and lie on
the normal section
locally orthogonal to
the subsatellite track.
The azimuth of this
section is ' 
/2.

Figure 2.11 
In Figure2.11 ,
Q is the
tabular point just
calculated and G_{j}
is the tie point
at distance,
L = 25
(j  11) km
from Q along the normal
section at azimuth
A = ' 
/2. Given
that both A
and L are known
together with the
coordinates of
Q, the
coordinates of tie
point G_{j}
can be
determined using
Clarke's formula.
This procedure can then
be repeated on the other
side of the track. The
alongtrack coordinate
of all 23 points in the
acrosstrack
direction are the same
as that of the
subsatellite point
through which the normal
section is drawn (by definition).
2.6.1.1.5.2.2.3 Image coordinates of
the scan pixel
The x and y coordinates may
in principle be calculated
for any scan pixel, but in
practice the coordinates
are only directly
calculated for a series of
tie points, represented by
every tenth pixel along each scan.
The first step is to identify
the value of i
corresponding to the
interval within which the
arc from P intersects the
ground track at right
angles. The angle between
the ground track and the arc
PX varies continuously as
the point X moves along the
ground track, and
therefore the correct
interval is identified using
the criterion that the angle
of intersection passes
through 90° within the interval.
2.6.1.1.5.2.2.3.1 Calculation of
the sides of the triangle
Once the correct interval
Q_{1}Q_{2}
has been identified, the
coordinates are
calculated by
trigonometry. The
latitude and
longitude of each point
P, Q_{1} and
Q_{2} are known,
and therefore the length
of each side of the
triangle
PQ_{1}Q_{2}
can be determined by
means of Robbins'
formula
eq.
2.97 (above).
This process is applied
in turn to each of the
three sides of the
triangle
PQ_{1}Q_{2}.
The triangle is then
fully determined and
any of its angles can be found.
Thus far our calculation
has been exact, to the
extent that sufficient
terms of the expansion
in equation
eq.
2.97
have been used,
given that the lengths
so determined will be
normal section lengths.
However, we know that the
length of the normal
section differs from
that of the geodesic by
a negligible amount for
sufficiently short
lines, and so we can
regard the lengths
calculated above as
geodesic lengths, and
can regard them as the
lengths of the auxiliary
plane triangle. We
can therefore use
Legendre's theorem
to derive the angles of
the geodesic triangle PQ_{1}Q_{2}.
2.6.1.1.5.2.2.3.2 The angle at Q
The next step is
therefore to compute the
angle at Q_{2}.
The area of the auxiliary
plane triangle is given
exactly by a standard
formula of plane trigonometry:
  eq 2.118 
  eq 2.119 
and so the spherical
excess is E = A/R2
By plane trigonometry the
angle at
Q_{2}' in
the plane triangle is
  eq 2.120 
hence the angle at
Q_{2} is this + E/3.
2.6.1.1.5.2.2.3.3 The x and y coordinates
The formulae of spherical
trigonometry are used to
to calculate the x and y coordinates.
Given the side
Q_{2}P
(q_{1}) and the
angle Q_{2}, the
rightangled triangle
PQ_{2}X ( See figure2.10 )
is fully determined,
and the arcs PX and
Q_{2}X can be
calculated. The arc PX
is determined by an
application of the sine rule.
sin
(PX) = sin
(PQ_{2}) sin Q_{2}
  eq 2.121 
The side Q_{2}X
is determined by an
application of the
cosine rule to the
rightangled triangle.
cos
(PQ_{2}) =
cos (PX) cos (Q_{2}X)   eq 2.122 
so
cos
(Q_{2}X) =
cos (PQ_{2})
/ cos (PX)   eq 2.123 
The arc lengths , are
determined by
multiplying the angular
lengths PX and
Q_{2}X
respectively by the
radius of the earth:
= R (PX)   eq 2.124 
= R (Q_{2}X)   eq 2.125 
The sign of the x
coordinate of P is
determined by inspecting
the sign of the angle
(  ) derived at
the last value of
i above. Then
the x and y coordinates
of P, expressed in km, are
x =
(sign of x)
  eq 2.126 
y =
25i + (25 
)   eq 2.127 
One final correction must
be made. All the
geolocation calculations
have been based on the
satellite coordinates
and other parameters
evaluated at the time of
the scan time tag. This
time corresponds to the
start of the scan; as
the scan proceeds the
satellite moves, and
at the end of the scan
it will have moved
approximately 1 km. The
movement is in the y
coordinate; thus the
true y coordinate of a
pixel measured time
t later than
the start of the scan
will be vt greater
than that calculated,
where v is the ground
trace speed. This
correction is added to
the y coordinate
calculated for each tie
point pixel.
2.6.1.1.5.2.2.4 Accuracies
