2.6.1.1.5.1 Instrument Pixel Geolocation
2.6.1.1.5.1.1 Physical Justification
The objective of geolocation is
to determine the coordinates on
the Earth's surface
corresponding to the centre of
each scan pixel. The
required coordinates are the
latitude and longitude of the
pixel on the reference
ellipsoid. (The radial
coordinate is then
automatically known from the
definition of the ellipsoid.)
The coordinates of the scan
pixel correspond to the
intersection of the line of
sight (the direction of the
instantaneous optic axis as
it leaves the scan mirror) with
the reference ellipsoid. The
problem, then, is to determine
this point of intersection,
given the satellite position
and attitude and the orientation
of the scan mirror.
2.6.1.1.5.1.1.1 Coordinates of the
scan pixel
To determine the coordinates
of the scan pixel, we work
in an Earthfixed reference
frame. This is a
righthanded Cartesian frame
of reference having its
origin at the centre of the
Earth. The Z axis is
directed along the rotation
axis towards the North pole,
and the X and Y axes lie
in the plane of the equator;
the X axis lies in the plane
of the Greenwich meridian,
and the Y axis completes the
righthanded set.
Suppose that at some instant
of time t, the
coordinates of the
satellite in the earthfixed
reference frame are X_{s}
, Y_{s}
, and Z_{s}
, and that the
instantaneous line of sight
of the ATSR optical
system is defined relative
to the same frame by the
direction cosines
l, m and
n. The line of
sight is then described
by the equations
  eq 2.23 
  eq 2.24 
  eq 2.25 
where the parameter represents the
distance between the
satellite and the point
(X, Y, Z).
The equation of the reference
ellipsoid is given by
  eq 2.26 
Here a is the
semimajor axis of the
ellipsoid (the equatorial
radius of the Earth) and
b is the semiminor
axis (the polar radius
of the Earth).
The point of intersection is
easily found by solving the
simultaneous equations as
follows. Substitution of the
parametric equations of
the line
(5.3.1  3) in (5.3.4)
gives
  eq 2.27 
This is a simple quadratic
equation in the parameter
;
multiplying out gives
  eq 2.28 
where
  eq 2.29 
  eq 2.30 
  eq 2.31 
The equation has two solutions
  eq 2.32 
Provided the argument of the
square root is positive,
which will always be the
case in practice, both
solutions are real and
positive, and the one
that we require is the
smaller of the two, which we
denote by
_{min}; this will be
the one corresponding to the
negative sign. The other
solution then defines the
point of emergence of
the line of sight at the far
side of the earth. (If the
quantity under the square
root is negative, the
solutions of the
equation are complex. This
case would arise if the line
of sight did not intersect
the ellipsoid, and will
never occur in the normal
course of geolocation of
AATSR data with the
satellite in yaw steering mode.)
The pixel coordinates are
then given by
  eq 2.33 
  eq 2.34 
  eq 2.35 
From the Cartesian
coordinates of the pixel we
can derive its longitude:
  eq 2.36 
and its geodetic latitude
  eq 2.37 
where
  eq 2.38 
This procedure solves for the
intersection point exactly.
2.6.1.1.5.1.1.2 Line of sight in the
satellite reference frame
In order to calculate the
pixel coordinates as above,
we must determine the
direction cosines of the
line of sight relative to
the Earthfixed frame of
reference X,
Y, Z. This
calculation must be repeated
for each pixel for which
geolocation is required.
(Note that strictly, the
X, Y,
Z coordinates of
the origin of the line of
sight should coincide
with the centre of the scan
mirror. In practice the
satellite centre of mass is
used. The error, in terms of
displacement on the surface,
is negligible.)
To determine the direction
cosines of the line of
sight, we proceed in two
main stages. First, from
knowledge of the angle
through which the scan
mirror has rotated, we
determine the direction of
the line of sight relative
to a frame of reference
fixed in the satellite; then
we use the attitude steering
law of the satellite and
knowledge of its position in
its orbit to relate the
direction to the Earthfixed
frame of reference. These
two aspects are discussed in
this and the following sections.
2.6.1.1.5.1.1.2.1 AATSR Scan geometry
Imagine a Cartesian frame
of reference fixed in
the AATSR instrument,
and orientated so that
in the nominal flight
attitude the Z axis
points towards nadir,
and the Y axis is
directed parallel to the
satellite velocity
vector, in the direction
of satellite motion. We
shall denote this
frame of reference by
the subscript b. Note
that relative to the
flight direction, the Y_{b}
axis points
backwards. The essential
features of the AATSR
scan geometry are
expressed in terms of
this frame.
In this reference frame,
the rotation axis of the
scan mirror, which
points forward in
flight, lies in the
(Y,
+Z)
quadrant of the
Y, Z
plane of this reference
frame, and is inclined
at an angle to the Z_{b}
axis.
Define a second reference
frame, the scan
reference frame, as the
frame of reference
derived from the first
by a rotation about the
common X axes through
the angle necessary to
bring the Z axis
parallel to the
instrument scan axis. It
will be denoted by the
subscript a.
The viewing direction may
be defined with respect
to this frame as
follows. The viewing
direction rotates in a
positive sense about the Z_{a}
axis, to which it
is inclined at angle
. This
means that the scan on
the surface is traced in
a clockwise direction,
as seen from above. Let
the scan rotation angle
be , defined to
be zero when the scan
direction is in the X_{a}
 Z_{a}
plane. With
respect to the scan
reference frame, the
direction cosines of the
line of sight are then
_{a}
= sin cos
µ
_{a}
= sin sin
_{a}
= cos
The components of any
vector defined in X_{a}
, Y_{a}
, Z_{a}
are related to
those of the same
vector defined with
respect to the
instrument axes X_{b}
, Y_{b}
, Z_{b}
by a linear
transformation M_{ab}
(). The
equations of this
transformation are as follows.
Suppose that x_{a}
, y_{a}
, z_{a}
are the components
of a vector
x
relative to the scan
reference frame, and
that x_{b}
, y_{b}
, z_{b}
are the components
of the same vector
relative to the
instrument reference
frame. M_{ab}
() is a
rotation of  about
the common X_{a}
, X_{b}
axes (figure2.5 ),
and so the
components are related by
  eq 2.40 
Therefore
  eq 2.41 

Figure 2.5 
Thus relative to the
instrument reference
frame, the direction
cosines of the line of
sight are
  eq 2.42 
2.6.1.1.5.1.1.2.2 Instrument misalignment
Imagine a set of
Cartesian axes X_{p}
, Y_{p}
, Z_{p}
fixed with respect
to the satellite. We
assume that the
directions of these axes
are defined with respect
to the structure of the
satellite, but in such a
way that in flight,
when the satellite is
being yawsteered, the
axis Z
_{p}
is nominally
directed towards nadir,
Y
_{p}
is parallel to the
satellite ground trace,
and X
_{p}
completes the
righthanded set. This
defines the platform
reference frame (p).
The reference frame X_{b}
, Y_{b}
, Z_{b}
defined above is
defined with respect to
the instrument. The
nominal orientation of
the AATSR instrument
should be such that,
when ENVISAT is flying
in its nominal attitude
in yaw steering
mode, the Z_{b}
axis points
towards the true nadir
and the Y_{b}
axis. In other
words the instrument
reference frame should
be parallel to the
platform frame after
integration of the
instrument into the
satellite. However in
practice the two frames
may differ by small
misalignment angles. The
misalignments are
defined by the
transformation between
the instrument
reference frame and the
platform frame.
Quite generally, the
relationship between two
different sets of
Cartesian axes can be
expressed in terms of
three consecutive
rotations about
different axes. Define
linear transformations M
_{z}(), M
_{y}(), M
_{x}(), as follows:
  eq 2.43 
  eq 2.44 
  eq 2.45 
M
_{z}(), M
_{y}(), M
_{x}() represent
elementary rotations of
, , about the
z, y
and x axes
respectively. The
transformation between
instrument and platform
frames is represented by
the product of these
elementary
transformations. Thus
the components of the
line of sight vector
expressed with reference
to the platform frame are
  eq 2.46 
This equation can be
regarded as defining the
misalignment angles
x,
y,
z. (We adopt
the convention that the
rotations are to be
applied in that order to
give the total
transformation to the
platform frame. Strictly
speaking the matrices
representing elementary
rotations about
different axes do not
commute, and so we
should specify the order
in which they are to be
applied. In practice the
angles are
sufficiently small that
any errors from this
source are small in
relation to the overall
attitude error budget
and the matrices can be
regarded as commuting to
a sufficient accuracy.)
In the above discussion
we have used frames of
reference in which the Z
axes point downwards.
This is convenient for a
nadir viewing
instrument; however in
order to discuss the
attitude transformations
and to relate our frames
of reference to those
defined in the Mission
Conventions Document we
need to transform to a
frame in which the Z
axis point upwards.
Finally, the matrix M_{ps}
transforms the
vector to the satellite
frame of reference. It
represents a simple
rotation of
180° about the
common Y_{s}
, Y_{p}
axes to bring the
Z axis parallel to the
outward vertical.
Relative to the
satellite frame of
reference the components
of x are
  eq 2.47 
so that
  eq 2.48 
The direction cosines
with respect to the
satellite frame are
therefore obtained by
multiplying the starting
vector by the matrix product
  eq 2.49 
The satellite frame that
we have defined here is
equivalent to the
Satellite Relative
Actual Reference Frame
defined in the mission
conventions document,
except that the frame
here is explicitly
imagined as fixed in the
satellite. Note that we
ignore mispointing throughout.)
2.6.1.1.5.1.1.3 Attitude Transformations
The Local Orbital Reference
Frame is the reference frame
with respect to which the
attitude of the satellite is
described. It is defined
in the ENVISAT
Mission Conventions
Document
Ref. [1.6 ]
(POISESAGS0561); its
origin is the centre of
mass of the satellite and
its basis vectors are the
three unit vectors
L,
R, and
T as follows.
 The unit vector
L is
directed along the
outward radius from the
centre of the earth to
the satellite centre of
mass. It is the
yaw axis.
 The unit vector
R is
perpendicular to
L, in
the plane containing
L and
the instantaneous
inertial velocity vector
of the satellite, and is
directed forwards,
approximately in the
direction of motion of
the satellite. It
represents the roll axis.
 Unit vector
T
completes the
righthanded set, so
that T
= R x
L.
T is in
the crosstrack
direction, and
represents the pitch axis.
This frame is defined with
respect to the orbit, not
the structure of the
satellite. It is an
instantaneous frame; that
is, it is defined at a
particular instant of time.
The attitude of the satellite
is specified relative to the
Local Orbital Reference
Frame by means of three
angles , , . These angles
define the rotations about
the roll, pitch and yaw
axes respectively which, if
applied in sequence to the
TRL frame, would bring its
axes parallel to the
satellite frame. The sign of
each rotation is to be
interpreted so that a
positive angle means that a
positive rotation about the
relevant axis, in the
conventional
righthanded sense, is
required to bring the
initial axes into
coincidence with the derived
set. Rotations about
different axes do not
commute, and so it is
strictly necessary to define
the order in which the
rotations are to be applied.
We adopt the convention that
the rotations are to be
applied in the order roll,
pitch, yaw.
Suppose that (t,
r, l) are
the components of a vector
in relative to the TRL axes,
and that
(t´,
r´,
l´) are the
components of the same
vector in the transformed
system, which we may denote
by T´R´L´.
(The frame
T´R´L´ is
essentially the 'Local
Relative Yaw Steering
Orbital Reference
Frame' defined in
the Mission
Conventions
Document
Ref. [1.6 ]
.) The rotation
matrices in pitch, roll and
yaw are identical to those
defined above for rotations
about X, Y
and Z axes
respectively. Thus the
overall transformation
can be expressed as
  eq 2.50 
From our definition of the
satellite attitude, the
transformed attitude frame
will be coincident with the
satellite fixed frame, apart
from a fixed rotation. The
latter appears because we
have defined the local
orbital reference frame so
that the roll axis
R points
forward, but the satellite
frame is defined so that the
corresponding axis points
backwards. (We have simply
followed the definitions
adopted by ESA for these
reference frames.)
Comparison of the
definitions of the two
frames shows that the
transformed frame
T´R´L´ is
related to the satellite
frame by a rotation through
180 degrees about the z
(L) axis. Specifically,
  eq 2.51 
We can introduce the matrix M
_{SA} to represent
this transformation:
  eq 2.52 
so that
  eq 2.53 
We now have all the
components of the
transformation from the
satellite reference frame to
the TRL frame. Equation
eq.
2.32
defines the
transformation from TRL to
T´R´L´. The
reverse transformation is
easily written
  eq 2.54 
This follows because each of
the matrices M_{y}
, M_{x}
, M_{z}
represents a pure
rotation, and the operation
inverse to any rotation is a
rotation equal in magnitude
but of opposite sign about
the same axis. Therefore
from equation
eq.
2.53
we have
  eq 2.55 
where M_{SA}
is the matrix given by
equation
eq.
2.52 .
The matrix M_{SA}
represents a rotation
about the vertical (L´
) axis. It therefore
reverses the direction of
the orthogonal (R´
and T´) axes, while
leaving the L´ axis
unchanged. Moreover it must
commute with the matrix M_{L}
(), since this
also represents a rotation
about the L´ axis,
and rotations about a common
axis commute. It is easy to
verify this directly.
However, M_{SA}
does not commute with
the other two rotation
matrices. Evidently a
rotation of about the
T´ axis is equivalent
to a rotation of  about the
T´ axis, and similarly
for rotations about R. (It
is perhaps easier to
visualise this if an
active interpretation of the
rotations is adopted, rather
than the passive
interpretation that is
strictly applicable the
present discussion.) Thus
one can verify by direct
multiplication that
M_{AS} M_{X}
() = M_{X}
() M_{AS}
and similarly that
M_{AS} M_{Y}
() = M_{Y}
() M_{AS}
Hence
  eq 2.58 
In equation
eq.
2.58
the sign of is negative,
while the signs of the other
two attitude angles are
positive. This is a
consequence of the
coordinate rotation
represented by M_{SA}
.
Finally, we must relate the
line of sight vector to the
inertial frame. Suppose that
the x, y,
and z
components of
T are t_{x}
, t_{y}
, t_{z}
respectively, and that
the components of
R and
L are (r_{x}
, r_{y}
, r_{z}
) and (l_{x}
, l_{y}
, l_{z}
) respectively. If the
components of the vector in
TRL are t,
r and l
then the vector is
In the inertial frame this
becomes, in component form
  eq 2.60 
The matrix is orthogonal
because the individual
vectors T,
R,
L are
normalized to unit
length. Note that the
explicit form of this matrix
depends on the position and
velocity of the satellite at
the time it is evaluated.
If we are given the
direction cosines of the
line of sight with respect
to TRL, then we can evaluate
the matrix equation to
derive the direction
cosines of the line of sight
with respect to the inertial frame.
Nothing prevents us from
choosing the inertial frame
as that whose axes
instantaneously coincide
with the earthfixed
reference frame at the
time in question, in which
case we can equate the
inertial coordinates that
we have derived to the
geostationary coordinates.
2.6.1.1.5.1.2 Algorithm Description
2.6.1.1.5.1.2.1 Summary
The geolocation algorithm
calculates the latitude and
longitude of each instrument
pixel. In principle this
would be done by applying
the transformations of section 2.6.1.1.5.1.1.1. to
each pixel. In practice, to
reduce the processing
overhead, they are carried
out for a subset of tie
point pixels, and the
coordinates of intermediate
pixels are determined by
linear interpolation in scan
number and scan angle. That
is, the pixel latitude
and longitude are regarded
as functions of scan number
and pixel number, and are
interpolated accordingly.
As discussed above, the
transformation between the
scan frame and the
Earthfixed frame can be
expressed in terms of a
series of consecutive
matrix transformations
applied to the line of sight
vector. However, the
implementation of this
algorithm can take advantage
of the fact that some of
these are catered for by the
ESA software target.
For each tie point pixel
p in each scan
s the direction of
the line of sight is
determined in the scan
reference frame. The
corresponding direction
cosines are determined,
transformed to the satellite
reference frame, and
converted back to define an
azimuth and elevation. The
TARGET subroutine is used to
derive the pixel
coordinates on the ellipsoid.
Given the pixel coordinates
of the tie point pixels,
linear interpolation with
respect to pixel number is
used to define the
coordinates of the
intermediate pixels on the
scan. The process is
repeated for both forward
and nadir view scans.
2.6.1.1.5.1.2.2 Algorithm Definition
The following steps are
carried out for each tie
point pixel on each scan s_{t}
{0,
INT_S,
2*INT_S, ...}. In
the general case these
points are
{0,
INT_P, 2*INT_P,
...,
MAX_NADIR_PIXELS  1}   eq 2.61 
on the nadir scan, and
{0,
INT_P, 2*INT_P,
...,
MAX_FRWRD_PIXELS  1}   eq 2.62 
on the forward scan. The
parameters
MAX_NADIR_PIXELS
and
MAX_FRWRD_PIXELS
are found in the Level 1B
Processor Configuration
File 6.6.40. . The interpolation
intervals INT_P
and INT_S are
defined in the Level 1B
Characterisation Data
File 6.6.15. . The adopted value
of INT_P is 10, and so in
practice the tie points are
{0, 10, 20,
30, ...,
MAX_NADIR_PIXELS
 1}   eq 2.63 
on the nadir scan, and
{0, 10, 20,
30, ...,
MAX_FRWRD_PIXELS
 1}   eq 2.64 
on the forward scan.
For each scan s_{t}
and for each tie point
pixel p_{t}
from the above sets,
the following steps are executed:
1. Determine line of
sight and its direction
cosines in the scan
reference frame.
The absolute pixel number
p is calculated from
p = + FIRST_NADIR_PIXEL_NUMBER
  eq 2.65 
or
p = + FIRST_FORWARD_PIXEL_NUMBER
  eq 2.66 
as appropriate, where the
values
FIRST_NADIR_PIXEL_NUMBER,
FIRST_FORWARD_PIXEL_NUMBER
are found also in the Level 1B
Characterisation Data
File 6.6.15. . The scan angle is
determined from
  eq 2.67 
and this value is substituted
in
Equation (5.3.18)
to determine the
unit vector along the line
of sight.
2. Transform to
satellite frame.
The direction cosines of the
line of sight are
transformed to the platform
frame according to
  eq 2.68 
where the halfangle of the
scan cone and the
misalignment correction
angles x, y, z are taken
from the Level 1B
Characterisation Data
File 6.6.15. . The interface to
the target subroutine
requires us to use a
slightly different set of
satellite axes to that
defined in Section
2.6.1.1.5.1.1. Instead of
Equation (5.3.27)
we therefore define
  eq 2.69 
and
  eq 2.70 
The azimuth and elevation of
the line of sight are
calculated with respect to
this frame by inverting the equations
_{s} = cos
(elevation) cos (azimuth)
µ_{s} = cos
(elevation) sin (azimuth)
_{s} = sin (elevation)
Thus
azimuth = atan2
(µ_{s},
_{s})
elevation = atan2 (
_{s}, sqrt(
_{s}
^{2} + µ_{s}
^{2}))
If azimuth < 0.0 then
azimuth = azimuth + 360.0.
Here atan2
represents the arc tangent
function of two arguments
defined in the conventional
way, atan2(y, x)
being the angle whose
tangent is (y/x) and whose
quadrant is defined by the
signs of x and y.
3. Use the
subroutine target to
determine the
geolocation parameters.
The subroutine target is
entered, with the line of
sight direction determined
by the azimuth and elevation
parameters defined above,
and with the orbital
parameters evaluated for the
time of the scan, to
determine the latitude and
longitude of the tie point
pixels. The required
outputs are the geodetic
pixel coordinates taken
from the results vector.
The longitude is converted to
lie in the range 180 to 180
by the subtraction or
addition of 360 if necessary.
2.6.1.1.5.1.2.3 Interpolation
The steps above have
determined the coordinates
of the tie point pixels.
Linear interpolation is used
to determine the
coordinates of the
intermediate pixels. The
process is repeated for both
forward and nadir views.
Linear interpolation with
respect to scan number
s is used to define
the latitude and longitude
of the intermediate pixels
s {s_{t}}.
For each scan s,
linear interpolation with
respect to pixel number is
used to define the latitude
and longitude of the
intermediate pixels
^{;}
.
In the case of longitude, the
interpolation must take
account of the fact that the
180 degree meridian may
intersect the interpolation interval.
2.6.1.1.5.1.3 Accuracies
(To be added:

Discussion of effect
of attitude
mispointing;
constant offset
indistinguishable
from misalignment.

Interpolation error;
main contribution to
absolute geolocation
error is
interpolation
between tie points
on same scan.)
