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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 Co-ordinates 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 x-y grid aligned to the instrument swath.

This process requires that the relationship between the map co-ordinates of the pixel and its latitude and longitude be defined. If the co-ordinates 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 co-ordinate of a pixel within the area is then given by its normal distance from the y axis (the ground track), while the y co-ordinate 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 co-ordinates. We can still define a curve of constant y co-ordinate, 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 co-ordinate through the pixel. The x co-ordinate of the pixel is then the distance of the pixel from the ground track measured along this curve, while the y co-ordinate is given by the distance of the intersection point Q from the origin, measured along the ground track. The origin of the y co-ordinate 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 co-ordinates 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 semi-minor axis (which will coincide with the axis of rotation of the Earth). The cross-section 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 semi-major and semi-minor axes of this ellipse, we can write its equation in the standard form

Equation 1 (1K) eq 2.71

where a is the semi-major axis and b is the semi-minor axis. The ratio (a - b)/a is the flattening, f. It is related to the eccentricity e by

Equation 2 (1K) eq 2.72

Figure 1 (3K)
Figure 2.6

Figure2.6 shows a point P (x', y') on the surface. The angle phi (1K)´ is the geocentric latitude of P given by

Equation 3 (1K) eq 2.73

The line PQ is the normal to the surface at P. The angle phi (1K) that this makes with the x axis (or the equatorial plane) is the geodetic latitude of P; it is related to the co-ordinates by

Equation 4 (1K) eq 2.74

The radius of curvature of the surface in the meridional plane at point P is given by

Equation 5 (1K) 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

Equation 6 (1K) 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 nu (1K), rho (1K) 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

Equation 7 (1K) eq 2.77

and its intercept Q at x = 0 is

Equation 8 (1K) eq 2.78

Thus the projection of PQ onto the y axis is

Equation 9 (1K) eq 2.79

and finally we have an expression for the coordinates of P in terms of N and phi (1K)

Equation 10 (part 1) (1K) , Equation 10 (part 2) (1K) eq 2.80

Define a right-handed 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 lambda (1K), the X, Y, Z coordinates of a point (x, y) in the plane are

Equation 11 (1K) eq 2.81

and so the geocentric Cartesian coordinates (X, Y, Z) of the point P are

eq 2.82

Equation 12 (part 1) (1K)

Equation 12 (part 2) (1K)

Equation 12 (part 3) (1K)

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 alpha (1K) 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 S1, and the intersection with the plane containing point 1 and the normal to the surface at point 2, which we may call S2. 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 S1 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 S1 and S2, somewhat as shown in figure2.7 . The angles between these curves are very small.

Figure 2 (2K)
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 (phi (1K) 1, lambda (1K) 1) and (phi (1K) 2, lambda (1K) 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 co-latitude of point P. The calculation of the arc NQ is more complicated.

Figure 3 (1K)
Figure 2.8

Suppose O is the centre of the Earth. We know that the z coordinate of O' is

delta (1K) = -e2 N1 sin phi (1K) 1

At the same time the z coordinate of Q is (Equation eq. 2.82 12)

N2 (1 - e2) sin phi (1K) 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

eq 2.83

N2 (1 - e2) sin phi (1K) 2 + e2 N1 sin phi (1K) 1

(Note the subscripts 1 and 2 refer to the points P and Q respectively.) The length ZQ is

N2 cos phi (1K) 2

and so we have

tan psi (1K) = cos phi (1K) 2 / ((1 - e2) sin phi (1K) 2 + e2(N1/N2) sin phi (1K) 1)eq 2.84

from which the angle psi (1K) 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 chi (1K).

Figure 4 (2K)
Figure 2.9

From the sine rule we get

sin A = sin Delta (1K) lambda (1K) sin psi (1K) cosec chi (1K) eq 2.85

Two applications of the cosine rule give

cos A = [cos psi (1K) cos phi (1K) 1 - sin phi (1K) 1sinpsi (1K) cos Delta (1K) lambda (1K)] cosec chi (1K)

Elimination of chi (1K)

cot A = cosec Delta (1K) lambda (1K) [cot psi (1K) cos phi (1K) 1 - sin phi (1K) 1 cosDelta (1K) lambda (1K)]eq 2.86

= sin phi (1K) 1 cosecDelta (1K) lambda (1K) [cot psi (1K) / tan phi (1K) 1 - cosDelta (1K) lambda (1K)]

where

Equation 17 (1K) eq 2.87

and

Equation 18 (1K) 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 co-ordinates 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 co-ordinate) from the point.

This is an example of one of the general problems that arises in surveying and geodesy, to find the co-ordinates of a point given that it is at a known distance, along a line at a known azimuth, from a point whose co-ordinates are known. Suppose that the two points are P1 and P2, and that their geodetic latitude and longitude are respectively (phi (1K) 1, lambda (1K) 1), (phi (1K) 2, lambda (1K) 2). The problem is to find (phi (1K) 2, lambda (1K) 2) given the distance L between P1 and P2, and the azimuth measured from North at P1, together with the latitude and longitude of P1. A formula that gives (phi (1K) 2, lambda (1K) 2) in terms of the known quantities is a direct formula.

To determine the x and y co-ordinates 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 co-ordinates (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 co-ordinates (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 chi (1K) (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 (nu (1K) 1 chi (1K)). 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 chi (1K). In effect this is what Robbins's formula does.

The radii of curvature in prime vertical at the points P, Q are as follows:

nu (1K) 1 = a/Equation 19 (1K) eq 2.89

nu (1K) 2 = a/Equation 20 (1K) eq 2.90

where a is the semi-major axis (the equatorial radius) of the Earth; cf. equation eq. 2.76 . We can then calculate the angle psi (1K) as in equation eq. 2.84 :

Equation 21 (1K) eq 2.91

Equation 22 (1K) eq 2.92

psi (1K) = arc cotEquation 23 (1K) eq 2.93

Calculate the geodetic correction coefficients

Equation 24 (1K) eq 2.94

Equation 25 (1K) eq 2.95

where

Equation 26 (1K) eq 2.96

The line segment between points P and Q is then given as a function of the angle chi (1K):

eqn_27 (1K) eq 2.97

plus terms in higher powers of chi (1K).

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

eqn_28 (1K) eq 2.98

eqn_29 (1K) eq 2.99

Calculate the radius in prime vertical at latitude phi (1K) 1,

nu (1K) 1 = a/Equation 30 (1K) eq 2.100

Then

Equation 31 (1K) eq 2.101

Refer again to figure2.9 . The quantity just calculated is the angular distance PQ' (chi (1K) in the diagram). Once this is known, the triangle can be solved by standard techniques for the longitude difference Delta (1K) lambda (1K) and for the angle psi (1K). Finally the latitude of Q can be determined from the latter as follows.

Equation 32 (1K) eq 2.102

Equation 33 (1K) eq 2.103

Equation 34 (1K) eq 2.104

Equation 35 (1K) 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 nu (1K) 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 along-track 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 across-track 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
chi (1K) 2 0.427 mm 0.895 mm 568.0 mm
chi (1K) 3 1.25 e-3 3.37 e-3 18.37
chi (1K) 4 6.57 e-7 2.25 e-6 0.106
chi (1K) 5 3.21 e-9 1.41 e-8 5.69 e-3

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/nu (1K) 1)2 0.427 mm 0.895 mm 568.0 mm
(L/nu (1K) 1)3 1.25 e-3 3.37 e-3 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 pi (1K) 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 pi (1K) by the amount

eq 2.106

E = (A + B + C) - pi (1K)

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 pi (1K) ;

eq 2.107

A' + B' + C' = pi (1K)

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 one-third of the spherical excess. Thus

eq 2.108

A' = A - E/3

and similarly

eq 2.109

B' = B - E/3

eq 2.110

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

Equation (1K) 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:

  1. Any trigonometric calculation on the ellipsoid may be reduced to a plane calculation if desired, by correcting the angles by E/3 as above.
  2. 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 co-ordinates of Scan Pixel

Both the calculation of the image co-ordinates of a scan pixel and the image pixel geolocation (next section) make use of a pre-calculated 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 co-ordinate of the pixel is then taken to be the distance PX of the pixel from the ground-track. The y co-ordinate is determined from the distance, measured along the ground-track, between the i'th tabular point and the point X, which represents the difference between the y co-ordinate 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 co-ordinate of Q[i] to give the y coordinate of P.

Figure 5 (2K)
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 along-track co-ordinate 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 look-up tables for use in the subsequent geolocation and image pixel co-ordinate determination modules. The tables comprise:

  1. Tables of the latitude, longitude, and y co-ordinate of a series of sub-satellite 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 co-ordinates 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 sub-satellite point is south of the equator. This table is derived with the aid of the ENVISAT orbit propagation subroutine.
  2. Tables of the latitudes and longitudes of a rectangular grid of points covering the satellite swath. These points are spaced by 25 km across-track, and at the granule interval defined above in the along-track direction. Thus the points in the centre of the column coincide with those described at (1) above, while there are 23 points in the across-track direction. Thus in the across-track direction the points extend 275 km on either side of the ground track. The co-ordinates 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 co-ordinates

The x-y (across-track and along-track) coordinates of each tie point pixel are derived from the instrument pixel latitude and longitude.

The x and y co-ordinates may in principle be calculated for each scan pixel, but in practice they are only directly calculated for a series of tie points. The co-ordinates 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 pre-calculated 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 co-ordinates 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 Along-track look-up tables

The orbit propagator is used to generate the look-up table that define a series of points on the ground track separated by a fixed time interval Delta (1K) T. The time step Delta (1K) 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 Delta (1K) T = 4.8 s.

Thus the orbit propagator is used to determine the latitude and longitude of the sub-satellite point at a sequence of times

t(k) = T0 + (k - K)Delta (1K)T, k isin (1K) {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 along-track co-ordinate of each tabular point is determined from

s(0) = 0.0eq 2.113

s(k) = s(k-1) + ds(k), k isin (1K) {1, 2, 3, ... }eq 2.114

y(k) = s(k) - s(K), k isin (1K) {1, 2, 3, ... }eq 2.115

The final step merely redefines the origin of the y co-ordinate to be the tabular point corresponding to T0. (The image scan y co-ordinate 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 Co-ordinates

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 co-ordinate, which is the y co-ordinate of Q. If G is any point on the normal section, the distance QG is equal by definition to the image x co-ordinate 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 co-ordinate and azimuth using Clarke's formula.

Each tabular point on the satellite ground track correspond to the centre of an image row. Image pixel co-ordinates 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 isin (1K) {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 beta (1K)' (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 sub-satellite point are calculated, again using an interval of exactly 25.0 km between the points. These points are in the across-track direction, and lie on the normal section locally orthogonal to the sub-satellite track. The azimuth of this section is beta (1K)' - pi (1K)/2.

Figure 6 (2K)
Figure 2.11

In Figure2.11 , Q is the tabular point just calculated and Gj is the tie point at distance, L = 25 (j - 11) km from Q along the normal section at azimuth A = beta (1K)' - pi (1K)/2. Given that both A and L are known together with the co-ordinates of Q, the co-ordinates of tie point Gj can be determined using Clarke's formula.

This procedure can then be repeated on the other side of the track. The along-track coordinate of all 23 points in the across-track direction are the same as that of the sub-satellite point through which the normal section is drawn (by definition).

2.6.1.1.5.2.2.3 Image co-ordinates of the scan pixel

The x and y co-ordinates may in principle be calculated for any scan pixel, but in practice the co-ordinates 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 Q1Q2 has been identified, the co-ordinates are calculated by trigonometry. The latitude and longitude of each point P, Q1 and Q2 are known, and therefore the length of each side of the triangle PQ1Q2 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 PQ1Q2. 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 PQ1Q2.

2.6.1.1.5.2.2.3.2 The angle at Q

The next step is therefore to compute the angle at Q2.

The area of the auxiliary plane triangle is given exactly by a standard formula of plane trigonometry:

Equation (1K) eq 2.118

Equation (1K) eq 2.119

and so the spherical excess is E = A/R2

By plane trigonometry the angle at Q2' in the plane triangle is

Equation (1K) eq 2.120

hence the angle at Q2 is this + E/3.

2.6.1.1.5.2.2.3.3 The x and y co-ordinates

The formulae of spherical trigonometry are used to to calculate the x and y co-ordinates.

Given the side Q2P (q1) and the angle Q2, the right-angled triangle PQ2X ( See figure2.10 ) is fully determined, and the arcs PX and Q2X can be calculated. The arc PX is determined by an application of the sine rule.

sin (PX) = sin (PQ2) sin Q2 eq 2.121

The side Q2X is determined by an application of the cosine rule to the right-angled triangle.

cos (PQ2) = cos (PX) cos (Q2X)eq 2.122

so

cos (Q2X) = cos (PQ2) / cos (PX)eq 2.123

The arc lengths xi (1K), eta (1K) are determined by multiplying the angular lengths PX and Q2X respectively by the radius of the earth:

xi (1K) = R (PX)eq 2.124

eta (1K) = R (Q2X)eq 2.125

The sign of the x co-ordinate of P is determined by inspecting the sign of the angle (beta (1K) - epsilon (1K)) derived at the last value of i above. Then the x and y co-ordinates of P, expressed in km, are

x = (sign of x) xi (1K) eq 2.126

y = 25i + (25 - eta (1K))eq 2.127

One final correction must be made. All the geolocation calculations have been based on the satellite co-ordinates 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 co-ordinate; thus the true y co-ordinate of a pixel measured time Delta (1K)t later than the start of the scan will be vDelta (1K)t greater than that calculated, where v is the ground trace speed. This correction is added to the y co-ordinate calculated for each tie point pixel.

2.6.1.1.5.2.2.4 Accuracies