LITE Profile Geolocation

• The geodetic latitude and longitude of the LITE footprint on the surface of the Earth.
• The geodetic latitude and longitude of the Space Shuttle nadir point on the surface of the Earth.
• The Space Shuttle altitude above the surface of the Earth.
• The off-nadir angle.

The Algorithm Description

• Step One: Extract Ephemeris Data - Extract STS-64 Space Shuttle Discovery ephemeris data from binary Postflight Attitude and Trajectory History (PATH) files.
  • Extract the following subset of PATH data that corresponds to the LITE measurement locations.
    • GMT
      • day
      • hour
      • minute
      • seconds
    • State vector, orbiter, Greenwich true of date (GTOD) (geographic) Cartesian coordinates.
      • X, km
      • Y, km
      • Z, km
    • Velocity vector, orbiter, Aries true of date Cartesian coordinates. (not used in LITE data processing)
      • XD, km/sec
      • YD, km/sec
      • ZD, km/sec
    • Position, orbiter, geodetic coordinates.
      • geodetic latitude, deg
      • longitude, deg
      • nadir height, km
    • Attitude, orbiter, Euler angles, body axes relative to GTOD axis.
      • yaw, deg
      • pitch, deg
      • roll, km

• Step Two: Define Geoid - Read the geoid undulation model. A geoid model of the Earth is formed by adding undulations to an ellipsoid whose flattening is 1.0/298.257--- and whose equatorial radius is approximately 6378.13655 km. This geoid model serves to approximate mean sea level.
  • File: tidefree.jgm3.osu91a.qrtdeg
  • Size: 5427945 bytes
  • Variable: geoidal undulations (array dimensions - 465 rows, 1441 columns)
  • Format: 10f8.3
  • Area:
    • Latitude: 58.0 to -58.0 (465 including both end points)
    • Longitude: 0.0 to 360.0 (1441 including both end points)
  • Interval: 0.25 X 0.25 degree
  • Unit: meter

• Step Three: Compute the Following LITE Location Parameters.
  • geodetic latitude of LITE footprint
  • longitude of LITE footprint
  • geodetic height of orbiter above footprint
  • geodetic latitude of orbiter (nadir point)
  • longitude of orbiter (nadir point)
  • geodetic height of orbiter (nadir point)

The Computation Process

1. Interpolate the PATH ephemeris data to LITE measurement times and extract the following:
   • GTOD position vector (X, Y, and Z locations of the orbiter),
   • attitude angles (pitch, yaw, and roll).

2. Define Euler angle transformation matrix, M. This matrix transforms vectors in the Space Shuttle Body Axis Coordinate System (BACS) into a system parallel to the GTOD system.

   Let the Space Shuttle attitude angles be defined as Y=yaw, P=pitch, R=roll.

   M =

   cos(P)*cos(Y)	sin(Y)*cos(P)	-sin(P)
   cos(Y)*sin(P)*sin(R)-sin(Y)*cos(R)	sin(Y)*sin(P)*sin(R)+cos(Y)*cos(R)	cos(P)*sin(R)
   cos(Y)*sin(P)*cos(R)+sin(Y)*sin(R)	sin(Y)*sin(P)*cos(R)-cos(Y)*sin(R)	cos(P)*cos(R)

3. Define the LITE pointing vector, LITE_pv, as (0,0,-1). This is a unit vector pointing out the orbiter cargo bay in the -Z direction of the Space Shuttle BACS. LITE_pv is in the direction of lidar pulse.

4. Multiply the transformation matrix, M, and LITE_pv, to form the unit vector, GTOD_pv, which is parallel to LITE_pv in the GTOD system.

   GTOD_pv = M * LITE_pv.

5. Determine where the lidar vector intersects the geoid. This is an iterative process and is performed as follows:

   NOTES:

   • return_radius: the distance from the center of the geoid to a predicted X, Y, Z position.
   • geoid_radius: the radius of the geoid with the same declination as the return_radius.
   • The predicted X, Y, Z position always lies along the vector that is in the same direction as GTOD_pv.

   The iteration is complete when the return_radius is within a user specified tolerance of the geoid_radius.

   1. Begin the iteration at the orbiter, which is located at the GTOD X, Y, and Z positions defined by the PATH ephemeris data, and initialize the range (distance) from the orbiter to the Earth to be zero.

      Initialize the following working parameters:

      range_value = 0.0
      x_pos = GTOD X position
      y_pos = GTOD Y position
      z_pos = GTOD Z position

   2. Define a new pointing vector, new_pv, in the direction of GTOD_pv, with magnitude range_value, and origin at the orbiter:

      new_pv = range_value * GTOD_pv.

   3. Update the GTOD X, Y, and Z position values along GTOD_pv that correspond to new_pv.

      x_new = x_pos + new_pv_x
      y_new = y_pos + new_pv_y
      z_new = z_pos + new_pv_z

   4. Update return_radius, so it now defines the magnitude of the vector which extends from the center of the geoid to the point x_new, y_new, z_new.

      return_radius = sqrt(x_new*x_new + y_new*y_new + z_new*z_new)

   5. Define the declination,

      declination = z_new/return_radius

   6. Calculate the geodetic latitude and longitude on the surface of the Earth for the nadir position of x_new, y_new, z_new.

   7. Determine the geoid undulation for the geodetic latitude and longitude.

   8. Compute the radius of the ellipsoid, ellipsoid_radius, for the declination.

   9. Compute the geoid radius, geoid_radius, as

      geoid_radius = ellipsoid_radius + undulation

   10. If the difference between return_radius and geoid_radius is greater than a specified tolerance, refine range_value and return to step 2.

   11. If the difference between return_radius and geoid_radius is less than a specified tolerance, the iteration is complete and the geodetic latitude and longitude of the LITE footprint are now defined.

6. Calculate the nadir geodetic coordinates (geodetic latitude, longitude and altitude) of the orbiter at the GTOD X, Y, and Z position. This calculation also provides the X, Y, and Z position of the nadir point on the surface of the geoid. The nadir X, Y, and Z positions are defined as x_surface, y_surface, and z_surface, respectively.

7. Calculate the off-nadir angle, the angle between GTOD_pv and the nadir altitude vector. The nadir altitude vector extends from the orbiter to the nadir point on the surface of the geoid with components defined as

   x_vector = x_surface - x_pos,
   y_vector = y_surface - y_pos,
   z_vector = z_surface - z_pos.

   The magnitude of the altitude vector, alt_mag, is defined as

   alt_mag = sqrt(x_vector*x_vector+y_vector*y_vector+z_vector*z_vector),

   and the off-nadir angle is defined as

   off-nadir angle = acosd( (x_vector*GTOD_pv_x+y_vector*GTOD_pv_y+z_vector*GTOD_pv_z)/alt_mag ).