ibul 2004
4 GRASS
; WG on
s of the
ics and
WG on
s of the
ics and
{/fimeter:
apping of
Altimeter
Planetary
- Manual.
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
[15] http://www.msss.com/mgcwg/mgm (accessed 2003)
[16] http://tes.asu.edu (accessed 2003)
APPENDIX A / ERROR ESTIMATION
Basically, a quantification of distortions that occur to a map
projection is given through Tissot's indicatrix. The semi-major
axes h and k of this infinite ellipse give the scales along
projected meridians and parallels and the areal scale
accordingly. GRASS considers these distortion scales for the
simple cylindrical database projection, which is a representation
of the spherical coordinate system. Therefore, it provides error-
free distance and area measurements. Taking a sphere, such
scales of a cylindrical projection are [12, formulae (4-2), (4-3)]:
1 dy
en (0)
r do
k |: dx Q)
* rcosq dA
where h, k, — spherical distortion scales
r = spherical radius
¢ = latitude
A = longitude
X, y 7 map projection coordinates
Since we use planetocentric latitudes that parameterize an
ellipsoidal surface, the effective scales of such a projection have
to be given for a biaxial ellipsoid. These scales read [12,
formulae (4-22), (4-23)]:
a — equatorial axis of the ellipsoid
e = eccentricity of the ellipsoid
¢ = latitude (any type)
(9, = planetographic latitude
À = longitude
Following that, a quantification of errors within GRASS
measurements can be given by the relation of the ellipsoidal
scales and the corresponding spherical ones that are considered
anyway. With the derivation of planetocentric with respect to
planetographic latitude [12, formula (3-28)]:
«ii eo je e (5)
where ©, = planetocentric latitude
these relative scales become:
h r à , 5 3/2 COS” @,
He Senne‘ sin? o, ) eos os (6)
Boal cos” @,
k r a 1/2 COS,
k'= — sod me Sin 0.) vos de (7)
k. à COS Q,
2 : 3
[^ qe. p cos @
sche? sin? o, ) x (8)
a cos” @,
While h' and k^? give the extreme values of distance
measurement errors in meridian or parallel direction, s' is the
deviation of areas (Fig. 2). These parameters — even that in
/2 5 12 ; .
(-e? sin” @ |^ dy (i-e? sin’ y dy de parallel direction — solely depend on latitudes and not on
g E ;
h, = = — = = — — (3) longitudes.
al -e] do, all -e!J de de,
(I e? sin? y dx For Mars, the scales h’ and k^ are almost equal (but not
k, = ne (4) identical — the difference would scale up with larger
25089, dA eccentricity). Therefore, distance measurement errors within
GRASS do not depend on azimuth in this case.
where h,, k, — ellipsoidal distortion scales
1.5
[p mS — Distance Deviation (h',k') eer ee
10: = I — Area Deviation (s') oT
£
=
g 05
g
ui
0.0 -
-0.5 T + T T T
-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90
Latitude in [Degree]
Figure 2. Overview of the error values for Mars, if ellipsoidal planetocentric latitudes are assumed to be spherical. Body definitions
follow the TAU 2000 conventions as given by table 2.
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