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for limited areas of approximately 40x40 km? of flat
terrain and stable flight conditions already allow to
obtain accuracies of about +1 pixel for ground con-
trol points.
The advantages of 2 dimensional polynomial
equations beside others are
- the didactic value for introducing into digital geo-
metric image processing,
- suitability for quick programming
- satisfying for limited areas of flat terrain
- support for approximate value determination
- support for blunder detection
Some disadvantages of 2 dimensional polynomials
are
- arbitrariness
- limited area of validation
- a blockadjustment based on arbitrary polynomial
equations of higher than first order shows an
extremely bad error propagation.
- restriction for 2 dimensions.
2.2 PHYSICAL PARAMETRIC APPROACH
In order to formulate strict geometric algorithms for
SAR- and SLAR radar imagery, a physical
parametric solution is envisaged, which allows to
calculate
- the global and local behaviour of the sensor posi-
tion and attitudes,
- 3dimensional ground control point coordinates and
- computation of imagecoordinates for 3 dimensional
output raster data (resampling).
Extended collinearity equations as derived by the
authors (see Konecny, Schuhr, 1986) fulfil this geo-
metric requirements for radar images. For the
abscisses values x' of the image coordinates the fol-
lowing constraint is valid:
a up DELTA fo ai is DELTAY - x (z - 2!
j
0 =-kx
)
137 oj
a (x -DELTAX-x' )+a (y.7 DELTAY — y! )+a (2.-—:2'
3134 oj 323 À
For the groundrange ordinate values y'gr follows:
a (x -DELTAX-x' )+a . (y - DELTAY - y' 3a (z - 2°
13 i oj 22) =i oj
oj 33] oj
23j oj
J
LL po -DELTAX-x' )+a (y - DEUTAY = y' )ia (2 - z'
31} i oj 32j i
Notice, the z value carries no index, because it repre-
sents the (unknown) (constant) terrainheight for the
groundrange image calculation, while DELTAX and
DELTAY depend on the individual terrainheight zi.
To achieve the measured y' value in the ground range
image, the near range distance r'o has to be
subtracted from the computed groundrange ordinate
value y'gr
Y-y gro
oj 333 oj
708
While for slant range ordinates follows
With sufficient approximation is valid:
DELTAX =
2 2 2 2 2 2
(xx) Hy!) ICS 721 )+(y_-y' )+(z -2' )-h
+ i. oj i.ej J|: i o) i oj i oj
(x ex")
-------------------- i oj
| xt) ay yt)
je Por y y oj
= F X (x. -x' )
1-0
DELTAY = F x (Y -y* )
io}
As usual
X1,y1,Z1 =3dimensional object point coordi-
nates
XO0j,yoj,Zo) -instantaneous sensor position
z =terrain height chosen for groun-
drange calculation
h =70'] - Z
kx, ky = equivalent focal length
al lj until a33j = instantaneous rotation coefficients,
which ,as sinusoidal functions, depend on roll, pitch
and yaw values as a function of x' ("= time"):
roll = omega = Qj = Qe + Qi - x'i +» Q2 - x'i?
pitch = phi = &j = $0 + $1.7 x'1 t $2 - x'i?
yaw = kappa= kj = ke + ki - x'i + k2 - x'i?
xo'j = x'900 t x'oi - x'i + »'g2 - x'j*
yo‘; = y'00 + y'oi - x'i + v'02 - x'i?
Zo) = 2'Q0 + 2'01 - x'i * z'o2 - x'i*?
This expressions are valid for the general formulation
of SLAR- and SAR- image geometry.
While deviations in the sensor position directly affect
the SAR image geometry, general attitude values
only, but not changes of attitudes do influence the
SAR image geometry, which is in opposite to the di-
rect influence of the unstable sensor behaviour in the
SLAR image geometry.
Linearized collinearity equations allow to derive
polynomial equations of equivalent content, which
are valid under particular flight behaviour assump-
tions. The following observation equations have been
derived from linearized equivalent radar collinearity
equations ‚for second order variations of the
orientation elements after the elimination of high
correlated terms and after transition to ground coor-
dinates. For a block consisting of overlapping radar
strips, the complete observation equations, which in-
clude the calculation of 3dimensional point
coordinates are
vx'iz AO + Al - yi + AZ - xi t AS - xi?
+ A4 - dxi - x'i measured
vy'l= Bo + Bl - vi + BZ2 - xi + B3 - xi?
TO B4-xi-yi * BO^yi* € B6-xi*-yi + B/-xi*<3/yi
*t BB-xi**4/yi
4 B9 2i/yi t B10-xi-z2i/yi *B11-xi?-zi/yi
* Bi2-dyi + B13-dzi - y'i measured
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