International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
images. Additionally, MOLA PEDR provides MOLA ranges 
measured by laser signals. MOLA ground coordinates in Mars 
body fixed system, X, Y and Z, are derived from latitude and 
longitude provided from MOLA PEDR as explained before. 
Exterior orientation parameters are the coefficients of the 
second polynomial sensor model. Among enumerated types of 
measurements in the bundle adjustment, image coordinates of 
tie points, MOLA ground coordinates and MOLA ranges retain 
high weights to strengthen the adjustments. Following will 
illustrate mathematical models and a priori standard deviations 
according to each type of measurements in the bundle 
adjustment. 
4.1 Image coordinates of MOLA and tie points 
Equation | presents the collinearity equations that are applied 
to every image point. Image coordinates x; and y; are related to 
ground coordinates, X, Y; and Z, and exterior orientation 
parameters in the collinearity equations. Since a MOC image is 
a linear pushbroom image, the exterior orientations, X^;, Y^; 
and Z^; and m, ... m5; are different at each scan line and x; 
should be 0 when the exterior orientation is precise. 
F, -KEQ X, Y, Z,) 
"mi (X, A) + (X -L)rm, (Z -Z;) 
max -X) mj -N)mz-Z) 
> 
  
  
F, -F(EQX, Y, Z) (1) 
Ama GG - XD -ms(Y -YX ems -2) 
zy + f - jJ d Y x | 0 
yf, gm 
NH X) mu em 2) 
s " e ; 
where, x; and y; are image coordinates of i" point 
X, Y, and Z; are ground coordinates of ; " point 
LU 1 I e p 
. . — . * . .. uU . A ae 
X.Y. and Z“;; are exterior orientation of 7" point ofj "image 
i ij ij c 
my; ... m5; are components of rotation matrix 
i igh > the 
/; is focal length of j ™ image 
Image coordinates of MOLA are calculated in the above 
cquations, while tie points are measured on the images. They 
are treated differently by assigning different a priori standard 
deviations. Image coordinates of tie points are as accurate as 
one pixel. In the bundle adjustment, the image coordinates 
variance of tie points is taken as the reference variance, oo, and 
the weight value of the image coordinates is one. 4 priori 
standard deviations of image coordinates of MOLA profiles are 
based on the previous analysis about the MOLA registration. 
4.2 MOLA ranges 
MOLA ranges are incorporated into the bundle adjustment as 
measurements. The ranges provided in the MOLA PEDR can 
be formularized by Euclidean distance as a function of camera 
position and the MOLA footprint as shown in Equation 2. Only 
simultaneous MOLA profile provides the range information, 
thus, the range measurement is applied only to the 
corresponding simultaneous MOC image in the bundle 
adjustment. 
824 
BEDS Yr Z5 Xni 2.) 
^ 
f range 
  
= range fs -X,)? «(i-Yy)-«j-Ziy «0 
u 
where, range is from MOLA PEDR file 
. x X ; ; : Ed . S AN 
X^, Y; and Z'; are exterior orientation of /" point of /" image 
T " . . th : 
X;, Y; and Z; are ground coordinates of i '' points 
Abshire et al. [1999] reported MOLA range uncertainty is about 
7 meters for a 400 km average range at nadir pointing. MOLA 
ranges are introduced in the bundle adjustment with 10 meters a 
priori standard deviation. 
4.3 Exterior orientation 
Exterior orientations extracted from SPICE at a certain time 
interval are used to model exterior orientations over a MOC 
image by a second order polynomial. The exterior orientations 
of each scan line can be estimated using the polynomial 
function according to the row of an image, x, as shown in 
Equation 3. 
: 7 C t 2 
X —-4gta KA Az Qi = dy + dj Xx * da X 
(3) 
: 9 C T o 3 2 
YS bg t5 xtb;x" 9j eye x rex x 
; 2 aC o. zt Por A 
Z6 x06 xtCyxX* Kj * fot fix* fax 
where X*;, Y*; and Z'; are the sensor position 
Og, o; and k^; are the sensor pointing 
eee f E the coefficients of the second order polynomial 
x is the line number of the image 
Exterior orientation elements are treated as weighted 
parameters in the bundle adjustment and each variable in 
Equation 3 will form an additional equation like Equation 4. 
F,o = EO -EO®°=0 (4) 
where, EO stands for the exterior orientation parameters and the 
polynomial coefficients. 
Based on our best knowledge on the accuracy of trajectory data, 
a priori standard deviation of exterior orientation is estimated 
as the level of 100 and 200 meters. An image with a small 
emission angle, a nadir image, is assigned 100 meters standard 
deviation while the off-nadir image with a large emission angle 
is assigned larger, 200 meters, standard deviation. 
4.4 Ground coordinates 
MOLA ground coordinates derived from MOLA profiles are 
input to the bundle adjustment with relatively small a priori 
standard deviations because the coordinates are considered as 
quite accurate data source. Therefore, a priori standard 
deviations of MOLA ground coordinates are set at the level of 
10 meter for each X, Y, and Z coordinates. Ground coordinates 
of tie points are treated as free unknowns, namely, those have 
practically infinity as a priori standard deviation. Ground 
points will form the following observation equations 
= d ye s 00 -— 
I ground XYZ - XYZ" =0 (5) 
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