ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
factors b= C,/C, can always be treated as a constant. 
According to our tests, in the majority of cases, the deviation of 
b from the mean value did not exceed 1 percent over the whole 
overlap area between two different interferograms. 
In order to obtain the absolute glacier velocity, the integration 
of velocity gradients in the output GINSAR product has yet to 
be performed. The absolute glacier velocity can be found by 
performing the line integral of its partial gradients V V_and VY, 
Vs y)e VY (by) VY. (y). 
V(x.y)- VV(x.1)* 37 VV, (x. 7) (14) 
with the subsequent or preliminary linear (line by line) high- 
pass filtering and averaging of results obtained in the azimuth 
and range directions as offered in (Scambos & Fahnestock, 
1998). Another interesting approach to the integration of 
velocity gradients is based on the adaptation of available 
algorithms for one- or two-dimensional phase unwrapping, but 
this method has not been tried yet. 
An alternative local solution that we found useful in our study is 
to solve an over-determined system of linear equations in the 
form of 
| - v 2 Ax with v! v 2 min (15) 
is the vector of 
observations V V. and VE with the dimension of 2(n 1} „V 
in a least-squares manner. In (14), / 
is the vector of residuals, 4 is the design matrix and x is the 
vector of unknowns. The total number of unknowns is equal to 
the number of pixels y = n°. 
In our case, the generally large matrix of normal equations 
N = A" 4 is sparse and banded with a bandwidth of . In order 
to solve such a matrix, we apply the modification of the 
Cholesky algorithm offered in (Schuh 1998). The number of 
operations is then reduced to 0.5. n* - 3? 4 0.5. n? & 5n. 
As we have only relative measurements between two pixel 
values, the equation system has a rank of 4; — 1, which means 
that one more independent measurement, e.g. one reference 
point with known (zero) velocity, is still necessary. 
3.3 Algorithmic Singularities 
It is worth noting that the basic equation (5) becomes invalid if 
the module of the phase difference between two neighbouring 
pixels in azimuth or/and in range direction exceeds z. In the 
interferential picture, such locations called singularities are 
related either with the phase noise or with steep topography and 
/ or highly accelerated motion. 
Apart from the phase noise influence, the module of the partial 
phase gradient will exceed z , if the terrestrial slope exceeds the 
critical value £,, , which can be determined from the following 
er? 
dé AB, -Ap 
equation 
The graph in Figure 3 shows, however, that depending on the 
length of the normal baseline and look angle (theta) the value of 
A - 328 
critical slope is usually not smaller than 50?. Such inclinations 
might be excluded from consideration. 
critical slope [deg] 
o N 
© o 
A 
o 
  
  
  
30 | 
1 50 100 150 200 250 300 | 
baseline length [m] 
Theta = 20 deg Theta = 23 deg Theta = 26 deg 
  
Figure 3. Critical slope versus normal baseline (ERS-1/2-SAR) 
In areas of high deformation along glacier walls and at glacier 
fronts with numerous crevasses and non-uniform motion of ice 
blocks, the motion phase gradient can also exceed z, if the 
velocity gradient between two neighbouring pixels is larger than 
0.25), Le. 
AV, =V 
ew =V,u—V,22/4=1415[cm/day]. (17) 
Other kinds of singularities in the form of narrow “traces” from 
the edges of interferometric fringes can also be seen in 
topograms. Such singularities are compensated by locally 
adding or subtracting a value of 27 to / from the topogram as 
follows 
—], if VQ,, ZN 
Vo, =Vo., +27 -n: n= 0, if—z <Vo,, «m. (18) 
+1, VO, SL 
A flow chart showing the sequence of principal stages of the 
GINSAR technique is given in Figure 4. The GINSAR 
algorithm has been implemented in the new RSG 4.0 
software package distributed by Joanneum Research. 
4. EXPERIMENTAL VERIFICATION AND FIELD 
VALIDATION 
The efficacy and robustness of our new algorithms have been 
verified by several independent experts who processed 
multitemporal (D)INSAR data using the GINSAR technique. 
Up to now, 27 GINSAR models have been processed and all 
results were quite satisfactory. The time required for processing 
one differential model did not exceed 4 hours. Visual analysis 
confirmed the fine detail of GINSAR products and revealed that 
the original ground resolution of INSAR data has been nearly 
preserved. 
The INSAR velocities of glaciers were measured several times 
using different interferograms and the results appeared quite 
consistent. The relative tachometric accuracy was verified by 
comparing the velocities of 7 test glaciers measured by 
conventional DINSAR and our alternative techniques. The 
mean difference between the velocities was estimated at +6.3