ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
  
  
Si) Sl a en): 
Ox Ox Ox 
oWig(s.»)). 2p») 2965»). , 1), (s 
Oy Oy Oy 
where W denotes the wrapping operator such as that 
Wio|]- o-2zk, kel. 
Unlike (Sandwell & Price 1998), we approximate the partial 
derivatives of interferometric phase in azimuth (x) and range (y) 
direction by differences as (Sharov & Gutjahr 2002) 
0g(x. y) g(x * ^x, y)- os. y). 
ENV ny SEE AN) mPa), 
ex 9. (x, y) Ax 
Ogp(x, y) «Ve (x,y)- gx y * y)- oos»). (6) 
Oy Ay 
where the shift values Ax and Ay usually equal 1 pixel. Thus, 
in our practice, the partial derivative of interferometric phase is 
calculated by subtracting an original interferential picture from 
a translated version of the same interferogram. The resultant 
gradient picture called topogram (Fig. 2, a) can be directly 
converted to the glacier slope map without phase unwrapping. 
The terrestrial slope value & is calculated on a pixel-by-pixel 
basis as follows 
Ap (7) 
JAR: + AR + Ap? 
where Ap is the pixel diagonal size on the ground. The height 
increments A/;, and AR, are defined as 
COSE = 
Ah. , = Cs. D : Vo., ? (8) 
where C(x, y) -025z !.A.Bi (x) R(y):siné((y) (9) 
is the conversion factor depending on the imaging geometry. 
Our topograms and slope maps are represented in the form of an 
RGB image. In the topogram, the first two layers represent 
partial phase gradients Vp _ and Ve . The third layer gives 
the full height increment representing the rate of increase of 
h(x, y) per unit distance, which is defined as 
Ah(x, y) = Ah, - Ax + Ah, - Ay- (10) 
The first two layers of the slope map represent the slope values 
in azimuth and range direction defined separately from the next 
equations 
e, =tan(Ah,/Ap,) and e, =tan"(Ah,/Ap,) (1D 
and the third layer shows the absolute slope value &. Therefore, 
third layers in topograms and slope maps can be compared with 
the corresponding results obtained from other interferograms 
more or less independently of their range and azimuth direction. 
Another significant advantage of the GINSAR technique is that 
the topogram can be scaled with any real, not necessarily 
integer factor, allowing for any linear combination between the 
topograms obtained from two different interferograms. Hence, 
we can easily compensate the topographic phase by differencing 
between two scaled and co-registered topograms. Assuming that 
Vo = Vo,» + NO ur and C : V0, = C, : V 9, 2? the 
operation of differencing between the third layers of two 
topograms can be formulated as follows 
F = C, "Mp ori -C, AX Ne (12) 
  
Figure 2. Topogram (a) and fluxogram (b) of Impetuous Glacier 
The resultant picture F (x, y) containing only the differential 
motion phase without topographic phase is called fluxogram 
(Fig. 2, b) following the definition given by I.Newton to 
differential calculus (fluxions, from Latin flüxus — flow, 
continual change). The fluxogram is represented in the form of 
a 4-layer image with the first two layers showing the difference 
between scaled partial motion increments in azimuth and range 
direction. The third layer shows the difference between full 
motion increments and the fourth layer gives the direction of 
differential motion calculated as the ratio between the first two 
layers. Although the fluxogram provides only the relative 
information about the glacier velocity, it can be directly applied 
to the analysis of the longitudinal strain rate o. The latter is 
usually calculated as a difference between two neighbouring 
velocity values divided by the distance between velocity records 
along the longitudinal transect (Forster et al., 1999). 
In order to solve the equation (12), e.g. with respect to Neo i, 
firstly, we assume that the relation between the velocity 
gradients a- VQ,» IV Pron remains constant over the 
whole glacier area. In contrast to the assumption made in 
(Meyer & Hellwich, 2001), such a constraint seems to be more 
reliable in the glacier environment, though it also has to be 
verified empirically. Secondly, we determine some reference 
values for Vg e.g. by analysing the frontal velocity 
ot 2? 
gradients derived from winter topograms on the basis of the 
transferential approach. Finally, the third layer of the fluxogram 
is converted to a new image product using the following relation 
ALT zAV, T, =F-(C, —aC,)". (13) 
where AV, is the velocity gradient in the first interferogram. It 
is interesting to note that the ratio between the conversion 
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