137 
X s — X s» X $ (t 1 0 ) 
(6) 
B = B 0 + t(t - t 0 ) 
(4) 
\ a x a i 
L = BX = I rol lift, b 2 
U C 2 
Linearized equation (2) will get the measurement error equation. 
Showed in equation (5). 
V x =A x X-L x 
(5) 
where 
Vx = 
A = 
V .V 
. V 
\ 
L x y 7 Z 
J 
«i» 
a X2 
a il7 
a 2\ 
a 22 
a 217 
_ a 3i 
°22 
^317 
X — [AX S0 ,AY S0 ,AZ S0 ,A(p 0 ,A(O 0 ,AK 0 ,AÀ X) , 
AX° s ,AYs,AZ° s ,A<p 0 ,Aco 0 ,A/c 0 ,AÀ 0 ,AB 0 ,AB 0 ,AO 0 ] t 
where Z, = coordinates matrix of GPS 
B = coordinates matrix of image 
X= conversion coefficient matrix 
For determining the conversion coefficient, there shall be at 
least 3 light points made by the comer reflectors in the SAR 
imagery. 
When more comer reflectors are available, the least square 
estimation algorithm could be used to obtain the conversion 
coefficient. The observation equation and the least square 
estimation are shown in equation (7) and equation (8). 
L = BX + A 
(7) 
There are 17 unknown parameters in equation (5), so 6 GPS 
ground control points needed at least to compute the orbital 
parameters. 
2.2.3 Geometric correction 
Geometric correction of InSAR data could be realized by using 
accurate orbital parameters, but this method cannot meet the 
demand of monitoring the surface subsidence in mining area 
accurately. Generally, it can only used to monitor the surface 
subsidence as a qualitative tool , and cannot achieve 
quantitative degree entirely (Ge, L., et al., 2003). 
Another approach is to deal with InSAR data geometric 
correction with GPS measurements. This work also needs 
comer reflectors or ground permanent scatterers (PSs). Using 
the geodetic coordinates of PSs (which can be obtained by GPS) 
and InSAR imageries’ pixel coordinates, precise InSAR 
geographic coding products could be created. 
The permanent scatterer InSAR (PS-InSAR) technique is an 
advanced InSAR technology (Bürgmann, R., et al., 2006). By 
selecting isolated ground points which exhibit extremely stable 
interferometric phase behavior in a certain period, the new 
approach overcomes most problems such as the resolution, 
temporal and spatial baseline limitations that hindered 
conventional InSAR processing. It has greatly improved the 
accuracy of InSAR to ground deformation monitoring, 
especially in the cities or the places where the rocks are well 
exposed. Sometimes the ground deformation can be measured 
to millimetre level accuracy under certain conditions (Zhang, J., 
et al., 2007). But PS-InSAR technique needs large amount of 
InSAR images, and it can only used to small areas for slow 
surface deformation of mining area generally. 
If the image coordinates of a reflector are assumed to be 
(Irowdcol)j and the corresponding geodetic coordinates is to be 
(G, at>G[ on ), the conversion between them with the same 
projection could be expressed in equation (6). 
where A = observation error vector 
X = (b t PbYb t PL (8) 
where p = £)“' 
Z) A = variance of vector A 
After conversion parameters are obtained, image coordinates 
can be converted one by one. The geometric corrected InSAR 
imagery then could be overlapped with other graphics to make 
detail analysis on the mining-induced surface subsidence. 
2.3 InSAR and GPS data’s interpolation 
Compared with cities, the mining area has a higher speed in 
surface subsidence. The repeated monitoring period (about a 
month) could meet the demand approximately. However, for 
determining the subsidence parameters, the repeat monitoring 
period must be shortened during the active stages of mining 
surface subsidence. In the case, the InSAR data must be 
interpolated with other data. 
As mentioned above, during monitoring mining subsidence, 
InSAR and GPS technology are very strong complementary in 
temporal and spatial domain. Therefore, it is very necessary to 
interpolate the InSAR data with GPS measurements by 
effective means. 
To interpolate InSAR and GPS data (Diao, J., 2005), the 
following steps could be applied. 
Firstly, select the InSAR image that was corrected with GPS 
data as the spatial distribution model, and the GPS data can be 
interpolated in the spatial domain at grid points. Least-square 
adjusted can be used here to get distribution model of 
subsidence deformation from the InSAR and GPS results. Then 
the GPS subsidence original results with the spatial 
interpolation could be obtained.