**9. Pseudo InSAR geophysical measurements: numerical results**

Consider a GeoTIFF file of Dilijan region in Caucasus, Armenia, located at the geographical coordinates 40° 44<sup>0</sup> <sup>27</sup>″ north and 44° 51<sup>0</sup> <sup>47</sup>″ east longitude. Consider 2 pass InSAR scenario. Coordinates of SAR positions in the moment of imaging are the following: master SAR position A, *xA* <sup>¼</sup> 0 m, *yA* <sup>¼</sup> <sup>300</sup>*:*<sup>3</sup> � 103 m, *zA* <sup>¼</sup> <sup>3</sup> � 105 <sup>m</sup> and slave SAR position B *xB* <sup>¼</sup> 0 m, *yB* <sup>¼</sup> <sup>300</sup> � 103 m, *zB* <sup>¼</sup> <sup>3</sup> � 105 m. Wavelength is 0.05 m. Distances at the moment of imaging from the SAR position A and SAR position B to each pixel on the surface are illustrated in **Figure 10a, b**. Interferogram wrapped phases and unwrapped phases are presented in **Figure 10c, d**, respectively.

Consider a three-pass InSAR scenario and a surface before (**Figure 11a**) and after (**Figure 11b**) displacement described by MATLAB function *peaks*. Coordinates of SAR positions in the moment of imaging are the following: master SAR position A, *xA* ¼ 350 km, *yA* ¼ 350 km, *zA* ¼ 800 km; slave SAR position B *xB* ¼ 351*:*5 km,

*Distances from SAR position A (a) and SAR position B (b) to each pixel on the surface in pseudo color map, interferogram wrapped phases (c) and unwrapped phases (d).*

*yB* ¼ 350 km, *zB* ¼ 800 km; and slave SAR position C *xC* ¼ 350 km, *yC* ¼ 351*:*2 km,

*Distances to the surface measured from SAR positions A (a), B (b), and C (c). AB interferogram (d), AC*

*interferogram (e) with surface displacement, and differential interferogram AB-AC (f).*

Distances to the surface at the moment of imaging as pseudo collar maps measured from SAR positions A, B, and C are presented in **Figure 12a–c**, respectively. AB interferogram without surface displacement and AC interferogram with surface displacement are presented in **Figure 12d, e**, respectively. Differential interfero-

The differential interferogram obtained by pixel subtraction of interferograms in **Figure 12d, e** is presented in **Figure 12f**. It illustrates the displacement of the surface. Only deformed part of the surface as differential fringes is depicted. The pseudo InSAR modeling can be applied to generate interferograms and differential interferograms based on real geophysical measurements and Geo TIFF maps of the observed surface.

A multi-pass InSAR system has been theoretically analyzed and numerically experimented. Geometry and kinematics of multi-pass InSAR scenario have been analytically described. Mathematical expressions for definition of current distance

*zC* ¼ 800 m. Wavelength is 0.03 m.

*InSAR Modeling of Geophysics Measurements DOI: http://dx.doi.org/10.5772/intechopen.89293*

gram AB-AC is presented in **Figure 12f**.

**10. Conclusions**

**13**

**Figure 12.**

**Figure 11.** *Surface (peaks) before (a) and after (b) displacement.*

**Figure 12.**

**9. Pseudo InSAR geophysical measurements: numerical results**

*Geographic Information Systems in Geospatial Intelligence*

**Figure 10.**

**Figure 11.**

**12**

Consider a GeoTIFF file of Dilijan region in Caucasus, Armenia, located at the geographical coordinates 40° 44<sup>0</sup> <sup>27</sup>″ north and 44° 51<sup>0</sup> <sup>47</sup>″ east longitude. Consider 2 pass InSAR scenario. Coordinates of SAR positions in the moment of imaging are the following: master SAR position A, *xA* <sup>¼</sup> 0 m, *yA* <sup>¼</sup> <sup>300</sup>*:*<sup>3</sup> � 103 m, *zA* <sup>¼</sup> <sup>3</sup> � 105 <sup>m</sup> and slave SAR position B *xB* <sup>¼</sup> 0 m, *yB* <sup>¼</sup> <sup>300</sup> � 103 m, *zB* <sup>¼</sup> <sup>3</sup> � 105 m. Wavelength is 0.05 m. Distances at the moment of imaging from the SAR position A and SAR position B to each pixel on the surface are illustrated in **Figure 10a, b**. Interferogram wrapped phases and unwrapped phases are presented in **Figure 10c, d**, respectively. Consider a three-pass InSAR scenario and a surface before (**Figure 11a**) and after (**Figure 11b**) displacement described by MATLAB function *peaks*. Coordinates of SAR positions in the moment of imaging are the following: master SAR position A, *xA* ¼ 350 km, *yA* ¼ 350 km, *zA* ¼ 800 km; slave SAR position B *xB* ¼ 351*:*5 km,

*Distances from SAR position A (a) and SAR position B (b) to each pixel on the surface in pseudo color map,*

*interferogram wrapped phases (c) and unwrapped phases (d).*

*Surface (peaks) before (a) and after (b) displacement.*

*Distances to the surface measured from SAR positions A (a), B (b), and C (c). AB interferogram (d), AC interferogram (e) with surface displacement, and differential interferogram AB-AC (f).*

*yB* ¼ 350 km, *zB* ¼ 800 km; and slave SAR position C *xC* ¼ 350 km, *yC* ¼ 351*:*2 km, *zC* ¼ 800 m. Wavelength is 0.03 m.

Distances to the surface at the moment of imaging as pseudo collar maps measured from SAR positions A, B, and C are presented in **Figure 12a–c**, respectively. AB interferogram without surface displacement and AC interferogram with surface displacement are presented in **Figure 12d, e**, respectively. Differential interferogram AB-AC is presented in **Figure 12f**.

The differential interferogram obtained by pixel subtraction of interferograms in **Figure 12d, e** is presented in **Figure 12f**. It illustrates the displacement of the surface. Only deformed part of the surface as differential fringes is depicted. The pseudo InSAR modeling can be applied to generate interferograms and differential interferograms based on real geophysical measurements and Geo TIFF maps of the observed surface.

### **10. Conclusions**

A multi-pass InSAR system has been theoretically analyzed and numerically experimented. Geometry and kinematics of multi-pass InSAR scenario have been analytically described. Mathematical expressions for definition of current distance vectors between SAR system and surface's pixels are derived. The basic InSAR parameters are defined. Analytical expressions to calculate pixel heights and pixel displacement have been derived. A model of linear frequency modulated SAR signal, reflected from the topographic surface, has been developed. An image reconstruction algorithm has been described. Numerical results verifying InSAR geometry, kinematics, and signal models are provided. Based on geometrical, kinematical, and signal models, numerical interferograms of a topographic surface have been created.

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**15**

A pseudo InSAR approach has been applied to model processes of interferograms and differential interferogram generation using GeoTIFF files and measurements of distances from SAR positions to each pixels of the observed surface at the moment of imaging. Based on distance vector description of the InSAR scenario, the interferometric phase and interferometric differential phase have been analytically described. Pseudo InSAR geophysical measurements and interferograms and differential interferogram generation have been illustrated by results of numerical experiments.

In conclusion, the results in the present work can be applied for analysis and modeling of SAR interferometric processes in scenarios with different geometric, kinematics, and geological structures as well as for generating pseudo SAR interferograms based on the geophysical measurements and topographic maps.
