**7. InSAR modeling: numerical results**

The SAR signal model and imaging algorithm are illustrated by results of numerical experiments. Consider three pass satellite SAR system with position coordinates at the moment of imaging as follows.

$$\begin{aligned} x\_0^1 &= 0 \text{ m}; y\_0^1 = 10.10^3 \text{ m}, z\_0^1 = 100.10^3 \text{ m}, x\_0^2 = 0 \text{ m}, y\_0^2 = 10, 1.10^3 \text{ m}, z\_0^2 = 100.10^3 \text{ m}, z\_0^3 = 100.10^3 \text{ m}, z\_0^4 = 100.10^3 \text{ m}, z\_0^5 = 100.10^3 \text{ m}, z\_0^6 = 100.10^3 \text{ m}, z\_0^7 = 100.10^3 \text{ m}, z\_0^8 = 100.10^3 \text{ m}, z\_0^9 = 100.10^3 \text{ m}, z\_0^9 = 100.10^3 \text{ m}, z\_0^8 = 100.10^3 \text{ m}, z\_0^9 = 100.10^3 \text{ m}, z\_0^9 = 100.10^3 \text{ m}, z\_0^9 = 100.10^3 \text{ m}, z\_0^8 = 100.10^3 \text{ m}, z\_0^9 = 100.10^3 \text{ m}, z\_0^9 = 100.10^3 \text{ m}, z\_0^8 = 100.10^3 \text{ m}, z\_0^9 = 100.10^3 \text{ m}, z\_0^9 = 100.10^3 \text{ m}$$

Coordinates of vector-velocity of the satellite are *vx* ¼ 0 m/s, *v <sup>y</sup>* ¼ �600 m/s, and *vz* ¼ 0 m/s. The surface observed by the SAR system is modeled by the following equation

$$x\_{\vec{\eta}} = \mathbf{3} \left(\mathbf{1} - \mathbf{x}\_{\vec{\eta}}\right)^2 \exp\left[-\mathbf{x}\_{\vec{\eta}^2} - \left(y\_{\vec{\eta}} + \mathbf{1}\right)^2\right] - \mathbf{1} \mathbf{0} \left(\frac{\mathbf{x}\_{\vec{\eta}}}{5} - \mathbf{x}\_{\vec{\eta}^1} - y\_{\vec{\eta}^1}\right) \exp\left(-\mathbf{x}\_{\vec{\eta}^2} - y\_{\vec{\eta}^2}\right) - \mathbf{1}, \tag{25}$$
 
$$- \frac{1}{3} \exp\left[-\left(\mathbf{x}\_{\vec{\eta}} + \mathbf{1}\right)^2 - y\_{\vec{\eta}^2}\right]$$

where *x*ij ¼ *iΔX*, *y*ij ¼ *jΔY*, *i* ¼ 1,*I*, *j* ¼ 1, *J*, *I* = 128 pixels; *J* = 128 pixels; *ΔX*; *ΔY*the spatial resolution of the pixels.

Normalized amplitude of reflected signals from every pixel *a*ij ¼ 0*:*001. The spatial resolution of the pixel are *ΔX* ¼ *ΔY* ¼ 2 m. Wavelength is 0.03 m. Carrier frequency is 3.10<sup>9</sup> Hz. Frequency bandwidth is *<sup>Δ</sup><sup>F</sup>* <sup>¼</sup> 250 MHz. Pulse repetition period is *Tp* <sup>¼</sup> <sup>25</sup>*:*10�<sup>3</sup> s. LFM pulse duration is *<sup>T</sup>* <sup>¼</sup> <sup>5</sup>*:*10�<sup>6</sup> s. Sample time duration is *<sup>Δ</sup><sup>T</sup>* <sup>¼</sup> 1, 95*:*10�<sup>8</sup> s. LFM sample number is *<sup>K</sup>* = 512. Emitted pulse number is

**Figure 2.** *The real (a) and imaginary (b) component of the SAR complex signal measured in the first SAR pass.*

*M* = 512. Digital geometry description and SAR signal modeling are performed based on the theory in Sections 3 and 4. The complex images are extracted from the SAR signal by applying correlation range compression and FFT azimuth compression. Based on a priori-known kinematical parameters of satellites and coordinates of reference point from the surface autofocusing phase correction of the SAR signals registered in the both passes can be implemented.

The complex SAR image's amplitude and phase obtained in the second SAR pass are depicted in **Figure 5**. It can be seen that the shape of the surface (the amplitude of the complex image) is similar to the shape of the surface obtained by the first SAR pass. In contrast, the phase structures of both complex images are different based on the different SAR positions in respect of the surface in the first and second

*The real (a) and imaginary (b) component of the SAR complex signal measured in the second SAR pass.*

By co-registration of the first and third SAR complex images, a complex SAR interferogram can be created with components in a coherent map and interfero-

The real and imaginary components of the SAR complex signal obtained in the

The complex SAR image's amplitude and phase obtained in the third SAR pass are depicted in **Figure 8**. The shape of the surface obtained in the third SAR pass is

*The amplitude (a) and phase (b) components of the SAR complex image obtained in the second pass.*

pass at the moment of imaging.

**Figure 4.**

**Figure 5.**

**9**

metric phase depicted in **Figure 6**.

third SAR pass is depicted in **Figure 7**.

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

The real and imaginary components of the SAR complex signal measured in the first SAR pass are depicted in **Figure 2**.

The complex SAR image's amplitude and phase obtained in the first SAR pass are depicted in **Figure 3**. The orientation of the surface's image (**Figure 3a**) in the frame is defined by the position of the SAR at the moment of imaging.

The real and imaginary components of the SAR complex signal measured in the second SAR pass are depicted in **Figure 4**.

**Figure 3.** *The amplitude (a) and phase (b) component of the SAR complex image obtained in the first pass.*

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

**Figure 4.** *The real (a) and imaginary (b) component of the SAR complex signal measured in the second SAR pass.*

The complex SAR image's amplitude and phase obtained in the second SAR pass are depicted in **Figure 5**. It can be seen that the shape of the surface (the amplitude of the complex image) is similar to the shape of the surface obtained by the first SAR pass. In contrast, the phase structures of both complex images are different based on the different SAR positions in respect of the surface in the first and second pass at the moment of imaging.

By co-registration of the first and third SAR complex images, a complex SAR interferogram can be created with components in a coherent map and interferometric phase depicted in **Figure 6**.

The real and imaginary components of the SAR complex signal obtained in the third SAR pass is depicted in **Figure 7**.

The complex SAR image's amplitude and phase obtained in the third SAR pass are depicted in **Figure 8**. The shape of the surface obtained in the third SAR pass is

**Figure 5.** *The amplitude (a) and phase (b) components of the SAR complex image obtained in the second pass.*

*M* = 512. Digital geometry description and SAR signal modeling are performed based on the theory in Sections 3 and 4. The complex images are extracted from the SAR signal by applying correlation range compression and FFT azimuth compression. Based on a priori-known kinematical parameters of satellites and coordinates of reference point from the surface autofocusing phase correction of the

*The real (a) and imaginary (b) component of the SAR complex signal measured in the first SAR pass.*

The real and imaginary components of the SAR complex signal measured in the

The complex SAR image's amplitude and phase obtained in the first SAR pass are depicted in **Figure 3**. The orientation of the surface's image (**Figure 3a**) in the frame

The real and imaginary components of the SAR complex signal measured in the

SAR signals registered in the both passes can be implemented.

is defined by the position of the SAR at the moment of imaging.

*The amplitude (a) and phase (b) component of the SAR complex image obtained in the first pass.*

first SAR pass are depicted in **Figure 2**.

*Geographic Information Systems in Geospatial Intelligence*

**Figure 2.**

**Figure 3.**

**8**

second SAR pass are depicted in **Figure 4**.

**Figure 6.**

*The coherent map (a) and interferometric phase (b) of the complex SAR interferogram created by the first and second SAR complex images.*

similar to the shape of the surface obtained by the first and second SAR passes. Comparing phase structures of the three complex SAR images, it can be noticed that they are different based on the different SAR's positions in respect to the surface at

*The coherent map (a) and interferometric phase (b) of the complex SAR interferogram created by the first and*

Under pixel co-registration of the first and third SAR complex images, a complex SAR interferogram can be created with components in a coherent map and

Due to precise under pixel co-registrations of the first and second and the first and third SAR complex images, the phase interferograms depicted in **Figures 6b**

Consider three-pass InSAR geometry (**Figure 1**). The vector distances from the

ij <sup>¼</sup> *<sup>R</sup><sup>S</sup>* � *<sup>R</sup>*ij,

denotes the ij-th

and **9b**, respectively, are characterized with the similar structures.

**8. Pseudo InSAR modeling of geophysical measurements**

SAR positions to each ij-th pixel from the region of interest are *R<sup>S</sup>*

where *<sup>S</sup>* <sup>¼</sup> *<sup>A</sup>*, *<sup>B</sup>*,*<sup>C</sup>* denotes the SAR position at the moment of imaging, *<sup>R</sup><sup>S</sup>* <sup>¼</sup>

denotes the SAR vector position, and *<sup>R</sup>*ij <sup>¼</sup> *<sup>x</sup>*ij, *<sup>y</sup>*ij, *<sup>z</sup>*ij h i*<sup>T</sup>*

pixel vector position. Coordinates of SAR positions in the moment of imaging are as follows: for a master SAR position A, *xA*, *yA*, *zA*; for a slave SAR position B, *xB*, *yB*,

After distance measurements from the master SAR position A and slave SAR positions B and C, respectively, to each ij-th pixel on the surface and co-registration of so obtained master image and slave images, the instrumental interferometric

> 2 *λ*

> > 2 *λ*

2 *λ*

*RA* ij � � � � � � � *<sup>R</sup>*ij � � � � � �

*RA* ij � � � � � � � *<sup>R</sup><sup>C</sup>* ij � � � � � �

*RA* ij � � � � � � � *<sup>R</sup><sup>C</sup>* ij,*d* � � �

� � � �,

� � � �

� � � �*:*

� � �

the moment of imaging.

*third SAR complex images.*

**Figure 9.**

*xS*, *yS*, *zS* � �*<sup>T</sup>*

**11**

interferometric phase depicted in **Figure 9**.

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

*zB*; and for a slave SAR position C, *xC*, *yC*, *zC*.

phase differences are calculated as follows

*RA* ij � � � � � � � *<sup>R</sup><sup>B</sup>* ij � � � � � � � � � <sup>2</sup>*π:* max

*RA* ij � � � � � � � *<sup>R</sup><sup>C</sup>* ij � � � � � � � � � <sup>2</sup>*π:* max

*RA* ij � � � � � � � *<sup>R</sup><sup>C</sup>* ij,*d* � � �

� � � � � � <sup>2</sup>*π:* max

ming, and 2D Goldstein branch cut phase unwrapping, can be applied.

In order to unwrap the interferometric phases, standard algorithms, MATLAB *unwrap* function, 2-D Costantini phase unwrapping based on network program-

• without pixel displacement

*ϕ*AB ij <sup>¼</sup> <sup>4</sup>*<sup>π</sup> λ*

*ϕ*AC ij <sup>¼</sup> <sup>4</sup>*<sup>π</sup> λ*

• with pixel displacement

*ϕ*AC ij,*<sup>d</sup>* <sup>¼</sup> <sup>4</sup>*<sup>π</sup> λ*

**Figure 7.** *The real (a) and imaginary (b) component of the SAR complex signal measured in the third SAR pass.*

**Figure 8.** *The amplitude (a) and phase (b) component of the SAR complex image obtained in the third SAR pass.*

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

#### **Figure 9.**

**Figure 6.**

**Figure 7.**

**Figure 8.**

**10**

*second SAR complex images.*

*Geographic Information Systems in Geospatial Intelligence*

*The coherent map (a) and interferometric phase (b) of the complex SAR interferogram created by the first and*

*The real (a) and imaginary (b) component of the SAR complex signal measured in the third SAR pass.*

*The amplitude (a) and phase (b) component of the SAR complex image obtained in the third SAR pass.*

*The coherent map (a) and interferometric phase (b) of the complex SAR interferogram created by the first and third SAR complex images.*

similar to the shape of the surface obtained by the first and second SAR passes. Comparing phase structures of the three complex SAR images, it can be noticed that they are different based on the different SAR's positions in respect to the surface at the moment of imaging.

Under pixel co-registration of the first and third SAR complex images, a complex SAR interferogram can be created with components in a coherent map and interferometric phase depicted in **Figure 9**.

Due to precise under pixel co-registrations of the first and second and the first and third SAR complex images, the phase interferograms depicted in **Figures 6b** and **9b**, respectively, are characterized with the similar structures.

## **8. Pseudo InSAR modeling of geophysical measurements**

Consider three-pass InSAR geometry (**Figure 1**). The vector distances from the SAR positions to each ij-th pixel from the region of interest are *R<sup>S</sup>* ij <sup>¼</sup> *<sup>R</sup><sup>S</sup>* � *<sup>R</sup>*ij, where *<sup>S</sup>* <sup>¼</sup> *<sup>A</sup>*, *<sup>B</sup>*,*<sup>C</sup>* denotes the SAR position at the moment of imaging, *<sup>R</sup><sup>S</sup>* <sup>¼</sup> *xS*, *yS*, *zS* � �*<sup>T</sup>* denotes the SAR vector position, and *<sup>R</sup>*ij <sup>¼</sup> *<sup>x</sup>*ij, *<sup>y</sup>*ij, *<sup>z</sup>*ij h i*<sup>T</sup>* denotes the ij-th pixel vector position. Coordinates of SAR positions in the moment of imaging are as follows: for a master SAR position A, *xA*, *yA*, *zA*; for a slave SAR position B, *xB*, *yB*, *zB*; and for a slave SAR position C, *xC*, *yC*, *zC*.

After distance measurements from the master SAR position A and slave SAR positions B and C, respectively, to each ij-th pixel on the surface and co-registration of so obtained master image and slave images, the instrumental interferometric phase differences are calculated as follows

• without pixel displacement

$$\begin{split} \phi\_{\text{ij}}^{\text{AB}} &= \frac{4\pi}{\lambda} \left( \left| R\_{\text{ij}}^{A} \right| - \left| R\_{\text{ij}}^{B} \right| \right) - 2\pi. \max \left[ \frac{2}{\lambda} \left( \left| R\_{\text{ij}}^{A} \right| - \left| R\_{\text{ij}} \right| \right) \right], \\ \phi\_{\text{ij}}^{\text{AC}} &= \frac{4\pi}{\lambda} \left( \left| R\_{\text{ij}}^{A} \right| - \left| R\_{\text{ij}}^{C} \right| \right) - 2\pi. \max \left[ \frac{2}{\lambda} \left( \left| R\_{\text{ij}}^{A} \right| - \left| R\_{\text{ij}}^{C} \right| \right) \right] \end{split}$$

• with pixel displacement

$$\phi\_{\rm ij,d}^{\rm AC} = \frac{4\pi}{\lambda} \left( \left| R\_{\rm ij}^{A} \right| - \left| R\_{\rm ij,d}^{C} \right| \right) - 2\pi. \max \left[ \frac{2}{\lambda} \left( \left| R\_{\rm ij}^{A} \right| - \left| R\_{\rm ij,d}^{C} \right| \right) \right].$$

In order to unwrap the interferometric phases, standard algorithms, MATLAB *unwrap* function, 2-D Costantini phase unwrapping based on network programming, and 2D Goldstein branch cut phase unwrapping, can be applied.
