**3. InSAR relief measurements**

The distances to ij-th pixel from SAR in *m*-th and *n*-th pass (*m* 6¼ *n*) at the moment of imaging can be defined by the cosine's theorem, i.e.,

$$\left| R\_{\rm ij}^{n} \right| = \left\{ \left| R\_{\rm ij}^{m} \right|^{2} + B\_{\rm mn}^{2} - 2B\_{\rm mn} \left| R\_{\rm ij}^{m} \right| \cos \left[ \frac{\pi}{2} - \left[ \theta^{m\_{\rm ij}} - a\_{\rm mn} \right] \right] \right\}^{\frac{1}{2}},\tag{5}$$

where *B*mn is the modulus of the baseline vector, *θ<sup>m</sup>*ij is the look angle, and *α*mn is a priory known tilt angle, the angle between the baseline vector and plane *Oxy*. The look angle *θ<sup>m</sup>*ij and height *h<sup>m</sup>* of an *ij*-th pixel on the surface with respect to *m*-th SAR position in the moment of imaging can be written as

$$\theta^{m\_{\vec{\}}} = a\_{\rm mn} + \arccos \frac{\left| R^{m}\_{\vec{\text{ij}}} \right|^{2} + B^{2}\_{\rm mn} - \left| R^{n}\_{\vec{\text{ij}}} \right|^{2}}{2B\_{\rm mn} \left| R^{m}\_{\vec{\text{ij}}} \right|},\tag{6}$$

$$z\_{\rm ij} = h^m - \left| R\_{\rm ij}^m \right| \cos \theta^{m\_{\rm ij}}.\tag{7}$$

The distance difference, *ΔR*mn ij � � � � � � <sup>¼</sup> *Rn* ij � � � � � � � *<sup>R</sup><sup>m</sup>* ij � � � � � �, can be expressed by the interferometric phase difference *ΔR*mn ij � � � � � � <sup>¼</sup> *<sup>λ</sup>* <sup>2</sup>*<sup>π</sup> Δϕ*mn ij . In case *R<sup>m</sup>* ij � � � � � � can be measured, i.e., *Rn* ij � � � � � � <sup>¼</sup> *Rm* ij � � � � � � <sup>þ</sup> *<sup>Δ</sup>R*mn ij � � � � � �, then

$$\theta^{m\_{\vec{\imath}}} = a\_{\rm mn} + \arcsin\left[\frac{B\_{\rm mn}}{2R\_{\rm ij}^{m}} - \frac{\lambda}{2\pi B\_{\rm mn}} \Delta\theta\_{\rm ij}^{\rm mm} \left(1 + \frac{\lambda}{4\pi R\_{\rm ij}^{m}} \Delta\theta\_{\rm ij}^{\rm mm}\right)\right],\tag{8}$$

$$z\_{\rm ij} = h^{m} - R\_{\rm ij}^{m} \cdot \cos\left\{a\_{\rm mn} + \arcsin\left[\frac{B\_{\rm mn}}{2R\_{\rm ij}^{m}} - \frac{\lambda}{2\pi B\_{\rm mn}} \Delta\theta\_{\rm ij}^{\rm mm} \left(1 + \frac{\lambda}{4\pi R\_{\rm ij}^{m}} \Delta\theta\_{\rm ij}^{\rm mm}\right)\right]\right\}.\tag{9}$$

## **4. InSAR measurements of relief displacement**

Consider a three-pass SAR interferometry (**Figure 1**). Let A and B be the two positions of imaging which can be defined by two passes of the same spaceborne SAR in different time (two pass interferometry). The third position C is defined by the third pass of the spaceborne SAR. The surface displacement, *Δz*ij, due, for instance, to an earthquake could derive from two SAR interferograms built before and after the seismic impact. The temporal baseline, the time scale over which the displacement is measured, must follow the dynamics of the geophysical phenomenon. Short-time baseline is applied for monitoring fast surface changes. Long temporal baseline is used for monitoring slow geophysics phenomena (subsidence). The interferometry phase before event is derived from complex images acquired by A and B SAR positions in the moment of imaging, while the interferometry phase after event is derived from complex images acquired by A and C SAR positions in the moment of imaging. The distances *R*<sup>1</sup> ij, *R*<sup>2</sup> ij, *R*<sup>3</sup> ij, and *R<sup>d</sup>*<sup>3</sup> ij after standard manipulations are written as.

displacement and *Rd*<sup>3</sup>

after *Δz*ij surface displacement.

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

Neglecting the term ð Þ *Δz*

*B*<sup>1</sup> sin *θ*ij � *α*<sup>1</sup>

ΔΦ <sup>¼</sup> <sup>4</sup>*<sup>π</sup> λ*

> *S t*ðÞ¼ <sup>X</sup> *M*

rate, and *T* is the time LFM pulse width.

*p*¼1

imaging can be written as.

*<sup>ϕ</sup>*AB <sup>¼</sup> <sup>4</sup>*<sup>π</sup> λ R*1 ij � *<sup>R</sup>*<sup>2</sup> ij � �, *<sup>ϕ</sup>*AC <sup>¼</sup> <sup>4</sup>*<sup>π</sup>*

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

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

ference ΔΦ*<sup>d</sup>* <sup>¼</sup> *<sup>ϕ</sup>*AC

pulses as follows

expressed as

**5**

where

ij is the slant range to from C SAR position to the observed pixel

*<sup>d</sup>* <sup>¼</sup> <sup>4</sup>*<sup>π</sup> λ R*1 ij � *<sup>R</sup>d*<sup>3</sup>

*B*<sup>2</sup> sin *θ*ij � *α*<sup>2</sup>

*B*2 *R*1 ij

ij < <1, then ΔΦ*<sup>d</sup>* <sup>¼</sup> ΔΦ <sup>þ</sup> <sup>4</sup>*<sup>π</sup>*

� � � *<sup>B</sup>*<sup>2</sup>

ij " #*:* (14)

! (13)

� � � *<sup>B</sup>*<sup>2</sup>

sin *α*<sup>2</sup>

<sup>2</sup> � *<sup>B</sup>*<sup>2</sup> 1 2*R*<sup>1</sup>

*:* (15)

ij ! (12)

ij in Eq. (10) can be rewritten as.

*λ*

þ *Δz* cos *θ*ij þ

sin *α*<sup>2</sup>

ij � � (11)

2 2*R*<sup>1</sup>

*<sup>λ</sup> Δz*ij cos *θ*ij,

Given the SAR wavelength λ, the phase differences proportional to range differences related to a particular pixel before and after displacement in the moment of

> *λ R*1 ij � *<sup>R</sup>*<sup>3</sup> ij � �, *<sup>ϕ</sup>*AC

1 2*R*<sup>1</sup> ij !; *<sup>ϕ</sup>*AC <sup>¼</sup> <sup>4</sup>*<sup>π</sup>*

> 2 2*R*<sup>1</sup> ij

" # !

*B*2 *R*1 ij

The displacement *Δz*ij is extracted from the differential interferometric phase dif-

� � � *<sup>B</sup>*<sup>1</sup> sin *<sup>θ</sup>*ij � *<sup>α</sup>*<sup>1</sup>

ΔΦ*<sup>d</sup>* � ΔΦ cos *θ*ij

The SAR transmits a series of electromagnetic waveforms to the surface, which are described analytically by the sequence of linear frequency modulation (chirp)

where *A* is the amplitude of the transmitted pulses, *Tp* is the pulse repetition period, *ω* ¼ 2*π:c=λ* is the angular frequency, *p* ¼ 1, *M* is the index of LFM emitted pulse, *<sup>M</sup>* is an emitted pulse number for synthesis of the aperture, *<sup>c</sup>* <sup>¼</sup> <sup>3</sup> � <sup>10</sup><sup>8</sup> m/s is the light speed in vacuum, *ΔF* is the LFM pulse bandwidth, *b* ¼ *π:ΔF=T* is the chirp

The SAR signal, reflected by ij-th pixel and registered in the *n*-th pass, can be

� � <sup>þ</sup> *b t* � pT*<sup>p</sup>*

� �<sup>2</sup> � � � � , (16)

� � � *<sup>B</sup>*<sup>2</sup>

*<sup>ϕ</sup>*AC <sup>þ</sup> *<sup>Δ</sup><sup>z</sup>* cos *<sup>θ</sup>*ij <sup>þ</sup>

*<sup>d</sup>* � *<sup>ϕ</sup>*AB. Considering *<sup>B</sup>*2*=R*<sup>1</sup>

*B*<sup>2</sup> sin *θ*ij � *α*<sup>2</sup>

*<sup>Δ</sup>z*ij <sup>¼</sup> *<sup>λ</sup>* 4*π*

*A* exp � *j ω t* � pT*<sup>p</sup>*

For surface displacement *z*ij can be written as

**5. SAR waveform and deterministic signal model**

2 *=*2*R*<sup>1</sup>

� � � *<sup>B</sup>*<sup>2</sup>

*B*<sup>2</sup> sin *θ*ij � *α*<sup>2</sup>

$$R\_{\rm ij}^2 \simeq R\_{\rm ij}^1 - B\_1 \sin \left(\theta\_{\rm ij} - \alpha\_1\right) + \frac{B\_1^2}{2R\_{\rm ij}^1},\\ R\_{\rm ij}^3 \simeq R\_{\rm ij}^1 - B\_2 \sin \left(\theta\_{\rm ij} - \alpha\_2\right) + \frac{B\_2^2}{2R\_{\rm ij}^1},\tag{10}$$

$$R\_{\rm ij}^{\rm d3} \simeq R\_{\rm ij}^3 - \Delta z \left(\cos \theta\_{\rm ij} + \frac{B\_2}{R\_{\rm ij}^1} \sin \alpha\_2\right) + \frac{\left(\Delta z\_{\rm ij}\right)^2}{2R\_{\rm ij}^1},$$

where *R*<sup>1</sup> ij, *R*<sup>2</sup> ij, and *R*<sup>3</sup> ij are the slant ranges from A, B, and C positions of SAR system to the observed pixel in the moment of imaging before the surface

**Figure 1.** *InSAR geometry and kinematics.*

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

**4. InSAR measurements of relief displacement**

*Geographic Information Systems in Geospatial Intelligence*

the moment of imaging. The distances *R*<sup>1</sup>

ij � *B*<sup>1</sup> sin *θ*ij � *α*<sup>1</sup>

*R*d3 ij ≃ *R*<sup>3</sup>

ij, and *R*<sup>3</sup>

� � <sup>þ</sup>

*B*2 1 2*R*<sup>1</sup> ij , *R*<sup>3</sup> ij ≃*R*<sup>1</sup>

ij � *Δz* cos *θ*ij þ

system to the observed pixel in the moment of imaging before the surface

tions are written as.

*R*2 ij ≃*R*<sup>1</sup>

where *R*<sup>1</sup>

**Figure 1.**

**4**

*InSAR geometry and kinematics.*

ij, *R*<sup>2</sup>

Consider a three-pass SAR interferometry (**Figure 1**). Let A and B be the two positions of imaging which can be defined by two passes of the same spaceborne SAR in different time (two pass interferometry). The third position C is defined by the third pass of the spaceborne SAR. The surface displacement, *Δz*ij, due, for instance, to an earthquake could derive from two SAR interferograms built before and after the seismic impact. The temporal baseline, the time scale over which the displacement is measured, must follow the dynamics of the geophysical phenomenon. Short-time baseline is applied for monitoring fast surface changes. Long temporal baseline is used for monitoring slow geophysics phenomena (subsidence). The interferometry phase before event is derived from complex images acquired by A and B SAR positions in the moment of imaging, while the interferometry phase after event is derived from complex images acquired by A and C SAR positions in

> ij, *R*<sup>2</sup> ij, *R*<sup>3</sup>

> > *B*2 *R*1 ij

!

sin *α*<sup>2</sup>

ij are the slant ranges from A, B, and C positions of SAR

ij, and *R<sup>d</sup>*<sup>3</sup>

ij � *B*<sup>2</sup> sin *θ*ij � *α*<sup>2</sup>

þ

� � <sup>þ</sup>

*Δz*ij � �<sup>2</sup> 2*R*<sup>1</sup> ij ,

ij after standard manipula-

*B*2 2 2*R*<sup>1</sup> ij

, (10)

displacement and *Rd*<sup>3</sup> ij is the slant range to from C SAR position to the observed pixel after *Δz*ij surface displacement.

Given the SAR wavelength λ, the phase differences proportional to range differences related to a particular pixel before and after displacement in the moment of imaging can be written as.

$$
\phi^{\rm AB} = \frac{4\pi}{\lambda} \left( R\_{\text{i}\text{j}}^1 - R\_{\text{i}\text{j}}^2 \right),
\phi^{\rm AC} = \frac{4\pi}{\lambda} \left( R\_{\text{i}\text{j}}^1 - R\_{\text{i}\text{j}}^3 \right),
\phi\_{\rm d}^{\rm AC} = \frac{4\pi}{\lambda} \left( R\_{\text{i}\text{j}}^1 - R\_{\text{i}\text{j}}^{\rm C\text{j}} \right) \tag{11}
$$

Neglecting the term ð Þ *Δz* 2 *=*2*R*<sup>1</sup> ij in Eq. (10) can be rewritten as.

$$\boldsymbol{\phi}^{\rm AB} = \frac{4\pi}{\lambda} \left( B\_1 \sin \left( \theta\_{\rm ii} - a\_1 \right) - \frac{B\_1^2}{2R\_{\rm ij}^1} \right); \quad \boldsymbol{\phi}^{\rm AC} = \frac{4\pi}{\lambda} \left( B\_2 \sin \left( \theta\_{\rm iii} - a\_2 \right) - \frac{B\_2^2}{2R\_{\rm ij}^1} \right) \tag{12}$$

$$\dots \quad \boldsymbol{\omega}\_{\rm AC} \left[ \boldsymbol{\qquad} \qquad \qquad \qquad \boldsymbol{\Lambda}^2 \qquad \qquad \qquad \qquad \qquad \boldsymbol{\Lambda}\_{\rm B} \tag{13}$$

$$\begin{aligned} \phi\_d^{\rm AC} &= \frac{4\pi}{\lambda} \left[ B\_2 \sin \left( \theta\_{\rm ii} - \alpha\_2 \right) - \frac{B\_2^2}{2R\_{\rm ij}^1} + \Delta z \left( \cos \theta\_{\rm ii} + \frac{B\_2}{R\_{\rm ij}^1} \sin \alpha\_2 \right) \right] \\ &\quad \phi^{\rm AC} + \Delta z \left( \cos \theta\_{\rm ii} + \frac{B\_2}{R\_{\rm ij}^1} \sin \alpha\_2 \right) \end{aligned} \tag{13}$$

The displacement *Δz*ij is extracted from the differential interferometric phase difference ΔΦ*<sup>d</sup>* <sup>¼</sup> *<sup>ϕ</sup>*AC *<sup>d</sup>* � *<sup>ϕ</sup>*AB. Considering *<sup>B</sup>*2*=R*<sup>1</sup> ij < <1, then ΔΦ*<sup>d</sup>* <sup>¼</sup> ΔΦ <sup>þ</sup> <sup>4</sup>*<sup>π</sup> <sup>λ</sup> Δz*ij cos *θ*ij, where

$$
\Delta\Phi = \frac{4\pi}{\lambda} \left[ B\_2 \sin\left(\theta\_{\text{i}\parallel} - \alpha\_2\right) - B\_1 \sin\left(\theta\_{\text{i}\parallel} - \alpha\_1\right) - \frac{B\_2^2 - B\_1^2}{2R\_{\text{i}\parallel}^2} \right]. \tag{14}$$

For surface displacement *z*ij can be written as

$$
\Delta z\_{\rm ij} = \frac{\lambda}{4\pi} \frac{\Delta \Phi\_d - \Delta \Phi}{\cos \theta\_{\rm ij}}.\tag{15}
$$

### **5. SAR waveform and deterministic signal model**

The SAR transmits a series of electromagnetic waveforms to the surface, which are described analytically by the sequence of linear frequency modulation (chirp) pulses as follows

$$S(t) = \sum\_{p=1}^{M} A \exp\left\{-j\left[a\left(t - \mathbf{p}\mathbf{T}\_p\right) + b\left(t - \mathbf{p}\mathbf{T}\_p\right)^2\right]\right\},\tag{16}$$

where *A* is the amplitude of the transmitted pulses, *Tp* is the pulse repetition period, *ω* ¼ 2*π:c=λ* is the angular frequency, *p* ¼ 1, *M* is the index of LFM emitted pulse, *<sup>M</sup>* is an emitted pulse number for synthesis of the aperture, *<sup>c</sup>* <sup>¼</sup> <sup>3</sup> � <sup>10</sup><sup>8</sup> m/s is the light speed in vacuum, *ΔF* is the LFM pulse bandwidth, *b* ¼ *π:ΔF=T* is the chirp rate, and *T* is the time LFM pulse width.

The SAR signal, reflected by ij-th pixel and registered in the *n*-th pass, can be expressed as

$$\mathcal{S}\_{\vec{\mathbf{i}}\dagger}^{\pi}(t) = a\_{\vec{\mathbf{i}}\dagger}(z\_{\vec{\mathbf{i}}\dagger}) \mathbf{rect} \frac{t - t\_{\vec{\mathbf{i}}\dagger}^{\pi}}{T} \exp\left\{-j\left[a\left(t - t\_{\vec{\mathbf{i}}\dagger}^{\pi}\right) + b\left(t - t\_{\vec{\mathbf{i}}\dagger}^{\pi}\right)^2\right]\right\} \tag{17}$$

$$\text{rect}\frac{t - t\_{\text{ij}}^n(p)}{T} = \left\{ \mathbf{1}, \mathbf{0} < \frac{t - t\_{\text{ij}}^n(p)}{T} \le \mathbf{1} \right\},\tag{18}$$

\_ *S n <sup>R</sup>* ^ *k*, *p* � � <sup>¼</sup> <sup>X</sup>

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

registered, and by Fourier transform

form of the range compressed signal, \_

the *n*-th pass data can be expressed as

\_ *I <sup>n</sup>* ^ *k*, *p*^ � � <sup>¼</sup> <sup>X</sup>

differential interferograms can be created.

**7. InSAR modeling: numerical results**

*x*3

coordinates at the moment of imaging as follows.

<sup>0</sup> <sup>¼</sup> <sup>10</sup>*:*10<sup>3</sup> m, *<sup>z</sup>*<sup>1</sup>

<sup>0</sup> <sup>¼</sup> 0 m, *<sup>y</sup>*<sup>3</sup>

exp �*x*ij<sup>2</sup> � *y*ij þ 1

� 1

*k* ¼ 1,*K*.

for each *<sup>p</sup>* <sup>¼</sup> 1, *<sup>M</sup>* and ^

for each *<sup>p</sup>*^ <sup>¼</sup> 1, *<sup>M</sup>*, ^

*x*1

*z*2

following equation

*<sup>z</sup>*ij <sup>¼</sup> 3 1 � *<sup>x</sup>*ij � �<sup>2</sup>

**7**

<sup>0</sup> <sup>¼</sup> 0 m; *<sup>y</sup>*<sup>1</sup>

<sup>0</sup> <sup>¼</sup> <sup>100</sup>*:*10<sup>3</sup> m,

the spatial resolution of the pixels.

\_ *S n <sup>R</sup>* ^ *k*, *p* � � <sup>¼</sup> <sup>X</sup>

*K*

\_ ^ *S n*

*K*

\_ ^ *S n*

*S n <sup>R</sup>* ^ *k*, *p*

The complex SAR image extracted from the *n*-th pass data preserves phases defined by distances from the satellite to each pixel at the moment of imaging. Based on pixel phases and image co-registration, a complex interferograms and

The SAR signal model and imaging algorithm are illustrated by results of numerical experiments. Consider three pass satellite SAR system with position

<sup>0</sup> <sup>¼</sup> <sup>100</sup>*:*10<sup>3</sup> m, *<sup>x</sup>*<sup>2</sup>

<sup>0</sup> <sup>¼</sup> 10, 2*:*10<sup>3</sup> m, *<sup>z</sup>*<sup>3</sup>

and *vz* ¼ 0 m/s. The surface observed by the SAR system is modeled by the

<sup>3</sup> exp � *<sup>x</sup>*ij <sup>þ</sup> <sup>1</sup> � �<sup>2</sup> � *<sup>y</sup>*ij2

� �<sup>2</sup> � � � <sup>10</sup> *<sup>x</sup>*ij

Coordinates of vector-velocity of the satellite are *vx* ¼ 0 m/s, *v <sup>y</sup>* ¼ �600 m/s,

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*-

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

<sup>5</sup> � *<sup>x</sup>*ij<sup>3</sup> � *<sup>y</sup>*ij<sup>5</sup>

*M*

*p*¼1 \_ *S n <sup>R</sup> <sup>p</sup>*, ^ *k* � � exp *<sup>j</sup>*

*k*¼1

ð Þ *<sup>k</sup>*, *<sup>p</sup>* exp jb *<sup>k</sup>* � ^

ð Þ *k*, *p :* exp *j*

where *K* is the full number of LFM samples, the range bins where SAR signal is

The range alignment and higher-order phase correction are beyond of the scope of the present work. The azimuth compression is accomplished by Fourier trans-

*k* � 1 � �*Δ<sup>T</sup>*

> 2*πk*^ *k Kn*

> 2*πpp*^ *M*

<sup>0</sup> <sup>¼</sup> 0 m, *<sup>y</sup>*<sup>2</sup>

<sup>0</sup> <sup>¼</sup> <sup>100</sup>*:*10<sup>3</sup> <sup>m</sup>*:*

� � exp �*x*ij2 � *<sup>y</sup>*ij2

h i , (25)

h i<sup>2</sup> � � (22)

max !, (23)

� �, (24)

<sup>0</sup> <sup>¼</sup> 10, 1*:*10<sup>3</sup> m,

� ��

� �. The complex image extracted from

*k*¼1

*k* ¼ 1,*K*.

where *a*ij *z*ij � � is the reflection coefficient of the pixel from the surface.

The parameter *a*ij *z*ij � � is a function of surface geometry; *t n* ijð Þ¼ *p R*1 ijð Þþ*<sup>p</sup> <sup>R</sup><sup>n</sup>* ijð Þ *p <sup>c</sup>* is the time propagation of the reflected signal from the ij-th scattering pixel registered in the n-th pass.

SAR signal reflected from the entire illuminated surface is an interference of elementary signals of scattering pixels and can be written as

$$\mathcal{S}''(t) = \sum\_{i} \sum\_{j} a\_{\text{ij}}(z\_{\text{ij}}) \text{rect} \frac{t - t\_{\text{ij}}^n}{T} \exp\left\{-j \left[o\left(t - t\_{\text{ij}}^n\right) + b\left(t - t\_{\text{ij}}^n\right)^2\right]\right\}.\tag{19}$$

The time dwell *t* of the SAR signal return for each transmitted pulse *p* can be expressed as *t* ¼ *t n* ijminð Þþ *<sup>p</sup> <sup>k</sup>ΔT*, where *<sup>k</sup>* <sup>¼</sup> *<sup>k</sup><sup>n</sup>* ijminð Þ *<sup>p</sup>* , *<sup>k</sup><sup>n</sup>* ijmaxð Þ *p* is the sample number of the SAR return measured on range direction in *n*-th pass, *kn* ijmin ¼ int *t n* ijminð Þ *p =ΔT* h i, *<sup>k</sup><sup>n</sup>* ijmax ¼ int *t n* ijmaxð Þ *p =ΔT* h i, *<sup>Δ</sup><sup>T</sup>* <sup>¼</sup> <sup>1</sup>*=*ð Þ <sup>2</sup>*Δ<sup>F</sup>* is the sample time width, and *k<sup>n</sup>* maxð Þ *p* is the number of the furthest range bin where SAR signal is registered in *n*-th pass. Hence, in discrete form SAR signal can be rewritten as

$$\dot{S}''(k, p) = \sum\_{i} \sum\_{j} a\_{\vec{\imath}j}(z\_{\vec{\imath}}) \text{rect} \frac{t - t\_{\vec{\imath}j}^{n}}{T}$$

$$\exp\left\{-j\left[o\left((k-1)\Delta T - t\_{\vec{\imath}j}^{n}(p)\right) + b\left((k-1)\Delta T - t\_{\vec{\imath}j}^{n}(p)\right)^{2}\right]\right\}^{-} $$

The expressions derived in Section 2 and Section 5 can be used for modeling the SAR signal return in case the satellites are moving rectilinearly in 3-D coordinate system.

### **6. SAR image reconstruction**

The complex image reconstruction includes the following operations: frequency demodulation, range compression, coarse range alignment, precise phase correction, and azimuth compression. The frequency demodulation is performed by multiplication of Eq. (20) with a complex conjugated function exp *<sup>j</sup> <sup>ω</sup>*ð Þ *<sup>k</sup>* � <sup>1</sup> *<sup>Δ</sup><sup>T</sup>* <sup>þ</sup> *b k* ½ � ð Þ � <sup>1</sup> *<sup>Δ</sup><sup>T</sup>* <sup>2</sup> n o � � .

Thus, the range distributed frequency demodulated SAR return in *n*-th pass for *p*-th pulse can be written as

$$\dot{\vec{S}}''(k,p) = \sum\_{i} \sum\_{j} a\_{\vec{\eta}}(z\_{\vec{\eta}}) \text{rect} \frac{(k-1)\Delta T - t\_{\vec{\eta}}^{\text{n}}}{T} \cdot \exp\left\{-j\left[\alpha t\_{\vec{\eta}}^{\text{n}}(p) + b\left((k-1)\Delta T - t\_{\vec{\eta}}^{\text{n}}(p)\right)^{2}\right]\right\}.\tag{21}$$

The range compression of the LFM demodulated SAR signal is performed by cross correlation with a reference function, exp jb½ � ð Þ *<sup>k</sup>* � <sup>1</sup> *<sup>Δ</sup><sup>T</sup>* <sup>2</sup> n o

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

*Sn*

where *a*ij *z*ij

*Sn*ðÞ¼ *<sup>t</sup>* <sup>X</sup>

be expressed as *t* ¼ *t*

ijminð Þ *p =ΔT* h i

width, and *k<sup>n</sup>*

int *t n* *i*

X *j*

*a*ij *z*ij � �rect

*n*

ijmax ¼ int *t*

\_ *S n*

exp � *j ω* ð Þ *k* � 1 *ΔT* � *t*

**6. SAR image reconstruction**

exp *<sup>j</sup> <sup>ω</sup>*ð Þ *<sup>k</sup>* � <sup>1</sup> *<sup>Δ</sup><sup>T</sup>* <sup>þ</sup> *b k* ½ � ð Þ � <sup>1</sup> *<sup>Δ</sup><sup>T</sup>* <sup>2</sup> n o � �

*p*-th pulse can be written as

*i* X *j a*ij *z*ij � �rect

\_ ^ *S n*

**6**

ð Þ¼ *<sup>k</sup>*, *<sup>p</sup>* <sup>X</sup>

, *k<sup>n</sup>*

the n-th pass.

The parameter *a*ij *z*ij

ijðÞ¼ *t a*ij *z*ij

� �rect

rect

*Geographic Information Systems in Geospatial Intelligence*

*t* � *t n* ij

*t* � *t n* ijð Þ *p*

elementary signals of scattering pixels and can be written as

*t* � *t n* ij

ijminð Þþ *<sup>p</sup> <sup>k</sup>ΔT*, where *<sup>k</sup>* <sup>¼</sup> *<sup>k</sup><sup>n</sup>*

*n*

ð Þ¼ *<sup>k</sup>*, *<sup>p</sup>* <sup>X</sup>

� �

*i*

*n* ijð Þ *p*

number of the SAR return measured on range direction in *n*-th pass, *kn*

ijmaxð Þ *p =ΔT* h i

registered in *n*-th pass. Hence, in discrete form SAR signal can be rewritten as

*a*ij *z*ij � �rect

The expressions derived in Section 2 and Section 5 can be used for modeling the SAR signal return in case the satellites are moving rectilinearly in 3-D coordinate system.

X *j*

The complex image reconstruction includes the following operations: frequency demodulation, range compression, coarse range alignment, precise phase correction, and azimuth compression. The frequency demodulation is performed by multiplication of Eq. (20) with a complex conjugated function

.

*n* ij *<sup>T</sup> :* exp � *<sup>j</sup> <sup>ω</sup><sup>t</sup>*

The range compression of the LFM demodulated SAR signal is performed by

ð Þ *k* � 1 *ΔT* � *t*

cross correlation with a reference function, exp jb½ � ð Þ *<sup>k</sup>* � <sup>1</sup> *<sup>Δ</sup><sup>T</sup>* <sup>2</sup> n o

Thus, the range distributed frequency demodulated SAR return in *n*-th pass for

*n*

ijð Þþ *p b k*ð Þ � 1 *ΔT* � *t*

� �<sup>2</sup> � � � �

*n* ijð Þ *p*

*:*

(21)

*<sup>T</sup>* exp � *<sup>j</sup> <sup>ω</sup> <sup>t</sup>* � *<sup>t</sup>*

(

� � is the reflection coefficient of the pixel from the surface.

� � is a function of surface geometry; *t*

time propagation of the reflected signal from the ij-th scattering pixel registered in

SAR signal reflected from the entire illuminated surface is an interference of

*<sup>T</sup>* exp � *<sup>j</sup> <sup>ω</sup> <sup>t</sup>* � *<sup>t</sup>*

maxð Þ *p* is the number of the furthest range bin where SAR signal is

The time dwell *t* of the SAR signal return for each transmitted pulse *p* can

*<sup>T</sup>* <sup>¼</sup> 1, 0 <sup>&</sup>lt;

*n* ij � �

> *t* � *t n* ijð Þ *p <sup>T</sup>* <sup>≤</sup>1<sup>j</sup>

� �<sup>2</sup> � � � �

*n* ij � �

ijminð Þ *<sup>p</sup>* , *<sup>k</sup><sup>n</sup>*

*t* � *t n* ij *T*

þ *b k*ð Þ � 1 *ΔT* � *t*

� �<sup>2</sup> � � � � *:* (20)

� �<sup>2</sup> � � � �

þ *b t* � *t*

*n* ij

*n* ijð Þ¼ *p*

þ *b t* � *t*

, *ΔT* ¼ 1*=*ð Þ 2*ΔF* is the sample time

*n* ijð Þ *p* *n* ij

ijmaxð Þ *p* is the sample

, (18)

*R*1 ijð Þþ*<sup>p</sup> <sup>R</sup><sup>n</sup>* ijð Þ *p <sup>c</sup>* is the

(17)

*:* (19)

ijmin ¼

$$\boldsymbol{\dot{S}}\_{R}^{\boldsymbol{n}}(\hat{\boldsymbol{k}},\boldsymbol{p}) = \sum\_{k=1}^{K} \dot{\boldsymbol{\dot{S}}}^{\boldsymbol{n}}(\boldsymbol{k},\boldsymbol{p}) \exp\left\{\mathbf{j}\mathbf{b}\left[\left(\boldsymbol{k}-\hat{\boldsymbol{k}}-\mathbf{1}\right)\boldsymbol{\Delta}\boldsymbol{T}\right]^{2}\right\}\tag{22}$$

where *K* is the full number of LFM samples, the range bins where SAR signal is registered, and by Fourier transform

$$\dot{S}\_{R}^{n}(\hat{k},p) = \sum\_{k=1}^{K} \dot{\hat{S}}^{n}(k,p) \cdot \exp\left(j\frac{2\pi k\hat{k}}{K\_{\text{max}}^{n}}\right),\tag{23}$$

for each *<sup>p</sup>* <sup>¼</sup> 1, *<sup>M</sup>* and ^ *k* ¼ 1,*K*.

The range alignment and higher-order phase correction are beyond of the scope of the present work. The azimuth compression is accomplished by Fourier transform of the range compressed signal, \_ *S n <sup>R</sup>* ^ *k*, *p* � �. The complex image extracted from the *n*-th pass data can be expressed as

$$\hat{I}^{\boldsymbol{\eta}}\left(\hat{k},\hat{p}\right) = \sum\_{p=1}^{M} \hat{S}\_{R}^{\boldsymbol{\eta}}\left(p,\hat{k}\right) \exp\left(j\frac{2\pi p\hat{p}}{M}\right),\tag{24}$$

for each *<sup>p</sup>*^ <sup>¼</sup> 1, *<sup>M</sup>*, ^ *k* ¼ 1,*K*.

The complex SAR image extracted from the *n*-th pass data preserves phases defined by distances from the satellite to each pixel at the moment of imaging. Based on pixel phases and image co-registration, a complex interferograms and differential interferograms can be created.
