InSAR Modeling of Geophysics Measurements

*Andon Lazarov, Dimitar Minchev and Chavdar Minchev*

## **Abstract**

In the present work, the geometry and basic parameters of interferometric synthetic aperture radar (InSAR) geophysics system are addressed. Equations of pixel height and displacement evaluation are derived. Synthetic aperture radar (SAR) signal model based on linear frequency modulation (LFM) waveform and image reconstruction procedure are suggested. The concept of pseudo InSAR measurements, interferogram, and differential interferogram generation is considered. Interferogram and differential interferogram are generated based on a surface model and InSAR measurements. Results of numerical experiments are provided.

**Keywords:** InSAR, geometry, signal modeling, SAR interferogram, SAR differential interferograms

### **1. Introduction**

Synthetic aperture radar (SAR) is a coherent microwave imaging instrument capable to provide for data all weather, day and night, guaranteeing global coverage surveillance. SAR interferometry is based on processing two or more complex valued SAR images obtained from different SAR positions [1–4]. The InSAR is a system intends for geophysical measurements and evaluation of topography, slopes, surface deformations (volcanoes, earthquakes, ice fields), glacier studies, vegetation growth, etc. The estimation of topographic height with essential accuracy is performed by the interferometric distance difference measured based on two SAR echoes from the same surface. Changes in topography (displacement), precise to a fraction of a radar wavelength, can be evaluated by differential interferogram generated by three or more successive complex SAR images [5, 6]. Demonstration of time series InSAR processing in Beijing using a small stack of Gaofen-3 differential interferograms is discussed in [7].

A general overview of the InSAR principles and the recent development of the advanced multi-track InSAR combination methodologies, which allow to discriminate the 3-D components of deformation processes and to follow their temporal evolution, are presented in [8]. The combination of global navigation satellite system (GNSS) and InSAR for future Australian datums is discussed in [9].

A high-precision DEM extraction method based on InSAR data and quality assessment of InSAR DEMs is suggested in [10, 11]. InSAR digital surface model (DSM) and time series analysis based on C-band Sentinel-1 TOPS data are presented in [12, 13]. DEM registration, alignment, and evaluation for SAR interferometry, deformation monitoring by ground-based SAR interferometry (GB-InSAR), a field

test in dam, and an improved approach to estimate large-gradient deformation using high-resolution TerraSAR-X data are discussed in [14–16]. *InSAR Time-Series Estimation of the Ionospheric Phase Delay: An Extension of the Split Range-Spectrum Technique* and InSAR data coherence estimation using 2D fast Fourier transform are performed in [17, 18].

*xn*ð Þ¼ *<sup>p</sup> <sup>x</sup><sup>n</sup>*

pulse; *V* ¼ *Vx*,*V <sup>y</sup>*,*Vz*

0, *y<sup>n</sup>*

<sup>0</sup>, and *zn*

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

*Rn*

**3. InSAR relief measurements**

*Rn* ij � � � � � � <sup>¼</sup> *<sup>R</sup><sup>m</sup>* ij � � � � � � 2 <sup>þ</sup> *<sup>B</sup>*<sup>2</sup>

The distance difference, *ΔR*mn

� � �, then

*<sup>θ</sup><sup>m</sup>*ij <sup>¼</sup> *<sup>α</sup>*mn <sup>þ</sup> arcsin *<sup>B</sup>*mn

ometric phase difference *ΔR*mn

*<sup>z</sup>*ij <sup>¼</sup> *hm* � *<sup>R</sup><sup>m</sup>*

*Rn* ij � � � � � � <sup>¼</sup> *Rm* ij � � � � � � <sup>þ</sup> *<sup>Δ</sup>R*mn ij � � �

**3**

where *x<sup>n</sup>*

<sup>0</sup> � *Vx*pT*p*, *<sup>y</sup>n*ð Þ¼ *<sup>p</sup> <sup>y</sup><sup>n</sup>*

velocity *V*. Modulus of the current distance vector *R<sup>n</sup>*

ijð Þ¼ *<sup>p</sup> xn*

ijð Þ *p* h i<sup>2</sup>

pass by calculation of the respective time delay and phase of the signal.

moment of imaging can be defined by the cosine's theorem, i.e.,

SAR position in the moment of imaging can be written as

*<sup>θ</sup><sup>m</sup>*ij <sup>¼</sup> *<sup>α</sup>*mn <sup>þ</sup> arcsin

ij � � �

� � � <sup>¼</sup> *<sup>λ</sup>*

ij � � �

ij *:* cos *<sup>α</sup>*mn <sup>þ</sup> arcsin *<sup>B</sup>*mn

� � � <sup>¼</sup> *Rn* ij � � � � � � � *<sup>R</sup><sup>m</sup>* ij � � � � �

2*Rm* ij

<sup>0</sup> � *<sup>V</sup> <sup>y</sup>*pT*p*, *<sup>z</sup>n*ð Þ¼ *<sup>p</sup> <sup>z</sup><sup>n</sup>*

the initial moment; *Tp* is the time repetition period; *p* is the number of the emitted

and *Vz* <sup>¼</sup> *<sup>V</sup>* cos *<sup>δ</sup>* are the components of vector velocity; cos *<sup>α</sup>*, cos *<sup>β</sup>*, and cos *<sup>δ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � cos <sup>2</sup>*<sup>α</sup>* � cos <sup>2</sup>*<sup>β</sup>* <sup>p</sup> are the guiding cosines; and *<sup>V</sup>* is the module of the vector

> <sup>þ</sup> *<sup>y</sup><sup>n</sup>* ijð Þ *p* h i<sup>2</sup>

Eq. (4) can be used to model a SAR signal from the *ij*-th pixel in the *n*-th SAR

The distances to ij-th pixel from SAR in *m*-th and *n*-th pass (*m* 6¼ *n*) at the

ij � � � � � � cos *<sup>π</sup>* <sup>2</sup> � *<sup>θ</sup>*½ Þ� *<sup>m</sup>*ij � *<sup>α</sup>*mn � � h i <sup>1</sup>

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

> *R<sup>m</sup>* ij � � � � � � 2 <sup>þ</sup> *<sup>B</sup>*<sup>2</sup>

ij � � � � � �

ij . In case *R<sup>m</sup>*

*Δϕ*mn ij 1 þ

� *<sup>λ</sup>* 2*πB*mn

mn � *Rn* ij � � � � � � 2

ij � � � � �

> *λ* 4*πR<sup>m</sup>* ij

ij " # ! , (8)

*Δϕ*mn ij *:* 1 þ

ij ( ) " # ! *:*

2*B*mn *R<sup>m</sup>* ij � � � � � �

mn � <sup>2</sup>*B*mn *Rm*

*<sup>z</sup>*ij <sup>¼</sup> *<sup>h</sup><sup>m</sup>* � *<sup>R</sup><sup>m</sup>*

<sup>2</sup>*<sup>π</sup> Δϕ*mn

� *<sup>λ</sup>* 2*πB*mn

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

� �*<sup>T</sup>* is the SAR vector velocity; *Vx* <sup>¼</sup> *<sup>V</sup>* cos *<sup>α</sup>*, *<sup>V</sup> <sup>y</sup>* <sup>¼</sup> *<sup>V</sup>* cos *<sup>β</sup>*,

h i<sup>2</sup> � �<sup>1</sup>

<sup>0</sup> are the SAR initial coordinates in the *n*-th pass, measured at

<sup>þ</sup> *<sup>z</sup><sup>n</sup>* ijð Þ *p*

ijð Þ *p* is defined by

2

<sup>0</sup> � *Vz*pT*p*, (3)

*:* (4)

2

, (6)

cos *θ<sup>m</sup>*ij*:* (7)

�, can be expressed by the interfer-

� can be measured, i.e.,

*Δϕ*mn

*λ* 4*πR<sup>m</sup>* ij

*Δϕ*mn

(9)

, (5)

In comparison with the results described in the aforementioned publications, the main goal of the present work is to suggest an analytical model of multi-pass InSAR geometry and derive analytical expressions of current distances between SAR's positions and individual pixels on the surface and to describe principal InSAR parameters: topographic height and topographic displacement from the position of InSAR modelling. The focus is on the two modelling approaches: first, by the definition of real scenario, geometry, and kinematics and SAR signal models and corresponding complex image reconstruction and interferogram and differential interferogram generation and, second, the process of pseudo SAR measurements and interferogram generation that is analytically described. Results of numerical experiments with real data are provided.

The rest of the chapter is organized as follows. In Section 2, 3D InSAR geometry and kinematics are analytically described. In Section 3 and Section 4, analytical expressions of InSAR relief measurements and relief displacement measurements are presented. In Section 5 and Section 6, SAR waveform, deterministic signal model, and image reconstruction algorithm are described. In Section 7, numerical results of InSAR modelling based on the geometry, kinematics, and signal models are provided. In Section 8 and Section 9, a pseudo InSAR modelling of geophysical measurements and numerical results are presented, respectively. Conclusion remarks are made in Section 10.

### **2. InSAR geometry and kinematics**

Assume a three-pass SAR system viewing three-dimensional (3-D) surface presented by discrete resolution elements, pixels. Each pixel is defined by the third coordinate *<sup>z</sup>*ij *<sup>x</sup>*ij, *<sup>y</sup>*ij � � in 3-D coordinate system Oxyz. Let A, B, and C, be the SAR positions of imaging. Between every SAR position, *C*<sup>2</sup> <sup>3</sup> ¼ 3 InSAR baselines can be drawn.

The basic geometric SAR characteristic is the time-dependent distance vector from SAR to each pixel on the surface in the *n*-th SAR pass at the *p*-th moment defined by

$$R\_{\vec{\text{ij}}}^{\mathfrak{n}}(p) = R^{\mathfrak{n}}(p) - R\_{\vec{\text{ij}}} = \left[ \varkappa\_{\vec{\text{ij}}}^{\mathfrak{n}}(p), \mathcal{y}\_{\vec{\text{ij}}}^{\mathfrak{n}}(p), \mathcal{z}\_{\vec{\text{ij}}}^{\mathfrak{n}}(p) \right]^T,\tag{1}$$

where *<sup>n</sup>* = 1–3 is the number of SAR passes and *<sup>R</sup><sup>n</sup>*ð Þ¼ *<sup>p</sup> <sup>R</sup>*<sup>0</sup>*<sup>n</sup>* <sup>þ</sup> *<sup>V</sup>:p:Tp* is the distance vector in the *n*-th SAR pass at the *p-*th moment, *R*<sup>0</sup>*<sup>n</sup>* is the initial distance vector in the *n*-th SAR pass, *R*ij is the constant distance vector of the ij- th pixel on the surface, and *x<sup>n</sup>* ijð Þ *<sup>p</sup>* , *<sup>y</sup><sup>n</sup>* ijð Þ *<sup>p</sup>* , and *<sup>z</sup><sup>n</sup>* ijð Þ *<sup>p</sup>* are the current coordinates of *<sup>R</sup><sup>n</sup>* ijð Þ *p* written by the expression.

$$\kappa\_{\vec{\imath}\vec{\jmath}}^{n}(p) = \kappa^{n}(p) - \kappa\_{\vec{\imath}\mathfrak{j}},\\\chi\_{\vec{\imath}\mathfrak{j}}^{n}(p) = \mathfrak{z}^{n}(p) - \chi\_{\vec{\imath}\mathfrak{j}},\\z\_{\vec{\imath}\mathfrak{j}}^{n}(p) = z^{n}(p) - z\_{\vec{\imath}\mathfrak{j}} \tag{2}$$

where *<sup>x</sup>*ij <sup>¼</sup> *<sup>i</sup>ΔX*, *<sup>y</sup>*ij <sup>¼</sup> *<sup>j</sup>ΔY*, and *<sup>z</sup>*ij <sup>¼</sup> *<sup>z</sup>*ij *<sup>x</sup>*ij, *<sup>y</sup>*ij � � is the pixel's discrete coordinates and *xn*ð Þ *<sup>p</sup>* , *<sup>y</sup><sup>n</sup>*ð Þ *<sup>p</sup>* , and *zn*ð Þ *<sup>p</sup>* are the SAR current coordinates in the *<sup>n</sup>*-th pass, defined by the following equation.

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

test in dam, and an improved approach to estimate large-gradient deformation using high-resolution TerraSAR-X data are discussed in [14–16]. *InSAR Time-Series Estimation of the Ionospheric Phase Delay: An Extension of the Split Range-Spectrum Technique* and InSAR data coherence estimation using 2D fast Fourier transform are performed

In comparison with the results described in the aforementioned publications, the main goal of the present work is to suggest an analytical model of multi-pass InSAR geometry and derive analytical expressions of current distances between SAR's positions and individual pixels on the surface and to describe principal InSAR parameters: topographic height and topographic displacement from the position of InSAR modelling. The focus is on the two modelling approaches: first, by the definition of real scenario, geometry, and kinematics and SAR signal models and corresponding complex image reconstruction and interferogram and differential interferogram generation and, second, the process of pseudo SAR measurements and interferogram generation that is analytically described. Results of numerical

The rest of the chapter is organized as follows. In Section 2, 3D InSAR geometry

Assume a three-pass SAR system viewing three-dimensional (3-D) surface presented

in 3-D coordinate system Oxyz. Let A, B, and C, be the SAR positions of

ijð Þ *<sup>p</sup>* , *<sup>y</sup><sup>n</sup>*

The basic geometric SAR characteristic is the time-dependent distance vector from SAR to each pixel on the surface in the *n*-th SAR pass at the *p*-th moment

where *<sup>n</sup>* = 1–3 is the number of SAR passes and *<sup>R</sup><sup>n</sup>*ð Þ¼ *<sup>p</sup> <sup>R</sup>*<sup>0</sup>*<sup>n</sup>* <sup>þ</sup> *<sup>V</sup>:p:Tp* is the distance vector in the *n*-th SAR pass at the *p-*th moment, *R*<sup>0</sup>*<sup>n</sup>* is the initial distance vector in the *n*-th SAR pass, *R*ij is the constant distance vector of the ij- th pixel on

ijð Þ¼ *<sup>p</sup> <sup>y</sup><sup>n</sup>*ð Þ� *<sup>p</sup> <sup>y</sup>*ij, *zn*

and *xn*ð Þ *<sup>p</sup>* , *<sup>y</sup><sup>n</sup>*ð Þ *<sup>p</sup>* , and *zn*ð Þ *<sup>p</sup>* are the SAR current coordinates in the *<sup>n</sup>*-th pass, defined

� �

<sup>3</sup> ¼ 3 InSAR baselines can be drawn.

ijð Þ *<sup>p</sup>* , *<sup>z</sup><sup>n</sup>* ijð Þ *p*

ijð Þ *<sup>p</sup>* are the current coordinates of *<sup>R</sup><sup>n</sup>*

, (1)

ijð Þ¼ *<sup>p</sup> <sup>z</sup><sup>n</sup>*ð Þ� *<sup>p</sup> <sup>z</sup>*ij (2)

is the pixel's discrete coordinates

ijð Þ *p* written

h i*<sup>T</sup>*

by discrete resolution elements, pixels. Each pixel is defined by the third coordinate

ijð Þ¼ *<sup>p</sup> <sup>R</sup><sup>n</sup>*ð Þ� *<sup>p</sup> <sup>R</sup>*ij <sup>¼</sup> *xn*

ijð Þ *<sup>p</sup>* , and *<sup>z</sup><sup>n</sup>*

and kinematics are analytically described. In Section 3 and Section 4, analytical expressions of InSAR relief measurements and relief displacement measurements are presented. In Section 5 and Section 6, SAR waveform, deterministic signal model, and image reconstruction algorithm are described. In Section 7, numerical results of InSAR modelling based on the geometry, kinematics, and signal models are provided. In Section 8 and Section 9, a pseudo InSAR modelling of geophysical measurements and numerical results are presented, respectively. Conclusion

in [17, 18].

experiments with real data are provided.

*Geographic Information Systems in Geospatial Intelligence*

remarks are made in Section 10.

*z*ij *x*ij, *y*ij � �

defined by

the surface, and *x<sup>n</sup>*

by the expression.

**2**

*xn*

by the following equation.

**2. InSAR geometry and kinematics**

imaging. Between every SAR position, *C*<sup>2</sup>

*Rn*

ijð Þ *<sup>p</sup>* , *<sup>y</sup><sup>n</sup>*

ijð Þ¼ *<sup>p</sup> <sup>x</sup><sup>n</sup>*ð Þ� *<sup>p</sup> <sup>x</sup>*ij, *<sup>y</sup><sup>n</sup>*

where *x*ij ¼ *iΔX*, *y*ij ¼ *jΔY*, and *z*ij ¼ *z*ij *x*ij, *y*ij

$$\mathbf{x}^{n}(p) = \mathbf{x}\_{0}^{n} - V\_{\mathbf{x}} \mathbf{p} \mathbf{T}\_{p}, \mathbf{y}^{n}(p) = \mathbf{y}\_{0}^{n} - V\_{\mathbf{y}} \mathbf{p} \mathbf{T}\_{p}, \ \mathbf{z}^{n}(p) = \mathbf{z}\_{0}^{n} - V\_{\mathbf{z}} \mathbf{p} \mathbf{T}\_{p},\tag{3}$$

where *x<sup>n</sup>* 0, *y<sup>n</sup>* <sup>0</sup>, and *zn* <sup>0</sup> are the SAR initial coordinates in the *n*-th pass, measured at the initial moment; *Tp* is the time repetition period; *p* is the number of the emitted pulse; *V* ¼ *Vx*,*V <sup>y</sup>*,*Vz* � �*<sup>T</sup>* is the SAR vector velocity; *Vx* <sup>¼</sup> *<sup>V</sup>* cos *<sup>α</sup>*, *<sup>V</sup> <sup>y</sup>* <sup>¼</sup> *<sup>V</sup>* cos *<sup>β</sup>*, and *Vz* <sup>¼</sup> *<sup>V</sup>* cos *<sup>δ</sup>* are the components of vector velocity; cos *<sup>α</sup>*, cos *<sup>β</sup>*, and cos *<sup>δ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � cos <sup>2</sup>*<sup>α</sup>* � cos <sup>2</sup>*<sup>β</sup>* <sup>p</sup> are the guiding cosines; and *<sup>V</sup>* is the module of the vector velocity *V*. Modulus of the current distance vector *R<sup>n</sup>* ijð Þ *p* is defined by

$$R\_{\vec{\eta}}^{\pi}(p) = \left\{ \left[ \varkappa\_{\vec{\eta}}^{\pi}(p) \right]^2 + \left[ \mathcal{Y}\_{\vec{\eta}}^{\pi}(p) \right]^2 + \left[ z\_{\vec{\eta}}^{\pi}(p) \right]^2 \right\}^{\frac{1}{2}}.\tag{4}$$

Eq. (4) can be used to model a SAR signal from the *ij*-th pixel in the *n*-th SAR pass by calculation of the respective time delay and phase of the signal.
