**2. Retrieval of surface displacement components through DInSAR and pixel-offset-based techniques**

In this section, the basic principles of differential SAR interferometry (DInSAR) for the detection of Earth's surface displacement are introduced. Moreover, the potential integration of DInSAR and pixel-offset-based measurement for the measurement of large deformation signals occurring in the case of large ruptures of Earth's crust is discussed.

### **2.1. Basics of InSAR technique**

One of the major applications of the SAR technology is represented by the SAR interferometry (InSAR) technique [1, 2], which relies on the measurements of the phase difference between two complex-valued SAR images gathered from a satellite/airborne platform at different times and from different orbital positions, so as to measure the geomorphological characteristics of the ground (such as the topography height and the modifications of the surface over time, due to earthquakes, volcano eruptions, or other geophysical phenomena). Historically, InSAR has been used for the measurement of ground topography. To understand how InSAR works, let us consider the imaging geometry shown in **Figure 1**, and let us suppose the first SAR image (referred to as master image) is acquired from the orbital position labeled to as M, and the second image (i.e., the slave image) is taken from the orbital position labeled to as S, located at a distance *b* (usually referred to as baseline) from M. By applying simple geometrical rules, it is possible to uniquely identify the location of each ground targets on the image as well as to get an estimate of their heights (namely, *z*) relative to the reference plane. As evident by inspection of **Figure 1**, if a same target (namely, T) is observed form two orbital positions (master and slave), with two corresponding ranges its distances from the first (namely, *r*) and the second position (namely, *r* + *δr*) can be correctly measured and the target height can be unambiguously determined. This is obtained by finding out the solution of the following system of two equations (see **Figure 1**):

and monitoring of Earth's crust modifications over time, both in academic and operative frameworks. DInSAR is mainly used to detect the temporal evolution of surface deformation through the generation of long-lasting displacement time-series. Several multi-temporal advanced DInSAR algorithms have been proposed in the literature [3–9]. At the present days, the availability of large archives of SAR images collected by several radar instruments operating with different wavelengths, and with complementary side-looking angle geometries has posed the problem to effectively combine the information associated with the different SAR datasets. In particular, the combination of multiple-platform line-of-sight (LOS) displacement time series can improve the ability to retrieve the three-dimensional (East-West, Up-Down, and North-South) components of the on-going surface displacement phenomena. Thus, it allows overcoming the main limitation of DInSAR, which is able only to measure the radar LOS projection of the displacement. This research field is of particular interest; and in the recent years, a few solutions have been proposed [10–22] based on the effective combination of multiple-orbit/multiple-angle DInSAR-based measurements, as well as on merging of DInSAR data products with external measurements (e.g., derived from processing GPS

In this chapter, first, the basic rationale of the multi-temporal DInSAR techniques for the generation of Earth's surface displacements maps (see Section 2) is summarized; and then, the characteristics of the principal combination techniques for multi-track/multiangle/multi-sensor SAR data recently proposed in the literature are discussed. In Section 3, the focus will be on the algorithm referred to as minimum acceleration combination technique (MinA) [23], which does not require the simultaneous process of very large sequence of differential SAR interferograms. The algorithm consists of a straightforward post-processing stage that involves the analysis of sequences of independently processed (potentially, also with different DInSAR toolboxes) multiple-platform LOS displacement time-series. Noteworthy, the adopted InSAR-combination scheme can be used in a wide context. Real SAR datasets are exploited to demonstrate the validity of the presented algorithm. Experimental results will be shown in Section 4. Conclusions and further perspec-

**2. Retrieval of surface displacement components through DInSAR and** 

In this section, the basic principles of differential SAR interferometry (DInSAR) for the detection of Earth's surface displacement are introduced. Moreover, the potential integration of DInSAR and pixel-offset-based measurement for the measurement of large deformation sig-

One of the major applications of the SAR technology is represented by the SAR interferometry (InSAR) technique [1, 2], which relies on the measurements of the phase difference between

nals occurring in the case of large ruptures of Earth's crust is discussed.

data).

56 Time Series Analysis and Applications

tives will be provided in Section 5.

**pixel-offset-based techniques**

**2.1. Basics of InSAR technique**

$$(r+\delta r)^2 = r^2 + b^2 - 2 \text{ r } b \sin(\theta - \alpha) \tag{1}$$

$$z = H - r\cos\theta\tag{2}$$

where *δr* and *δr* + *r* represent the radar ranges from each antenna to the target point, ϑ represents the radar side-looking angle, *α* the angle of the baseline relative to the horizontal, and *z* is the scatterer height above the flat-earth reference. *H* is the height of the sensor above the reference surface and b is the distance between the antennas, which is referred to as baseline. The ability in successfully reconstructing the unknown topography (z) is strictly dependent on the capability to precisely measure the slant-range difference *δr*, which represents one of the known terms of the system of Eqs. (1) and (2).

**Figure 1.** SAR interferometric configuration. The dashed lines show that radar signal paths for the first interferogram pair formed by antennas at M and S.

To understand how InSAR works, let us consider again the imaging geometry, let us assume the radar system has infinite bandwidth; under this condition the master and slave complex-valued SAR images (pixel-by-pixel) can be mathematically represented as follows:

$$\hat{\gamma}\_1 = \gamma\_1 \exp\left[-j\frac{4\pi}{\lambda}\mathbf{r}\right] \tag{3}$$

$$\dot{\gamma}\_2 = \gamma\_2 \exp\left[-j\frac{4\pi}{\lambda}(\mathbf{r} + \delta\mathbf{r})\right] \tag{4}$$

where γ<sup>1</sup> and γ<sup>2</sup> are the complex reflectivity functions of the master and slave scene, respectively, and λ denotes the operative radar wavelength. An interferogram is formed on a pixelby-pixel basis by starting from two complex-valued co-registered SAR images, as outlined in the following. For each pixel, the phase difference between the two SAR images is computed, by multiplying the first image (master) by the complex conjugate of the second image (slave) and, then, by extracting the relevant phase.

From Eqs. (3) and (4), the interferometric phase is obtained, as follows:

$$\arg\left[\hat{\gamma}\_1 \,\hat{\gamma}\_2^\*\right] = \arg\left[\mathbf{j}\frac{4\pi}{\lambda}\,\mathbf{\delta}\mathbf{r}\right] + \arg\left[\boldsymbol{\gamma}\_1\right] - \arg\left[\boldsymbol{\gamma}\_2\right] \tag{5}$$

where the asterisk denotes the complex conjugate operation, and the symbol  arg[∙] refers to the operator that extracts the phase of a complex number, which is restricted to the [− π, π] interval. However, by assuming the scattering mechanism on the ground is the same (arg[γ<sup>1</sup> ] = arg[γ<sup>2</sup> ]) between the two passages of the sensor over the illuminated area (mutually coherent observations), the measured phase difference ψ ~k (where k identifies a specific interferometric pair of a multiple baseline configuration) depends upon on the range difference δr, only:

$$\tilde{\psi}^{\;k} = \arg\left[ \exp\left( \mathbf{j} \frac{4\pi}{\lambda} \,\delta \mathbf{r} \right) \right] \tag{6}$$

The observed interferometic phase ψ ~k is 2π ambiguous, and the obtained image is called an interferogram. Since the ambiguity of the phase, which is measured modulo 2π, the information on range difference δr is retrieved from the interferogram by applying the phase unwrapping operation [24, 25], thus estimating the inherent absolute interferometric phase ψk , given by:

$$
\psi^{\mu} = \frac{4\pi}{\lambda} \,\delta \mathbf{r} \tag{7}
$$

By considering the standard interferometric configuration depicted in **Figure 1** and a few mathematical calculations detailed in [26], it is possible to relate the computed interferometric phase difference to the (unknown) height topography as:

$$\psi^{\mathbf{k}} \approx \psi\_0^{\mathbf{k}} + \frac{\partial \psi^{\mathbf{k}}}{\partial \mathbf{z}} \Delta \mathbf{z} = -\frac{4\pi}{\lambda} \mathbf{b}^{\mathbf{k}} \sin(\mathfrak{S}\_0^{\mathbf{k}} - \alpha^{\mathbf{k}}) - \frac{4\pi}{\lambda} \frac{\mathbf{b}\_{\perp}^{\mathbf{k}}}{\mathbf{r} \sin \mathfrak{S}\_0^{\mathbf{k}}} \tag{8}$$

where ∆z is the surface height variation above the flat-earth reference plane, ϑ<sup>0</sup> k is the sidelooking angle of each point in the image, assuming zero local height, b<sup>⊥</sup> <sup>k</sup> <sup>=</sup> <sup>b</sup><sup>k</sup> cos(ϑ<sup>0</sup> <sup>k</sup> − α<sup>k</sup> ) represents the projection of the baseline in the direction perpendicular to the line-of-sight from the radar to the target. The first term ψ<sup>0</sup> <sup>k</sup> <sup>=</sup> \_\_\_ 4π <sup>λ</sup> <sup>b</sup><sup>k</sup> sin(ϑ<sup>0</sup> <sup>k</sup> − α<sup>k</sup> ) in Eq. (8) accounts for the phase contribution corresponding to the flat-Earth (z = 0), which is the term that is present in the absence of any height elevation on the ground. From Eq. (8), it is clear that the sensitivity of the InSAR measurement can be improved by increasing the baseline. However, the perpendicular baseline, cannot exceed the limiting value (critical baseline) for which the variation of the phase difference across one single ground range resolution element is 2π.

To understand how InSAR works, let us consider again the imaging geometry, let us assume the radar system has infinite bandwidth; under this condition the master and slave complex-valued SAR images (pixel-by-pixel) can be mathematically represented as

<sup>1</sup> <sup>=</sup> <sup>γ</sup><sup>1</sup> exp[−j \_\_\_ 4π

tively, and λ denotes the operative radar wavelength. An interferogram is formed on a pixelby-pixel basis by starting from two complex-valued co-registered SAR images, as outlined in the following. For each pixel, the phase difference between the two SAR images is computed, by multiplying the first image (master) by the complex conjugate of the second image (slave)

where the asterisk denotes the complex conjugate operation, and the symbol  arg[∙] refers to the operator that extracts the phase of a complex number, which is restricted to the [− π, π] interval. However, by assuming the scattering mechanism on the ground is the same

ferometric pair of a multiple baseline configuration) depends upon on the range difference

<sup>=</sup> arg[exp(<sup>j</sup> \_\_\_ 4π

interferogram. Since the ambiguity of the phase, which is measured modulo 2π, the information on range difference δr is retrieved from the interferogram by applying the phase unwrapping operation [24, 25], thus estimating the inherent absolute interferometric phase ψk

By considering the standard interferometric configuration depicted in **Figure 1** and a few mathematical calculations detailed in [26], it is possible to relate the computed interferometric

<sup>λ</sup> bk sin(ϑ<sup>0</sup>

<sup>k</sup> − αk

) <sup>−</sup> \_\_\_ 4π λ

<sup>λ</sup> (<sup>r</sup> <sup>+</sup> δr)

are the complex reflectivity functions of the master and slave scene, respec-

]) between the two passages of the sensor over the illuminated area (mutually

~k

<sup>2</sup> <sup>=</sup> <sup>γ</sup><sup>2</sup> exp[−j \_\_\_ 4π

From Eqs. (3) and (4), the interferometric phase is obtained, as follows:

~ <sup>k</sup>

~k

] <sup>=</sup> arg[<sup>j</sup> \_\_\_ 4π

1 γ̂ 2 ∗

coherent observations), the measured phase difference ψ

<sup>λ</sup> r] (3)

<sup>λ</sup> δr] + arg[γ1] − arg[γ2] (5)

(where k identifies a specific inter-

, given

<sup>k</sup> (8)

<sup>λ</sup> δr)] (6)

<sup>λ</sup> δr (7)

b⊥ k \_\_\_\_\_\_\_\_\_\_\_ r sinϑ<sup>0</sup>

is 2π ambiguous, and the obtained image is called an

] (4)

follows:

where γ<sup>1</sup>

(arg[γ<sup>1</sup>

δr, only:

by:

γ̂

γ̂

and, then, by extracting the relevant phase.

and γ<sup>2</sup>

58 Time Series Analysis and Applications

arg[γ̂

ψ

The observed interferometic phase ψ

ψk ≈ ψ<sup>0</sup>

ψk = \_\_\_ 4π

phase difference to the (unknown) height topography as:

<sup>k</sup> + ∂ψk \_\_\_

∂z <sup>∆</sup> <sup>z</sup> <sup>=</sup> <sup>−</sup>\_\_\_ 4π

] = arg[γ<sup>2</sup>

SAR interferometry nowadays is mostly used for the detection of surface changes occurring over time. In such a case, when a slight change of the surface across the two SAR acquisition times occurs in the imaged scene, an additive term arises in the interferometric phase, which is associated with the radar line-of-sight (LOS) component of the surface displacement, in addition to a phase term that depends on topography. By the inspection of the imaging geometry depicted in **Figure 2**, we get:

$$
\Delta\boldsymbol{\psi}^{\rm k} = \frac{\partial\boldsymbol{\psi}^{\rm k}}{\partial\mathbf{z}}\Delta\mathbf{z} + \frac{\partial\boldsymbol{\psi}^{\rm k}}{\partial\mathbf{d}^{\rm k}\_{\rm Los}}\Delta\,\mathbf{d}^{\rm k}\_{\rm Los} = -\frac{4\pi}{\lambda}\frac{\mathbf{b}^{\rm k}\_{\perp}}{\mathbf{r}\sin\mathfrak{S}^{\rm k}\_{\rm o}}\Delta\,\mathbf{z} + \frac{4\pi}{\lambda}\Delta\,\mathbf{d}^{\rm k}\_{\rm Los} \tag{9}
$$

where ∆ <sup>d</sup>Los k represents the projection of the surface displacement vector onto LOS (range) direction pertinent to the kth interferometric pair. Note that the presence of the flat earth phase contribution was neglected, for the sake of convenience.

In order to measure the interferometric phase term related to the surface displacement, it is thus essential to remove the interferometric phase contribution pertinent to the topography in Eq. (9). Specifically, a differential SAR interferogram is formed by synthesizing the topographic phase from an available digital elevation model (DEM) of the area (using the so

**Figure 2.** Differential SAR interferometry imaging geometry.

so-called back-geocoding process) and by subtracting, pixel-by-pixel, these synthetic fringes to the corresponding InSAR phase, thus leaving only the terms associated with the displacement. Accordingly, computed differential SAR interferograms can be expressed as:

$$
\Delta \psi^{\mathbf{k}} = \Delta \psi^{\mathbf{k}}\_{\text{disp}} + \Delta \psi^{\mathbf{k}}\_{\text{topo}} + \Delta \psi^{\mathbf{k}}\_{\text{orb}} + \Delta \psi^{\mathbf{k}}\_{\text{prop}} + \Delta \psi^{\mathbf{k}}\_{\text{scat}} \tag{10}
$$

where k ∈ {1, .., M} specifies the considered interferometric pair (master/slave);

∆ ψdisp <sup>k</sup> <sup>=</sup> \_\_\_ 4π <sup>λ</sup> ∆ dLos k represents possible displacement of the scatterer between observations, where dLOS is the projection of the relevant displacement vector on the line of sight;

∆ ψtopo <sup>k</sup> <sup>=</sup> \_\_\_ 4π λ b⊥ k \_\_\_\_\_\_\_\_\_\_\_ r sinϑ<sup>0</sup> <sup>k</sup> <sup>∆</sup>~z represents the residual-topography induced phase due to a non-perfect knowledge of the actual height profile (i.e., the DEM errors ∆~z);

∆ ψorb k accounts for possible inaccurate orbital information;

∆ ψprop k denotes the phase components due to the variation of propagation conditions (pertinent to the change in the atmospheric and ionospheric dielectric constant) between the two master/ slave acquisitions;

∆ ψscat <sup>k</sup> accounts for change in scattering behavior [13].

#### **2.2. Combination of ascending/descending displacement maps**

Availability of InSAR results computed from SAR data obtained from ascending and descending orbits allows also for the separation of the East-West (E-W) and the vertical components of the detected deformation. In particular, for all the pixels that are common to both radar geometries, the sum and the difference of LOS-projected deformations computed for the ascending and the descending orbits are calculated. In particular, the sum of the ascending/ descending LOS-projected displacement measurements is related to the vertical component of the ground deformation, whereas the difference of the ascending/descending components gives an estimate for the E-W component of the deformation. Also, because modern spaceborne radar systems are mounted on-board satellites that fly nearly polar orbits, the North-South (N-S) component of the deformation cannot be reliably measured. To explain the rationale for the retrieval of the E-W and the vertical components of the deformation, the following assumptions are made: (i) ascending and descending radar LOS directions (*dLOS* (*Asc*) and *dLOS* (*Desc*) , respectively) lay wholly in the east–z plane and (ii) the sensor side-looking angle is approximately the same along the ascending and descending orbits. Both these assumptions are acceptable. If we refer to the same homologous pixel imaged by the ascending and descending orbits, the E-W and Up-Down components of the measured surface deformation can be estimated from the ascending/descending LOS measurement (e.g., the LOS-projected rates of deformation) as follows:

$$d\_{\rm LOS}^{(Loo)} \approx \frac{d\_{\rm LOS}^{(Doo)} - d\_{\rm LOS}^{(Loo)}}{2 \sin 8} \tag{11}$$

$$d\_{\rm LOS}^{\{l\_{\rm I}\}} \approx \frac{d\_{\rm LOS}^{\{l\_{\rm LOS}\}} + d\_{\rm LOS}^{\{l\_{\rm LOS}\}}}{2 \cos \theta} \tag{12}$$

Geometric scheme to interpret the deformation component is portrayed in **Figure 3**. Finally, it is worth emphasizing that a fundamental advantage of InSAR technology, with respect to o GPS networks, resides in its dense sampling grid of the displacement field.

## **2.3. The advanced multi-temporal small baseline subset (SBAS) technique**

so-called back-geocoding process) and by subtracting, pixel-by-pixel, these synthetic fringes to the corresponding InSAR phase, thus leaving only the terms associated with the displace-

<sup>k</sup> + ∆ψorb

k denotes the phase components due to the variation of propagation conditions (pertinent to the change in the atmospheric and ionospheric dielectric constant) between the two master/

Availability of InSAR results computed from SAR data obtained from ascending and descending orbits allows also for the separation of the East-West (E-W) and the vertical components of the detected deformation. In particular, for all the pixels that are common to both radar geometries, the sum and the difference of LOS-projected deformations computed for the ascending and the descending orbits are calculated. In particular, the sum of the ascending/ descending LOS-projected displacement measurements is related to the vertical component of the ground deformation, whereas the difference of the ascending/descending components gives an estimate for the E-W component of the deformation. Also, because modern spaceborne radar systems are mounted on-board satellites that fly nearly polar orbits, the North-South (N-S) component of the deformation cannot be reliably measured. To explain the rationale for the retrieval of the E-W and the vertical components of the deformation, the following assumptions are made: (i) ascending and descending radar LOS directions (*dLOS*

, respectively) lay wholly in the east–z plane and (ii) the sensor side-looking angle

<sup>2</sup> sin*<sup>ϑ</sup>* (11)

<sup>2</sup> cos*<sup>ϑ</sup>* (12)

is approximately the same along the ascending and descending orbits. Both these assumptions are acceptable. If we refer to the same homologous pixel imaged by the ascending and descending orbits, the E-W and Up-Down components of the measured surface deformation can be estimated from the ascending/descending LOS measurement (e.g., the LOS-projected

> (*East*) <sup>≈</sup> *dLOS* (*Desc*) − *dLOS* (*Asc*) \_\_\_\_\_\_\_\_\_

(*Up*) <sup>≈</sup> *dLOS* (*Desc*) + *dLOS* (*Asc*) \_\_\_\_\_\_\_\_\_

<sup>k</sup> + ∆ψprop

k represents possible displacement of the scatterer between observations, where

<sup>k</sup> <sup>∆</sup>~z represents the residual-topography induced phase due to a non-perfect

<sup>k</sup> + ∆ψscat

<sup>k</sup> (10)

(*Asc*)

ment. Accordingly, computed differential SAR interferograms can be expressed as:

<sup>k</sup> + ∆ψtopo

where k ∈ {1, .., M} specifies the considered interferometric pair (master/slave);

dLOS is the projection of the relevant displacement vector on the line of sight;

knowledge of the actual height profile (i.e., the DEM errors ∆~z);

**2.2. Combination of ascending/descending displacement maps**

k accounts for possible inaccurate orbital information;

<sup>k</sup> accounts for change in scattering behavior [13].

∆ψk = ∆ψdisp

∆ ψdisp <sup>k</sup> <sup>=</sup> \_\_\_ 4π

∆ ψtopo <sup>k</sup> <sup>=</sup> \_\_\_ 4π λ b⊥ k \_\_\_\_\_\_\_\_\_\_\_ r sinϑ<sup>0</sup>

∆ ψorb

∆ ψprop

∆ ψscat

and *dLOS* (*Desc*)

rates of deformation) as follows:

*dLOS*

*dLOS*

<sup>λ</sup> ∆ dLos

60 Time Series Analysis and Applications

slave acquisitions;

In the following, the small baseline subset (SBAS) algorithm is presented. SBAS was developed in 2002 by a team of researchers from National Council Research (CNR) of Italy [19]. To introduce the rationale of the algorithm, let us consider a set of *Q* single-look-complex (SLC) SAR images acquired by a radar instrument over a certain area of interest. One of these images is selected and assumed as the reference (master) image, with respect to which all available SAR images are properly co-registered. The set is characterized by the corresponding acquisition times {t<sup>1</sup> , …, tQ} and the inherent perpendicular baselines vector {b⊥<sup>1</sup> , …, b⊥<sup>Q</sup>} estimated with respect to the reference image. Application of the standard SBAS technique starts with the generation of a set of *M* of small baseline multi-look (differential) interferograms. On these interferograms, the retrieval of the original (unwrapped) phase signals from the modulo-2π measured (wrapped) phases is carried out. The expression of the *k*th interferometric phase is as follows:

**Figure 3.** Combination scheme of ascending and descending displacement fields.

$$
\Delta\boldsymbol{\psi}^{k} = \frac{4\pi}{\lambda} \Delta \operatorname{d}^{k}\_{\text{LOS}} - \frac{4\pi}{\lambda} \frac{\operatorname{b}^{k}\_{\perp}}{\operatorname{r}\sin\mathfrak{S}^{k}\_{\text{0}}} \Delta \operatorname{Z} + \Delta\psi^{k}\_{\text{orb}} + \Delta\psi^{k}\_{\text{prop}} + \Delta\psi^{k}\_{\text{scat}} \tag{13}
$$

The system of Eq. (13) can be re-organized in a matrix form as:

$$\mathbf{A} \cdot \Psi = \Delta \Psi \tag{14}$$

wherein A is the incidence-like matrix directly related to the selected set of small baseline (SB) interferometric data-pairs. Let us now manipulate the system of Eq. (14) to replace the unknown phase vector *Ψ* with the mean phase velocities between adjacent time acquisitions (see [6]). Accordingly, the new unknowns become:

$$\mathbf{v} = \begin{bmatrix} v\_1 = \frac{\Psi^a - \Psi^a}{t\_2 - t\_1}, \dots, v\_{Q^2} = \frac{\Psi^{Q \cdot 1} - \Psi^{Q \cdot 2}}{t\_{Q^{\cdot 1}} - t\_{Q^{\cdot 2}}} \end{bmatrix}^T \tag{15}$$

and the system (14) can be re-formulated as follows:

$$\mathbf{B} \cdot \mathbf{v} = \Delta \Psi \tag{16}$$

wherein **B** is the incidence matrix of the linear transformation in Eq. (15), whose detailed expression can be found in [6]. It is worth remarking that, depending upon the selection of small baseline interferometric data pairs, it is possible that SAR data could be separated into some independent subsets separated one another by large baselines. Mathematically, this leads the rank deficiency of matrix **B** and, accordingly, the system (16) can have infinite possible solutions. In order to figure it out a solution among the infinite ones, the singular value decomposition (SVD) method is applied. This allows us to evaluate the pseudo-inverse of the matrix **B**, which gives the minimum norm least-squares (LS) solution of the system (16). In this context, the minimum-norm constraint on the velocity vector **v** allows mitigating the presence of temporal discontinuities in the final result, so as to guarantee a physically sound solution. Finally, an additional integration step is necessary to compute the solution from the estimated vector **v**. After solving the system (16) an estimate of the spurious terms due to the presence of some residual topographic artifacts in the generated interferograms is usually performed [6]. Atmospheric phase screen (APS) is also estimated and filtered out [6]. The quality of retrieved LOS time-series is finally evaluated pixel-by-pixel by calculating the values of the temporal coherence factor, defined in [26].

#### **2.4. Pixel-offset (PO) technique for the estimation of large rupture deformations**

In areas where large and/or rapid deformation phenomena occur, the exploitation of the differential interferograms, thus the generation of displacement time series, can be however strongly limited by the presence of very high fringe rates, which in turn introduce additional difficulties in the phase unwrapping step and may lead to significant misregistration errors. A pictorial representation of how the interferometric phase degrades as the displacement amount increases is given in **Figure 4**. Nevertheless, the information on the occurred displacements might be preserved in the amplitude of the investigated data pair, considering the offset of the image's pixels in range and azimuth directions.

Generation of Earth's Surface Three-Dimensional (3-D) Displacement Time-Series by Multiple-Platform SAR Data http://dx.doi.org/10.5772/intechopen.71329 63

∆ψk = \_\_\_ 4π

62 Time Series Analysis and Applications

**<sup>v</sup>** <sup>=</sup> [

<sup>λ</sup> ∆ dLOS

The system of Eq. (13) can be re-organized in a matrix form as:

(see [6]). Accordingly, the new unknowns become:

and the system (14) can be re-formulated as follows:

ues of the temporal coherence factor, defined in [26].

of the image's pixels in range and azimuth directions.

**2.4. Pixel-offset (PO) technique for the estimation of large rupture deformations**

In areas where large and/or rapid deformation phenomena occur, the exploitation of the differential interferograms, thus the generation of displacement time series, can be however strongly limited by the presence of very high fringe rates, which in turn introduce additional difficulties in the phase unwrapping step and may lead to significant misregistration errors. A pictorial representation of how the interferometric phase degrades as the displacement amount increases is given in **Figure 4**. Nevertheless, the information on the occurred displacements might be preserved in the amplitude of the investigated data pair, considering the offset

<sup>k</sup> − \_\_\_ 4π λ

*<sup>v</sup>*<sup>1</sup> <sup>=</sup> *<sup>Ψ</sup>*<sup>2</sup> <sup>−</sup> *<sup>Ψ</sup>*<sup>1</sup> \_\_\_\_\_\_ *t* <sup>2</sup> − *t* 1

b⊥ k \_\_\_\_\_\_\_\_\_\_\_ r sinϑ<sup>0</sup> <sup>k</sup> ∆ z

**A** ⋅ Ψ = ∆Ψ (14)

wherein A is the incidence-like matrix directly related to the selected set of small baseline (SB) interferometric data-pairs. Let us now manipulate the system of Eq. (14) to replace the unknown phase vector *Ψ* with the mean phase velocities between adjacent time acquisitions

**B** ⋅ **v** = ∆Ψ (16)

wherein **B** is the incidence matrix of the linear transformation in Eq. (15), whose detailed expression can be found in [6]. It is worth remarking that, depending upon the selection of small baseline interferometric data pairs, it is possible that SAR data could be separated into some independent subsets separated one another by large baselines. Mathematically, this leads the rank deficiency of matrix **B** and, accordingly, the system (16) can have infinite possible solutions. In order to figure it out a solution among the infinite ones, the singular value decomposition (SVD) method is applied. This allows us to evaluate the pseudo-inverse of the matrix **B**, which gives the minimum norm least-squares (LS) solution of the system (16). In this context, the minimum-norm constraint on the velocity vector **v** allows mitigating the presence of temporal discontinuities in the final result, so as to guarantee a physically sound solution. Finally, an additional integration step is necessary to compute the solution from the estimated vector **v**. After solving the system (16) an estimate of the spurious terms due to the presence of some residual topographic artifacts in the generated interferograms is usually performed [6]. Atmospheric phase screen (APS) is also estimated and filtered out [6]. The quality of retrieved LOS time-series is finally evaluated pixel-by-pixel by calculating the val-

<sup>~</sup> <sup>+</sup> ∆ψorb

, …,*vQ*−<sup>2</sup> <sup>=</sup> *<sup>Ψ</sup><sup>Q</sup>*−<sup>1</sup> <sup>−</sup> *<sup>Ψ</sup><sup>Q</sup>*−<sup>2</sup> \_\_\_\_\_\_\_\_ *t <sup>Q</sup>*−<sup>1</sup> − *t <sup>Q</sup>*−<sup>2</sup> ] *T*

<sup>k</sup> + ∆ψprop

<sup>k</sup> + ∆ψscat

<sup>k</sup> (13)

(15)

**Figure 4.** Pictorial representation of the effects due to different amounts of deformations. (a) Plot of the deformation time-series showing the temporal evolution of displacement in the Sierra Negra Caldera, as computed from a sequence of ENVISAT DInSAR interferograms. (b)–(d) Differential interferograms relevant to different deformation regimes and time epochs: when the deformation rate is low the information conveyed in the phase is fully preserved (b). As deformation rate increases the fringe spatial frequency increases (c), and in the occurrence of the large rupture of terrain the fringe rate is so high that the corresponding interferometric phase (d) is completely corrupted by decorrelation noise, thus making the use of phase not effective. This figure is a re-adaptation of **Figure 1** originally presented in [27].

Pixel-offset (PO) is a technique that attempts to find the same distinctive features within subscenes of two images relevant to the same target area. In remote sensing, this is usually performed by considering either the Fourier shift theorem [28], or normalized cross-correlation (NCC) algorithms [29]. In SAR applications, PO allows co-registering image pairs or, while already co-registered, identifying the residual shifts related to the motion of distinctive features with accuracies in the order of 1/20th of pixel.

In this scenario, the availability of a sequence of full resolution SAR data pairs, already coregistered, was assumed. For this purpose, the NCC algorithm, which is widely used for SAR images, is applied. This step, carried out on across-track and along-track directions, provides for each pixel an estimation of range and azimuth shifts, finally leading to two offset maps for each data pair. The performed NCC analysis might be evaluated through an estimator of the "goodness" of the retrieved offsets.

At this stage, in order to obtain the corresponding time series, the SBAS strategy is applied (see Section 2.3) by substituting the amplitude-driven information to the phase-driven displacement measurements. This operation was performed to the sequences of smoothed range and azimuth pixel-offset maps. Hence, the small baseline constraint in the data pair's selection, implicit in the SBAS strategy, is convenient in order to maximize the amount of the exploitable pixels. Notice that this algorithm, which is known in the literature to as pixel offset SBAS (POSBAS) [27], is particularly attractive in the case of large deformation, because it allows us to have an estimate of the North-South deformation components, with accuracy in the order of some centimeter, whereas (as said before) the information related to North-South displacement is almost absent in the conventional DInSAR interferometric phase, being the sensors' flight trajectories almost parallel to the North-South direction. The results of the performed experiments are shown in Section 4.

## **3. Minimum acceleration (MinA) algorithm**

In this section, the minimum acceleration (MinA) combination algorithm [23] used for the extraction of displacement time-series of the Up-Down, East-West, and North-South components is detailed. To introduce the right mathematical framework, let us assume the availability of K independent sets of multiple-platform SAR data collected at ordered times *<sup>t</sup>*(*j*) <sup>=</sup> [ *t* 0 (*j*) , *t* 1 (*j*) , …,*t Qj* −1 (*j*) ] *<sup>T</sup>* <sup>∀</sup>*<sup>j</sup>* <sup>=</sup> 1, …,*K* over the same area on the ground, consisting of *Qj* distinctive time epochs, respectively. The MinA algorithm [23] requires the preliminary generation from each single SAR data-track of the inherent LOS-projected deformation time-series. This task can be achieved by independently applying either the SBAS technique [6] or other alternative multi-temporal DInSAR approaches [4–9] to the available K SAR datasets. During this preliminary stage, the residual topography as well as the atmospheric phase screen (APS) artifacts corrupting differential SAR interferograms might be estimated and successfully filtered out from the generated LOS-projected displacement time-series [7–16]. The so-obtained LOS-projected time-series of deformation along with other ancillary information, such as the maps of temporal coherence (quantifying the goodness of obtained time-series) as well as the LOS mean deformation velocity maps are geocoded to a common spatial grid of points where to apply the subsequent combination stage. During this preliminary stage the location of high coherent targets is also identified. Henceforth, let **d**(*j*) <sup>=</sup> [ *d*0 (*j*) , *d*1 (*j*) , …,*dQj* −1 (*j*) ] *<sup>T</sup>* ∀*j* = 1, …,*K* be the geocoded LOS-projected deformation time-series relevant to a generic pixel P that belongs to the group of high-coherent pixels common to all the available K SAR datasets.

LOS-projected time-series of deformation are expressed with respect to the instants *t* 0 (*j*) , ∀*j* = 1, …,*K*, which are singularly taken as reference for each dataset, that is to say *d*0 (*j*) = 0, ∀*j* = 1, …,*K*. Let us now describe the algorithm works, and *Q* = ∑ *j*=1 *K Qj* be the total number of the available SAR images collected at the "whole" ordered times *<sup>T</sup>* <sup>=</sup> <sup>∪</sup>*<sup>j</sup>*=1 *<sup>K</sup> <sup>t</sup>*(*j*) <sup>=</sup> [*T*<sup>0</sup> , *T*1 , …,*TQ*−1] *T* .

Let us start by observing that a generic LOS-projected deformation measurement, namely *dLOS*, can be related to its inherent 3-D components, namely [*dE* − *<sup>W</sup>*, *dU* − *<sup>D</sup>*, *dN* − *<sup>S</sup>* ] *<sup>T</sup>*, as [18, 22]:

$$d\_{\rm LOS} = d \cdot \hat{\mathbf{i}}\_{\rm LOS} = \sin \theta \cos \varphi \, d\_{\rm East-Wat} - \cos \theta \, d\_{\rm tlp-Down} + \sin \theta \sin \varphi \, d\_{\rm Net-South} \tag{17}$$

where **i**̂ *LOS* is the LOS-direction versor. Note that θ and ϕ represent the radar side-looking and the satellite heading angles, respectively; the imaging geometries for ascending/descending data-tracks are shown in **Figure 5**. Extending to our case what originally proposed in [6] and subsequently adapted in [22], let us relate the available LOS deformations *d*(*j*) (*P*),  *j* = 1, …, *K*

Generation of Earth's Surface Three-Dimensional (3-D) Displacement Time-Series by Multiple-Platform SAR Data http://dx.doi.org/10.5772/intechopen.71329 65

**Figure 5.** SAR data acquisition geometries for descending (a) and ascending (b) orbits, respectively.

exploitable pixels. Notice that this algorithm, which is known in the literature to as pixel offset SBAS (POSBAS) [27], is particularly attractive in the case of large deformation, because it allows us to have an estimate of the North-South deformation components, with accuracy in the order of some centimeter, whereas (as said before) the information related to North-South displacement is almost absent in the conventional DInSAR interferometric phase, being the sensors' flight trajectories almost parallel to the North-South direction. The results of the

In this section, the minimum acceleration (MinA) combination algorithm [23] used for the extraction of displacement time-series of the Up-Down, East-West, and North-South components is detailed. To introduce the right mathematical framework, let us assume the availability of K independent sets of multiple-platform SAR data collected at ordered times

*<sup>T</sup>* <sup>∀</sup>*<sup>j</sup>* <sup>=</sup> 1, …,*K* over the same area on the ground, consisting of *Qj*

*d*0 (*j*) , *d*1 (*j*) , …,*dQj* −1 (*j*) ]

*LOS* = sin*ϑ* cosϕ *dEast*−*West* − cos*ϑ dUp*−*Down* + sin*ϑ* sinϕ *dNorth*−*South* (17)

*j*=1 *K Qj*

time epochs, respectively. The MinA algorithm [23] requires the preliminary generation from each single SAR data-track of the inherent LOS-projected deformation time-series. This task can be achieved by independently applying either the SBAS technique [6] or other alternative multi-temporal DInSAR approaches [4–9] to the available K SAR datasets. During this preliminary stage, the residual topography as well as the atmospheric phase screen (APS) artifacts corrupting differential SAR interferograms might be estimated and successfully filtered out from the generated LOS-projected displacement time-series [7–16]. The so-obtained LOS-projected time-series of deformation along with other ancillary information, such as the maps of temporal coherence (quantifying the goodness of obtained time-series) as well as the LOS mean deformation velocity maps are geocoded to a common spatial grid of points where to apply the subsequent combination stage. During this preliminary stage the location

geocoded LOS-projected deformation time-series relevant to a generic pixel P that belongs to

LOS-projected time-series of deformation are expressed with respect to the instants

Let us start by observing that a generic LOS-projected deformation measurement, namely

*LOS* is the LOS-direction versor. Note that θ and ϕ represent the radar side-looking and the satellite heading angles, respectively; the imaging geometries for ascending/descending data-tracks are shown in **Figure 5**. Extending to our case what originally proposed in [6] and

, ∀*j* = 1, …,*K*, which are singularly taken as reference for each dataset, that is to say

the group of high-coherent pixels common to all the available K SAR datasets.

(*j*) = 0, ∀*j* = 1, …,*K*. Let us now describe the algorithm works, and *Q* = ∑

of the available SAR images collected at the "whole" ordered times *<sup>T</sup>* <sup>=</sup> <sup>∪</sup>*<sup>j</sup>*=1

*dLOS*, can be related to its inherent 3-D components, namely [*dE* − *<sup>W</sup>*, *dU* − *<sup>D</sup>*, *dN* − *<sup>S</sup>*

subsequently adapted in [22], let us relate the available LOS deformations *d*(*j*)

distinctive

*<sup>T</sup>* ∀*j* = 1, …,*K* be the

be the total number

, *T*1 , …,*TQ*−1] *T* .

*<sup>T</sup>*, as [18, 22]:

(*P*),  *j* = 1, …, *K*

*<sup>K</sup> <sup>t</sup>*(*j*) <sup>=</sup> [*T*<sup>0</sup>

]

performed experiments are shown in Section 4.

64 Time Series Analysis and Applications

*<sup>t</sup>*(*j*) <sup>=</sup> [ *t* 0 (*j*) , *t* 1 (*j*) , …,*t Qj* −1 (*j*) ]

*t* 0 (*j*)

*d*0

where **i**̂

*dLOS* = *d* ⋅ *i*

^

**3. Minimum acceleration (MinA) algorithm**

of high coherent targets is also identified. Henceforth, let **d**(*j*) <sup>=</sup> [

(for each high coherent pixel) to their unknown 3-D components. This leads writing a system of Q-K independent linear equations with respect to the M = 3(Q-1) unknowns representing the East-West (E-W), Up-Down (U-D), and North-South (N-S) deformation velocities components between adjacent time-acquisitions, namely *VE* = [ *VE*<sup>1</sup> , *VE*<sup>2</sup> , ..., *VEQ*−<sup>1</sup> ]*<sup>T</sup>*, *VU* = [ *VU*<sup>1</sup> , *VU*<sup>2</sup> , ..., *VUQ*−<sup>1</sup> ]*T* and *VN* = [ *VN*<sup>1</sup> , *VN*<sup>2</sup> , ..., *VNQ*−<sup>1</sup> ]*<sup>T</sup>*. This system of linear equations can be expressed using matrix formalism as follows

$$\mathbf{B} \cdot \begin{bmatrix} \mathbf{V}\_{\mathrm{E}} \\ \mathbf{V}\_{\mathrm{U}} \\ \mathbf{V}\_{\mathrm{N}} \end{bmatrix} = \begin{bmatrix} \Gamma^{1} (d^{(1)} - d\_{0}^{(1)}) \\ \Gamma^{2} (d^{(2)} - d\_{0}^{(2)}) \\ \cdots \\ \Gamma^{k} (d^{(\infty)} - d\_{0}^{(\infty)}) \end{bmatrix} = \mathbf{\bar{d}} \tag{18}$$

where *Γ<sup>j</sup> j* = 1, …, *K* are the values of temporal coherence associated to the *K* different datasets (representing a quality factor of the obtained LOS displacement time-series) and **B** is the incidence-like matrix of the linear transformation that converts LOS-projected measurements into their inherent 3-D components. It is defined by taking into account (1) as:

$$\mathbf{B} = \begin{bmatrix} \mathbf{B}^{(1)} \varGamma^{(1)} \sin \mathfrak{S}^{(1)} \cos \mathfrak{q}^{(1)} & -\mathbf{B}^{(1)} \varGamma^{(1)} \cos \mathfrak{S}^{(1)} & \mathbf{B}^{(1)} \varGamma^{(1)} \sin \mathfrak{S}^{(1)} \sin \mathfrak{q}^{(1)} \\\\ \mathbf{B}^{(2)} \varGamma^{(2)} \sin \mathfrak{S}^{(2)} \cos \mathfrak{q}^{(2)} & -\mathbf{B}^{(2)} \varGamma^{(2)} \cos \mathfrak{S}^{(2)} & \mathbf{B}^{(2)} \varGamma^{(2)} \sin \mathfrak{S}^{(2)} \sin \mathfrak{q}^{(2)} \\\\ \cdots & \cdots & \cdots \\\\ \mathbf{B}^{(\ell)} \varGamma^{(\ell)} \sin \mathfrak{S}^{(\ell)} \cos \mathfrak{q}^{(\ell)} & -\mathbf{B}^{(\ell)} \varGamma^{(\ell)} \cos \mathfrak{S}^{(\ell)} & \mathbf{B}^{(\ell)} \varGamma^{(\ell)} \sin \mathfrak{S}^{(\ell)} \sin \mathfrak{q}^{(\ell)} \end{bmatrix} \tag{19}$$

wherein **B**(*j*) , *j* = 1, …, *K* is the jth incidence-like matrix of the linear transformation that relates displacement time-series with velocity deformation rates between consecutive time intervals for the jth SAR data set. Derivation of that incidence-like matrix is detailed in the paper [6]. Note that this matrix is the same as the one used in SBAS and discussed in the previous section.

The system (18) and (19) has fewer linear independent equations (*Q-K*) than unknowns (*M*), thus it is an under-determined system that does not admit a unique solution. The matrix **B**  of the system has singular values that gradually decay to zero, thus rendering any solution

much sensitive to noise level corrupting the vector **d**˜. It represents a canonical example of a linear discrete ill-posed problem whose meaningful solution can be obtained by replacing the "original" linear system (18) and (19) by a nearby system that is less sensitive to perturbations of the right-hand side of the system, and considers the solution of this new system as a good approximation of the original one. This operation is known as regularization and can be performed using truncated singular value decomposition (TSVD) [30], Tikhonov regularization [31], maximum entropy principle [32]. TSVD practically consists in decomposing the matrix **B** with SVD and truncating (putting to zeros) the small singular values, in such a way that they do not dominate the solution leading to spurious oscillations. In turn, Tikhonov regularization consists in replacing the solution of the system (2) and (3) by the following minimization problem:

$$\min\_{\mathbf{V}\in\mathbb{R}} \left\{ \left\| \mathbf{B}\cdot\mathbf{V} - \mathbf{\bar{d}} \right\|\_{2} + \alpha^{2} \left\| \mathbf{V} \right\|\_{2} \right\} \tag{20}$$

for a suitable value of the regularization parameter α, which can effectively be found using (for instance) L-curve method [31]. The goal of L-curve is to search for a regularization parameter α for which the solution has an optimal balance between the minimization of residual norm ‖**<sup>B</sup>** <sup>⋅</sup> **<sup>V</sup>** <sup>−</sup> **<sup>d</sup>**˜ ‖ <sup>2</sup> and the "weight" of the minimum-norm velocity regularization ‖**V**‖<sup>2</sup> .

A similar regularized problem was proposed within the MSBAS algorithm [22], even though in that case the relevant system of linear equations was derived from a sequence of unwrapped multiple-tracks differential SAR interferograms. In the case of MinA algorithm, the regularization problem is achieved differently, by adding to the original system (18) and (19) other equations imposing the condition that the (unknown) 3-D (E-W, U-D, N-S) displacement time-series are with minimum curvature, that is to say the velocity deformation differences (for all the 3-D components) between consecutive time intervals is minimal. Such conditions can formally be expressed by adding to Eq. (2) the following set of 3(*Q*-2) additional equations:

$$\begin{cases} V\_{E\_{\shortrightarrow}} - V\_{E\_{\shortrightarrow}} \ i = 1, \dots, \mathbb{Q} - 2 \\ V\_{\shortrightarrow} - V\_{\shortrightarrow} \ i = 1, \dots, \mathbb{Q} - 2 \\ V\_{N\_{\shortrightarrow}} - V\_{N\_{\shortparallel}} \ i = 1, \dots, \mathbb{Q} - 2 \end{cases} \tag{21}$$

Accordingly, the regularized system of linear equations becomes:

$$
\begin{bmatrix} \mathbf{B} \\ \mathbf{C} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{V}\_{\text{E}} \\ \mathbf{V}\_{\text{U}} \\ \mathbf{V}\_{\text{N}} \end{bmatrix} = \begin{bmatrix} \mathbf{d} \\ \mathbf{0} \end{bmatrix} \tag{22}
$$

where **C** is the incidence-like matrix related to the minimum-acceleration-regularization linear transformation. The solution of Eq. (6) is finally obtained in the LS sense by applying Truncated SVD to the matrix <sup>=</sup> ( **B <sup>C</sup>**). Once the problem in Eqs. (21) and (22) is solved, the Generation of Earth's Surface Three-Dimensional (3-D) Displacement Time-Series by Multiple-Platform SAR Data http://dx.doi.org/10.5772/intechopen.71329 67

much sensitive to noise level corrupting the vector **d**˜. It represents a canonical example of a linear discrete ill-posed problem whose meaningful solution can be obtained by replacing the "original" linear system (18) and (19) by a nearby system that is less sensitive to perturbations of the right-hand side of the system, and considers the solution of this new system as a good approximation of the original one. This operation is known as regularization and can be performed using truncated singular value decomposition (TSVD) [30], Tikhonov regularization [31], maximum entropy principle [32]. TSVD practically consists in decomposing the matrix **B** with SVD and truncating (putting to zeros) the small singular values, in such a way that they do not dominate the solution leading to spurious oscillations. In turn, Tikhonov regularization consists in replacing the solution of the system (2) and (3) by the following

for a suitable value of the regularization parameter α, which can effectively be found using (for instance) L-curve method [31]. The goal of L-curve is to search for a regularization parameter α for which the solution has an optimal balance between the mini-

A similar regularized problem was proposed within the MSBAS algorithm [22], even though in that case the relevant system of linear equations was derived from a sequence of unwrapped multiple-tracks differential SAR interferograms. In the case of MinA algorithm, the regularization problem is achieved differently, by adding to the original system (18) and (19) other equations imposing the condition that the (unknown) 3-D (E-W, U-D, N-S) displacement time-series are with minimum curvature, that is to say the velocity deformation differences (for all the 3-D components) between consecutive time intervals is minimal. Such conditions can formally be expressed by adding to Eq. (2) the following set of 3(*Q*-2)

*i* = 1, …,*Q* − 2

*i* = 1, …,*Q* − 2

*<sup>i</sup>* <sup>=</sup> 1, …,*<sup>Q</sup>* <sup>−</sup> 2

⎧ ⎪ ⎨ ⎪ ⎩

Accordingly, the regularized system of linear equations becomes:

*VEi*+1 − *VEi*

*VUi*+<sup>1</sup> − *VUi*

*VNi*+<sup>1</sup> − *VNi*

> **B <sup>C</sup>**] <sup>⋅</sup>

**B**

⎡ ⎢ ⎣

**VE VU VN**

where **C** is the incidence-like matrix related to the minimum-acceleration-regularization linear transformation. The solution of Eq. (6) is finally obtained in the LS sense by applying

⎤ ⎥ ⎦ = [ **d**˜

*<sup>V</sup>*∈ℝ {‖**<sup>B</sup>** <sup>⋅</sup> **<sup>V</sup>** <sup>−</sup> **<sup>d</sup>**˜ ‖<sup>2</sup> <sup>+</sup> *<sup>α</sup>*<sup>2</sup> ‖**V**‖2} (20)

and the "weight" of the minimum-norm velocity

**<sup>0</sup>**] (22)

**<sup>C</sup>**). Once the problem in Eqs. (21) and (22) is solved, the

(21)

minimization problem:

66 Time Series Analysis and Applications

regularization ‖**V**‖<sup>2</sup>

additional equations:

min

mization of residual norm ‖**<sup>B</sup>** <sup>⋅</sup> **<sup>V</sup>** <sup>−</sup> **<sup>d</sup>**˜ ‖ <sup>2</sup>

[

Truncated SVD to the matrix <sup>=</sup> (

.

**Figure 6.** MinA diagram block. K independent sets of LOS measurements of the surface ground displacement, as obtained by processing data from K multi-angle/multi-sensor SAR systems also using independent DInSAR toolboxes, are combined (taking into account the associated quality reconstruction maps). Combination is based on the application of minimum-acceleration constraints on the achievable Up-Down, East-West, and North-South time-series of deformation. Use of external data is helpful for better constraining the solutions.

3-D velocity deformation components are independently time-integrated to recover the relevant 3-D displacement time-series.

As earlier said, a similar system of equations was also derived within the MSBAS algorithm [22] but, at variance with MinA, it relies on searching for a minimum-velocity-norm (MN) solution considering the Tikhonov regularization. Moreover, MSBAS requires the simultaneous inversion of several (a few hundreds or more) unwrapped interferograms for the retrieval of the 3-D components of deformation, and the achieved time-series were however still affected by possible topographic and atmospheric artifacts (although considering several hundreds of interferograms various sources of noise and related artifacts are averaged and only partly filtered out) that need to be subsequently filtered out in a post-processing phase. Accordingly, even though the adopted combination strategy shares some similarities with the MSBAS algorithm, MinA does not require simultaneous processing of multi-platform/multiangle SAR datasets and can be applied with no restrictions at all on the method (permanent scatterers and/or small baseline-oriented) used for the retrieval of LOS DInSAR time-series, making its field of applicability extremely wide.

Noteworthy, the MinA algorithm can also be extended to include azimuth- and range-pixeloffset (AZPO and RGPO) time-series, as computed using the PO-SBAS method (or alternative solutions). This case is very suitable when large deformation phenomena have to be monitored, being the accuracy of these methods is on the order of 10 cm (or larger). In this case, the strategy here adopted can be extended using AZPO and RGPO time-series of deformation, instead of the LOS deformation measurements, and applying the minimum-acceleration (MA) regularization.

The diagram block of the MinA algorithm is shown in **Figure 6**.
