# On the Analysis of the Phase Unwrapping Process in a D-InSAR Stack with Special Focus on the Estimation of a Motion Model

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## Abstract

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## 1. Introduction

#### 1.1. The Phase Unwrapping Problem-State of the Art

_{1}-norm minimization problem of the deviation between the estimated and unknown phase gradients of the unwrapped phases under the above assumption that both differ by an integer multiple of $2\pi $. This assumption prevents the errors from spreading effects [16]. Considering the structure, the problem could be transformed into a network, searching for the minimum cost flow, defined by the unknown phase ambiguity factor.

#### 1.2. Outline

## 2. Phase Unwrapping in 2D Using the MCF Approach

_{1}-norm for the error criterion

_{1}-norm problem

_{1}-norm phase unwrapping problem with integer variables. Therefore, the problem is considered in some more detail.

**b**consists of integer values due to the rounding operator. Moreover, the matrix ${\mathit{B}}_{\mathrm{spatial}}^{T}$ is totally unimodular. With these properties the solution of the LP relaxation is already an integer solution [21]. Consequently, the constrained weighted L

_{1}-norm phase unwrapping problem described by (9) to (11) can be solved with a standard LP solver without considering the integer condition. This can be done for example with the simplex method [22] or the interior point method [23].

#### 2.1. Solution as Network Flow Problem

**b**expressed by the right-hand side vector of the triangle constraint. The potential of the node belonging to the outer face is represented as the sum of the outlying arcs of the primal graph. There is a flow along each arc $\u2206{x}_{kl}$ of the dual graph represented by the phase ambiguity factor ${k}_{\u2206{x}_{kl}}^{\u2206{t}_{\alpha \beta}}$ which is weighted with a cost value ${p}_{\u2206{x}_{kl}}^{\u2206{t}_{\alpha \beta}}$. Hence, the minimum cost flow problem is defined as

**l**and an upper

**u**boundary for the flow. Assuming integer values for the costs and the lower and upper capacities, the values for the variables will be integers. Consequently, the phase unwrapping problem can be solved using network flow algorithms like the out-of-kilter algorithm [25] or the relaxation method [26]. Since this formulation depends on the dual graph, it is only possible if the primal graph is planar. Thus, redundant arcs cannot be used in this formulation.

#### 2.2. Solution as Parametric Adjustment

_{1}-norm phase unwrapping problem

_{1}-norm problem can be solved with an LP solver neglecting the integer constraints. The solution of the LP relaxation is already an integer.

_{1}-norm problem can also be seen as a network flow problem. Therefore, the primal graph in Figure 1 is regarded. Compared to the previously described formulation as a constrained L

_{1}-norm problem, this formulation has the advantage that the generation of the dual graph is not necessary. It depends on the primal graph and therefore, redundant arcs which cut each other are possible.

_{1}-norm problem is not recommended compared to the constrained formulations.

## 3. Phase Unwrapping in 3D Using the Extended MCF Approach

**M**.

## 4. Analysis

#### 4.1. Simulation Scenario

#### 4.2. EMCF vs. MCF

#### 4.3. Estimation of the Motion Model

**M**from (20) which depends on the parameters ${v}_{\u2206{x}_{kl}}$ and $\u2206{h}_{\u2206{x}_{kl}}$. The EPC function is continuous with values between zero and one. A value of zero means that model and observation do not fit and a value of one means that model and observation fit optimal. With a higher noise level the maximum value of the function decreases and it gets difficult to find the global maximum. However, the idea is to define this (${v}_{\u2206{x}_{kl}},\u2206{h}_{\u2206{x}_{kl}}$)-pair as optimal which maximizes the EPC, equal to [1].

#### Application to Simulated Data

## 5. Application to Real Data

^{©}. The unwrapped phase gradients are included in RSG again for the final SBAS analysis which results in a mean deformation velocity map shown in Figure 10.

## 6. Discussion: Analysis for Different Scenarios

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Spatial Delaunay triangulation for an example of four pixels. To ensure that the solution is rotation-free, the sum of the unwrapped phase gradients in one triangle has to be zero.

**Figure 2.**Dual graph of Figure 1. The constraints of the primal problem define the new nodes with node potential

**b**. The arcs connect these nodes by cutting the arcs of the primal graph.

**Figure 3.**Temporal triangulation related to the small baseline subset (SBAS) method. The white dots represent the individual SAR scenes at the individual acquisition times with corresponding orthogonal spatial baseline relating to the master scene. The black lines represent the interferograms.

**Figure 4.**Temporal relationship for one spatial arc. The thickly marked line is analysed in time. To ensure that the solution is rotation-free in the temporal plane, the sum of the three phase gradients has to be zero.

**Figure 5.**Workflow for generating simulated interferometric phases. The simulated SAR images consist of a ground settlement, a topographical error and a noise part. On this basis, interferograms are generated, which in turn are provided with noise.

**Figure 6.**Coherent pixels of simulated ground settlement depression for one interferogram with a temporal baseline of $\u2206{t}_{\alpha \beta}$ = 315 days and an orthogonal spatial baseline of $\u2206{b}_{\perp ,\alpha \beta}$ = −217.645 m.

**Figure 7.**Percentage of correctly unwrapped phase gradients depending on the temporal baseline. The results of the extended minimum cost flow (EMCF) algorithm are shown as black bars. The gray bars represent the results of the MCF algorithm without considering the temporal information.

**Figure 8.**Cost values and corresponding ensemble phase coherence (EPC) values for two simulated spatial gradients. The black triangles in the left figures symbolize the optimal point resulting from the iterative search process where the costs of the temporal linear program (LP) are minimized. For comparison, the black star is the reference solution. The other symbols in the right figures represent the optimum point from maximizing the EPC. The gray square is obtained using the Nelder–Mead method, the gray triangle using the grid search process and the light gray point using simulated annealing. The black point represents the starting solution.

**Figure 9.**Percentage of correctly unwrapped phase gradients depending on the temporal baseline. The motion model is estimated with different approaches. The black bars show the results of the conventional way by minimizing the costs of the temporal LP in an iterative search process. The dark gray bars show the results using the Nelder–Mead method to estimate the motion model by maximizing the EPC, the middle gray bars when using the grid search method and the light gray bars when using simulated annealing.

**Figure 10.**Mean deformation velocity map of the Lower-Rhine-Embayment based on ERS 1/2 data from May 1992 to December 2000. The deformation model is estimated using the conventional and the alternative approach. The highlighted test regions 1 to 6 are examined in more detail as a time series in Figure 11.

**Figure 11.**Deformation time series of five points lying in each of the six highlighted test region shown in Figure 10. The black triangles are the results using the conventional approach and the light gray points are the results using the alternative way. For comparison the black squares indicate the data from the closest leveling point.

**Figure 12.**Percentage of correctly unwrapped phase gradients depending on the noise level. The motion model is estimated with different approaches, see different gray levels already described in Figure 9. The additional white bars show the results using a modified search algorithm combining simulated annealing and the Nelder–Mead method.

**Figure 13.**Relative frequency of occuring EPC values using simulated annealing. The darker the color of the histogram, the higher the noise level. It varies from $0.1$ rad to $0.6$ rad.

**Figure 14.**Cost values and corresponding EPC values for one spatial arc of the simulated D-InSAR stack with a noise level of 0.6 rad. The black triangle in the left figure shows the results of the conventional way. The black star is the reference solution. The gray square in the right figure shows the results using the Nelder–Mead method for maximizing the EPC, the gray triangle when using the grid search method, the light gray point when using simulated annealing and the white triangle when using a modified search algorithm combining simulated annealing and the Nelder–Mead method. The black point represents the starting solution which is necessary for some algorithms.

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**MDPI and ACS Style**

Esch, C.; Köhler, J.; Gutjahr, K.; Schuh, W.-D. On the Analysis of the Phase Unwrapping Process in a D-InSAR Stack with Special Focus on the Estimation of a Motion Model. *Remote Sens.* **2019**, *11*, 2295.
https://doi.org/10.3390/rs11192295

**AMA Style**

Esch C, Köhler J, Gutjahr K, Schuh W-D. On the Analysis of the Phase Unwrapping Process in a D-InSAR Stack with Special Focus on the Estimation of a Motion Model. *Remote Sensing*. 2019; 11(19):2295.
https://doi.org/10.3390/rs11192295

**Chicago/Turabian Style**

Esch, Christina, Joël Köhler, Karlheinz Gutjahr, and Wolf-Dieter Schuh. 2019. "On the Analysis of the Phase Unwrapping Process in a D-InSAR Stack with Special Focus on the Estimation of a Motion Model" *Remote Sensing* 11, no. 19: 2295.
https://doi.org/10.3390/rs11192295