# One-Step Three-Dimensional Phase Unwrapping Approach Based on Small Baseline Subset Interferograms

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## Abstract

**:**

## 1. Introduction

#### 1.1. General Aspects and Motivation

#### 1.2. Scientific Context

#### 1.3. Outline

## 2. State-of-the-Art Extended Minimum Cost Flow Approach

#### 2.1. Problem Formulation

#### 2.2. Temporal Phase Unwrapping

**M**.

#### 2.3. Spatial Phase Unwrapping

## 3. Modified Extended Minimum Cost Flow Approach

#### 3.1. Estimation of the Motion Model Parameters

**M**from Equation (3) which depends on the parameters ${v}_{\Delta {x}_{kl}}$ and $\Delta {h}_{\Delta {x}_{kl}}$. The EPC function is continuous with values between zero and one, see the exemplary shown EPC function for one simulated phase gradient in Figure 2. A value of zero means that model and observation do not fit and a value of one corresponds to an optimal fit. It can be seen that the function shows a clear maximum. The idea is to find this maximum of the EPC function and to use the corresponding parameters as optimal motion model parameters to calculate the modified observations, equal to [3]. Especially with noisy data, where the global maximum is not as pronounced, a modified algorithm that combines simulated annealing and the local Nelder–Mead search algorithm is the best way to find the maximum [22]. If the maximum found by simulated annealing falls below a certain threshold, it is assumed that the motion model cannot be estimated reliably enough. Instead of using a possibly too extreme motion model, which is very unreliable and may lead to phase unwrapping errors, the maximum around an appropriate approximate value is locally searched using Nelder–Mead. In contrast to the conventional iterative approach, this alternative approach has the advantage that the motion model can be estimated independently of the temporal phase unwrapping. Consequently, the temporal phase unwrapping only needs to be solved once for each phase gradient. This offers an improved run time. Furthermore, it is no longer necessary to define a discrete search space for the parameters. More information on the analysis and comparison of the alternative and conventional methods can be found in [22].

#### 3.2. Choice of the Weights

## 4. One-Step Three-Dimensional Phase Unwrapping Approach

#### 4.1. Problem Formulation

#### 4.2. Temporal Inconsistency

#### 4.3. Application to Simulated Data

#### 4.3.1. Simulation Scenario

#### 4.3.2. Results of Closed Loop Simulation

## 5. Application to Real Data

#### 5.1. Data Basis

#### 5.2. Temporal Consistency

#### 5.3. Smoothness in Space

#### 5.4. Single Pixel Evaluation

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Temporal triangulation related to the Small BAseline Subset (SBAS) method with a set ${\mathcal{M}}^{\prime}$ of SAR images and a set ${\mathcal{N}}^{\prime}$ of interferograms. 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. The spatial and temporal baseline information comes from the ERS 1/2 data from May 1992 to December 2000.

**Figure 2.**EPC function exemplary for one phase gradient depending on the error of the scene topography and the deformation velocity variation.

**Figure 3.**Exemplary structure of global constraint matrix for a small D-InSAR stack consisting of six interferograms between which a total of three temporal constraints can be generated and five spatial gradients, between which two spatial constraints must be fulfilled.

**Figure 4.**Exemplary structure of a global constraint matrix with slack variables for a small D-InSAR stack consisting of six interferograms between which a total of three temporal constraints can be generated and five spatial gradients between which two spatial constraints must be fulfilled.

**Figure 5.**Percentage of correctly unwrapped phase gradients depending on the noise level added per SAR image. The green bars show the results of the conventional EMCF approach, the dark blue bars are the results using the alternative EMCF algorithm and the orange bars are the results using the one-step three-dimensional approach.

**Figure 6.**Mean deformation velocity map of the Lower-Rhine-Embayment based on ERS 1/2 data from May 1992 to December 2000 for pixels with a coherence value greater 0.7 in at least 95% of interferograms and estimated using the conventional EMCF approach. The highlighted test regions 1 to 3 are examined in more detail as time series in Figure 8.

**Figure 7.**Difference of RMS values of the deformation time series estimated with (

**a**) the conventional EMCF approach minus RMS of one-step three-dimensional approach and (

**b**) the alternative EMCF approach minus RMS of one-step three-dimensional approach.

**Figure 8.**Deformation time series of five pixels lying in each of the three highlighted test regions shown in Figure 6. (

**a**) shows the pixels in Mohnheim at the Rhine, (

**b**) in Koslar and (

**c**) in Odenkirchen. The results using the conventional EMCF approach are shown as green triangles, using the alternative EMCF as dark blue points and using the one-step three-dimensional approach as orange points. The black squares indicate the data from the closest leveling point.

**Table 1.**Total number of temporal inconsistencies for different noise levels. In addition to the reference, the number is listed for different phase unwrapping methods.

Method | Total Number of Temporal Inconsistencies for Different Noise Levels [rad] | ||||
---|---|---|---|---|---|

0.2 | 0.4 | 0.6 | 0.8 | 0.9 | |

reference | 59 | 59 | 59 | 59 | 59 |

conventional EMCF approach | 23 | 23 | 46 | 30,705 | 142,090 |

alternative EMCF approach | 33 | 19 | 88 | 28,336 | 65,599 |

one-step three-dimensional approach | 14 | 14 | 14 | 14 | 14 |

**Table 2.**Number of total temporal inconsistencies for ERS 1/2 data. The number is listed for different phase unwrapping methods after the first and the second phase unwrapping step in the SBAS workflow.

Method | Total Number of Temporal Inconsistencies after | |
---|---|---|

1st Phase Unwrapping | 2nd Phase Unwrapping | |

conventional EMCF approach | 16,309 | 2947 |

alternative EMCF approach | 15,711 | 2810 |

one-step three-dimensional approach | 10,491 | 938 |

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**MDPI and ACS Style**

Esch, C.; Köhler, J.; Gutjahr, K.; Schuh, W.-D.
One-Step Three-Dimensional Phase Unwrapping Approach Based on Small Baseline Subset Interferograms. *Remote Sens.* **2020**, *12*, 1473.
https://doi.org/10.3390/rs12091473

**AMA Style**

Esch C, Köhler J, Gutjahr K, Schuh W-D.
One-Step Three-Dimensional Phase Unwrapping Approach Based on Small Baseline Subset Interferograms. *Remote Sensing*. 2020; 12(9):1473.
https://doi.org/10.3390/rs12091473

**Chicago/Turabian Style**

Esch, Christina, Joël Köhler, Karlheinz Gutjahr, and Wolf-Dieter Schuh.
2020. "One-Step Three-Dimensional Phase Unwrapping Approach Based on Small Baseline Subset Interferograms" *Remote Sensing* 12, no. 9: 1473.
https://doi.org/10.3390/rs12091473