# IGS-CMAES: A Two-Stage Optimization for Ground Deformation and DEM Error Estimation in Time Series InSAR Data

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Modelling for InSAR Phase

#### 2.2. Proposed Method

#### 2.2.1. Definition of Optimization Problem

#### 2.2.2. CMAES

#### 2.2.3. Iterative Grid Search (IGS)

**search cost**) is an essential metric for assessing a stochastic optimizer’s performance. Especially when higher precision is required, dense grid search is extremely insufficient. Moreover, from an optimization point of view, determining whether the true global optima reached is a fundamental challenge as a stochastic optimization algorithm. As a consistency check, the algorithm can be run from several different random starting points to ensure the result of each run converges to the global optimal. Rather than using randomly selected starting points, in this work, we hypothesize that using a low-precision grid search can select a set of initial candidate solutions that are potentially close to the global optima. To support our assumption, we plot the loss landscapes with different sampling steps sizes by setting $\kappa =[1,3,5,8]$ as shown in Figure 4.

#### 2.2.4. IGS-CMAES

Algorithm 1: IGS-CMAES for deformation rate and DEM error estimation. |

## 3. Results

#### 3.1. Experimental Setup

#### 3.1.1. Simulation Data

#### 3.1.2. Real-World Data

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of phase jump and compassion real and imaginary MSE (RI-MSE) and mean square error (MSE) objective functions.

**Figure 2.**Illustration of proposed two-stage iterative grid search with covariance matrix adaptation evolution strategy (IGS-CMAES) method.

**Figure 3.**Loss landscape of objective function RI-MSE (Equation (7)) with two different real-world spatial and temporal baselines. Site (

**A**) and Site (

**B**) are two different real-world sites.

**Figure 4.**The landscape of loss function. Each row represents the loss landscapes that corresponds to real–world baselines with synthetic 0 deformation and 0 DEM error, and each column shows the results under different step sizes of grid sampling. The linear deformation rate x–axis and DEM error y–axis form a 2D solution space for an InSAR pixel location. The value is calculated by Equation (7) with selected real-world baselines.

**Figure 5.**A simple illustration of selecting candidate solutions by skipping the estimates (Estimate 2, 3) that are too close to any picked candidate solution (Estimate C).

**Figure 8.**Visualization of estimated linear deformation rate and DEM on R1–R7 real-world stack’s PS pixels. IGS-CMAES is applied on wrapped interferogram directly. Reference results are generated using 3vGeomatics processing pipeline with unwrapped phase.

**Table 1.**The key parameters of covariance matrix adaptation evolution strategy (CMAES) used in this work.

Parameter | Definition | Value |
---|---|---|

S | Number of candidate solutions at each iteration | 30 |

${\sigma}_{0}$ | Initial step size | 0.01 |

$\mu $ | Number of selected top ranked solutions | 7 |

$\tau $ | Threshold value to terminate the optimization | ${10}^{-11}$ |

c | Learning rate for updating evolution path | 0.5 |

${c}_{1}$ | Learning rate for updating covariance matrix | 0.5 |

${c}_{\sigma}$ | Learning rate for updating step size | 0.5 |

**Table 2.**Quantitative assessment for simulated data with several widely used local optimizers and their IGS-extended version.

Baseline | Categories | LS | IGS- LS | Nelder- Mead | IGS- Nelder- Mead | CG | IGS- CG | BFGS | IGS- BFGS |
---|---|---|---|---|---|---|---|---|---|

R1 | ${m}_{r}$-RMSE (cm/year) | 12.2382 | 2.9481 | 12.4695 | 2.0734 | 12.5197 | 1.3484 | 12.4414 | 1.2893 |

${h}_{e}$-RMSE (m) | 107.0381 | 0.0011 | 105.6981 | 0.0001 | 107.6607 | 0.0000 | 106.5213 | 0.0000 | |

R2 | ${m}_{r}$-RMSE (cm/year) | 13.2260 | 2.3247 | 13.3805 | 2.5936 | 13.6642 | 1.3439 | 13.6011 | 1.3143 |

${h}_{e}$-RMSE (m) | 107.2283 | 0.0016 | 106.3737 | 0.0002 | 107.4976 | 0.0000 | 107.5918 | 0.0000 | |

R3 | ${m}_{r}$-RMSE (cm/year) | 13.0308 | 1.9413 | 13.1002 | 1.6877 | 13.3366 | 1.4307 | 13.2444 | 1.0897 |

${h}_{e}$-RMSE (m) | 107.8291 | 0.0016 | 106.5126 | 0.0002 | 108.3554 | 0.0000 | 108.7696 | 0.0000 | |

R4 | ${m}_{r}$-RMSE (cm) | 12.7119 | 1.9372 | 13.1455 | 2.2679 | 13.1560 | 0.8951 | 13.2614 | 2.0387 |

${h}_{e}$-RMSE (m) | 107.5101 | 0.0032 | 106.5326 | 0.0002 | 108.2290 | 0.0000 | 108.4366 | 0.0000 | |

R5 | ${m}_{r}$-RMSE (cm/year) | 12.6042 | 2.1055 | 12.6375 | 1.9330 | 12.8295 | 0.8136 | 12.8457 | 1.5362 |

${h}_{e}$-RMSE (m) | 106.7188 | 0.0022 | 105.9384 | 0.0002 | 109.1707 | 0.0000 | 107.6000 | 0.0000 | |

R6 | ${m}_{r}$-RMSE (cm/year) | 12.4571 | 1.9549 | 12.5720 | 1.7121 | 12.5872 | 0.6817 | 12.6906 | 0.7807 |

${h}_{e}$-RMSE (m) | 105.3393 | 0.0062 | 106.0447 | 0.0003 | 108.2960 | 0.0000 | 109.1020 | 0.0000 | |

R7 | ${m}_{r}$-RMSE (cm/year) | 12.5991 | 14.9993 | 12.6070 | 14.1472 | 12.7588 | 6.7126 | 12.6561 | 3.2887 |

${h}_{e}$-RMSE (m) | 106.6014 | 0.0006 | 106.0582 | 0.0001 | 108.0011 | 0.0000 | 12.6561 | 0.0000 |

**Table 3.**Quantitative comparison for simulated data using IGS-CMAES with two other global optimizers.

Baseline | Categorie | IGS-CMAES | Grid-Search | Dual-SA |
---|---|---|---|---|

R1 | ${m}_{r}$-RMSE | 0.0284 | 0.0967 | 0.6336 |

${h}_{e}$-RMSE | 0.0000 | 0.5611 | 31.7574 | |

L1-UWPD | 0.0424 | 0.1191 | 1.3113 | |

ACC | 99.94% | 100% | 96.88% | |

NFev | 2725.03 | 20800 | 4109.38 | |

R2 | ${m}_{r}$-RMSE | 0.1137 | 0.0967 | 0.1792 |

${h}_{e}$-RMSE | 0.0000 | 0.5610 | 6.7533 | |

L1-UWPD | 0.0424 | 0.1051 | 0.1279 | |

ACC | 99.94% | 100% | 99.35% | |

NFev | 2585.18 | 20800 | 4103.06 | |

R3 | ${m}_{r}$-RMSE | 0.1991 | 0.0967 | 0.4899 |

${h}_{e}$-RMSE | 0.0000 | 0.5613 | 15.5361 | |

L1-UWPD | 0.0791 | 0.1180 | 0.7239 | |

ACC | 99.61% | 100% | 96.33% | |

NFev | 2687.90 | 20800 | 4112.38 | |

R4 | ${m}_{r}$-RMSE | 0.2276 | 0.0967 | 0.0588 |

${h}_{e}$-RMSE | 0.0000 | 0.5617 | 7.5336 | |

L1-UWPD | 0.0848 | 0.0806 | 0.0534 | |

ACC | 99.56% | 100% | 99.69% | |

NFev | 2500.69 | 20800 | 4089.58 | |

R5 | ${m}_{r}$-RMSE | 0.2844 | 0.0967 | 0.1556 |

${h}_{e}$-RMSE | 0.0000 | 0.5617 | 5.8893 | |

L1-UWPD | 0.1065 | 0.0864 | 0.0853 | |

ACC | 99.44% | 100% | 99.52% | |

NFev | 2520.12 | 20800 | 4093.36 | |

R6 | ${m}_{r}$-RMSE | 0.0000 | 0.0967 | 0.0000 |

${h}_{e}$-RMSE | 0.0000 | 0.5617 | 0.0000 | |

L1-UWPD | 0.0000 | 0.0591 | 0.0000 | |

ACC | 100% | 100% | 100.00% | |

NFev | 2381.48 | 20,800 | 4070.47 | |

R7 | ${m}_{r}$-RMSE | 1.6145 | 7.763333 | 13.902 |

${h}_{e}$-RMSE | 0.0000 | 0.5609 | 104.1441 | |

L1-UWPD | 1.1259 | 5.5327 | 27.8864 | |

ACC | 94.88% | 70% | 39.03% | |

NFev | 3576.92 | 20800 | 4121.54 |

Baseline | NFev | RI-MSE Proposed (Rad) | RI-MSE Reference (Rad) | WPR IGS-CMAES (Rad) | WPR Reference (Rad) | ${\mathit{m}}_{\mathit{r}}$-RMSE (cm) | ${\mathit{h}}_{\mathit{e}}$-RMSE (m) |
---|---|---|---|---|---|---|---|

R1 | 4023.67 | 0.305124 | 0.307999 | 0.431716 | 0.435595 | 0.114381 | 0.061021 |

R2 | 3960.59 | 0.306338 | 0.307771 | 0.438660 | 0.439584 | 0.080547 | 0.053405 |

R3 | 3872.19 | 0.238092 | 0.242657 | 0.371183 | 0.374736 | 0.109501 | 0.052438 |

R4 | 3510.90 | 0.120080 | 0.119850 | 0.238335 | 0.238453 | 0.034719 | 0.048583 |

R5 | 3563.92 | 0.120687 | 0.122859 | 0.249803 | 0.250186 | 0.059676 | 0.122966 |

R6 | 3496.90 | 0.161789 | 0.161793 | 0.321250 | 0.321475 | 0.025550 | 0.102940 |

R7 | 3817.64 | 0.191073 | 0.210659 | 0.326159 | 0.352141 | 0.190598 | 0.061347 |

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**MDPI and ACS Style**

Sun, X.; Zimmer, A.; Mukherjee, S.; Ghuman, P.; Cheng, I. IGS-CMAES: A Two-Stage Optimization for Ground Deformation and DEM Error Estimation in Time Series InSAR Data. *Remote Sens.* **2021**, *13*, 2615.
https://doi.org/10.3390/rs13132615

**AMA Style**

Sun X, Zimmer A, Mukherjee S, Ghuman P, Cheng I. IGS-CMAES: A Two-Stage Optimization for Ground Deformation and DEM Error Estimation in Time Series InSAR Data. *Remote Sensing*. 2021; 13(13):2615.
https://doi.org/10.3390/rs13132615

**Chicago/Turabian Style**

Sun, Xinyao, Aaron Zimmer, Subhayan Mukherjee, Parwant Ghuman, and Irene Cheng. 2021. "IGS-CMAES: A Two-Stage Optimization for Ground Deformation and DEM Error Estimation in Time Series InSAR Data" *Remote Sensing* 13, no. 13: 2615.
https://doi.org/10.3390/rs13132615