# A Multi-Scale Spatial Difference Approach to Estimating Topography Correlated Atmospheric Delay in Radar Interferograms

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## Abstract

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## 1. Introduction

## 2. Model and Estimation Approach

#### 2.1. Model

#### 2.2. Estimation Approach

## 3. Synthetic Test

## 4. Correcting Real Interferogram

#### 4.1. Sierra Nevada Mountains

#### 4.2. 2016 Menyuan Earthquake

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Original (

**top**) and differential topography (

**first column**) and interferogram (

**second column**) with scale factors of 1 km, 2 km, 3 km, 4 km, and 5 km, respectively. The last column is the scatter plots of phase differences and topographic height differences. The scatter plots are diluted 300 times, and the direction of difference is azimuth 0°. The estimated values of ${K}_{1}$, ${K}_{2}S$, and correlation coefficient R for each scale factor are shown at the bottom right corners of the scatter plots. The final estimate values of K

_{1}and K

_{2}S are 2.50 rad/km and 0.1 rad/km, respectively, which are equal to the values set in the synthetic test (2.5 rad/km and 0.1 rad/km). In comparison, the K

_{1}value calculated using full interferogram–topography correlation is 1.81 rad/km.

**Figure 2.**Values of the correlation coefficient R (column 1), ${K}_{1}$ (column 2), and ${K}_{2}S$ (column 3, blue dots, the red line is the fitting line of the blue dots) after the difference of multiple scales in four directions; the final ${K}_{1}$ is estimated according to the proposed method. The interval of the difference scale factor is 0.25 km, and the first scale factor is 0.025 km. The maximum absolute value of ${K}_{2}$ is 0.10 rad/km (

**A**), the corresponding azimuth is 0°, and the related ${K}_{1}$ is 2.50 rad/km. The ${K}_{2}$ values of other azimuth (

**B**–

**D**) are all less than 0.1 rad/km.

**Figure 3.**Reference map of the Sierra Nevada Mountains with maximum elevation up to 4.13 km. The blue frames are the coverage of the Sentinel-1A data.

**Figure 4.**A schematic description of the construction of the synthetic interferometry: (

**A**) Topography; (

**B**) Topographically correlated tropospheric delays; (

**C**) Large bilinear ramp are computed as described in the text; (

**D**) Turbulent signals; (

**E**) deformation signals that we project them to phase and combine them to form the (

**F**) final synthetic interferogram. In this example, the Fried parameter ${r}_{0}$ is 5 km, the inner scale ${l}_{0}$ is 10 m, the outer scale ${L}_{0}$ is 30 km, the ${K}_{1}$ is 2.5 rad/km, and the ramp amplitude (${K}_{2}$) is 0.1 rad/km.

**Figure 5.**Comparison of the transfer functions estimated by using full interferogram–topography correlation, BP, MSSD, and the ${K}_{2}$ estimated by our method. There are 20 realizations of synthetic interferograms in each plot, with different turbulence signals, amplitudes, and ramp directions. The input ${K}_{1}$ is 2.5 rad/km (first column, blue dashed lines). The input ${K}_{2}$ is 0.1 rad/km (

**A**,

**B**,

**E**,

**F**) and 0.01 rad/km (

**C**,

**D**,

**G**,

**H**) respectively (second column, blue dashed lines).

**Figure 6.**(

**A**) The topography of the Sierra Nevada mountains; (

**B**) the original interferogram; (

**C**) the Interferogram of the Mono Lake area (Despite the mask processing, a large number of noise signals remain in this area); (

**D**) the layover area in the interferogram; (

**E**) the layover area in SAR image.

**Figure 7.**The values of the correlation coefficient R (column 1), ${K}_{1}$ (column 2), and ${K}_{2}S$ (column 3, blue dots, the red line is the fitting line of the blue dots) after the difference of the interferogram in four directions, and the final ${K}_{1}$, estimated according to the method described in the text. The maximum absolute value of ${K}_{2}$ is 0.070 rad/km (

**D**), the corresponding azimuth is 135°, and the related ${K}_{1}$ is −0.200 rad/km. The ${K}_{2}$ values of other azimuth (

**A**–

**C**) are all less than 0.070 rad/km.

**Figure 8.**Scatter plots (phase difference vs. topographic height difference, blue dots) in various spatial scales in the Sierra Nevada Mountains with an azimuth of 135°. This comparison shows the influence of error on correlation coefficient R in different scales.

**Figure 9.**Comparison between the interferograms in the Sierra Nevada mountains before and after correction using the MSSD approach; (

**A**) the original interferogram (

**B**) the corrected interferogram obtained through the full interferogram–topography approach (

**C**) the corrected interferogram obtained through the BP approach and (

**D**) that obtained by MSSD; (

**E**,

**F**) the ramp component and the sum of topography-related component and ramp component, both acquired by the MSSD approach. Notice that the phase gradient in (

**D**) is reduced after correction. Correlation coefficients of 9 sub-regions(0–8) are shown in Table 2.

**Figure 10.**Correlation coefficients of 9 sub-regions; only the absolute values of the coefficients corrected by MSSD in the sub-regions 2, 4, and 6 are larger than those corrected by the full interferogram–topography correlation approach or BP approach (Table 2).

**Figure 11.**Reference map of the 21 January 2016 Menyuan Earthquake superimposed on topographic relief, with maximum elevation up to 5.17 km. The star shows the location of the 2016 Menyuan event. The red lines denote the active faults. The blue frames are the coverage of the Sentinel-1A data.

**Figure 12.**Comparison between the interferograms (wrapped) in the 2016 Menyuan coseismic displacement example before and after correction using the MSSD approach; (

**A**–

**D**) the model-retained interferograms before and after correction; (

**E**–

**H**) the local view of the epicenter area; (

**I**–

**L**) the model-removed interferograms before and after correction.

**Figure 13.**Correlation coefficients of 9 sub-regionss: (

**A**) model-retained interferogram; (

**B**) the model-removed interferogram.

**Table 1.**Statistical values of transfer functions estimated by three approaches: average value (AVG) and standard deviation (S.D.).

Group | A | B | C | D | E | F | G | H | |
---|---|---|---|---|---|---|---|---|---|

${K}_{1}$ (MSSD) | AVG | 2.503 | 2.505 | 2.500 | 2.492 | 2.500 | 2.499 | 2.500 | 2.500 |

S.D. | 0.016 | 0.013 | 0.016 | 0.019 | 0.002 | 0.002 | 0.003 | 0.003 | |

${K}_{1}$ BP | AVG | 2.499 | 2.507 | 2.498 | 2.507 | 2.500 | 2.500 | 2.500 | 2.500 |

S.D. | 0.025 | 0.019 | 0.023 | 0.019 | 0.000 | 0.000 | 0.000 | 0.000 | |

${K}_{1}$ (Full-Igram) | AVG | 1.603 | 3.734 | 2.426 | 2.569 | 1.665 | 3.656 | 2.413 | 2.620 |

S.D. | 0.200 | 0.192 | 0.268 | 0.252 | 0.031 | 0.032 | 0.033 | 0.024 | |

${K}_{2}$ (MSSD) | AVG | 0.101 | 0.095 | 0.011 | 0.011 | 0.100 | 0.093 | 0.010 | 0.010 |

S.D. | 0.005 | 0.003 | 0.008 | 0.003 | 0.001 | 0.000 | 0.001 | 0.000 |

Sub Region | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

Original | −0.878 | −0.416 | 0.391 | −0.545 | −0.111 | 0.183 | 1.808 | 0.234 | 0.153 |

Full-Igram | −0.743 | −0.280 | 0.532 | −0.406 | 0.024 | 0.319 | 2.030 | 0.374 | 0.284 |

BP | −0.710 | −0.249 | 0.513 | −0.416 | 0.058 | 0.345 | 2.589 | 0.361 | 0.352 |

MSSD | −0.128 | 0.044 | 0.528 | −0.052 | 0.026 | −0.001 | 2.151 | −0.004 | −0.248 |

Sub Area | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

Orignal | 2.138 | 2.320 | 2.431 | 1.975 | 1.538 | 1.976 | 1.856 | 1.824 | 1.802 |

Full-Igram | 0.219 | 0.198 | 0.068 | 0.064 | −0.158 | 0.119 | −0.103 | −0.180 | −0.178 |

BP | 0.205 | 0.235 | 0.161 | 0.049 | −0.226 | 0.090 | −0.107 | −0.173 | −0.177 |

MSSD | 0.108 | 0.080 | −0.057 | 0.061 | −0.166 | 0.117 | 0.010 | −0.063 | −0.066 |

Sub Area | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

Orignal | 2.138 | 2.315 | 2.430 | 1.966 | 1.717 | 1.974 | 1.855 | 1.825 | 1.802 |

Full-Igram | 0.191 | 0.190 | 0.093 | 0.027 | −0.033 | 0.082 | −0.127 | −0.195 | −0.199 |

BP | 0.184 | 0.206 | 0.136 | 0.019 | −0.064 | 0.069 | −0.129 | −0.193 | −0.198 |

MSSD | 0.083 | 0.049 | −0.086 | 0.032 | −0.004 | 0.095 | −0.008 | −0.079 | −0.083 |

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## Share and Cite

**MDPI and ACS Style**

Yu, Z.; Huang, G.; Zhao, Z.; Huang, Y.; Zhang, C.; Zhang, G.
A Multi-Scale Spatial Difference Approach to Estimating Topography Correlated Atmospheric Delay in Radar Interferograms. *Remote Sens.* **2023**, *15*, 2115.
https://doi.org/10.3390/rs15082115

**AMA Style**

Yu Z, Huang G, Zhao Z, Huang Y, Zhang C, Zhang G.
A Multi-Scale Spatial Difference Approach to Estimating Topography Correlated Atmospheric Delay in Radar Interferograms. *Remote Sensing*. 2023; 15(8):2115.
https://doi.org/10.3390/rs15082115

**Chicago/Turabian Style**

Yu, Zhigang, Guoman Huang, Zheng Zhao, Yingchun Huang, Chenxi Zhang, and Guanghui Zhang.
2023. "A Multi-Scale Spatial Difference Approach to Estimating Topography Correlated Atmospheric Delay in Radar Interferograms" *Remote Sensing* 15, no. 8: 2115.
https://doi.org/10.3390/rs15082115