# Multi-Temporal Small Baseline Interferometric SAR Algorithms: Error Budget and Theoretical Performance

## Abstract

**:**

## 1. Introduction

## 2. The SB InSAR Framework

## 3. Relative Error Bounds of Ground Deformation Measurements via the SB Methods

## 4. How Do the Relative Error Bounds Depend on the Perpendicular Baseline Threshold of the SB Interferograms?

## 5. How Does the Temporal Coherence Get Valuable Information on the SBAS-InSar Products Error?

## 6. Experimental Results

## 7. Discussion

## 8. Conclusions and Future Perspectives

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Examples of SB InSAR data pair distributions in the temporal/perpendicular baseline domain. (

**a**) SAR data collected by the ERS/ENVISAT sensors in Yellowstone, U.S. (track 41, frame 2709) region. The applied thresholds on the maximum geometrical and temporal baselines are 800 m and three years, respectively. (

**b**) A triangular-shaped network of SB interferograms related to a SAR data set collected by the ENVISAT sensor (ascending orbit, VV polarization) in the area of Pearl River Delta, China, from 2006 to 2010.

**Figure 2.**Pictorial representation of the lack of temporal phase consistency among a set of three images A, B, and C. The terms $\mathsf{\Delta}{n}_{i}^{uncor}$ and ${K}_{i}^{uncorr}$ are the time-uncorrelated phase alterations and the time-uncorrelated phase unwrapping mistakes, respectively.

**Figure 3.**Vectorial representation of the least-squares problem discussed in Equation (4). The grey area identifies the co-domain of the linear transformation given by Equation (4).

**Figure 4.**Study of the perpendicular baseline distribution of sets of SAR images collected by the first-generation SAR sensors. (

**a**–

**e**) Representation in the temporal/perpendicular baseline plane of relevant sets of SAR data acquired by the ERS and ENVISAT sensors on the five selected test-site areas of Afar, Ethiopia, Abruzzi, Central Italy, San Andreas Fault region, U.S., the Mt. Etna zone, Sicily Island, and the metropolitan area of the city of Naples in Italy. The data span the time interval between 1992 and 2010. (

**f**) Theoretical density probability function (pdf) (red line) and the empirical (black line) density probability function of the whole potential set of interferometric perpendicular baselines. Note that the theoretical pdf is drawn using Equation (21). In contrast, the empirical pdf is computed by the data by merely drawing the (normalized) histogram of the SAR dataset interferometric perpendicular baselines, see [58] for details on the probability distribution function of a random variable.

**Figure 5.**Theoretical Distribution of the perpendicular baselines of the InSAR data. (

**A**) Normal distribution of the perpendicular baseline of InSAR data pairs. (

**B**) Distribution of the perpendicular baselines of the InSAR data pairs below the critical baseline boundary. (

**C**) Distribution of the perpendicular baseline of the selected small baseline interferograms obtained by imposing a maximum allowed absolute perpendicular baseline equal to Bmax.

**Figure 6.**The plot of the term $\mathrm{\Pi}-1$ in Equation (23) vs. the maximum perpendicular baseline of the selected SB interferograms and various values of the standard deviation of the distribution of the perpendicular baseline of the whole possible InSAR data pairs.

**Figure 7.**The plot of the unwrapped phase’s relative errors vs. the selected maximum allowed perpendicular baseline of the SB interferograms, for different values of the ground mean displacement rate of the imaged scenes.

**Figure 8.**Upper bounds of the relative error of the measurement model parameters vs. the maximum perpendicular baseline of the SB interferogram, considering different mean ground displacement values of the observed SAR pixel on the terrain.

**Figure 9.**Sets of SB interferograms relevant to South California’s area obtained by processing a group of ENVISAT/ASAR images. The SB InSAR data pairs are identified by imposing a constraint on the maximum perpendicular baseline of (

**a**) 1100 m, (

**b**) 800 m, (

**c**) 600 m, (

**d**) 400 m, and (

**e**) 200 m. Dates shown in this Figure are expressed with the format (day, month, year).

**Figure 10.**2003–2010 Mean deformation velocity map of South California obtained using the SBAS method to the group of SB interferograms identified in Figure 9e and corresponding to a maximum value of the InSAR perpendicular baselines equal to 200 m. The map is in radar coordinates, and the ground deformation is superimposed on an amplitude SAR image of the area. Shown deformation values are saturated between +/− 10 mm/year.

**Figure 11.**Experimental SBAS Results. (

**a**) Comparing the temporal coherence values of the five performed runs of the SBAS inversion obtained by progressively relaxing the constraint on the maximum allowed perpendicular baseline of the interferograms, from 1100 m to 200 m; (

**b**) zoomed view of (

**a**) the interval of very high temporal coherence values (higher than 0.9).

**Figure 12.**Map of the Relative Errors of the SB unwrapped phases relevant to the test-case with SB interferograms with a maximum perpendicular baseline of 200 m.

**Figure 13.**(

**a**) Temporal coherence map of South California area (test-site case with the maximum perpendicular baseline of 200 m), (

**b**) map of the absolute error (upper bound) of the ground mean displacement rate after SBAS inversion; the map does not take into account the effects of APS and the residual topography of the area.

**Figure 14.**The plot of the ground displacement rate absolute error of the SBAS measurements vs. the temporal coherence.

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**MDPI and ACS Style**

Pepe, A.
Multi-Temporal Small Baseline Interferometric SAR Algorithms: Error Budget and Theoretical Performance. *Remote Sens.* **2021**, *13*, 557.
https://doi.org/10.3390/rs13040557

**AMA Style**

Pepe A.
Multi-Temporal Small Baseline Interferometric SAR Algorithms: Error Budget and Theoretical Performance. *Remote Sensing*. 2021; 13(4):557.
https://doi.org/10.3390/rs13040557

**Chicago/Turabian Style**

Pepe, Antonio.
2021. "Multi-Temporal Small Baseline Interferometric SAR Algorithms: Error Budget and Theoretical Performance" *Remote Sensing* 13, no. 4: 557.
https://doi.org/10.3390/rs13040557