# Sinkhole Scanner: A New Method to Detect Sinkhole-Related Spatio-Temporal Patterns in InSAR Deformation Time Series

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background on Sinkhole Types and Shapes

## 3. Methods

#### 3.1. Multi-Burst TSInSAR Processing

#### 3.2. Sinkhole Simulation

#### 3.3. Sinkhole Scanner

**Inverted Gaussian shape solution:**

**A**is the design matrix and

**x**is the vector of unknowns. The stochastic model ${Q}_{\mathit{dd}}$ is designed to describe the observation errors. By relating the functional model in Equation (3) with Equation (2), we specify the functional model of the inverted Gaussian model as

**A**and the unknown parameters

**x**have the size of $bm\times 2$ and $2\times 1$, respectively. The estimates of the extent of the Gaussian function $\widehat{\zeta}$ and the linear velocity $\widehat{v}$ can be determined once $\widehat{\mathbf{x}}$ is estimated. The corresponding stochastic model $\underset{bm\times bm}{{Q}_{\mathit{dd}}}$ describes the deformation errors in logarithmic form. To simplify the simulation, one can use a scaled identity matrix, i.e., ${Q}_{\mathit{dd}}={\sigma}^{2}\xb7{R}_{bm\times bm}={\sigma}^{2}\xb7{\mathbb{I}}_{bm\times bm}$, which implies deformation of the b points are uncorrelated in time and space and the deformation variance is invariant over time. Here, ${\sigma}^{2}$ is the variance of unit weight, ${R}_{bm\times bm}$ is the cofactor matrix, and ${\mathbb{I}}_{bm\times bm}$ is the identity matrix.

**Cylinder shape solution:**

**A**, with size ${b}^{\prime}m\times 2$ is the design matrix and

**x**, with size $2\times 1$ is the vector of unknowns.

**Conical shape definition:**

**A**and

**x**are the same as in the previous Cylinder shape case as well.

#### 3.4. Moving Window Operation

#### 3.5. Hypothesis Testing Method

## 4. SAR Data and Test Site Description

## 5. Results

#### 5.1. TSInSAR Results

#### 5.2. Sinkhole Scanner Results

**Case: Simulation**

**Case: Scanning Simulated Data**

**Case: Scanning Real Data**

#### 5.3. Hypothesis Testing Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A sinkhole at an early development stage [1]. The surface S subsides due to the presence of a dissolution zone C underneath the surface. Side views of (

**b**) Gaussian, (

**c**) Cylindrical, and (

**d**) Conical simulated sinkholes. (

**e**) Top view of the sinkhole spot with 30 randomly distributed InSAR CCS, and rendered structure of sinkholes with these 30 CCS for (

**f**) Gaussian, (

**g**) Cylindrical, and (

**h**) Conical shape.

**Figure 3.**Conical shaped sinkhole geometry. The sinkhole has the center at O and subsides by time ${t}^{\prime}$ to point ${B}^{\prime}$, a surface measurement point ${P}_{i}$ located at $({x}_{i}^{\prime},{y}_{i}^{\prime})$ subsides to point D at time ${t}^{\prime}$.

**Figure 4.**(

**a**) Sentinel-1A spatial coverage and test site location superimposed on an ellipsoidal height DEM map, (

**b**) zoomed-in optical image, and (

**c**) areal image taken after the sinkhole occurred, covering the area shown by the white quadrilateral in subfigure (

**b**), (

**d**) overlapped burst area (red rectangle), azimuth merge line, and cropped area used for the study.

**Figure 5.**CCS distribution and deformation velocity map over (

**a**) the cropped study area and (

**b**) the subset corresponding to the white box in subfigure (

**a**), in radar coordinates. Deformation trend in WGS84 coordinate system (

**c**), and zoomed in over the sinkhole area (

**d**) corresponding to the white box in subfigure (

**c**). Note that the cluster of red CCS over the mining site is near the sinkhole site. The pink triangle shows the reference point location.

**Figure 6.**Sinkhole simulation and scanning with Gaussian shape. (

**a**) CCS distribution and contour plot for simulated sinkhole depth at the last simulation epoch, simulation Scenario L, (

**b**) the equivalent simulated sinkhole deformation velocity map, (

**c**) Posterior variance after scanning using Scanner 1 for Gaussian sinkholes over the simulated data, (

**d**) zoomed in image corresponding to the black box in (

**c**).

**Figure 7.**Sinkhole scanning using Gaussian shape for real sinkholes using 4 window sizes: (

**a**) 2 km, (

**b**) 1 km, (

**c**) 500 m, and (

**d**) 100 m, with a zoomed-in image as inset on the top-right of (

**d**). The red ellipse represents the sinkhole rim.

**Figure 8.**Original InSAR estimated velocities (

**a**) and modelled velocities (

**b**) using Sinkhole scanning with Gaussian shape for real sinkholes for window size 100 m, and a scatter-plot plotted between the two (

**c**).

**Figure 9.**Anomalous deformation behaviour near the sinkhole site. Deformation velocity map (

**a**) and associated deformation time series (

**b**). Significant deformation velocity and deep deformation values can be observed for areas near the sinkhole site.

**Figure 10.**Test ratio statistic for Heaviside (

**a**) and Breakpoint (

**b**) anomaly models for deformation time series of PS points in Figure 9.

Inverted Gaussian Model | Cylindrical Model | Conical Model | |
---|---|---|---|

Model equation | $F(x,y,t)=$$I\left(t\right)\xb7{e}^{-\left(\frac{{x}^{2}+{y}^{2}}{2\xb7{\zeta}^{2}}\right)}$ | $F\left(t\right)=I\left(t\right)$, $\forall {P}_{i}\mid {{x}_{i}^{\prime}}^{2}+{{y}_{i}^{\prime}}^{2}\le {r}^{2}$ | $F(x,y,t)=I\left(t\right)\left(1-\frac{\sqrt{({x}^{2}+{y}^{2})}}{r}\right)$$\forall {P}_{i}\mid {{x}_{i}^{\prime}}^{2}+{{y}_{i}^{\prime}}^{2}\le {r}^{2}$ |

Model parameters | $\zeta ,v$ | $v,c$ | $v,c$ |

Data | Date Range | Source | Description |
---|---|---|---|

Sentinel-1 SLC SAR | April 2015–December 2018 | Alaska Satellite Facility (ASF) | Polarization: VV, Nr. images: 75 |

Acquisition mode: IW, Track nr.: 1 | |||

Sentinel-1 Orbit data | April 2015–December 2018 | Copernicus Sentinels POD Data Hub | Orbit type: Precise |

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

Simulation grid size (m) | 10 × 10 |

Extent of grid ($x,y$) (km) | 63.73 ×43.84 |

Starting depth of sinkhole (mm) | 0.5 |

Maximum depth of center (mm) | 100 |

Temporal baseline (min, max) (year) | (0, 3.65) |

Epochs | 10 |

Uncertainty (e) (mm) | 10 |

Velocity of center (mm · yr${}^{-1}$) | 25 |

Simulation Space | ||||
---|---|---|---|---|

Scenario S | Scenario L | |||

Window size (m) | 100 × 100 | 2000 × 2000 | ||

Stride (m) | 100 × 100 | 2000 × 2000 | ||

${\zeta}_{x}$, ${\zeta}_{y}$ | (5,5) | (50, 50) | ||

Scanning Space | ||||

Scanner 1 | Scanner 2 | Scanner 3 | Scanner 4 | |

Window size (m) | 2000 × 2000 | 1000 × 1000 | 500 × 500 | 100 × 100 |

Stride (m) | 2000 × 2000 | 1000 × 1000 | 500 × 500 | 100 × 100 |

Scanner | Total Grids | Nr. of Scanned Grids | Scanned Area (km${}^{2}$) |
---|---|---|---|

Scanner 1 | 651 | 499 | 1996 |

Scanner 2 | 2709 | 1142 | 1142 |

Scanner 3 | 10,922 | 2819 | 704.75 |

Scanner 4 | 274,320 | 1964 | 19.64 |

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**MDPI and ACS Style**

Kulshrestha, A.; Chang, L.; Stein, A.
Sinkhole Scanner: A New Method to Detect Sinkhole-Related Spatio-Temporal Patterns in InSAR Deformation Time Series. *Remote Sens.* **2021**, *13*, 2906.
https://doi.org/10.3390/rs13152906

**AMA Style**

Kulshrestha A, Chang L, Stein A.
Sinkhole Scanner: A New Method to Detect Sinkhole-Related Spatio-Temporal Patterns in InSAR Deformation Time Series. *Remote Sensing*. 2021; 13(15):2906.
https://doi.org/10.3390/rs13152906

**Chicago/Turabian Style**

Kulshrestha, Anurag, Ling Chang, and Alfred Stein.
2021. "Sinkhole Scanner: A New Method to Detect Sinkhole-Related Spatio-Temporal Patterns in InSAR Deformation Time Series" *Remote Sensing* 13, no. 15: 2906.
https://doi.org/10.3390/rs13152906