# The Potential of the Least-Squares Spectral and Cross-Wavelet Analyses for Near-Real-Time Disturbance Detection within Unequally Spaced Satellite Image Time Series

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## Abstract

**:**

## 1. Introduction

- Can SHP representing both inter-annual (gradual) and intra-annual (seasonal) variations be determined within the time series?
- How reliable is SHP for detecting a disturbance in MP?
- How much the weights obtained from the observational uncertainties can increase the probability of a disturbance detection in MP?
- How additional data sets, such as temperature and precipitation, can be used for assessment?

## 2. Materials and Methods

#### 2.1. Study Region

#### 2.2. Data Sets and Pre-Processing

#### 2.2.1. Satellite Imagery

#### 2.2.2. Validation Data Sets

#### 2.3. Monitoring Method

#### 2.3.1. Stable History Period Selection

#### 2.3.2. Near-Real-Time Disturbance Detection

#### 2.4. Least-Squares Cross-Wavelet Analysis as an Assessment Method

## 3. Results

#### 3.1. Simulation Experiment for Validation of the Monitoring Method

#### 3.2. Disturbance Detection in the Study Region

#### 3.3. Assessment of the Results Using the Coherency Analyses

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Study region: A forested region in northern Alberta, Canada. The enlarged section shows a view of the study region in April 2020, provided by the Environmental Systems Research Institute (ESRI).

**Figure 2.**A weighted average of EVI images as a proxy for the vegetation coverage during 2013–2019 with a photograph by the Government of Alberta taken in May 2019. The details of the numerically labeled locations in the EVI image are shown in Table 1. The top-right panels show parts of a satellite image clipped to the polygon shapefiles of the burned areas in locations 3 and 4. Pixels A, B, and C are also selected to further investigate the changes in 2019.

**Figure 3.**Flowchart of the monitoring algorithm. The disturbance detection process using ALLSSA and LSSA is independently applied to each per-pixel EVI time series. One may also use other spectral indices instead of EVI in this algorithm, such as NDVI. An appropriate weight may be assigned to each pixel value based on the uncertainties in the presence of cloud/shadow, snow, ice, smoke, and haze using a cloud-masking algorithm, such as FMask. Please note that the process of selecting SHP corresponding to each pixel is performed only once, and the new observations during MP will be concatenated to the end of time series as they become available.

**Figure 4.**A simulated unequally spaced time series with consistent data gaps (yellow backgrounds) and error bars. The estimated components in SHP (mid-2016 to mid-2019) by the OLS and ALLSSA season-trend models are shown with red and blue solid lines, respectively. MP (gray background) started right before an abrupt change with magnitude $-0.2$ occurs. After three new incoming observations, the abrupt change is detected at the time location shown by green vertical line.

**Figure 5.**An example of the near-real-time disturbance detection from the residual series. (

**a**) the residual series of the time series shown in Figure 4 which has an abrupt change located at the beginning of MP (gray background), (

**b**–

**d**) the weighted least-squares spectra for $d=1,2,3$, respectively, with the critical values at $99\%$ confidence level (red lines), and (

**e**) the maximum percentage variance of the spectral peaks in the spectrum corresponding to each d, three of which are shown in panels (

**b**–

**d**). A disturbance is detected after three newly acquired observations, circled in panel (

**e**).

**Figure 6.**The probability of change detection (y-axis) against the noise level (x-axis) for magnitudes listed in Table 2. The weights are defined and considered after noise level $0.025$ (see vertical dashed lines). The results of the weighted and unweighted methods for each d are shown by solid and dashed lines, respectively. The top left panel represents the method specificity, and the other three panels represent the method sensitivity. In general, the results of the weighted method show significantly higher true-positive and true-negative rates than the results of the unweighted method as the noise level increases.

**Figure 7.**Statistical results for the study region obtained from analyzing per-pixel EVI time series, where each pixel has a resolution of 300 m. (

**a**) the estimated date map of the most recent disturbances before 2019 for visualizing SHP, (

**b**) spatial variation of the noise level for SHP (i.e., the standard deviation of the residual series in SHP), (

**c**) the estimated dates of detected disturbances in MP (April–November, 2019), and (

**d**) the change magnitude. Please note that the panels only show the results for the areas where the weighted average EVI shown in Figure 2 is greater than $0.2$. Also, panels (

**b**–

**d**) are obtained only for the areas with at least two years SHP, i.e., for the areas whose most recent disturbance dates, shown in panel (

**a**), are before 2017. The detailed analyses of pixels A, B, and C are demonstrated in Figure 8.

**Figure 8.**The near-real-time change detection results for pixels A, B, and C displayed in Figure 7. The history period is from 2013, and the expected values (forecast) in MP (April–November, 2019; gray background), using the observations in SHP, are shown with blue dashed lines. The green vertical lines in panels (

**a**,

**b**) show the detected disturbances after one and four observations in MP, respectively. The observational uncertainties are shown by the black bars. The periods with large data gaps have yellow backgrounds. The green dashed lines in panel (

**c**) show another season-trend fit when all Landsat 8 images are considered, denoted by “ALLSSA +” in the legend. The additional observations are in red with relatively much larger error bars.

**Figure 9.**The near-real-time change detection results using NDVI time series corresponding to pixels A, B, and C displayed in Figure 7. The history period is from 2013, and the expected values (forecast) in MP (April–November, 2019; gray background), using the observations in SHP, are shown with blue and red dashed lines. The green vertical lines in panels (

**a**,

**b**) show the detected disturbances after one and four observations in MP, respectively. The observational uncertainties are shown by the black bars. The periods with large data gaps have yellow backgrounds. In the legend, “ALLSSA −” means that SHP in panel (

**b**) is determined by ALLSSA after ignoring the observation shown by arrow. A view of the clipped satellite image for pixel B, acquired on the 6 of November 2018, is also shown in panel (

**b**). No disturbance is detected during MP in panel (

**c**).

**Figure 10.**The coherency analyses between EVI and weather time series. Panels (

**a**–

**c**) show the results of time series corresponding to pixels A, B, and C shown in Figure 7, respectively. Temperature and precipitation are expressed in degree Celsius and millimeters, respectively. The stochastic surfaces in LSCWS at $99\%$ confidence level are in gray, and the white arrows represent the phase differences following the trigonometric circle.

**Table 1.**Validation data within the study region provided by the Government of Alberta. Each location (before 2019), is shown by an asterisk (labeled) in Figure 2, impacted by wildfire.

Label | Date of Wildfire | Latitude | Longitude | Area Burned (ha) | Cause |
---|---|---|---|---|---|

1 | 2014-05-10 | ${58}^{\circ}{26}^{\prime}{30}^{\u2033}$ | $-{117}^{\circ}{13}^{\prime}{14}^{\u2033}$ | 0.7 | Railroad |

2 | 2015-04-28 | ${58}^{\circ}{31}^{\prime}{20}^{\u2033}$ | $-{117}^{\circ}{04}^{\prime}{59}^{\u2033}$ | 0.5 | Resident |

3 | 2015-05-30 | ${58}^{\circ}{15}^{\prime}{01}^{\u2033}$ | $-{117}^{\circ}{20}^{\prime}{36}^{\u2033}$ | 7.0 | Recreation |

4 | 2015-06-27 | ${58}^{\circ}{24}^{\prime}{53}^{\u2033}$ | $-{117}^{\circ}{17}^{\prime}{31}^{\u2033}$ | 246.9 | Lightning |

5 | 2016-04-23 | ${58}^{\circ}{32}^{\prime}{03}^{\u2033}$ | $-{117}^{\circ}{19}^{\prime}{12}^{\u2033}$ | 0.5 | Recreation |

6 | 2017-09-28 | ${58}^{\circ}{18}^{\prime}{41}^{\u2033}$ | $-{117}^{\circ}{09}^{\prime}{39}^{\u2033}$ | 0.5 | Recreation |

7 | 2018-05-08 | ${58}^{\circ}{26}^{\prime}{23}^{\u2033}$ | $-{117}^{\circ}{09}^{\prime}{49}^{\u2033}$ | 0.4 | Agriculture |

2019-05-12 | ${58}^{\circ}{17}^{\prime}{16}^{\u2033}$ | $-{117}^{\circ}{16}^{\prime}{47}^{\u2033}$ | 350,135.0 | Lightning |

**Table 2.**Parameters for simulation of vegetation time series: amplitude of the asymmetric Gaussian function, noise level, magnitude of the abrupt change, and number of new observations to quantify detection delay (d).

Parameters | Values | |
---|---|---|

amplitude | $0.1,0.2,0.3,0.4,0.5$ | |

noise level | $0.005,0.01,0.015,\dots ,0.15$ | |

magnitude | $0.0,-0.1,-0.2,-0.3$ | |

d | $1,2,3,4,5,6$ |

Pixel | Time Series | Frequency | Amplitude | Phase | Intercept | Slope |
---|---|---|---|---|---|---|

A | Temperature | $1.00$ | $17.57\pm 0.42$ | $-1.55\pm 0.00$ | $-0.08\pm 0.59$ | $-0.03\pm 0.15$ |

A | Precipitation | $1.00$ | $28.03\pm 3.21$ | $-1.59\pm 0.00$ | $36.56\pm 4.52$ | $-1.25\pm 1.13$ |

$2.01$ | $13.49\pm 3.20$ | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1.64\pm 0.24$ | ||||

B | Temperature | $1.00$ | $17.99\pm 0.42$ | $-1.55\pm 0.00$ | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.24\pm 0.59$ | $-0.02\pm 0.15$ |

B | Precipitation | $1.00$ | $18.30\pm 2.43$ | $-1.68\pm 0.03$ | $27.86\pm 3.41$ | $-0.78\pm 0.85$ |

$2.01$ | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}9.98\pm 2.41$ | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1.70\pm 0.24$ | ||||

C | Temperature | $1.00$ | $18.05\pm 0.41$ | $-1.54\pm 0.00$ | $-0.34\pm 0.58$ | $-0.02\pm 0.15$ |

C | Precipitation | $1.00$ | $21.03\pm 2.84$ | $-1.62\pm 0.01$ | $31.79\pm 3.99$ | $-1.28\pm 0.99$ |

$2.01$ | $11.95\pm 2.82$ | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1.62\pm 0.24$ |

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Ghaderpour, E.; Vujadinovic, T.
The Potential of the Least-Squares Spectral and Cross-Wavelet Analyses for Near-Real-Time Disturbance Detection within Unequally Spaced Satellite Image Time Series. *Remote Sens.* **2020**, *12*, 2446.
https://doi.org/10.3390/rs12152446

**AMA Style**

Ghaderpour E, Vujadinovic T.
The Potential of the Least-Squares Spectral and Cross-Wavelet Analyses for Near-Real-Time Disturbance Detection within Unequally Spaced Satellite Image Time Series. *Remote Sensing*. 2020; 12(15):2446.
https://doi.org/10.3390/rs12152446

**Chicago/Turabian Style**

Ghaderpour, Ebrahim, and Tijana Vujadinovic.
2020. "The Potential of the Least-Squares Spectral and Cross-Wavelet Analyses for Near-Real-Time Disturbance Detection within Unequally Spaced Satellite Image Time Series" *Remote Sensing* 12, no. 15: 2446.
https://doi.org/10.3390/rs12152446