# Change Detection within Remotely Sensed Satellite Image Time Series via Spectral Analysis

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## Abstract

**:**

## 1. Introduction

- The detection of jumps (sudden changes) in the trend component of an unequally spaced time series;
- Accounting for uncertainties in the time series values (observational uncertainties) to improve the estimation of trend and seasonal components, and jump locations; and
- The characterization of the gradual and sudden changes in the ecosystem by estimating the jump direction and magnitude.

## 2. Materials and Methods

#### 2.1. Study Regions

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

#### 2.3. Weighted Vegetation Time Series

#### 2.4. Jumps Upon Spectrum and Trend (JUST)

#### 2.5. Breaks for Additive Seasonal and Trend (BFAST)

#### 2.6. Validation through Descriptive Statistics

- Jump error for the K time series: the number of incorrectly detected jump locations divided by K, i.e., normalized.
- Root Mean Square Error (RMSE) for the jump magnitude when the jump location is correctly detected:$$\begin{array}{c}\hfill \mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{m}\sum _{k=1}^{m}{\left(\right)}^{{\widehat{\mathrm{MAG}}}_{k}}2}},\end{array}$$
- Mean Normalized Residual Norm (MNRN): compute the Normalized Residual Norm (NRN) for each of the K time series, then find their average. For a given time series, NRN is defined as the weighted L2 norm of estimated residual series divided by the weighted L2 norm of the original series. More precisely:$$\begin{array}{c}\hfill {\displaystyle \mathrm{NRN}=\left(\right)open="("\; close=")">\sum _{j=1}^{n}w\left({t}_{j}\right){r}^{2}\left({t}_{j}\right)/\left(\right)open="("\; close=")">\sum _{j=1}^{n}w\left({t}_{j}\right){f}^{2}\left({t}_{j}\right)},\end{array}$$

## 3. Results and Discussion

#### 3.1. Simulation Experiment

#### 3.1.1. Simulation of Time Series with Unknown Seasonality

#### 3.1.2. Simulation of Time Series with Two Noises of the Same Type

#### 3.1.3. A Simulated EVI Time Series with Multiple Jumps

#### 3.2. Detecting and Characterizing Jumps in the NDVI Time Series for the First Study Region

#### 3.3. Detecting and Characterizing Jumps in the Landsat 8 Image Time Series for the Second Study Region

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ALLSSA | Anti-Leakage Least-Squares Spectral Analysis |

BFAST | Breaks For Additive Seasonal and Trend |

CCDC | Continuous Change Detection and Classification |

DBEST | Detecting Breakpoints and Estimating Segments in Trend |

EVI | Enhanced Vegetation Index |

JUST | Jumps Upon Spectrum and Trend |

LSWA | Least-Squares Wavelet Analysis |

LSWAVE | Least-Squares Wavelet (software) |

MNRN | Mean Normalized Residual Norm |

MODIS | Moderate Resolution Imaging Spectroradiometer |

NASA | National Aeronautics and Space Administration |

NDVI | Normalized Difference Vegetation Index |

OLS | Ordinary Least-Squares |

OLS-MOSUM | Ordinary Least-Squares Residuals-Based Moving Sum |

PQA | Pixel Quality Assessment |

RMSE | Root Mean Square Error |

STL | Seasonal-Trend decomposition procedure based on Loess |

TOA | Top Of Atmosphere |

USGS | U.S. Geological Survey |

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**Figure 1.**Study regions: the first region is a forested region in New South Wales, Australia, the same region studied in [8]. The second region is a forested region in California, U.S. The magnified section shows a view of the second study region in January 2020 by Google Earth. The transparent areas in the magnified section are generated by adding a layer containing the polygon shapefiles for the burned areas in October 2017 obtained from [43]. The topographic maps are generated by the SimpleMappr online tool [46].

**Figure 2.**An illustration of obtaining per-pixel weighted EVI time series. EVI images are acquired over unequally spaced time intervals due to availability of valid satellite imagery that provide unequally spaced EVI image time series. In this illustration, each EVI image consists of 100 pixels, and so in total there are 100 unequally spaced time series whose sizes may not be the same due to several reasons, such as cloud contamination and high aerosol content. The cloud score for each spatial window of size nine is used to define an appropriate weight for the pixel located at the window center (see the shaded gray pixels around the pixel in a solid color for each image).

**Figure 3.**(

**a**) an illustration of jumps in the trend component of three different NDVI time series. $\mathrm{DIR}$ and $\mathrm{MAG}$ denote the true direction and magnitude of the jump located at ${t}_{9}$, respectively, and (

**b**) the translating window of size $R=21$ observations, translating (sliding) by $\mathsf{\delta}=6$ observations. JUST has detected a jump within the first window (see the red arrow) and another jump within the second window (see the green arrow). The solid red and green lines are the estimated linear trends (each trend has two pieces) corresponding to the segments within the red and green windows, respectively.

**Figure 4.**Flowchart of JUST. The weights are optional inputs and in general can also be entered in the form of a matrix $\mathbf{P}$, the inverse of the covariance matrix associated with the time series. The constituents of known forms are the linear trends by default ($\mathsf{\Gamma}$). Symbol $|\xb7|$ denotes the absolute value. The flowchart of ALLSSA used in JUST is in Figure 3.4 of [22], or see [31].

**Figure 5.**Statistical results for the (

**a**) detection of the true jumps, (

**b**) magnitudes of the true jumps, and (

**c**) residual series using the OLS-based method and JUST shown by dashed and solid lines, respectively. The results are for a collection of 1000 unequally spaced two-year-long time series per each noise level.

**Figure 6.**Same caption as Figure 5 but for collections of three-year-long time series.

**Figure 7.**Simulation of an EVI time series for a two-year period, (

**a**) the simulated seasonal component given by Equation (10) (the symmetric Gaussian function is shown by dashed lines for comparison), (

**b**) the trend components with magnitude $-0.2$ given in Table 2, (

**c**) the noise ${\mathsf{\eta}}_{1}$ with $0.1$ level and noise ${\mathsf{\eta}}_{2}$ with $0.4$ level, (

**d**) the weights defined by ${\mathsf{\eta}}_{2}$, and (

**e**) the simulated EVI time series that is the sum of the seasonal, trend, and ${\mathsf{\eta}}_{1}$ components minus the absolute value of ${\mathsf{\eta}}_{2}$. To help comparisons, the solid bars with the same data range are displayed on the right-hand side of the plot.

**Figure 8.**Statistical results for the (

**a**) detection of true jumps, (

**b**) magnitudes of the true jumps, and (

**c**) residual series using the unweighted and weighted JUST shown by dashed and solid lines, respectively. The initial ${\mathsf{\eta}}_{1}$ noise level is $0.1$. These results are for a collection of 1000 unequally spaced two-year-long time series per each noise level, one of such time series is illustrated in Figure 7e.

**Figure 9.**Same caption as Figure 8 but for collections of three-year-long time series.

**Figure 10.**Decomposition of a noisy unequally spaced EVI time series into trend, seasonal, and remainder components. (

**a**) the simulated EVI time series, (

**b**) the weights associated with the time series values derived from the ${\mathsf{\eta}}_{2}$ noise, and (

**c**–

**e**) the estimated and simulated results for trend, seasonal, and remainder components, shown in blue and red, respectively. The solid bars with the same data range are displayed on the right-hand side of the plot to help comparison.

**Figure 11.**Decomposition of an equally spaced NDVI time series into trend, seasonal, and remainder components. (

**a**) NDVI time series (harvest), and (

**b**–

**d**) the JUST and BFAST results for trend, seasonal, and remainder components, shown in blue and red, respectively. To help comparisons, the solid bars with the same data range are displayed on the right-hand side of the plot.

**Figure 12.**Decomposition of a NDVI time series (drought, insect attacks) with several missing values into trend, seasonal, and remainder components, see the caption of Figure 11 for more details.

**Figure 13.**The results of analyzing the Landsat 8 weighted EVI time series since 2013. (

**a**) The weighted average of all EVI images as a proxy for vegetation coverage within the second study region during 2013–2020, (

**b**) the year of jump with the largest absolute value of magnitude ($\left|\mathrm{MAG}\right|<0.05$ is shown in green), (

**c**) the magnitude of jump, and (

**d**) the direction of jump. Panels (

**b**–

**d**) are only for the areas where the weighted average of EVIs is greater than 0.2. The more detailed analyses of EVI time series corresponding to pixels A, B, and C in panel (

**c**) are demonstrated in Figure 14, Figure 15 and Figure 16, respectively.

**Figure 14.**Decomposition of a real-world unequally spaced EVI time series into trend, seasonal, and remainder components. (

**a**) EVI time series corresponding to pixel A shown in Figure 13c, (

**b**) the weights associated with the time series values derived from the PQA band, and (

**c**–

**e**) the estimated trend, seasonal, and remainder components. The solid bars with the same data range are displayed on the right-hand side of the plot to help comparison.

Inputs | Description | Default |
---|---|---|

t | Time values | |

f | Time series values | |

$\mathbf{P}$ | Weight matrix | None |

R | Window size | Sampling rate tripled: $3M$ |

$\mathsf{\delta}$ | Translation step | Sampling rate: M |

$\mathsf{\Omega}$ | Cyclic frequencies | $1,2,3,4$ |

Constituents | Known forms | Linear trend |

$\mathsf{\alpha}$ | Significance level | $0.01$ |

Outputs: the estimated jump locations, and their directions and magnitudes; | ||

the estimated trend, seasonal, and remainder components |

**Table 2.**Parameter values for simulation of the linear trends in Equation (8) to compare JUST with the OLS-based method for jump detection.

${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | MAG |
---|---|---|---|---|

$0.05$ | $0.05$ | $0.300$ | $0.250$ | $-0.05$ |

$0.05$ | $0.05$ | $0.350$ | $0.250$ | $-0.10$ |

$0.05$ | $0.05$ | $0.375$ | $0.175$ | $-0.20$ |

$0.05$ | $0.05$ | $0.400$ | $0.100$ | $-0.30$ |

**Table 3.**Parameter values for simulation of the trend in Equation (8) to compare JUST with BFAST.

${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | MAG |
---|---|---|---|---|

$-0.05$ | $0.14$ | $0.30$ | $0.00$ | $0.0$ |

$-0.05$ | $0.04$ | $0.30$ | $0.05$ | $-0.1$ |

$-0.05$ | $0.07$ | $0.30$ | $-0.10$ | $-0.2$ |

Noise Level (${\mathsf{\eta}}_{1}$) | ||||||
---|---|---|---|---|---|---|

$0.048$ | $0.096$ | $0.144$ | $0.192$ | $0.240$ | ||

Method | $\mathbf{MAG}$ | Jump Error | ||||

JUST | $0.0$ | $0.877$ | $0.951$ | $0.970$ | $0.961$ | $0.976$ |

BFAST | $0.0$ | $1.000$ | $1.000$ | $0.994$ | $0.992$ | $0.991$ |

JUST | $-0.1$ | $0.000$ | $0.066$ | $0.305$ | $0.517$ | $0.703$ |

BFAST | $-0.1$ | $1.000$ | $0.998$ | $0.989$ | $0.952$ | $0.954$ |

JUST | $-0.2$ | $0.000$ | $0.001$ | $0.030$ | $0.170$ | $0.264$ |

BFAST | $-0.2$ | $0.389$ | $0.481$ | $0.572$ | $0.605$ | $0.638$ |

Method | $\mathbf{MAG}$ | RMSE | ||||

JUST | $0.0$ | $0.007$ | $0.019$ | $0.029$ | $0.047$ | $0.057$ |

BFAST | $0.0$ | N/A | N/A | $0.044$ | $0.055$ | $0.072$ |

JUST | $-0.1$ | $0.006$ | $0.012$ | $0.017$ | $0.023$ | $0.032$ |

BFAST | $-0.1$ | N/A | $0.034$ | $0.051$ | $0.063$ | $0.066$ |

JUST | $-0.2$ | $0.006$ | $0.012$ | $0.018$ | $0.024$ | $0.030$ |

BFAST | $-0.2$ | $0.047$ | $0.045$ | $0.049$ | $0.054$ | $0.058$ |

Noise Level (${\mathsf{\eta}}_{1}$) | ||||||
---|---|---|---|---|---|---|

$0.048$ | $0.096$ | $0.144$ | $0.192$ | $0.240$ | ||

Method | $\mathbf{MAG}$ | Jump Error | ||||

JUST | $0.0$ | $0.913$ | $0.946$ | $0.968$ | $0.982$ | $0.986$ |

BFAST | $0.0$ | $0.974$ | $0.973$ | $0.972$ | $0.970$ | $0.983$ |

JUST | $-0.1$ | $0.000$ | $0.078$ | $0.397$ | $0.621$ | $0.762$ |

BFAST | $-0.1$ | $0.047$ | $0.268$ | $0.487$ | $0.628$ | $0.731$ |

JUST | $-0.2$ | $0.000$ | $0.000$ | $0.026$ | $0.148$ | $0.294$ |

BFAST | $-0.2$ | $0.000$ | $0.000$ | $0.031$ | $0.130$ | $0.226$ |

Method | $\mathbf{MAG}$ | RMSE | ||||

JUST | $0.0$ | $0.009$ | $0.019$ | $0.035$ | $0.042$ | $0.050$ |

BFAST | $0.0$ | $0.016$ | $0.025$ | $0.033$ | $0.039$ | $0.060$ |

JUST | $-0.1$ | $0.007$ | $0.012$ | $0.017$ | $0.024$ | $0.029$ |

BFAST | $-0.1$ | $0.008$ | $0.015$ | $0.021$ | $0.028$ | $0.035$ |

JUST | $-0.2$ | $0.006$ | $0.012$ | $0.018$ | $0.024$ | $0.029$ |

BFAST | $-0.2$ | $0.009$ | $0.014$ | $0.019$ | $0.025$ | $0.030$ |

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**MDPI and ACS Style**

Ghaderpour, E.; Vujadinovic, T.
Change Detection within Remotely Sensed Satellite Image Time Series via Spectral Analysis. *Remote Sens.* **2020**, *12*, 4001.
https://doi.org/10.3390/rs12234001

**AMA Style**

Ghaderpour E, Vujadinovic T.
Change Detection within Remotely Sensed Satellite Image Time Series via Spectral Analysis. *Remote Sensing*. 2020; 12(23):4001.
https://doi.org/10.3390/rs12234001

**Chicago/Turabian Style**

Ghaderpour, Ebrahim, and Tijana Vujadinovic.
2020. "Change Detection within Remotely Sensed Satellite Image Time Series via Spectral Analysis" *Remote Sensing* 12, no. 23: 4001.
https://doi.org/10.3390/rs12234001