# Gap-Filling of MODIS Time Series Land Surface Temperature (LST) Products Using Singular Spectrum Analysis (SSA)

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## Abstract

**:**

## 1. Introduction

## 2. Study Area, Data

#### 2.1. Study Area

#### 2.2. Data (Satellite Time Series Images)

## 3. Methodology

#### Singular Spectrum Analysis (SSA)

^{T}are eigenvectors of matrix XTX. If we plot the singular values in descending order, one can often distinguish between an initial steep slope, representing a signal, and a (more or less) flat floor, representing the noise level [30]. Then, any subset of the d eigenvalues (EOFs), 1 ≤ d ≤ m, for which related eigenvalues are positive, provides the best representation of the matrix X as a sum of matrices X

_{i}, i = 1, 2, …, d [32].

_{In′}have the form of a Hankel matrix in an ideal case, and consequently fit the trajectory matrices.

_{In′}matrices should be transformed into the form of a Hankel matrix to fit the trajectory matrices. This step is known as diagonal averaging. In this sense, the original matrix can be reconstructed as the sum of these matrices:

_{t}

^{n′}is the result of the diagonal averaging of the matrix X

_{In′}.

_{i}and y

_{i}represent original and reconstructed data, respectively.

## 4. Results

#### 4.1. Spatio-Temporal Distribution of the Missing Data

#### 4.2. Determining the Parameters of SSA Algorithm

^{2}) increased significantly between the original data and reconstructed signal by SSA algorithm. Then, by increasing the number of components from two to eight, the R

^{2}value was not significantly increased. Figure 6a illustrates the effect of different window sizes on the reconstruction of the same time series with four components. By increasing the window size, the R

^{2}value between the original data and reconstructed data was reduced. Using small window sizes to reconstruct a time series with long-distance missing data has no favorable results, and using large window sizes not only reduces the accuracy of reconstruction, it also increases the time of processing. Thus, in the present study, a window size of 60 was used to process the time series. In the following, the accurate determination of the number of significant components and window sizes by SSA algorithm are fully described.

#### 4.3. Determining the Significant Periodic Components in the SSA Algorithm

#### 4.4. Data Basis and Null Hypothesis Basis Monte Carlo Test Results

^{2}) with a pure sinusoid for better visualization. The top graph of Figure 9 shows the result of the data-based Monte Carlo test.

#### 4.5. Assessment of Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Time series with 730 images (

**a**), trend of temperature changes along one pixel (

**b**), and magnification of a part of time series (

**c**).

**Figure 3.**The missing data percentage of each image (temporal dispersion) in the annual LST time series during the day.

**Figure 4.**The missing data percentage of each image (temporal dispersion) in the annual LST time series during the night.

**Figure 5.**Percentage of gaps in each pixel of an annual LST time series with 730 images (

**a**) and its histogram (

**b**).

**Figure 6.**R

^{2}values between the original data and singular spectrum analysis (SSA) reconstruction in a time series with 730 data with different window sizes (

**a**) and different numbers of components (

**b**).

**Figure 7.**Singular values spectrum of data with a window size of 60 images showing three modes (note: the vertical axis shows the log-transformed variance of three modes).

**Figure 9.**Monte Carlo SSA based on data (

**a**) and null hypothesis empirical orthogonal functions (EOFs) test (

**b**).

**Figure 10.**SSA T-EOFs of components 1, 2, and 3 values with image window size 60 (

**a**), and SSA T-EOFs of components 3 with image window size 730 (

**b**).

**Figure 11.**Root mean square error (RMSE) map between the original data and reconstructed data by SSA algorithm in an annual LST time series with 730 images (

**a**) and histogram of the RMSE map (

**b**).

**Figure 12.**An LST image of the studied region before reconstruction (

**a**) and after reconstruction (

**b**) by SSA.

**Figure 13.**A time series along a pixel together with the reconstruction results of SSA (

**a**) and the magnification of a part of a time series (

**b**).

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**MDPI and ACS Style**

Ghafarian Malamiri, H.R.; Rousta, I.; Olafsson, H.; Zare, H.; Zhang, H. Gap-Filling of MODIS Time Series Land Surface Temperature (LST) Products Using Singular Spectrum Analysis (SSA). *Atmosphere* **2018**, *9*, 334.
https://doi.org/10.3390/atmos9090334

**AMA Style**

Ghafarian Malamiri HR, Rousta I, Olafsson H, Zare H, Zhang H. Gap-Filling of MODIS Time Series Land Surface Temperature (LST) Products Using Singular Spectrum Analysis (SSA). *Atmosphere*. 2018; 9(9):334.
https://doi.org/10.3390/atmos9090334

**Chicago/Turabian Style**

Ghafarian Malamiri, Hamid Reza, Iman Rousta, Haraldur Olafsson, Hadi Zare, and Hao Zhang. 2018. "Gap-Filling of MODIS Time Series Land Surface Temperature (LST) Products Using Singular Spectrum Analysis (SSA)" *Atmosphere* 9, no. 9: 334.
https://doi.org/10.3390/atmos9090334