# Large-Area Gap Filling of Landsat Reflectance Time Series by Spectral-Angle-Mapper Based Spatio-Temporal Similarity (SAMSTS)

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Satellite Gap-Filling Literature Review

## 3. Data

#### 3.1. Landsat 8 Data

#### 3.2. Cropland Data Layer

## 4. Test Areas

## 5. Gap-Filling Methodology

#### 5.1. Overview

_{gap}) with one or more missing pixels in the target image, an alternative similar segment (${S}_{gap}^{alt}$), which has valid non-missing pixel values in the target image, is identified using the cluster information. Then, for each missing pixel in segment S

_{gap}, its alternative similar pixel ${p}_{gap}^{alt}$ is identified within ${S}_{gap}^{alt}$ and is used to fill the gap pixel’s values.

_{r}) is incorporated in the steps of the time series segmentation, segment clustering, and identification of alternative similar segments and pixels in the target image. This measure is based on the spectral angle mapper (SAM) that is used conventionally to determine the spectral similarity between two pixels by calculating the cosine of the angle subtended between their points in feature space and the feature space origin [73,74,75]. Previously, we refined SAM to allow the comparison of two single-pixel Landsat multispectral time series with missing temporal observations [13] as:

_{z}

_{∈[1…n]}, and n (≥2) is the number of feature space dimensions, i.e., the product of the number of spectral bands and the number of images in the time series; $\chi ({x}_{z}^{a},{x}_{z}^{b})$ is a step-function returning 1 if both ${x}_{z}^{a}$ and ${x}_{z}^{b}$ are non-missing observations and returning 0 otherwise. Thus, SAM

_{r}

_{0}measures the similarity between two pixels’ time series considering only their temporally-corresponding non-missing observations. SAM

_{r}

_{0}is bounded in the range [0,1], tends to 1 as the time series values of the two pixels become similar, and is 1 when the time series values are identical. However, where there is a high proportion of missing observations, then only a small number of temporally-corresponding observations will be available, which will reduce the reliability of SAM

_{r}

_{0}. To minimize the impact of this issue, the SAM

_{r}is used:

_{r}

_{0}$(\stackrel{\rightharpoonup}{a},\stackrel{\rightharpoonup}{b})$ is defined by Equation (1), n′ is the number of temporally-corresponding observations between $\stackrel{\rightharpoonup}{a}$ and $\stackrel{\rightharpoonup}{b}$, n is the number of feature space dimensions, i.e., the product of the number of spectral bands and the number of images in the time series (5 × 26), and the term $\frac{1}{{n}^{\prime}}{\displaystyle \sum _{z=1}^{n}\left|{x}_{z}^{a}-{x}_{z}^{b}\right|}\chi ({x}_{z}^{a},\hspace{0.17em}{x}_{z}^{b})$ is the mean difference between the two time series’ temporally-corresponding observations. Like SAM

_{r}

_{0}, the SAM

_{r}is bounded in the range [0,1] and equals 1 when the time series are identical.

_{50%}in Equation (2) is set as the percentage of non-missing tile 30 m pixel observations over the 26 weeks multiplied by the product of the number of weeks (26) and the number of bands (five Landsat bands, Section 3.1) times 0.5. For example, for the California study area, 47.5% of the tile 30 m pixel observations over the 26 weeks are missing (Table 1, middle column) and so obs

_{50%}is 34 (derived as ((100 − 47.5)/100) × 26 × 5 × 0.5). Thus, the misalign$(\stackrel{\rightharpoonup}{a},\stackrel{\rightharpoonup}{b},ob{s}_{50\%})$ penalty term occurs for the California gap-filling when the five-band 26-week vectors $\stackrel{\rightharpoonup}{a}$ and $\stackrel{\rightharpoonup}{b}$ have less than 34 temporally-corresponding values. If there are more temporally-corresponding values, i.e., n′ is high, then the penalty term is not invoked.

_{r}is used to compare not only single-pixel time series, but also to compare the mean time series values of pixels that are associated together after the application of a segmentation algorithm. For brevity, we define the term “segment signature” as the mean reflectance values for each of the five Landsat bands over the 26 weeks (i.e., a vector of 130 × 1 mean values). The mean values are derived considering all the non-missing pixels in the segment. The SAM

_{r}is then calculated as Equation (2) with $\stackrel{\rightharpoonup}{a}$ and $\stackrel{\rightharpoonup}{b}$ defined by segment signatures or single pixel time series. We also define the term “cluster signature” as the mean of the segment signatures in a cluster. The cluster signature is a vector of 130 × 1 reflectance values.

#### 5.2. The Spectral-Angle-Mapper-Based Spatio-Temporal Similarity (SAMSTS) Gap-Filling Algorithm

_{r}time series calculations are undertaken comparing the segment containing the gap pixel (cross, falling in the red segment with diagonal hatching) with alternative similar segments that belong to the same cluster (diagonal hatching), and the most similar alternative segment that maximizes SAM

_{r}is selected; and (4) for each gap pixel in the gap segment, its corresponding alternative pixel is individually searched only among the candidate pixels in the corresponding most similar alternative segment. These four steps are described below in detail through Section 5.2.1, Section 5.2.2 and Section 5.2.3

#### 5.2.1. Time Series Image Segmentation

_{r}is used with a simple region-growing image segmentation method [76]. Each pixel is associated with a spatially-adjacent pixel (considering an 8-connected pixel neighborhood) if the SAM

_{r}value of the two pixels’ time series is >0.9995. The 0.9995 threshold is purposefully set high to ensure that pixels belonging to the same segment are highly similar, i.e., have a nearly identical 26 week temporal evolution in the five-band reflectance. This threshold was empirically determined through tests on the input data for the three test areas. Since the agricultural study areas typically have a higher degree of spatio-temporal complexity than most other landscapes [31,51], we anticipate that this threshold will be suitable for general cases when six months of Landsat data are used.

_{r}between the pixel time series and the segment signature is derived, and for segments composed of more than one pixel, the standard deviation of these SAM

_{r}values is derived. Two neighboring segments are considered sufficiently similar to be merged if either segment’s SAM

_{r}standard deviation value is less than twice the value of 1 – SAM

_{r}derived between their segment signatures (as 1 – SAM

_{r}quantifies the degree of disimilarity of two signatures). The merging is undertaken in an iterative manner. In each iteration, only unmerged segments are eligible to be merged, and the merged segments are taken as a new segment in the next iteration. A maximum of three iterations was found to be sufficient to avoid over-merging. Large numbers of small-sized segments may exist in the merged segmentation results, primarily due to mixed pixel effects occurring along object edges in the Landsat 30 m data. The segment signatures of small segments are more likely to be affected by mixed pixel effects than large ones. In particular, segments composed of one, two, or three pixels have long perimeters relative to their areas, and so are usually mixed. Therefore, as illustrated in Figure 3, the merged segments are divided into two size groups (>3 pixels and ≤3 pixels) and the groups are treated separately in the following clustering and gap-filling steps to speed up the processing.

#### 5.2.2. Segment Clustering

_{r}measure that can handle time series with missing observations. The K-means approach requires initialization with K observations, i.e., K observations that can be treated as candidate clusters. In this study, the candidate clusters are selected in an iterative manner starting with one randomly-selected segment. Then in each iteration, every unselected segment’s signature is compared with the signature(s) of the currently selected segment(s), and if the SAM

_{r}values are smaller than 0.96, the segment is selected as a new candidate cluster. In this way, the number of candidate clusters typically increases with each iteration. If no new candidate clusters are found and there are fewer than 300 candidate clusters, the process is stopped. If there are more than 300, the process is restarted, but the SAM

_{r}0.96 threshold is decreased by 0.01 so that fewer candidate clusters will be selected. We note that 300 is considerably larger than the number of CDL classes in the 5000 × 5000 30 m pixel test areas (Figure 1), but this is needed to account for subtle land cover and surface differences, and the thresholds set ensure that close to (but less than) 300 clusters can be obtained. All the segments are then clustered using the K-means clustering approach initialized with the K candidate clusters. The SAM

_{r}metric is used as the K-means distance measure as it can handle missing Landsat time series observations [13].

_{r}standard deviation of each cluster is derived based on the SAM

_{r}value between a cluster signature and the signature of each segment in the cluster; the SAM

_{r}between each pair of clusters is also calculated. Two clusters are merged conservatively if they are mutually the most similar cluster to each other and their cluster SAM

_{r}standard deviations are relatively small. Here, small is defined as less than 1/2 of the 1 – SAM

_{r}value derived between their cluster signatures (note that 1 – SAM

_{r}reflects the dissimilarity between two cluster signatures). The merging is repeated up to five times until no merging takes place. After this step, the number of clusters are typically reduced by more than 50%, i.e., to fewer than 150 clusters.

_{r}. For Landsat time series, M should be sufficiently large to capture perturbations in the time series due to non-surface variations imposed by the remote sensing process (e.g., differences in illumination and observation angles, atmospheric contamination, sensor calibration/degradation changes, sensor noise) while being small enough to be computationally efficient. In this study, M was set as 10 as we found M higher than this did the not provide improved results and reduced the computational efficiency.

#### 5.2.3. Identification of Alternative Segments and/or Pixels to Fill Gaps

## 6. Analysis Methodology

#### 6.1. Gap-Filling Assessment Metrics

_{fill}(i, j) is the spectral root-mean-square difference at pixel location (i, j) between the original pixel reflectance ${p}_{k}^{original}\left(i,j\right)$ and the gap-filled ${p}_{k}^{filled}\left(i,j\right)$ reflectance values considering five Landsat bands, i.e. the green (0.56 μm), red (0.66 μm), near-infrared (NIR) (0.87 μm), and the two shortwave infrared (SWIR) (1.61 μm and 2.20 μm) bands.

_{preceding}(i, j) and RMSD

_{subsequent}(i, j) are the spectral root-mean-square differences at pixel location (i, j) between the original pixel reflectance value ${p}_{k}^{original}(i,j)$ and the preceding temporally-closest observation reflectance value ${p}_{k}^{preceding}(i,j)$, and the subsequent temporally-closest observation reflectance value ${p}_{k}^{subsequent}(i,j)$, respectively. As some gap-filling methods may simply take the temporally-closest non-missing pixel observations, regardless of whether the closet observation occurred before or after the gap, the closest RMSD was also derived as:

#### 6.2. Gap-Filling Experiments

#### 6.2.1. SAMSTS Gap Filling

#### 6.2.2. Comparison with Sinusoidal Harmonic Model-Based Gap-Filling

_{0}is a coefficient describing the mean of f(t) over the time series; a

_{m}and b

_{m}are coefficients for harmonic component m with M ≥ 1 components; L is the length of the time period; and t is time. The model can reproduce periodic time series, but higher model orders (larger M) are required to fit more complex time series and may result in spurious oscillations, particularly with noisy data and/or if there are gaps in the time series where change occurred [35,82,83]. Zhu et al. [38] applied the sinusoudal model as shown in Equation (7) to several years of Landsat data, but used an additional coefficient to model inter-annual changes. In this study, as only six months of Landsat data were used, the inter-annual change coefficient was not used. The six-months of Landsat data were sensed from 28 April 2013 (DOY = 118) to 26 October 2013 (DOY = 299) and so L was set to 182 (299 – 118 + 1). Zhu et al. [38] recommended that there should be at least three times more valid surface observations than the number of model coefficients, and did not run the model but output the median observation value if there were fewer than six observations. Therefore, in this study, the sinusoidal model shown in Equation (7) was implemented with M = 2 (i.e., five Fourier coefficients) when the number of valid observations was ≥15 and with M = 1 when there were 5 to 14 observations, and the median observation value was output if there were fewer than five observations. The sinusoidal harmonic model was applied independently at each pixel location in the 25 600 × 600 30 m pixel simulated gaps over the California, Minnesota and Kansas tiles, and was applied to the Landsat time series independently for each of the five Landsat bands and used to predict the reflectance on the dates of the simulated gaps.

## 7. Results

#### 7.1. Detailed Gap-Filling Example Demonstration

_{fill}), red (RMSD

_{preceding}) and green (RMSD

_{subsequent}). The SAMSTS RMSD

_{fill}histogram has a greater frequency of smaller RMSD values than the RMSD

_{preceding}and RMSD

_{subsequent}histograms. Thus the SAMSTS performs better than simple closest temporal pixel substitution.

_{preceding}(g), RMSD

_{fill}(h) and RMSD

_{subsequent}(i), and their comparisons illustrate gap-filling errors and sensitivities. This subset is dominated by circular center-pivot irrigation agricultural fields and within some of the circular fields, “pie slice” circular sectors are evident that are associated with harvesting [84]. The majority the RMSD

_{preceding}pixel values (78.1%) and the RMSD

_{subsequent}pixel values (65.3%) are greater than the corresponding RMSD

_{fill}values. The mean RMSD

_{fill}value is 0.022 (median is 0.016), and 7.2% of the pixels have RMSD

_{fill}values > 0.05, and 0.7% have RMSD

_{fill}values > 0.1. The largest SAMSTS gap-filling errors occur where there were abrupt surface changes due to agricultural harvesting that is evident when the preceding (d) and subsequent (f) weeks are compared. The mean RMSD

_{subsequent}value (0.045) is slightly greater than the mean RMSD

_{preceding}value (0.042) due to the greater amount of harvesting that occurred between weeks 38 and 40, compared to the harvesting between weeks 36 and 38. However, despite the rapid surface changes, much of the Kansas SAMSTS gap-filled image (e) is consistent with the original image (b).

#### 7.2. Large Area Gap-Filling Results

_{preceding}(g), RMSD

_{fill}(h), and RMSD

_{subsequent}(i) maps.

_{preceding}(g) and RMSD

_{subsequent}(i) values. As discussed above, these errors are primarily due to surface changes before or after the target image acquisition. In addition, residual atmospheric contamination and bi-directional effects due to differences in illumination and observation angles [23,57] may also cause temporal differences.

_{fill}values do not obviously increase further from the spatially-closest valid observations, as observed in other ASP gap filling methods [50]. The majority of the RMSD

_{fill}values are smaller than the corresponding RMSD

_{preceding}tile pixels values (62.5%, 69.6%, 72.3% for California, Minnesota and Kansas, respectively), RMSD

_{subsequent}tile pixel values (68.8%, 75.6%, 55.2% for California, Minnesota and Kansas, respectively), and RMSD

_{temporally_closest}pixel values (61.4%, 69.1%, 71.6% for California, Minnesota and Kansas, respectively).

_{temporally_closest}, for each study site. The SAMSTS gap filling method (blue line) has generally smaller errors than the closest temporal pixel substitution approaches. As with the detailed gap-filling results, the gap-filling errors are greatest for the Kansas, then Minnesota, then California tile data. The majority of the RMSD

_{fill}errors are less than 0.05 with mean values less than 0.02, i.e., less than the 3% OLI reflectance calibration accuracy [85]. The mean tile RMSD values provide a straightforward way to assess the gap filling. The smallest mean RMSD values are consistently observed for the SAMSTS gap-filling method with mean RMSD

_{fill}values of 0.014 (California), 0.016 (Minnesota) and 0.018 (Kansas). The mean RMSD

_{preceding}, RMSD

_{subsequent}, and RMSD

_{temporally_closest}values vary considerably, but are greater than the mean RMSD

_{fill}values by at least 61% (California), 69% (Minnesota), and 55% (Kansas). For the California and Minnesota tiles, the mean RMSD

_{temporally_closest}value is smaller than the mean RMSD

_{preceding}and RMSD

_{subsequent}values, which is expected as the closest observation is selected from either the subsequent or preceding observation. The Kansas tile has a higher mean RMSD

_{temporally_closest}value (0.031) than the mean RMSD

_{subsequent}value (0.027), but is only marginally higher than the RMSD

_{preceding}value (0.030). This is because the Kansas target week (Figure 9b) is in the harvesting season with considerable preceding and subsequent surface reflectance changes (Figure 2).

## 8. Discussion

## 9. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Alternative Similar Segment Identification Algorithm

_{gap}, the best alternative similar segment, termed ${S}_{gap}^{alt}$ is selected. The segment signature, i.e., the vector of 130 × 1 (five bands by 26 weeks) mean reflectance values, is denoted $\stackrel{\rightharpoonup}{S}$. The algorithm to identify ${S}_{gap}^{alt}$ is as follows, where k is initially set as 2, and the number of M most similar clusters is set as 10:

- (i)
- Start a spatial search from S
_{gap}examining the spatially nearest segments. - (ii)
- For each candidate segment ${S}_{gap}^{candidate}$, check whether it has any valid non-missing observations on the t-th temporal image. If not, skip it and flag it as processed to be excluded in further searches; if yes, and if its first k nearest clusters (k ≤ M, M = 10) overlap with S
_{gap}’s first k nearest clusters, then SAM_{r}(${\stackrel{\rightharpoonup}{S}}_{gap}^{}$, ${\stackrel{\rightharpoonup}{S}}_{gap}^{candidate}$) is calculated, and ${S}_{gap}^{candidate}$ is flagged as processed. The largest SAM_{r}value is recorded as $SA{M}_{gap}^{candidate\_max}$ with respect to S_{gap}, and the corresponding ${S}_{gap}^{candidate}$ is recorded as ${S}_{gap}^{candidate\_optimal}$ with respect to S_{gap}. - (iii)
- Stop the search and go to step (vii) if one of the following conditions are met; otherwise continue to step (iv).
- ①
- If $SA{M}_{gap}^{candidate\_max}$ > 0.990 and at least 100 candidate segments are processed, i.e., 100 candidate segments have been inspected based on SAM
_{r}(${\stackrel{\rightharpoonup}{S}}_{gap}^{}$, ${\stackrel{\rightharpoonup}{S}}_{gap}^{candidate}$); - ②
- If $SA{M}_{gap}^{candidate\_max}$ > 0.980 and more than 5000 candidate segments have been processed;
- ③
- If $SA{M}_{gap}^{candidate\_max}$ > 0.970 and the whole image space has been searched with k = M, i.e., available segments in all the M nearest clusters have been inspected.

- (iv)
- If the whole image space is searched and k < M, increment k by 1 and go to step (i) to restart the search with the new k and only considering the unprocessed segments.
- (v)
- If the whole image space is searched and k = M, go to step (i) to restart the search considering all unprocessed segments, and still increment k by 1.
- (vi)
- If the whole image space is searched and k = M + 1, which means all available segments have been considered, stop the search.
- (vii)
- Record ${S}_{gap}^{candidate\_optimal}$ as S
_{gap}’s alternative similar segment ${S}_{gap}^{alt}$.

## Appendix B. Alternative Similar Pixel Identification Algorithm

_{gap}with respect to the t-th temporal image that contains gaps. By definition, S

_{gap}has gaps on the t-th temporal image and ${S}_{gap}^{alt}$ has valid observations on the t-th temporal image. For a gap pixel p

_{gap}in segment S

_{gap}, its alternative similar pixel ${p}_{gap}^{alt}$ is searched in ${S}_{gap}^{alt}$ among the candidate pixels with valid observations on the t-th temporal image. The algorithm to identify the ${p}_{gap}^{alt}$ of a gap pixel p

_{gap}in S

_{gap}is as follows:

- (i)
- From ${S}_{gap}^{alt}$, extract all the pixels with valid observation on the t-th temporal image. Randomly sample a maximum of 100 pixels from the extracted pixels.
- (ii)
- For each p
_{gap}in S_{gap}, calculate SAM_{r}(${\stackrel{\rightharpoonup}{p}}_{gap}^{}$, ${\stackrel{\rightharpoonup}{p}}_{gap}^{candidate}$) where ${p}_{gap}^{candidate}$ is from the up-to-100 sampled pixels obtained in steps i). The ${p}_{gap}^{alt}$ with respect to p_{gap}is identified as the one with the maximum SAM_{r}.

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**Figure 1.**The locations of the three WELD/ARD 150 × 150 km (5000 × 5000 30 m pixel) test tiles and the corresponding 2013 CDL data. (

**a**) California (38.22991560° to 39.16110308°N, 120.67484015° to 122.81778573°W, WELD tile h02v08/ARD tile 002008) main CDL classes are grassland/pasture (24.2%), forest (13.5%), shrub/scrub (11.9%), developed (11.3%), rice (6.4%), fallow/idle cropland (5.8%), water (3.0%), grapes (2.9%), herbaceous wetlands (2.5%), alfalfa (2.4%), and corn (2.0%); (

**b**) Minnesota (43.36640358° to 44.72180606°N, 94.33364760° to 96.19691668°W, WELD tile h17v06) main CDL classes are corn (44.7%), soybean (31.9%), grassland/pasture (6.6%), developed (5.9%), and herbaceous wetlands (5.0%); (

**c**) Kansas tile (36.62626467° to 37.90044279°N, 99.56286386° to 101.34813598°W; WELD tile h13v12) main CDL classes are grassland/pasture (49.1%), winter wheat (18.7%), fallow/idle cropland (8.5%), corn (8.4%), sorghum (7.3%), and developed (4.0%). The red square in (

**c**) shows the location of a 15 × 15 km (500 × 500 30 m pixel) area subset for detailed gap filling demonstration described in Section 7.1. Please refer to Figure 2 for the CDL color legend.

**Figure 2.**Mean CDL class-specific weekly (weeks 18–43, 2013) NDVI values derived from the three WELD/ARD test tiles (Figure 1) California (

**top**), Minnesota (

**middle**), and Kansas (

**bottom**). Only values for the CDL classes that cover more than 2% of the tile are shown. Weeks with no data are not illustrated, but the plotted lines are shown dashed.

**Figure 4.**Overview of the SAMSTS gap filling algorithm. (

**a**) The segmentation map obtained from the time series; (

**b**) The clustered segment (denoted by different hatching); (

**c**) Given a segment with gaps (e.g., the red segment), its alternative segment is searched for considering only the segments in the same cluster (i.e., the purple, cyan and blue segments). The missing-observation-adaptive similarity metric SAM

_{r}is used in the segmentation, clustering, and alternative similar segment search.

**Figure 5.**Kansas 500 × 500 30 m pixel subset (centered at 37.75755020°N, 100.78698061°W, Figure 1c red square shows the subset tile location) results. 2010 CDL subset (

**a**), Landsat 8 false-color (1.61 μm, 0.87 μm, 0.66 μm) images of the original (

**b**), the preceding (

**d**), and subsequent (

**f**) weeks and the gap-filled version of the original data (

**e**) and associated RMSD images (

**g**–

**i**) (colored: 0 ≤ dark blue ≤ 0.05; 0.05 < light blue ≤ 0.08; 0.08 < green ≤ 0.11; 0.11 < yellow ≤ 0.13; 0.13 < orange ≤ 0.15; 0.15 < red < 0.2; brown ≥ 0.2). The two red circles denote two circular center-pivot irrigation fields where partial harvesting occurred between weeks 36 and 38 and then complete harvesting occurred between weeks 38 and 40. The yellow arrow denotes a small area that was flooded between weeks 36 and 38.

**Figure 6.**Example NDVI time series for the flooded pixel (denoted by the yellow arrows in Figure 5, located at 37.70121869°N, 100.82220237°W), and the corresponding NDVI time series for the selected alternative similar pixel. For clarity, the NDVI, rather than five-band reflectance time series, is shown.

**Figure 7.**California tile (5000 × 5000 30 m pixels) gap filling experiments, 25 600 × 600 30 m pixel areas, shown by white squares in the target week (

**b**), were removed to simulate gaps and then filled by the SAMSTS method (

**e**), and by closest preceding (

**d**) and subsequent (

**f**) pixel substitution. Associated RMSD images are shown (

**g**–

**i**) colored as for Figure 5. The temporally-closest non-missing preceding and subsequent observations did not always belong to the same image and were acquired from one to five weeks before (

**a**) and after (

**c**) the target week, colored as black (1), dark gray (2), light gray (3), and white (4 or 5).

**Figure 8.**Minnesota tile (5000 × 5000 30 m pixels) gap filling experiments, 25 600 × 600 30 m pixel areas, shown by white squares in the target week (

**b**), were removed to simulate gaps and then filled by the SAMSTS method (

**e**), and by closest preceding (

**d**) and subsequent (

**f**) pixel substitution. Associated RMSD images are shown (

**g**–

**i**) colored as for Figure 5. The temporally-closest non-missing preceding and subsequent observations did not always belong to the same image and were acquired from one to five weeks before (

**a**) and after (

**c**) the target week, colored as black (1), dark gray (2), light gray (3), and white (4 or 5).

**Figure 9.**Kansas tile (5000 × 5000 30 m pixels) gap filling experiments, 25 600 × 600 30 m pixel areas, shown by white squares in the target week (

**b**), were removed to simulate gaps and then filled by the SAMSTS method (

**e**), and by closest preceding (

**d**) and subsequent (

**f**) pixel substitution. Associated RMSD images are shown (

**g**–

**i**) colored as for Figure 5. The temporally-closest non-missing preceding and subsequent observations did not always belong to the same image and were acquired from one to five weeks before (

**a**) and after (

**c**) the target week, colored as black (1), dark gray (2), light gray (3), and white (4 or 5).

**Figure 11.**Sinusoidal harmonic model gap-filling results for the 25 600 × 600 30 m pixel simulated gap areas for the California (

**a**), Minnesota, (

**b**) and Kansas (

**c**) tiles. The original images before gap removal are illustrated in Figure 7b, Figure 8b, and Figure 9b. The associated RMSD gap-filling values are shown in (

**d**–

**f**) colored as for Figure 7, Figure 8 and Figure 9.

**Table 1.**Summary of missing weekly pixel observations (26 weeks from 28 April to 26 October 2013) in the three test tiles. Note that in the same week different Landsat-8 orbits can overpass the eastern and western sides of a 5000 × 5000 30 m tile.

Test Sites | Number of weeks (Out of 26) with at Least One Valid 30 m Pixel (n) in the Study Area | Percentage of Missing WELD Tile 30 m Pixel Observations over the 26 Weeks | Percentage of Missing Weekly WELD Tile 30 m Pixel Observations Computed over the n Weeks |
---|---|---|---|

California | 22 | 47.5% | 40.6% |

Minnesota | 22 | 54.2% | 45.9% |

Kansas | 20 | 46.2% | 30.1% |

Test Sites | All (Crop and Non-Corp) | Crop | Non-Crop | |||
---|---|---|---|---|---|---|

SAMSTS | Harmonic | SAMSTS | Harmonic | SAMSTS | Harmonic | |

California | 0.014 | 0.019 | 0.020 | 0.0300 | 0.012 | 0.016 |

Minnesota | 0.016 | 0.025 | 0.015 | 0.026 | 0.018 | 0.021 |

Kansas | 0.018 | 0.023 | 0.023 | 0.036 | 0.015 | 0.016 |

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**MDPI and ACS Style**

Yan, L.; Roy, D.P.
Large-Area Gap Filling of Landsat Reflectance Time Series by Spectral-Angle-Mapper Based Spatio-Temporal Similarity (SAMSTS). *Remote Sens.* **2018**, *10*, 609.
https://doi.org/10.3390/rs10040609

**AMA Style**

Yan L, Roy DP.
Large-Area Gap Filling of Landsat Reflectance Time Series by Spectral-Angle-Mapper Based Spatio-Temporal Similarity (SAMSTS). *Remote Sensing*. 2018; 10(4):609.
https://doi.org/10.3390/rs10040609

**Chicago/Turabian Style**

Yan, Lin, and David P. Roy.
2018. "Large-Area Gap Filling of Landsat Reflectance Time Series by Spectral-Angle-Mapper Based Spatio-Temporal Similarity (SAMSTS)" *Remote Sensing* 10, no. 4: 609.
https://doi.org/10.3390/rs10040609