A Kalman Filter-Based Method to Generate Continuous Time Series of Medium-Resolution NDVI Images
"> Figure 1
<p>Location of study sites in North America (<b>A</b>) and in Africa (<b>B</b>). Grey polygons indicate the swath of the Landsat scenes.</p> "> Figure 2
<p>Flow chart of the Kalman filter approach. At each time step, the transition model, <span class="html-italic">A</span>, projects the estimate from a previous state (time update). The time update combines the estimates of Submodels 1 and 2, <math display="inline"> <semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi>k</mi><mn>1</mn></mrow> <mo>−</mo> </msubsup> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi>k</mi><mn>2</mn></mrow> <mo>−</mo> </msubsup> </mrow> </semantics> </math>, respectively, to produce a single <span class="html-italic">a priori</span> estimate, <math display="inline"> <semantics> <mrow> <msubsup> <mi>X</mi> <mi>k</mi> <mo>−</mo> </msubsup> </mrow> </semantics> </math>. If available, a Landsat observation, <span class="html-italic">Z<sub>k</sub></span>, provides a new estimate for the state (measurement update). The final estimate of the state, <math display="inline"> <semantics> <mrow> <msub> <mover accent="true"> <mi>X</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </semantics> </math>, is the weighted average of the time update (transition model) and the measurement update (Landsat observation), with the weights inversely proportional to their respective uncertainties. If a new Landsat observation is not available, the estimate of the state at that time step is solely the result of the measurement update from previous time step. The model is alternatively run in forward and backward mode (filtering mode) and the corresponding estimates subsequently combined (smooth mode).</p> "> Figure 3
<p>MODIS, Landsat NDVI validation, synthetic NDVI images and spatially explicit residuals corresponding to one time step (<span class="html-italic">k</span> + 1) from the last Landsat observation for the 13 × 13-km subset of the site (A1–A5) and the 30 × 30-km subset of the Manitoba site (B1–B5). Prediction residual (A4 and B4) values are color coded: −0.05 ≤ purple, −0.05 < blue ≤ 0, 0 ≤ green < 0.2, 0.2 ≤ yellow < 0.4, 0.4 ≤ orange < 0.6, red ≥ 0.6. Plates A5 and B5 show residuals higher than ±0.25. (<b>A</b>) Zambezia site; (<b>B</b>) Arizona site.</p> "> Figure 3 Cont.
<p>MODIS, Landsat NDVI validation, synthetic NDVI images and spatially explicit residuals corresponding to one time step (<span class="html-italic">k</span> + 1) from the last Landsat observation for the 13 × 13-km subset of the site (A1–A5) and the 30 × 30-km subset of the Manitoba site (B1–B5). Prediction residual (A4 and B4) values are color coded: −0.05 ≤ purple, −0.05 < blue ≤ 0, 0 ≤ green < 0.2, 0.2 ≤ yellow < 0.4, 0.4 ≤ orange < 0.6, red ≥ 0.6. Plates A5 and B5 show residuals higher than ±0.25. (<b>A</b>) Zambezia site; (<b>B</b>) Arizona site.</p> "> Figure 4
<p>Scatterplot of 67 image-level predicted and temporal mean normalized residuals for the four study sites: red (Kansas); blue (Mozambique); green (Arizona); black (Manitoba). Symbols indicate the number of time steps away of the synthetic image from the last Landsat observation: one time step observation, <span class="html-italic">k</span> + 1, (square); two time steps, <span class="html-italic">k</span> + 2, (triangle); three time steps, <span class="html-italic">k</span> + 3, (circle). Points with very high mean normalized values in the temporal axis indicate the presence of undetected clouds.</p> "> Figure 5
<p>Scatterplot of pixel-level predicted and temporal normalized residuals corresponding to one time step (<span class="html-italic">k</span> + 1) from the last Landsat observation for the Kansas site (k = DOY 160; k + 1 = 176). The larger number of points above the 1:1 line and the larger range of normalized temporal residuals values indicate that a reliable synthetic NDVI image has been generated. The normalized predicted residuals of 48% of the pixels in the image were below 0.1; 28% between 0.1 and 0.2; 13% between 0.2 and 0.3; 13% between 0.3 and 0.4; and 5% were higher than 0.4.</p> "> Figure 6
<p>(<b>A</b>) Time series of NDVI image subsets (9 km × 9 km) from the MODIS NDVI (MOD13Q1) and combined mode Kalman simulation NDVI images for the Kansas site. The time series includes eight 16-day periods starting in early May (DOY 120) and finishing in late August (DOY 241). The dates in grey indicate the time steps for which Landsat images were used as model observations. (<b>B</b>) Time series of NDVI image subsets (48 km × 48 km) from the MODIS NDVI (MOD13Q1) and combined mode Kalman simulation NDVI images for the Mozambique site. The time series includes four 16-day periods starting in June (DOY 193) and finishing in late September (DOY 241). The date in grey indicates the time steps for which a Landsat image was used as the model observation.</p> "> Figure 6 Cont.
<p>(<b>A</b>) Time series of NDVI image subsets (9 km × 9 km) from the MODIS NDVI (MOD13Q1) and combined mode Kalman simulation NDVI images for the Kansas site. The time series includes eight 16-day periods starting in early May (DOY 120) and finishing in late August (DOY 241). The dates in grey indicate the time steps for which Landsat images were used as model observations. (<b>B</b>) Time series of NDVI image subsets (48 km × 48 km) from the MODIS NDVI (MOD13Q1) and combined mode Kalman simulation NDVI images for the Mozambique site. The time series includes four 16-day periods starting in June (DOY 193) and finishing in late September (DOY 241). The date in grey indicates the time steps for which a Landsat image was used as the model observation.</p> "> Figure 7
<p>NDVI pixel-level values for combined mode Kalman filter implementation time series for (<b>A</b>) Kansas, (<b>B</b>) Mozambique, (<b>C</b>) Manitoba and (<b>D</b>) Arizona for a variable number of Landsat images as model observations. The black line shows the NDVI time series retrieved from the Kalman filter implementation. The grey area indicates model uncertainties (±standard deviation from the model estimate). The red line shos the average seasonal NDVI pattern extracted from the MODIS NDVI 16-day composites (MOD13Q1). The blue dots indicate the time steps for which Landsat images where used as model observations. The red dots indicate the time steps for which Landsat images where not used as model observations. The images display snapshots of MODIS and synthetic Landsat images. The black circle in the NDVI snapshots indicates the location of the pixel shown in the time series.</p> "> Figure 7 Cont.
<p>NDVI pixel-level values for combined mode Kalman filter implementation time series for (<b>A</b>) Kansas, (<b>B</b>) Mozambique, (<b>C</b>) Manitoba and (<b>D</b>) Arizona for a variable number of Landsat images as model observations. The black line shows the NDVI time series retrieved from the Kalman filter implementation. The grey area indicates model uncertainties (±standard deviation from the model estimate). The red line shos the average seasonal NDVI pattern extracted from the MODIS NDVI 16-day composites (MOD13Q1). The blue dots indicate the time steps for which Landsat images where used as model observations. The red dots indicate the time steps for which Landsat images where not used as model observations. The images display snapshots of MODIS and synthetic Landsat images. The black circle in the NDVI snapshots indicates the location of the pixel shown in the time series.</p> "> Figure 8
<p>Sensitivity of temporal level mean uncertainties to the number of observations (Landsat scenes) used to generate the temporal sequence: (<b>A</b>) Kansas; (<b>B</b>) Mozambique; (<b>C</b>) Manitoba; (<b>D</b>) Arizona. Temporal sequence level mean uncertainties were calculated from all temporal sequences generated with the same number of Landsat observations in Monte Carlo simulations. NDVI uncertainties are4 expressed as the standard deviation of the NDVI value for that state. Forward mode recursion (blue line); backward mode recursion (red line); combined mode recursion (black line).</p> "> Figure 9
<p>NDVI pixel values for combined mode Kalman filter implementation time series for (<b>A</b>) Kansas, (<b>B</b>) Mozambique, (<b>C</b>) Manitoba and (<b>D</b>) Arizona for forward, backward and combined recursive modes. The black line shows the NDVI time series retrieved from the Kalman filter implementation. The grey area indicates model uncertainties (±standard deviation from model estimate). The red line shows the average seasonal NDVI pattern extracted from the MODIS NDVI 16-day composites (MOD13Q1). The blue dots indicate the time steps for which Landsat images where used as model observations. The blue dots indicate the time steps for which Landsat images were used as model observations.</p> ">
Abstract
:1. Introduction
2. Study Areas and Data
2.1. Study Areas
ID | Country | Province/State | Path/Row | Scene Center (Lat, Long) | Sensor |
---|---|---|---|---|---|
1 | U.S. | Kansas | P029R033 | 38.92, −98.92 | TM |
2 | Mozambique | Zambezia | P166R072 | −17.20, 36.42 | ETM+ |
3 | Canada | Manitoba | P033R021 | 55.93, −97.92 | TM, ETM+ |
4 | U.S. | Arizona | P036R036 | 34.80, −110.90 | TM |
ID | Province/State | Number of Scenes | Year | Acquisition Dates (DOY) |
---|---|---|---|---|
1 | Kansas | 10 | 2001 | 32; 128; 160; 176; 192; 224; 240; 256; 272; 288 |
2 | Zambezia | 6 | 2009 | 110; 158; 174; 270; 302; 318 |
3 | Manitoba | 4 | 2004 | 101; 149; 154; 237 |
4 | Arizona | 15 | 2004 | 10; 26; 42; 74; 90; 106; 122; 154; 170; 186; 250; 266; 282; 330; 346 |
2.2. Landsat Data
2.3. MODIS Data
ID | Country | Province/State | MODIS Tile | MOD13Q1 Date Ranges (Julian Day/Year) | Number of Composites |
---|---|---|---|---|---|
1 | U.S. | Kansas | h10v05 | January–December 2001 | 23 |
2 | Mozambique | Zambezia | h21v10 | January–December 2009 | 23 |
3 | Canada | Manitoba | h12v03 | January–December 2010 | 23 |
4 | U.S. | Arizona | h08v05 | January–December 2004 | 23 |
3. Methods
3.1. Kalman Filter Implementation
3.2. Accuracy Analysis
4. Results
Number Observations | Forward Mode | Backward Mode | Combined Mode | ||||||
---|---|---|---|---|---|---|---|---|---|
Mean | SD | Max | Mean | SD | Max | Mean | SD | Max | |
1 | 0.306 | 0.04 | 0.438 | 0.315 | 0.04 | 0.442 | 0.141 | 0.01 | 0.195 |
3 | 0.229 | 0.05 | 0.438 | 0.246 | 0.04 | 0.411 | 0.103 | 0.02 | 0.190 |
5 | 0.177 | 0.04 | 0.420 | 0.195 | 0.04 | 0.391 | 0.081 | 0.01 | 0.171 |
7 | 0.144 | 0.04 | 0.395 | 0.163 | 0.05 | 0.374 | 0.068 | 0.01 | 0.156 |
9 | 0.130 | 0.06 | 0.233 | 0.161 | 0.09 | 0.323 | 0.067 | 0.03 | 0.106 |
Number Observations | Forward Mode | Backward Mode | Combined Mode | ||||||
---|---|---|---|---|---|---|---|---|---|
Mean | SD | Max | Mean | SD | Max | Mean | SD | Max | |
1 | 0.200 | 0.03 | 0.328 | 0.189 | 0.02 | 0.308 | 0.088 | 0.01 | 0.126 |
3 | 0.143 | 0.04 | 0.237 | 0.130 | 0.04 | 0.259 | 0.055 | 0.02 | 0.108 |
5 | 0.116 | 0.07 | 0.210 | 0.094 | 0.06 | 0.184 | 0.039 | 0.02 | 0.068 |
Number Observations | Forward Mode | Backward Mode | Combined Mode | ||||||
---|---|---|---|---|---|---|---|---|---|
Mean | SD | Max | Mean | SD | Max | Mean | SD | Max | |
1 | 0.209 | 0.03 | 0.325 | 0.207 | 0.03 | 0.349 | 0.198 | 0.02 | 0.345 |
3 | 0.157 | 0.03 | 0.325 | 0.150 | 0.02 | 0.320 | 0.146 | 0.02 | 0.308 |
5 | 0.129 | 0.03 | 0.325 | 0.126 | 0.03 | 0.270 | 0.123 | 0.02 | 0.267 |
7 | 0.102 | 0.02 | 0.325 | 0.100 | 0.02 | 0.230 | 0.099 | 0.02 | 0.229 |
9 | 0.084 | 0.02 | 0.234 | 0.090 | 0.02 | 0.227 | 0.090 | 0.02 | 0.226 |
Number Observations | Forward Mode | Backward Mode | Combined Mode | ||||||
---|---|---|---|---|---|---|---|---|---|
Mean | SD | Max | Mean | SD | Max | Mean | SD | Max | |
1 | 0.265 | 0.07 | 0.388 | 0.274 | 0.09 | 0.376 | 0.112 | 0.02 | 0.152 |
2 | 0.210 | 0.11 | 0.340 | 0.228 | 0.12 | 0.375 | 0.084 | 0.04 | 0.132 |
5. Discussion
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Sedano, F.; Kempeneers, P.; Hurtt, G. A Kalman Filter-Based Method to Generate Continuous Time Series of Medium-Resolution NDVI Images. Remote Sens. 2014, 6, 12381-12408. https://doi.org/10.3390/rs61212381
Sedano F, Kempeneers P, Hurtt G. A Kalman Filter-Based Method to Generate Continuous Time Series of Medium-Resolution NDVI Images. Remote Sensing. 2014; 6(12):12381-12408. https://doi.org/10.3390/rs61212381
Chicago/Turabian StyleSedano, Fernando, Pieter Kempeneers, and George Hurtt. 2014. "A Kalman Filter-Based Method to Generate Continuous Time Series of Medium-Resolution NDVI Images" Remote Sensing 6, no. 12: 12381-12408. https://doi.org/10.3390/rs61212381