# Reconstructing Cloud Contaminated Pixels Using Spatiotemporal Covariance Functions and Multitemporal Hyperspectral Imagery

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}) pollution over Tehran [32], and estimating hourly average nitrogen dioxide (NO

_{2}) concentrations [33] over Milan, have achieved more reliable and accurate results compared to using more traditional covariance functions.

## 2. Materials and Methods

#### 2.1. Study Site

^{2}, growing a range of crops and vegetables across 47 center-pivots. The arid climate of this region is reflected by the 35 °C average daily maximum over the summer, and an average total annual precipitation of 90 mm, which is concentrated during the months of December through to April [34]. The site was chosen due to a range of ongoing hydrometeorological and remote sensing studies, which are supported by a dense network of ground-based infrastructure [35,36].

#### 2.2. Hyperion Data Pre-Processing

^{2}·sr

^{−1}·nm

^{−1}) and pre-processed to eliminate bad bands before conversion from at-sensor radiance to surface reflectance. The Fast Line-of-sight Atmospheric Analysis of Spectral Hypercubes (FLAASH) tool in ENVI 5.1, which is based on the Moderate-resolution atmospheric transmission (MODTRAN) model [38], was used to perform the atmospheric correction. Water vapor was calculated using the U.S standard atmosphere model, setting water vapor absorption bands around 1135 nm, based on the non-tropical moisture conditions [39,40]. In addition, an aerosol optical depth at 550 nm was retrieved considering the standard MODTRAN maritime model. From Table 1, the 10 to 20 km visibility suggests the presence of larger particles. Here, the maritime aerosol is the best option since it gives a more realistic wavelength dependence and the desert model is not available in FLAASH. Visibility was estimated per scene via the Koschmieder Equation (1) [41], using aerosol optical depth (AOD) at 550 nm from the MOD04 product (see Table 1):

^{−1}is the Rayleigh scattering coefficient for air at the surface at 550 nm. E550 is estimated by dividing AOD by an effective layer thickness of 2 km [40,41].

_{550}(2 km). Multiple scattering in short wavelengths (blue-red) is corrected by using the high-scatter DISORT model [44].

^{2}and covering a total of 4446 “cloud” pixels (Figure 1).

#### 2.3. Spatiotemporal Statistics Approach

#### 2.4. Block Sampling Scheme

#### 2.5. Spatiotemporal Covariance Function

_{1},λ

_{1},t

_{1}),…,(s

_{i},λ

_{b},t

_{j}), corresponding to the center point of a pixel at location s, band λ, and time t. It is assumed that Z(s,λ) has a mean function, μ(s,t) = E(Z(s,t)), and its covariance structure is stationary in space and time. Thus, Cov(Z(s,t),Z(s + h,t + u)) = C(h,u), where the covariance depends on the space and time lags, h and u. If u = 0, the covariance function, C(h,0), is purely spatial. Likewise, if h = 0, the covariance function, C(0,u), is considered purely temporal [54]. In contrast to separable models, non-separable functions have a different structure. One such example is that proposed by Gneiting (2) [55], which takes into account space-time interactions when h > 0 and u > 0:

^{2}is the variance of the process, φ(t) is any completely monotone function and represents the spatial structure of the covariance, and ψ(t) is any positive function with a completely monotonic derivative and represents the temporal structure of the covariance.

#### 2.6. Exploratory Data Analysis

#### 2.7. Spatiotemporal Covariance Model Fitting and Kriging

_{s},k

_{t},ψ

_{s},ψ

_{t}), given the observed data Z* = (Z*(s

_{1};t

_{1}),…,Z*(s

_{i};t

_{j})) expressed as (7):

_{ε}

^{2}. If the aim is to predict the gap positions, Y($s$

_{0};t

_{0}), for the image at time t

_{8}from the observed data, Z*, the simple Kriging predictor, $\widehat{Y}$, is defined as (12):

#### 2.8. Evaluation of Results

_{8}) over four different free-cloud scenes, t

_{1}, t

_{2}, t

_{3}, and t

_{4}. The predictions are performed using a subset of three concentric blocks, in order to assess the accuracy of the model under high percentages of clouds, different block sizes, and different times. The defined sizes were 30 × 30 (area 1), 60 × 60 (area 2), and 90 × 90 (area 3) pixels, with 95%, 75%, and 55% of cloud cover levels, respectively (see Figure 8) (see Table 3). Each round of evaluation involves masking one block per time and per band, then fitting the model using the cloud-free data series and retrieving the prediction on the subset of masked pixels.

## 3. Results

#### 3.1. Spatiotemporal Covariance Model

_{8}, a spatiotemporal covariance function was fitted for predicting reflectance across cloud affected pixels. The sensitivity of the model was evaluated based on the separability, $\eta $, which is the only free-parameter, considering three different scenarios. In accordance with Gneiting [54], as $\eta $ increases, the space–time interaction strengthens. Table 4 presents the estimated parameters obtained by the composite maximum likelihood estimator.

_{s},k

_{t}) define the smooth fitting of the curve, and how the scale parameters define the ranges in time (ψ

_{t}) and distance (ψ

_{s}), where the models first flatten out to reach a sill. The lower AIC value reached by the last model in Table 4 (when $\eta =1$), suggests that the non-separable method performs the best fit, which means that for this dataset, a model that accounts for time–space interactions is suitable. Both spatial and temporal variograms provide information about how much two sampled pixels vary in reflectance percentage, depending on the distance and time between them. Following the spatial autocorrelation principle [53], pixels that are closer should have similar values, and as they become farther apart, they should be more dissimilar. In the case of boundary areas where two or more land cover types converge, the spatial autocorrelation will be low and the estimation will rely mostly on the temporal component of the model. At some distance and lapse of time, in this case, 200 meters and 2 weeks, the differences between pixels become fairly constant and the spatial and temporal variograms flattens out into a maximum variance of 0.91 and 0.09, respectively (see Figure 9b). This means that from 0 to 200 meters of spatial distance, and considering up to 2 weeks of temporal difference, the pixel reflectance can still be considered autocorrelated.

#### 3.2. Reflectance Predictions

_{8}(top right panel), zooming in on the initial cloud affected area. A visual inspection of the results shows how cloudy areas with different shapes and sizes were filled with predicted reflectance values over heterogeneous land covers, including grass, maize (at maturity stage), bare-field soil, and desert soil. Clear spatial patterns and features in the original image are retained: For instance, the uniformity in circle pivot edges, the linear features of some trails, and the center pivot locations.

#### 3.3. Evaluation of Cloud-Free Imagery

_{1}, t

_{2}, t

_{3}, and t

_{4}, by superimposing an artificial cloud and performing the prediction on each one of these. Figure 11 presents the achieved root mean square error (RMSE) and relative root mean square error (rRMSE) per band (rows), for each time and sampled area (columns). As can be seen, low RMSE and rRMSE are achieved for most of the bands. Those in the visible spectrum from 420 to 700 nm and in the shortwave infrared from 2300 to 2350 nm, show the lowest RMSE values (0.5%–3% of absolute reflectance), which amount to an rRMSE of between 0.3% and 2%. In contrast, the largest RMSE (13%–16% of absolute reflectance) were estimated for shortwave infrared channels, corresponding to an rRMSE between 6% and 7% for those bands centered on 1500 nm, and between 3% and 4% for those centered on 2000 nm. It is evident that the accuracy of the model varies depending on the block size, but not on the cloud percentage, since there is an incremental RMSE as the block size grows while the cloud area decreases (Figure 8). Comparing the residual results in time, there is a slight incremental trend from t

_{1}to t

_{4}predictions, although that trend strengthens as the area size increases. For area 1, the minimum RMSE (0.5% of absolute reflectance) is achieved when t

_{1}is modeled, and the maximum RMSE of around 11% and 12% when t

_{4}is considered. For the larger prediction blocks (area 2 and area 3), higher residuals start to appear at a range of 13% to 14% and 14% to 16%, respectively. Consistent with the tendency observed in the RMSE, the prediction errors show overall low variances for the evaluated scenarios with some high rRMSE values for those bands centered on the 2000 nm wavelength (Figure 11b).

## 4. Discussion

#### 4.1. Spatiotemporal Statistical Model for Predicting Reflectance

#### 4.2. Assessment of the Spectral Consistency

#### 4.3. Block Sampling Strategy for Predictions in Heterogeneous Areas

#### 4.4. Computational Efficiency

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## References

- Gamon, J.A.; Qiu, H.; Sanchez-Azofeifa, A. Ecological applications of remote sensing at multiple scales. In Functional Plant Ecology, 2nd ed.; Pugnaire, F., Valladares, F., Eds.; CRC Press: Boca Raton, FL, USA, 2007; Volume 1, pp. 655–675. [Google Scholar]
- Ustin, S.L.; Gamon, J.A. Remote sensing of plant functional types. New Phytol.
**2010**, 186, 795–816. [Google Scholar] [CrossRef] [Green Version] - Malenovský, Z.; Mishra, K.; Zemek, F.; Rascher, U.; Nedbal, L. Scientific and technical challenges in remote sensing of plant canopy reflectance and fluorescence. J. Exp. Bot.
**2009**, 60, 2987–3004. [Google Scholar] [CrossRef] [Green Version] - Dale, L.; Thewis, A.; Boudry, C.; Rotar, L.; Dardenne, P.; Baeten, V.; Fernández, J.A. Hyperspectral imaging applications in agriculture and agro-food product quality and safety control: A review. Appl. Spectrosc. Rev.
**2013**, 48, 142–159. [Google Scholar] [CrossRef] - Preliminary Assessment of the Value of Landsat 7 ETM+ SLC-off Data. Available online: https://landsat.usgs.gov/sites/default/files/documents/SLC_off_Scientific_Usability.pdf (accessed on 8 April 2019).
- Tanre, D.; Deschamps, P.Y.; Devaux, C.; Herman, M. Estimation of Saharan aerosol optical thickness from blurring effects in thematic mapper data. J. Geophys. Res.
**1988**, 93, 15955–15964. [Google Scholar] [CrossRef] - Adler-Golden, S.M.; Robertson, D.C.; Richtsmeier, S.C.; Ratkowski, A.J. Cloud effects in hyperspectral imagery from first-principles scene simulations. In Proceedings of the SPIE Defense, Security, and Sensing, Orlando, FL, USA, 27 April 2009. [Google Scholar]
- Validation of On-Board Cloud Cover Assessment Using EO-1. Available online: https://eo1.gsfc.nasa.gov/new/extended/sensorWeb/EO-1_Validation On-board Cloud Assessment_Rpt.pdf (accessed on 8 April 2019).
- Ju, J.; Roy, D.P. The availability of cloud-free Landsat ETM Plus data over the conterminous United States and globally. Remote Sens. Environ.
**2007**, 112, 1196–1211. [Google Scholar] [CrossRef] - Lin, C.H.; Tsai, P.H.; Lai, K.H.; Chen, J.Y. Cloud removal from multitemporal satellite images using information cloning. IEEE Trans. Geosci. Remote Sens.
**2013**, 51, 232–241. [Google Scholar] [CrossRef] - Shen, H.; Xinghua, L.; Cheng, Q.; Zeng, C.; Yang, G.; Li, H.; Zhang, L. Missing Information Reconstruction of Remote Sensing Data: A Technical Review. IEEE Geosci. Remote Sens. Mag.
**2015**, 3, 61–85. [Google Scholar] [CrossRef] - Helmer, E.; Ruefenacht, B. Cloud-free satellite image mosaics with regression trees and histogram matching. Photogramm. Eng. Remote Sens.
**2005**, 71, 1079–1089. [Google Scholar] [CrossRef] - Benabdelkader, S.; Melgani, F. Contextual spatiospectral postreconstruction of cloud-contaminated images. IEEE Geosc. Remote Sens.
**2008**, 5, 204–208. [Google Scholar] [CrossRef] - Melgani, F. Contextual reconstruction of cloud-contaminated multitemporal multispectral images. IEEE Trans. Geosci. Remote Sens.
**2006**, 44, 442–455. [Google Scholar] [CrossRef] - Shen, H.; Wu, J.; Cheng, Q.; Aihemaiti, M.; Zhang, C.; Li, Z. A spatiotemporal fusion based cloud removal method for remote sensing images with land cover changes. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2019**, 12, 862–874. [Google Scholar] [CrossRef] - Chang, N.B.; Bai, K.; Chen, C.F. Smart information reconstruction via time-space-spectrum continuum for cloud removal in satellite images. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2015**, 8, 1898–1912. [Google Scholar] [CrossRef] - Wang, B.; Ono, A.; Muramatsu, K.; Fujiwarattt, N. Automated detection and removal of clouds and their shadows from Landsat TM images. IEICE Trans. Inf. Syst.
**1999**, 82, 453–460. [Google Scholar] - Gabarda, S.; Cristóbal, G. Cloud covering denoising through image fusion. Image Vis. Comput.
**2007**, 25, 523–530. [Google Scholar] [CrossRef] [Green Version] - Roerink, G.J.; Menenti, M.; Verhoef, W. Reconstructing cloudfree NDVI composites using Fourier analysis of time series. Int. J. Remote Sens.
**2010**, 21, 1911–1917. [Google Scholar] [CrossRef] - Mariethoz, G.; McCabe, M.F.; Renard, P. Spatiotemporal reconstruction of gaps in multivariate fields using the direct sampling approach. Water Resour. Res.
**2012**, 48, W10507. [Google Scholar] [CrossRef] - Jha, S.K.; Mariethoz, G.; Evans, J.P.; McCabe, M.F. Demonstration of a geostatistical approach to physically consistent downscaling of climate modeling simulations. Water Resour. Res.
**2013**, 49, 245–259. [Google Scholar] [CrossRef] [Green Version] - Zhang, C.; Li, W.; Travis, D.J. Restoration of clouded pixels in multispectral remotely sensed imagery with cokriging. Int. J. Remote Sens.
**2009**, 30, 2173–2195. [Google Scholar] [CrossRef] - Meng, Q.M.; Borders, B.E.; Cieszewski, C.J.; Madden, M. Closest spectral fit for removing clouds and cloud shadows. Photogramm. Eng. Remote Sens.
**2009**, 75, 569–576. [Google Scholar] [CrossRef] - Cheng, Q.; Shen, H.; Zhang, L.; Yuan, Q.; Zeng, C. Cloud removal for remotely sensed images by similar pixel replacement guided with a spatio-temporal MRF model. ISPRS J. Photogramm. Remote Sens.
**2014**, 92, 54–68. [Google Scholar] [CrossRef] - Cerra, D.; Müller, R.; Reinartz, P. Cloud removal in image time series through unmixing. In Proceedings of the 8th International Workshop on the Analysis of Multitemporal Remote Sensing Images, Annecy, France, 22–24 July 2015. [Google Scholar]
- Xu, M.; Pickering, M.; Plaza, A.J.; Jia, X. Thin cloud removal based on signal transmission principles and spectral mixture analysis. IEEE Trans. Geosci. Remote Sens.
**2016**, 54, 1659–1669. [Google Scholar] [CrossRef] - Feng, W.; Chen, Q.; He, W.; Gu, G.; Zhuang, J.; Xu, S. A defogging method based on hyperspectral unmixing. Acta Opt. Sin.
**2015**, 35, 115–122. [Google Scholar] - Yin, G.; Mariethoz, G.; McCabe, M.F. Gap-filling of landsat 7 imagery using the direct sampling method. Remote Sens.
**2017**, 9, 12. [Google Scholar] [CrossRef] - Shekhar, S.; Jiang, Z.; Ali, R.Y.; Eftelioglu, E.; Tang, X.; Gunturi, V.M.V.; Zhou, X. Spatiotemporal data mining: A computational perspective. ISPRS Int. J. Geo-Inf.
**2015**, 4, 2306–2338. [Google Scholar] [CrossRef] - Gneiting, T.; Genton, M.G.; Guttorp, P. Geostatistical space-time models, stationarity, separability and full symmetry. In Statistical Methods for Spatio-Temporal Systems, 1st ed.; Finkenstadt, B., Held, L., Isham, V., Eds.; Chapman and Hall/CRC: New York, NY, USA, 2006; Volume 1, pp. 151–175. [Google Scholar]
- De Iaco, S.; Palma, M.; Posa, D. A general procedure for selecting a class of fully symmetric space-time covariance functions. Environmetrics
**2016**, 27, 212–224. [Google Scholar] [CrossRef] - Omidi, M.; Mohammadzadeh, M. A new method to build spatio-temporal covariance functions: Analysis of ozone data. Stat. Pap.
**2015**, 57, 689–703. [Google Scholar] [CrossRef] - De Iaco, S.; Myers, D.E.; Posa, D. Nonseparable space-time covariance models: Some parametric families. Math. Geol.
**2002**, 34, 23–42. [Google Scholar] [CrossRef] - Historical Weather for 2015 in Al-Kharj Prince Sultan Air Base, Saudi Arabia. WeatherSpark, 2015. Available online: http://weatherspark.com/history/32768/2015/Al-Kharj-Riyadh-Saudi-Arabia (accessed on 8 April 2019).
- Houborg, R.; McCabe, M.F. Adapting a regularized canopy reflectance model (REGFLEC) for the retrieval challenges of dryland agricultural systems. Remote Sens. Environ.
**2016**, 186, 105–112. [Google Scholar] [CrossRef] - El Kenawy, A.M.; McCabe, M.F. A multi-decadal assessment of the performance of gauge and model based rainfall products over Saudi Arabia: Climatology, anomalies and trends. Int. J. Climatol.
**2016**, 36, 656–674. [Google Scholar] [CrossRef] - Folkman, M.A.; Pearlman, J.; Liao, L.B.; Jarecke, P.J. EO-1/Hyperion hyperspectral imager design, development, characterization, and calibration. Proc. SPIE
**2001**, 4151, 40–51. [Google Scholar] [Green Version] - Perkins, T.; Adler-Golden, S.M.; Matthew, M.W. Speed and accuracy improvements in FLAASH atmospheric correction of hyperspectral imagery. Opt. Eng.
**2012**, 51, 111707. [Google Scholar] [CrossRef] - Felde, G.W.; Anderson, G.P.; Gardner, J.A.; Adler-Golden, S.M.; Matthew, M.W.; Berk, A. Water vapor retrieval using the FLAASH atmospheric correction algorithm. In Proceedings of the SPIE Defense and Security, Orlando, FL, USA, 12 August 2004. [Google Scholar]
- Rochford, P.; Acharya, P.; Adler-Golden, S.M. Validation and refinement of hyperspectral/multispectral atmospheric compensation using shadowband radiometers. IEEE Trans. Geosci. Remote Sen.
**2005**, 43, 2898–2907. [Google Scholar] [CrossRef] - Griffin, M.K.; Burke, H.K. Compensation of hyperspectral data for atmospheric effects. Linc. Lab. J.
**2003**, 14, 29–54. [Google Scholar] - Houborg, R.; McCabe, M.F. Impacts of dust aerosol and adjacency effects on the accuracy of Landsat 8 and RapidEye surface reflectances. Remote Sens. Environ.
**2017**, 194, 127–145. [Google Scholar] [CrossRef] - Adler-Golden, S.M.; Matthew, M.W.; Berk, A.; Fox, M.J.; Ratkowski, A.J. Improvements in aerosol retrieval for atmospheric correction. In Proceedings of the IGARSS 2008—2008 IEEE International Geoscience and Remote Sensing Symposium, Boston, MA, USA, 7–11 July 2008. [Google Scholar]
- Stamnes, K.; Tsay, S.C.; Wiscombe, W.; Jayaweera, K. Numerically stable algorithm for Discrete-Ordinate-Method Radiative Transfer in multiple scattering and emitting layered media. Appl. Opt.
**1988**, 27, 2502–2509. [Google Scholar] [CrossRef] - Staenz, K.; Neville, R.A.; Clavette, S.; Landry, R.; White, H.P. Retrieval of surface reflectance from Hyperion radiance data. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Toronto, ON, Canada, 24–28 June 2002. [Google Scholar]
- Pearlman, J.S.; Barry, P.S.; Segal, C.C.; Shepanski, J.; Beiso, D.; Carman, S.L. Hyperion, a space-based imaging spectrometer. IEEE Trans. Geosci. Remote Sens.
**2003**, 41, 1160–1173. [Google Scholar] [CrossRef] - Matthew, M.W.; Adler-Golden, S.M.; Berk, A.; Felde, G.; Anderson, G.P.; Gorodetzkey, D.; Paswaters, S.; Shippert, M. Atmospheric correction of spectral imagery: Evaluation of the FLAASH algorithm with AVIRIS data. In Proceedings of the Applied Imagery Pattern Recognition Workshop, Washington, DC, USA, 17–23 October 2003. [Google Scholar]
- Green, A.A.; Berman, M.; Switzer, P.; Craig, M.D. A Transform for ordering multispectral data in terms of image quality with implications for noise removal. IEEE Int. Geosci. Remote Sens.
**1988**, 26, 65–74. [Google Scholar] [CrossRef] - Matthew, M.W.; Adler-Golden, S.M.; Berk, A.; Richtsmeier, S.C.; Levine, R.Y.; Bernstein, L.S.; Acharya, P.K.; Anderson, G.P.; Felde, G.W.; Hoke, M.P.; et al. Status of atmospheric correction using a modtran4-based algorithm. In Proceedings of the SPIE AeroSense 2000, Orlando, FL, USA, 23 August 2000. [Google Scholar]
- Cochran, W.G. Sampling Techniques, 3rd ed.; John Wiley & Sons: New York, NY, USA, 1977. [Google Scholar]
- Wang, J.F.; Haining, R.P.; Cao, Z.D. Sample surveying to estimate the mean of a heterogeneous surface: Reducing the error variance through zoning. Int. J. Geogr. Inf. Sci.
**2010**, 24, 523–543. [Google Scholar] [CrossRef] - Wang, J.F.; Stein, A.; Gao, B.; Ge, Y. A review of spatial sampling. Spat. Stat.
**2012**, 2, 1–14. [Google Scholar] [CrossRef] - Montero, J.M.; Fernández, G.; Mateu, J. Spatial and Spatio-Temporal Geostatistical Modeling and Kriging, 1st ed.; John Wiley & Sons: Chichester, UK, 2015; pp. 178–265. [Google Scholar]
- Cressie, N.; Huang, H. Classes of nonseparable, spatio-temporal stationary covariance functions. J. Am. Stat. Assoc.
**1999**, 94, 1330–1340. [Google Scholar] [CrossRef] - Gneiting, T. Nonseparable, stationary covariance functions for space-time data. J. Am. Stat. Assoc.
**2002**, 97, 590–600. [Google Scholar] [CrossRef] - Tukey, J. On the comparative anatomy of transformations. Ann. Math. Stat.
**1957**, 28, 602–632. [Google Scholar] [CrossRef] - Package CompRandFld. R Package Version 1.0.3-4. Available online: https://cran.r-project.org/package=CompRandFld (accessed on 8 April 2019).
- R Core Team. R: A Language and Environment for Statistical Computing. R Package Version 3.2.2. Available online: http://www.R-project.org (accessed on 8 April 2019).
- Zimmerman, D.L.; Zimmerman, M.B. A comparison of spatial semivariogram estimators and corresponding ordinary kriging predictors. Technometrics
**1991**, 33, 77–91. [Google Scholar] [CrossRef] - Curriero, F.; Lele, S. A Composite Likelihood Approach to Semivariogram Estimation. J. Agric. Biol. Environ. Stat.
**1999**, 4, 9–28. [Google Scholar] [CrossRef] - Padoan, S.; Bevilacqua, M. Analysis of random fields using CompRandFld. J. Stat. Softw.
**2015**, 63, 1–27. [Google Scholar] [CrossRef] - Cressie, N.; Wikle, C. Statistics for Spatio-Temporal Data, 1st ed.; John Wiley & Sons: Chichester, UK, 2011. [Google Scholar]
- Zhang, C.; Li, W.; Travis, D. Gaps-fill of SLC-off Landsat ETM+ satellite image using a geostatistical approach. Int. J. Remote Sens.
**2007**, 28, 5103–5122. [Google Scholar] [CrossRef] - Pringle, M.J.; Schmidt, M.; Muir, J.S. Geostatistical interpolation of SLC-off Landsat ETM+ images. ISPRS J. Photogramm. Remote Sens.
**2009**, 64, 654–664. [Google Scholar] [CrossRef] - Webster, R.; Oliver, M.A. Geostatistics for Environmental Scientists, 1st ed.; John Wiley & Sons: Chichester, UK, 2007. [Google Scholar]
- Porcu, E.; Bevilacqua, M.; Genton, M. Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. J. Am. Stat. Assoc.
**2016**, 111, 888–898. [Google Scholar] [CrossRef] - Genton, M.; Kleiber, W. Cross-covariance functions for multivariate geostatistics. Stat. Sci.
**2015**, 30, 147–163. [Google Scholar] [CrossRef] - Jun, M. Non-stationary cross-covariance models for multivariate processes on a globe. Scand. J. Stat.
**2011**, 38, 726–747. [Google Scholar] [CrossRef] - Thome, K.J.; Biggar, S.F.; Wisniewski, W. Cross comparison of EO-1 sensors and other Earth resources sensors to Landsat-7 ETM+ using Railroad Valley Playa. IEEE Trans. Geosci. Remote Sens.
**2003**, 41, 1180–1188. [Google Scholar] [CrossRef] - Datt, B.; Jupp, D.L.B. Hyperion Data Processing Workshop, Hands-On Processing Instructions; CSIRO Office of Space Science & Applications Earth Observation Centre: Canberra, Australia, 2004. [Google Scholar]
- Cressie, N.; Johannesson, G. Fixed rank kriging for very large spatial data sets. J. R. Stat. Soc. Ser. B Stat. Methodol.
**2008**, 70, 209–226. [Google Scholar] [CrossRef]

**Figure 1.**Location of the region of interest covered by each Hyperion swath at eight different times. Color composition for all images is described by an infrared false color, with R: Band 45 (862.01 nm) G: Band 23 (650.66 nm) B: Band 13 (548.92 nm). Crop circles correspond to alfalfa, corn, and grass fields (approximately 800 m in diameter) at different growth stages.

**Figure 3.**Cloudy areas masked (blue) from the t

_{8}scene are divided into 15 different regular blocks accounting for different cloud cover percentages.

**Figure 4.**Reflectance histograms per band for a block with 22% of cloud cover (A

_{1}), based on the image t

_{8}.

**Figure 5.**Comparison between multitemporal reflectance histograms (

**a**) for band 1 (426 nm), and standardized reflectance histograms (

**b**) after applying a transformation.

**Figure 6.**Directional empirical semi-variograms captured on September 18. (

**a**) Semi-variograms for band 13 centered at 549 nm, reached the sill at a range of 240 m. (

**b**) Semi-variograms for band 23 centered at 650 nm, reached the sill at a range of 180 m.

**Figure 7.**Multi-temporal linear Pearson correlation matrix for all datacubes. Grayscale indicates the correlation level.

**Figure 8.**Artificial cloud superimposed over cloud-free areas, which is then used to assess the spatiotemporal model performance at four different times (t

_{1}, t

_{2}, t

_{3}, t

_{4}).

**Figure 9.**Spatiotemporal variograms for band 45 (862.01nm). (

**a**) 3D empirical variogram. (

**b**) 3D fitted variogram. (

**c**) Fitted separable and non-separable temporal variogram. (

**d**) Fitted separable and non-separable spatial variogram.

**Figure 10.**(

**a**) True color composite of the reconstructed image (before and after), together with a comparison between the cloud resolved (predicted average) and cloud-free (true average) spectral profile for different land covers, including (

**b**) green grass, (

**c**) desert, (

**d**) bare soil, (

**e**) maize, and (

**f**) green grass. Gray bars indicate non-observed wavelength ranges.

**Figure 11.**Evaluation results of the gap-filling spatiotemporal model showing: (

**a**) color-scale image of the RMSE (% of absolute reflectance units) for each one of the 166 bands (rows) per prediction time (columns) and arranged in ascending order from left to right according to the sampled area. (

**b**) Color-scale image of the corresponding rRMSE achieved (% units).

**Figure 12.**Evaluation of the spectral accuracy of the gap-filling spatiotemporal model, showing (

**a**) spectral angle difference per retrieved time per each sampled area, and (

**b**) MSA metric per retrieved time per each sampled area.

Time (t) | Date | DOY | Time (UTC + 3) | Cloud (%) | AOD τ _{550} | Visibility (km) | Sensor Azimuth (°) | Sensor Zenith (°) | Sensor Look Angle (°) |
---|---|---|---|---|---|---|---|---|---|

t_{1} | 18-Sep-15 | 261 | 8:54:05 | 2 | 0.405 | 18 | 112.475 | 153.778 | 26.222 |

t_{2} | 26-Sep-15 | 269 | 8:48:59 | 2 | 0.194 | 36 | 115.518 | 162.033 | 17.967 |

t_{3} | 4-Oct-15 | 277 | 8:43:00 | 2 | 0.366 | 20 | 118.321 | 171.426 | 8.5741 |

t_{4} | 12-Oct-15 | 285 | 8:37:22 | 4 | 0.460 | 16 | 120.731 | 181.554 | -1.554 |

t_{5} | 20-Oct-15 | 293 | 8:31:39 | 4 | 0.564 | 13 | 122.827 | 168.255 | 11.745 |

t_{6} | 28-Oct-15 | 301 | 8:25:51 | 9 | 0.530 | 14 | 124.538 | 158.672 | 21.328 |

t_{7} | 3-Nov-15 | 307 | 8:45:51 | 4 | 0.490 | 15 | 130.147 | 159.946 | 20.054 |

t_{8} | 11-Nov-15 | 315 | 8:39:59 | 20 | 0.587 | 13 | 130.944 | 169.782 | 10.218 |

Block | Clouds % | Block | Clouds % |
---|---|---|---|

A1 | 22 | A9 | 25 |

A2 | 30 | A10 | 28 |

A3 | 23 | A11 | 90 |

A4 | 17 | A12 | 18 |

A5 | 20 | A13 | 20 |

A6 | 95 | A14 | 32 |

A7 | 83 | A15 | 5 |

A8 | 10 |

Block | Size | % of Clouds |
---|---|---|

Area 1 | 30 × 30 | 95 |

Area 2 | 60 × 60 | 75 |

Area 3 | 90 × 90 | 55 |

Separability $\mathit{\eta}$ | Power Space ${\mathit{k}}_{\mathit{s}}$ | Power Time ${\mathit{k}}_{\mathit{t}}$ | Scale Space ${\mathit{\psi}}_{\mathit{s}}$ | Scale Time ${\mathit{\psi}}_{\mathit{t}}$ | Spatial Sill | Temporal Sill | AIC |
---|---|---|---|---|---|---|---|

0 | 0.25 | 2 | 400 | 6 | 0.96 | 0.16 | 6330.44 |

0.5 | 1.50 | 1 | 290 | 4 | 0.93 | 0.11 | 6311.78 |

1 | 1.50 | 0.25 | 200 | 2 | 0.91 | 0.09 | 6305.72 |

Author | Covariance Function | Domains | Fitting Method | Kriging |
---|---|---|---|---|

Zhang et al. [61] | Exponential | Spatial (2D) + Temporal (1D) | Full likelihood | Ordinary, Co-Kriging |

Pringle et al. [62] | Double spherical | Spatial (2D) + Temporal (1D) | Simulated annealing | Co-Kriging |

Here proposed | Gneiting | Spatio-temporal (3D) | Composite marginal likelihood | Simple |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Angel, Y.; Houborg, R.; McCabe, M.F.
Reconstructing Cloud Contaminated Pixels Using Spatiotemporal Covariance Functions and Multitemporal Hyperspectral Imagery. *Remote Sens.* **2019**, *11*, 1145.
https://doi.org/10.3390/rs11101145

**AMA Style**

Angel Y, Houborg R, McCabe MF.
Reconstructing Cloud Contaminated Pixels Using Spatiotemporal Covariance Functions and Multitemporal Hyperspectral Imagery. *Remote Sensing*. 2019; 11(10):1145.
https://doi.org/10.3390/rs11101145

**Chicago/Turabian Style**

Angel, Yoseline, Rasmus Houborg, and Matthew F. McCabe.
2019. "Reconstructing Cloud Contaminated Pixels Using Spatiotemporal Covariance Functions and Multitemporal Hyperspectral Imagery" *Remote Sensing* 11, no. 10: 1145.
https://doi.org/10.3390/rs11101145