Abstract. Moderate resolution satellite sensors including MODIS (Moderate Resolution Imaging Spectroradiometer) already provide more than 10 yr of observations well suited to describe and understand the dynamics of earth's surface. However, these time series are associated with significant uncertainties and incomplete because of cloud cover. This study compares eight methods designed to improve the continuity by filling gaps and consistency by smoothing the time course. It includes methods exploiting the time series as a whole (iterative caterpillar singular spectrum analysis (ICSSA), empirical mode decomposition (EMD), low pass filtering (LPF) and Whittaker smoother (Whit)) as well as methods working on limited temporal windows of a few weeks to few months (adaptive Savitzky–Golay filter (SGF), temporal smoothing and gap filling (TSGF), and asymmetric Gaussian function (AGF)), in addition to the simple climatological LAI yearly profile (Clim). Methods were applied to the MODIS leaf area index product for the period 2000–2008 and over 25 sites showed a large range of seasonal patterns. Performances were discussed with emphasis on the balance achieved by each method between accuracy and roughness depending on the fraction of missing observations and the length of the gaps. Results demonstrate that the EMD, LPF and AGF methods were failing because of a significant fraction of gaps (more than 20%), while ICSSA, Whit and SGF were always providing estimates for dates with missing data. TSGF (Clim) was able to fill more than 50% of the gaps for sites with more than 60% (80%) fraction of gaps. However, investigation of the accuracy of the reconstructed values shows that it degrades rapidly for sites with more than 20% missing data, particularly for ICSSA, Whit and SGF. In these conditions, TSGF provides the best performances that are significantly better than the simple Clim for gaps shorter than about 100 days. The roughness of the reconstructed temporal profiles shows large differences between the various methods, with a decrease of the roughness with the fraction of missing data, except for ICSSA. TSGF provides the smoothest temporal profiles for sites with a % gap > 30%. Conversely, ICSSA, LPF, Whit, AGF and Clim provide smoother profiles than TSGF for sites with a % gap < 30%. Impact of the accuracy and smoothness of the reconstructed time series were evaluated on the timing of phenological stages. The dates of start, maximum and end of the season are estimated with an accuracy of about 10 days for the sites with a % gap < 10% and increases rapidly with the % gap. TSGF provides more accurate estimates of phenological timing up to a % gap < 60%.

Moderate resolution satellite sensors including MODIS already provide more than 10 yr of observations well suited to describe and understand the dynamics of the Earth surface. However, these time series are incomplete because of cloud cover and associated with significant uncertainties. This study compares eight methods designed to improve the continuity by filling gaps and the consistency by smoothing the time course. It includes methods exploiting the time series as a whole (Iterative caterpillar singular spectrum analysis (ICSSA), empirical mode decomposition (EMD), low pass filtering (LPF) and Whittaker smoother (Whit)) as well as methods working on limited temporal windows of few weeks to few months (Adaptive Savitzky-Golay filter (SGF), temporal smoothing and gap filling (TSGF) and asymmetric Gaussian function (AGF)) in addition to the simple climatological LAI yearly profile (Clim). Methods were applied to MODIS leaf area index product for the period 2000–2008 over 25 sites showing a large range of seasonal patterns. Performances were discussed with emphasis on the balance achieved by each method between accuracy and roughness depending on the fraction of missing observations and the length of the gaps. Results demonstrate that EMD, LPF and AGF methods were failing in case of significant fraction of gaps (%Gap > 20%), while ICSSA, Whit and SGF were always providing estimates for dates with missing data. TSGF (respectively Clim) was able to fill more than 50% of the gaps for sites with more than 60% (resp. 80%) fraction of gaps. However, investigation of the accuracy of the reconstructed values shows that it degrades rapidly for sites with more than 20% missing data, particularly for ICSSA, Whit and SGF. In these conditions, TSGF provides the best performances significantly better than the simple Clim for gaps shorter than about 100 days. The roughness of the reconstructed temporal profiles shows large differences between the several methods, with a decrease of the roughness with the fraction of missing data, except for ICSSA. TSGF provides the smoothest temporal profiles for sites with %Gap > 30%. Conversely, ICSSA, LPF, Whit, AGF and Clim provide smoother profiles than TSGF for sites with %Gap < 30%. Impact of the accuracy and smoothness of the reconstructed time series were evaluated on the timing of phenological stages. The dates of start, maximum and end of the season are estimated with an accuracy of about 10 days for the sites with %Gap < 10% and increases rapidly with %Gap. TSGF provides the more accurate estimates of phenological timing up to %Gap < 60%

The smoothing problem for continuous systems is treated in a state space representation by means of variational calculus techniques. The smoothing problem is introduced in an criterion by means of an artificial discontinuity that splits the problem in term of forward and backward filtering problems. Hence, the smoother design is realized in three steps. First, a forward filter is developed. Secondly, a backward filter is developed taking into account the backward Markovian model. The third step consists of combining the two previous steps in order to compute the smoothed estimate. An example shows the efficiency of this proposed smoother.

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