Quantifying how long burned landscapes remain spectrally visible in satellite imagery here termed burn-scar persistence is essential for understanding post-fire recovery and ecological resilience. Even with the continuity of modern Earth-observation archives, incomplete observation windows caused by cloud cover, orbital gaps and sensor outages often prevent full monitoring of fire-affected surfaces. The resulting temporal truncation leads to censoring, where the beginning or end of a persistence episode is unobserved, and thus complicates direct estimation of recovery durations. Similar difficulties arise in other persistence-type phenomena such as snow-cover longevity (Brown and Mote, 2009), floodwater retention (Sai Bhageerath et al., 2025) and orbital-debris decay (Ashruf et al., 2024), where observational incompleteness can bias lifetime inference (Gross and Clark, 1975). Post-fire vegetation dynamics in remote sensing have typically been described using spectral indices such as the Normalized Difference Vegetation Index (NDVI) and the Normalized Burn Ratio (NBR) derived from MODIS and Landsat imagery. Many studies fit smooth exponential or logistic trajectories to these indices to represent disturbance and recovery (Kennedy et al., 2010; Roy and Boschetti, 2009; Verbesselt et al., 2010), yielding visually intuitive but essentially deterministic descriptions. However, such trajectory-based models rarely account explicitly for statistical censoring or for radiometric noise in the underlying satellite observations (Kennedy et al., 2010; White et al., 2022) and may therefore underestimate actual recovery durations when recovery extends beyond the observation window. A probabilistic lifetime formulation, by contrast, can incorporate partial observation directly through survival-analysis principles, for example using nonparametric methods such as those of Kaplan and Meier (1958) and Turnbull (1976).

Within this probabilistic setting, the Weibull distribution provides a flexible model for persistence times: its shape parameter characterizes whether recovery accelerates or decelerates, while its scale parameter defines a characteristic duration (Murthy et al., 2004). Standard parameter estimation via maximum likelihood, however, is known to be sensitive to contamination and model deviations (Huber, 1981; Hampel et al., 1986; Huber and Ronchetti, 2009). Satellite data frequently contains outlying or mixed pixels arising from residual cloud, geolocation errors or misclassification, and these anomalies can distort the estimated hazard and bias ecological interpretation. To mitigate such effects, robust estimation methods have been proposed. The weighted-likelihood estimator was developed in the robustness literature to downweight discordant observations through case-specific likelihood weights (Markatou, 2000), and related minimum-distance and divergence-based approaches have been synthesized by Basu et al. (2011). The underlying principles of bounded influence and adaptive weighting were formalized in the robust M-estimation framework of Huber (1981) and Hampel et al. (1986) and further extended in later treatments (Huber and Ronchetti, 2009; Rousseeuw and Leroy, 1987). Building on these advances, the present study develops a Weibull lifetime model for burn-scar persistence under the doubly censored calendar-time structure induced by a finite observation window, estimated via a Huber-type weighted-likelihood estimator. The framework explicitly accounts for incomplete persistence episodes while moderating the impact of anomalous pixels, thereby improving stability without sacrificing interpretability. Using Landsat surface-reflectance time series at Monitoring Trends in Burn Severity (MTBS) sampling locations within a large interior-Alaska fire scar, with an analysis window from 2010 to 2018, the framework aims to combine statistical rigor with physical realism and to provide a unified tool for extracting meaningful persistence metrics from noisy, incomplete satellite observations.

The analysis uses Landsat surface-reflectance time series extracted at Monitoring Trends in Burn Severity (MTBS) sampling locations within a large fire scar in interior Alaska (≈65.5° N, 145–146° W; USDA Forest Service and U.S. Geological Survey, 2024, MTBS; see also the MTBS methodological documentation in Fire Ecology). The MTBS points provide fixed 30 m reference locations inside the mapped burned area, allowing long NDVI histories to be reconstructed without spatial tracking. For each point, all available Landsat 5 TM, Landsat 7 ETM+, and Landsat 8 OLI observations from early 2000 onward were collected and converted to the Normalized Difference Vegetation Index (NDVI). Observations flagged as cloudy, heavily hazy, or snow-covered were discarded, leaving irregular but typically multi-decadal NDVI series for roughly one hundred burned pixels over the period 2000–2020. Burn-scar persistence was defined at each pixel as the time during which vegetation remained in a spectrally degraded post-fire state. To adapt this definition to local conditions, a “healthy vegetation” baseline was computed per pixel as the 80th percentile of its NDVI record. Percentile-based baselines and relative recovery thresholds are commonly used in Landsat time-series studies to define disturbance and recovery in a manner robust to interannual variability (e.g., Pickell et al., 2016; White et al., 2022; Smith-Tripp et al., 2024). A 5-point running median of NDVI was then used to identify disturbance and recovery. A burn episode was declared when the smoothed NDVI fell below 60 % of the baseline and remained below this threshold for several consecutive acquisitions; the first such crossing time was taken as the ignition date. Recovery was defined as the first subsequent time when smoothed NDVI returned to at least 85 % of the baseline, indicating that pre-fire greenness had effectively re-established. Similar relative-threshold definitions have been employed in post-fire recovery studies using Landsat time series, where recovery is operationally defined as return to a specified fraction of pre-disturbance spectral conditions (Pickell et al., 2016; White et al., 2022).

For each pixel, the dominant episode was taken as the longest contiguous interval below the burn threshold, with onset and recovery dates denoted by Strue and Etrue. To mimic the finite monitoring windows typical of climate and satellite archives (Kennedy et al., 2010; White et al., 2022), a common analysis window was imposed across all pixels. In this study, [A,B] spans 1 January 2010 to 31 December 2018, capturing several post-fire growing seasons while remaining well within the Landsat record. The analysis window (2010–2018) was selected to represent a finite observation period and to ensure that the data exhibit a mixture of fully observed and censored persistence episodes. While the full Landsat record spans 2000–2020, using the entire period would lead to a higher proportion of fully observed episodes and fewer censored cases. In contrast, the selected window provides a more balanced combination of fully observed and right-censored observations, which is consistent with the censored lifetime framework adopted in this study. The choice of window is therefore intended to reflect incomplete monitoring conditions rather than to restrict the available data. Episodes that both ignite and recover within [A,B] contribute to fully observed lifetimes, with persistence T=Etrue-Strue. Episodes that ignite within the window but remain spectrally degraded beyond B are right-censored, since only a lower bound on their persistence is observed. A small number of pixels are already in a degraded state at A and have not recovered by B; these give rise to doubly censored episodes whose ignition and recovery both lie outside the analysis window and whose persistence therefore exceeds the 3286 d span of [A,B]. After applying these steps, the final dataset consists of fully observed episodes, right-censored episodes, and a small number of long-lived doubly censored episodes, all expressed as persistence times in days. This configuration reflects key features of satellite-derived impact indicators – finite windows, irregular sampling, and moderate radiometric noise (Kennedy et al., 2010; Verbesselt et al., 2010; White et al., 2022) – and motivates a censored lifetime formulation in which complete and incomplete episodes are estimated jointly rather than discarding partially observed persistence intervals. Although the NDVI thresholds used to define disturbance (60 % of the pixel-specific baseline) and recovery (85 % of the baseline), as well as the 80th-percentile baseline itself, are necessarily heuristic, they are aligned with operational definitions of spectral recovery in Landsat-based disturbance studies. Simple sensitivity checks using alternative thresholds (e.g., 55 %–65 % for disturbance and 80 %–90 % for recovery) yielded qualitatively similar persistence distributions and Weibull parameter estimates, suggesting that the main conclusions are not driven by a single arbitrary choice of cut-off. Additional quantitative sensitivity analyses are provided in Appendix A (Table A2).

Satellite archives rarely capture the full life cycle of ecological disturbances. In the case of NDVI-derived burn scars, some pixels are already in a degraded post-fire state when the analysis window opens, while others remain degraded when it closes. As a result, the complete trajectory from ignition to recovery is not always observed. This incomplete visibility naturally leads to censoring, a setting that parallels classical survival and reliability models (Gross and Clark, 1975; Lawless, 2003). We interpret burn-scar persistence as a lifetime variable. For each burned pixel, the persistence lifetime begins at the true onset of spectral degradation and ends when NDVI returns to a near-baseline level. Let Strue denote the true onset time of degradation and Etrue the true recovery time. The persistence time is therefore T=Etrue-Strue, T>0, measured in days from ignition to recovery. The satellite data are observed only within a fixed temporal window [A,B], defined in Sect. 2. Because this window does not necessarily cover the full interval [Strue,Etrue], several censoring patterns arise in calendar time:

Fully observed episodes: Strue≥A and Etrue≤B. The entire persistence interval lies within the window, and the observed duration equals the true lifetime.

To model persistence, we assume that T follows a Weibull distribution with shape parameter b>0 and scale parameter θ>0. The cumulative distribution function is

1F(t)=1-e-tθb and the corresponding density is 2f(t)=bθtθb-1e-tθb as in standard Weibull lifetime models (Murthy et al., 2004). For a fully observed persistence time ti, the likelihood contribution is f(ti). For a right-censored episode observed up to time ci, the contribution is the survival probability 1-F(ci). Accordingly, the likelihood used in this study is the standard right-censored lifetime likelihood familiar from survival analysis (Lawless, 2003; Turnbull, 1976). Framing burn-scar persistence as a censored lifetime process provides a transparent bridge between satellite data limitations and established statistical tools for incomplete time-to-event data. This formulation also enables principled extensions, including robust estimation and alternative distributional choices, which are developed in the following section. In Fig. 1, schematic NDVI trajectories illustrate fully observed, right-censored, and window-spanning episodes within the observation window [A,B].

In summary, although some episodes are left- or doubly censored in calendar time, the statistical inference is conducted entirely on the persistence-time scale, where incomplete episodes enter the likelihood as right-censored observations. This distinction ensures internal consistency between the conceptual description and the formal likelihood formulation.

This section describes the estimation procedures used to model burn-scar persistence under censoring and to incorporate robustness through a weighted-likelihood estimator (WLE). The framework has three components: (i) formulation of the Weibull likelihood under general censoring; (ii) introduction of adaptive weighting to achieve robustness against outliers and model deviations; and (iii) quantification of uncertainty using parametric bootstrap resampling.

Table 1 summarizes the observed and censored burn-scar episodes in the MTBS sample (n=90). In the MTBS sample, 49 episodes (54 %) are fully observed and 35 (39 %) are right-censored on the persistence-time scale. An additional 6 episodes (7 %) remain degraded throughout the analysis window. These latter cases are doubly censored in calendar time but contribute as right-censored observations on the persistence-time scale. Nearly half of all episodes are therefore incomplete, underscoring the need for an explicit censored-lifetime formulation rather than relying on raw observed durations.

The results show that treating NDVI-based burn-scar persistence as a censored lifetime yields a coherent and physically meaningful description of recovery. In the MTBS sample, more than 40 % of episodes are incomplete either right-censored within the analysis window or degraded throughout it so that an approach based solely on raw observed durations would disregard a substantial portion of the available information. Within the Weibull family, the shape parameter summarizes whether the recovery hazard increases or decreases with time since fire, while the scale parameter defines a characteristic persistence horizon. Both quantities behave in an ecologically plausible manner, with median recovery times around 2.7–2.9 years and 90th-percentile durations of roughly five to six years. A key limitation of the present analysis is the modest sample size (n=90 episodes, of which only six span the entire observation window), which limits precision in characterizing the extreme upper tail of the persistence distribution and constrains exploration of more complex model structures. The definition of disturbance and recovery involves several NDVI thresholds: a pixel-specific baseline defined as the 80th percentile of the NDVI record, a 60 % drop below this baseline to flag disturbance, and an 85 % return threshold for recovery. These choices are necessarily heuristic but align with established practice in Landsat-based post-fire recovery studies. Sensitivity analyses with alternative thresholds (e.g., 55 %–65 % for disturbance and 80 %–90 % for recovery) yielded qualitatively similar persistence distributions and Weibull parameter estimates, indicating that the main findings are not driven by a single arbitrary specification. Methodologically, the weighted-likelihood estimator (WLE) provides clear robustness gains relative to standard maximum likelihood. Bootstrap diagnostics show that a small number of very long-lived scars inflate the variance of the MLE. Under the WLE, adaptive Huber weights moderate the influence of these episodes while leaving the bulk of the distribution essentially unchanged. This behavior is consistent with the theoretical properties of bounded-weight estimators, which stabilize likelihood curvature and reduce sensitivity to atypical observations while retaining the large-sample properties of likelihood-based inference. The data-driven selection of the tuning constant further enhances stability by adapting the robustness level to the observed lifetime distribution rather than fixing it a priori. Comparisons with alternative parametric forms reinforce these conclusions. Although the lognormal model attains the lowest AIC and BIC values and the gamma model performs competitively, neither offers the same combination of monotone hazard interpretation and well-developed robust censored-likelihood methodology available within the Weibull family. The WLE–Weibull fit closely follows the Kaplan–Meier and Turnbull curves over the central range of the data and yields a recovery hazard consistent with established patterns of post-fire regrowth. We therefore retain the Weibull family, estimated via WLE, as the primary modelling framework and regard the lognormal and gamma fits as benchmarks that confirm the overall stability of the substantive conclusions. Taken together, the results indicate that the proposed framework achieves three aims: i.

It accommodates fully observed and right-censored persistence episodes within a unified lifetime model.

This study develops a robust lifetime-based framework for analyzing burn-scar persistence from NDVI time series, explicitly accommodating the censored structures that arise from the finite temporal extent of satellite archives. By modelling persistence as a Weibull lifetime and incorporating adaptive Huber weighting into the likelihood, the method combines a physically interpretable description of recovery with statistically stable estimation in the presence of noise, misclassified pixels, and unusually persistent scars.

Applied to 90 MTBS persistence episodes, the Weibull model captures the principal features of the recovery distribution, with median recovery times of approximately 2.7–2.9 years and substantially longer durations in the upper tail. The weighted-likelihood estimator stabilizes inference by reducing the influence of a small number of extreme episodes while preserving the central tendency of the distribution. Comparisons with nonparametric survival curves indicate that the robust Weibull fit aligns closely with the empirical lifetime structure over the central range of the data. Although the lognormal model attains slightly smaller AIC/BIC values and the gamma model performs competitively, neither offers the same combination of monotone hazard interpretation and an established robust censored-likelihood framework. In this sense, the Weibull–WLE formulation provides a balanced compromise between interpretability and robustness.

The framework is generic and can be extended to other satellite-based disturbance and restoration processes, such as vegetation browning, drought impacts, land-use transitions, and flood-recovery cycles. In the context of fire, the same methodology could be applied to major events outside interior Alaska – for example, the 2019–2020 Australian bushfires or large Amazonian drought fires – by adapting only the NDVI extraction and baseline definitions. Future work may incorporate spatial pooling across pixels, covariate effects such as climate forcing or fire severity, and hierarchical structures that combine multiple fire events within a unified statistical model. Overall, the proposed approach provides a principled and practical tool for extracting recovery timescales from incomplete and noisy remote-sensing observations, enabling more consistent characterization of post-fire ecosystem dynamics across diverse biomes and fire regimes.