Harmonic modeling

Our approach is based on the assumption that univariate time series of reflectance observations for a band or index of interest can be relatively well-characterized by a harmonic regression model. This approach is particularly suited for monitoring using vegetation indices in regions/ecosystems with strong phenological signals.

We use the following model form:

where y is the predicted vegetation index (e.g. Tasseled Cap Greenness), x is the ordinal date of each observation; a₀ is the modeled intercept; b₀ is the modeled slope; N is a set of integers specifying the frequency, j, of the Fourier series harmonics (e.g. N = {1, 3}, corresponding to 12-month and 4-month harmonics); a_j are the sine coefficients and b_j are the cosine coefficients estimated at each frequency; T is the number of days in a year (T = 365.25); and ε_t is the residual error term for each observation.

Condition Scores

Condition scores are calculated as the difference between the observed and predicted reflectance values for a given acquisition date, and are normalized by the Root Mean Square Error (RMSE) of the baseline model used to generate the predicted value:

Thus, scores represent the magnitude of change in reflectance relative to the uncertainty in baseline model fit.

Ensemble Approach

Previous implementations of this workflow have relied on a single baseline model for prediction and monitoring such that a single condition score is generated for each acquisition date. However, selection of a suitable baseline period may be challenging in frequently disturbed landscapes and noise in reflectance observations may impact the quality of models fit to different time periods. Therefore, the latest version of our workflow utilizes an ensemble approach that combines condition scores estimates from multiple baselines. Baseline models are fit using a moving window and the n models preceding the specified monitoring period are used to generate a set of n condition score estimates for each acquired image during the specified monitoring period. These scores can then be averaged across all dates within the monitoring period to produce a more robust estimate of potential condition change. Our final assessment products combine results across Landsat Paths and include per-pixel mean and standard deviation of scores as well as number of observations used for monitoring.