A per-pixel, non-stationary mixed model for empirical line atmospheric correction in remote sensing

Atmospheric correction is a key stage in the processing of remotely sensed data. The empirical line method (ELM) is used widely to correct at-sensor radiance or DN to at-surface reflectance. It is based on a simple linear relationship between those two variables. Effective application of the model requires that it is estimated in a precise and unbiased fashion. The usual approach is to use ordinary least squares (OLS) regression to model the relationship between the average reflectance and radiance for a small number (3 to 8) of ground targets (GTs) and then to apply the regression on a per-pixel basis to the image. This leads to a mismatch between the scale at which the model is estimated and the scale at which the model is applied. Further, this approach wastes information and can lead to inconsistent estimators. These problems are addressed in the new approach presented here. The model was estimated on a per-pixel rather than per-GT basis. This yielded consistent, precise estimators for the ELM, but placed stronger requirements on the modeling. Specifically spatial autocorrelation and non-constant variance (heteroskedasticity) in the model residuals needed to be addressed. This was undertaken using the linear mixed model (LMM), which is a model-based expression of the geostatistical method. Of particular interest is the use of a non-stationary LMM to address the heteroskedasticity.The approach taken in this paper is of significance for a broader set of remote sensing applications. Regression and geostatistics are often applied, based typically on a stationary model. This paper shows how heteroskedasticity can be assessed and modeled using the non-stationary LMM. Heteroskedasticity is present in other remote sensing applications hence the non-stationary modeling approach, demonstrated here, is likely to be beneficial.