Per-Pixel Uncertainty Quantification and Reporting for Satellite-Derived Chlorophyll-a Estimates via Mixture Density Networks

Abstract

Mixture density networks (MDNs) have emerged as a powerful tool for estimating water-quality indicators, such as chlorophyll-a (Chl <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> ) from multispectral imagery. This study validates the use of an uncertainty metric calculated directly from Chl <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> estimates of the MDNs. We consider multispectral remote sensing reflectance spectra ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R_{\text {rs}}$ </tex-math></inline-formula> ) for three satellite sensors commonly used in aquatic remote sensing, namely, the ocean and land colour instrument (OLCI), multispectral instrument (MSI), and operational land imager (OLI). First, a study on a labeled database of colocated in situ Chl <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R_{\text {rs}}$ </tex-math></inline-formula> measurements clearly illustrates that the suggested uncertainty metric accurately captures the reduced confidence associated with test data, which is drawn for a different distribution than the training data. This change in distribution maybe due to: 1) random noise; 2) uncertainties in the atmospheric correction; and 3) novel (unseen) data. The experiments on the labeled in situ dataset show that the estimated uncertainty has a correlation with the expected predictive error and can be used as a bound on the predictive error for most samples. To illustrate the ability of the MDNs in generating consistent products from multiple sensors, per-pixel uncertainty maps for three near-coincident images of OLCI, MSI, and OLI are produced. The study also examines temporal trends in OLCI-derived Chl <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> and the associated uncertainties at selected locations over a calendar year. Future work will include uncertainty estimation from MDNs with a multiparameter retrieval capability for hyperspectral and multispectral imagery.