# Quantifying Uncertainties in OC-SMART Ocean Color Retrievals: A Bayesian Inversion Algorithm
## 1 Introduction  
Remote sensing of the ocean is now routinely performed by using instruments deployed on satellite platforms [1; 2] to measure the radiance emerging at the top of the atmosphere and then apply an atmospheric correction (AC) algorithm [3] to remove all but the water-leaving radiance component [4] or the closely related remote sensing reflectance (\(R_{\mathrm{rs}}\)). From the \(R_{\mathrm{rs}}\) determined in the AC step, various ocean color products can be inferred, such as chlorophyll-a concentration [5], suspended particle matter [6], turbidity, and phytoplankton abundance [7]. Water quality is crucial to life in nearby regions [8; 9], so it is of the utmost importance that measurement data with low bias and low uncertainty can be obtained. The uncertainty associated with a measurement characterizes the dispersion of the values that could reasonably be attributed to the property being measured [10; 11; 12]. In this paper, the metric used to express uncertainty associated with a measurement will be the standard deviation. Several AC algorithms have have been developed to estimate the water-leaving radiance and achieved reasonable results in open ocean areas where the water-leaving radiances at near-infrared (NIR) wavelengths are negligible. Significant issues remain however; in coastal waters, which are not black in the NIR, negative water-leaving radiances are frequently produced by traditional AC algorithms [13; 14]. In addition, aerosols in coastal areas often dramatically differ from those encountered over the open ocean, so any realistic AC approach should take into account the diversity of atmospheric and oceanic conditions encountered in such complex environments. Turbid waters, whitecaps, and sunlight can also lead to unreliable results. Various attempts to improve AC algorithms have been proposed, such as (i) using a turbid water flag to switch between correction algorithms depending on the region [15], (ii) changing the bio-optical model in an iterative manner [16], (iii) determining atmosphere-ocean properties simultaneously [17], (iv) employing a multi-band approach [18], (v) incorporating hyperspectral data [19], (vi) optimizing for sunlightthrough the use of a polynomial function [20], and (vii) using a coupled ocean/atmosphere inversion scheme based on neural networks [21]. AC algorithms that produce the highest quality results often use several of these improvements in conjunction to form a more complete model of the ocean and atmosphere. Traditional AC approaches typically do not provide uncertainty estimates, which should be included, however, to gain confidence in retrieved results. Recently, some studies have discussed uncertainty issues based on optimal estimation (Bayesian inversion) [22; 23; 24]. Algorithms based on neural networks, such as C2 RCC [25] and SWARM [26], provide uncertainties based on a comparison of radiative transfer model simulations and IOP inversions of the training datasets. These uncertainties reflect the difference between estimated ocean color retrievals and training simulations. Another approach is to use an ensemble of neural networks [27] with differing initializations to perform spectral inversion. These approaches for quantifying retrieval uncertainty rely heavily on the capability of algorithms to produce good match-ups between simulations and training data, as poor match-ups will lead to larger uncertainties. In two recent papers [28; 29], the authors discuss steps to quantify pixel-level uncertainties using a derivative approach to propagate uncertainties through the \(R_{\text{rs}}\) retrieval process, primarily focusing on uncertainties arising from atmospheric correction. This approach allows one to account for uncertainties from a wide range of sources, but, in practice, not all source uncertainty information is available and several estimates and assumptions about the system need to be made.