A semi‐analytical Bayesian estimate retrieval algorithm for the inversion of remote‐sensing reflectance in optically deep and shallow waters

The inverse problem in remote sensing of aquatic environment consists in retrieving optically significant constituents (OSCs) from a spectral measurement of the remote sensing reflectance ().Optically significant constituent includes chlorophyll a concentration (), a proxy of phytoplankton biomass, absorption of colored detrital matter (); a proxy of organic carbon and detrital matter, and particulate backscatter (); a proxy of total suspended particulate matter. In optically shallow waters, apart from OSCs, water depth () and benthic albedo () are retrieved due to their influence on the spectral shape and magnitude. The state‐of‐the‐art methods invert from a semi‐analytical formulation of the forward radiative transfer model using Gauss–Newton or gradient‐descent–based optimization routines. However, these methods at times are inadequate to deal with the “ill‐posed” nature of inversion, where multiple combinations of OSCs, and can produce almost identical spectra. Here, a Bayesian probabilistic inversion scheme named semi‐analytical Bayesian estimate retrieval (SABER) is proposed and developed to simultaneously retrieve OSCs and values along with inversion occurred uncertainty (Bayesian Credible Interval [BCI 95 ]) in optically complex waters following the Markov Chain Monte Carlo (MCMC) sampling approach. The model prior distribution (Prior) was fitted as Log‐Normal , obtained from a comparison of three distributions in the Positive real domain () and the model likelihood was fitted as Gaussian . The least ambiguous estimates of OSC and were obtained from Maximum‐a‐posterior (MAP) estimates, associated with the highest MCMC sampled posterior probability density. Semi‐analytical Bayesian estimate retrieval was tested with both synthetic datasets and global in situ observations across eight distinct optical water types, and for both optically deep and shallow water cases. The model performance was compared to the quasi‐analytical algorithm (QAA) in terms of mean uncertainty of OSC (SABER: 38.23%; QAA: 48.86%) and to the Hyperspectral Optimization Process Exemplar (HOPE) model for retrieval (SABER:7.05%; HOPE: 18.67%). Among the inversion retrieved variables, the highest inversion uncertainty was obtained for across all the OWTs, whereas estimates had the least. The benefits and limitations of the MCMC‐based inversion are finally discussed concerning its application to remote sensing of seawater constituents and bathymetry retrievals in optically complex waters. The source code of SABER is open‐sourced and available in a compilable package format at https://github.com/homas01123/SABER_fast .