4SLUT Selfcalibrated Supervised Simplified Shallow LUT Once the optical calibration is achieved, 4SM now operates a simplified LUT, attempting to account for the spatial variations of the water's optical properties, as described by Jerlov, 1976. 
Mapping of the ratio Kblue/Kgreen of an effective twoways diffuse attenuation coefficients for irradiance This is done in a very basic process, by spectral matching using a lookup table with the following four input parameters:  spectral 2K: derived from the ratio Kblue/Kgreen of the effective twoways spectral diffuse attenuation coefficients 2K,
 using from Jerlov's optical classification of marine waters
 over the range Kblue/Kgreen=0.27 to Kblue/Kgreen=1.94
 spectral LB: BOA bottom reflectance
 spectral Lw: BOA water volume reflectance
 Z: bottom depth Z over the range 030 m

The simplified RTE L=Lw+(LBLw)/exp(2K*Z) where  LB is the BOA shallow bottom reflectance as if at null depth,
 normalized on a scale of 0200
 Lw is the BOA waterleaving water volume reflectance over optically deep waters,
 normalized on a scale of 0200
 2K is an effective twoways diffuse attenuation coefficient in units of 1/meter for remote sensing radiance
 Z is the shallow bottom depth in units of meter
The simplified RTE is dimensionless  L, LB and Lw are dimensionless reflectances/radiances/DN
 2K*Z is a dimensionless product
==>consequently: no need for formal atmospheric correction of the image data 
 Note that only models based on Lee's analytical radiative transfer equation (RTE) actually account for the spatial variations of Lw, 2K and LB over the scene.
 This involves inherent optical properties, and requires a detailed/exhaustive lookup table to specify, in physical units, all possible variations of the optical parameters of the atmosphere, the water column, and the bottom substrate reflectance.
 Water column correction then proceeds by spectral matching to derive spectral 2K, spectral LB, and Z (and many more, in physical units) at the current pixel.

4SM simplified LUT for spectral matching  We have shown (Favoretto and Morel, 2017) that, together with data published by Jerlov (1976) , the image itself contains enough information to calibrate a simplified RTE (Maritorena et al, 1994) to derive both LB in relative units and Z in meters by inverting the simplified RTE, without the need for (i.e. ahead of access to) any field data.
 As of 2021, further to the above calibration of the simplified RTE,
 4SM now builds a simplified LUT to store the variations of LB, Lw, 2K, and Z ,
 which possibly combine into the spectral waterleaving bottom reflected spectrum observed in the remote sensing image at the current shallow pixel.
 Water column correction then proceeds by matching the waterleaving spectrum observed at the current shallow pixel against ~11 millions lookup table spectra:
 greenish, neutral or reddish signature
 ?140 values of Kblue/Kgreen, from 0.3 to 1.94
 310 values of Z from 0 to 310 dm
 200 values of LB, from 200 to 1

derive spectral 2K from the ratio Kblue/Kgreen  This new feature of 4SM is now fairly well developped, and needs to be tested and possibly confirmed by independent workers for publication.
 In the 4SM calibration, Lw and the ratio Kgreen/Kblue are estimated for the clearest waters observed at the scene.
 Jerlov, 1976 and Kirk, 1994 have shown that the ratio Kgreen/Kblue varies in a known/predictable way through the complete suite of water types, from Oceanic I to Oceanic 3, to Coastal 1 to Coastal 9 types:
 4SM uses this remarkable feature to derive spectral K for the clearest waters observed at the scene.
Kirk: "The reflectance spectra of oceanic waters vary in a roughly systematic way. A family of curves, of progressively changing shape, determined mainly by the phytoplankton concentration, is observed. Thus, for any given oceanic water, specification of the ratio of radiances or radiance reflectances at any two wavelengths, should in effect specify the whole radiance reflectance curve, and therefore the optical character of the water.” (Kirk) 
What of the spatial variations of 2K over the scene?  Until recently, these variations were only accounted for in a very crude manner in 4SM.
 As of 2021, the LUT development now the value of the ratio Kblue/Kgreen (stored in the LUT) which achieves the best spectral match for any combination of Z, spectral Lw, and spectral LB.
 But there are at least two weak links:
 the spectral Lw.
 the computing time: over 8 millions spectra in the LUT, ~4000 shallow pixels per second on my laptop for the LSI study case: this requires optimization!

Lw: the weak link from Oceanic I to Coastal 9 water types of Jerlov  As for Lw, this water volume reflectance is highest (Lw~=12% for the UltraBlue band) for Oceanic I water type (like the upwelling waters of Sargasso Sea).
 We can only assume that the increase of CDOM (yellow substances: decay of biomass production, along with other biological causes) entails the decrease of the water volume reflectance Lw,
 such that the Lw term should become extinct altogether at some point through this "familly of water types".
 As of february 2021, we seem to be comfortable assuming
 that Lw in the bluegreen range decreases regularly through oceanic water types (blue waters),
 so that all Coastal waters exhibit quasi null water volume reflectance (brown waters).
 This cannot be overlooked, and needs to be investigated using Hydrolight.

4SM: estimating spectral 2K in unit of m1, from the image, using the ratio Kblue/Kgreen and Jerlov's data, ahead of any field work, worldwide The ratio Kblue/Kgreen is listed below for 10 water types Landsat 8 wavelength in nanometers, 2K in m1 2K440 2K480 2K560 2K655 K480/K560 4SMresampled_curve_for_Jerlov_O1 0.04039 0.03960 0.14680 0.74384 0.26974 4SMresampled_curve_for_Jerlov_O1A 0.05599 0.05280 0.15560 0.76384 0.33931 4SMresampled_curve_for_Jerlov_O1B 0.07599 0.06960 0.16560 0.77383 0.42026 4SMresampled_curve_for_Jerlov_O2 0.14637 0.12719 0.19880 0.82582 0.63980 4SMresampled_curve_for_Jerlov_O3 0.28995 0.23158 0.26240 0.91980 0.88256 4SMresampled_curve_for_Jerlov_C1 0.58790 0.32798 0.29400 0.94420 1.11557 4SMresampled_curve_for_Jerlov_C3 0.89985 0.55196 0.42400 0.97579 1.30180 4SMresampled_curve_for_Jerlov_C5 1.29578 0.83194 0.61800 1.12376 1.34619 4SMresampled_curve_for_Jerlov_C7 2.02765 1.36791 0.92000 1.31972 1.48686 4SMresampled_curve_for_Jerlov_C9 3.37939 2.36384 1.22000 1.58366 1.93757 
