use a LUT in 4SM to map the diffuse attenuation coefficient Kd
work done in 2021 at 15 m GSD
Using the Panchromatic band for water column correction
to derive water depth and spectral bottom signature:

Landsat 8 OLIP bandset used for this work

Purple=1Blue=2Green=3PAN=4Red=5NIR=6 and SWIR1=7

   peer-reviewed and published in 2017    

Mobley 2004 : CDOM concentration varies with the tidal cycle!

Assuming homogeneous waters is not an option any longer:
these spatial variations of the optical properties of the water over the scene 

must be accounted when operating the simplified RTE.

Mapping of the ratio Kblue/Kgreen
of an effective two-ways 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 two-ways 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 0-30 m
                   The simplified RTE                  
  • LB is the BOA shallow bottom reflectance as if at null depth, 
    • normalized on a scale of 0-200
  • Lw is the BOA water-leaving water volume reflectance over optically deep waters,  
    • normalized on a scale of 0-200
  • 2K is an effective two-ways 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
Analytical methods: LUT and spectral matching
  • 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 look-up table to specify, in physical units, all possible variations of atmospheric and underwater optical parameters, and also of bottom substrate reflectance.
  • Water column correction then proceeds by spectral matching to derive spectral 2K, spectral LB, and Z (and more, in physical units) at the current pixel (ALUT: Hedley et al., 2009).
4SM LUT and 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, without the need for (i.e. ahead of) any field data.
  • As of 2021, further to the above calibration of the simplified RTE,
    • 4SM now builds a look-up table to store the variations of LB, Lw, 2K, and Z,
    • which possibly combine into the spectral water-leaving bottom reflected spectrum observed in the remote sensing image at the current pixel.
  • Water column correction then proceeds by matching the water-leaving spectrum observed at  the current pixel against several millions of look-up table spectra.
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 independant workers for publication.
  • In 4SM, 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:
    • 4SM uses this remarkable feature to derive spectral K  for the clearest waters observed at the scene.
"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.
  • The LUT development now derives a 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 two weak links:
    • the spectral Lw. 
    • the computing time: over 4 millions spectra, 160 shallow pixels per second on my laptop, requires optimization (CUDA, cloud, ...).
Lw: the weak link 
from Oceanic I to  Coastal 9 water types of Jerlov
  • As for Lw, this water volume reflectance is highest (Lw~=0.12 for the UltraBlue band) for Oceanic I water type (like upwelling waters of Sargasso).
  • 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 blue-green 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 m-1 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 m-1
                                                                           2K440     2K480     2K560    2K655     K480/K560
4SM-resampled_curve_for_Jerlov_O1    0.04039  0.03960  0.14680  0.74384  0.26974
4SM-resampled_curve_for_Jerlov_O1A  0.05599  0.05280  0.15560  0.76384  0.33931
4SM-resampled_curve_for_Jerlov_O1B  0.07599  0.06960  0.16560  0.77383  0.42026
4SM-resampled_curve_for_Jerlov_O2    0.14637  0.12719  0.19880  0.82582  0.63980
4SM-resampled_curve_for_Jerlov_O3    0.28995  0.23158  0.26240  0.91980  0.88256
4SM-resampled_curve_for_Jerlov_C1    0.58790  0.32798  0.29400  0.94420  1.11557
4SM-resampled_curve_for_Jerlov_C3    0.89985  0.55196  0.42400  0.97579  1.30180
4SM-resampled_curve_for_Jerlov_C5    1.29578  0.83194  0.61800  1.12376  1.34619
4SM-resampled_curve_for_Jerlov_C7    2.02765  1.36791  0.92000  1.31972  1.48686
4SM-resampled_curve_for_Jerlov_C9    3.37939  2.36384  1.22000  1.58366  1.93757

Mapping of the ratio Kblue/Kgreen   see legend
blue_to_white_tones for Oceanic waters
over the range 0.3 to 0.8
white_to_red_tones for Coastal waters
over the range 0.8 to 1.9

Lee Stocking Island, Bahamas


Nosy Be, Madagascar 


Lizard Island, GBR, Australia


SanLorenzoChannel, BajaCalifornia, Mexico



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