December 2021: I give up
Further progress in the LUT approach
shall require investigation of Lw

Backscattering and Absorption Coefficients

Inherent optical properties of Jerlov water types

Did you say HOMOGENEOUS?
Kd in French Polynesia   Maritorena 1996
Diffuse attenuation coefficient  for downwelling irradiance measured in situ
  • 5 stations over the ocean:  the water type of Jerlov varies
    • from Kblue/Kgreen=0.46 this is OIB+0.2 type
    • to     Kblue/Kgreen=0.80 this is OII+0.7 type
  • 11 stations over Moorea lagoon:  the water type of Jerlov varies
    • from Kblue/Kgreen=0.63 this is OII type
    • to     Kblue/Kgreen=0.89 this is OIII type
  • 5 stations over Takapoto lagoon: the water type of Jerlov varies
  • from Kblue/Kgreen=0.70 in the center of the lagoon this is OII+0.2
  • to three intermediate water types, 
  • to Kblue/Kgreen=0.79 this is OII+0.6 inside a small confined part of the lagoon 
PAR in the Western Caribbean    Karpouzli and Malthus, 2003
"the results of studies where single measurements of ‘average’ attenuation have been used to depth-correct remotely sensed imagery should be interpreted
with a high degree of caution

CDOM in the Bahamas   Mobley 2004
Absorption coefficient measured at Lee Stocking Island varies with the tidal cycle 


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

must be accounted for when operating the simplified RTE.

Self-calibrated 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,
Mapping of the ratio Kblue/Kgreen
of the 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 in units of 1/meter,
    • 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: shallow bottom depth over the range 0-30 meters
                   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
no need for formal atmospheric correction
of the image data into physical units!
Analytical methods: LUT for 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 the optical parameters of the atmosphere,
    • of the optical parameters of the water column, and
    • of the bottom substrate reflectance.
  • Water column correction then proceeds by spectral matching
    • to derive spectral K, 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).
    • Consequently, one may derive both spectral 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 first builds a simplified LUT 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 shallow pixel.
    • 4SM then proceeds by matching the observed water-leaving spectrum derived at  the current shallow pixel against ~11 millions look-up 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 developed, 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 predictable way through the complete suite of water types, from Oceanic I to Oceanic III types, then from 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.”  
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 derives the value of the ratio Kblue/Kgreen (stored in the LUT) which allows to achieve the best spectral match for any combination of spectral 2K, Z, spectral Lw, and spectral LB stored in the LUT.
  • 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, the 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 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 a low-to-null water volume reflectance (green to brown-black waters).
This needs to be investigated using Hydrolight.
 Blue waters over the ocean 
Ultrablue and Blue bands shine       Green, Pan and Red bands are low
 Green waters inside the lagoon 
UltraBlue and Blue bands are low    Green, Pan, and Red bands     shine
The purpose of the LUT approach is, among other things, 
to account for such variations of the optical properties of the shallow waters

 upon operating the water column correction, pixel-wise

is extinct


is slightly depressed

glows strongly


glows slightly


Mapping 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, The Bahamas

Nosy Be, Madagascar 

Lizard Island, GBR, Australia


Florida Bay, USA


Clipperton Atoll, France


Fakarava Atoll, French Polynesia

Majuro and Arno atolls,
The Marshall Islands

Zakinthos Island, Greece

San Lorenzo Channel, Baja California

La Parguera , Puerto Rico, USA



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