Lw and La
How to estimate
the water volume reflectance Lw
the path radiance La
and the Soil Line

 
Ls-Lsw=(LsB-Lsw)/e2K*Z     ......................(TOA)
L=Lw+(LB-Lw)/e2K*Z  .........................................(BOA)
LB=Lw+(L-Lw)*e2K*Z     .....................................(BOA)


The importance of the BOA Lw term only shows up for dark/deep bottoms.
Using depth soundings over sandy substrates at shallow/modereate depths
for calibrating the model 
is likely to result in bad underestimation
of depths retrieved over dark substrates,

as experienced by Lyons, Phinn and Roelfsema in 2011
.
The role of the Soil Line
Beyond Lyzenga: the beauty of a "ratio method"


 
1 - NO NEED for field data, nor for atmospheric correction
2 - this is demonstrated in this website, using a variety of hyper/multi spectral data
 
Requirements are
1 - homogeneous water body and atmosphere
2 - some coverage of optically deep water
3 - some coverage of dry land
 
Problems are
1 - the precision on estimated depth is found wanting, because the noise-equivalent change in radiance  of accessible data is too high for shallow water column correction work 
2 - radiance data should be preprocessed by the provider at level 1 in order to improve S/N ratio
3 - exponential decay: the deeper/darker the bottom, the poorer the performances
 
So
I keep digging
until suitable data
become available
 
Vegetation Line


Lyzenga et al's model
Lyzenga's

Lyzenga et al 2006: figure 15
for the Blue vs Green pair
I have rotated this image
It now shows Xblue vs Xgreen
Kblue=0.0561  Kgreen=0.736   Kblue/Kgreen=0.762
Isobath lines are clearly curved.
  • red points are ~2-4 m deep over a very wide range of bottom types
  • green points are 6-8 m deep over a very wide range of bottom types
  • blue points are 10-12 m deep over a very wide range of bottom types
4SM

4SM optical model
for the Blue vs Green pair

It shows Xblue vs Xgreen
Kblue/Kgreen=0.76

Thick blue: the brightest pixels line is shown.
All iso-bottom lines are straight
and parallel to the BPL line.
Star: the brightest bottom type at null depth
is represented by the big star symbol.
The authors write:

"Clearly, there is a wide variation of
the signals within each of these depth intervals,
which could be caused by variations in either the water optical properties or the bottom composition
"

 
From right to left,
isobath lines are shown
for 0, 5, 10, 15, 20 and 25 m.

The difference Lwblue-Lwgreen   
determines their curvature,
even in the absence of

"variations in either the water optical properties or the bottom composition".
  For the Red vs NIR case,
because Lw is next to null in both bands,
the Z=0 isobath goes through
the origin of this plot,
and all isobath lines
are straight and parallel lines.

 








   
illustrations from Dry Tortugas Reef , Florida Keys
This tuning is most critical in 4SM
Please refer to 4SM summary
Please refer to 4SM reflectance

Area sampled
for the bi-dimensional histograms
includes Dry Tortugas and Mooney Harbor

 

Land areas are shown in yellow
Deep water area is shown in red

Land areas in Dry Tortugas area
for bands 2, 3, 5 and 6
(blue, green, red, NIR)
from bright beach coral sands/rubbles
to dark mangrove vegetation.
Includes a small area over optically deep waters.
Please note
where the deep water pixels plot, 

relative to land pixels.

 

Land areas in Dry Tortugas area
for bands 1, 2, 4 and 6

(purple, blue, PAN, NIR)
from bright beach coral sands/rubbles
to dark mangrove vegetation.
Includes a small area over optically deep waters.

Please note
where the deep water pixels plot, 

relative to land pixels.
Knowing that
  • Lsw=La+Lw
  • LswNIR=LaNIR
  • brighter pixels are coral sand/rubble
  • darker pixels are green vegetation:
    • high in green
    • low in blue
    • low in red
we now have enough information
to specify the Soil Line for all pairs of bands

 
Choosing the values for La (or Lw)
is a tradeoff
which must result in a consistent system
in accordance with the basics of physical realities
In a bi-dimensional histogram,
the Soil Line runs
  • from a bright point LsM 
    • which represents the bright coral sand/rubble
  • to a dark point La
    • which represents a black body
The Soil Line
  • leaves greenish land pixels on its green side
  • leaves blueish   land pixels on its blue   side
  • etc

Optical model over
Land areas in Dry Tortugas area

for bands 6, 5, 3, 2
from bright beach coral sands/rubbles
to dark mangrove vegetation.

Optical model over
Land areas in Dry Tortugas area
for bands 6, PAN, 2 and 1

Note that the PAN band is fuzzy
(is it a co-registration problem?)
For OLI data,
LM, La and Lw values are easily converted 
into RM, Ra and Rw reflectance:
we get the following for this calibration:
  • RM    0.279 0.356 0.415 0.437 0.516 0.696 0.584
    • this represents average coral sand in the Bahamas
  • Rw    0.033 0.026 0.006 0.000 0.000 0.000 0.000
    • this may be deemed a bit low for such water, in view of published scholar knowledge,
    • we must remember that it represents near-nadir viewing rather average cosine viewing
  • Ra    0.097 0.075 0.049 0.048 0.033 0.017 0.014
    • this is available for further use in formal atmospheric correction

 
 
Use of the Soil Line
Upon inversion of the simplified radiative transfer equation, he SOIL Line is used as a reference at null depth:
  • spectrally neutral water column corrected shallow pixels shall plot on the Soil Line
  • greenish water column corrected shallow pixels shall plot on the green side of the Soil Line
  • etc
What is important
  • is not so much that the values adopted are exact
  • but rather that they fit reasonably well in the overall optical picture for the specific illumination conditions of the current scene
What is at stake
  • is that the darker shallow bottoms must be modeled under reasonable conditions
    • while the brighter shallow bottoms shall yield much more reliable results anyway
  • is that the potential bor fottom typing is maximized
  • Neither the red line or the green line or any line in between shall yield optimal depth retrievals
    • for a Blue vs Green two bands case (above), retrievals can be underestimated by up to several meters
  • One needs to account for the fact that the Soil Line is a curve in the linearized bi-dimensional space
    • this is what 4SM does!
 

For the two bands case,
Lyzenga's proposition   Z=A + aX1 + bX2
amounts to using existing depth sounding points
to specify a straight Soil Line
in the linearized bi-dimensional space:
X1=m0+m1*X2
where
Z=0 
m0=-A/a
m1=-b/a

X1=ln(LB1-Lw1) at null depth
X2=ln(LB2-Lw2) at null depth
LB
1=LsB1-La1    BOA radiance
LB
2=LsB2-La2    BOA radiance
THIS IGNORES THE ROLE
OF WATER VOLUME REFLECTANCE
  • if depth points mainly represent bright bottoms, then we get the red line
    • retrieved depths over bright bottoms shall be very good
    • retrieved depths over dark bottoms shall be badly under-estimated
  • if depth points mainly represent dark bottoms, then we get the green line
    • retrieved depths over dark bottoms shall be very good
    • retrieved depths over bright bottoms shall be badly under-estimated
  • So, some investigators process bright bottoms and dark bottoms separately!
  • Stumpf and Holdereid (2003) just give up altogether, and engineer an operational workaround for use by NOAA
  • Lyons, Phinn and Roelfsema conclude in 2011 that neither Lyzenga's nor Stumpf's are suitable for seagrass bottoms
WHAT 4SM DOES
  • Use the image to specify the radiance of the brightest shallow bottom at null depth LsM
  • Use the image to specify the deep water radiance
  • Use the image to split Lsw and LsM into
    • Lsw=La+Lw
    • LsM=La+LM
  • Now the Soil Line is specified
    • TOA: it runs from La to LsM
    • BOA: it runs from 0   to LM
  • Provided the slope and intercept of the Soil Line remain unchanged, modifying Lw or LM does not affect the retrieval of shallow depth
    • LM can be specified to represent average coral sand reflectance: this shall yield scene-independant water column corrected reflectances in view of time series studies
    • this is the beauty of a "ratio method"!!
WHAT IS IMPORTANT
  • Spectral La and Lw do not need to be exact
  • Rather, what is important is that 
    • the spectral deep water radiance Lsw be as exact as possible
    • the slope and intercept of the Soil Line in the natural bi-dimensional space be as exact as possible.
 







The Vegetation Line


The Vegetation Line
  • unlike the Soil Line which has an intercept, the Vegetation Line strikes through the origin of this plot
Path radiance La in 4SM
  • this is generalized to all pairs of visible bands, and used in 4SM to estimate the spectral path radiance La
    • which represents the TOA radiance of a black body in a deglinted image
Water volume reflectance Lw in 4SM
  • therefore, the spectral Water Volume Reflectance is estimated as Lw = Lsw - La
The Soil Line in 4SM
  • this is where depth Z may be assumed to be null
  • most shallow substrates at null depth are likely to range from bright non-vegetated sands all the way to pitch dark dense vegetation, as illustrated above
  • so, for all pairs of visible bands,  the Soil Line in 4SM idealy extends from bright non-vegetated substrates (i.e. coral sands) to black body (origin)


Vegetation Line at Caicos Bank, Bahamas