This attempts to answer
most current objections, like

• Field data shall always be needed for calibration
• Atmospheric correction must be applied prior to shallow water processing
• Depth over dark bottoms is well known to be badly underestimated
• One should be aware that Kup is much higher than Kdown
• One should be wary of multiple reflection of the in-water photon path
• Radiances versus irradiances: the RTE was established for irradiances
• NOAA make extensive use of a their own new "Log ratio method"

 references

Preamble

Shallow water refers to areas in the imagery where the bottom contrast Ls-Lsw is significant , whether negative or positive : this is to mean that the radiance measured at the remote sensor includes a significant contribution by the light that has been reflected on the sea-bottom when compared with the contribution by the light that has been backscattered by the water column (or even reflected by the water surface).


Optically deep water refers to areas in the imagery where such contribution caused by the existence of a shallow bottom may safely be assumed to be non-significant.

Operational K_wl : the wavebands of a multi/hyperspectral remote sensing image are rated in terms of the operational two-ways water attenuation coefficient, denoted K_wl, which is specific to each waveband's wavelength (a delicate concept!).


Optical water type : spectral values of K also depend on the optical properties (i. e. turbidity) of the particular water mass under study.


Visible spectrum : bathymetry modeling takes advantage of the very large variation of the attenuation properties of light in marine/coastal waters across the visible part of the solar spectrum.


Water Column Correction : the purpose of optical modeling is to "un-mix" depth from spectral reflectance of a shallow bottom: 4SM modeling allows for estimating and accounting for the variations of the spectral reflectance of the bottom in the process of estimating the depth in shallow areas.
 

  
The simplified optical model  
for remote sensing shallow water radiance at Top Of Atmosphere (TOA)
is referred to by O'Neil and Miller (1989)
as the " most familiar form of shallow water reflectance model".

Ls = Lsw + (LsB-Lsw) / exp(K*Z)

• Ls=L+La         is the radiance measured at the sensor
• Lsw=Lw+La is expected to be constant over the scene
• LsB=LB+La  is the radiance if the bottom were at null depth (Z=0), i. e. the bottom reflectance
• K                  is an operational two-ways attenuation coefficient for remotely sensed radiance
• Z                  is the depth of the shallow bottom

  " In conclusion, the approximate formula can be safely adopted in operation when interpreting or predicting the reflectance of shallow waters, in particular if Lw and K have been estimated from remotely sensed data. " (equation 9a of Maritorena et al. (1994), pp 1689-1703).


The above statement actually refers to irradiance reflectances (R, A and 2Kd) , whereas we are dealing with remote sensing radiances here (La, Lw, LB and K). It remains to be seen whether or not it may safely be re-written, as we do here, for remote sensing radiances for an operational purpose. This question in particular bears on what K value is adopted, and therefore on the error on computed depths.

We think that this most important and debated question is properly addressed by the following statement of Jerlov (1976): " It should be borne in mind that, in the surface region 0-10 m, the irradiance attenuation coefficient Kd for high solar elevation is close to the absorption coefficient a ."

We have come to the conclusion that a similar statement applies to upwelling remote sensing shallow water radiances: most of the photon paths which reach to the remote sensor have experienced a direct path from the bottom upwards, thus accounting for an attenuation coefficient which is just a little higher than the absorption coefficient a. Indeed, any upwelling photon path that departs significantly from the vertical shall be refracted even further away from the vertical upon crossing the water-air interface, and therefore stands a very low probability of being captured by the remote sensor. According to our own experience, and in contrast with prevailing opinions which consider the fate of upwelling shallow water irradiance, it would appear that operational 2K for remote sensing radiance over shallow bottoms is very close to 2Kd of Jerlov, i. e. just a little higher than the absorption coefficient a.

See in the following paragraph a complete discussion of that most important point.

This might explain why the observed 4SM performance in computed depths exceeds by far the performance predicted by Maritorena et al. (1994).

 

BPL: spectral K values
are estimated from the image itself
through the concept of the Brightest Pixels Line

• BPL concept : for a given pair of bands i and j with K i<K j, it is assumed that the outer exponential shape in a bidimensional scatter plot is most likely to be caused by the brightest end member bottom substrate that exists in the area of interest.
• Sample the BPL in the imagery through an automatic process for all pairs of wavebands available: see natural BPL and SL in scatter plot of band i vs band j.
• Measure all ratios Ki/Kj: through the linearization of the BPL pixels, the ratio Ki/Kj is measured for all possible pairs of available wavebands i and j: see linearized BPL and SL in scatter plot of linearized data band i vs band j.
• A consistent system of ratios : it is important to verify the internal consistency of all these ratios. Spectral K values are estimated from the image itself in such a way that their ratios must achieve the values observed for all pairs of wavebands in the spectral image: see tarawa_polygonred.jpg and tarawa_m2.eps.png .
   
• Seed value for operational K : at this stage, spectral operational K may be derived for all wavebands available by just providing a seed value of K at any of these wavebands. Any seed value may be used, so that spectral operational K can be fully specified and the "water column correction" may be conducted over shallow areas.In practice, two approachesare proposed:
  • Alternative A: K~=2Kd this is done in line with the statement by J. Kirk that "a family of curves, of progressively changing shape, determined mainly by phytoplancton 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".
    • A first step consists in estimating one interpolated marine water type of Jerlov for which one of these ratios - conveniently for example the ratio Kblue/Kgreen matches the observed value: this yields an estimate of Kblue and Kgreen.
    • A second step consists in deriving all other K values using the observed ratios and either Kblue or Kgreen: see mb4.sh.jpg for a 17-bands hyperspectral case study.
    • Wavelength vs waveband : in this approach, a precise wavelength must be specified for each waveband.This is a major source of uncertainty.
    • Radiance vs irradiance : then we need to figure out how/why properties observed for downwelling irradiances may possibly be applied to two-ways water-leaving near-nadir upwelling radiances.
    • Kd ~= a : as stated by Jerlov " the spirit of this classification is that the irradiance attenuation coefficient Kd for any wavelength can be expressed as a linear function of a reference wavelength. It should be borne in mind that, in the surface region 0-10 m, the irradiance attenuation coefficient Kd for high solar elevation is close to the absorption coefficient a ".
    • K ~= 2*Kdown : we have observed that 4SM yields operational spectral K values which are in satisfactory agreement with sea truth of computed depths. Therefore, we have to assume here that, due to refraction of water-leaving radiance, the photon paths which manage to enter the field of view of the remote sensor are mostly those which had a near-nadir in-water upwelling path.
      • In other words, and for upwelling remote sensing radiances, Kup would be quite close to a .
      • In other words also, and in spite of warnings by Kirk in respect of Kup , as oblique upwelling in-water photon paths stand a very poor chance of being captured by the remote sensor, scattering would appear to play a second or third order role in the formation of operational K for shallow water remote sensing radiances.
    • Hence the observation that we can use an operational K~=2Kd of Jerlov for shallow water modeling.
  • Alternative B: K_nir : we are experimenting on a seed value of K_nir for a quasi-absolute_calibration , provided of course that a NIR band is available and that the imagery contains a fair coverage of very shallow clean sandy bottoms away from any breaking waves.
    • If/when this is confirmed, we should reach to a quasi-absolute calibration of spectral operational K for remote sensing radiances.
       
    • In other words, scattering would appear to play a second or third order role in the formation of operational K for shallow water remote sensing radiances in a NIR waveband.
    • This second approach seems to be promising, as K_nir would appear to be very stable from Oceanic to Coastal water types, because absorbption coefficient a_nir is much higher than scattering  coefficient b_nir in all Case I waters.

SL: atmospheric path radiance La
is estimated from the image itself
through the concept of the average Soil Line

• Null depth : in 4SM, the average spectral Soil Line is used as a reference of pixels at null depth (Z=0).
• Brightest to darkest : the spectral radiometric model for the average Soil Line is observed using the dry land parts of the image, preferably non-vegetated, from brightest to darkest.
• Natural/Linearized: the Soil Line runs as a straight line in a bidimensional histogram of natural data. It runs as a curved line in most bidimensional histograms of linearized data.
• Lsw=La+Lw : this concept of the average Soil Line starts from the spectral radiance of a black body on land (i.e. the atmospheric path radiance La) and runs through an ideal average very bright non-vegetated dry land pixel like a bright sandy beach.
  
  The key relationship here is Lsw=La+Lw.
  Lsw, the deep water radiance, is observed from the image.
  Lw, the color of optically deep water, is assumed to be null in the Red to NearInfraRed region of the visible spectrum: this is referred to as the "black pixel assumption" in the litterature.
  Lw is assigned a value which allows an acceptable match of the Soil Line with the average Soil Line observed in the image itself.
  For clear blue waters, it is usually observed that Lw_blue >> Lw_green

CALIBRATION DIAGRAM and
SHALLOW WATER OPTICAL MODELING

• Multiple reflection==> non-linearities? In spite of all the complexities of underwater optics for irradiance, it is demonstrated that the above simplified RTE for remote sensing radiances in its operational form
  • Ls=Lsw+(LsB-Lsw)/exp(K*Z) does behave in a linear manner across the visible spectrum .
• See in particular for a SPOT XS image that multiple reflection at extremely shallow depths does not appear to be a problem : this is demonstrated by the very linear shape of the BPL at depths of just a very few decimeters in the X[1-2] vs X[3] scatter plot of a very bright BPL in tarawa-subset.
• Fully framed: once all aspects of the calibration diagram are satisfactory , the spectral model is fully "framed" and ready for inversion .
• Depths over dark bottoms: accounting for the water volume reflectance: 
  • ?model inversion accounts for the volume reflectance of the shallow water column through the term 
    • Lw - Lw / exp (K*Z) .
  • In their "simple ratio method", Polcyn et al.'s well known condition that "a pair of bands may be found..." amounts to assuming that the Soil Line is a straight line in a linearized scatter plot of band i vs band j.
  • We know that this only holds true when Lw is null in both bands i and j, i.e. in the red-nir range of the solar spectrum.
  • The result of that shortcoming is that depths computed through the "simple ratio" algorithm are badly underestimated over dark bottoms.
  • For any pair of bands that involves a Blue or a Green band, the linearized Soil Line is a distinctly curved line.
  • Water volume reflectance is properly accounted for in 4SM,
    • unlike in NOAA's "Log ratio" method which overestimates depths over darker bottoms.
  • Computing Z in meters and LB in DNs : for the current pixel, modeling then consists in searching iteratively for a depth value Z which yields a spectral bottom signature LsB=Lsw+(Ls-Lsw)*exp(K*Z) that is consistent with the average spectral Soil Line observed in the image.
  • Final Z : we lack the experience of numerous sea truth experiments for a variety of spectral images in diverse environments.
    Therefore, all computed depths still need to be multiplied by a final depth correcting factor CoefZ to be derived from some sea truth when it becomes available.This does not affect the computed bottom reflectances though.
    
     

4SM References
 

• Jerlov N., “ Marine Optics”, Elsevier Scientific Publishing Co, Amsterdam, 1976.
• Kirk J. T., " Light and Photosynthesis in Aquatic Ecosystems", Cambridge U. Press, Cambridge, 1994.
• Lyzenga D. R., “ Passive remote sensing techniques for mapping depth and bottom features”, Applied Optics, vol. 13, no 3, 379-383, 1978.
• Malthus T. and Karpouzli E., “ Integrating field and high spatial resolution satellite-based methods for monitoring shallow submersed aquatic habitats in the Sound of Eriskay, Scotland, UK”, Int. J. Remote Sensing, 2003, vol. 24, 2585-2593.
• Maritorena S., Morel A. and B. Gentilly B., " Diffuse reflectance of oceanic shallow waters: influence of water depth and bottom albedo", Limn. and Ocean., Vol. 39, No 7, pp. 1689-1703, 1994.
• Maritorena S., " Remote sensing of water attenuation in coral reefs: a case study in French Polynesia", Int. J. Remote Sensing, Vol. 17, No 1, pp. 155-166, 1996.
• Morel Y. and Lindell T., “ Passive multispectral bathymetry mapping of Negril Shores, Jamaica ", Fifth International Conference on Remote Sensing for Marine and Coastal Environments, San Diego, California, 5-7 October 1998.
• Philpot W. D., " Bathymetry mapping with passive multispectral imagery", Applied Optics, Vol. 28, No. 8, pp. 1569-1578, April 1989.
• Polcyn F. C. , Brown W. L. , and Sattinger I. J., " The measurement of water depth by remote sensing techniques", Report 897326-F, Willow Run Laboratories, U. of Michigan, Ann Arbor, 1970.
• Stumpf R. P. and K. Holderied K.," Determination of water depth with high resolution satellite imagery over variable bottom types", Limnol. Oceanogr. Vol 48, 547-556, 2003.