4SM summary detailed

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4SM assumptions
The SOIL line
The Brightest Pixels Line
The ratio K_i/K_j
A ratio method
A seed value for K in units of m^-1
Water column correction
Retrieved depth in units of m
Seatruth
Where is the catch?

**4SM details**
SOME COMMON SIMPLIFYING ASSUMPTIONS The water body is homogeneous: each waveband i is assumed to have a specific operational two-ways attenuation coefficient Ki for radiance The water body and the atmosphere are homogeneous: each waveband i is assumed to exhibit a deep water radiance value Lsw i over a deep water area in the image Backscatter of the deep water column Lw (water volume reflectance) Atmospheric path radiance is noted La We assume that we may write Lsw=La+Lw and that the glint at sea-surface is negligible Then in 4SM we introduce some more assumptions as follows.
On the beach SL: THE SOIL LINE ASSUMPTION A clean and fine-grained coral sand pixel on the beach is bright in both bands i and j in the image: LsBi and LsBj A black body pixel on the beach is black in both bands in the image: Lai and Laj From the brightest pixel to the darkest pixel on the beach,we may consider the linear radiometric model of the Soil Line (SL) All intermediate pixels on the beach ideally plot along the linear SL in a ~diagonal position in a bidimensional histogram natural SL Of course, this is a sheer reduction of spectral diversity of natural substrates please visit 4SM La and Lw
Shallow bottoms BPL: THE BRIGHTEST PIXELS LINE ASSUMPTION Let the tide come : *Ls=Lsw + (LsB-Lsw)/exp(KZ) Ls decreases from Ls=LsB (Z=0) all the way to Ls=Lsw (Z=infinite) as Z increases at all visible and near-infrared wavelengths This model accounts for the backscatter of the shallow water column (i.e. color of the water) because Lsw=La+Lw Let Ki <Kj , and examine the scatter plot Lsi= f( Lsj) plot of natural RTE ==> the brightest pixels plot along an exponential-shaped bulge: the Brightest_Pixels_Line (BPL) : this is the "exponential decay" ==> The BPL may be thought of as the outer physical limit of the scatter plot ot the natural_BPL Is the BPL assumption trustworthy?** ?
THE RATIO K_i/K_j Let us linearize this model: *X = ln(LsB-Lsw) = XB - KZ** View the scatter plot of Xi= f( Xj) plot of Linearized RTE HowNice! The brightest pixels now plots along a linear BPL : the exponential decay is linearized linearized BPL The ratio Ki/Kj may be measured as the slope of the linearized BPL
A RATIO METHOD : a consistent system of ratios for a 3-bands image i, j and k, one gets a consistent system of 3 ratios Ki/Kj Ki/Kk ==> Kj/Kk = Ki/Kj / Ki/Kk for a 4-bands image i, j, k and l, one gets a consistent system of 6 ratios and so on
A seed value A seed value may be introduced for Ki at any particular wavelength in order to specify spectral K , from Ki to Kn , for all wavebands available i to n This seed value does not need to be realistic: it might as well be taken as Ki=any_value, as all other values are then specified through the series of ratios This choice of a seed value only affects the depth computed in meters, while "water column correction" actually depends on the ratios among spectral K values to compute spectral bottom signature.
SHALLOW WATER COLUMN CORRECTION Invert the RTE: *LB = Lw + (Ls-Lsw)exp(KZ)* The water column correction of a shallow pixel may be achieved by bringing it back to where it belongs on the Beach, i.e. on the Soil Line: LB = Lw + (Ls-Lsw)exp(KZ) This is done by increasing Z until the "averaged" spectral LB sits on the Soil Line see Computing Depth Z and Retrieved Depth Z Water Column Correction: a "low tide spectral view" of all shallow water pixels has been achieved Spectral diversity of natural shallow substrates is restored to some extent in this process More wavebands with bottom detection result in a more realistic spectral bottom signature
RETRIEVED DEPTH IN METERS Computed Depth Z still needs to be calibrated in meters. In 4SM, this last step is done : by interpolating a seed value from Jerlov's table of diffuse attenuation coefficients of Marine/Coastal Water Types (case 1 waters) No fielddata is needed Spectral K obtained through this process are very close to reality, by some trick of the upwelling radiance vs irradiance scenario: see discussion of the Q factor page 370 of Voss et al. , 2003: "The spectral upwelling radiance distribution in optically shallow waters"
SEA TRUTH Some sea truth is still needed to fine-tune this calibration of Z: a CoefZ must be estimated from seatruth depth data through linear regression a tide correction may be applied *Zfinal = TideHeight + CoefZ Zcomputed**

3 : go to Summary further: why does it possibly work?

**4SM : where's the catch?**
Semi-analytical methods like EOMAP, SAMBUCA, ALUT, DigitalGlobe's, Hope, ...	4SM empirical method	Other empirical methods like Polcyn's, Lyzenga's, Stumpf's
Inverting the model is an optimization process which needs some form of assumption as regard water column corrected spectral reflectance signature	Inverting the model is an optimization process which needs some form of assumption as regard water column corrected spectral radiance signature	Coefficients A, B and C in Polcyn's magic formula or Coefficients m0 and m1 in Stumpf's magic formula are a form of of assumption as regard water column corrected spectral reflectance/radiance signature Just make Z=0 and rewrite to get the equation of a straight line
They iterate the inversion of the semi-analytical RTE -by increasing Z in a LUT- until an occurrence is found that yields the closest spectral matching with the observed signature at the current pixel	4SM iterates the inversion of the simplified RTE -by increasing Z- until the water column corrected spectral reflectance for the current pixel is deemed to match satisfactorily some form of the spectral "Soils Line" where Z is null	No iteration is needed to compute Z
For this they need spectral value for 2K, the diffuse attenuation coefficient at specified wavelengths for the bansdet a database of spectral signatures for all endmembers bottom types that possibly exist at the site the a, b, c, ... coefficients for the mixture of endmembers they want to unmix at the current pixel	For this I need the spectral ratio K_i/K_j observed in the image, Lyzenga's way, for all pairs of bands i, j, k, ... and a seed value derived from Jerlov's data, so that spectral value for 2K, the diffuse attenuation coefficient , is estimated at all specified wavelengths for the bansdet a spectral Soils Line model derived from observed spectral signature of bareland in the image at null depth	For this they need a dataset of depth points over the whole shallow depth range to represent all major shallow bottom types that exist at the site The least we can say is that such dataset is difficult and costly to collect and tricky to reduce The actual dataset is quite often limited to a few depth soundings that feature on some outdated existing nautical or bathymetric chart
Because these are unknown, and because their database is a discrete collection of pure endmembers at null depth, they choose that particular quantitative mixture of all possible endmembers with all possible 2K values which yields a spectral signature that best fits the current pixel. This they call "spectral matching" of the observed signature with zillions of LUT occurrences	Because these are all derived from the image itself and Jerlov's data through 4SM's own calibration process, the inversion then is a simple matter of increasing Z as explained above.	Complaints If sandy bottoms prevail in this dataset, then depths estimated over vegetated/dark bottoms shall most probably be underestimated, possibly very badly (a common complaint!!!) Conversely, if dark bottoms prevail in this dataset, then depths estimated over bright bottoms shall most probably be overestimated, possibly very badly.
Involves multidimensional matrixes, entails horrific computing time	Very fast results in very attractive performances	Very fast
But their process also accounts for -and maps- spatial variations of water optical quality on the fly this would appear to be a distinct advantage this has a very high cost though!! see also 4SM vs ALUT see also 4SM vs DigitalGlobe	Because using Lyzenga's trick in 4SM pertains to the brightest bottoms in the clearest waters in the scene, 4SM cannot account properly for areas where waters are less clear: they shall show badly in the true color composite screen display of water column corrected bottom signatures. This provides a means for the practioner to either devise specific conditions for the processing of areas affected or flag them as artifacts see this artifact on Ikonos at Dubai Specific conditions that sort of artifact can be quite important and entails very bad underestimation of depth for pixels processed by the N-bands case in many cases, it is much less sensitive in the Blue-Green bands and disappears outright when using only those bands no wonder then that empirical methods -unlike 4SM- generaly use only the Blue and Green bands, which represents a distinct loss of accuracy in estimated depth for all those faithful pixels the recent development of the "4SM_PAN_solution" offers a nice way around this limitation	Like in 4SM, water optical properties are assumed to be constant This does not "unmix" the influence of variable depth from that of variable bottom signature, as computed depth is the only output Nobody worries about bad or fancy results garbage in ==>> garbage out until these methods ultimately are shelved and poeple start investigating into semi-analytical methods
	go to Summary Further why does it possibly work