Semianalytical 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 semianalytical 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 The actual dataset is quite often limited to a few depth soundings that feature on some outdated existing nautical or bathymetric chart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!!
 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   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 semianalytical methods

 go to Summary Further why does it possibly work  