We all know Z = A + a*X_{1} + b*X_{2} + ... k*X_{N} where, for each waveband at wavelength WL_{i }
We all know that this amounts to introducing an assumption regarding the spectral signature of the shallow bottom Lyzenga's proposition works reasonably well over fairly bright bottoms, which are likely to be well represented in the depth sounding dataset used for calibration. Another way to write this is by removing the path radiance La Z = A + a*X_{1} + b*X_{2} + ... k*X_{N} where, for each waveband at wavelength WL_{i }
Let Z=0 for the twobands case: this can be rearranged into a straight line X1=m0+m1*X2 where intercept m0=A/a and slope m1=b*/a But Lyzenga's proposition fails to ackowledge for the role of water volume reflectance Lw This is why it is well documented to yield underestimated depths Z over dark bottoms. Just look at this

This is illustrated at http://www.watercolumncorrection.com/4smpresentation.php 
In 4SM, we acknowledge for some extra evidence which are plainly obvious in most image data

This is illustrated below : X_{blue} vs X_{green} for the atoll of Cliperton (Ikonos) for Negril shores in Jamaica (TM) 
Estimating spectral K in m^{1}

Estimating spectral Lw

Calibration diagram The strength of 4SM is that, unlike other empirical ratio methods, this approach keeps reasonable track of the basic of atmospheric and underwater optics, through the calibration diagram. 