I agree that the result would have been very similar without the NN but the calculation (optimization problem for every pixel) has been considerably speeded up by the NN .

Neural networks, that reminds me of the 2012 Roesel paper: Melt ponds on Arctic sea ice determined from MODIS satellite data using an artificial neural network.

I do not doubt their results, do not misunderstand me, but the method is a prime example where a trivial straight forward calculation would have yielded exact results without the computing overhead and uncertainty of neural networks.

Lars,

We are talking about equation 2 in the paper, relating area fractions of melt ponds , ice and open water (A

_{M}, A

_{I}, A

_{W}) to reflectances in three MODIS wavelenths and an additional constraint that the sum of area fractions is 1.

The paper says:

The set of linear Eq. (2) contains three unknowns

(AW,AM,AI) in four equations, therefore the equations are

overdetermined. That means more than one exact solution is

possible and thus, we consider the linear Eq. (2) as an optimization

problem which needs to be solved in a least-square

sense. For the solution of these equation we use a quasiNewton

approximation method (Broyden-Fletcher-GoldfarbShanno

method).

That is over the top. Solving an overdetermined system is the basis of all analysis of physical measurements that involve multiple readings **). Any student physics may be asked to solve it in his first year exam "measuring in physics".

You can consider it as an optimization problem, but to solve it (in the least square sense) there exist efficient direct methods.

There are complications however, the paper says:

With the assumption of a three class mixture model and

the selection of three surface types, we find, that especially

the surface types open water and melt ponds are almost linearly

dependent, therefore the set of linear Eq. (2) is not well

conditioned. To comply with the physical principles, it is

necessary to constrain the interval of the solution between

zero and one for each class. (...)

This I understand, but again I do not see the need for the complicated way the problem is solved.

Restricting the area fractions between 0 and 1 makes the optimization of the type "mixed integer". In this case with only 3 unknowns it is very simple to reduce them to several simpler equations:

For each of the fractions (I,M,W) consider three possibilities: 0, 1 or in between. That is 27 cases at most, of which most do not make sense. Left are a handful of cases (7 but some are trivial) that can be solved efficiently, lowest least square is the optimum.

**): That second line in the quoted text should be "That means that in general no exact solution exists". The opposite of what is written.