In real geophysical applications (either meteorology or oceanography), there is usually no viscosity. In this case, assuming the observability condition, the BFN algorithm is well posed, and theorem 3.6 and proposition 3.2 show that the solution tends to the observation trajectory everywhere, and not only on the support of 
[29].
From a numerical point of view, we can observe that even with discrete and sparse observations in space, the numerical solution is corrected everywhere. We also observed that, with a not too large viscosity coefficient, the behaviour of the algorithm remains unchanged [21].