Well when we normalize a wave function we are correcting it by a constant such that it physically makes sense.
I mean it's physically nonsense to talk about a wave function that isn't normalized.
But at the same time the normalization is contained in the wave function itself so you won't have \frac{\psi}{const} you just have \psi which when we integrate the square of the wave function over all space equals 1.
To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.