Introduction and definitions

This is to look for a unification of concepts developed in theory.  Our reference work is \cite{hydrology} however, this matter has been developed since the sixties in Chemical engineering. Notably, here, we want to bridge \cite{Nauman_1969} to recent work which milestone can be considered \cite{Botter_2011}. Let's start with definitions:
Therefore, after these definitions, we have two distinct distribution to care of: the distribution of travel times and the distribution of residence times. Notably, if we collect outgoing particles at the boundary of the control volume,  i.e. the discharge (or evapotranspiration that we neglect here for simplicity), and we have their injection time distribution, for them \(t_{ex}=t\) and we are measuring their travel time. 
The definition of residence times opens also to a new definition. Particles inside the control volume can have a 
Let's assume then that we can label each particle with its injection time and exit time (because eventually we found a way to forecast future).  We can define then, relative to the control volume, the quantity:
We can normalize it over the total volume \(\)\(V\) to obtain:
Based on the above probability, we can define the marginals: