Friday, November 10, 2017

About Benettin et al. 2017, equation (1)

Gianluca (Botter) in his review of Marialalaura (Bancheri) Ph.D. Thesis brought to my attention the paper Benettin et al. 2017. A great paper indeed, where a couple of ideas are clearly explained:

  • SAS functions can be derived from the knowledge of travel and residence times probability
  • a virtual-experiment where they show that traditional pdfs (travel times pdf) can be seen an the ensamble of the actual time-varying travel times distributions.

The paper is obviously relevant also for the hydrological contents it explains, but it is not the latter point the one which I want  to argue a little. I want here just to argue about the way they present their first equation.

SAS stands for StorAge Selection functions and they are defined, for instance in Botter et al. 2011 (with a little difference in notation) as:
$$
\omega_x(t,t_{in}) = \frac{p_x(t-t_{in}|t)}{p_S(t-t_{in}|t)} \ \ \ (1)
$$
as the ratio between the travel time probability related to output $x$ (for instance discharge or evapotranspiration) and the residence time probability.
In the above equation (1)
  •  $\omega_x$ is the symbol that identifies the SAS
  • $t$ is the clock time
  • $t_{in}$ is the injection time, i.e. the time when water has entered the control volume
  • $p_x(t-t_{in}|t)$ with $x \in \{Q, ET, S\}$  is the probability that a molecule of water entered in the system at time $t_{in}$ is inside the control volume, $S$, revealed as discharges, $Q$, or evapotranspiration, $ET$

Equation (1) in Benettin et al. is therefore written as
$$
\frac{\partial S_T(T,t)}{\partial t} + \frac{\partial S_T(T,t)}{\partial T} = J(t) - Q(t) \Omega_Q(S_T(T,t),t)-ET(t) \Omega_{ET}(S_T(T,t),t) \ \ \ \ (2)
$$
Where:

  • $T$ is residence time (they call  water age but this could be a little misleading because the water age of water in storage could be, by their own theory different in storage, discharge, evapotranspiration)
  • $S_T$ is the age-ranked storage, i.e. “the cumulative volumes of water in storage as ranked by their age” (I presume the word “cumulative”  implies some integration. After thinking a while and looking around, also to paper van der Velde et Al. 2012, I presume the integration is over all the travel times up to $T$ which, because the variable of integration in my notation is $t_{in}$ means that $t_{in} \in [t,t-T]$  )
  • $J(t)$ is  the precipitation rate at time $t$
  • $Q(t)$ is the discharge rate at time $t$
  • $\Omega_x$ are the integral of the integrated SAS function which are more extensively derived below.

In fact, this (2) should be just an integrated version (integrated over $t_i$) of equation (9) of Rigon et al., 2016:
$$
\frac{ds(t,t_{in})}{dt} = j(t,t_{in}) - q(t,t_{in}) -et(t,t_{in})
\ \ \ \ (3)
$$
where:
  • $s(t,t_{in})$ is the water stored in the control volume at time $t$ that was injected at time $t_{in}$
  • $j(t,t_{in})$ is the water input which can have age $T=t-t_i$
  • $q(t,t_{in})$ is the discharge that exits the control volume at time $t$ and entered the control volume at time $t_{in}$
  •  $et(t,t_{in})$ is the evapotranspiration that exits the control volume at time $t$ and entered the control volume at time $t_{in}$
In terms of the SAS and the formulation of the problem given in Rigon et al. (2016), the $\Omega$s can be defined as follows:
\begin{equation}
\Omega_x(T,t) \equiv \Omega_x(S_T(T,t),t) := \int_{t-T}^t \omega_x(t,t_i) p_S(t-t_i|t) dt_i = \int_0^{p_S(T|t)} \omega_x(P_S,t) dP_S
\end{equation}
Where the equality ":=" on the l.h.s is a definition, so the $\Omega$s ($\Omega_Q$ and $\Omega_{ET}$) are this type of object. The identity $\equiv$ stresses that the dependence on $t_in$ is mediated by a dependence on the cumulative storage $S_T$ and $T$ is the travel time. As soon as $T \to \infty$, $\Omega \to 1$ (which is what written in equation (2) of Benettin's paper). This is easily understood because by definition ${\omega_x(t,t_i) p_S(t-t_i|t)} \equiv {p_x(t-t_i|t)}$ are probabilities (as deduced from (1)).
An intermediate passage to derive (2) from (3) requires to make explicit the dependence of the age-ranked functions from the probabilities. From definitions, given in Rigon et al., 2016. It is
$$
\frac{d S(t) p_S(t-t_{in}|t)}{dt} = J(t) \delta (t-t_in) - Q(t) p_Q(t-t_{in}|t) - ET(t) p_{ET}(t-t_{in}|t)
$$
which  is Rigon et al. equation (14).
Now integration over $t_i \in [t-T, t]$ can be performed to obtain:
$$
S_T(t_{in},t):= \int_{t-T}^t s(t_{in},t) dt_{in}
$$
and, trivially,
$$
J(t) = J(t) \int_{t-T}^t \delta(t-t_{in}) dt_{in}
$$
while for the $\Omega$s I already said.
The final step is finally to make a change of variables that eliminate $t_{in}$ in favor of $T := t-t_{in}$. This actually implies the last transformation. In fact:
$$
\frac{dS(t,T(t_{in},t))}{dt} =\frac{\partial S(t,T(t_{in},t))}{\partial t} + \frac{\partial S(t,T(t_{in},t))}{ \partial T}\frac{\partial T}{ \partial t} = \frac{\partial S(t,T(t_{in},t))}{\partial t} + \frac{\partial S(t,T(t_{in},t))}{ \partial T}
$$
since $\partial T/\partial t$ =1. Assembling all the results, equation (2) is obtained.

Note:
Benettin et al., 2017 redefines the probability $p_S$ as “normalized rank storage … which is confined in [0,1]” which seems weird with respect to the Authors own literature. In previous papers this $p_S$ was called backward probability and  written as $\overleftarrow{p}_S(T,t)$. Now probably they have doubt we are talking about probability.  In any case, please read it again: "normalized rank storage … which is confined in [0,1]”. Does not sound unnatural is not a probability ? Especially when you repetitively estimate averages with it and comes out with “mean travel times”?Operationally, it IS a probability. Ontologically the discussion about if there is really random sampling or not because there is some kind of convoluted determinism in the travel times formation can be interesting but it brings to a dead end. On the same premised we should ban the word probability from the stochastic theory of water flow, that, since Dagan has been enormously fruitful.

This long circumlocution,  looks to me like the symbol below

or TAFKAP, which was used by The Artist Formerly Known As Prince when he had problems with his record company.
In any case, Authors should pay attention in this neverending tendency to redefine the problem rather beacause it can look what Fisher (attribution by Box, 1976) called mathemastry. This is fortunately not the case of the paper we are talking about. But then why not sticking with the assessed notation ?

The Authorea version of this blog post can be found here.

References 

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