Saturday, December 16, 2017

Marialaura Bancheri defense

The Ph.D. Thesis of Marialaura Bancheri is already available in a previous post. On december 14, she finally defended it. This is the video of her performance. Her topics are: research reproducibility, GEOFRAME, reservoir based modelling (or semidistributed modelling) of the hydrological cycle, travel times theory re-interpreted in the perspective of reservoirs modelling.

I have no doubt that it could be very useful to all who are interested in our recent work and to all those that try to interpret catchment scale behavior through travel times. Marialaura was an outstanding student, is an exceptional team manager, and she is looking for an appropriate post-doc position.

Wednesday, December 13, 2017

Monday's discussion on evapotranspiration - Part II - The soil-plants fluxes

The first post treated transpiration from the point of view of the atmosphere control volume. There is a “below” though. Below is composed by leaves, trunks/stems, roots. Roots, in turn, are being inserted in soil from where they sip water and nutrients.
Water in soil is understood to be moved by Richards equation (with all the possible variations or extensions), essentially a Stokesian flow (therefore laminar) in the bundle of soil pores.
Plants do not have a pumping heart and therefore has been since long time argued how they can move water up until the tallest leaves that, can be as high as 150 m above soil level. Some plants do not have either a real “vascular” system in the sense we mean for animals, with arteries and veins. They have indeed specialised interconnected cells to move water up, called collectively xylem, and specialised interconnected cells to move around sucrose and the products of photosynthesis (especially to fruits and roots) called phloem.

So the xylem is the place were to look for ascending water. But how water moves in it ? Since Hales (1727), reported in Holbrook and Zwieniecki (2005), the theory invoked was the cohesion-tension one, which is well illustrated in the introduction of e.g. Holbrook and Zwieniecki (2005), which is open (on Amanazon). Other references include Tyree (2003), which is satisfying from the conceptual point of view but not from the point of view of equations. From this side, possibly Steudle (2001) and Strook et al., (2014) are better. Also Pickard (1981) remains a good reference.
The problems to be understood in xylem water movement is how cohesion-tension works. Under normal conditions, atmosphere is very arid and, for instance at normal temperatures, assuming a 50% of specific humidity of air, it correspond to a pressure of -100MPa (e.g. Jensen at al, 2016), while at roots is usual conditions, water is at much higher pressure, ~ -1.5MPa, meaning, that the gradient of pressure along a plant of ten can be as high are 10 MPa/m (see also Nobel, 2009).
Therefore water is “pulled” and we have to face with the counterintuitive idea that water resist to a tension. For liquids to resist to tensile forces, it is necessary that no bubble is nucleated inside the liquid that disrupt the liquid continuity (creating emboli, e.g. Fsher, 1948). Eventually mechanisms for refilling the vessels have also to be required for understanding the real functioning of plants. This is actually matter of research.
Very much attention to the physics of the process, is also paid in the recent review by Jensen et al, (2016). There also the phloem flux is covered with quite detail and reference therein is large and up-to date. Reading the papers I cited, that are just a few in my collection, can be a starting point for understanding the problem, and this is an advise that I am experimenting myself.
Personally, being highly ignorant of plants physiology, I also require to study it overall. A reference I am following is a classic textbook, Taiz and Zeiger (2002), but a more physical-chemical-mathematical approach ca be found in Nobel (2009).
My first look at the above papers make me remain with the idea that too details hide a possible, more integrated and macroscopic treatment of the matter, at level of single tree, without having necessarily to cope with each cellular movements of water. In fact a look to plants functioning as a whole, is what we, hydrologists are looking for.
Concentrating on plants does not mean we have the whole picture, since soil-plant(s) interactions must be accounted for. We already said that, especially in this case, Richards equation is considered the equation describing water flow in soil. Richards equation, however, is a partial differential equation, ideally written at the Darcy scale, while soil-water-plant interactions happen at the smallest scale of roots. Pickard (1981) gives a description of roots structure but this is therefore not enough to understand well what happens. Soil scientists are bold, and therefore they use a sort of brute-force attack to the problem, where the Darcy scale is ignored and Richards equation is used at small scale where one root link can be associated “mechanistically” to an elementary control volume. A good and up-to-date illustration of this approach is given, for instance in Schröder (2013) Ph.D. Thesis. The only trick used to differentiate the usual approach for adapting it to root interactions is to add two type of conductivities. But please read Schröder (2013) and Huber et al. (2014) to have full and detailed account of it. Companion to this approach is the use of some root model, for instance as Root Typ (Pagès et al., 2004). The latter model are useful also alone, cause the information they contain of roots architecture and density, factors that certainly any theory cannot neglect.


So, I hope to have indicated some initial lectures of which you find the reference below. Below below you also find a bunch of other references, some from the same Authors, that could probably be a good second lecture.

References

A wild bunch of references

  • Aroca, R., Porcel, R., & Ruiz-Lozano, J. M. (2011). Regulation of root water uptake under abiotic stress conditions. Journal of Experimental Botany, 63(1), 43–57. http://doi.org/10.1093/jxb/err266
  • Bouda, M., & Saiers, J. E. (2017). Dynamic effects of root system architecture improve root water uptake in 1-D process-based soil-root hydrodynamics. Advances in Water Resources, 1–53. http://doi.org/10.1016/j.advwatres.2017.10.018
  • Carminati, A., Moradi, A. B., Vetterlein, D., Vontobel, P., Lehmann, E., Weller, U., et al. (2010). Dynamics of soil water content in the rhizosphere. Plant and Soil, 332(1-2), 163–176. http://doi.org/10.1007/s11104-010-0283-8
  • Couvrer, V. (2017, October 30). Emergent properties of plants hydraulic architecture: a modelling study. 
  • Debenedetti, P. G. (2012). Stretched to the limit. Nature Physics, 1–2. 
  • Delory, B. M., Baudson, C., Brostaux, Y., Lobet, G., Jarden, du, P., Pagès, L., & Delaplace, P. (2015). archiDART: an R package for the automated computation of plant root architectural traits, 1–20.
  • Fiscus, E. L. (1975). The Interaction between osmotic- and pressure-induced water flow in plats roots, 55, 917–922. 
  • Fisher, J. C. (1948). The Fracture of Liquids. Journal of Applied Physics, 19(11), 1062–1067. http://doi.org/10.1063/1.1698012
  • Hartvig, K. (2016). Osmotically driven flows and maximal transport rates in systems of long, linear, porous pipes. arXivfluid, 1–18. 
  • Hildebrandt, A., Kleidon, A., & Bechmann, M. (2016). A thermodynamic formulation of root water uptake. Hydrology and Earth System Sciences, 20(8), 3441–3454. http://doi.org/10.5194/hess-20-3441-2016
  • Hodge, A., Berta, G., Doussan, C., Merchan, F., & Crespi, M. (2009). Plant root growth, architecture and function. Plant and Soil, 321(1-2), 153–187. http://doi.org/10.1007/s11104-009-9929-9
  • Holbrook, N. M., Burns, M. J., & Field, C. B. (1995). Negative Xylem Pressures in Plants: A Test of the Balancing Pressure Technique. Science, 270(5239), 1–3. 
  • Huber, K., Vanderborght, J., Javaux, M., & Vereecken, H. (2015). Simulating transpiration and leaf water relations in response to heterogeneous soil moisture and different stomatal control mechanisms. Plant and Soil, 394(1-2), 1–18. http://doi.org/10.1007/s11104-015-2502-9
  • Huber, K., Vanderborght, J., Javaux, M., Schröder, N., Dodd, I. C., & Vereecken, H. (2014). Modelling the impact of heterogeneous rootzone water distribution on the regulation of transpiration by hormone transport and/or hydraulic pressures. Plant and Soil, 384(1-2), 93–112. http://doi.org/10.1007/s11104-014-2188-4
  • Iversen, C. M., McCormack, M. L., Powell, A. S., Blackwood, C. B., Freschet, G. T., Kattge, J., et al. (2017). A global Fine-Root Ecology Database to address below-ground challenges in plant ecology. New Phytologist, 215(1), 15–26. http://doi.org/10.1111/nph.14486
  • Janbek, B., & Stokie, J. (2017). Asymptotic and numerical analysis of a porous medium model for transpiration-driven sap flow in trees. arXivfluid, 1–24. 
  • Javaux, M., Couvreur, V., Vanderborght, J., & Vereecken, H. (2013). Root Water Uptake: From Three-Dimensional Biophysical Processes to Macroscopic Modeling Approaches. Vadose Zone Journal, 12(4), 0–16. http://doi.org/10.2136/vzj2013.02.0042
  • Javaux, M., Schröder, T., Vanderborght, J., & Vereecken, H. (2008). Use of a Three-Dimensional Detailed Modeling Approach for Predicting Root Water Uptake. Vadose Zone Journal, 7(3), 1079–1088. http://doi.org/10.2136/vzj2007.0115
  • Jensen, K. H., Berg-Sørensen, K., Bruus, H., Holbrook, N. M., Liesche, J., Schulz, A., et al. (2016). Sap flow and sugar transport in plants. Reviews of Modern Physics, 88(3), 320–63. http://doi.org/10.1103/RevModPhys.88.035007
  • Jorda, H., Perelman, A., Lazarovitch, N., & Vanderborght, J. (2017). Exploring Osmotic Stress and Differences between Soil–Root Interface and Bulk Salinities. Vadose Zone Journal, 0(0), 0–13. http://doi.org/10.2136/vzj2017.01.0029
  • Kalbacher, T., Delfs, J.-O., Shao, H., Wang, W., Walther, M., Samaniego, L., et al. (2011). The IWAS-ToolBox: Software coupling for an integrated water resources management. Environ Earth Sci, 65(5), 1367–1380. http://doi.org/10.1007/s12665-011-1270-y
  • KALDENHOFF, R., RIBAS-CARBO, M., SANS, J. F., LOVISOLO, C., HECKWOLF, M., & UEHLEIN, N. (2008). Aquaporins and plant water balance. Plant, Cell and Environment, 31(5), 658–666. http://doi.org/10.1111/j.1365-3040.2008.01792.x
  • Kuhlmann, A. (2011, November 14). Influence of soil structure and root water uptake on flow in the unsaturated zone. (I. Neuweiler, Ed.). Stuttgart University. 
  • Ma, L., Chen, H., Li, X., He, X., & Liang, X. (2016). Root system growth biomimicry for global optimization models and emergent behaviors. Soft Computing, 21(24), 1–18. http://doi.org/10.1007/s00500-016-2297-5
  • Maherali, H. (2017). The evolutionary ecology of roots. New Phytologist, 215(4), 1295–1297. http://doi.org/10.1111/nph.14612
  • Medlyn, B. E., De Kauwe, M. G., Lin, Y.-S., Knauer, J., Duursma, R. A., Williams, C. A., et al. (2017). How do leaf and ecosystem measures of water-use efficiency compare? New Phytologist, 216(3), 758–770. http://doi.org/10.1111/nph.14626
  • Nelson, P. (2002). Biological Physics: Energy, Information, Life (pp. 1–532).
  • Nobel, P. (2017). Physicochemical and environmental plant physiosology (pp. 1–8).
  • Pickard, W. F. (1981). The ascent of sap in plants. Progr. Biophys. Molec. Biol., 37, 181–229. 
  • PITTERMANN, J. (2010). The evolution of water transport in plants: an integrated approach. Geobiology, 8(2), 112–139. http://doi.org/10.1111/j.1472-4669.2010.00232.x
  • Rand, R. H. (1983). Fluid Mechanics of Green Plants. Annu. Rev. Fluid Mech., 15(1), 29–45. http://doi.org/10.1146/annurev.fl.15.010183.000333
  • Rockwell, F. E., Holbrook, N. M., & Stroock, A. D. (2014). The Competition between Liquid and Vapor Transport in Transpiring Leaves. Plant Physiology, 164(4), 1741–1758. http://doi.org/10.1104/pp.114.236323
  • Sack, L., Ball, M. C., Brodersen, C., Davis, S. D., Marais, Des, D. L., Donovan, L. A., et al. (2016). Plant hydraulics as a central hub integrating plant and ecosystem function: meeting report for “Emerging Frontiers in Plant Hydraulics” (Washington, DC, May 2015). Plant, Cell and Environment, 39(9), 2085–2094. http://doi.org/10.1111/pce.12732
  • Sane, S. P., & Singh, A. K. (2011). Water movement in vascular plants: a primer. Journal of the Indian Institute of Science, 91(3), 233–243. 
  • Schlüter, S., Vogel, H. J., Ippisch, O., & Vanderborght, J. (2013). Combined Impact of Soil Heterogeneity and Vegetation Ty e on the Annual Water Balance at the Field Scale. Vadose Zone Journal, 12(4), 0–17. http://doi.org/10.2136/vzj2013.03.0053
  • Schneider, C. L., Attinger, S., Delfs, J. O., & Hildebrandt, A. (2010). Implementing small scale processes at the soil-plant interface - the role of root architectures for calculating root water uptake profiles. Hess, 279–290. 
  • Schröder, N. (2013, November 14). Three-dimensional Solute Transport Modeling in Coupled Soil and Plant Root Systems. 
  • Schwartz, N., Carminati, A., & Javaux, M. (2016). The impact of mucilage on root water uptake-A numerical study. Water Resources Research, 52(1), 264–277. http://doi.org/10.1002/2015WR018150
  • Severino, G., & Tartakovsky, D. M. (2014). A boundary-layer solution for flow at the soil-root interface. Journal of Mathematical Biology, 70(7), 1645–1668. http://doi.org/10.1007/s00285-014-0813-8
  • Somma, F., Hopmans, J. W., & Clausnitzer, V. (1998). Transient three-dimensional modeling of soil water and solute transport with simultaneous root growth, root water and nutrient uptake. Plant and Soil, 201, 281–293. 
  • Sperry, J. S., Hacke, U. G., Oren, R., & Comstock, J. P. (2002). Water deficits and hydraulic limits to leaf water supply. Plant, Cell and Environment, 25(2), 251–263. http://doi.org/10.1046/j.0016-8025.2001.00799.x
  • Steudle, E. (2000a). Water uptake by roots: effects of water deficit. Journal of Experimental Botany, 51(350), 1351–1542. 
  • Steudle, E. (2000b). Watter uptake by plant roots: an integration of views. Plant and Soil, 226, 45–56. 
  • Steudle, E., & Henzler, T. (2005). Water channels in plants: do basic concepts of water transport change ? Journal of Experimental Botany, 46(290), 1067–1076. 
  • Steudle, E., & Peterson, C. A. (1998). How does water get through roots ? Journal of Experimental Botany, 49(322), 775–788. 
  • Stroock, A. D., Pagay, V. V., Zwieniecki, M. A., & Michele Holbrook, N. (2014). The Physicochemical Hydrodynamics of Vascular Plants. Annu. Rev. Fluid Mech., 46(1), 615–642. http://doi.org/10.1146/annurev-fluid-010313-141411
  • THOMPSON, M. V., & Holbrook, N. M. (2003). Application of a Single-solute Non-steady-state Phloem Model to the Study of Long-distance Assimilate Transport. Journal of Theoretical Biology, 220(4), 419–455. http://doi.org/10.1006/jtbi.2003.3115
  • Thompson, M. V., & Holbrook, N. M. (2003). Scaling phloem transport: water potential equilibrium and osmoregulatory flow, 1–17.
  • Twenty-five years modeling irrigated and drained soils: State of the art. (2007). Twenty-five years modeling irrigated and drained soils: State of the art. Agricultural Water Management, 92(3), 111–125. http://doi.org/10.1016/j.agwat.2007.05.013
  • Tyree, M. T. (2003). The ascent of water. Nature, 423(26 June 2003), 923. 
  • Vadez, V., Kholova, J., Medina, S., Kakkera, A., & Anderberg, H. (2014). Transpiration efficiency: new insights into an old story. Journal of Experimental Botany, 65(21), 6141–6153. http://doi.org/10.1093/jxb/eru040
  • Vrugt, J. A., Hopmans, J. W., & Simunek, J. (2001). Calibration of a two-dimenional root water uptake model. Soil Science Society of America Journal, 1–11. 
  • WINDT, C. W., VERGELDT, F. J., DE JAGER, P. A., & van AS, H. (2006). MRI of long-distance water transport: a comparison of the phloem and xylem flow characteristics and dynamics in poplar, castor bean, tomato and tobacco. Plant, Cell and Environment, 29(9), 1715–1729. http://doi.org/10.1111/j.1365-3040.2006.01544.x
  • Zarebanadkouki, M., Meunier, F., Couvreur, V., Cesar, J., Javaux, M., & Carminati, A. (2016). Estimation of the hydraulic conductivities of lupine roots by inverse modelling of high-resolution measurements of root water uptake. Annals of Botany, 118(4), 853–864. http://doi.org/10.1093/aob/mcw154

Friday, December 1, 2017

Krigings paper

Finally we submitted the Kriging paper. Interpolation of hydrological quantities is a necessity in hydrological modeling. Since the beginning of last century, various techniques were implemented to obtain it:
or other types of interpolation 

We prefer Kriging. This paper accounts for the implementation of Krigings inside the JGrass-NewAGE system


The paper can be seen here. However, you can find all the material of the paper on the OSF platform. We tried to share everything from code to data and even the simulation we have performed. Therefore, in principle, any reader could try our software and reproduce our result.  

Thursday, November 30, 2017

Journal Papers using GEOtop

This is the growing list of papers built upon GEOtop in its various versions.

Thursday, November 23, 2017

Monday's discussion on evapotranspiration - Part I - The vapor budget

Last Monday at lunch, I and my students discussed about evapotranspiration. I already talk about it in various comments here. However, the starting point was the impression, coming from one of my student that the topic of transpiration is still in its infancy. I agree with him and I offered my synthesis.

  1. In hydrology we use Dalton’s law (here, slide 21) or the derived equations named Penman-Monteith and Priestley Taylor (forgetting all the empirical formulas).
  2. Dalton’s law  puts together, diffusive vapor flux, vapor storage and turbulent transport.
  3. We should have a water vapor budget equation instead, written for some control volume where all the stuff is at its right place.

There is no difficult to recognize that control volume is limited in our case by the open atmosphere and the surfaces that are emitting vapor as consequence of  their thermodynamics. There is also no difficult to recognize that such limiting surfaces can be very complicate, as those of a canopy, for instance, which is also varying in shape and form with time. Out of these surfaces comes a vapor flux which is dependent on the thermodynamics and physiology existing below them, the holes in number and distribution through which the vapor is emitted, the water availability (and dynamics) in the storage below the surface and, last but not least, the vapor content in the control volume which, as in Dalton’s law commands the driving force.

It is not easy actually to account well for all of these factors. We can, maybe, for a single leaf. It is more complicate for the canopy of a single tree. It is even more difficult for a forest. Unless some goddess  acts to simplify the vapor budget, over the  billions of details, and reduces all to some tretable statistics (we can call this statistics the Holy Grail of evapotranspiration - or the whole Hydrology itself) 

Assume we can deal with it, and  we have the fluxes right. Then the vapor budget seems cristalline simple to obtain, the variation of vapor in the control volume is given by the incoming vapor flux, minus the outcoming vapor flux, minus, in case, the vapor condensation. The good old mass conservation. Unfortunately, also the output flux is not that easy to estimate, because the transport agent is atmospheric turbulence, which is affected by the non-linearities of Navier-Stokes (NSeq) equations, and its interactions  with the complex boundary represented by the terrain/vegetation surfaces. All of this involves a myriads of spatio-temporal scales and degrees of freedom which are not easy to simplify. 

Therefore, literature treats the evaporation as a flux, forgets the real mechanics of fluxes and simplifies turbulence according to similarity theory, essentially due to Prandtl work at the beginning of the last century with the additions of Monin-Obukhov theory.  However, in real cases, the hypotheses of similarity theory are easily broken and the velocities distributions are rarely those expected. All of this makes largely unreliable the transport theory applied in a pedestrian way (as we do). See References below and here a quite informed lecture to get a deeper view.

Summarizing,  the transport is complicate because turbulence interacting with complex surfaces is complicate (probably would be better to say "complex"). Numerically is a problem whose solution is still open (a full branch of science, indeed), and we do not know how to model the rustling of leaves (“and the icy cool of the far, far north, with rustling cedars and pines). 
In fact, the models we use for what we usually call potential evapotranspiration, are an extreme simplification of the wishlist.

Finally, the above picture forgets the role of air and soil temperature (thermodynamics). We were thinking, in fact, only to the mass budget and the momentum budget (the latter is what NSeq is),  but there is no doubt that evaporation and transpiration are commanded also, and in many ways, by the energy budget. Turbulence itself is modified by temperature gradients, but also water vapor tension which concurs to establish the quantity of water vapor ready to be transported at vapor emitting surfaces. A good news is that energy is conserved as well, but this conservation includes the phase of the matter transported (the so-called latent heat). So necessarily, in relevant hydrological cases, we have to solve besides the mass and momentun conservation, the energy conservation itself.


Looking for simplified versions of mass, momentum and energy budget, would require a major rethinking of all the derivations and new impulse to proper measurements that, however, some authors already started (e.g., for instance Schymansky, Or and coworkers,  here).

References

Tuesday, November 14, 2017

Open Science

Nothing really original in this post. I just recollect what already said in the FosterOpenScience web pages. Their definition is:

"Open Science represents a new approach to the scientific process based on cooperative work and new ways of diffusing knowledge by using digital technologies and new collaborative tools (European Commission, 2016b:33). The OECD defines Open Science as: “to make the primary outputs of publicly funded research results – publications and the research data – publicly accessible in digital format with no or minimal restriction” (OECD, 2015:7), but it is more than that. Open Science is about extending the principles of openness to the whole research cycle (see figure 1), fostering sharing and collaboration as early as possible thus entailing a systemic change to the way science and research is done." The wikipedia page is also useful.

This approach get along with the one of doing reproducible research which I already talked about several times. I do not have very much to add to what they wrote, but I also want to make you note that "there are in fact multiple approaches to the term and definition of Open Science, that Fecher and Friesike (2014) have synthesized and structured by proposing five Open Science schools of thought" .

In our work a basic assumption is that openness require also the appropriate tools, and we are working hard to produce them and use those other that make a scientific workflow open.

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