## 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:

\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

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

## Tuesday, October 31, 2017

### Meledrio, or a simple reflection on Hydrological modelling - Part VI - A little about calibration

The normal calibration strategy is to split the data we want to reproduce into two setz:

• one for the calibration phase
• one for the "validation" phase
Let's assume that we have an automatic calibrator. It usually:
• generates a set of model's parameters,
• estimates with the rainfall-runoff hydrological model and any given set of parameters the discharges,
• compares what computed with what is measured by using a goodness of fit indicator
• keeps the set of parameter that gives the best performances
• repeats the operation a huge number of times (and use some heuristics for searching the best set overall)

This  set of parameters is the one used for "forecasting" and

• is now used against the validation set to check its performances.
However, my  experience (with my students who usually perform it) is that the best parameter set in the calibration procedure, is not usually the best in validation procedure. So I suggest, at least as a trial and for further investigations to:

• separate the initial data set into 3 parts (one for first calibration, one for selection, and one for validation).
• Among the 1% (or x% where x is let at your decision) of best performing in the calibration phase  is selected (called the behavioural set). Then 1% (one over 10^4) best performing in the selection phase is further sieved.
• This 1 per ten thousand is chosen to be used in the validation phase
The hypothesis to test is that this three steps way to calibrate returns usually better performances in validation than the original two step steps one.

## Sunday, October 29, 2017

### Open Science Framework - OSF

And recently I discovered OSF, the Open Science Framework. My students told me that there exists many of them, of this type of on-line tools that make leverage of the cloud to store and helps groups to manage their workflow. However, OSF seems particularly well suited to work for scientists’ group, since it contains links various science-oriented features, like connections to Mendeley, Figshare, Github and others. An OSF “project” can contain writings, figures, codes, data. All of this can be uploaded for free in their servers or being maintained in one of your cloud storage like Dropbox or GoogleDrive.

For starting, you can take one our of your time to follow one of their YouTube video, like the one below.

Their web page contains also some useful guides that make the rest (do not hesitate to click on icons: they contain useful material!). The first you can start with is the one about the wiki, a customizable initial page that appear in any project or sub-project. There are some characteristics that I want to emphasize here. Startin a new project is easy, and when you have learn how to do it, you almost have learn all of it. Any project can have subprojects, called “components”. Each component behaves like a project by itself, so when dealing with it, you do not have to learn something really new. Any (sub)project can be private (the default) or public, separately, and therefore your global workflow can contain private and public stuff.

Many people are working on OSF. For instance Titus Brown’s Living in a Ivory Basement blog also has some detailed review of it. They also coded a command line client for downloading files from OSF which can be further useful.

## Wednesday, October 25, 2017

### Return Period

Some people, I realised, have problems with the concept of return period. This is the definition in wikipedia (accessed October 25th, 2017):
A return period, also known as a recurrence interval (sometimes repeat interval) is an estimate of the likelihood of an event, such as an earthquake, flood[1], landslide[2], or a river discharge flow to occur.
It is a statistical measurement typically based on historic data denoting the average recurrence interval over an extended period of time, and is usually used for risk analysis (e.g. to decide whether a project should be allowed to go forward in a zone of a certain risk, or to design structures to withstand an event with a certain return period). The following analysis assumes that the probability of the event occurring does not vary over time and is independent of past events.
Something that wikipedia does not include is rainfall intensity. The first paragraph, should be then something like:
"A return period of x time units, also known as a recurrence interval (sometimes repeat interval) is an estimate of the likelihood of an event, such as an earthquake, flood[1], landslide[2], rainfall intensity, a river discharge flow or any observable, to occur (or be overcome) on average every x time units."

Return period clearly involves a statistical concept, which is traced back to a probability, and a time concept, that is the sampling time.
Let us assume we have a sequence of data, for which, at moment, the sampling time is unknown, composed by a discrete number, $n$, of data.
The empirical cumulative distribution function (ECDF) of the data is a representation of the empirical statistics for those data. Let $ECDFc$ be the complementary empirical cumulative distribution function, meaning $ECDFc(h) \equiv 1 - ECDF(h)$.
Let h* be one of the possible values of these data (not necessarily present in the sequence but included in the range of experimental values). We are interested in the probability of $h^*$ being overcome. If $m$ is the number of time $h^*$ is matched or overcome, then
$$ECDFc(h^*)= m/n$$
$$ECDF(h^*) = 1 - m/n$$
We can, at this point assume that ECDF resembles some probability function, but this is a further topic we do not want to talk about here. What we want to stress i s that ECDFs (probabilities) are not automatically associated to a time. All the data in the sequence refers to different picks of a random variable, and these picks are not necessarily time-ordered or can be happened all at the same time. So the “frequencies" that can be associated to the above events are not time frequencies.
Now introduce time by saying that, for instance, each datum was sampled at regular time step $\Delta t$, what before I called “time units”, and, for practical reasons we are not interested to the ECDF of the data but to know how frequently (in clock time sense) it is repeated. So, we can say that the total time of our record is
$$T = n\, \Delta t$$
and in this time span, the number of time, h* is overcome is (by construction)
$$m=ECDFc(h^*)*n$$
On average, along the record obtained, the time frequency on which values greater than $h^*$ are obtained is the empirical return period:
$$T_r:=\frac{T}{m} =\frac{n *\Delta t}{ECDFc(h^*)*n} = \frac{\Delta t}{ECDF(h^*)}$$
So, the empirical return period of $h^*$ is inversely proportional to the complementary ECDF($h^*$) but, properly there is a “$\Delta t$” to remind that it is given in time units. One basic assumption in our definition is that the underneath probability is well defined, which is not if climate change is in action. This is a delicate and well discussed topic*, but again, not the core of this page.

There is a crucial initial step, the sampling of data which affects the final result. If the data in the sequence are, for instance annual maxima of precipitation, then the return period is given in years. If the data were daily precipitation totals, then the return period is given in days. And so on. Because usually the time unit has value “1” (but dimension of a time), the numeric value of the return period is just the inverse of the ECDFc. We should not forgot, however, that the equation contains a mute dimension. We are talking about times, not dimensionless numbers (probabilities).

Being Bayesian, probably you can introduce this in a different way. I let you as an exercise to do it.

** On the topic of stationarity, please give a look to:

Milly, P. C. D., Betancourt, J., Falkenmark, M., Hirsch, R. M., Kundzewicz, Lettenmaier, D. P., & Stouffer, R. J. (2008). Stationarity Is Dead: Whither Water Management? Science, 319, 1–2.

Montanari, A, and Koutsoyiannis, D,  Modeling and mitigating natural hazards: Stationarity is immortal!, Water Resources Research, 50 (12), 9748–9756, doi:10.1002/2014WR016092, 2014.

Serinaldi, F., & Kilsby, C. G. (2015). Stationarity is undead: Uncertainty dominates the distribution of extremes. Advances in Water Resources, 77(C), 17–36. http://doi.org/10.1016/j.advwatres.2014.12.013

## Sunday, October 22, 2017

### Simple models for hydrological hazard mapping

This contains the second talk I gave to high-school teacher at MUSE for the Life Project FRANCA. My intention was to show (under a lot of simplification assumptions) how hydrological models work, and give a few hints on which type of hydraulics models of sediment transport can be useful.
Clicking on the figure above you can access the slides (in Italian but with a little time, I will provide a translation). In their simplicity, the slides are a storyboard for action that could be taken in the SteepStream project to provide an estimation of hazards of Meledrio river basin (and the other two selected).

## Friday, October 20, 2017

### On some Hydrological Extremes

This is the talk given at MUSE for the Life FRANCA Project. Life FRANCA has the objective to communicate with people about hydrological hazards and risk. In particular the Audience in this case was composed by high school teachers.

Clicking on the Figure you will be redirected to the presentation.

## Wednesday, October 18, 2017

### Using Colorblind friendly Plots

Brought to my attention by Michele Bottazzi. I rarely think to this. Instead it is important. Please refers to this Brian Connelly post:

Click on the figure to be redirected. BTW, this was the 500th post!🎉

## Tuesday, October 17, 2017

### TranspirAction

This post contains the presentation given by Michele Bottazzi. His presentation look forward to dig into the forecasting of transpiration from plants (and evaporation from soils) through concentrated parameters modelling. His findings will have a counterpart in our JGrass-NewAGE system.
The figure illustrate his willing to find a new, modern, way to scale up leaf theories to canopy and landscape. The starting point is one recent work by Schymanski and Or but it will go, hopefully, far beyond it. Click on the Figure to access his presentation.

### An ML based meta modelling infrastructure for environmental meodels

This is the presentation Francesco gave for his admission to the third year of Ph.D. studies. He summarizes his work done so far and foresees his work during the next year.
Francesco's work is a keystone of the work in our group, since he sustains most of informatics and pur commitment to OMS3. Besides of this two are his major achievements: the building of the Ne3 infrastructure (an infrastructure inside an infrastructure!)  which allows an enormous flexibility to our modelling, and the new road opened towards modeling discharges through machine learning techniques. But there are other connections he opens that are visible through his talk. Please clisk on the figure to access the presentation.

## Sunday, October 15, 2017

### A few topics for a Master thesis in Hydrology

After the series about Meledrio I thought that each one of the post actually identifies at least one Thesis topic:

Actually, each one of them could be material for more than one Thesis, depending the direction we want to take. All the Theses topics assume that JGrass-NewAGE is the tool used for investigations.
Actually there are some spinoff of those topics:
• Using machine learning to set part of model inputs and/or
• Doing hydrological modeling with machine learning
• Preprocessing and treating (via Python or Java) satellite data as input of JGrass-NewAGE (a systematisation of some work made by Wuletawu Abera on Posina cacthment and/or Blue Nile)
• Implementation of the new version of JGrass-NewAGE on val di Sole
• Using satellite data, besides geometric features, to extract river networks
• Snow models intercomparison (GEOtop and those in JGrass-NewAGE, with reference to work done by Stefano Tasin and Gabriele Massera)
Other to other Hydrological topics:
• Mars (also here) and planetary Hydrology (with GEOtop or some of its evolutions which account for different temperature ranges and other fluid fluxes)
• Copying with Evapotranspiration and irrigation at various scales
• Copying the carbon cycle to the hydrological cycle (either in GEOtop or in JGrass-NewAGE)
Other possible topics regarding water management:
• Hypothesis on the management of reservoir for optimal water management in river Adige.
• Managing Urban Waters Complexity
Other possible topics regards, on a more theoretical (mathematical-physical) side:
On the side of informatics:
For who wants to work with us on the Master thesis, the rules to follow are those for Ph.D. students, even if to a minor extent. See here:

## Saturday, October 14, 2017

### Meledrio, or a simple reflection on hydrological modelling - Part V

Another question related to discharges is, obviously their measure. Is discharge measure correct ? Is the stage-discharge relation reliable ? Why do not give intervals of confidence for the measures ? Yesterday, a colleague of mine, told me. A measure without an error band is not a measure. That is, obviously an issue. But today reflection is on a different question.  We have a record of discharges. It could look like this (forgive me the twisted lines):
Actually, what we imagine is the following:
I.e. we think it is all water. However, a little of reflection should make us think that, a more realistic picture is:
Meaning that part of the discharge volume is actually sediment transported around. This open the issue on how to quantify it. Figure enlighten than during some floods, actually the sediment could be a consistent part of the volume, and, if we are talking of small mountain catchments like Meledrio, it could be the major part of the discharge. Hydraulics and sediment transport, so far, was used separately from hydrology and hydrology separated from sediment transport, but what people see is both of them (water and sediment).
This actually could be not enough. The real picture could be, actually like this:
Where we have some darker water. The mass transport phenomena, in fact, could affect part of the basin during intense storms, but the liquid water could not be able to sustain all this transport. Aronne Armanini suggested to me that, in that case, debris flow can start and be stopped somewhere inside of the basin. Te water content they have, instead, could be equally likely released to the streams and boosting furthermore the flood.  Isn't it interesting ? Who said that modeling discharges is an assessed problem ?

## Friday, October 13, 2017

### Meledrio, or a simple reflection on hydrological modelling - Part IV

An issue that often is risen is about the complexity of models. Assuming the same Meledrio basin, which is the model we can think to be the simpler for getting quantitatively the water budget ?
The null-null hypothesis model is obviously using the past averages to get the future. Operatively:
• Get precipitation and discharge
• Precipitation is  separated by temperature (T) in rainfall (T>0) and snowfall. Satellite data can be used for the separation.
• Take their average (maybe monthly average)
• Take their difference.
• Assume that the difference is  50% recharge and 50% ET

My null hypothesis is the following. I kept it simple but not too simple:
• Precipitation, discharge and temperature are the measured data
• Their time series are split into 2 parts (one for calibration and one for validation)
• Precipitation is measured and separated by temperature (T) in rainfall (T>0) and snowfall (T<0). Satellite data can be used alternatively for the separation. These variable can be made spatial by using a Kriging (or
• Infiltration is estimated by SCS-CN method. SCS parameters  interval are set according to soil cover, by distinguishing it in qualitatively 4 classes of CN (high infiltrability, medium high, medium low, low). In each subregion, identified by soil cover, CN is let vary in the range allowed by its classification. Soil needs to have a maximum storage capacity (see also ET below). Once this has been exceeded water goes to runoff.
• Discharge is modeled as a set of parallel linear reservoirs. One for HRU (Hydrologic Response Unit).
• Total discharge is simply the summation of all the discharges of the HRUs.
• CN and mean residence time (the parameter in linear reservoirs) are calibrated to reproduce total discharge (so a calibrator must be available)
• A set of optimal parameters is selected.
• Precipitation that does not infiltrates is separated into evapotranspiration, ET, and recharge.
•  ET is estimated with Priestly-Taylor (so you need an estimator for radiation) corrected by a stress factor, linearly proportional to the water storage content. PT alpha coefficient is taken at its standard value, i.e 1.28
• What is not ET is recharge.  Please notice that there is a feedback between recharge and ET because of the stress factor.
• If present, snow is modeled through Regina Hock model (paper here), in case, calibrated trough MODIS.
The Petri Net representation of the model (no snow) can be figured out to be as follows:

The setup this model, therefore is not so simple, indeed, but not overwhelmingly complicate.

Any other model has to do better than this. If successful, it become hp 1.
A related question is how we measure goodness of fitting and if we can distinguish the performances of one model from another one. That is, obviously, another issue.

## Thursday, October 12, 2017

### Meledrio, or a simple reflection on hydrological modelling - Part III

Well, this is not exactly Meledrio.  It starts a little downstream of it. In fact, we do not have discharge data in Meledrio (so far) and we want to anchor our analysis to something measured. So we have a gauge station in Malè. A gauge station for who does not know it, measure just water levels (stages) and them convert to water discharge through a stage-discharge relation (see USGS here). Anyway, a sample signal is here:
The orange lines represent discharge simulated with one of our models (uncalibrated at this stage). The blue line is the measured discharge (meaning the measured stage after having applied an unknown stage-discharge relationship, because the guys who should did not gave us it). But look at little more closer:
We could have provided a better zooming, however, the argument of discussion is: what the hell is all that noise in the measured signal ? It is natural ? It is error of measurements ? Is due to some human action ?
Having a better zoom, one could see that that signal is almost a square wave going up and in few hours, and therefore the suspected cause are humans.
Next question: how can we calibrate the model that does not have this unknown action inside to reproduced the measured signal ?
Clearly the question is ill-posed and we should work the other way around. Can we filter out in the measured signal the effect of humans ?
Hints: we could try to analyze the measured signal first. Analyzing actually could mean, in this case, to decompose it, for instance in Fourier series or Wavelets and wipe away the square signal (a hint in hints), reproducing an "undisturbed signal" to cope with.
Then we could probably calibrate the the model to the cleaned data. Ah! You do not know what calibration means ? This is another story.

P.S. - This is actually part of a more general problem, which is measurement treatments. Often we, naively, treat them as true values. Instead they are not and should pre-analyzed for consistency and validate before. MeteoIO is a tool that answers to part of the requests. But, for instance, it does not treat the specific question above.