Representation of Hydrological Dynamical Systems Using the extended Petri Nets

Finally we came out with the submission of the paper on Petri Nets. This was the topic of various posts collected under the name of reservoirology.  You  can see the submitted paper by clicking on figure below.

The paper deals with the graphical representation of lumped-parameter hydrological models or, as we called them, Hydrological Dynamical Systems (HDSys). We were not satisfied with previous representations of such models and we thought that figures in literature do not convey the right information to the readers, usually being not enough explicative. At the same time, we streamlined the process to document appropriately the models for reproducibility. Insufficiently explained HDSys are not reproducible and this is bad for hydrology. Then we setup a one-to-one relationship between graphics and mathematics and this could help the passage from the initial ideas about processes to their representation in formulas. At that point, we asked ourselves if our graphic tools were suitable to represent models with the ambition to explore the interactions between hydrology and ecosystems, and therefore account for their co-evolution. We obtained a positive answer adding a graphical feature to visualize how state variables regulate the models' parameters through quantities called controllers. Once the representation was complete, we could observe the analogy of our representation with those used in other sciences, as, for instance, theoretical biology. This open the way to connect hydrological work to the graphical methods in the non-linear systems theory.

Who wants to browse the history of the review, they can find it here:

• The first version of the paper
• The Associate editor observations
• The reviewers observations (major) and our rebuttal
• The second version of the paper
• The supplementary material
• The reviewers observations (minor) and our rebuttal
• The third version of the paper
• The revised supplementary material

Now accepted for publication in WAter Resources Research.

Concentrations

Some notes on concentration of solutes in Hydrological Dynamical Systems  (a.k.a. lumped, a.k.a. compartmental ) Hydrological models.

The slides can be seen by clicking on the Figure above.  These slides pretend to be be quite intuitive in content but, obviously to people  who have some familiarity with the topic treated. A good lecture for  their introduction is Rigon et al. 2016. For understanding the graphs the blog posts entitled “Reservoirology”, especially Reservoirology #3 are also a preparatory lecture.

Categories of systems of equations in ODEs based hydrological systems

This is the continuation of the reservoirology topic saga.  Especially preparatory for understanding  is the Reservoirology #3 post. Here we show that the same topological structure of Petri Nets can address various physical concepts related to a hydrological system conceptualised as a set of reservoir and therefore solvable as a set of ODEs.
The main ODE system, however, does not reveal all of the system. A finer inspection can be obtained by investigating travel times and concentration of tracers (being the theoretical point of view, the first, the experimental one, the second).

To fix some concept we wrote the short presentations above. Or just click here. The pdf contains itself links to other posts and literature.

On Complex Network Representation and Computation of Hydrological Quantities

Francesco Serafin participated for us to the AGU fall meeting presenting part of his work. They gave him a poster, actually the poster whose image is below. It is about the representation of Hydrological models as a network of interactions.

You can access a more readable version of the poster by clicking on the Figure. A similar, more recent presentation, given for IEMSS 2018 at Fort Collins can be found below:

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 ?

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.

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.

Meledrio, or a simple reflection on hydrological modelling - Part II

In the previous studies made on the hydrology of Meledrio some ancillary data are produced. For instance:

Soil Use

Geo-lithology-Lithology

Usually also other maps are produced, for instance soil cover (which, in principle, could be different from soil use).  The problem I have is that, usually, I do not know what to do with these data.  There are actually two questions related to maps of such kind.

• The first is,  are these characteristics are part of the model (see, for instance, the previous post)?. 
• The second is, if the models somewhat contains a quantity, or a parameter,  that can be affected by the mapped characteristics, but the is not directly the characteristic,  how the parameter can be deduced ? In other words there is a (statistical) method to relate soil use to models parameters ?  

I confess that the only systematic trial to obtain this type of inference that I know are the pedotransfer functions. Whilst the concept could be exported to more general models' attributes, however they refer to very specific models that contains hydraulic conductivity or porosity as a parameter and not to other models, for instance those based on reservoirs, where hydraulic conductivity usually is not explicitly present.
Another typology of sub-models where something similar exists is the SCS-CN model.  Specific models, sometimes can contain specific conversion tables produced either by Authors than practictioners (SWAT, for instance).  In SCS-CN, the tables of soil categories are associated with values of the Curve Number parameters, and people pretend to believe that the association is reliable. But it is fiction not science.

In a time when reviewers say that modelling discharges is not enough to assess the validity of a hydrological model, at the same time they allows holes in the peer review process where papers make an unscrupulous use of the same concept.  

There is actually a whole new science branch, hydropedology, that seems devoted to the task to transform maps of soil properties into significant hydrological numbers (mine is the brutal interpretation of it, obviously hydropedology has the scope to understand, not only to predict), and I add below some relevant reference.  However, the analysis are fine and interesting food to thoughts, but the practical matter is still scanty. Probably for two facts: because normal statistical inference is not enough sophisticated to obtain important results (beyond pedotransfer functions) and because (reservoir type of) models have parameters that are too much involved to be interpreted as a simple function of a mapped characteristics. An opportunity for machine learning techniques ?

References

Lin, H., Bouma, J., Pachepsky, Y., Western, A., Thompson, J., van Genuchten, R., et al. (2006). Hydropedology: Synergistic integration of pedology and hydrology. Water Resources Research, 42(5), 2509–13. http://doi.org/10.1029/2005WR004085

Pacechepsky, Y. A., Smettem, K. R. J., Vanderborght, J., Herbst, M., Vereecken, H., & Wösten, J. (2004). Reality and fiction of models and data in soil hydrology (pp. 1–30).

Vereecken, H., Schenpf, A., Hoopmans, J. V., Javaux, M., Or, D., Roose, J., et al. (2016, May 13). Modeling Soil Processes: Review, Key Challenges, and New Perspectives. http://doi.org/10.2136/vzj2015.09.0131

Vereecken, H., Weynants, M., Javaux, M., Pachepsky, Y., Schaap, M. G., & Genuchten, M. T. V. (2010). Using Pedotransfer Functions to Estimate the van Genuchten–Mualem Soil Hydraulic Properties: A Review. Vadose Zone Journal, 9(4), 795–27. http://doi.org/10.2136/vzj2010.0045

Terribile, F., Coppola, A., Langella, G., Martina, M., & Basile, A. (2011). Potential and limitations of using soil mapping information to understand landscape hydrology. Hydrology and Earth System Sciences, 15(12), 3895–3933. http://doi.org/10.5194/hess-15-3895-2011

Meledrio, or a simple reflection on hydrological modelling - Part I

The problem is well explained by the following figure, which represents the statistics of slopes in Meledrio basin.

The overall distribution is bimodal, that make us to suspect that something was going on. In fact, this below is the Google view of the basin.

It clearly show that the hydrographical right side of the basin (on the left in figure) is the one that has steeper slopes, and the left side the one that has the lower ones. This is definitely shown by the slope map

(Please observe that the map is reversed with respect the Google view, since there we were looking to the basin from North). Different slopes, would be associated in our mind with different runoff and subsurface water velocities. This would clearly be accounted for in a model like GEOtop but not (at least explicitly) by a system of reservoirs, especially when we calibrate all the reservoirs all together. A possible partition of the basin in the Jgrass-NewAGE system is represented below

Because the single Hydrologic Response Units are mostly on one side of the catchment, they could be said to be in a area which is homogeneous from the point of view of slope statistics. Therefore, when we treat it as a collection of reservoirs, in principle we could parameterise them differently, according to their slope. In practice, however, we do not have enough measurements to be able to do this separate calibration and we look at the basin homogeneously.  Are we not missing something ?
Well, we are. The first thinking would be to try to add to our reservoir the knowledge gained from geomorphology, and assume that the mean travel time, or some relevant parameter connected to it, depends proportionally to (mean) slope (or some of its power) and inversely to the distance water has to across to get out of the HRU. This is obviously possible, and maybe we could easily try it.
In general, however,  hydrologists who are not stupid, do not care of it. Why ? The reasons can be that our assumption that slopes count is blurred by the heterogeneity of the other factors that concur to form the hydrologic response. However, the magnitude of the heterogeneity can be different at different scales and could be really nice to do some investigations in this direction.

Reservoirology #4: the case of Richards 1d

This is the follow up of Reservoirology #3 post, where (there) I used  Petri nets to represent ordinary differential equations (ODEs). Here, instead, I try to use the same graphical formalism to represent Partial Differential Equations (PDEs), by extending a little the graphics.

Let's say that the number of specifications needed is larger here and, therefore, a lot of ancillary information has to be conveyed through Tables (and some interpretation). However, click on the figure above if you are curious to know more.

Reservoirology #3

This is a revision of the previous post on the same topic. There I tried to develop my own algebra of symbols to represent coarse grained (spatially integrated) hydrological system. Later on I understood that Petri networks were already there and useful to obtain the same result. The graphs obtained in such a way where, besides, studied in several places, and many contributes of literature convergent from other disciplines, can be used for hydrological scopes.

The presentation (click on the figure) completely substitutes the old one. Who liked it, will like better this.  When you will be done with this post, consider that there is also Reservoirology #4.

THIS MATERIAL IS NOW REFINED AND PUBLISHED IN WATER RESOURCES RESEARCH. YOU CAN FIND THE PAPER HERE.

Reservoirology #2

Because, as we already noticed, often modelling the hydrological cycle is studying the mutual interactions of "reservoirs" of something (which I named "reservoirology"), I tried with my students to have a clean way to represent reservoirs. As we said in the slides below, we try to represent them in a way that, if it is not in a one to one correspondence with the equations that can be derived from the graphs, we are as close as possible to it. This post, left for saving the history, is, however, superseded by the new post reservoirology #3.

Please find above this graphic collection and help us to improve it. One thing to notice is that our representation is mappable, with different expressivity of the concepts to the Petri Networks algebra, which puts in our hands several studies in mathematics, computer sciences, biology and other sciences. 

Reservoirology

The current idea in many modern modelling systems (at catchment scale) is that the hydrology of control volumes can be reduced to a set of interconnected reservoirs (Fenicia et al., 2008; Tague et al., 2010; Clark et al., 2008; Hrachowitz et al., 2013). Each one of these reservoir can be thought as “well mixed”, meaning that each water reservoir acts like a chemical reactor where all what comes in is uniformly distributed across the reservoir and perfectly mixes (and instantaneously too) with everything already present. Some of these reservoirs have a geographical identification (as happened in the geomorphologic unit hydrograph, eg. Rinaldo and Rodriguez-Iturbe, 1996), some others have a functional reason and we could call them “embedded”, or with no-geographical reference. They are used especially to disentangle processes, to attribute the right travel time to water, and, more prosaically, to get right quantitative adjustements to the various outputs, discharge, first of all. 

This way of working is quite necessary (but remind Todini’s adage: first they had a linear reservoir -Sherman, 1932 -, then they introduced a sequence of linear reservoirs - Nash, 1957, after, they made the reservoirs non linear, Dooge, 1959; finally was a mess, Sagawara, 1967) but the separation (or the composition, it depends from the point of view) of the domain in parts is somewhat that remains to be demonstrated.

In using these simplifications  nobody of the researchers mentioned goes through a direct simplification of coupled (hydrological) equations at the finer scale and coarse grain them^1, but directly adopts the paradigm of reservoirs and validate it, nowadays technologically, by using a set of GOFs (goodness of fit) indicators. 

This system is, IMO, intrinsically weak but I, like the others, adopted it. A statistical theory is missing, and I believe that it will arise from the travel time theories. 

In the revamp of  these reservoirs' theories, the attention to the spatial distribution of the reservoirs should find again its place. Distinguishing, “vertically”, reservoirs as canopy, surface, vadose zone and groundwater storage is a necessity that is widely recognised and should be deployed. Reservoirs “lateral” aggregation (by using convolutions or following new paradigms) is the follow up (see also Rigon et al., 2016a). Embedded chains of reservoirs should be also necessary, as invoked by those who study small catchments experimentally (e.g. Birkel, 2011). All should be investigated carefully, and complexity added after implementing “ad hoc” experiments, or using appropriate datasets. 

This scenery would not be complete if these models would limit themselves to consider just the forecasting of discharges. They should also convey a reasonable set of processes to close the water budget. Next would be to include appropriate simplifications of the energy budget, a necessary companion. The latter, however, was never tried so far in coarse grained models.

Notes

^1 Perhaps Paolo Reggiani et al, 1998 tried it in a generic way, making experience on Gray’s previous work, and Todini did it, his own way with Topkapy, i.e. Liu and Todini, 2002; see also Todini, 2007 and my talk here. Paolo's work is certainly to be reconsidered.

References

Birkel, C., Soulsby, C., & Tetzlaff, D. (2014). Developing a consistent process-based conceptualization of catchment functioning using measurements of internal state variables. Water Resources Research, 50(4), 3481–3501. http://doi.org/10.1002/2013WR014925

Clark, M. P., A. G. Slater, D. E. Rupp, R. A. Woods, J. A. Vrugt, H. V. Gupta, T. Wagener, and L. E. Hay (2008), Framework for Understanding Structural Errors (FUSE): A modular framework to diagnose differences between hydrological models, Water Resour. Res., 44, W00B02, doi:10.1029/2007WR006735.

Dooge J,  (1959) - This reference was suggested by Ezio Todini, but I did not find it (the one talking of non-linear reservoirs).

Fenicia F, Savenije HHG, Matgen P, Pfister L, 2008. Understanding catchment behavior through stepwise model concept improvement. Water Resour. Res. 44(1): W01402. ISSN 0043-1397. doi:10.1029/2006WR005563. 

Gray, W., Lijennse, A, Kolar, R.L, Blain, C.A., Mathematical tools for changing spatial scales in the analysis of physical systems, CRC Press, Boca Raton, 1994

Hrachowitz, M., Savenije, H., Bogaard, T. A., Tetzlaff, D., & Soulsby, C. (2013). What can flux tracking teach us about water age distribution patterns and their temporal dynamics? Hydrology and Earth System Sciences, 17(2), 533–564. http://doi.org/10.5194/hess-17-533-2013

Liu and Todini (2002), Towards a comprehensive physically-based rainfall-runoff model, Hydrology and Earth System Sciences, 6(5), 859–881

Nash, J.E., 1958, The form of the instantaneous unit hydrograph, IUGG General Assembly of Toronto, Vol III, IAHS pub. no. 45, 1141-121. 

Reggiani, P., M. Sivapalan, and S. M. Hassanizadeh (1998), A unifying

framework of watershed thermodynamics: 1 Balance equations for mass, momentum, energy and entropy and the second law of thermodynamics, Adv. Water Resour., 22(4), 367–398.

Rinaldo A & Rodríguez-Iturbe I, 1996. Geomorphological theory of the hydrological response. Hydrol. Process. 10(6): 803–829. ISSN 1099-1085. doi:10.1002/(SICI)1099- 1085(199606)10:6<803::AID-HYP373>3.0.CO;2-N. 

Rigon R., Bancheri M., Formetta G., & de Lavenne, A. (2015). The geomorphological unit hydrograph from a historical-critical perspective. Earth Surface Processes and Landforms, n/a–n/a. http://doi.org/10.1002/esp.3855

Rigon R., Bancheri M, Green T., Age-ranked hydrological budgets and a travel time description of catchment hydrology,Hydrol. Earth Syst. Sci. Discuss., doi:10.5194/hess-2016-210, in review, 2016.

Sherman, L. K., Streamflow from rainfall by the unit hydrograph method, Eng. News-Record 108, 501-505, 1932. 

Sugawara, 1967, The flood forecasting by a seried storage type model, IAHS Publication no. 85, 1-6

Tague, C., & Dugger, A. L. (2010). Ecohydrology and Climate Change in the Mountains of the Western USA - A Review of Research and Opportunities. Geography Compass, 4(11), 1648–1663. http://doi.org/10.1111/j.1749-8198.2010.00400.x

Todini, E. (2007). Hydrological catchment modelling: past, present and future. Hess, 11(1), 468–482.