Tuesday, July 22, 2014

Patterns for the application of modern informatics to the integration of PDEs: the case of the Boussinesq Equation

Today Francesco Serafin graduated finishing his master in engineering. In brief, the scope of his thesis was to implement a series of classes, eventually ported to OMS to solve, in particular, the groundwater Boussinesq equation, but with a more large scope. I let to explain more to an excerpt of Francesco's introduction.

"Mathematical models play a fundamental role in many scientific and engineering fields in today’s world. They are used for example in geotechnics to evalute the hillslope stability, in weather science to predict weather trends and produce weather reports, in structural design to study the resistance to stress, and in fluid dynamics to compute fluid flows and air flows.

Consequently mathematical models are evolving all the time: more and more new numerical methods are being invented to solve the Partial Differential Equations (PDE)s that describe physical problems with increasing precision, and more and more complex and efficient processor units are being created to reduce the computational time.
Therefore, the code into which the mathematical models are translated has to be “dynamic” in order to be easily updated on the basis of the continuous developments (Formetta et al. (2014) [16]).
On the other hand, completely different physical problems are often de- scribed using similar PDEs. For this reason, the numerical methods which provide solutions to different problems can be the same. This suggest the implementation of an IT infrastructure that hosts a standard structure for solving PDEs and that can serve various disciplines with the minimum of hassles.

This work is focused on the application of what is envisioned above, with the main purpose of the creation of an abstract code for implementing every type of mathematical model described by PDEs.

We work on hydrological topics but we hope to design a structure of general interest. Obviously the final goal of any work of this type is to find a proper numerical solver, and therefore, part of the thesis is devoted to the analysis of the problem under scrutiny, and the description of the solution found."

Tuesday, July 8, 2014

Quickness and exactitude

I put here an internal review of one of our manuscript, because, I hope, it can be useful in general. The topic is evaluating the rainfall runoff of a small catchment (but I hoped it was en estimation of the global hydrological cycle, even if without evapotranspiration measurements).

"The paper is written in a good English (finally). However good English does not mean is a good paper. It lacks of focus and is not concise (lack of exactitude and quickness, see at the end of the post). Objectives are not clear, and the novelties of the paper not evident. However, I am not desperate to obtain at the end something reasonable: but this just because I know the amount of work behind it, and, in part, the row-material.

Making a rainfall-runoff model cannot be usually considered an exercise at the frontier of our science (citing conversations with Ignacio Rodriguez-Iturbe. However, it could be, as testified by Gunther Bloschl's ERC). It, making rainfall-runoff, I mean, certainly can bring information about a certain basin. However, in our case, the works of N* and M* already filled this space. So what it is the goal of this paper ?
The initial idea was to assess the uncertainty in prediction of discharges by using appropriate statistical techniques. In particular, the idea was to assess the uncertainty inherent to rainfall extrapolation from point measurements to spatial measurements. 
This task has been only partially fulfilled. For the following reasons: errors due to instruments precision were not included (just the hypothesis of perfect functioning measures was applied);  the way rainfall has been included in the model (is not yet clear if average rainfall, one point for each hillslope was used, average rainfall volume for any information or other approximations were utlised: and no sensitivity analysis with respect to the way distribute rainfall was squeezed into the model was performed); the interplay between rainfall and discharge forecasting is not well developed, at least as it could be, i.e. explaining how it works inside the whole procedure is not explained well.  
Therefore the overall rainfall prediction analysis is incomplete, and I expect it would be completed for the thesis. 
The technical novelty we apply in this work is that we use a calibration tool (LUCA) to assess variograms, and we do it at hourly time step, while others do usually at daily time step. A few questions here: how much this approach improves rainfall estimates ? i.e., taking uncalibrated variograms and/or constant variograms (not varying in time) how much difference do we get ? How much this affects the forecasting of the volumes of water? Which comprehensive effect has this on the forecasting of the discharges ?

It could be that all of these approximation have negligible effects on the forecasting of discharges. But this would be indeed good to know and an achievement, which was not obtained so far. 

A second topic of interest was the simulation of the whole hydrological cycle, and a tentative to close the hydrological budget with the Priestley-Taylor simulation of evapotranspiration. This simulations were done but not shown at all in the manuscript. Why not ? Do the simulated discharges and the  simulated ET sum to the total volume of rainfall ? If not, which interpretation do we have about the missing mass ?  Are we able to assess the uncertainty in predictions of each single component of the hydrological cycle obtained with this method? Are we able to observe interannual variability (both in discharges and evapotranspiration, and, if the case, in storage) ? Is this variability estimate reliable, at least as a gross budget ?

Having missed to answer to each one of the questions above the paper results a wandering around that breaks our karma (citation from Vijay K. Gupta).  Please save us with more rigor. 

Regarding quickness and exactitude, I suggest the reading of Italo Calvino's Six Memos for the next Millennium.^1^2

^1 - Here a video seminar on the Six Memos by Paolo Granata
^2 - Hainging around, in a digression maybe, and unfortunately in Italian, the Discorso sulla Matematica (Talk on Mathematics) inspired and guided by Calvino's lectures, written by Gabriele Lolli

Monday, July 7, 2014

The History of Noise

 Discovered in the page of Angelo Vulpiani, a preminent Italian physicist, I believe this paper on noise by L. Cohen  is an  amusing paper that can redirect to more technical readings.
Noise, Brownian motions (a nice  paper by Cecconi et al.), stochastic equations were never touched in my blog before, but they have indeed an important role Hydrology, and one of the warhorses of my friend and master Ignacio Rodriguez-Iturbe (see Random Functions and Hydrology). As Amilcare Porporato said, part of the recent Ecohydrological way is based upon the writing of Master equations, and solving them, under the appropriate assumptions about  noise.
The paper indeed treats noise as the product of atoms and molecules, or, from a deeper perspective, as produced by quantum effects.
Noise in hydrology has problably a different origin, in chaos from one side and on heterogeneity from the other (well heterogeneity is, in a sense, randomness, so the phrase is quite tautological).

Friday, July 4, 2014

Cartoon guide to statistics

Since I am supporting the idea that Hydrologists should know very well statistics, it is with pleasure that I discovered in Rbloggers these two cartoon guides to statistics.

The first is the Cartoon guide to statistics by Gomick and Smith from which the figure above is an excerpt.

"Witty, pedagogical and comprehensive, this is the best book of the bunch! It provides a historical perspective and covers quite advanced topics such as confidence intervals, regression analysis and probability theory. The book contains a fair deal of mathematical notation but still manages to be accessible." MarkR

The second one is the Manga guide to statistics by Shin Takahashi and illustrated by Iroha Inoue.

"As opposed to the Cartoon Guide to Statistics the Manga Guide reads more like a standard comic book with panels and a story line. The story centers around the schoolgirl Rui who wants to learn statistics to impress the handsome Mr. Igarashi. To her rescue comes Mr. Yamamoto, a stats nerd with thick glasses. The story and the artwork is archetypal manga (including very stereotype gender roles) but if you can live with that it is a pretty fun story."MarkR

Sunday, June 29, 2014

Residence time approaches to the hydrological budgets

The natural evolution of geomorphic unit hydrograph approach to the hydrologic response is the analysis of residence time of water for any of the processes in the hydrological budget. Indeed,  there exists something already done in this direction of research, and can be found in the work of Andrea Rinaldo and collaborators. Gianluca Botter talked about the topic in his speech reported here, in a recent post. Without the claim to be very general, very deep, or very informed, I am collecting here some papers of the group on the subject.
Residence time is important under several aspects. The more direct application of theories per residence time seems to be the estimation of pollutants transport around the catchments, but the use of isotopic tracers to determine the age of water, immediately move their applications also the  understanding of the dynamics of runoff formation with mixing between various "waters". If plants are included, also evapotranspiration can become part of the game thus modifying what we expect (See also the post here with related references). Why not, then, make a step forward and use the theory also for temperature (as a passive tracer) ?
This could disclose a way to follow the entropy production and fluxes in the hydrological cycle at catchment scale: a topic in itself.



References

Benettin, P., A. Rinaldo, and G. Botter (2013), Kinematics of age mixing in advection-dispersion models, Water Resour. Res., 49, 8539–8551, doi:10.1002/2013WR014708.

E. Bertuzzo, M. Thomet, G. Botter, A. Rinaldo, Catchment-scale herbicides transport: Theory and application, Advances in Water Resources 52 (2013), p. 232–242

Botter, G., E. Bertuzzo, and A. Rinaldo (2010), Transport in the hydrologic response: Travel time distributions, soil moisture dynamics, and the old water paradox, Water Resour. Res., 46, W03514, doi:10.1029/2009WR008371.

McDonnell, J., et al. (2010), How old is the water ? Open questions in catchment transit time conceptualization, modelling and analysis, Hydrol. Processes, 24(12), 1745–1754.

McGuire, K. J., and J. J. McDonnell (2006), A review and evaluation of catchment transit time modelling, J. Hydrol., 330, 543–563.

Niemi, A. J. (1977), Residence time distribution of variable flow processes, Int. J. Appl. Radiat. Isot., 28, 855–860.

Rinaldo, A. and Rodriguez-Iturbe, I., Geomorphological theory of the hydrologic response, Hydrol Proc., vol 10, 803-829, 1996

Rinaldo, A., K. J. Beven, E. Bertuzzo, L. Nicotina, J. Davies, A. Fiori, D. Russo, and G. Botter (2011), Catchment travel time distributions and water flow in soils, Water Resour. Res., 47, W07537, doi:10.1029/2011WR010478.

Friday, June 27, 2014

Long wave radiation

I have already dedicated some posts and a paper to radiation. Radiation is deemed necessary to drive evapotranspiration and snow models. However, our previous efforts were dedicated mainly to shortwave radiation. Especially Giuseppe Formetta, however, was pushing to have a solid parametrisation also for long wave radiation (a.k.a as infrared radiation). The preliminary results are shown in the talk here given at the iEMSs conference in S.Diego.

In the talk we used Ameriflux measurements to calibrate a quite long list of parameterisations. No new theory, but testing of theories developed by others, in that kind of agnostic approach suggested by the use of modelling by component, supported by OMS and used in the JGrass-NewAGE system. The same topic in a poster by Marialaura Bancheri, my youngest Ph.D., here.

Friday, June 20, 2014

Four academic brothers (of mine)

I have many academic brother since Andrea Rinaldo is very prolific in generating first class researchers. I have even more I consider the inheritance of Ignacio Rodriguez-Iturbe, my postdoc advisor at (that time at) Texas A&M Unversity. Of the many three agreed to send me the presentations they gave at the Honour doctorate of Andrea Rinaldo, and you can find them with a little comment here below. 

The older (of the three) brother, Marco Marani, from Padova University and Duke, presented a work on the soil-water-plants continuum. He emphasize the role of roots in modifying the soil water distribution, otherwise controlled by Darcy flows. However, he also studied and talked about the influence of the soil-plants-atmosphere continuum. The presentation is here. The couple of references cited are: Volpe et al., 2013 and Manoli et al., 2014

Gianluca Botter talked about the travel time distribution approach to catchment scale transport. A topic that intersects also the “old water paradox” querelle, but is, in general, pretty effective in getting the distribution of pollutants. This approach has a long story that put its roots, in Gedeon Dagan’s work, as well as in Rodriguez-Iturbe geomorphic unit hydrograph. Andrea own papers on Mass response function with Sandro Marani can also be considered at the foundations of this presentation. 
Among the reference, recent papers on the topic are Botter et al., 2010 and Benettin et al., 2013. The presentation is here

Enrico Bertuzzo (GS) covered instead the new topic of water borne  diseases and their spreading along rivers. The way Enrico and coworkers analysed the problem, certainly inherited many notions and ideas sprout the early studies on river networks structure by Andrea (I had a part in it), but also on recent and domain specific achievements and findings. In the presentation he cited just one paper, but the research outcomes on the topic are certainly copious and exciting. The presentation is here. 

Andrea D’Alpaos (GS)  talked about tidal networks, their formation, their shapes, their similarity or dissimilarity from river networks. All of it in a blend of equations, analysis in the field and lab experiments. Another fascinating topic that was started with Andrea.  The presentation is here.

Overall is interesting to judge the differentiation of topics and methods used by the authors, expressing that each developed his on research personality and attitude.