Advances in permafrost modelling: application of the Nested Newton algorithm for solving the heat equation

This is to enlighten the very good job Enrico Borinato did in its Master Thesis in Environmental Engineering. He worked especially on top of the PhD work by Niccolò Tubini and the suggestions and supervision of Prf. Stephan Gruber, expanding in various directions his.  Please find below the Abstract of the Thesis. Clicking on the figure, you can access the manuscript. 

Abstract

Permafrost is a product of cold climatic conditions and is widespread in high-latitude and high-elevation environments. Permafrost is a key component of the cryosphere through its influence on energy exchanges, hydrological processes, natural hazards and carbon budgets. With the increasing awareness of climate change and global warming also the interest on relation climate-permafrost has been rising. A correct knowledge of permafrost behavior allows us to predict the evolution of environments characterized by perennially frozen ground and thus the impacts due to climate changes. Hence the importance to develop the right tools to study this phenomenon. 

One aim of this work is to implement the problem of the heat conduction with phase change in a numerical code. The novelty is in using a new type of Newton's algorithm (Casulli & Zanolli, 2010) in combination with finite element method implemented in FEniCS, an open-source computing platform for solving partial differential equations. 

Then also different analysis on thermal conductivity parameterization and soil freezing characteristic curve have been studied using FreeThaw1D, a Java code in which some soil thermal conductivity models have been implemented.

The GEOframe-Prospero ET model

 As an outcome of Michele Bottazzi dissertation finally we have this first paper. It is a validation of the Prospero model of evapotranspiration. What is that special in the Prospero Model ? At the end it is a Jarvis-Type ET model but what then. Is that new ? Part of the novelty, obviously, comes from it to be in the GEOframe system and  to be used conjointly with  the 50 or more components available. Part of its novelty is in being an OMS3 component which allows for being inserted in a spatial hydrologic unit or in many of them, connected by Net3 to return the semi-distributed transpiration all over a large catchment.  These features already distinguish it from other more normal implementation of ET estimations. 

However, part of the novelty is in its equations. It get the solution of the vegetation energy budget using the method first envisioned by Schimansky and Or and returning as prognostic variables, besides transpiration, the sensible heat, temperature and water vapor pressure. 

This paper validate the model in five grassland sites of the FLUXNET networks, differentiated by being in distinct climate classes. For the results of the application, please read the paper preprint by clicking the figure above. One result I can anticipate. At least for grassland, Prospero works decently well, even without calibration of the parameters, This is quite a result in its simplicity.

4 Lysimeter GEO users

  In ten minutes or so I am meeting the first organised group of possible Lysimeter GEO users that are not just my students of the Hydrology Class. It is time, I guess for collecting all the material we organised around it.  Lysimeter GEO is the informatics companion of a real Lysimeter, like the one in Figure.

Installations should follow the standards of the GEOframe systems (edition Rossano in these days)

I assume that Richards theory is known. Otherwise, one should give a look to this old but still valid presentation (if English speaking). Otherwise the most recent class of Hydrology can be useful for this learning task. You can find it here.

Figure is from Nehemy, et al. 2020. “Tree Water Deficit and Dynamic Source Water Partitioning.”  https://doi.org/10.1002/hyp.14004.

Lysimeter GEO uses WHETGEO 1D (the creature of Niccolò Tubini) for estimating infiltration in waters. Information on how using WHETGEO can be found in the lab classes of the Hydrology course (unfortunately  the video material is in Italian but, all the documentation provided trough Jupyter Notebooks is in English. Particularly relevant, when you go to download the code (see below) is giving a look to the Notebook_Zero which give the general information.  A paper on WHETGEO is in the process of writing. As soon as we will have a decent draft, it will be shared with people on this and other posts. 

Lysimeter GEO currently implements the Priestley-Taylor, Penman-Monteith-FAO and the Prospero model for the estimation of evapotranspiration. All of them are well documented in Michele's (Bottazzi) Ph.D. Thesis. It is producing some publications too, and when they will be in preprint, they will be shared. Applications of the GEOframe models of evapotranspiration are available at the GEOframe Winter School pages (here). 

All of it is put together with the Lysimeter GEO model. Concetta D'Amato prepared a couple of seminar for its use which are available at this WATZON project page. The code and material used by Concetta in her presentation is available here.

All of us is committed to promote the use of our tools. Everything is open source, well designed, well documented. Do not esitate to ask us for support, if needed. WHETGEO, Prospero and Lysimeter GEO are on Github. 

Yet Another Webinar and Paper on Machine Learning and Hydrology

 Just Yesterday I linked a webinar of Machine Learning and Hydrology, and today I am linking a new one which is the companion of a paper just appeared in Water Resiurces Research. Here it is the video below and below below, the paper.

Reference

Nearing, Grey S., Frederik Kratzert, Alden Keefe Sampson, Craig S. Pelissier, Daniel Klotz, Jonathan M. Frame, Cristina Prieto, and Hoshin V. Gupta. 2020. “What Role Does Hydrological Science Play in the Age of Machine Learning?” Water Resources Research, November. https://doi.org/10.1029/2020wr028091.

Machine Learning and Process Based Modelling

 I had the occasion to see a presentation given by Saman Razavi on this topic and I found it educative.  It can be found on YouTube and you can see it below. It contains a quick historical review and exemplify the different abilities of the two type of models. Therefore I decided to share it with you.

The example Dr. Razavi does in the projections is interesting but, wanting to reproduce it, how did he obtained the target future curve ?  Dott. Razavi also addressed to his recent paper that is the basis for the talk and where you can find further details.

References

Razavi, Saman. 2021. “Deep Learning, Explained: Fundamentals, Explainability, and Bridgeability to Process-Based Modelling.” Earth and Space Science Open Archive. Earth and Space Science Open Archive. https://doi.org/10.1002/essoar.10506045.1.

Translating MARRmoT representation of models into the Extended Petri Net

 MARRmoT is a toolbox containing several Rainfall-Runoff models. The paper and the description of the toolbox is here. The operation is a smart initiative to include many of the commonly used models in a single place and let the users decide which model is more suitable to their cases. 

Our main interest here is actually devoted to the supplemental material present in MARRmoT where the Authors present the 46 models of the version 1.2. The presentation is clear and well illustrated trough the traditional use of reservoirs. Here I made an exercise for my students in transforming the MARRmoT representation into the  EPN one. EPN, as deducible from the title are  diagrams that allows to represent any Dynamical System (in mathematical or system and control science meaning) with graphics which have a one-to-one correspondence with the equations. To learn briefly how they works, please refer to  these other posts.  We wrote a couple of papers, which are cited below,  describing them and showing how to exploit some of their capabilities. 

From the educational point of view I realized that the students are quite helped and in using them they have a better understanding, and, moreover, a method, to grasp what the lumped hydrological models contains. So, anyone interested, can  find the presentation by clicking on the Figure above and my short talk in the video below.

I gave as an exercise to the class the duty to transform the remaining MAARmoT models into the EPN, and we'll see soon the results.

References

Bancheri, Marialaura, Francesco Serafin, and Riccardo Rigon. 2019. “The Representation of Hydrological Dynamical Systems Using Extended Petri Nets (EPN).” Water Resources Research 55 (11): 8895–8921.

Rigon, Riccardo, and Marialaura Bancheri. 2020. “On the Relations between the Hydrological Dynamical Systems of Water Budget, Travel Time, Response Time and Tracer Concentrations.” Hydrological Processes, no. hyp.14007 (December). https://doi.org/10.1002/hyp.14007.

Rigon, Riccardo, and Marialaura Bancheri. 2020. “On the Relations between the Hydrological Dynamical Systems of Water Budget, Travel Time, Response Time and Tracer Concentrations., Supplemental material” Hydrological Processes, no. hyp.14007 (December). https://doi.org/10.1002/hyp.14007.

Chain derivation rule is not valid when you go discrete

 One interesting fact that is not clear to most of people is that the chain derivation rule is not valid when we go from the continuous domain of calculus to the discrete domain of numerics. This means, for instance that:

$$\frac{\partial \theta(\psi(t))}{\partial t} \neq \frac{\partial \theta(\psi(t))}{\partial \psi} \frac{\partial \psi(t)}{\partial t} $$

where, for instance, $\theta$ is the volumetric water contente, $\psi$ is soil suction, $t$ is time.  The arcane is explained because when you go to discretize the differential we have:

$$ \frac{\theta^{n+1}-\theta^n}{\psi^{n+1}-\psi^n} $$

where the upper scripts $n+1$ and $n$ indicate respectively the $n+1$ and $n$ time step.  Both on the numerator and denominator we have a dependence on time step $n+1$ which is unknown and the way they are approximated can bring to an inconsistent estimation of the derivative.  As a result,  conservation equations are said written in conservative form when they contain are written as:

$$ \frac{\partial \# }{\partial t}= - \nabla \cdot * $$

where $\#$ stands for any state variable to be conserved and $*$ for any flux expression. $\nabla \cdot$ is the divergence operator. 

This issue was brought to light by the mathematician P.L. Roe in his 1981 paper, and to us by colleagues Vincenzo Casulli and Michael Dumbser and is clearly illustrated in Tubini et al. (2021) below.  

References

Roe, P. L.: Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 43, 357–372, 1981

Tubini, N., Gruber, S., and Rigon, R.: A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation, The Cryosphere Discuss. [preprint], https://doi.org/10.5194/tc-2020-293, in review, 2020.