Modelling the River Po (Poster @SII Hydrology days 2021)

 Wow, this will be the first meeting I will attend in more than two years. The are the Italian Hydrological Society (SII) days 2021 and will be held in Naples. We will be presenting a couple of interesting thing. This is the first poster on the modelling of the largest river basin in Italy, which is the topic of the poster here below. The poster should be self-explaining. 

Clicking on the above figure, you can download the high resolution pdf. 

Notes for an incoming hydrology book, Contemporary Hydrology - a Preface

 What’s on Earth is hydrology ? It is the science that studies the movements of water, from the atmosphere to the ground and into the ground and then back to the atmosphere again. This book does not aim to cover all the variety of issues and topics the movement of water causes but is mainly concerned with keeping the material simple and possibly concise. Insights are left to notes and to appropriate literature.

Everyone sees water moving on the Earth surface, maybe excluding those who lives in deserts, and it looks like the it should be easy to get a quantitative mathematical description of water movements: but this is not the case! Even the channelized flow, the simplest of the water flows, shows a great complexity, due to the turbulent nature of the flow and the variety of boundaries in which the water is “constrained”. I have put constrained among quotes “” because water is never really constrained. It goes everywhere, sometimes very slowly, compared to the subjective time of our human actions. Other  flows needs to cope with the medium where water flows, for instance the porous soils, the underground, the plants vessels.

More than that, water on Earth is usually liquid, solid, in form of ice or snow, vapor and each of the phase of water  interacts with and transform into the others. The complexity of phases is just one side of the multifaceted dynamics of water because the elementary physics or thermodynamics of water movements at the centimeter or smaller spatial scale (but how the small is small if, for instances, pores in soil can be micrometers and intercellular space in leaves nanometers ? ) dynamically organizes when we look at the movements at the meter scale and reorganizes at the decameters scale and reorganizes again at the hectare, and, again and again, at kilometers, up to the dimensions of continents. In this reorganization some feature of the elementary motion is loosed while some other feature emerges as fundamental. We could say that the core of contemporary hydrology is in getting a reason of these scales behavior in heterogeneous, i.e. affected by  a great variety of small causes, environment. Galilean science is the light to move among the variety. However hydrologists, the scientists who study and evolve Hydrology’s understanding, rarely can setup experiments (they do, obviously, in the labs) which are really answering the questions, because simplifications and distortion, implicit in other physical sciences branches do not work when you have to deal with a transpiring plant or when the inanimate (?) but complex channel networks forms. Hydrology is mostly observed and recorded, in a sequence of single cases, each one at best slightly different from the others and often very different from the others. Therefore from where we start ? A few certainties we have: mass conservation is the first. We give it for granted but it was recognized more than two centuries ago and its status was even more uncertain in hydrology up to Dalton’s research. The second is the energy conservation, which is in fact more elusive, since the difficulty to estimate some of its parts is bigger than the one to measure mass flow. Those two statements seem so obvious when  enunciated but when we come to measure the mass budget and the energy budget closure troubles begin. Historically, actually, their were not the center of scientists’ research, because this was concentrated merely on fluxes determination, like the river discharge, the evaporative transport, or groundwater flow, at most. However, let’s say, that this book is very much concerned with the closure of these budgets, than concentrating on any of the fluxes, on which actually various sub-disciplines flourished in the past.

Newtonian physics rules, when messing with hydrology, but also the explication of the second law of the dynamics is largely simplified, even in the recent treatments, because of the pitfalls it hides when dealing with simple mechanical systems but that reveals in their whole complexity when the case of water fluxes is the topic.

The nature of the measured hydrological quantities is usually presenting a  degree of randomness, even if their physics is known to be deterministic. This apparent randomness is certainly due to the high number of degree of freedom that the fluid nature of the water, either liquid or gaseous, brings, but also of the uncountable histories that water encounters in its travel. Probably it is not true randomness, is maybe random-mess, because of the structure of the medium water crosses.

If then it is true that determinism is at the core of the hydrological thermodynamics (or maybe saying “thermodynamics” we are contradicting ourselves?) the analysis of the matter lives at the faint light of statistics for almost every phenomenon.  Therefore, almost always hydrologists have hard times to deduce  patterns from very uncertain information. All of these issues in fact are those that makes this science one of the most fascinating of the contemporary science.

What I did in research the last five years, 2016-2021

 In the last ten years I focused on building a reliable system for doing hydrology by computer. This systems learns from the implementation of the process-based GEOtop and is based on the framework developed by ARS/USDA called OMS3. The new system is called GEOframe . 

I have been concerned with the fact that many results claimed on the basis of computer simulations were, in fact, not properly reliable and verifiable, do to lack of software engineering, description of the internals of the tools and availability to researchers. Moreover, also the goodness experimental science is quite dependent on the capability and reliability of models.  In my experience with the model GEOtop so far, when there have been discrepancies between data and the model, most of the time the model was right and the experiment imprecise. Sometimes though was the model GEOtop (or other that we used) wrong and we worked to improve it. Models have to be robust, reliable, realistic and their results reproducible (R4). The new system GEOframe, which is built on these premises, is not a model though. It can fit several modelling solutions and it is actually is agnostic with respect to the methods. It is designed to offer a platform to compare different modelling strategies, lumped modelling, process-grid-based modelling and whatever but avoiding to redo any time the unnecessary.

We used some ML technique in the process of calibration and we implemented also a ANN framework (in OMS3) but that part is underdeveloped at that moment. (If we expand to much, we bleed).  

The older part of the GEOframe system contains lumped based types of models. However, we have recently implemented a solver of Richards 1D [0] and 2D (paper in writing) and 3D (software in deployment). The latter implement a new algorithm for integration of non linear PDE systems which always converges and can naturally switch between groundwater, vadose zone and surface water. Soil can be hot, warm and frozen  without problems (the latter is in deployment). Besides this I worked underground in having a better estimation of evaporation and transpiration. I published very little with respect the amount of work I did on these subjects, but disentangling the theory, the misconceptions, the scale issue was (is) not very easy, and took its time. I have a first (not completely satisfactory paper, from my point of view,  on this topic, just published on Water [1] , but better ones are going to be written in the Fall 2021 and in the 2022. Finally I worked on disentagling the theory of travel time residence time. We have some paper on it, since 2016 [2,3,4,5] which are also connected to a new way to categorize and, before of it, representing  lumped-semi-distributed  models [6] in order to be able to produce some quantitative reasoning about models structure. I did not pursued very much applied work but with GEOframe growing we were able to produce some nice applications on the Posina catchment [6] (~110 km2), Blue Nile (~175000 km2) [7]. These applications, could be the basis for a comparison of traditional and ML methods. We have also ongoing  the modelling of the largest river basin in Italy, the Po river (~75000km2) and the Nera catchment (closed at some hundreds of square kilometers),  being  very interesting because affected by karst.  Those  latter two catchments could be possible candidates for applying ML techniques and doing performance comparisons, once we have collected the appropriate data. Notably in the last work (actually since the implementation of GEOtop) working on the catchment meant for me working on the water budget of which the discharge is just one element completed by evaporation, transpiration and heat turbulent transfer. I viewed the use of the energy budget necessary to describe irrigation needs by crops and vegetation in this changing climate era, and in general as a tool to support a more complete view of the hydrological cycle.

Building a story or how a "narrative" is important in science I: in writing

 I found the below figure which I do not know the Author which I think can be useful to understand both what it is implied in writing a scientific paper and what is a theory, with respect to more simple analysis of data (or models, BTW). The first four arrangements of the data have some interest but, they do not capture much our interest. To do a gory example, is like to take an animal or a tree, separate them in parts and analyzing them from the point of view of the atoms it is made.  This information is real but it does not say anything crucial about the living being.  The being important think is realized when all the material is put together again (it would be great if the separation operation would be reversible) and it is analyzed in its "holistic" form and function. 

So, it is for the topic of a paper. Usually we have a problem and we dissect it using a "reductionist" approach, but we are not able to put again the whole together and make sense of it. The reductionist action is usually not trivial at all, as the figure could, on the contrary, suggest, and the Authors are usually exhausted after having applied the techniques for doing it which could be experimentally and/or mathematically very complex.  But writing the paper needs the vision of the whole, the story to tell, and the discovery, in the rough matter,  of the sense. It is not a bottom-up action but more a top-down one, where hypothesis and deduction comes before than induction.  

Therefore in writing a paper, we should first have an idea of the whole functioning, trying to get a working a logic from the elements we have in hands and see if they fit. At the first trial they will probably don't. Then we have to go back, refining our theory and trying again (and again).  After a while, the narrative, if we are patient and lucky, works. It is not necessary that it fully works. It is necessary that the final results is an appealing theory that can be further tested (or, better, falsified) in other cases and an improvement with respect to the actual knowledge. The steps forward are usually small. 

GEOframe Summer School 2021 (moved to early Autumn for this year)

September 27 - September 28, 2021/ October 4 - October 7, 2021

Scientific Committee: Prof. Riccardo Rigon, Ph.D.; Prof. Giuseppe Formetta, Ph.D; Ing. Niccolò Tubini, Ing. Concetta d’Amato, Ing. Marialaura Bancheri, Ph.D.

Organizing Institutions:

Department of Civil, Environmental and Mechanical Engineering, University of Trento
Center Agriculture Food Environment, University of Trento
Institute for Agricultural and Forest Systems in the Mediterranean, National Research Council, Ercolano NA, Italy

CONTENTS

The Earth’s Critical Zone (CZ) is defined as the heterogeneous, near surface environment in which complex interactions involving rock, soil, water, air, and living organisms regulate the natural habitat and determine the availability of life-sustaining resources (National Research Council, 2001). Clear interest in studying the CZ is spurred on by ever-increasing pressure due to the growth in human population and climatic changes.
Main topics will embrace the water flow (and heat transport) in porous media, the soil-plant-atmosphere continuum, and inverse problems. The aim of the course is to enable participants to run their own simulations with the GEOframe tools prepared to simulate the critical zone. They are process-based (e.g. Fatichi et al, 2016) tools, whose ambition is to simulate the processes of infiltration, heat transport and evaporation and transpiration. The GSS2021 deals mainly with the 1D tools and introduces the 2D ones called WHETGEO (1D and 2D), GEOframe-Prospero and LysGEO.
Besides the lectures and the hands-on sessions, the Summer School is the occasion for discussion and experience exchange among senior scholars and young researchers.
The School will be online on the Zoom platform.

PARTICIPANTS' BACKGROUND

Admissions are reserved to up to 30, PhD students and postdoctoral students, young researchers willing to learn the use of the GEOframe tools envisioned for the study of infiltration, energy budget, vegetation transpiration, water budget with process-based models

All students are asked to upload a CV and a motivation letter when applying.

WORKLOAD AND CREDITS
The Summer School which is to be held in English, consists of 6 hours/day of activities for 6 days. The first two days, 27, 28 of September the installation of the GEOframe-OMS system tools and the general characteristics of the system. Lectures will be brief, dedicated to informatics and most of the time will be used for supporting participants’ installations.
The other four days will cover simulation of infiltration with WHETGEO-1D and 2D, with Prospero Transpiration model, and with the LysGEO model. There will be lectures on the hydrological processes implemented and applications to use cases.

LOCATION
Due to the Covid-19 emergency all the activities will be held via Zoom.

PARTICIPATION COSTS

The cost is free for Students of the Hydrological Modelling Classes at the University of Trento, for Ph.D. students of the University of Trento DICAM and C3A programs, for the participants of the WATZON PRIN project and for all who wants to participate without having a certificate of GEOframe proficiency. Subscription to the class is necessary to receive the information to participate. For those who want the certificate, the Course costs 180 Euros. In any case the certificate is issued after the presentation of a small project of simulations for which appropriate tutoring will be given during and after the School.

CONTACTS

For further information write to: abouthydrology@google.com or to the Secretary of the Class dott. Lorena Galante, lorena.galante@unitn.it

OTHER INFORMATION

The GSS2021 talks and labs will be recorded and made publicly available during the School for self-training through the GEOframe blog (http://geoframe.blogspot.com).

Foreseen schedule

The details of the program are still to be defined

September 27-28:

These days are dedicated to those who never approached the GEOframe system and pursue the understanding of how it works. Who already knows how GEOframe works or have already installed it for different purposes than those of this School, can skip them

• Introduction to the Object Modelling System and GEOframe Infrastructures (Verona 2022 environment)
• Installation of OMS and GEOframe Verona
• Brief introduction to Jupyter notebooks and Python
• Few examples and Problem solving

October 4:

This morning is mostly dedicated to fill theory of the processes investigated by this School on GEOframe, meaning infiltration in soil, the basics of Richards/Richardson equation to which follow some exercises. The afternoon will be used to discuss issues related to the application of different boundary conditions, different parameterizations of the soil water retention curves.

Morning session

• The Richardson-Richards equation
• The equation and its parts, and three form of the equation
• Soil Water Retention Curves
• Hydraulic conductivity models
• Numerical issues to keep in mind

Afternoon session

• Practical session on Richardson-Richards equatio
• one homogeneous layer
• stratified layers
• playing with boundary conditions
• Presenting the results with Jupyter Notebooks

October 5:

This day is dedicated to discuss the problem of the surface boundary condition.

Morning session

• Surface boundary condition and numerical issues
• Practical session simulating:
• Horton process
• Dunnian process
• Presenting the results with Jupyter Notebooks

Afternoon session

• Individual exercises with support

October 6:

This day is dedicated to the bi-dimensional case of the Richardson-Richards equation and to present the radiation energy budget.

Morning session

• Installing the software for building unstructured grids
• Manage 2D unstructured grids.
• Practical session on WHETGEO-2D on some pre-prepared cases

Afternoon session

• Theory of radiation energy budget
• Practical session on computing the radiation energy budget

October 7:

Day four is dedicated to the LysGEO model, evaporation and transpiration modelling and their coupling with R2.

Morning session

• Evapotranspiration theory and equations in the Prospero model
• Use of GEOframe - ET tools practices

Afternoon session

• LysGEO theory
• Practical session on LysGEO:
• Comparison between potential ET and actual ET
• Set different stress factors
• Introducing vegetation traits

Specific Documentation

The specific documentation regards papers and thesis written on the GEOframe components used in this School. Other literature, of general interest, is provided within the presentations given during the course. Practical documentation for any of the tasks is provided by means of Jupyter Notebooks, of which the general ones are reported below.



Some essential about the Object Modelling System

• Bottazzi, M, Ph.D. Thesis
• David, O., Ascough II, J. C., Lloyd, W., Green, T. R., Rojas, K. W., Leavesley, G. H., & Ahuja, L. R. (2013). A software engineering perspective on environmental modeling framework design: The Object Modeling System. Environmental Modelling & Software, 39, 201-213.
• Tubini, N., Ph.D. Thesis (available soon)
• Bottazzi, M., Bancheri, M., Mobilia, M., Bertoldi, G., Longobardi, A., & Rigon, R. (2021). Comparing Evapotranspiration Estimates from the GEOframe-Prospero Model with Penman–Monteith and Priestley-Taylor Approaches under Different Climate Conditions. Water, 13(9), 1221.
• Tubini, Niccolò, and Riccardo Rigon. 2021. “Implementing the Water, HEat and Transport Model in GEOframe WHETGEO-1D v.1.0: Algorithms, Informatics, Design Patterns, Open Science Features, and 1D Deployment.” https://doi.org/10.5194/gmd-2021-163.

Karst

 Karsts exist and cover approximately 15% of the Earth surface. Therefore it can happen very easily that in your hydrological analysis you across a Karst catchment. A general knowledge  of karst environment can be gained by reading the White, 2002 paper but, online you can find also the book by Ford and Williamson, 2007.They are quite comprehensive readings, not necessarily focused on the hydrology of the karst systems and how they can be modeled.

White (2000) propose a conceptual map of karst hydrology that is better represented later in Hartmann et al., Figure 4 (see below).

Precipitation falling into a karstic system can be divided into:

• Allogenic recharge: precipitation that falls on non-carbonatic portion of the catchment and enter  the carbonate aquifer to the swallets
• Disperse-diffuse infiltration directly happening on the karst surface and from there through the soil or the fractures
• Internal runoff, falling into sinkholes drains
• Flow from perched aquifers. Rainfall is collected by there aquifer and subsequently captured by vertical shaft or widen fractures in the vadose zone. 

One important point  for hydrologists is to be able to recognize the karst geomorphology, e.g. Waele 2011,  to be able to automatically detect it and to treat differently from the rest of the catchment. Hofierka et al., 2018 offers a modern view (and very nice maps) on how to detect the areas presenting sink-holes (dolines) and so from IDAR topographic data. The topic of sinkholes detection, actually is sufficiently covered in literature, see, for instance,  other recent references are Pardo-Igúzquiza, 2013, Wu et al., 2016,  Zunpanoi et al., 2019.  It i s not clear to me at present if sinkholes  are  the only detectable manifestation of karst, but what I believe is that the decrease superficial erosion in karst area should also be recognizable, together with a disconnected or absent river network but on these specific topics, I did not find suitable references. 

Where you have karst, you also  have springs. Even if they are not usually detectable by objective methods to find them (e.g. Geology Stack Exchange), many of them are already known from geological surveys and therefore a smart use of geological maps can help. It is self evident but hydrologist often rush to extract catchment and network characteristics without taking care of them in advance, as they should. 

Karst formation are usually mapped,  geologists do their work  since long time ago, and we should use appropriately their information.

Coming to us, hydrologists measure rainfall and discharges and observe that the spring discharges of catchments affected by karst may look quite insensitive to rainfall variations. The direct way to investigate  the response of these catchment is to make leverage on tracer and tracers theory, as for instance reviewed in Hartmann et al., 2014, and shown in Zhang et al., 2021, or Nanni et al., 2020. These techniques, however, are well consolidated and known. For the desktop hydrologist,  something can be tried out with techniques of analysis of correlation between rainfall and discharge. Two notable contributions are Fiorillo and Doglioni, 2010 and Jukic and Denic-Jukic, 2015.  And another, more hydrological-hydrological, oriented to the determination of some characteristics time of catchments (not necessarily karst) is Giani et al., 2021.  More techniques for analyzing these time series, can be found in one previous post of this blog. 

From a practical point of view, what said so far would them urge the astute hydrologist to look for karst when delineating the catchments and subsequently do a careful time series analysis. 

Finally, one like me wants to model the water budget. Even this case there exist a quite developed literature. The paper by White (2000) let us envision the typer of models which can be more process-based-groundwater oriented (Rooji, 2007, Hartmann et al, 2014)  or lumped, i.e. built on reservoir type of models (also Hartmann et al., 2014). 

In the  more process-based type of models, the issue is put properly at work together the Darcian flow and the turbulent flow whose path is unknown and buried, hidden to our eyes which both contribute to the final flow. Because of the structure of the karst network, which is three-dimensional, threshold type of functioning can happen make the modelling more complex (Hartmann et al., 2014).

Lumped, ordinary differential equation  (ODEs) type of modelling is simpler but to be not too simple,  much heuristic has to be used to implement models that return reasonable behavior.  The high heterogeneity of the medium sometimes could help in simplifying the picture but just the real cases applications can discern what is acceptable. In both cases, obviously, the problem of parameter identification is the more important one.  There are several good examples of lumped model, well summarized by the paper of Hartmann et al, 2014  or, from a more practical point of view by the KarstMod model, Mazzilli et al., 2019 (please find its manual here). A careful reading of Hartmann et al 2014, can bring easily to a general conceptual model of karst in term of reservoirs. Rimmer et al, 2012, gives a few use case example of simple working models.  Butscher and Huggenberger, 2008 and Tritz et al, 2011 are some deployments of these models that can give some general guidance. 

The high heterogeneity of the medium sometimes could help in simplifying the picture but just the real cases applications can discern what is acceptable.

In both cases, of using process-based modelling or lumped models, obviously, the problem of parameter identification is an important one.  Worldwide Karst data were made available by Olarinoye et al., 

If you do not have enough time, the best is to read White (2000),  De Waele, 2011 and Hartmann et al, 2014 papers first. If you have time for reading just one paper, read Hartmann et al. 2020. 

References

Butscher, Christoph, and Peter Huggenberger. 2008. “Intrinsic Vulnerability Assessment in Karst Areas: A Numerical Modeling Approach.” Water Resources Research 44 (3). https://doi.org/10.1029/2007wr006277.

Ford, Derek, and Paul Williams. 2007. Karst Hydrogeology & Geomorphology. Wiley.

Hartmann, A., N. Goldscheider, T. Wagener, J. Lange, and M. Weiler. 2014. “Karst Water Resources in a Changing World: Review of Hydrological Modeling Approaches.” Reviews of Geophysics 52 (3): 218–42.

Hofierka, Jaroslav, Michal Gallay, Peter Bandura, and Ján Šašak. 2018. “Identification of Karst Sinkholes in a Forested Karst Landscape Using Airborne Laser Scanning Data and Water Flow Analysis.” Geomorphology 308 (May): 265–77.

De Waele, Jo, Francisco Gutiérrez, Mario Parise, and Lukas Plan. 2011. “Geomorphology and Natural Hazards in Karst Areas: A Review.” Geomorphology 134 (1-2): 1–8.

Fiorillo, Francesco, and Angelo Doglioni. 2010. “The Relation between Karst Spring Discharge and Rainfall by Cross-Correlation Analysis (Campania, Southern Italy).” Hydrogeology Journal 18 (8): 1881–95.

Giani, G., M. A. Rico‐Ramirez, and R. A. Woods. 2021. “A Practical, Objective, and Robust Technique to Directly Estimate Catchment Response Time.” Water Resources Research 57 (2). https://doi.org/10.1029/2020wr028201.

Mazzilli, N., V. Guinot, H. Jourde, N. Lecoq, D. Labat, B. Arfib, C. Baudement, C. Danquigny, L. Dal Soglio, and D. Bertin. 2019. “KarstMod: A Modelling Platform for Rainfall - Discharge Analysis and Modelling Dedicated to Karst Systems.” Environmental Modelling and Software[R] 122 (103927): 103927.

Nanni, T., P. M. Vivalda, S. Palpacelli, M. Marcellini, and A. Tazioli. 2020. “Groundwater Circulation and Earthquake-Related Changes in Hydrogeological Karst Environments: A Case Study of the Sibillini Mountains (central Italy) Involving Artificial Tracers.” Hydrogeology Journal 28 (7): 2409–28.

Olarinoye, Tunde, Tom Gleeson, Vera Marx, Stefan Seeger, Rouhollah Adinehvand, Vincenzo Allocca, Bartolome Andreo, et al. 2020. “Global Karst Springs Hydrograph Dataset for Research and Management of the World’s Fastest-Flowing Groundwater.” Scientific Data 7 (1): 59.

Pardo-Igúzquiza, Eulogio, Juan José Durán Valsero, and Peter A. Dowd. 2013. “Automatic Detection and Delineation of Karst Terrain Depressions and Its Application in Geomorphological Mapping and Morphometric Analysis.” Acta Carsologica 42 (1). https://doi.org/10.3986/ac.v42i1.637.

Rimmer, Alon, and Andreas Hartmann. 2012. “Simplified Conceptual Structures and Analytical Solutions for Groundwater Discharge Using Reservoir Equations.” Water Resources Management and Modeling 2: 217–38.

Rooij, Rob de. 2007. “Towards Improved Numerical Modeling of Karst Aquifers: Coupling Turbulent Conduit Flow and Laminar Matrix Flow under Variably Saturated Conditions.” Université de Neuchâtel. https://doc.rero.ch/record/8817.

Tritz, Sébastien, Vincent Guinot, and Hervé Jourde. 2011. “Modelling the Behaviour of a Karst System Catchment Using Non-Linear Hysteretic Conceptual Model.” Journal of Hydrology 397 (3): 250–62.

White, William B. 2002. “Karst Hydrology: Recent Developments and Open Questions.” Engineering Geology 65 (2): 85–105.

Wu, Qiusheng, Chengbin Deng, and Zuoqi Chen. 2016. “Automated Delineation of Karst Sinkholes from LiDAR-Derived Digital Elevation Models.” Geomorphology 266 (August): 1–10.

Zhang, Zhicai, Xi Chen, Qinbo Cheng, and Chris Soulsby. 2021. “Using StorAge Selection (SAS) Functions to Understand Flow Paths and Age Distributions in Contrasting Karst Groundwater Systems.” Journal of Hydrology 602 (November): 126785.

Zumpano, V., L. Pisano, and M. Parise. 2019. “An Integrated Framework to Identify and Analyze Karst Sinkholes.” Geomorphology 332 (May): 213–25.

Causal Inferences and times series

 In the last post,  In the last post, moved by the necessity to compare time series, I browsed literature and my library of papers to find solutions to my needs (essentially I tried to understand if two time series are related by a time lag). In the search I found other things and my literature grew beyond the original scope. One direction that actually I had frequented previously was the one of distinguishing causal connections beyond just correlations. In my previous searches, I has been fascinated by the work by Judea Pearl and part of the findings inherited from his work. The theory of Pearl has been expressed in various part, including the 2000 paper and some books, that you can find in the references. His teachings were directly absorbed by Hannart and Noveau, themselves good statisticians working in climatology, who produced some papers (2016, 2017) using Judea's theory and notation.

The idea can be generalized fro two to multiple time series, as Eichler (2013) actually shows. Eichler actually is know to have produced such analysis in 2003. A trend of more recent paper on the topic are represented by  Jacob Runge (GS)  work , who also have the merit to have implemented and shared his TiGraMITe package. He also has got a prestigious ERC research program on this topic called Causal Earth. On the concepts he wants to develop in the ERC, he gave talks and produced may interesting papers, among those one in Nature and a second on in Science affiliated Journal (see below)

Because we like to do calculation not just read of write abstractly, we can find relief in the Causality handbook that can be a viable (open source) way to put in practice some of the ideas seen in the previous papers. A final, honorable mention goes also the the San Lian (2014) paper.

References

Dahlhaus, Rainer, and Michael Eichler. 2003. “Causality and Graphical Models in Time Series Analysis.” Oxford Statistical Science Series, 115–37.

Eichler, Michael. 2013. “Causal Inference with Multiple Time Series: Principles and Problems.” Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences 371 (1997): 20110613.

Hannart, A., J. Pearl, F. E. L. Otto, P. Naveau, and M. Ghil. 2016. “Causal Counterfactual Theory for the Attribution of Weather and Climate-Related Events.” Bulletin of the American Meteorological Society 97 (1): 99–110.

Hannart, A., and P. Naveau. 2017. “Probabilities of Causation of Climate Change.” arXiv.v, December, 1–54.

Pearl, Judea, 2000. “Models, Reasoning and Inference.” Cambridge, UK: CambridgeUniversityPress.

Runge, Jakob, Sebastian Bathiany, Erik Bollt, Gustau Camps-Valls, Dim Coumou, Ethan Deyle, Clark Glymour, et al. 2019. “Inferring Causation from Time Series in Earth System Sciences.” Nature Communications 10 (1): 2553.

Runge, Jakob, Peer Nowack, Marlene Kretschmer, Seth Flaxman, and Dino Sejdinovic. 2019. “Detecting and Quantifying Causal Associations in Large Nonlinear Time Series Datasets.” Science Advances 5 (11): eaau4996.

San Liang, X. 2014. “Causality between Time Series.” arXiv [stat.ME]. arXiv. http://arxiv.org/abs/1403.6496.