Richards as a limit: a derivation of Richards' equation from the Continuum Kinetic soil Equation

Two quantities that have no business being related: the spectral gap of a pore network — a purely structural number, computed from the connectivity, with no flow solved anywhere — and the Stokes permeability of the same network, computed by actually solving viscous flow under a pressure drop. Drain the network step by step and they vanish at the same water content. Not approximately. Identically, both zero, at the same θ. That coincidence is the subject of this post: it is the point where a macroscopic constitutive law stops existing, and the theory says so by itself.

In July I posted the first of the two papers — the kinetic theory of the pore-occupancy g(r) — and promised the companion "shortly." Here it is:

Richards' equation as a hydrodynamic limit: Chapman–Enskog reduction of the continuum kinetic equation for unsaturated soil water
R. Rigon, arXiv:XXXX.XXXXX [physics.flu-dyn], 24 pages, 1 figure, 8 appendices, with numerical Supplemental Material.

It goes to Physical Review E, like its companion. The first paper said: θ is not enough, the state is g(r). This one says the other half, and it is the half that makes the first one respectable.

The argument in one sentence

Richards' equation is not an assumption of soil physics. It is a theorem — the solvability condition of a kinetic equation whose redistribution operator has exactly one invariant.

That is the whole paper. Everything below is a consequence.

Two limits that hydrology has always taken together

The reason Richards' equation has never been derived, only motivated, is that the passage from pores to fields conflates two entirely different operations. The paper's first move is simply to take them apart.

• The spatial limit, ε = L/Λ → 0, shrinks the representative volume to a point. It is pure kinematics: it says nothing whatsoever about time scales. What comes out is a closed continuum kinetic equation, ∂g/∂t + ∇·F = 𝒞[g], with F a pore-resolved flux that is still entangled — a transport coefficient and a driving gradient multiplied together inside one kernel, inseparable.
• The temporal limit is where the physics is. Redistribution among pores is fast; the forcing is slow; their ratio is the Damköhler number Da. When Da ≪ 1 the soil is pinned near local equilibrium, and one can expand — a Chapman–Enskog reduction, exactly as one passes from Boltzmann to Navier–Stokes.

Keeping them apart is what makes the structure visible, and it is why the paper can be honest about where each classical assumption enters.

The one operator everything depends on

Linearise the redistribution operator about equilibrium and you get 𝒥, and then the whole reduction is a statement about 𝒥. Three properties, and nothing else, do all the work:

1. 𝒥 is self-adjoint — in the mass inner product, ⟨u,v⟩ = ∫ u v f dr. And here is the small piece of algebra I find most satisfying in the paper: self-adjointness and conservation of water are literally the same statement. Detailed balance K(r,r′) f(r) = K(r′,r) f(r′) holds identically because the mobility is symmetric, and that single fact gives you both.
2. Its kernel is one-dimensional. There is exactly one thing redistribution cannot change: water. The Boltzmann gas has five invariants and therefore five macroscopic equations. Viscous pore flow has one invariant, and therefore one macroscopic equation.
3. It has a spectral gap — a slowest relaxation rate λ₁ > 0 — which is what makes the expansion asymptotic at all.

Then the derivation is almost mechanical. Take the f-moment of the first-order equation: redistribution drops out (that is the null space), and what remains is a condition on the source. That condition — the Fredholm alternative, the unglamorous requirement that the first-order correction should exist at all — is mass conservation. It is Richards' equation.

Five things that fall out, which I did not put in

1. Hydraulic conductivity is a transport coefficient. K is not a constitutive input. It is the first-order Chapman–Enskog coefficient, K = φ⟨κ|𝒥⁻¹|S⟩ — the exact structural counterpart of viscosity in the kinetic theory of gases. Which means, among other things, that K is a property of the medium's relaxation spectrum and is independent of the forcing, in the same sense that viscosity does not depend on the shear rate you apply.

2. The classical formulas are approximations of that coefficient. Take the mean field of it and you get back the standard Mualem–Burdine integral. Resum the serial paths — the fact that large pores must push through small-pore bottlenecks — and out comes Mualem's heterogeneity penalty exp(−4σ²), not as an empirical factor but as a Neumann series that collapses to a harmonic mean. I did not expect that to work as cleanly as it did.

3. Dual-permeability models are derived, not posited. This is the result with the widest practical reach. Give the operator a bimodal pore-size distribution and its relaxation spectrum splits into two bands separated by a gap. Apply Chapman–Enskog within each band, keep the slow cross-band relaxation, and two coupled Richards equations fall out — Gerke–van Genuchten, with Weiler's IN3M as the three-band case and mobile–immobile as the limit where one band is conductively dead. And the exchange coefficient Γ_w, which everybody fits, is computed from the cross-band connectivity. What was a modelling choice becomes a theorem with a formula.

4. The theory predicts its own breakdown — by two different routes. Raise the forcing and Da → 1: sharp fronts, the expansion fails, and you are in the preferential-flow regime the first paper described. But lower the water content and something else happens: at the percolation threshold the spectral gap closes, ‖𝒥⁻¹‖ diverges, and the closed conductivity ceases to exist. These are genuinely two different exits from the Richards regime — one by fast forcing, one by loss of connectivity — and the theory locates both. That is the figure at the top: the gap and the permeability going to zero together.

5. Hysteresis and dynamic capillarity have an operator address. The non-commutativity [𝒲,𝒟] ≠ 0 from the first paper turns out, at the operator level, to be the curvature of the projection onto local equilibrium — the same object that carries dynamic capillarity. I am not claiming to have solved hysteresis. I am claiming to know where in the mathematics it lives.

The part I am least able to hide behind

Point 4 above is the paper's most exposed claim, so it is the one I made numerical. The Supplement does three parameter-free computations, on soils I can name:

• A loam (unimodal, median 8 μm, draining around −1.9 m). Diagonalise the operator: one exact invariant (λ₀ ≈ 10⁻¹⁷), and the modes turn out to be localized by pore radius, each relaxing at the local rate of its own radius, to within 1%. They are not standing waves on the band — I had assumed they were, wrote it into an early draft, and the numerics said no. Ordering the modes by rate is ordering the pores by size: slow modes on small pores.
• The same loam, drained on a 24³ pore network. Gap and permeability vanish together at θ_c ≈ 3.5×10⁻³. Below it, no spanning cluster: the water is there, it simply cannot go anywhere.
• A structured soil (matrix at 4 μm plus macropores at 40 μm). The spectrum splits into two bands with a slow, sign-changing mode across the split — the exchange mode, appearing on its own. And projecting the operator onto the two bands gives an exchange rate that converges to the true one as the bands decouple. The split radius the operator produces falls at ψ ≈ −1 m ≈ −10 kPa, which is where soil physicists have been drawing the macropore boundary by hand for decades.

Where this connects

Nothing here replaces anything. Capillary-bundle models are the diagonal limit; Mualem and Burdine are the mean field; critical-path analysis is the spectral limit near θ_c; Gerke–van Genuchten is the two-band projection. The classical results are not overturned — they are located, each one identified as a particular approximation to a single operator inversion. That is the most useful thing a derivation can do for a field: not to declare the old results wrong, but to say precisely what they are approximations of, and therefore when they will fail.

What I am not claiming

Again, better said by me than to me.

This is a formal Chapman–Enskog reduction, in the sense the phrase carries in kinetic theory: the expansion is organised in powers of Da and closed order by order, but I do not prove convergence, and the higher-order remainders are not bounded rigorously. The closures come from the companion paper and are physical, not derived from molecular dynamics. The numerics are on synthetic networks, not on imaged soils. And the linearisation that makes 𝒥 an operator at all is exactly that — a linearisation, with the nonlinearity pushed into successive sources.

There is also one honest limitation inside the numerics that I have written into the Supplement rather than left for a referee to find: away from the threshold, the spectral gap of a growing cluster is increasingly dominated by its size rather than its connectivity, so the correlation between gap and permeability is only meaningful near θ_c. What is unambiguous — and all the argument needs — is that they vanish together, and that is exact rather than statistical.

Materials

• Paper: arXiv:XXXX.XXXXX (link when it lands) (The provisional pdf here until acceptance on arXiv)
• Companion paper: arXiv:2607.09416 — the kinetic theory this reduces
• The first post: If not Richards, what else ?
• Numerical Supplement + Jupyter notebook: every number and figure above is reproducible; the notebook runs top to bottom in a few seconds.
• "The Real Book": a companion document I wrote for myself and then decided to keep — the entire derivation worked step by step, blackboard style, nothing skipped, with boxes reminding the reader of the linear algebra (Fredholm alternative, pseudo-inverse, graph Laplacians, Rayleigh quotients) and a glossary. If the paper looks forbidding, start there. It is the gentlest way in.

As always: comments, objections and counterexamples are welcome — especially the counterexamples. This paper makes a falsifiable structural claim (Γ_w computable from connectivity, gap closure at θ_c) and I would rather find out early.

Water In Soil - A MOOC

I'm happy to share that my new MOOC, SOIL – The Hydrology of Soil, is now live on the University of Trento's MOOC platform. It's free, open, and self-paced, and it's aimed at anyone who wants to properly understand how water moves through unsaturated soil — not just as a set of formulas to memorize, but as a coherent chain of physical reasoning.

What the course is about

The course covers the hydrology of unsaturated soils and builds up, step by step, the mathematical tools needed to describe water flow through them — culminating in the Richards equation, the cornerstone of unsaturated flow theory.

It's organized into five chapters, each combining short video episodes, further readings, hands-on activities, and self-assessment quizzes:

1. What is Soil — the basic quantities used to describe water content and structure in soil.
2. The Energy of Water in Soil — how water's energy is distributed, capillary pressure, and the construction of soil-water retention curves.
3. Darcy and Buckingham's Laws — from Darcy's law in saturated soil to Buckingham's extension for unsaturated conditions, and how hydraulic conductivity is derived.
4. Soil Water Budget — the mass budget, conservation laws, and the derivation of Richards' equation itself (plus its groundwater counterpart).
5. Solving Richards' Equation and Its Limits — pedotransfer functions, numerical solution strategies, macropores, and where the classical theory breaks down.

What you'll be able to do by the end

By the end of the course, you should be able to explain the physical meaning of the water retention curve and the Richards equation and the reasoning that connects them, solve simple unsaturated flow problems by choosing the right constitutive relationships, and critically evaluate when the Richards-equation framework is — and isn't — a valid description of what's happening in real soil.

Who it's for

If you work in hydrology, agronomy, environmental engineering, or soil science — or you're a student who wants to go beyond a black-box use of Richards' equation — this course is built for you. The course can be seen as an introduction to the theory of the WHETGEO model.  WHETGEO provides an open source tool for solving 1D and 2D Richards equation. 

Try it

The course is free to enroll: mooc.unitn.it/course/view.php?id=32

If not Richards, what else ?

Two soils with the same water content θ are not in the same hydraulic state. The pore-occupancy g(r) — the fraction of pores of radius r that are water-filled — distinguishes them, while θ, being an integral of g, cannot. The navy step is the reference equilibrium g_eq = H(r* − r): water fills the small pores first. Everything else on the plot is a state that Richards' equation is blind to.

In May I posted the talk I gave at EGU 2026 in Vienna, and promised the two papers “in a couple of weeks after EGU.” It took a little longer than that — it always does — but the first one is now public:

The Statistical Physics of Unsaturated Soil Water: kinetic theory and non-commutative pore-water dynamics
R. Rigon, arXiv:2607.09416 [cond-mat.stat-mech], 22 pages, 9 figures, 2 appendices.

It is going to be submitted to Physical Review E. The companion paper — the Chapman–Enskog derivation that recovers Richards' equation as a hydrodynamic limit — follows shortly, and I will post it here when it lands.

The argument in one sentence

Unchanged from the talk, and worth repeating because everything else is a consequence of it:

Richards' equation is not wrong; it is the equilibrium limit of a deeper kinetic theory — in the same sense that Navier–Stokes is the hydrodynamic limit of Boltzmann's equation for a gas.

Mario Putti asked me, twenty years ago, “if not Richards, what else?” This is my attempt at an answer, and it arrives only after many years spent trying to solve Richards' equation properly — first with GEOtop, later with WHETGEO. You have to take an equation seriously for a long time before you earn the right to say what it is missing.

What the theory actually says

The state variable is not θ. It is the pore-occupancy g(r, x, t): the fraction of pores of radius r that are water-filled at position x and time t. Water content is recovered as a moment of it, θ[g] = φ ∫ g(r) f(r) dr — which is precisely the point: θ is an integral of g, so it throws information away. Two soils with the same θ, one wetted by rain (which fills pores by areal exposure, favouring the large ones) and one drained to the same θ (which empties the large ones first), are in genuinely different hydraulic states. They will conduct water differently, and they will respond to the next rainfall differently. Richards' equation cannot see the difference. That is the figure above, and that is the whole motivation.

The theory is built by passing through three scales, and I think this is the part hydrologists will find easiest to trust, because each step is ordinary physics:

• Microscale. A single water transfer between two pores is set by a Hagen–Poiseuille rate and driven by the difference of pore chemical potentials Φ(r, r′) — capillary and gravitational here, but open to adsorptive, osmotic, or thermal refinement without touching the structure of the theory.
• Mesoscale. Averaging over a representative volume gives a master equation — a gain–loss (Boltzmann-type) kinetic equation whose terms relax the occupancy toward its equilibrium, with a connectivity kernel C(r, r′) that encodes which pores can actually talk to which.
• Macroscale. A Chapman–Enskog reduction gives back Richards' equation in the quasi-static limit Da → 0.

Everything the theory needs as input is a geometric property of the pore network — measurable from micro-CT. Nothing is calibrated against macroscopic hydrological data. I want to be blunt about how unusual that is, and how exposed it leaves me: the theory makes parameter-free predictions, and parameter-free predictions can be wrong in public.

Four things that fall out, which I did not put in

This is the part I care about. These were not assumptions; they are consequences.

1. Matric potential and hydraulic conductivity exist only in the limit. ψ and K are not primitive quantities of the theory. They emerge at Da → 0, and K is derived from the connectivity kernel rather than postulated. Below the percolation threshold, K vanishes — not as a fitting choice, but because the water phase stops spanning the medium. Field capacity gets a geometric meaning: θ_FC ≈ θ_c.

2. Hysteresis is geometry, not memory. It is the holonomy of a forcing bundle — a geometric phase, arising from the non-commutativity [W, D] ≠ 0 of the wetting and drying operators. Wetting fills by areal exposure; drying empties by capillary ordering; the two operations do not commute, so a closed loop in the forcing does not return you to where you started. Independent-domain and Preisach models posit bistable pores and reproduce the loop. Here the loop is derived, and it comes with a falsifiable prediction: the loop area scales as H ∼ I² with the forcing intensity. Domain models are rate-independent and predict no such thing. That is a clean experimental discriminant, and I would very much like someone to go and measure it.

3. Preferential flow is not a separate process. It is what the same equation does when Da > 1. The molecular-chaos (Stosszahlansatz) closure that underlies the kinetic equation fails exactly when pore occupancies become correlated near the percolation threshold — and that correlated, channelized regime is fingering and preferential flow. So the Richards / preferential-flow dichotomy dissolves into a continuous, Da-controlled crossover. We do not need two domains and a phenomenological exchange term; we need one equation and an honest look at its Damköhler number.

4. Out of the quasi-static limit, g(r) is irreducible. No single scalar — not θ, not ψ — is a complete description. And, as I discovered while revising the manuscript (a lesson in the value of being asked a hard question at the right moment): this is true even at equilibrium. With gravity present in a finite volume, the equilibrium occupancy is not the sharp step H(r* − r) at all; it is a smeared step, because a large pore low in the profile can stay filled while a smaller pore higher up has already drained. Each pore holds water within its own Jurin rise. The retention curve — the last place where the classical scalar picture was supposed to be exact — is not exact either.

Where this connects

The framework absorbs rather than replaces. Capillary-bundle models are its diagonal limit; critical-path models are its spectral limit; Hassanizadeh–Gray is a thermodynamically consistent extension, here resolved pore-class by pore-class; phase-field methods are gradient flow on a free energy, here with explicit network connectivity; dual-permeability models are the Da > 1 regime, without the phenomenology. The same machinery, with capillary pressure replaced by freezing-point depression, is the freezing-soil problem I have worked on with Niccolò Tubini and John Mohd Wani.

This is not a parallel universe to Richards. It contains it.

What I am not claiming

I would rather say this myself than have it said to me. The paper is a construction, not a rigorous reduction from molecular dynamics: the closures are posited on physical grounds and judged by their consequences. The full kinetic equation has not yet been solved on a real soil — the numerics live in the companion paper and in the supplementary demonstrations. And the single most obvious next step is also the hardest and the most interesting one:

directly observing g(r).

Micro-CT can see it. Nobody, as far as I know, has yet used it to test a kinetic theory of soil water. If you work with imaging of pore-scale water and this sounds like a collaboration, write to me.

Materials

• Paper: arXiv:2607.09416
• The EGU 2026 talk (slides, storyboard, and the notebooks behind the figures): the May post — still the gentlest way in, if the paper looks forbidding.
• Code: will be added on GitHub; the OpenPNM notebooks that generate the supporting figures are already in the OSF repository linked from the talk.
• The companion paper: "Richards as a Limit"  where Richards equation is derived from the main general equation. Here.
• My MOOC (Massive Open Online Course) on Water in Soil. This, especially the part related to the energy of water in soil, can be though as an introduction to these more advanced topics. 

Comments, objections, and counterexamples are all welcome — especially the counterexamples. A theory that cannot be attacked is not saying anything.

Extended PETRI Net examples from the MARRMoT models collection

Five years ago, when we wrote that any lumped-parameter hydrological model can be represented as an Extended Petri Net, the statement had the flavor of a theorem asserted with a couple of worked examples. Now it has the flavor of a proof by exhaustion. The presentation below, prepared with Marialaura Bancheri and Anna De Nardi, contains the EPN translation of all forty-six conceptual models of the MARRMoT collection (Knoben et al., 2019), from the single-bucket Collie River Basin 1 up to SACRAMENTO, PRMS and CLASSIC. Anna carried out the bulk of this work as her graduation exercise, completing the task I had assigned to my class back in 2021, and I think the result deserves to be seen in its entirety.

The presentation can be found by clicking on the above image. 

I will not explain here what an EPN is: the definitive references remain Bancheri, Serafin and Rigon (2019), which introduces the formalism and its exact correspondence with the ordinary differential equations of the water budget, and Rigon and Bancheri (2021), which shows how the same topology carries, almost for free, the travel time, response time and tracer dynamics. A gentler entry point, with slides and a video tutorial, is this older post, and the conceptual background on the equivalences among the various hydrological dynamical systems is discussed here.

What the collection adds is something the papers could not give: the experience of seeing forty-six models side by side under a single graphical grammar. Some things become obvious that the original equations, or the traditional bucket sketches, keep hidden. Family resemblances jump out — the MOPEX series, the Flex variants, the Tank models reveal themselves as small mutations of a shared skeleton, and one starts to suspect that the space of conceptual models is much smaller than the number of their names suggests. Complexity becomes measurable at a glance: you can literally count places, transitions, splitters and see where a model concentrates its assumptions. And the pathologies show up too — when the wiring of a model resists a planar, readable drawing, as it happens with SACRAMENTO, that is telling you something about the model, not about the drawing. In this sense the EPN works as a diagnostic instrument, not merely an illustration.

There is also a forward-looking reason to care. Once a model is a graph with typed nodes, it is data: it can be stored, compared, composed with other graphs, translated automatically into code — which is what we pursue in the GEOframe/OMS3 world — and it connects naturally with the compositional, category-theoretic view of open systems I discussed apropos of stock and flow diagrams. The forty-six drawings below are therefore not an endpoint but a dataset.

All the previous material on the topic is collected under the EPN label of this blog, starting from the original announcement of the WRR paper.

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. https://doi.org/10.1029/2019WR025099

Rigon, Riccardo, and Marialaura Bancheri. 2021. "On the Relations between the Hydrological Dynamical Systems of Water Budget, Travel Time, Response Time and Tracer Concentrations." Hydrological Processes 35 (1). https://doi.org/10.1002/hyp.14007

Knoben, W. J. M., J. Freer, K. J. A. Fowler, M. C. Peel, and R. A. Woods. 2019. "Modular Assessment of Rainfall-Runoff Models Toolbox (MARRMoT) v1.0: An Open Source, Extendable Framework Providing Implementations of 46 Conceptual Hydrologic Models as Continuous Space-State Formulations." Geoscientific Model Development. https://gmd.copernicus.org/articles/12/2463/2019/

What is a DARTH, and which is its way to implement a digital twin of Hydrology ?

So what, exactly, is a DARTH? Not the software, not the cloud — the thing itself. This post is my attempt to answer, in thirteen tenets.

Digital twins are everywhere now, at least we see several contributions that name themselves such. They appear in the great international programmes, in the roadmaps of agencies, in the slides of almost every keynote about the future of Earth science. Many of them are magnificent feats of engineering: high-resolution solvers, elastic cloud, fast emulators. And yet, watching talk after talk, I kept feeling that something was missing. Technology is not the same thing as a vision. A high-resolution engine and a cloud are not, by themselves, a scientific instrument.

The precedent I keep returning to is FAIR. Years ago the FAIR principles changed the way our community treats data — not by inventing any new technology, but by writing down, compactly and citably, what good stewardship requires: that data be findable, accessible, interoperable, reusable. Their power was in the act of definition. We need something similar for twins. But a twin is neither only data nor only software. It is a model bound to a real basin, kept in correspondence with it, and used to make claims about its past, present, and possible futures. So I sat down and tried to write what makes such a twin trustworthy.

I came out with seven points. Talking them over with colleagues, the seven became thirteen, and they arranged themselves naturally into five movements — from what a twin is, through the record it keeps and the machinery that runs it, to the commons that builds it and the ends it serves.

I · Representation — what a twin is

1. A multi-resolution, multi-process, representation of the hydrological cycle, built from interchangeable modelling solutions that close mass and energy budgets and quantify their own error — at least in a subjective, Bayesian sense — propagated across scales.

2. Bound to a real system: calibrated, validated, and — where data allow — kept synchronised with the basin through data assimilation. This binding to reality is what distinguishes a twin from a mere simulator.

3. Transparent — no black boxes. Every component is inspectable, and even the machine-learning parts carry a stated structure and an interpretable role.

II · Data — the record it keeps

4. All data, consumed and produced, are geo-registered, versioned, and FAIR.

5. Every result is reproducible end to end — code, parameters, environment, and data lineage recorded as provenance. FAIR governs the data; this governs the computation.

III · Software — the machinery that runs it

6. Model parts are interchangeable by construction, so alternative descriptions can compete as falsifiable hypotheses.

7. Deployable over the web and, through shared standards and interfaces, interoperable across infrastructures — components and twins from different groups composing together, running from laptop to HPC.

8. Open source: built on, released as, and developed in the open, and free of lock-in.

IV · Community — the commons that builds it

9. A shared infrastructure, not isolated codes, that serves the widest community and grows with knowledge.

10. Organised, by construction, for participation and cooperation — for the collective action of scientists.

11. Stewarded: it carries its own documentation, training, and transparent governance, so that it outlives any single project or generation of its makers.

V · Purpose — the ends it serves

12. General-purpose by design — prediction, scenarios, decisions, education, or pure inquiry alike.

13. Bounded by an ethical purpose: to serve the public good and the stewardship of water and land, and to do no harm.

Please observe that Number 1, excludes practically all exiting model, which are certainly not multi-resolution, neither multiprocess. Therefore, for now, we have to consider DARTHs as an "organizing metaphor", i.e. a working program.  If you make me choose the tenets I care most about, they are these. Number 2, the binding to observation: a model that never meets a measurement is a beautiful animation, not a twin. Number 3, no black boxes: I am not against machine learning — I use it — but a learned component still has to say what it is and what role it plays. Number 7, interoperability among infrastructures: no platform is a twin of the planet on its own; the digital Earth appears only when many of them interoperate. One Earth, many infrastructures. And number 13, the ethical bound, which I added last and on purpose — openness and transparency are not only engineering choices, they are how we stay accountable.

Here is the part I most want to insist on: these are not wishes. The means to honour most of them already exist, in the open. The Basic Model Interface (CSDMS) makes components interchangeable and couplable; the OGC API – Processes family lets us deliver and chain models as web services across infrastructures; and the Object Modeling System with its cloud companion CSIP turns model components into open services at scale. GEOframe and GEOtop are simply our own way of walking that path. What no standard hands you for free is the last rung — transparency, honest uncertainty, components that behave as hypotheses rather than as fitted curves. That rung is the science, and it stays our job.

I am posting this as a draft, deliberately, because arguing about it in the open is rather the point of tenet 10. We are also writing it up — in the spirit of the FAIR paper — as a short article, anchored to those existing standards. If you think I have a tenet wrong, or that I have missed one, please tell me. That is how a manifesto earns the name.

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The one-page manifesto (PDF): here.  ·  The short paper, in preparation: draft.  ·  The original blueprint: Rigon et al., HESS Opinions: Participatory Digital eARth Twin Hydrology systems (DARTHs), HESS 26, 4773–4800 (2022).

DARThs 2026 for Vth Summer School of the IAEG

I was invited to explain the idea of Digital Twins of the Earth System at the IAEG Summer School in Aosta. This time it was in person — a welcome change from the Chandigarh and Bern talks, which were both online — and the occasion was a good reason to rework and rearrange material I had used before.

The talk followed this outline:

• Earth Digital Twins — the general idea
• Models
• New ways to do software
• Steps towards DARThs
• Working for Science
• Enabling Technologies
• Open Science and participation

(For anyone who wants the definition itself — what a DARTh actually is — I set it out separately, as thirteen tenets, in an earlier post, after my talk)

From my by-now rather large collection I had also prepared a few further sets of slides covering the processes of interest (the flesh to add to the bone). I did not manage to show them, for lack of time, but they follow the same thread, and I am leaving them here for anyone of IAEGists curious to see where we think this is heading:

• GEOtop EO — an older presentation on using GEOtop to anticipate the location and timing of shallow landslides: an example of a piece of existing software that could grow into a DARTh.
• Some results from the Morfeo Project (2008), along the same line.
• DARThs 2027 — where we stand today, and the perspective under which we are working.
• A message to the young — a few words for early-career researchers.
• Catchment processes (or the processes we need to implement)

I should be honest about that last group. The closing part of the DARThs 2027 presentation is still under construction: it needs more thinking, and a deeper reading of several resources I cite but with which I have no direct experience yet. It is, frankly, a work in progress — and I would rather say so than pretend otherwise. If any of it is useful to you, or if you see where it should go next, I would be glad to hear from you.

Because you are probably interested in landslides and, consequently on Richards equations, and its generalizations, you can find material here:

• The various post I did in the last fifteen years
• The WHETGEO model integrating Richards
• An incoming new theory for the movement of water in the vadose zone under disequilibrium
• My Massive Open Online Course on water in soil (incoming, July 10)

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Slides: the presentation (and the additional sets mentioned above).
Video: the recording of the talk will be added here once it is available — [to come].

A 30-Year 1-km Daily Precipitation and Air Temperature Dataset for the Po River District (Italy)

 Mountains are unkind to gridded data. The processes that matter most in Alpine terrain — orographic rainfall, elevation-dependent temperature, the sharp contrasts between a valley floor and a ridge a few kilometres away — live at scales that most continental products simply cannot see. A 10-km cell averages away exactly the gradients a hydrologist needs.

So we built something finer. Together with colleagues from the University of Trento, the C3A centre, Fondazione Edmund Mach, the Po River Basin District Authority, and led by Hossein Salehi, we have produced a 1-km, daily precipitation and temperature dataset covering the entire Po River District over the full 1991–2020 climatological reference period. To our knowledge, no publicly available product previously combined all three of those properties — kilometre resolution, daily steps, and a continuous 30-year record — for this basin.

The dataset is now openly available on Zenodo under a CC BY 4.0 licence.

Why the Po District is a hard test

The Po River District is arguably one of Europe's most topographically demanding hydrological systems. It spans roughly 83,000 km² and stretches from sea level in the Adriatic delta to nearly 4,800 m in the Alpine headwaters. That relief produces strong spatial contrasts in both rainfall and temperature, and it governs snow accumulation, melt, runoff, and water availability across the region. Any interpolation method that ignores elevation is doomed from the start.

How it was made

The dataset rests on a harmonised, multi-source observational network: 1,583 precipitation stations and 1,555 temperature stations surviving quality control, drawn from the regional ARPA agencies and supplemented with the EEAR-Clim dataset. We deliberately extended a 20-km buffer beyond the district boundary and pulled in stations there too, so that interpolation near the edges — and across watershed divides — would not be starved of data.

Interpolation was carried out in the GEOframe new kriging framework:

• Precipitation uses Ordinary Kriging.
• Temperature uses Detrended Kriging, with elevation as the trend variable. A daily linear regression estimates the lapse rate for each day, removes the elevation signal before kriging, then restores it at every grid cell. The lapse rate is therefore allowed to breathe with the synoptic conditions rather than being fixed at a textbook value.

A nice methodological detail: at every daily time step, the empirical semivariogram is re-fitted against five candidate models (Exponential, Gaussian, Linear, Power, Spherical), with parameters optimised by Particle Swarm Optimisation, and the best-fitting model is selected for that day. The structure of the field is re-estimated daily rather than imposed once.

Does it work?

Leave-one-out cross-validation across the whole 30-year record says yes.

• Precipitation: mean KGE above 0.84 (All-Days) and 0.82 (Wet-Days), with mean absolute errors of 1.28 mm and 3.05 mm respectively. As expected from kriging's conditional bias and smoothing, wet-day skill drops a little and shifts slightly toward underestimation of peaks.
• Temperature: mean KGE of 0.88, correlation of 0.98, MAE of 1.14 °C, RMSE of 1.5 °C, and essentially no bias (+0.02 °C). The elevation detrending earns its keep.

The honest caveats are spatial. Errors concentrate in the high-relief northern sectors and along the southern boundary, and skill declines measurably with altitude — for precipitation, MAE rises by roughly 0.54 mm per 1000 m on all days and 0.87 mm per 1000 m on wet days. This is a station-density story more than a method story: the mountains are where the gauges thin out. We also note that no wind-undercatch correction was applied, so high-elevation snowfall is likely underestimated — something to keep in mind for Alpine applications.

Reassuringly, the annual diagnostics show gradual, continuous improvement over the three decades (precipitation KGE climbing from ~0.76–0.78 in the early 1990s to ~0.83–0.84 by the mid-2010s) with no abrupt step changes — evidence that progressive network densification, not methodological artefacts, drives the trend. The dataset is temporally coherent.

What the resolution buys you

Benchmarking against E-OBS (~10 km) and EMO (1 arcmin) makes the case for going fine. Two extremes are particularly telling:

• During Storm Alex (October 2020), the maximum three-day accumulation reaches 482 mm in the 1-km product over western Piedmont, against only 277 mm in E-OBS — a loss of roughly 43% of the peak signal at coarser resolution.
• For minimum annual temperature, the 1-km field reaches −11.1 °C where E-OBS reads only −4.7 °C and EMO −9.8 °C. Pixel-averaging quietly amputates the cold tail of high-altitude climate.

In short: kilometre-scale topographic forcing can only be represented faithfully when the grid is commensurate with the physiographic gradients doing the forcing.

Getting the data

The product is distributed as annual NetCDF files (CF conventions), one per variable per year — 60 files in total — readable with xarray, netCDF4, R's ncdf4/terra, or QGIS/ArcGIS. It is intended as spatially consistent meteorological forcing for hydrological and ecohydrological models, not as calibration ground truth.

• Dataset (Zenodo): https://doi.org/10.5281/zenodo.19207256 — CC BY 4.0
• Code (GEOframe Krigings): https://github.com/geoframecomponents/Krigings
• Method reference: Bancheri et al. (2018), Geoscientific Model Development but a new method was actually implemented on that basis, and will be submitted soon (e.g. Andreis et al., 2026) 

It is a small piece of infrastructure, but the kind we keep needing: a clean, open, high-resolution climatic backdrop against which the hydrology of a complex basin can actually be modelled.

Reference

• Hossein Salehi, Daniele Andreis, Sohaib Baig, Gaia Roati, Marco Brian, Francesco Tornatore, Giuseppe Formetta, and Riccardo Rigon, A 30-Year 1-km Daily Precipitation and Air Temperature Dataset for the Po River District (Italy), submitted to ESSD.