Please you can find what we present and do at the EGU 2026 General Assembly in Wien.
The high resolution pdf is here.
Below one by one the contributions:
- Marianna Tavonatti & Al. (On Tuesday) Pico Presentation
Thursday, April 30, 2026
Monday, April 27, 2026
GEOSPACE Validation Paper: Application and Testing in the "Spike II" Lysimeter Experiment
We have just submitted a new paper — A flexible open-source modular framework for ecohydrological modeling: Application and validation of GEOSPACE-1D — by Concetta D'Amato, Paolo Benettin, Andrea Rinaldo, and Riccardo Rigon. It is the natural companion and follow-up to the GEOSPACE framework paper published in Geoscientific Model Development earlier this year, which described the design principles and modular architecture of GEOSPACE. That paper introduced the framework; this one puts it to work.
The validation is built around the "Spike II" experiment, a carefully instrumented lysimeter study carried out in 2018 on the EPFL campus in Lausanne. Four soil columns — a willow tree (L2), two grass-covered lysimeters (L1 and L4), and a bare-soil system (L3) — were monitored over two months, providing high-quality weight-based evapotranspiration estimates alongside measurements of soil water content, pressure, and drainage. These data allow a thorough assessment of the model across contrasting vegetation types and soil configurations.
The core of the paper addresses three questions: Can GEOSPACE reproduce observed ecohydrological dynamics across such diverse conditions? What are the practical advantages of its modular structure? And does it enable novel analyses — the kind that open new scientific doors rather than merely close validation loops?
On performance: GEOSPACE reproduces soil water pressure dynamics, depth-resolved water content, bottom drainage, and evapotranspiration fluxes across all four lysimeters with R² values of 0.87, 0.81, 0.83, and 0.73 for L2, L1, L4, and L3 respectively. Mean residual biases are negligible throughout. The slightly lower performance over bare soil reflects a known structural limitation of the Penman–Monteith formulation for soil evaporation under conditions where thermal inertia matters — an honest diagnosis rather than a defect to be papered over.
On modularity: the willow lysimeter was simulated with three alternative evapotranspiration formulations — GEOET-Prospero-PM, GEOET-Priestley-Taylor, and GEOET-Penman-Monteith FAO — keeping the soil component (WHETGEO) and the partitioning solver (BrokerGEO) identical across all three runs. All formulations close the cumulative water balance (~600 mm over the experiment), but the Prospero-PM formulation captures sub-daily peak dynamics with roughly half the residual spread of the other two. The calibrated Priestley-Taylor α = 4.16 and FAO Kc = 3.9 — both well above standard values — are informative precisely because they expose structural limitations of simplified formulations when applied to a high-transpiration system dominated by stomatal control.
On novel capabilities: the model computes root water uptake (RWU) for every control volume at every time step, yielding a full depth–time distribution of uptake intensity. The willow shifts its water sourcing dynamically in response to moisture depletion and atmospheric demand, with the mean uptake depth varying over time in a way that closely mirrors the measured root density profile. This kind of depth-resolved diagnostic is directly relevant to isotope-based ecohydrology, where xylem water provides only a bulk integrated signal — GEOSPACE's spatial resolution of the RWU can help interpret what that bulk signal actually means.
The paper grew out of Concetta D'Amato's PhD work at the Center Agriculture Food Environment (C3A) at the University of Trento, supported by the WATZON COST Action and the PRIN 2017 WATZON project. Readers interested in the longer history of GEOSPACE and its components can find much of the background documented here on AboutHydrology: see the posts on GEOSPACE and WHETGEO, and in particular the earlier post on Concetta's PhD thesis and the exploration of the SPAC.
The source code is on GitHub at https://github.com/geoframecomponents/GEOSPACE-1D, with a frozen version on Zenodo. All simulation data are openly available. GEOSPACE continues to grow.
Waiting for the official preprint, you can download it here. Here instead, find the supplemental material.
Monday, March 30, 2026
Three Decades of Snow Water Equivalent Dynamics in the Po River Basin, Italy: Trends and Implications
Seasonal snowpack is a key component of the mountain cryosphere, acting as a vital natural reservoir that regulates runoff downstream in snowfed basins.
In mid- and low-elevation mountain regions such as the European Alps, snow processes, such as accumulation and ablation, are highly sensitive to climate change, having direct implications for hydrological forecasting and water availability. In this study, we present the analysis of a 30-year (1991–2021) long dataset of snow water equivalent (SWE) in the Po River District, Italy, which includes parts of the Alps and Apennines. The data is available at a 500 × 500 m2 spatial resolution and at a daily temporal scale (Dall’Amico et al., 2025). This data was generated using the “J-Snow” modeling framework, which integrates the physically based GEOtop model with in situ snow height observations and earth observation snow cover products such as MODIS. Our results show that the long-term (30 year) basin-wide mean annual SWE volume equals 3.34 Gm3. The elevation-wise statistical analysis of key snow volume and duration metrics shows that the most pronounced snow water equivalent losses occur below 2000 m a.s.l. Below this threshold, both snow volume metrics and duration metrics show a significant decrease, indicating decrease in snow water storage and earlier melt. Above this elevation, the snow volume metrics show increasing trend while as the duration metrics continue to show a shortened (decreasing trend) snow season except at the highest elevations (> 2500 m). The findings of this study highlight the changes to the mountain seasonal snow storage and the timing of snow disappearance across the Italian Alps. This combined effect highlights a fundamental shift in the hydrological regime of the Po River Basin, with significant implications for water availability and management under ongoing climate change. The data used in this paper are those freely available in Dall'Amico et al., 2025.
References
In mid- and low-elevation mountain regions such as the European Alps, snow processes, such as accumulation and ablation, are highly sensitive to climate change, having direct implications for hydrological forecasting and water availability. In this study, we present the analysis of a 30-year (1991–2021) long dataset of snow water equivalent (SWE) in the Po River District, Italy, which includes parts of the Alps and Apennines. The data is available at a 500 × 500 m2 spatial resolution and at a daily temporal scale (Dall’Amico et al., 2025). This data was generated using the “J-Snow” modeling framework, which integrates the physically based GEOtop model with in situ snow height observations and earth observation snow cover products such as MODIS. Our results show that the long-term (30 year) basin-wide mean annual SWE volume equals 3.34 Gm3. The elevation-wise statistical analysis of key snow volume and duration metrics shows that the most pronounced snow water equivalent losses occur below 2000 m a.s.l. Below this threshold, both snow volume metrics and duration metrics show a significant decrease, indicating decrease in snow water storage and earlier melt. Above this elevation, the snow volume metrics show increasing trend while as the duration metrics continue to show a shortened (decreasing trend) snow season except at the highest elevations (> 2500 m). The findings of this study highlight the changes to the mountain seasonal snow storage and the timing of snow disappearance across the Italian Alps. This combined effect highlights a fundamental shift in the hydrological regime of the Po River Basin, with significant implications for water availability and management under ongoing climate change. The data used in this paper are those freely available in Dall'Amico et al., 2025.
References
Dall’Amico, Matteo, Stefano Tasin, Federico Di Paolo, Marco Brian, Paolo Leoni, Francesco Tornatore, Giuseppe Formetta, John Mohd Wani, Riccardo Rigon, and Gaia Roati. 2025. “30-Years (1991-2021) Snow Water Equivalent Dataset in the Po River District, Italy.” Scientific Data 12 (1): 374. https://doi.org/10.1038/s41597-025-04633-5.
Wani, John Mohd, Kelly E. Gleason, Matteo Dall’Amico, Federico Di Paolo, Stefano Tasin, Gaia Roati, Marco Brian, Francesco Tornatore, and Riccardo Rigon. 2025. “Three Decades of Snow Water Equivalent Dynamics in the Po River Basin, Italy: Trends and Implications.” EGUsphere. https://doi.org/10.5194/egusphere-2025-5520.
From Snow Depth to Streamflow: Reducing Snowfall Uncertainty in Alpine Headwaters with Sentinel-1 based snow depth retrievals
In mountainous regions, the sparse distribution of precipitation gauges at high elevations is a major source of uncertainty in snowfall estimation. This matters beyond the local scale: uncertainties originating in headwater areas propagate through hydrological modelling, affecting the estimation of all water balance components downstream. Yet establishing dense gauge networks in complex mountain terrain remains logistically and economically challenging — which makes it worthwhile to ask whether remote sensing can fill the gap.
This study assimilates Sentinel-1 C-band snow depth observations into the snow module of the GEOframe hydrological model, coupled with a snow-density scheme, to jointly update snow depth, snow water equivalent (SWE), and snowfall estimates. The method is applied to two key Alpine catchments: the Aosta River catchment and the headwaters of the Piemonte catchment in the upper Po River basin. Both are critical contributors of snowmelt-driven discharge to the Po Valley — sustaining its agricultural water supply — and both suffer from limited high-elevation gauge coverage.
Results show that assimilating satellite-derived snow depth systematically increases snowfall estimates across elevation gradients relative to the model's partitioned snowfall, and substantially improves simulated river discharge during the snowmelt season. Notably, similar improvements persist in years without active data assimilation, suggesting that the approach has a lasting positive influence on model state and performance.
This work¹ has been submitted to The Cryosphere and is currently under review.
References
Azimi, S., Girotto, M., Rigon, R., Roati, G., Barbetta, S., and Massari, C.: From Snow Depth to Streamflow: Reducing Snowfall Uncertainty in Alpine Headwaters with Sentinel-1 Based Snow Depth Retrievals, EGUsphere [preprint], https://doi.org/10.5194/egusphere-2026-793, 2026.
Girotto, Manuela, Giuseppe Formetta, Shima Azimi, Claire Bachand, Marianne Cowherd, Gabrielle De Lannoy, Hans Lievens, et al. 2024. “Identifying Snowfall Elevation Patterns by Assimilating Satellite-Based Snow Depth Retrievals.” The Science of the Total Environment 906 (167312): 167312. https://doi.org/10.1016/j.scitotenv.2023.167312.
Monday, March 9, 2026
Where do we stand
Aristotle had it all wrong.
Dalton, Horton, Sherman and Leopold played the starting gong.
Eagleson, Rodriguez-Iturbe went for a grand theory, in which they believed.
Ignacio (Vujica teaches) dated with randomness.
It is (dis-)organized complexity, Jim Dooge said.
Richards, Richardson, Harlan and Freeze insisted on using PDEs.
Horton said the runoff is infiltration excess,
Dunne said that it is saturation excess,
Hewlett and Hibbert said that overland flow is not necessary.
Tracer research screwed it all up.
Darcy and Buckingham — it is all a matter of gradients, they thought.
Beven and Germann set up a mountain of doubts.
And many, I forgot, I do not know.
(Klemeš complains.)
Now we do not really know what we know,
except that we know more than before,
better data we have,
satellites see it all (but what you see, you do not believe).
Modelers give numbers without caring,
machine learning thinks it can do all without understanding —
and because we did not have it when we thought we did,
they probably sing the right song.
Musical coda
Saturday, February 7, 2026
Digressions on the Euler Characteristic (M₃): Unlocking Pore Network Topology for Non-Equilibrium Infiltration
This post extends the discussion of [Minkowski functionals] by exploring the deepest topological aspects of M₃—the Euler characteristic.
In our previous posts on Minkowski functionals, we established that M₀ (volume), M₁ (surface area), and M₂ (curvature) are purely geometric measures. They tell us about size, shape, and form. But M₃—the Euler characteristic—is fundamentally different. It's a topological invariant that captures something geometry alone cannot: how the pore space is connected.
Consider two soil samples with identical pore size distributions, identical surface areas, and identical mean curvatures. One could be a tree-like network where every pore has a unique path to every other pore. The other could be riddled with loops—alternative pathways that provide redundancy. The first three Minkowski functionals (M₀, M₁, M₂) might be identical, but M₃ would reveal the profound topological difference.
This distinction isn't academic. It's critical for understanding non-equilibrium infiltration, hysteresis, and the dynamics of water movement that Richards' equation cannot capture.
The Euler characteristic for a 3D pore network can be expressed in two equivalent ways:
Graph-theoretic form:
$$\chi = V - E + F$$
where V is the number of vertices (pore nodes), E is the number of edges (connections), and F is the number of faces (enclosed regions).
Topological form (via Betti numbers):
$$\chi = \beta_0 - \beta_1 + \beta_2$$
where:
While Vogel and Roth (2001) pioneered the use of the Euler characteristic for soil pore characterization, the decomposition into Betti numbers reveals structure that χ alone obscures:
β₀ (Connected Components):
When you progressively drain water from soil, β₀ tracks how the water phase fragments. High β₀ means disconnected water clusters—poor hydraulic connectivity. This is crucial during drought or in the late stages of drainage.
β₁ (Independent Loops):
This is perhaps the most important quantity for non-equilibrium flow. Loops create:Alternative flow paths: If one pore throat becomes blocked (by air during drainage, or by swelling clay), water can route around it
Hysteresis mechanisms: The ink-bottle effect is fundamentally about loops—water trapped in a loop during drainage can only escape when all connecting throat pressures exceed the entry pressure
Air entrapment during imbibition: Loops trap air bubbles that persist even after the network is nominally saturated
β₂ (Enclosed Voids):
In most soils, β₂ ≈ 0 (isolated voids are rare), but in fractured or macroporous media, enclosed voids contribute to non-productive porosity and affect effective saturation.
The Critical Gap: C(r→r') vs. χ
As we discussed in earlier posts, the connectivity matrix C(r→r') describes the probability that a pore of size r connects to a pore of size r'. This is essential information—it tells us about local, pairwise connectivity. But C(r→r') is fundamentally a first-order descriptor. It cannot distinguish between:
In our previous posts on Minkowski functionals, we established that M₀ (volume), M₁ (surface area), and M₂ (curvature) are purely geometric measures. They tell us about size, shape, and form. But M₃—the Euler characteristic—is fundamentally different. It's a topological invariant that captures something geometry alone cannot: how the pore space is connected.
Consider two soil samples with identical pore size distributions, identical surface areas, and identical mean curvatures. One could be a tree-like network where every pore has a unique path to every other pore. The other could be riddled with loops—alternative pathways that provide redundancy. The first three Minkowski functionals (M₀, M₁, M₂) might be identical, but M₃ would reveal the profound topological difference.
This distinction isn't academic. It's critical for understanding non-equilibrium infiltration, hysteresis, and the dynamics of water movement that Richards' equation cannot capture.
The Euler characteristic for a 3D pore network can be expressed in two equivalent ways:
Graph-theoretic form:
$$\chi = V - E + F$$
where V is the number of vertices (pore nodes), E is the number of edges (connections), and F is the number of faces (enclosed regions).
Topological form (via Betti numbers):
$$\chi = \beta_0 - \beta_1 + \beta_2$$
where:
- β₀ = number of connected components (separate pore clusters)
- β₁ = number of independent loops (redundant pathways)
- β₂ = number of enclosed voids (trapped air bubbles)
While Vogel and Roth (2001) pioneered the use of the Euler characteristic for soil pore characterization, the decomposition into Betti numbers reveals structure that χ alone obscures:
β₀ (Connected Components):
When you progressively drain water from soil, β₀ tracks how the water phase fragments. High β₀ means disconnected water clusters—poor hydraulic connectivity. This is crucial during drought or in the late stages of drainage.
β₁ (Independent Loops):
This is perhaps the most important quantity for non-equilibrium flow. Loops create:Alternative flow paths: If one pore throat becomes blocked (by air during drainage, or by swelling clay), water can route around it
Hysteresis mechanisms: The ink-bottle effect is fundamentally about loops—water trapped in a loop during drainage can only escape when all connecting throat pressures exceed the entry pressure
Air entrapment during imbibition: Loops trap air bubbles that persist even after the network is nominally saturated
β₂ (Enclosed Voids):
In most soils, β₂ ≈ 0 (isolated voids are rare), but in fractured or macroporous media, enclosed voids contribute to non-productive porosity and affect effective saturation.
The Critical Gap: C(r→r') vs. χ
As we discussed in earlier posts, the connectivity matrix C(r→r') describes the probability that a pore of size r connects to a pore of size r'. This is essential information—it tells us about local, pairwise connectivity. But C(r→r') is fundamentally a first-order descriptor. It cannot distinguish between:
- Linear chains : r₁ → r₂ → r₃ (tree-like, vulnerable)
- Loops : r₁ ⇄ r₂ ⇄ r₃ ⇄ r₁ (robust, redundant)
Both might have similar C(r→r') statistics, but their hydraulic behavior under non-equilibrium conditions would be dramatically different.
The Euler characteristic χ captures this higher-order topology . But even χ is incomplete—it's a single scalar. The full story requires the Betti numbers as functions of pore size :
$$\beta_1(r_{\min}) = \text{number of loops using pores } r \geq r_{\min}$$
This size-resolved topology tells us which pore size classes contribute to network connectivity and robustness.
Persistent Homology: A New Frontier
Recent work in computational topology has introduced persistent homology to porous media analysis (Jiang et al., 2018; Moon et al., 2019). This technique tracks how Betti numbers evolve as you "fill" the pore space from small to large pores:Birth : The pore size at which a topological feature (component, loop, void) first appears
Death : The pore size at which it disappears (merges with another feature)
Persistence : Death - Birth (a measure of robustness)
Features with high persistence are geometrically significant; those with low persistence are noise.
For soil hydrology, this means:
During Drainage (Decreasing Saturation):Large loops "die" first (they contain large pore bodies that empty early)
Small loops persist longer (they're held by capillarity in small throats)
The transition χ = 0 often marks the percolation threshold where the network fragments
During Imbibition (Increasing Saturation):Small pores fill first, creating many disconnected clusters (high β₀)
As larger pores fill, clusters merge (β₀ decreases, β₁ increases as loops form)
Air becomes trapped in loops that cannot drain (high β₁ of the air phase)
Connection to Permeability: Beyond Kozeny-Carman
Scholz et al. (2012) demonstrated experimentally that permeability scales with the Euler characteristic:
$$k \propto |\chi|^\alpha \cdot f(\phi)$$
This is remarkable because it's independent of the percolation threshold —unlike Katz-Thompson and other models that require identifying a critical pore size.
Why does this work? Because χ directly encodes:Connectivity (via β₀ - β₁): How many pathways are available?
Network topology: Tree-like (low β₁) vs. loop-rich (high β₁) structures
Liu et al. (2017) extended this to 3D and found that void ratio must also be included. But the fundamental insight remains: topology, not just geometry, controls flow .
Application to Non-Equilibrium Infiltration
This brings us to the central challenge: developing a pore-network theory that replaces Richards' equation. Richards assumes:Local equilibrium : θ(ψ) and K(ψ) are unique, path-independent functions
Instantaneous capillary-gravity balance : No dynamic lag
Continuum description : Pore-scale structure doesn't matter
All three assumptions break down during rapid infiltration, preferential flow, and hysteretic cycling. A pore-network approach can address these limitations, but only if we properly account for topology .
The Role of Loops in Non-Equilibrium Dynamics
The Euler characteristic χ captures this higher-order topology . But even χ is incomplete—it's a single scalar. The full story requires the Betti numbers as functions of pore size :
$$\beta_1(r_{\min}) = \text{number of loops using pores } r \geq r_{\min}$$
This size-resolved topology tells us which pore size classes contribute to network connectivity and robustness.
Persistent Homology: A New Frontier
Recent work in computational topology has introduced persistent homology to porous media analysis (Jiang et al., 2018; Moon et al., 2019). This technique tracks how Betti numbers evolve as you "fill" the pore space from small to large pores:Birth : The pore size at which a topological feature (component, loop, void) first appears
Death : The pore size at which it disappears (merges with another feature)
Persistence : Death - Birth (a measure of robustness)
Features with high persistence are geometrically significant; those with low persistence are noise.
For soil hydrology, this means:
During Drainage (Decreasing Saturation):Large loops "die" first (they contain large pore bodies that empty early)
Small loops persist longer (they're held by capillarity in small throats)
The transition χ = 0 often marks the percolation threshold where the network fragments
During Imbibition (Increasing Saturation):Small pores fill first, creating many disconnected clusters (high β₀)
As larger pores fill, clusters merge (β₀ decreases, β₁ increases as loops form)
Air becomes trapped in loops that cannot drain (high β₁ of the air phase)
Connection to Permeability: Beyond Kozeny-Carman
Scholz et al. (2012) demonstrated experimentally that permeability scales with the Euler characteristic:
$$k \propto |\chi|^\alpha \cdot f(\phi)$$
This is remarkable because it's independent of the percolation threshold —unlike Katz-Thompson and other models that require identifying a critical pore size.
Why does this work? Because χ directly encodes:Connectivity (via β₀ - β₁): How many pathways are available?
Network topology: Tree-like (low β₁) vs. loop-rich (high β₁) structures
Liu et al. (2017) extended this to 3D and found that void ratio must also be included. But the fundamental insight remains: topology, not just geometry, controls flow .
Application to Non-Equilibrium Infiltration
This brings us to the central challenge: developing a pore-network theory that replaces Richards' equation. Richards assumes:Local equilibrium : θ(ψ) and K(ψ) are unique, path-independent functions
Instantaneous capillary-gravity balance : No dynamic lag
Continuum description : Pore-scale structure doesn't matter
All three assumptions break down during rapid infiltration, preferential flow, and hysteretic cycling. A pore-network approach can address these limitations, but only if we properly account for topology .
The Role of Loops in Non-Equilibrium Dynamics
For each pore i with saturation S_i(t):
$$\frac{\partial S_i}{\partial t} = \frac{1}{V_i} \sum_{j \in \mathcal{N}(i)} Q_{ij}(S_i, S_j, \psi_i, \psi_j, \text{Topology})$$
The flow Q_ij depends not just on local states (S_i, S_j, ψ_i, ψ_j) but also on: Whether (i,j) is part of a loop : If yes, alternative paths exist—the flow can redistribute when one path becomes unfavorable
The saturation state of loops containing (i,j) : A fully saturated loop behaves differently from a partially saturated one
Contact line pinning : In rough or angular pores, interfaces can be pinned by geometry, creating metastable states
This requires tracking:
$$\beta_1^{\text{filled}}(t) = \text{number of loops with all pores filled at time } t$$
As infiltration progresses, new loops become active. During drainage, loops trap water. The dynamics are fundamentally topology-dependent .
The Missing Measurement
Current persistent homology applications to porous media compute β₁(r_min) globally. But for infiltration dynamics, we need:
$$\beta_1(r_{\min}, r_{\max}, S) = \text{loops using pores in } [r_{\min}, r_{\max}] \text{ at saturation } S$$
This would tell us:
Practical Implications
Hysteresis is fundamentally about loops. The ink-bottle effect, snap-off during imbibition, and air entrapment all require topological understanding. Empirical hysteresis models (like Scott 1983 or Luckner et al. 1989) lack physical basis. A topology-based approach could predict hysteresis from pore structure alone.
In macroporous or structured soils, flow doesn't occur uniformly—it follows preferred pathways. These pathways are defined by topology: which loops provide the path of least resistance? Traditional continuum models cannot represent this; pore networks can, if we track β₁(r,S).
During infiltration, trapped air creates additional resistance. But how much air gets trapped? Where? This depends entirely on loop structure. High β₁ in large pores means air can be trapped in those loops even after the matrix is saturated.
Upscaling:
Can we derive effective Richards-like equations from pore-network topology? Perhaps. If we can relate:
$$K(S) = K_{\text{sat}} \cdot f(S, \chi(S), \beta_1(S), \ldots)$$
then topology provides the missing link between pore structure and continuum behavior.
Methodological Challenges
Implementing this vision requires solving several problems:
Current algorithms compute Betti numbers for geometric complexes. We need algorithms that track topology by pore size class and by saturation state . This is "attributed" persistent homology—a frontier in computational topology.
During infiltration, topology evolves. We need:
$$\beta_1(r, t) = \text{loops at pore size } r \text{ and time } t$$
This requires combining persistent homology with time-dependent network analysis—an open problem.
A representative elementary volume (REV) for structured soil might contain 10⁹ pores. Current persistent homology algorithms scale poorly beyond 10⁶ elements. Optimizations are needed.
We need dynamic X-ray CT experiments that track topology during infiltration/drainage cycles. Some pioneering work exists (Armstrong & Berg 2013; Schlüter et al. 2016), but systematic topology analysis is lacking.
The Path Forward
The integration of algebraic topology into soil hydrology is still in its infancy. The foundational work exists:
But critical gaps remain:
Loop statistics L_n(r₁,...,r_n) connecting C(r→r') to β₁(r)
Size-resolved persistent homology for two-phase flow
Dynamic topology tracking during infiltration
Topology-dependent constitutive relations for pore-network models
Filling these gaps would enable a truly mechanistic alternative to Richards' equation—one that captures non-equilibrium dynamics, hysteresis, and preferential flow from first principles.
Topology as the Fourth Dimension
We often think of soil characterization in three dimensions:Size (pore size distribution)
Shape (surface area, curvature)
Space (spatial correlations)
But there's a fourth dimension, one that geometry alone cannot capture: Topology .
M₃ and its decomposition into Betti numbers provide access to this dimension. They tell us not just what pores exist, but how they connect —and that connectivity determines everything about non-equilibrium flow.
As we move beyond Richards' equation toward pore-network theories of infiltration, topology will be as important as geometry. The Euler characteristic is our window into this hidden structure—but we're only beginning to look through it.
References
$$\frac{\partial S_i}{\partial t} = \frac{1}{V_i} \sum_{j \in \mathcal{N}(i)} Q_{ij}(S_i, S_j, \psi_i, \psi_j, \text{Topology})$$
The flow Q_ij depends not just on local states (S_i, S_j, ψ_i, ψ_j) but also on: Whether (i,j) is part of a loop : If yes, alternative paths exist—the flow can redistribute when one path becomes unfavorable
The saturation state of loops containing (i,j) : A fully saturated loop behaves differently from a partially saturated one
Contact line pinning : In rough or angular pores, interfaces can be pinned by geometry, creating metastable states
This requires tracking:
$$\beta_1^{\text{filled}}(t) = \text{number of loops with all pores filled at time } t$$
As infiltration progresses, new loops become active. During drainage, loops trap water. The dynamics are fundamentally topology-dependent .
The Missing Measurement
Current persistent homology applications to porous media compute β₁(r_min) globally. But for infiltration dynamics, we need:
$$\beta_1(r_{\min}, r_{\max}, S) = \text{loops using pores in } [r_{\min}, r_{\max}] \text{ at saturation } S$$
This would tell us:
- Which pore size classes create redundant pathways?
- At what saturation does the network transition from tree-like (β₁ ≈ 0) to loop-rich (β₁ >> 0)?
- How does the air phase topology evolve during imbibition?
Practical Implications
Hysteresis is fundamentally about loops. The ink-bottle effect, snap-off during imbibition, and air entrapment all require topological understanding. Empirical hysteresis models (like Scott 1983 or Luckner et al. 1989) lack physical basis. A topology-based approach could predict hysteresis from pore structure alone.
In macroporous or structured soils, flow doesn't occur uniformly—it follows preferred pathways. These pathways are defined by topology: which loops provide the path of least resistance? Traditional continuum models cannot represent this; pore networks can, if we track β₁(r,S).
During infiltration, trapped air creates additional resistance. But how much air gets trapped? Where? This depends entirely on loop structure. High β₁ in large pores means air can be trapped in those loops even after the matrix is saturated.
Upscaling:
Can we derive effective Richards-like equations from pore-network topology? Perhaps. If we can relate:
$$K(S) = K_{\text{sat}} \cdot f(S, \chi(S), \beta_1(S), \ldots)$$
then topology provides the missing link between pore structure and continuum behavior.
Methodological Challenges
Implementing this vision requires solving several problems:
Current algorithms compute Betti numbers for geometric complexes. We need algorithms that track topology by pore size class and by saturation state . This is "attributed" persistent homology—a frontier in computational topology.
During infiltration, topology evolves. We need:
$$\beta_1(r, t) = \text{loops at pore size } r \text{ and time } t$$
This requires combining persistent homology with time-dependent network analysis—an open problem.
A representative elementary volume (REV) for structured soil might contain 10⁹ pores. Current persistent homology algorithms scale poorly beyond 10⁶ elements. Optimizations are needed.
We need dynamic X-ray CT experiments that track topology during infiltration/drainage cycles. Some pioneering work exists (Armstrong & Berg 2013; Schlüter et al. 2016), but systematic topology analysis is lacking.
The Path Forward
The integration of algebraic topology into soil hydrology is still in its infancy. The foundational work exists:
- Vogel & Roth (2001) : Introduced χ(r) for pore connectivity
- Scholz et al. (2012) : Connected χ to permeability
- Jiang et al. (2018), Moon et al. (2019) : Applied persistent homology to predict flow properties
- Lucas et al. (2020) : Multi-scale connectivity analysis
But critical gaps remain:
Loop statistics L_n(r₁,...,r_n) connecting C(r→r') to β₁(r)
Size-resolved persistent homology for two-phase flow
Dynamic topology tracking during infiltration
Topology-dependent constitutive relations for pore-network models
Filling these gaps would enable a truly mechanistic alternative to Richards' equation—one that captures non-equilibrium dynamics, hysteresis, and preferential flow from first principles.
Topology as the Fourth Dimension
We often think of soil characterization in three dimensions:Size (pore size distribution)
Shape (surface area, curvature)
Space (spatial correlations)
But there's a fourth dimension, one that geometry alone cannot capture: Topology .
M₃ and its decomposition into Betti numbers provide access to this dimension. They tell us not just what pores exist, but how they connect —and that connectivity determines everything about non-equilibrium flow.
As we move beyond Richards' equation toward pore-network theories of infiltration, topology will be as important as geometry. The Euler characteristic is our window into this hidden structure—but we're only beginning to look through it.
References
- Jiang, Z., Wu, K., Couples, G., Van Dijke, M.I.J., Sorbie, K.S., and Ma, J. (2018). Pore geometry characterization by persistent homology theory. Water Resources Research , 54, 4150–4163.
- Liu, Z., Herring, A., Robins, V., and Armstrong, R.T. (2017). Prediction of permeability from Euler characteristic of 3D images. Proceedings of the International Symposium of the Society of Core Analysts , SCA2017-016.
- Lucas, M., Schlüter, S., Vogel, H.-J., and Vereecken, H. (2020). Revealing pore connectivity across scales and resolutions with X-ray CT. European Journal of Soil Science , 72, 546–560.
- Moon, C., Mitchell, S.A., Heath, J.E., and Andrew, M. (2019). Statistical inference over persistent homology predicts fluid flow in porous media. Water Resources Research , 55, 9592–9603.
- Scholz, C., Wirner, F., Götz, J., Rüde, U., Schröder-Turk, G.E., Mecke, K., and Bechinger, C. (2012). Permeability of porous materials determined from the Euler characteristic. Physical Review Letters , 109, 264504.
- Vogel, H.-J., and Roth, K. (2001). Quantitative morphology and network representation of soil pore structure. Advances in Water Resources , 24, 233–242.
Friday, February 6, 2026
The Hydrology 2026 Lab
The lab component makes up nearly half of the course, following the motto:
"Learning by doing."
Throughout the lab, you will conduct at least three key numerical experiments:
- Time Series Analysis – Exploring various data elaborations with various Jupyter Notebooks and a little of Python
- Intensity-Duration-Frequency (IDF) Curves – Estimating rainfall intensity over different time scales with various Jupyter Notebooks and a little of Python, as well
- Infiltration Experiments – Investigating soil absorption dynamics using the WHETGEO system
- Evaporation & Transpiration Experiments – Understanding water loss processes in different conditions using the GEOSPACE ssytem
Resources
🔹[Vimeo Showcase – General Lab Videos]
🔹[OSF Repository – Lab Materials]
📌 Detailed videos and materials for each experiment are listed below.
📅 26 February 2026 – Installation and Introduction to working with Jupyter and Notebooks
- A little introduction to Jupyter Notebook by Concetta D'Amato (YouTube2020, Data).
- A little of introduction
- Elementary operations
- Vimeo2025
- Vimeo2024 part1
- Vimeo2024 part2
📅 6 March 2026 Reading a file
📅 9 March 2026 How to read and plot data
- Reading an Excel File
- Reading a timeseries of daily rainfall organized in a Table whose rows contains the days and columns different years
- Data (Daily Precipitation od San Martino di Castrozza from 1921 to 1990)
- Notebooks:
- Same thing as above (Vimeo2022)
- Chat GPT use suggestions (Vimeo2024)
- Other ChatGPT suggestions (Vimeo2024)
- Grouping the data by year (Vimeo 2022)
- A better plot of the time series
- Monthly mean precipitation (Vimeo2022)
📅 12 March 2026 - San Martino Reprise
- Same data as above but with some modified notebooks
Interpolating the Gumbel distribution to annual precipitation maxima
- with using the mean and the variance (Notebook, Vimeo2026, Vimeo2025, Vimeo2024, Vimeo2023)
- with the maximum likelihood (Notebook, Vimeo2026, Vimeo2025 I, Vimeo2025 II, Vimeo2024, Vimeo2023)
- with the minimum square methods (Notebook, Vimeo2026, Vimeo2025 ,Vimeo2024, Vimeo2023)
- Il test di Pearson (Notebook, Vimeo2026, Vimeo2025, Vimeo2024, Vimeo 2023)
- Finally interpolating the curves (Notebook, Vimeo2025, Vimeo2024, Vimeo2023)
Where to find the data
- Finding the data for the Time Series Analysis (Vimeo2025)
📅 13 April 2026
Below, you'll discover the resources for the second part of the lab, which encompass instructions for utilizing Object Modelling System version 3 (OMS3) models such as WHETGEO1D and GEOET. Within the OMS Projects, you'll find a directory named Jupyter_Notebook, housing sets of notebooks designed to guide you through handling both input and output data for these programs.
OMS and further Anaconda Installations
A general introduction to some OMS3 concepts
- A Short introduction to OMS3 (Vimeo2026, Vimeo2025, Vimeo2024)
- The Working Environment (Vimeo2026, Vimeo2025, Vimeo2024)
📅 16 April 2026
- Executing OMS from Jupyter Notebooks
- Tutorial installation Geoframe-Vincennes (Vimeo2026)
- Hopefully obsolete:
- The OMS3 Console
- Download Utilities For the Java searching Notebook
- Additional instructions to run the console (Vimeo2025 I , Vimeo2025 II)
📅 28 April 2026
Introduction to Infiltration with WHETGEO 1D. WHETGEO specifically focuses on simulating the coupled processes of water and heat transport, which are crucial for understanding: Soil moisture dynamics, Evapotranspiration processes, Energy balance at the land surface, Groundwater-surface water interactions, Snow and ice processes (where temperature is critical)
- Material
- Executing Hello World (again)
- Creation of the WHETGEO - 1D Grid (Vimeo2026, Vimeo2025, Vimeo2024)
- Time Series Formatter (Vimeo2024)
- The Simulation File (Vimeo2026, Vimeo2025, Vimeo2024)
- Instructions for Infiltration Exercise (Vimeo2026-1, Vimeo2026-2)
- The WHETGEO - 1D sim file explained (from minute 5 to 24 ca, Vimeo2024)
- Ways to visualize the output (Vimeo2026, Vimeo2025, Vimeo2024)
- Summary of the instructions (Vimeo2025,Vimeo2024)
- Building a WHETGEO simulation from the scratch (Vimeo 2022)
- A reference only for the brave: Tubini and Rigon, GMD 2022
- An example of project by Lorenzo Dal Sasso
- A summary of the action to do for running a WHETGEO simulation (by Marianna Tavonatti)
📅 21 May 2026 - Radiation estimation
is preliminary to any Evapotranspiration estimation since our models use it as input. The estimation of radiation, as it results from the theory, needs a few GIS operations to determine the sky view factor and the aspect of the terrain in the point of interest. These operation for the present exercise are skipped and pre-analyzed data are provided.
- Radiation project to download
- Radiation calculation (Vimeo2026 part I, Vimeo2026 part II Vimeo2024)
- Radiation calculation with Clearness (Vimeo2026, Vimeo2024)
📅 4 June 2026 - Estimation of Evapotranspiration
Please below find Notebooks and video related to the estimation of radiation. Theradiationin this class, will be functional to the estimation of evapotranspiration with the Priestley-Taylor and Penman-FAO models,
Introduction to the estimation of evapotranspiration
Musical Coda
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