A double failure is a giant failure and probably a sin of pride.I see now that I was wrong/ it was hubris all along

I've shared various posts about the STRADIVARI project over recent months. Now that both versions, submitted to the ERC Advanced Grant and FIS3 programs, were not selected for funding, I feel free to upload them to my collection of unsuccessful proposals. The panel evaluations, when they arrive, will provide valuable learning opportunities.

I'm fully aware these projects were ambitious, perhaps overly so. I sinned of pride. Yet I would be far more troubled if the panels judged them as "incremental" science, which would reveal a superficial analysis. While science advances steadily, hydrological science in particular suffers from persistent methodological weaknesses in model construction that prevent us from addressing fundamental questions properly. Too many simulation-based papers rely on flawed algorithms, imprecise conceptualizations, vague process descriptions, equations applied beyond their validity ranges,  missing feedbacks between coupled processes, imprecise and not well controlled data inputs (especially when going global).

The root cause lies in how models are conceptualized and built. Until our community recognizes this fundamental issue, we cannot progress toward answering the discipline's core questions. The current modeling paradigm perpetuates a cycle where narrative sophistication masks conceptual and  physical/mathematical  inadequacy.

When one attempts serious methodological critique, it's too easily dismissed as contrarianism, nastiness, or the rant of an angry dog rather than engaged with substantively. Flawed or myopic established approaches routinely appear in top journals without appropriate analysis, often relying on cherry-picked validation through calibration exercises. Let's be clear: many models do not work properly, and their results lack accountability. They are built on fragile foundations and superficial verification, yet their publication convinces project reviewers that fundamental problems have been solved when they haven't. This reaches its apex in global models where hyperresolution is married with hyperignorance (Beven et al., 2015), and in ancillary sciences studying the "Total Environment," where results on water quantities, fluxes and quality are presented as established fact when they remain deeply uncertain. 

To Cosma Shalizi's sharp observation (adapted to our context): the public discussion of hydrological models and processes often becomes "polluted by maniacal cultists with obscure ties to decadent plutocrats." I am obviously kidding. However, today there are many people  ".. who think they can go from the definition of conditional probability, via Harry Potter fanfic, to prophesying that an AI god will judge the quick and the dead, and condemn those who hindered the coming of the Last Day to the everlasting simulated-but-still-painful fire." I fight against this but I am, maybe, obsolete. 

This resistance and blindness to foundational critique, choosing institutional legacy over rational evaluation, ultimately impedes scientific progress. The STRADIVARI projects represented an attempt to break this cycle through component-based, rigorously coupled Earth system modeling, allowing hypothesis testing and cooperative work. Whether deemed too ambitious or not, the underlying questions it raised remain urgent and unresolved (and I am not the only one to write it. Voices are many).

Here the projects, useful to learn how to write a project differently:

• FIS3 project
• ERC PART 1
• ERC PART 2
• Timeline

Oh Notation ! About notation in Transit Time literature

What written here is nothing new. Is just what already present in previous papers and post, just said in a different manner to put a bridge among what people write in literature and what we think should be written.

 Time Variables and Their Definitions

The foundation of travel time theory rests on two fundamental time variables: $T$ (transit time) and $t$ (actual or clock time). Transit time is defined as the time a water parcel takes to cross a control volume or domain, expressed mathematically as $T = t_{ex} - t_i$, where $t_{ex}$ is the exit time and $t_i$ is the injection (entry) time.

If we observe the system at time $t$ and have measured or estimated the transit time $T$, we are effectively positioned at the outlet of the domain, considering water parcels exiting at that moment. These exiting parcels entered the domain at various previous times $t_i$, extending theoretically back to $-\infty$, meaning $T$ ranges from 0 (for parcels entering at time $t$ and exiting instantaneously) to $\infty$ (for parcels that entered in the distant past). While this represents an idealization—since the distant past is practically unknowable—it provides a useful theoretical framework. Please also observe that the injection time $t_i$ is a discrete variable, since the rainfall happens at discrete time steps and the same happens to any transported material, even if it is usually approximated as a continuous one. (Note:We cannot  distinguish ages from data at a very fine resolution. Therefore blurring the concept could not be wrong).  

The Transit Time Distribution

Transit time follows a probability distribution that can be conceptualized as the distribution of a random variable $T$. This distribution is time-dependent and conditional on the observation time $t$. The proper notation is $p_Q(T|t)$, though the literature often uses $p_Q(T,t)$, which can be misleading as it suggests a bivariate distribution when it's actually a conditional one. Some authors use $\overset{\leftarrow}{p}_Q(T,t)$ or place a backward arrow above $p$ to indicate that this probability refers to past events—hence the term "backward probability" or "backward transit time distribution."

As noted in Benettin et al. (2022)'s comprehensive review on transit time estimation and your previous AboutHydrology posts on residence time approaches, this backward-looking perspective is crucial for understanding catchment memory and the age composition of streamflow. In literature $p_Q(T|t)$ is treated as a continuous variable, but for the reason $t_i$ is a discrete variable, it is a discrete time distribution.  

The Complexity of Transit Time as a Variable

Using $T$ as the primary variable introduces conceptual complexity. Since $T = t - t_i$ with $t$ being the observation time (the conditioning variable), $T$ inherently depends on both the current time and the injection time $t_i$. The injection time $t_i$ is the true independent variable in this framework.

This distinction has important mathematical consequences: when expressed in terms of $t_i$, the mass conservation law remains an ordinary differential equation, but when formulated using $T$, it becomes a partial differential equation—a transformation that significantly complicates the mathematical treatment, as discussed in Botter et al.'s (2011) work on the master equation.

Residence Time Distribution

A distinct but related concept is the residence time distribution, denoted as $p_S(T_r|t)$, where the residence time at observation time $t$ is $T_r = t - t_i$. Unlike transit time, residence time considers all water parcels currently within the domain, not just those exiting. We assume we can label parcels by their age (their injection time $t_i$).

This distribution also looks backward from the current time $t$ and is often represented in literature as $p_S(T,t)$ or $\overset{\leftarrow}{p}_S(T,t)$, potentially causing confusion with transit time notation. Crucially, this distribution characterizes the age composition of water currently stored in the domain at time $t$—it makes no predictions about the future but provides a snapshot of the past up to the present moment.

The Link Between Distributions: StorAge Selection Functions

While $p_Q$ refers to water exiting the domain and $p_S$ refers to water stored within it, these distributions are naturally related—what exits must have been stored. The relationship between these probabilities is mediated by the StorAge Selection (SAS) function, denoted as $\omega(T,t)$, which describes how water of different ages is preferentially selected for discharge.

The SAS framework, as elaborated in recent work including studies on the contribution of groundwater to catchment travel time distributions, provides:

$$p_Q(T|t) = \omega(T,t) \cdot p_S(T|t)$$

The Special Case of Complete Mixing

There exists a special case where residence time and transit time distributions coincide: when parcels exiting the domain are uniformly sampled from the population of ages within the domain. This represents the "complete mixing" or "random sampling" assumption, where $\omega(T,t) = 1$ for all $T$.

Under this condition:

• Transit time and residence time collapse in a unique variable and the distributions become equal: $p_Q(T|t) = p_S(T|t)$
• All mass conservation equations simplify and become linear in the probability distributions
• The system behavior resembles that of a well-mixed reactor

As discussed in my posts about celerity versus velocity and the travel time problem, this simplification, while mathematically convenient, rarely holds in real catchments where preferential flow paths and incomplete mixing dominate the hydrological response.

Here's the complete reference list including your own AboutHydrology posts on the topic:

References

Published Literature

Benettin, P., Rodriguez, N. B., Sprenger, M., Kim, M., Klaus, J., Harman, C. J., van der Velde, Y., et al. (2022). Transit Time Estimation in Catchments: Recent Developments and Future Directions. Water Resources Research, 58(11). https://doi.org/10.1029/2022wr033096

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

Botter, G., Bertuzzo, E., & Rinaldo, A. (2011). Catchment residence and travel time distributions: The master equation. Geophysical Research Letters, 38, L11403. doi:10.1029/2011GL047666

Botter, G. (2012). Catchment mixing processes and travel time distributions. Water Resources Research, 48, W05545. doi:10.1029/2011WR011160

Comola, F., Schaefli, B., Rinaldo, A., & Lehning, M. (2015). Thermodynamics in the hydrologic response: Travel time formulation and application to Alpine catchments. Water Resources Research, 51(2), 1671-1687. doi:10.1002/2014WR016228

Cornaton, F., & Perrochet, P. (2006). Groundwater age, life expectancy and transit time distributions in advective-dispersive systems: 1. Generalized reservoir theory. Advances in Water Resources, 29(9), 1267-1291. doi:10.1016/j.advwatres.2005.10.009

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

McGuire, K. J., & McDonnell, J. J. (2006). A review and evaluation of catchment transit time modelling. Journal of Hydrology, 330, 543-563.

Niemi, A. J. (1977). Residence time distribution of variable flow processes. International Journal of Applied Radiation and Isotopes, 28, 855-860.

Rigon, Riccardo, Marialaura Bancheri, and Timothy R. Green. 2016. “Age-Ranked Hydrological Budgets and a Travel Time Description of Catchment Hydrology.” Hydrology and Earth System Sciences 20 (12): 4929–47. https://doi.org/10.5194/hess-20-4929-2016.

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.

Rinaldo, A., & Rodriguez-Iturbe, I. (1996). Geomorphological theory of the hydrologic response. Hydrological Processes, 10, 803-829.

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

van der Velde, Y., Torfs, P. J. J. F., van der Zee, S. E. A. T. M., & Uijlenhoet, R. (2012). Quantifying catchment-scale mixing and its effect on time-varying travel time distributions. Water Resources Research, 48, W06536. doi:10.1029/2011WR011310

AboutHydrology Blog Posts

Rigon, R. (2014). Residence time approaches to the hydrological budgets. AboutHydrology Blog. http://abouthydrology.blogspot.com/2014/06/residence-time-approaches-to.html

Rigon, R. (2016). Celerity versus velocity and the travel time problem. AboutHydrology Blog. http://abouthydrology.blogspot.com/2016/06/celerity-vs-velocity.html

Rigon, R. (2023). A Commentary on transit (travel) times theory. AboutHydrology Blog. http://abouthydrology.blogspot.com/2023/01/a-commentary-on-transit-travel-times.html

Rigon, R. (2025). Transit Time and Residence Time Distributions: Fundamental Time Variables in Catchment Hydrology. AboutHydrology Blog. [This post]

Alps Are Losing Snow

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.

This study provides the first comprehensive long-term (1991-2021) analysis of snow water equivalent changes in the Po River District, Italy, one of Europe’s second most climate sensitive regions. Our findings show stark elevation-dependent changes in snow water storage and duration with profound and immediate implications for water security and climate adaptation.

Using a high-resolution (500m, daily) dataset from 1991-2021 (Dall'Amico et al., 2025), we observed two primary findings: First, we observed a profound loss of snow volume and decrease in duration below 2000 meters, with some low-elevation bands losing over 30% of their total snow-water storage. In contrast, high-elevation zones (>2500 m) are experiencing increased accumulation, but a continued shortened snow season. However, the increase in snow water storage at high elevations requires careful interpretation due to methodological constraints and systematic overestimation of high elevation precipitation by ERA5. Second, we show that this shorter snow season is not just an artifact of earlier spring melt, but is primarily driven by a delayed onset of snow accumulation in early winter. 

These elevation-dependent changes and loss of the seasonal snowpack highlight a fundamental shift in the hydrological regime of the Po River Basin, with significant implications for the timing and volume of runoff and the future availability of water in the region.Therefore, the Po River Basin is moving from a stable to a more volatile system.

Note: To know more, click on the picture (study area) above to read our preprint.

References:

Dall’Amico, M., Tasin, S., Di Paolo, F. et al. 30-years (1991-2021) Snow Water Equivalent Dataset in the Po River District, Italy. Sci Data 12, 374 (2025). https://doi.org/10.1038/s41597-025-04633-5

Roots2025 - A presentation of the GEOSPACE system

This is the presentation I am giving at the Roots2025 event . It talks about the GEOSPACE infrastructured to study the soil-plant-atmosphere interactions.  Being a very compressed presentation I cannot go to all the details which are better grasped by reading the references below or browsing the various contributions that can be found in this blog under the keyword  GEOSPACE. 

GEOSPACE infrastructure is very modular and its peculiarity is that it is based on "components" that are joined with a scripting language just before being executed. The system managing such components is OMS3.  GEOSPACE integrated two big subprojects, WHETGEO the subsystem that deals with soil and infiltration and GEOET the system that contains various solutions for estimating evaporation and transpiration. To get the slides, clik on the figure above. To get the code look at the Gitub repository. Video lectures (here) or to a certain extent here. To get more information read the following references.

References

D’Amato, Concetta. n.d. “Exploring the Soil-Plant-Atmosphere Continuum: Advancements, Integrated Modeling and Ecohydrological Insights.” Ph.D., Università di Trento.

D’Amato, Concetta, and Riccardo Rigon. 2025. “Elementary Mathematics Helps to Shed Light on the Transpiration Budget under Water Stress.” Ecohydrology: Ecosystems, Land and Water Process Interactions, Ecohydrogeomorphology 18 (2). https://doi.org/10.1002/eco.70009.

D’Amato, Concetta, Niccolò Tubini, and Riccardo Rigon. 2025. “A Component-Based Modular Treatment of the Soil–Plant–Atmosphere Continuum: The GEOSPACE Framework (v.1.2.9).” Geoscientific Model Development 18 (20): 7321–55. https://doi.org/10.5194/gmd-18-7321-2025.

Tubini, Niccolò, and Riccardo Rigon. 2022. “Implementing the Water, HEat and Transport Model in GEOframe (WHETGEO-1D v.1.0): Algorithms, Informatics, Design Patterns, Open Science Features, and 1D Deployment.” Geoscientific Model Development 15 (1): 75–104. https://doi.org/10.5194/gmd-15-75-2022.

Bonus Reference (unpublished so far)

Tubini, N., and R. Rigon. n.d. “WHETGEO-2D: A Framework to Solve 2D Partial Differential Equation Domain within GEOframe System. The Richardson-Richards Equation.” http://abouthydrology.blogspot.com/2022/08/whetgeo-2d-open-source-tool-fo-solving.html.

Research in Arctic Permafrost. A new start

In the rapidly changing Arctic, understanding permafrost behavior is critical for infrastructure, ecosystems, and climate science. Marianna Tavonatti's master's thesis at the University of Trento has delivered some achievements that points towards the advance our understanding of Canadian Arctic permafrost dynamics and provide essential tools for climate adaptation.

Permafrost—permanently frozen ground—covers 24% of the Northern Hemisphere and is rapidly thawing due to climate change. This thesis focused on the Canadian Arctic near the Inuvik-Tuktoyaktuk Highway, using advanced computer modeling to understand how permafrost responds to changing temperatures over time scales from decades to seasons. 

Permafrost - https://www.maggiebaylor.com/permafrost

Marianna Thesis covers: 

1. GEOtop Model Implementation: First comprehensive application of the GEOtop model for Canadian Arctic permafrost, successfully validated against real ground temperature data.

2. Historical Analysis: 73-year simulation (1950-2023) revealing clear evidence of accelerating permafrost warming and active layer deepening.

3. Future Projections: Advanced climate scenarios showing significant future changes in permafrost stability with direct implications for infrastructure planning.

4. Advanced Theoretical Framework: Enhanced understanding of frozen soil physics, improving how we model phase changes in complex soil systems.

5. Methodological Innovation: Created an integrated GlobSim-GEOtop modeling chain applicable to Arctic regions worldwide.

6. Practical Applications: Provided quantitative data essential for Arctic infrastructure design and climate adaptation strategies.

7. Scientific Contributions: Advanced climate change understanding with implications for carbon cycling and global climate feedbacks.

However, the best things is to read the thesis that you can get by clicking on the figure. 

Marianna's work establishes a foundation for future advancement in Arctic permafrost science. The research identifies specific opportunities for enhanced spatial modeling, improved climate projections, and expanded ecosystem coupling—providing a clear roadmap for continued innovation in this critical field.

As the Arctic continues to experience rapid environmental change, research like Marianna's becomes increasingly vital for understanding system responses and supporting sustainable development in one of Earth's most climate-sensitive regions. 

About erosion and hillslope evolution: a lost master thesis rediscovered

 This 2003/2004 Master thesis by Martina Brotto deserves some more visibility. It presents an exploration into how slopes evolve over time through erosion processes and tackles one of the fundamental challenges in geomorphology: understanding and predicting how hillsides change their shape through the complex interplay of soil production and erosion.

What makes this research particularly interesting is its departure from traditional landscape evolution models. While most existing models assume that erosion is limited only by the transport capacity of flowing water or wind (what scientists call "transport-limited" processes), Brotto introduces a more realistic approach. Her model considers that erosion can also be limited by how quickly rock breaks down into soil and how much material is actually available to be moved ("detachment-limited" processes). This distinction might seem technical, but it's crucial for understanding real-world erosion, especially in areas where bedrock is close to the surface or where soil production rates are slow.

The heart of the research lies in developing two complementary mathematical models. The first focuses on diffusive erosion processes – the slow, continuous movement of soil particles down slopes through countless small disturbances like frost action, animal activity, and the impact of raindrops. Using sophisticated numerical techniques including the conjugate gradient method, Brotto shows how these processes gradually smooth out irregularities in the landscape, creating the gentle, rounded hills we often see in nature. Her simulations, some extending over 20,000 years of landscape evolution, reveal how factors like the initial slope angle and the rate of soil creep influence the final landform shape.

Beyond this introduction is the good thing to do is to read the thesis. You can find it by clicking on the Figure above (one beautiful painting by Vincent Van Gogh). 

Giulia Merler using GEOSPACE for simulations on a wineyard

Giulia Merler's Bachelor thesis investigates the effects of climate change on Trentino vineyards using GEOSPACE simulation software. While this research is preliminary and requires further validation, it reveals compelling insights into how global warming affects vine health.

For a high-resolution view of the poster, simply click on the image below. All supporting materials and data are accessible via the QR code provided.

The findings may surprise those unfamiliar with viticulture: elevated temperatures place significant stress on grapevines, posing a serious threat to wine production. While experts in the field may find this unsurprising, seeing the impact quantified through sophisticated modeling tools brings the reality into sharper focus and provides valuable data for future planning. This work pairs with the one by Marco Feltrin that can be found here.