Uncertain quantities are often described through statistical samples. Can samples for numerical weather forecasts be generated dynamically? At a great expense, they can. With statistically constrained perturbations, a cloud of initial states are created and then integrated forward in time. By now, this technique has become ubiquitous in weather and climate research and operations. Ensembles are widely used, with demonstrated value.

The atmosphere evolves in a multidimensional phase space. Does a cloud of ensemble solutions encompass the evolution of the real atmosphere? Theoretically, random perturbations in high dimensional spaces have negligible projection on any direction, including the error in the best estimate, therefore consistently degrading it. As the bulk of the perturbation variance lies in the null-space of error, samples in multidimensional space do not contain reality.

An evaluation suggests that initial and short-range forecast error and ensemble perturbations are random draws from a high dimensional domain we call the subspace of possible error. Error in any initial condition is a result of stochastic observational and assimilation noise, while perturbations explore other, mostly independent directions from the subspace of possible error that may have resulted from other configurations of stochastic noise. What benefits may arise from the deterministic projection of such noise?

How to cite: Toth, Z., Feng, J., Zhang, J., and Pena, M.: A Foray of Dynamics into the Realm of Statistics: A Review of Ensemble Forecasting, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-6791, https://doi.org/10.5194/egusphere-egu24-6791, 2024.

How to cite: Toth, Z. and Jankov, I.: Initial-Value vs. Model-Induced Forecast Errors: A New Perspective, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-4179, https://doi.org/10.5194/egusphere-egu24-4179, 2024.

Heatwaves represent a major health hazard, as was the case for the 2003 summer heat wave, responsible for more than 70 000 deaths in western Europe. The interplay of mean flow, quasi-periodic and random fluctuations — associated with the westerly jet, Rossby and gravity waves, and eddies — of the large-scale temperature field results in complex heat wave–related amospheric conditions. Understanding how the different physical processes interact is thus crucial for prediction of heat wave events. 
In this study, we use the triple decomposition of turbulent flow (Hussain and Reynolds, JFM, 1972) to compute mean, quasi-periodic and random energy fluctuations of the temperature field, that is the 1-point energy budget of temperatures. This decomposition takes into account all interactions between the zonal jet, Rossby waves, gravity waves, and eddies. Both spectral and dynamical systems analyses are applied to the computed terms. More specifically, the concept of extremal length (Ahlfors, Vol. 371, AMS, 2010) is integrated into the equations to quantify how each term of the energy budget equations contributes to the "trapping" of temperature anomalies over Europe.      
Results show that, amid positive sea surface temperatures and negative soil moisture anomalies, during the first half of August, i.e., the hottest days of the heat wave, quasi-periodic oscillations of polar air increased, resulting in meridional migration of cold air over Canada, and subsequent mixing with warmer air coming from North America. This mixing triggered baroclinic instabilities that led to production of turbulent eddies, which by August 5th suddenly stopped their eastward progression, creating a cyclonically stalled regime over the Mid-Atlantic; this stationary cyclone interacted positively with North African warm air propagating northward over Europe, thus sustaining the dry conditions over France and much of Western Europe. The cyclonic block finally disappeared, stopping the warm air advection from North Africa, with temperatures falling just after that. 
The study reveals that interactions between quasi-periodic and random processes of production, diffusion and dissipation of a scalar field’s energy play an important role in the evolution of a major heatwave. Hence, the 2003 event was not just the result of a superposition of Rossby waves and eddy anomalies. Extremal length analysis thus reveals that the zonal advection of temperature anomalies was blocked by interactions between quasi-periodic and random production and diffusion processes. 
This study highlights the complex turbulent interactions that lead to major heat wave events, and the fact that each such event is thus unique.

How to cite: Fossa, M., Danaila, L., and Ghil, M.: Temperature at 500hPa, energy budget and the 2003 summer heatwave over Western Europe: a triple-decomposition approach, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-17846, https://doi.org/10.5194/egusphere-egu24-17846, 2024.

The Arctic and Antarctic sea ice covers form key components of Earth's climate system. Their precipitous declines are impacting the polar marine environment and its ecosystems, with ripple effects felt far beyond the polar regions. As a material sea ice exhibits composite structure on many length scales. A principal challenge is how to use information on small scale structure to find the effective or homogenized properties on larger scales relevant to climate and ecological models. From tiny brine inclusions to rich ice pack dynamics on oceanic scales, and from microbes to polar bears, we'll consider recent advances in modeling sea ice and the ecosystems it hosts. In the spirit of MPE 2013, we’ll focus on the broad range of mathematics and physics being used. Percolation theory and statistical physics, fractal geometry, spectral analysis and random matrix theory, advection diffusion processes, topological data analysis, and uncertainty quantification for dynamical systems will arise naturally in considering various sea ice structures and organisms. This work is helping to advance how sea ice is represented in climate models, and to improve projections of the fate of Earth’s sea ice packs and the ecosystems they support.

How to cite: Golden, K.: Mathematics of Sea Ice and its Ecosystems, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-14101, https://doi.org/10.5194/egusphere-egu24-14101, 2024.

The Atlantic Meridional Overturning Circulation (AMOC) is considered a tipping element of the earth system featuring bistability: for a given external forcing, a strong and a weak circulation state coexist as competing attracting states of the system. In the presence of random fluctuations, noise-induced transitions between the competing states are possible, posing a risk of abrupt AMOC tipping even without crossing a critical forcing threshold. It is thus crucial to better understand the stability landscape of the earth system with a multistable AMOC, particularly the properties of the boundary separating the basins of attraction of the strong and weak AMOC states. For weak noise, transitions are expected to cross the basin boundary at so-called edge states or "Melancholia states", typically chaotic saddles which are attracting on the boundary but asymptotically unstable. Here we find an edge state between the two stable AMOC states in an earth system model of intermediate complexity, PlaSim-LSG. Our approach is based on an edge-tracking technique that allows to construct a pseudo-trajectory on the chaotic saddle. We characterize the climatic and dynamical properties of this edge state and map out its location in different projections of state space. Near the edge state, the AMOC strength exhibits strong transient oscillations which we link to the ongoing physical processes. We relate our findings to the theory of unstable chaotic sets and discuss implications for the predictability of potential AMOC tipping in the future.

How to cite: Börner, R., Mehling, O., von Hardenberg, J., and Lucarini, V.: Oscillatory Melancholia state of the Atlantic Meridional Overturning Circulation in an intermediate-complexity climate model, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-18006, https://doi.org/10.5194/egusphere-egu24-18006, 2024.

There is growing concern that various large-scale elements of the Earth system may undergo catastrophic transitions under future climate change. But already at present-day conditions there is a risk of spontaneous transitions to an undesired state induced by stochastic fluctuations, given that any of those elements occupy a multi-stable regime. For many climate sub-systems this potential present-day multi-stability is still uncertain. For instance, it cannot be ruled out that there is a regime of a collapsed Atlantic Meridional Overturning Circulation (AMOC) that is stable under present-day conditions. Assuming such an undesired stable state exists, there also exists an additional unstable state called the edge state. This state anchors the basin boundary separating the desired and undesired regimes, and it lies at the heart of the path taken by the system during a noise-induced transition between the two stable states.

In this work such an edge state lying between the stable regimes of a vigorous and a collapsed AMOC is computed for the first time in a global ocean model using an edge tracking algorithm. The physical characteristics that set this state apart from the usually observed stable regimes are analyzed. This can be useful to detect if a spontaneous collapse of the AMOC induced by stochastic climate variability is underway, or to detect so-called rate-induced tipping. It may be especially helpful if the system is close to the tipping point where the desired state loses stability. Here the desired stable state and the edge state become increasingly similar, but a transition towards the undesired state may nevertheless be detected early-on if specific signatures of the edge state are recognized.

How to cite: Lohmann, J.: Characterization of Edge States as Gateway to a Collapse of the Atlantic Ocean Circulation, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-18809, https://doi.org/10.5194/egusphere-egu24-18809, 2024.

Low-order climate models have played an important role in understanding the low-frequency variability of the atmospheric circulation and how it can be affected by trends in anthropogenic forcing. A simple quasi-geostrophic model of the midlatitudes’ circulation (Lorenz, Tellus, 1984, 1990) is studied from the perspective of the theory of nonautonomous dynamical systems (NDS: e.g., Ghil et al., Physica D, 2008).

We start with a study of the model’s behavior in the absence of time dependent forcing and determine in this case its steady states. A bifurcation analysis is carried out in order to identify distinct regime behavior types – stationary, periodic and chaotic – in the model’s parameter space. Next, we study the nonautonomous system with a meridional temperature gradient that varies seasonally, according to changes in insolation. The snapshot attractor (Tel et al., JSP, 2020) of the seasonally forced model is compared with the standard forward attractor of the autonomous model for two distinct epochs of the year, at peak summer and peak winter. In both cases, the effects of the change in forcing are reflected in a clear change of shape of the attractor. Predictability is lost in both cases: the summer attractor loses its periodicity when the forcing is seasonal. The winter one favors energy transport through one of the two wave components included in the model. For the same value of the forcing, the structure of the attractor in the autonomous case is quite different from that in the nonautonomous one.

Finally, the meridional forcing is subjected to climate trends, both positive and negative, since the jet intensity changes in opposite directions at low and high altitudes (Lee et al., Nature, 2019). The analysis of the snapshot attractor of the system under climate trends suggests that the model does not follow the geostrophic assumption in certain ranges of the forcing, as the average zonal flow does not always show the expected dependence on the equator-to-pole temperature contrast. On the other hand, the energy transported by the eddies does follow the sign of the climatic trend. Overall, distinct effects are observed. Chaotic behavior can be completely suppressed in favor of a regularly periodic one and vice-versa. At the same time, circulation patterns can change, suddenly disappear, and be restored.

In general, the snapshot attractor proved to be a robust tool in studying the internal variability of the midlatitude circulation, as well as the changes arising in it from anthropogenic forcing trends. The distinct regimes of behavior are being examined more closely by advanced spectral analysis methods (Ghil et al., Rev. Geophys., 2002) to better understand the effects of climate trends on low-frequency variability.

How to cite: Maraldi, B., Dijkstra, H., and Ghil, M.: Intraseasonal atmospheric variability under climate trends, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-21689, https://doi.org/10.5194/egusphere-egu24-21689, 2024.

Linking free and forced variability is one of the key challenges in climate science. As the climate system is an out of equilibrium one, the standard application of the fluctuation-dissipation theorem is out of scope. It has been shown in the past that it is possible to construct response operators that can be used to perform climate change projections using a more general formulation of response theory for nonequilibrium systems. Nonetheless, such operators lack the key property of interpretability: one cannot separate the contribution to the total response coming from different modes of natural variability of the system. We show here in a few low-dimensional models how this issue can be taken care of by taking advantage of the Koopman formalism. One can then write the response operator as a sum of terms each associated with a specific mode of variability. The obtained results also shed light on previous findings by Hasselmann and colleagues and on recently proposed data-driven methods aimed at deriving response operators from data.

How to cite: Moroney, J., Lucarini, V., and Zagli, N.: Building Response Operators Using Koopman Formalism: the Link between Free and Forced Variability, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-18089, https://doi.org/10.5194/egusphere-egu24-18089, 2024.

The introduction of random perturbations by noise in partial differential equations has proven extremely useful to understand more about long-time behaviour in complex systems like atmosphere and ocean dynamics or global temperature. Considering additional transport by noise in fluid models has been shown to induce convergence to stationary solutions with enhanced dissipation, under specific conditions. On the other hand the presence of simple additive forcing by noise helps to find a stationary distribution (invariant measure) for the system and understand how this distribution changes with respect to changes in model parameters (response theory). I will discuss these approaches with a multi-layer quasi-geostrophic model as example.

How to cite: Carigi, G.: Long-time behaviour of stochastic geophysical fluid dynamics models., EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-1133, https://doi.org/10.5194/egusphere-egu24-1133, 2024.

Many physical, chemical, financial, and ecological processes show the presence of a threshold, which may affect their dynamics. For instance, chemical reactions occur when the energy reaches the activation value; for insurance companies and banks, ruin may happen when the balance drops down a minimum amount; in fluvial hydraulics, sediment transport, and morphodynamic processes start when bed shear stresses (i.e., flow discharge) overcome a critical threshold. Other processes may show the presence of additional boundaries. For stochastic processes, whose dynamics depends on the frequency and magnitude of random fluctuations, it is interesting to know the average time the process takes to reach one of the critical boundaries starting from a known value. This quantity is known in the literature as Mean First Passage Times (MFPTs). The quantification of the MFPTs is usually performed by considering one threshold, only. When two or multiple thresholds are present, one may consider the MFPTs of reaching either one of the thresholds, without having passed the other ones. Such a selective condition is referred to in the literature as splitting probability. In this work, we consider stochastic processes governed by a typical Langevin equation with deterministic drift and random instantaneous jumps (white shot noise). We perform a statistical-trajectory-analysis starting from a point between two thresholds and derive exact relationships of the splitting probabilities and the MFPTs of crossing one threshold, only, based on process-dependent dimensionless parameters. Such formulations are then explicitly given for the cases of constant and linear drift functions and both positive and negative jumps. We test the derived formulations against data from MonteCarlo simulations, by varying the process parameters, the starting point, and the values of the thresholds. The comparison shows very good agreement and confirms the correctness of the derived relationships. Additionally, the analysis highlights the role played by the dimensionless parameters. Then, data from flow measurements in a river are considered and we successfully test the formulations against the duration of the raising limb of high-stage events. The derived formulations can be readily applied to calculate both the duration of the raising and the falling limbs of flow events, which are important quantities for engineering applications, as well as for modeling purposes in river eco-morphodynamics.

How to cite: Calvani, G. and Perona, P.: The selective boundary crossing in stochastic processes driven by white shot noise, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-17713, https://doi.org/10.5194/egusphere-egu24-17713, 2024.

How to cite: Toppaladoddi, S. and Wells, A.: A stochastic model for the turbulent ocean heat flux under Arctic sea ice , EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-8590, https://doi.org/10.5194/egusphere-egu24-8590, 2024.

In this study, we introduce a variety of ocean eddy parameterizations and discuss how they affect the representation of mesoscale turbulence in ocean models. They ultimately all aim at reducing overdissipation at eddy-permitting resolutions by utilizing the inverse energy cascade and energy conversions between potential and kinetic energy.

Mesoscale eddies play a crucial role in the global oceans. They transport tracers and heat, cascade energy across scales and interact with the mean currents and the atmosphere. However, their representation at resolutions close to the Rossby radius of deformation is insufficient. Such eddy-permitting, i.e. barely eddy resolving grids are still commonly applied for decadal climate simulations and will remain state-of-the-art at high latitudes for years to come. These simulations generally suffer from an excessive dissipation of kinetic energy, leading to reduced eddy variability, eddy formation and eddy-mean flow interactions.

Reducing overdissipation via optimized viscous closures is one way forward. Another option is to reinject some of the overdissipated energy back into the resolved flow via so called kinetic energy backscatter parameterizations. We will investigate different methods how to complement our own viscous and backscatter schemes with stochastic components, to account for unresolved chaotic variations of dissipative processes and for scale interactions across the resolution limit. For this purpose, we use data informed approaches such as linear inverse models to generate stochastic patterns based on high resolution reference simulations of idealized channel and double gyre configurations. Our results show that incorporation of such schemes can help to substantially improve the kinetic energy and mean flow characteristics. Furthermore, the varied application of the noise can reveal pathways of energy conversion between potential and kinetic energy, shedding light on the simulated energy cascades at such model resolutions. Aside from the learned construction of the stochastic patterns based on high resolution data, these new schemes come at a small additional computational cost, especially compared to higher resolution simulations. When tuned with caution, they provide a means to incorporate model uncertainty and to reduce systematic biases in ocean models.

How to cite: Juricke, S., Bagaeva, E., Danilov, S., Franzke, C., and Oliver, M.: Ocean eddy parameterizations: Stochastic and deterministic approaches for kinetic energy injection, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-19974, https://doi.org/10.5194/egusphere-egu24-19974, 2024.

The route to chaos in a truncated rotating shallow-water model has been investigated by constructing an autonomous five-mode Galerkin truncated system with complex variables. Two distinct transitions to chaos were observed as the energy injected into the system increased. The first transition is characterized by forming a continuous sequence of bifurcations that follow the usual Feigenbaum path. The second transition, occurring for high values of injected energy, exhibits a sharp transition between quasi-periodic states and chaotic regimes. The first chaotic regime arises since nonlinear interactions are principally dominated by inertial terms, while the second one is related to the increasing importance of free surface elevation in the overall process. By rewriting the system in terms of phase and amplitude, for each variable truncated system, it has been found that phases are locked at the initial value for a certain period of time, followed by a sudden transition due to a simple rotation of $\pm \pi$, even when amplitudes show a chaotic dynamic. The time duration of phase locking decreases as the injected energy increases, and, for high values of injected energy, even phases reach a chaotic regime. This behaviour is observed since, in the nonlinear term of the equations, phases appear through linear combinations of triads of different modes. When the duration of locking periods is different for each mode, the superposition of multiple $\pi$ phases jumps, making the dynamics of the coupled phase triads stochastic, even for small values of the injected energy.

How to cite: Carbone, F. and Dutykh, D.: Route to chaos and resonant triads interaction in a truncated Rotating Nonlinear shallow-water model, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-13180, https://doi.org/10.5194/egusphere-egu24-13180, 2024.

El Niño-Southern Oscillation (ENSO) is a complex climatic phenomenon with significant impacts on global weather patterns and ecosystems. Improving ENSO predictability is therefore an issue of high societal value. However, Global Circulation Models present severe biases when predicting ENSO, and their skill remains comparable to that of vastly simpler empirical models such as Linear Inverse Models (LIMs). LIMs, however, rely on linear dynamics, and they have inherent limitations in capturing the behavior of non-linear phenomena. In this context, Koopman operator theory has emerged as a powerful mathematical framework, offering a novel perspective for analyzing complex non-linear systems, such as ENSO. In this study, we investigate the potential of Koopman operator theory to enhance ENSO forecasting accuracy. Leveraging 2000 years of tropical SST pre-industrial CESM data, we have assessed the skill of the Niño 3.4 index forecasts using the Koopman framework, and compared it to the benchmark set by LIMs. Our analysis includes sensitivity testing of both methods across various parameters, such as retained variability and data length used for operator computations. Our findings reveal nuances in the robustness of Koopman Operator estimates, particularly evident when using shorter training periods, contrasting with more stable LIM counterparts. However, a notable breakthrough emerges as we demonstrate the higher skill of Koopman multimodel ensembles, showcasing consistent improvements over linear models. The comparative analysis highlights the potential of Koopman operator theory in advancing ENSO forecasting beyond linear models. The utilization of Koopman multimodel ensembles emerges as a promising strategy, demonstrating enhanced forecasting capabilities. Yet, challenges in robustness persist, particularly in shorter data spans, signaling avenues for further refinement. Overall, these findings underscore the significance of the Koopman framework and lay the groundwork for future research aimed at refining methodologies for more accurate predictions in complex climatic systems.

How to cite: Lorenzo Sánchez, P., Newman, M., Navarra, A., Albers, J., and Subramanian, A.: Koopman operator theory for enhanced Pacific SST forecasting, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-2554, https://doi.org/10.5194/egusphere-egu24-2554, 2024.

Understanding Earth's terrestrial biosphere dynamics is vital for comprehending our planet's environmental health and sustainability. Recently, the frequency and intensity of extreme climate events have risen, significantly impacting the biosphere. Given the advancements of recurrent neural networks in modeling complex, nonlinear dynamics, we explore the capability of recurrent neural network models to model and predict the impacts of extreme events on biosphere dynamics. In this work, we compare four different recurrent network architectures, each with distinct features: 1) Recurrent Neural Networks (RNNs); 2) Long Short-Term Memory-based networks (LSTMs), known for their efficacy in handling long-term dependencies; 3) Gated Recurrent Unit-based networks (GRUs), which offer a simplified yet effective alternative to LSTMs; and 4) Echo State Networks (ESNs), which are distinguished by fixed internal weights and training based on simple linear regression. Our study found that while recurrent network architectures show similar performance under standard conditions, Echo State Networks (ESNs) show slightly superior performance, particularly under extreme events. However, we also identify limitations in current models under extreme conditions, underscoring the need for specialized approaches to enhance predictive accuracy in these circumstances.

How to cite: Martinuzzi, F., Mahecha, M. D., Camps-Valls, G., Montero, D., Williams, T., and Mora, K.: Impact Predictability: Exploring Extremes in Biosphere Dynamics with Recurrent Neural Networks, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-9569, https://doi.org/10.5194/egusphere-egu24-9569, 2024.

Medicanes, tropical-like cyclones in the Mediterranean Sea, pose unexpected challenges to unprepared areas due to their projected increases in intensity. To address these challenges, we proposed: 1) the development of an automatic tracking method in the absence of a comprehensive tracking database for Medicanes; 2) the implementation of a forecasting model for extreme cyclones utilizing artificial intelligence techniques. This is especially beneficial when traditional numerical models struggle to account for nonlinear interactions. We use a K-means algorithm and mean sea level pressure reanalysis data to track storm centers, determining maximum wind speed and position throughout each case’s lifetime. This information categorizes our dataset into storm-like and extreme Medicanes, and facilitates the extraction of spatiotemporal data from infrared satellite images. These features enable us to predict the final classification of Medicanes (whether they are storm-like or extreme) 6 to 36 h before peak wind speed, using an optimized combination of Convolutional Neural Network and Random Forest binary classification methods. By training and testing on Mediterranean data from 1984 to 2020, we successfully diagnosed between 72% and 87% of extreme Medicanes in the studied cases, depending on the lead-time. Our study is the first to employ artificial intelligences for both tracking and forecasting Medicanes, offering a foundational approach to enhance Medicanes preparedness and awareness.

How to cite: Martinez-Amaya, J., Nieves, V., and Muñoz-Marí, J.: Advancements in Medicanes Tracking and Forecasting Using Artificial Intelligence, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-119, https://doi.org/10.5194/egusphere-egu24-119, 2024.

Numerical weather prediction and climate models encounter challenges in accurately representing flow regimes in the stably stratified atmospheric boundary layer and the transitions between them. This leads to an inadequate depiction of regime occupation statistics and, therefore, to biases in forecasts of near-surface temperature. To explore inherent uncertainties in modeling regime transitions, the response of the near-surface temperature inversion to transient small-scale phenomena is analyzed based on a stochastic modeling approach. A sensitivity analysis is conducted by augmenting a single-column model for the atmospheric boundary layer with deterministic perturbations accounting for small-scale fluctuations in the wind and temperature dynamics and with a stochastic stability function to account for turbulent bursts. The model is a tool to systematically investigate what types of unsteady flow features may trigger abrupt transitions in the mean boundary layer state. Previous research showed that incorporating enhanced mixing, a common practice in numerical weather prediction models, blurs the two regime characteristics of the stably stratified atmospheric boundary layer. Simulating intermittent turbulence through a stochastic stability function is shown to provide a potential workaround for this issue. Including key uncertainty in models could lead to a better statistical representation of the regimes in long-term climate simulation. 

How to cite: Kaiser, A., Vercauteren, N., and Krumscheid, S.: Capturing the Variability of the Nocturnal Boundary Layer through Localized Perturbation Modeling, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-8958, https://doi.org/10.5194/egusphere-egu24-8958, 2024.

Land surface models provide self-consistent estimates of the water stored in various components of the land system, i.e. in the soil, vegetation and snow, and of water fluxes. Assimilation of satellite-based data helps to update these estimates to some extent, but it has some limitations when human processes, such as irrigation, are missing in the modeling system. Furthermore, the simulated water distribution heavily depends on the choice of meteorological input and other model choices. In this study, we aim to quantify the relative contribution of (i) Sentinel-1 data assimilation for soil moisture and snow updating, (ii) meteorological input, and (iii) modeling irrigation over the Po river basin in Italy.

The Po river network channels the discharge of snow melt water from the Alps and Apennines, combined with surface and deep subsurface runoff from the hillslopes and valley. During the summer, the river network supplies irrigation water to the large agricultural area in the Po river valley. The Po basin is thus a unique testbed to study various water budget components in an environment with pronounced seasonal water storage dynamics and human water management.

More specifically, we assimilate 1-km Sentinel-1 data into the Noah-MP land surface model coupled to an irrigation module and the hydrological modeling and analysis platform (HyMAP) as runoff routing module. The Noah-MP simulations are forced with meteorological data from either the fifth generation ECMWF atmospheric reanalysis (ERA5) or the Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2) for the years 2015-2023. Sentinel-1 snow depth retrievals are assimilated over the mountains in the winter, whereas Sentinel-1 backscatter are mainly assimilated in the valley during the spring, summer and fall. The state updates are applied to snow depth, snow water equivalent, and soil moisture. These updates subsequently trigger updates in estimates of irrigation, leaf area index, discharge and other variables, resulting in a self-consistent re-analysis of the entire water budget of the Po basin. The impact of the Sentinel-1 data assimilation relative to that of the activation of irrigation modeling is quantified using independent in situ and remotely sensed measurements of soil moisture, leaf area index, snow depth, evaporation, irrigation and discharge.

How to cite: De Lannoy, G., Busschaert, L., Modanesi, S., Dunmire, D., Brangers, I., Lievens, H., Heyvaert, Z., Massari, C., Getirana, A., and Bechtold, M.: Relative contribution of high-resolution Sentinel-1 data assimilation and modeling choices to improve regional water budget estimates, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-17015, https://doi.org/10.5194/egusphere-egu24-17015, 2024.

In this contribution, we will analyse simple data assimilation schemes that not only estimate the underlying states of a dynamical system but simultaneously reconstruct unknown components of the dynamics. We focus on quasigeostrophic and transport-diffusion equations (for instance for atmospheric aerosols or tracer gases) and reconstruct forcings or surfacd fluxes, along with the underlying dynamical states. Tracer gases and aerosols play an important role in the dynamics of the atmosphere; aerosols for instance act as condensation nuclei and thus have a major influence on precipitation, while tracer gases such as ozone, methane, or CO₂ impact the radiative transfer and are thus linked to important atmospheric phenomena such as the ozone hole and the energy budget of the planet ("greenhouse effect"), respectively. Furthermore, gases as well as aerosols (especially in the lower troposphere) are common pollutants with strong and potentially adverse effects on the environment, human activity, and health. We discuss two algorithms that both apply in the context of the quasigeostrophic as well as the transport-diffusion equations.

How to cite: Bröcker, J., Carigi, G., Kuna, T., and Martinez, V. R.: Data assimilation and the reconstruction of surface fluxes in quasigeostrophic and transport equations., EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-1148, https://doi.org/10.5194/egusphere-egu24-1148, 2024.

Iterative ensemble smoothers, originally developed for parameter estimation in petroleum applications, are effective data assimilation methods in coupled, unstable dynamical systems. In this study, we demonstrate this using a coupled multiscale model based on two Kuramoto-Sivashinsky equations with different spatial and temporal scales, representing two subsystems of an earth-system model. The cross-covariance between the variables of the two subsystems reflects how each subsystem influences the other, leading to unexpected structures that reveal interesting physics of the coupled system. The setup of the data assimilation allows simultaneous updating of both systems, leading to consistent estimates.

A comparative study illustrates the properties of iterative ensemble smoothers and assimilation updates over finite-length assimilation windows. We demonstrate the increased accuracy of the smoothers’ solution compared to that of the standard ensemble Kalman filter and the fast convergence of the iterations related to the efficient handling of nonlinearities by the nonlinear space-time ensemble.

Localisation, whether distance-based or adapted to spatial correlations, can be used in iterative ensemble smoothers to effectively deal with limited ensemble sizes. We discuss the effects of the spatial distribution and temporal frequency of available observations and illustrate how data gaps in one of the two subsystems affect the coupled estimate. Looking forward, we present the possibilities and benefits of a potential implementation of this approach in coupled earth-system models.

How to cite: Vossepoel, F., Evensen, G., and van Leeuwen, P. J.: Data assimilation approaches with iterative ensemble smoothers in coupled nonlinear multiscale models, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-12751, https://doi.org/10.5194/egusphere-egu24-12751, 2024.

Nonlinearities in numerical models for the geosciences and in observation operators that map model states to observation space have become so strong that they can no longer be ignored. The particle flow filter (PFF) is a fully nonlinear and efficient sequential Monte Carlo filter that removes the weight degeneracy problem in particle filters by iteratively transporting the equal-weighted particles from the prior to the posterior distribution. The deterministic version of the PPF has been successfully applied to high-dimensional systems and is unbiased in the limit of an infinite number of particles. However, with a small number of particles, the ensemble spread can be biased low, especially in the observed part of the state space. This can be partly alleviated by using a so-called matrix-valued kernel in the algorithm, but the fundamental issue remains. To address this challenge, we propose a novel approach, the Stochastic Particle Flow Filter (SPFF), which includes a Gaussian noise in the Stein Variational Gradient Descent dynamics, the amplitude and covariance of which follow directly from theory. With this additional repulsive force between particles, the SPFF guarantees an unbiased posterior pdf, even with a finite number of particles.

We demonstrate the performance of the SPFF using detailed experiments with the 1000-dimensional Loreanz-96 model. Our results demonstrate that SPFF successfully avoids particle collapse of the marginal distributions and accurately captures the evolutions of particles Additionally, and initially unexpectedly, the SPFF exhibits faster convergence than the deterministic PFF and thus improves analysis accuracy compared to the PFF with a matrix-valued kernel at the computational cost. We also show results of its performance on a high-dimensional ocean model demonstrating that we, as a community, are very close to solving the nonlinear data assimilation problem.

How to cite: Yeh, H.-L. and van Leeuwen, P. J.: Unbiased fully nonlinear data assimilation: the Stochastic Particle Flow Filter, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-14208, https://doi.org/10.5194/egusphere-egu24-14208, 2024.

This study aims at proposing a new framework to perform ensemble-based estimations of dynamical trajectories of a geophysical fluid flow system. To perform efficient estimations, the ensemble members are embedded in a set of evolving reproducing kernel Hilbert spaces (RKHS) defining a family of spaces.

The method proposed here is designed to deal with very large scale systems such as oceanic or meteorological flows, where it is out of the question to explore the whole attractor, neither to run very long time simulations. Instead, we propose to learn the system locally, in phase space, from an ensemble of trajectories.

The novelty of the present work relies on the fact that the feature maps between the native space and the RKHS manifold are transported by the dynamical system. This creates, at any time, an isometry between the tangent RKHS at time t and the initial conditions. This has several important consequences. First, the kernel evaluations are constant along trajectories, instead to be attached to a system state. By doing so, a new ensemble member embedded in the RKHS manifold at the initial time can be very simply estimated at a further time. This framework displays striking properties. The Koopman and Perron-Frobenius operators on such RKHS manifold are unitary, uniformly continuous (with bounded generators) and diagonalizable. As such they can be rigorously expended in exponential forms.

This set of analytical properties enables us to provide a practical estimation of the Koopman eigenfunctions. In the proposed strategy, evaluations of these Koopman eigenfunctions at the ensemble members are exact. To perform robust estimations, the finite-time Lyapunov exponents associated with each Koopman eigenfunction (which are easily accessible on the RKHS manifold as well) are determined. On this basis, we are able to filter the kernel by removing contributions of the Koopman modes that exceed the predictability time. We show that it leads to robust estimations of new unknown trajectories. This framework allows us to write an ensemble-based data assimilation problem, where constant-in-time linear combinations coefficients between ensemble members are sought in order to estimate the QG flow based on noisy swath observations.

The methodology is demonstrated on a multilayer quasi-geostrophic model representative of the Gulf Stream area in the North Atlantic at a 10 km resolution and considering 100 training ensemble members. We show the ability of the method to estimate trajectories knowing the initial condition of a new ensemble member. Moreover, ensemble-based data assimilation is performed based on realistic swaths of altimetry observations.

How to cite: Tissot, G., Jaouen, M., and Mémin, E.: Ensemble forecasts in reproducing kernel Hilbert space family: Application on a multilayer quasi-geostrophic numerical simulation, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-10075, https://doi.org/10.5194/egusphere-egu24-10075, 2024.

The forecasting of earthquake occurrence remains a significant challenge in seismology, primarily due to uncertainties in understanding the current state of stress, strength, and the parameters controlling the slip behaviour of faults. Among these parameters, friction parameters are crucial as they control the earthquake recurrence interval and their nucleation size. There is a critical gap in effectively integrating observational data with physics-based models, particularly in the face of parameter bias. This study addresses how ensemble data assimilation methods can be optimized to address these challenges and reduce uncertainties in fault-slip estimates.

Our objective is to enhance the accuracy of earthquake forecasting by incorporating model error into ensemble data assimilation methods, thus improving the estimation of critical state variables such as shear stress and slip velocity. We employed the Ensemble Kalman (EnKF), Adaptive Gaussian Mixture (AGMF), and Particle Flow (PFF) filters, which are integrated with earthquake sequence models. These methods were applied in two stages: 1) Perfect model experiments using 1-D Burridge-Knopoff models to assess the benefits of including model error in estimating non-periodic and chaotic behaviours in low-dimensional systems. 2) Application in a meter-scale direct-shear laboratory setup, assimilating measurements from shear-strain gauges near the fault, to examine the effects of varying normal stress profiles on the fault on the estimates and the impact of including model error.

The perfect model experiments demonstrated improved estimation of shear stress, slip velocity, and the state variable (θ), particularly in estimating non-periodic sequences and chaotic behaviour using stable periodic solutions when confronted with small parameter biases. In the laboratory setup, variations in normal stress profiles significantly influenced the information content. Sensor placement relative to the fault and seismic phase was found to critically impact the observations' informational value, with sensor proximity to the fault being a critical aspect and the information content being higher around the coseismic phase.

This research provides valuable insights into the intricate process of earthquake forecasting, underscoring the role of data-assimilation techniques in enhancing our understanding and forecasting abilities in this field.

How to cite: Diab-Montero, H. A., Li, M., van Dinther, Y., and Vossepoel, F. C.: Ensemble Data Assimilation with Model Error Inclusion for Estimating Fault-Slip Occurrence in Large-Scale Laboratory Experiments, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-11543, https://doi.org/10.5194/egusphere-egu24-11543, 2024.

State-of-the-art data assimilation systems, such as the 4DVar system of the European Centre for Medium-Range Weather Forecasts (ECMWF), are highly successful in producing state estimates of the atmosphere constrained by millions of observations. However, existing systems require substantial approximations, e.g. in forward operators, and employ conventional models in their optimization loop. This limits the amount of information that can be extracted from observations, e.g. the assimilation of visible satellite channels is still challenging. The current approach also separates the use of observations into a data assimilation step and a forecasting one, with observations only being used indirectly for forecasting, for example for tuning of parametrizations and for evaluation.

Here, we explore the possibility to train large machine learning models for data assimilation and forecasting directly from observations. In particular, we build a generative transformer neural network that models the joint the probability distribution p(y,x) over output states y for an input x. The input are observations from a temporal window, e.g. 6h or 12h, and the output y can either be an estimate of the state within the window or a short-term forecast, e.g. for another 12h. Different observations are processed by different embedding networks but then fused in the backbone transformer network. To obtain an integrated and consistent representation of the atmospheric state that corresponds to the different input data streams, we train with a variation of the masked token model training objective from natural language processing that impels the network to learn the correlation between the different input streams and channels. To properly represent the statistical nature of the estimation of y given x, our network provides an ensemble prediction as a nonparametric model for the probability distribution over y.

We present results for a network trained with a substantial amount of data, including different satellite observations (such as AMSU-A microwave sounders from NOAA 15-19 and the METOP satellites as well as IASI), radiosondes, and ground station-based measurements. The skill for both data assimilation and forecasting is analyzed and compared to ECMWF’s operational 4DVar system. We also ablate the effect different observations have on the skill of the network output.

How to cite: Lessig, C., Lean, P., McNally, T., Alexe, M., Lang, S., Chantry, M., and Dueben, P.: Towards a machine learning model for data assimilation and forecasting directly trained from observations, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-11724, https://doi.org/10.5194/egusphere-egu24-11724, 2024.

Mechanistic or dynamical models based on governing equations are ubiquitous throughout science and engineering. Such models, also referred to as “simulators”, are typically characterized by a forward mapping from some set of inputs or parameters to one or more output quantities of interest. In many cases, the inputs required for the forward model are either unknown or represent approximations of unresolved processes. Bayesian inference provides a natural framework for constraining this model uncertainty using observed data. Such a framework is especially valuable in cryospheric application domains such as permafrost research, where direct observations of many quantities of interest, e.g. subsurface temperature and soil moisture, are only sparsely available. Mechanistic models based on known physics therefore play an indispensable role in filling these gaps. However, virtually all methods for Bayesian inference require repeated evaluation of the forward model which is often computationally challenging, especially for dynamical systems. As a result, computational requirements of statistical inference very quickly become intractable even for systems of only moderate complexity. The burgeoning field of “simulation-based inference” (SBI) aims to leverage modern computational methods from machine learning (ML) and data assimilation (DA) to overcome these challenges and facilitate large scale uncertainty quantification in complex scientific models. In this work, we show how SBI can be seen as a unifying theoretical framework that bridges the gap between existing DA methods (e.g. variants of the ensemble Kalman filter) and full-fledged Bayesian inference with the goal of facilitating hybrid statistical-physical modeling of complex systems. We present a novel set of software tools for making SBI more accessible to researchers along with benchmarks of several methods drawn from both the ML and DA literature. Two of these benchmarks are based on use cases from Arctic land surface modeling: degree day approximation of snowmelt and geothermal inversion of historical climate change from boreholes in Arctic permafrost. We highlight the contributions that SBI can make to solving such inverse problems and discuss further potential applications in the cryosphere and beyond.

How to cite: Groenke, B., Aalstad, K., Pirk, N., Westermann, S., Zscheischler, J., Gallego, G., and Boike, J.: Simulation-based inference as a paradigm for scientific machine learning in the cryosphere and beyond, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-16847, https://doi.org/10.5194/egusphere-egu24-16847, 2024.