The critical cut length beyond which a crack starts to propagate depends on the mechanical behavior of the weak layer and the overlying slab and on the geometric configuration of the snowpack considered. Therefore, in order to estimate the fracture energy wf from a PST or to even predict the critical cut length of a given snowpack stratigraphy, it is necessary to model the deformation of the slab. Specifically, such a model is required to quantify the potential energy (gravity and elastic) released by the slab as it loses basal support from the weak layer during crack propagation. In this work, we used two slab models: the WEAC numerical model by , and the polynomial model by . The WEAC model represents the current state-of-the-art and has been used to invert PST results into pseudo-measurements of wf. Both models are then used to quantify ac from simulated weak layer fracture energy and slab stratigraphy. In this case, the analytical model by , despite its simplicity, is particularly valuable due to its computational efficiency, making it well-suited for integration into large-scale simulations such as land surface models.

SURFEX-Crocus is a coupled system including a land surface model, SURFEX-ISBA with the Crocus snow model . Crocus belongs to the most detailed class of snow models, including SNOWPACK . It explicitly models the snowpack stratigraphy and describes the snow microstructure by its density, sphericity and specific surface area. This model was initially developed for local, single-point, simulations to support the operational forecasting of avalanche hazard by Météo-France. Indeed, its description of the stratigraphy and its layering management enables to follow the evolution of the properties of each layer and, for example, derived indicators of mechanical stability to estimate avalanche hazard . In this study, we used the version 9.0 of SURFEX-Crocus .

Figure  shows the min-cut as a function of the ice volume fraction and grain type. The min-cut increases with ice fraction ϕ, following a power-law relationship. The min-cut theoretically becomes zero below the percolation threshold (i.e. the critical ice fraction at which ice bonds first form a continuous network across the system). Therefore, this power-law relationship, often found for snow properties e.g., must account for a percolation threshold ϕt. In addition, for a given density, faceted crystals and depth hoar are characterized by a lower min-cut compared to other grain types. Therefore, an empirical fit should both account for density and grain type. We chose the following multiplicative form: 4ζ=fgrain typeϕ-ϕt1-ϕtm with ϕt and m parameters to be fitted, and f a scaling factor depending on grain type.

To ensure a smooth transition between grain types and to provide a physically meaningful description of the microstructural determinants of the min-cut, the scaling factor f is defined as a continuous function of microstructural descriptors. Here, we use the two microstructural descriptors employed by the Crocus model to characterise snow types: sphericity Ψ and specific surface area (SSA) s. Since grain types in Crocus are defined over specific intervals of these two variables (as reported in Table  in Appendix ), the fitting procedure accounts for these ranges, making this part of the parameterization dependent on the Crocus grain-type classification.

We used a logistic function for f and expressed the min-cut ζ as: 5ζ=11+exp⁡-α0-αΨΨ-αss-αΨsΨsϕ-ϕt1-ϕtm A least-square minimisation yields the following parameters: α0 = -2.51, αΨ = 6.51, αs = 0.16 kg m⁻², αΨs = -0.40 kg m⁻², ϕt = 0.04, and m = 2.7. Figure  shows the fitted relationships (Eq. ) across the range of sphericity and SSA represented in the Crocus model for three grain types: melt forms (MF), faceted crystals (FC), and depth hoar (DH).

In this section, we estimate the relationship between the weak layer fracture energy wf and the min-cut ζ. Using the WEAC model (Sect. ), we invert the critical cut length from PSTs to derive wf (dominated by mode I, see Sect. ) from both observed and simulated snow profiles. These estimates are then compared to the min-cut ζ predicted using Eq. (). To compute ζ, we used either manually measured density combined with the range of SSA/sphericity values associated to the observed grain type (see Table  in Appendix A), or the modeled density, SSA and sphericity from the corresponding weak layer in the Crocus profile.

In this section, we evaluate the parameterization of the fracture energy wf (Eq. ) from several perspectives. First, we assess whether the prediction of the measured ac is improved by using the proposed parameterization compared to a constant wf. Next, we analyze the temporal evolution of predicted critical cut lengths from modeled profiles throughout a season. Finally, we examine the potential of predicted critical cut lengths to discriminate between weak layers and more stable layers within a profile.

We estimated the fracture energy of snow by relating it to parameter of snow layers that are both available from observations and models, through a parameterization developed from microstructural properties measured using X-ray microtomography. A central contribution of this work is the use of the min-cut. This approach builds partly on the idea introduced by and is in line with research seeking to relate macroscopic mechanical variables to microstructural properties derived from microtomography. For example, used finite-element simulations and multiple tomographic images to infer macroscopic properties such as the elastic modulus, while similarly developed a parameterization of the effective elastic tensor of snow. More recently, combined tomographic images with neural-network methods to predict compressive strength and stiffness.

In the present study, we rely on an extensive microtomography dataset covering a broad range of snow types and densities, from precipitation particles at around 75 kg m⁻³ to melt forms up to 550 kg m⁻³, as shown in Fig. . This approach avoids the use of discontinuous parameterizations and provides a unified relationship (Eq. ) valid across all snow types. The fitting procedure, a weighted least squares, yields a percolation threshold ϕt = 0.04, corresponding to a snow density of approximately 35 kg m⁻³. This value lies slightly below the range of the snow dataset used, yet remains close to the density range of naturally deposited fresh snow (30 to 40 kg m⁻³, ), for which a non-zero min-cut is expected. It should also be noted that the characterization of different min-cut between grain types (for a given density) is captured in the fit using prognostic variables specific to the Crocus model, namely SSA and sphericity. This part of the parameterization is therefore model-dependent and would need to be recalibrated using the corresponding variables and their associated value ranges for any other snowpack model to which it is transferred. The power-law component (i.e. density dependency), by contrast, is model-independent. Ultimately, this parameterization may be applied consistently across all layers of a snow profile (whether field-measured or simulated with Crocus) to compute wf. This value can then be incorporated into a slab model to estimate the critical cut length.

Two key assumptions are made in our approach. First, we assume that the min-cut can be related to a macroscopic quantity, in particular the snow density. Results presented in Sect.  support this assumption and show that for a given density, layers composed of faceted crystals and depth hoar consistently exhibit lower min-cut values, with depth-hoar layers showing the lowest values among the grain types considered (see Fig. ). These observations motivated the introduction of additional microstructural descriptors (grain sphericity and SSA) into the parameterization, such that the min-cut scales depending on grain type and decreases specifically when the grain type approaches the characteristics of faceted crystals and depth hoar. These findings align with results, who show that density captures the first-order variability of mechanical properties such as stiffness and compressive strength, while microstructural attributes are essential for distinguishing the mechanical behavior of persistent weak-layer grain types, including faceted crystals and depth hoar.

The second assumption is that the min-cut scales linearly with fracture energy. While this assumption is not directly supported by experimental studies or fracture theory, the min-cut quantifies the minimum amount of microstructure that must be disconnected, and it may reasonably be interpreted as a first-order proxy for fracture energy. Under this interpretation, it could be viewed as a lower-bound estimate, as additional energy dissipation associated with microstructural rearrangement, local stabilization mechanisms, or non-optimal fracture paths is not accounted for. This hypothesis could not be confirmed, as the linear fit between min-cut and fracture energy provided only a moderate correlation at most (R = 0.48; see Fig. ).

Several factors may explain the relatively low correlations obtained in the linear regression between wf and min-cut, some of which may be related to measurement and modeling uncertainties (discussed in the following section of the discussion), but could also point to a lack of complexity in the proposed parameterization, as the min-cut alone may not fully capture the variability of fracture energy. Nonetheless we demonstrate the clear advantages of using this relationship rather than a constant wf, as shown by the results in Sect.  and discussed in a further section.

Using modeled rather than measured slab and weak layer characteristics consistently showed better performance in the correlation between wf and the min-cut (Fig. ) and to the R and for RMSE scores for critical cut lengths (Fig. ). Lower scores obtained with measured profiles, and the weak correlations between wf and the min-cut, can partly be attributed to uncertainties in measuring and modeling slab and weak layer properties, primarily density.

The temporal evolution of these properties is shown in Fig. . While the modeled weak layer density increases smoothly over time, the measured density fluctuates unrealistically by about ±50 kg m⁻³ between 2-week intervals. These values were obtained using density cutters or tubes , which are known to have uncertainties up to 10 % (i.e. when compared to micro-CT measurements), increasing for layers only a few centimeters thick (concerning most of the followed weak layers), generally overestimating low-density and underestimating high-density snow layers . Added to this uncertainty are the observer's subjective choices regarding the delineation of the different layers, as well as the hectometric variability of the snowpack, all of which make it difficult to monitor the different layers consistently and reliably over the course of a season. Similarly, the evolution of slab characteristics (aggregating all layers) shows greater oscillations in measured than in modeled profiles. Figure  also indicates that Crocus densities are systematically lower than measured ones, an already documented model bias such as in , that could be explained by an underestimation of compaction processes. Critical cut lengths from PSTs, used in both modeled and measured cases, also carry uncertainty, similarly to those related to the weak layer. They result from the observer's choice of cut, spatial metric variability that can be pronounced for thin layers such as weak layers that can be seen on uncertainties in Figs.  and , and by design in the PST protocol for large values approaching the 150 cm column length.

Because the WEAC model depends on numerous parameters (e.g., density, layer thickness, measured critical length), these uncertainties compound, potentially by adding up one another (i.e., underestimating critical length while overestimating weak layer density) thereby applying an increased penalty to the scores.

All of these uncertainties, whether they concern measurements or modeling, make it difficult to estimate a relationship between wf and the min-cut and probably explain the better scores obtained when using modeled profiles, which, although biased, remain temporally stable and internally consistent. These considerations underline the need for improved field measurement techniques for weak layers and continued refinement of parameter evolution laws in macroscopic snow models.

The parameterization of wf remains satisfactory in several respects, even though the correlations between wf and the min-cut values presented in Sect.  (Fig. ) are only moderate. First, the values of wf inferred through inversion of the WEAC model lie between 0.01 and 0.7 J m⁻² for weak layers. This range is consistent with previous studies, which generally report weak-layer fracture energies between 0.01 and approximately 1.5 J m⁻² . This range is also consistent with the total energy release rate values where mode I is dominant (including collapse and in-plane shear), of 0.4–0.8 J m⁻² as reported in , which was retrieved by performing PSTs over buried surface hoar. This agreement provides confidence that the parameterization derived from our fit will provide physically realistic values of wf.

Second, the temporal evolution of wf derived from our simulations is consistent with the observations of , who documented time evolutions of mechanical properties (i.e. elastic modulus and specific fracture energy) in weak layers. Their results show that fracture energy is generally not expected to decrease over time because sintering increases bonding between grains, except under strong temperature-gradient. A similar pattern is visible in Fig. , where many weak layers exhibit relatively stable wf values around 0.1 J m⁻² for several weeks before increasing toward the end of the season. This late-season increase is consistent with the stabilization trends described by .

The results in Sect. , comparing modeled and measured critical cut lengths, further highlight the benefits of using the wf parameterization developed in Sect.  and , rather than using a constant wf. For both observed profiles (Fig. ) and simulated profiles (Fig. ), statistical scores (i.e., R, RMSE) are better when the parameterized wf is used.

In addition, employing a constant wf severely restricts the range of critical cut lengths that the WEAC model can reproduce, preventing any values below 20 or above 60 cm. These findings indicate that wf plays a major role in the dynamics of critical cut length modeling within the WEAC model, and emphasizes the importance of even slight variations during the season, as well as its increase at the end of the season acting the stabilization of the weak layer.

Finally, although acceptable scores can still be obtained when using a constant wf (Fig. ), it remains applied to pre-identified weak layers. The narrow range of critical cut lengths produced under a constant wf limit the model's applicability for detecting weak layers within full profiles as in Fig.  or for tracking their stabilization over time as in Fig. . Overall, these results clearly demonstrate the advantages of using a parameterized wf over a constant value within the WEAC model.

A qualitative comparison with (based on a stress-strength approach) reveals consistent results across several aspects. The stratigraphic profiles of ac (their Fig. 9) exhibit a pattern similar to those obtained here, with a progressive increase of ac with depth and local minima clearly associated with weak layers. The temporal evolution (their Fig. 11) is likewise comparable: critical cut lengths remain relatively stable over most of the season before increasing substantially towards its end, consistent with the progressive stabilization of the profile. Similarly, report degraded performance when using manually observed profiles instead of simulated ones, in line with the score differences observed here between observed and Crocus-simulated profiles. This overall consistency across two independent modelling frameworks is encouraging and further supports the physical coherence of the present results. Part of this agreement between the strength-based approach of and the energy-based framework of this study could be theoretically motivated. Indeed, cohesive-zone formulations may bridge the two approaches by removing the crack-tip stress singularity and relating the effective fracture toughness to the constitutive properties of the weak layer (e.g., its strength, elasticity, and effective thickness) . Under the small-scale yielding assumption (i.e., when the process zone remains small compared to the elastic characteristic length of the slab–weak-layer system), such stress-based formulations and the linear elastic fracture mechanics approach are expected to agree .

Overall, the approach proposed in this study builds on the state-of-the-art WEAC slab model and a min-cut parameterization derived from a large and diverse set of snow microstructure samples, suggesting more robust transferability to a broader range of snowpack conditions compared to . The inherent limitations of the linear relationship between min-cut and wf are, discussed in Sect. .

In the study sections dedicated to the evaluation of the parameterization of wf, two anticrack models of different complexity were compared: Heierli's model and the state-of-the-art WEAC model. The two models differ primarily in their treatment of slab and weak-layer deformation. The WEAC model explicitly accounts for the deformation of a stratified slab as well as that of the weak layer. Its geometry is free on both sides, designed specifically to represent the configuration of a PST. In contrast, the Heierli model is based on an idealized geometry in which a homogeneous slab is supported on one side, and describes the deformation of slab sections that are no longer supported by the weak layer. Owing to these simplifications, the Heierli model depends solely on the weak layer fracture energy wf and does not require additional parameters describing the weak layer that are in any case difficult to obtain through measurements or modeling.

It should be noted that in this study, the Heierli model is implemented using an elastic modulus taken from the formulation of recently modified by , with an equivalent elastic-modulus expression derived from the mode-I formulation of WEAC (see Sect. ). These additions enhance the representation of layering effects, even though they are not explicitly included in the deformation model. Such additions are absent from previous implementations described by , , , and they allow the Heierli model to get results that are more consistent with those of the WEAC model. Despite the differences in complexity and underlying assumptions, the temporal evolution of ac for weak layers (Fig. ) and the profiles of ac for selected dates (Fig. ) show similar results for the two models.

Ultimately, although the overall performance of the WEAC model is slightly superior to that of the Heierli model (notably in Fig. ), the discrepancies remain small. This suggests that either model could be used in operational applications based on snow-model outputs, particularly given the relative simplicity of the Heierli formulation and the smaller number of parameters required for its implementation.

The work presented in this article, although making the best use of the tools and datasets available, is subject to a number of limitations that are worth discussing. The parameterization of wf using the min-cut as an intermediate variable substantially reduces the dimensionality of the problem as it integrates both density and morphological effects into a single physically interpretable quantity. This allows calibration over a much larger and more diverse dataset than the PST dataset alone, improving robustness across snow types. The wf min-cut relationship is nonetheless validated on a restricted set of FC and DH weak layers, though its application to full stratigraphic profiles (Fig. ) yields physically consistent ac values across layer types, suggesting reasonable potential broader applicability.

Regarding the limitations of the wf dataset itself, it is mainly restricted to Mode-I-dominant conditions, as illustrated in Fig. b). However, following the work of , who performed a large number of PSTs under various geometric configurations and loading conditions, demonstrate that such mode mixity conditions (i.e. crack propagation dominated by mode I, approx. > 75 %) are representative of typical avalanche-triggering configurations (including “remotely”) with slope angles of 0 to 40. Nonetheless, the snowpack in a 30 to 40° actual avalanche start zone may differ from a research site in terms of snow stratigraphy (i.e. due to wind redistribution, settlement, radiation, and spatial variability). This represents a limitation, although it is common in most research on snowpack stability, due to the difficult access and monitoring conditions in these areas.

We also note that the PST dataset covers only a limited range of weak layer types, namely faceted crystals (FC) and depth hoar (DH). The fact that surface hoar (SH) is not represented in Crocus limits the validation of the parameterization for this grain type, and no weak layers composed of fresh snow, such as precipitation particles (PP) or decomposing and fragmented particles (DF), are included in the dataset. It should further be noted that the two study sites, located approximately 3 km apart in Switzerland, are representative of Alpine snowpacks, which may differ substantially from other snowpack types, such as maritime snowpacks found in parts of the mountain ranges of North America, or continental and Arctic snowpacks characteristic of higher-latitude environments.

It would therefore be valuable to extend the validation of the proposed parameterization by applying the method to other PST datasets, in order to test the transferability and robustness of the relationship across different snowpack types (oceanic, continental, and Arctic snowpacks), a larger range of grain types including surface hoar SH, non-persistent weak layers such as PP and DF, and ideally PSTs performed on non-weak layers as a reference, as well as on geometric configurations favoring greater mode mixity. The proposed wf parameterization should thus be regarded as an effective, PST-based formulation, subject to the limitations of the dataset on which it was built, rather than as an intrinsic mechanical law. Its operational applicability to any avalanche release zone remains to be demonstrated, particularly because the test design of the PST (i.e., the introduction of an unsupported part of the slab) may not be representative of the processes involved in avalanche formation on steep slopes.

Ultimately, the critical cut length modeling proposed in this study remains insufficient to assess crack propagation on its own, as it addresses only one specific stage: the onset of crack propagation, assuming that failure initiation has already occurred. As illustrated in Fig. , which shows the evolution of ac along a snow profile, particularly low critical cut lengths are found near the surface. However, near-surface fresh snow may lack sufficient bonding to constitute a cohesive slab , and crack propagation may be prevented by slab fracture even when the computed critical cut length is small . This highlights the need to combine the present indicator with additional mechanical criteria in future work: a strength-to-stress ratio for failure initiation, thereby satisfying the coupled criterion of and a tensile strength criterion for crack arrest e.g.. Such a combined analysis would connect the present work to operational triggering likelihood, and would contribute to providing more effective tools for operational risk management.