Cone penetration tests have long been used to characterize snowpack stratigraphy. With the development of sophisticated digital penetrometers such as the SnowMicroPen, vertical profiles of snow hardness can now be measured at a spatial resolution of a few micrometers. By using small penetrometer tips at this high vertical resolution, further details of the penetration process are resolved, leading to many more stochastic signals. An accurate interpretation of these signals regarding snow characteristics requires advanced data analysis. Here, the failure of ice connections and the pushing aside of separated snow grains during cone penetration lead to a combination of (a) diffusive noise, as in Brownian motion, and (b) jumpy noise, as proposed by previous dedicated inversion methods. The determination of the Kramers–Moyal coefficients enables differentiating between diffusive and jumpy behaviors and determining the functional resistance dependencies of these stochastic contributions. We show how different snow types can be characterized by this combination of highly resolved measurements and data analysis methods. In particular, we show that denser snow structures exhibited a more collective diffusive behavior supposedly related to the pushing aside of separated snow grains. On less dense structures with larger pore space, the measured hardness profile appeared to be characterized by stronger jump noise, which we interpret as related to breaking of a single cohesive bond. The proposed methodology provides new insights into the characterization of the snowpack stratigraphy with micro-cone penetration tests.

Snow is an essential component of our environment and can significantly impact our lives: from the wishful dream of a white Christmas to the misfortune of avalanche accidents. Having a closer look at snow, one discovers many microstructural patterns and realizes that snow on the ground undergoes constant evolution

Here, we consider the measured fluctuating hardness as a consequence of summing up the interactions between the penetrometer tip and individual snow particles. We describe this penetration process in analogy to the well-known Brownian motion

A direct comparison of our stochastic approach with the works based on shot noise is out of the scope of this paper.

The article is organized as follows. In Sect. 2, we summarize the stochastic analysis method and show how it is possible to distinguish between diffusive and jump noise. In Sect. 3, the method is applied to centimetric snow samples whose microstructure is also captured by tomography. In Sect. 4, as a proof of concept, the SMP profile of a natural snowpack is analyzed with this technique.

A stochastic process

A diffusive process

KM coefficients for the Langevin equation are defined as

For the SMP data considered, the drift term

Typically, when the signal of a stochastic process presents sharp changes at some instant (discontinuity events), higher-order Kramers–Moyal coefficients (especially

For jump-diffusion processes, the drift and diffusion coefficients (

The estimate of the drift coefficient is the same for the diffusion process (Eq.

To improve the estimation of KM coefficients

For the SMP measurements considered, we also assume that for a chosen small depth interval (

In this section, we illustrate how diffusive and jump noises affect the stochastic fluctuations on a synthetic example. An Ornstein–Uhlenbeck (OU) process

Three synthetic time series of the OU jump-diffusion process were generated for

Normalized time series of Ornstein–Uhlenbeck processes with only diffusion, only jump (drift-jump) and jump-diffusion terms

For a jump-diffusion process, another parameter that we considered was the ratio of diffusion and jump noise

Normalized time series of OU jump-diffusion processes with

In this section, our main aim is to show how the jump-diffusion model can be used to distinguish snow types from hardness profiles measured with the SMP. Firstly, small snow samples whose microstructure was also fully characterized by tomography before being measured by the SMP were used to test the developed methodology. Secondly, as a proof of concept, we analyzed one penetration profile of a snowpack measured in the field and we provided the subsequent profile of microstructural parameters. Last, the results were discussed.

Three-dimensional view of the microstructure of some representative samples: precipitation particles (PP1), depth hoar (DH1), rounded grains (RG1) and large rounded grains (RGlr1). The ice matrix is shown in gray; the pore space is transparent. Sub-samples shown are cubic with a side length of 3

We tested several snow samples composed of four different natural snow types, namely precipitation particles (PP), depth hoar (DH), rounded grains (RG) and large rounded grains (RGlr) as classified in

Overview of the detailed properties of the snow samples used. Snow types are classified according to the international classification of snow on the ground

Setup of micro-cone penetration test

Segments of snow hardness profiles of PP1, DH1, RG1 and RGlr1. These four different types occur as natural snow types. Precipitation particles (PP) have the smallest trend and fluctuation force; large rounded grains (RGlr) are the largest, while depth hoar (DH) and rounded grains (RG) have similar trends and fluctuation forces between those of PP and RGlr. The first 4

To work out the significance of advanced stochastic features for snow, we focused on the fluctuations in the hardness profiles. Each profile was first detrended. The trend

The results do not change significantly if the kernel widths are changed between 0.14 and 0.66

To estimate errors, we divided the detrended and normalized data into different sub-samples. Given are two PP, three DH, three RG and six RGlr measurement profiles. These profiles were separated into smaller segments, which finally gives four PP, five DH, five RG and six RGlr samples. We estimated the KM coefficients of each sample using Eq. (

Detrended snow hardness profiles of four different snow types, PP, DH, RG and RGlr. For the snow type RGlr, we only show the samples RGlr4, RGlr5 and RGlr6, as described in Table

According to the description provided in Sect.

State-dependent drift

As shown in Fig.

Jump amplitudes

The jump amplitudes,

Summary of the results for the correlation length scales

Next, measurements from a field campaign are presented

The snow hardness profile of a field measurement is shown in the top left of Fig.

The snow hardness profile of a field measurement together with its fourth-order KM coefficients

Our work is based on the proposed analogy of Brownian motion and the SMP penetration process, as illustrated in Fig.

We start the discussion with the mean values of

The drift terms

The results for

The amplitudes of jump noise,

Besides the features of the different terms in the stochastic processes, the contributions of the diffusive and the jump noise can be compared by the dimensionless quotient of

In Sect.

In conclusion, we observe that the advanced stochastic analysis of SMP measurements of snow layers allows differentiation of snow types. The diffusive and jump-noise contribution can be quantified and gives new insights into the stochastic behaviors of the cone penetration test in snow. For different snow types, we find an interesting mixture of diffusive- and jump-like noise. We propose the interpretation that the dominant diffusive process is due to the pushing aside of many snow grains, whereas the breaking of ice structures leads to dominant jump noise. Our results show that the denser structures typical of DH and RGlr lead to a more collective diffusive behavior, whereas for the highly porous snow structures of PP and RG, the single breaking events lead to a relatively strong jump noise. For this interpretation, all

Finally, we would like to point out that our characterization of a complex material, snow, by a penetration process should have the potential to be generalized to, for example, biological tissue or ground layers. Last but not least, we would like to point out that our work provides additional insight into analyzing and modeling the complex nature of snow types, complementing existing methods.

All the data and codes can be found under

PPL did the preliminary analysis of the data and simulations and wrote the main part of the manuscript. PH and IP performed the experiments. JP had the initial idea, and JP, MW and MRRT supervised the work. All the authors interpreted the results and helped in preparing and editing the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank Christian Behnken and Hauke Hähne for helpful discussions.

This research has been supported by the European Space Agency (grant no. 4000112698/14/NL/LvH) and the Agence Nationale de la Recherche (grant no. ANR10 LABX56).

This paper was edited by Melody Sandells and reviewed by Henning Löwe and Adrian McCallum.