The undrained shear strength, usually written as su, describes how much resistance a waterlogged sediment offers when it is loaded faster than pore water can drain. This parameter is central to slope stability analysis, especially for fine-grained soils, because it controls whether a submerged clay layer will hold or yield under the stress of an earthquake shake or an accumulating sediment drape. Measuring it directly is expensive and laborious. The most sophisticated empirical approach is the Cone Penetration Test with pore pressure measurement, or CPTu, in which an instrumented cone is pushed steadily into the lake floor while sensors record the resistance continuously with depth. Between 2018 and 2020, 152 such tests were performed at eleven sites across Lake Lucerne, reaching an average penetration of about 3.9 metres and a maximum of 7.4 metres, limited by the length of the equipment.

Lake Lucerne is an ideal natural laboratory for this kind of work. The fjord-type peri-alpine lake covers 114 square kilometres, comprises seven sub-basins separated in places by submerged moraine ridges, and reaches a maximum depth of 214 metres. Its documented history of seismically and aseismically triggered mass movements is exceptionally rich, and decades of previous research have supplied dense bathymetric, seismic, sedimentological and geotechnical datasets. The lake floor sediments on the lateral slopes typically consist of Holocene silty clays, referred to as lacustrine sediments, overlying Late Glacial clays and clayey silts, referred to as glaciolacustrine sediments, with glacial deposits beneath. Slope failures revealed in bathymetric and seismic data commonly initiate within the glaciolacustrine unit, making the lacustrine and glaciolacustrine sequence the failure-prone drape on non-deltaic slopes. Deltaic slopes, in contrast, contain interbedded Holocene sediments of mixed grain size deposited near the mouths of inflowing rivers such as the Reuss, Muota and Engelberger Aa.

Previous work had generalised the depth behaviour of shear strength with a power-law model, su equals gamma times z to the power alpha, where z is depth below the lake floor, gamma represents the shear strength at one metre depth and alpha governs how rapidly strength increases with depth. This model is consistent with the mechanics of gravitationally deposited granular materials and performed better than a linear fit, avoiding overestimation of strength at large depths. But the generalised models used a single pair of coefficients per sediment type, ignoring the spatial variability within the same lithology and neglecting site-specific variations observed in the empirical data. The new study addresses exactly this gap by making the coefficients alpha and gamma sensitive not only to lithology but also to the morphology of the lake floor.

The core innovation lies in the choice of predictors. The researchers evaluated a suite of topographic and morphometric parameters derived from a high-resolution one-metre multi-beam bathymetric map: water depth, slope angle, surface roughness, surface curvature, distance to the nearest river estuary and distance to the shore. Each predictor was computed at fourteen spatial scales, from ten to 550 metres, capturing morphometric variability from local bed-forms to broad basin geometry. Small scales better reflect roughness and slope related to local sediment disturbance, intermediate scales capture curvature along slope segments, and larger scales integrate long-term sedimentation patterns. Correlation analysis using scatter plots and Pearson coefficients then identified which parameters and at which scales best explained the empirically derived alpha and gamma values at each CPTu location.

The resulting multivariate regressions reveal physically meaningful relationships. Increasing water depth is generally associated with more continuous sedimentation and less disturbance, producing higher strength, while shallow settings are more frequently reworked and typically weaker. Steeper slopes tend to accumulate thinner sediment covers and may experience sediment bypass and erosion, whereas gentle slopes favour normally consolidated deposits. High roughness signals disturbance or heterogeneous sedimentation and is commonly linked to lower and more variable strength, while concave slopes usually carry thicker sediment covers with higher strength than convex ones. Distance to shore and to estuaries controls sediment supply and grain size: nearshore and deltaic environments experience rapid deposition, cyclic loading and under-consolidation, leading to low and spatially variable strength, while deeper and more distant areas host finer sediments with more uniform consolidation. Finer-grained lacustrine and glaciolacustrine sediments tend to show higher alpha and lower gamma values, meaning they start soft near the surface but strengthen steeply with depth, whereas coarser deltaic sediments begin stiffer but strengthen more gradually.

The best-performing regression models explained a substantial portion of the variability, with determination coefficients ranging from 0.23 for gamma of lacustrine sediments up to 0.91 for gamma of deltaic sediments. Moderate values are expected for natural sedimentary systems, where a substantial fraction of strength variability is controlled by local processes not resolved by indirect predictors. To capture this unexplained variation, the researchers computed the residuals between predicted and observed coefficients at every measurement location and analysed their spatial structure through empirical semi-variograms, fitted with Gaussian models. These semi-variograms quantify the spatial autocorrelation of the residuals, yielding key parameters such as the range, the distance over which residuals remain correlated, which varied from about 60 metres for deltaic gamma up to roughly 605 metres for the same parameter, and the nugget, which indicates measurement error or small-scale variability below the sampling resolution.

Ordinary kriging was then applied to interpolate the residuals across the entire subaqueous slope area of the lake. This geostatistical method uses the fitted semi-variogram models to build a covariance matrix, determine kriging weights for each map pixel and estimate residual corrections at unsampled locations. Where measurements exist, the correction equals the observed residual; elsewhere, it represents the kriging-estimated adjustment based on nearby measured residuals. Adding these corrections to the first-order regression predictions produced the final corrected maps of alpha and gamma for lacustrine, glaciolacustrine and deltaic sediments, constrained to physically reasonable ranges. From these coefficient maps, the team generated basin-wide predictions of undrained shear strength at one and five metres below the lake floor, using both simplified stratigraphic assumptions and the real stratigraphic model accounting for sediment thickness derived from reflection seismic profiles.

The resulting strength maps are strikingly consistent with the sedimentological and geomorphological features of the lake. Lacustrine and glaciolacustrine deposits in the north-western basins appear softer and more fine-grained, with lower alpha, gamma and strength values, than those in the south-eastern basins, where sedimentation is strongly influenced by input from three major inflowing rivers. The north-western basins are shallower and gentler, favouring hemi-pelagic sedimentation, while the south-eastern areas receive coarser riverine material. To test the practical significance of the maps, the researchers performed a static one-dimensional infinite slope stability analysis, computing the Factor of Safety at the base of the failure-prone sediment drape using three different sources of shear strength: a constant value of eight kilopascals, the previous generalised power-law models and the new site-specific estimates. The site-specific approach provided the highest spatial resolution and an intermediate case between the other two, predicting more unstable areas on shallow slopes with thin sediments and higher stability on deeper slopes with thicker covers, a pattern that appears more realistic given the absence of recent failures in those deeper regions.

The authors caution that the maps should be interpreted as relative indicators rather than deterministic predictions. Areas where the computed Factor of Safety falls below one do not necessarily indicate imminent failure; such values primarily reflect simplified assumptions, uncertainty in the estimated strength and the inability of the model to capture three-dimensional effects, strength anisotropy and localised strengthening mechanisms. Artefacts may appear in morphologies under-represented in the calibration dataset, such as steep or stiff slopes, and the dataset contains almost no data from gas-rich basin sediments. Addressing these gaps would require additional CPTu investigations. Future improvements could include testing alternative spatial scales, exploring non-linear regression forms, applying machine-learning techniques or propagating uncertainty through probabilistic frameworks. The regression-kriging framework is not limited to shear strength and could be extended to other geotechnical parameters, provided sufficient calibration data and appropriate proxy variables are available. Where sufficient geotechnical data exist, the procedure can be fully replicated in other lakes, and for data-sparse basins with similar sediment types, the multivariate regressions developed here may still be applied. The ultimate goal is ambitious but concrete: basin-wide assessment of subaqueous slope stability and estimation of potentially unstable sediment volumes in Lake Lucerne and comparable lacustrine or marine settings, forming part of an integrated approach to assess and mitigate the hazards of subaqueous slope failures.

Subject of Research: Application of regression-kriging for lake-wide mapping of sediment shear strength on subaqueous slopes in Lake Lucerne

Article Title: Application of regression-kriging for lake-wide mapping of sediment shear strength on subaqueous slopes in Lake Lucerne

Article References: Shynkarenko, A., Bergamo, P., Kremer, K., & Fäh, D. (2026). Application of regression-kriging for lake-wide mapping of sediment shear strength on subaqueous slopes in Lake Lucerne. Swiss Journal of Geosciences, 119, 133-163. https://doi.org/10.3897/sjg.119.195900

Image Credits: AI Generated

DOI: 10.3897/sjg.119.195900

Keywords: Application, regression-kriging, lake-wide, mapping, sediment, shear, strength, subaqueous, slopes, Lake, Lucerne, scientific research