The sliding laws examined in this study (Sect. ) are thought to represent sliding over different subglacial beds (Fig. b–f). To infer which of these sliding laws is most probable, we first derive the basal drag (τb) and sliding speed (ub; Figs.  and S1) from inverse methods using the Wavelet-based Adaptive-grid Vertically-integrated Ice-sheet-model WAVI;Sect. . The effective pressure (N) can then be estimated by rearranging the sliding laws.

The VGS theory (Sect. ) provides a model of acoustic propagation in granular material. Substituting the estimated effective pressure into this model and using independent estimates for grain diameter (dg) and porosity (ϕ) from sediment cores and Smith, unpublished data, provides an estimate of acoustic impedance for each sliding law. The predicted acoustic impedance is then compared to acoustic impedance measurements collected at five sites on Pine Island Glacier (PIG) in Antarctica Fig. ; by calculating the misfit χΘi2 according to 1χΘi2=1Nd∑jNdZΘi,j-Zj*2σj2, where Nd=300 is the number of data points (60 per site, 120 m apart), and ZΘi,j are the acoustic impedance predictions under a given sliding law i and the model parameters Θi (grain diameter and porosity, along with any additional sliding-law-specific parameters; further details in Sect.  and ). Data points are treated as independent: a sub-sampled data set (every 10th data point) generally yields the same conclusions (Figs. S2 and S3). While there is evidence that PIG is largely underlain by deformable sediments , the exact values of Θi are uncertain. Therefore, the misfit χΘi2 is systematically assessed across what is considered to be a reasonable parameter space (Sect. ). The model parameters do not vary spatially. Zj* and σj are the acoustic impedance observations and their uncertainties. As an example, all metrics involved in predicting the acoustic impedance and calculating the misfit χΘi2 (ub, τb, N, ZΘi, Z*, σ) are shown for one set of parameter values (dg=0.063 mm, ϕ=0.43, Coulomb friction coefficient μ=0.49) and the Coulomb sliding law in Fig. S4.

However, inferring the best-candidate sliding law based solely on the minimum misfit is inadequate, as it does not take into account any prior assessment of the probability of the parameter values used. Instead, we use Bayesian model selection (Sect. ) to identify the most probable sliding law based on all misfits within the parameter space (likelihood function in Fig. ). In this framework, the a priori probability of each model, and of particular parameter values within each model, is specified by prior distributions. Using Bayes' rule, these prior probabilities are updated using seismic data to provide posterior probabilities. Ultimately, this allows us to compute the normalized posterior probability of each sliding law, given the seismic observations collected on PIG (Eq. ). An advantage of the Bayesian approach is that Occam's razor is automatically applied: overly flexible models with a large range or dimension of parameter space are penalized relative to simpler, less flexible models with fewer parameters or tighter bounds upon parameters.

The effective pressure required as input for the VGS theory is determined based on the basal sliding laws described here. Usually, these laws are expressed so that basal drag is a function of sliding speed and effective pressure. To compute effective pressures, these relationships must be inverted, either by explicitly rearranging the equations or by numerical root-finding. For all sliding laws and sites, we ensure the effective pressure does not exceed the ice overburden pressure.

Strictly speaking, the VGS theory used to predict acoustic impedance only applies to granular material (Sect. ). However, while the formation of cavities, for example, is most appropriate for undeformable bed protrusions, larger rock fragments embedded in granular sediment or even fine-grained deformable sediment might play a similar role . Therefore, whenever we are using a sliding law initially developed for hard bedrock (Sect.  and ), we assume a granular, relatively undeformable material that can not support tangential friction at its interface with the ice (here referred to as rigid bed).

Basal drag and basal sliding speed are derived using the ice sheet model WAVI, which is vertically integrated but retains an implicit velocity-depth profile . Data assimilation methods are used to initialize the model into a present-day state (approximately 2015): spatially varying two-dimensional fields of ice stiffness and basal drag are calculated by matching modelled surface velocities with observations of surface velocities , accumulation rates , and thinning rates . Internal ice temperatures are provided from a thermal solve of the BISICLES ice sheet model . Full details of the inverse method are detailed in , and the resulting basal sliding speed and basal drag are shown in Fig. S1. In this inversion, the basal drag is identified using the Weertman sliding law. However, the sliding relationship that links basal drag and basal speed can be re-parameterized in terms of any of the selected sliding laws that we test here, as long as neither the basal speed nor the basal drag are altered in this process.

The Viscous Grain-Shearing (VGS) theory is used to relate seismic observations to effective pressure (Fig. ). According to the VGS theory, the elastic deformation under effective pressure that generates frictional resistance also stiffens the sediment and increases the speed of propagation of sound waves. Changes in the speed of sound alter the acoustic impedance (Z=ρscp), the product of the compressional wave speed in the sediment (cp) and density (ρs). In turn, the acoustic impedance controls the reflection coefficient of seismic energy from the base of the ice sheet. The acoustic propagation model predicts the compressional wave speed (cp=ψ[N,dg,ϕ,fs]) as a function of effective pressure (N), grain diameter (dg), porosity (ϕ), and seismic frequency (fs). The link between the compressional wave speed and effective pressure predicted by the acoustic model provides an avenue to test whether a given sliding law applies at any location. All other parameters of the acoustic propagation model have been calibrated using acoustic observations of the ocean floor.

The governing equation for the compressional wave speed is 12cp=c0Re1+ζiωTqgωτp-1/2, where c0=κ0ρ0 is the sound speed in the absence of grain-to-grain interactions, κ0=ϕκp+1-ϕκg-1 the bulk modulus of the medium, and ρ0=ϕρp+(1-ϕ)ρg the bulk density of the medium. The dimensionless grain-shearing coefficient is 13ζ=γp+4/3γsρ0c02, where γp=γp0NdgN0dg01/3 and γs=γs0NdgN0dg02/3 are the compressional and shear rigidity coefficients, respectively. N0=(1-ϕ0)(ρg-ρp)gz0 is the reference effective pressure. The function 14gωτp=1+1iωτp-1+q accounts for the effect of the viscosity of the molecularly thin layer of pore fluid between contiguous grains (ν). Molecularly thin films become progressively more viscous as they are squeezed, and, therefore, ν differs significantly from the viscosity of the bulk fluid . The compressional viscoelastic time constant τp is defined as τp=ν/E, where E is a spring constant . The values of τp used in the VGS theory are visual fits to the SAX99 experiments . However, the measurements were taken in 18 to 19 m deep water . Therefore, the exerted overburden pressure is ∼2 orders of magnitude smaller (less squeezed) than under PIG ice thickness of 1500 to 2500 m in tributaries; e.g.,. While it is apparent that the viscosity of molecularly thin layers increases with the applied pressure (or loading) pL, the exact relationship between pL, the thickness of the thin film, and the viscosity ν is not straightforward e.g.,. Assuming ν∝pL, we set τp=0.012s 2 orders of magnitude larger than the value in. However, future studies should further explore the adaptation of the VGS theory from oceanographic to glacial contexts.

ω=2πf is the angular frequency, i=-1, and Re returns the real part of a complex number. All other parameters are listed in Table .

We compare the different sliding laws using Bayes' Rule: 15P(Mi|D,I)=P(D,I|Mi)P(Mi)P(D,I), where D represents the data (acoustic impedance observations), I represents the inverted ub and τb, and Mi represents the model for sliding law i together with the VGS theory. However, the situation here slightly differs from the routine application of Bayes' rule for inferring model parameters within a single model and is more akin to Bayesian model selection. The main difference for the model selection framework is that the probability space is extended to cover multiple models, each of which has its own parameter space. Since the number of parameters differs between models (e.g., two for the fixed effective pressure scenarios and four for the Zoet-Iverson sliding law) and we aim to compare the posterior probabilies of models P(Mi|D,I), not the joint posterior probability of models and parameters P(Θi,Mi|D,I), we marginalize over the model parameters Θi to retrieve P(D,I|Mi): 16aP(D,I|Mi)=∫ΘiP(D,I|Θi,Mi)P(Θi|Mi)dΘi16b=∫ΘiP(D|I,Θi,Mi)P(I|Θi,Mi)P(Θi|Mi)dΘi. Assuming the error of the data follows a Gaussian distribution, the likelihood of the acoustic impedance data given the model, its parameters, and the inverted ub-τb is calculated according to 17P(D|I,Θi,Mi)=exp⁡-0.5χΘi2. Therefore, the posterior probability of each model Mi is 18P(Mi|D,I)=∫Θiexp⁡-0.5χΘi2P(I|Θi,Mi)P(Θi|Mi)dΘiP(Mi)∑j=1n∫Θjexp⁡-0.5χΘj2P(I|Θj,Mj)P(Θj|Mj)dΘjP(Mj). The prior information from the inverted ub-τb (not used to constrain P(Θi|Mi)) can be directly incorporated into an updated prior using Bayes’ rule: 19P(Θi|I,Mi)=P(I|Θi,Mi)P(Θi|Mi)P(I|Mi), where P(I|Mi)=∫ΘiP(I|Θi,Mi)P(Θi|Mi)dΘi is a normalization term. Eq. () can then be written as 20P(Mi|D,I)=∫Θiexp⁡-0.5χΘi2P(Θi|I,Mi)dΘiP(Mi|I)∑j=1n∫Θjexp⁡-0.5χΘj2P(Θj|I,Mj)dΘjP(Mj|I), where we use a prior P(Mi|I)=1/n that considers each sliding law equally probable, with n being the number of sliding laws considered. Posterior probabilities calculated using P(Mi)=1/n, i.e. without the normalization through P(I|Mi) in Eq. (), are shown in Fig. S6.

Finally, the prior distributions for all model parameters P(Θi|Mi) need to be defined. The prior distributions for all individual parameters are shown in Fig. . The combination of multiple individual priors creates a model's parameter space Θi and determines the model prior P(Θi|Mi). Since the parameter space differs between the models (number of individual parameters (dimensions) as well as number of tested parameter values), we ensure ∫ΘiP(Θi|Mi)dΘi=1 for all models. This normalization reflects the fact that once a model has been chosen, the parameters of that model must lie somewhere within its parameter space with certainty. This is self-evident and automatically applies Occam's Razor, penalizing models with a larger parameter space compared to less flexible models. The key idea of Occam's Razor is that a balance between goodness of fit and model flexibility is desirable, but we emphasise that no special manipulations are required to enforce this balance in the Bayesian approach.

When constructing the parameter space Θi, the prior distributions of individual parameters are treated as independent of one another. Although physical relationships among some of these parameters have been described in the literature, the formulation of a coupled prior remains challenging, as these relationships are often convoluted by other properties. For instance, the porosity is generally inversely related to the mean (or median) grain size, but this relationship is convoluted by, e.g., the particle size uniformity e.g.,. As the Bayesian model selection framework already downweights extreme parameter combinations (e.g., high porosity and large grain size) through the chosen independent prior distributions, and because the minimum misfit and most probable parameters are generally consistent with, e.g., the porosity-grain size relationship described in the literature e.g.,, we do not expect a significant change in the posterior probabilities.

Various literature estimates inform the examined parameter ranges and corresponding prior distributions. The porosity prior (Fig. a) is derived from borehole data and seismic experiments from Ice Stream B and C, West Antarctica , borehole data from Trapridge Glacier, Yukon Territory, Canada , marine sediment cores from the Amundsen Sea Embayment Table S1;, sediment recovered from beneath Rutford Ice Stream, West Antarctica (Table S1; Smith, unpublished data), as well as the porosity of sands and glass beads used to validate the VGS theory and references therein. As the latter do not directly relate to a glacial context, we assign these higher porosities a lower probability. The porosity estimates from seismic experiments assume no significant dependence on effective pressure and are employed as an independent comparison rather than to directly inform the prior.

The grain diameter prior (Fig. b) is based on sediment cores collected in the Amundsen Sea Embayment, particularly Pine Island Bay Table S1; and the Rutford ice stream (Table S1; Smith, unpublished data). We differentiate between Clay (<1/256 mm), Silt (≥1/256 mm and ≤1/16 mm), and Sand (>1/16 mm). The prior is then derived from the relative fractions of these grain-size classes.

The transition speed coefficient (CZI) values reported in the initial publication of the Zoet-Iverson sliding law range from 56.36 to 363.52 MPa⁻¹ m a⁻¹ . A later study using the same bed material (Horicon till sourced from the same location) but with plowing clasts removed uses the same parameters given in Table S1 of except for a smaller clast radius R=0.0045 m (instead of R=[0.015,0.030] m), leading to CZI=1120.17 MPa⁻¹ m a⁻¹ Fig. S4 in. Given these significant uncertainties and that CZI depends on several other uncertain parameters, a log-uniform prior covering the range 3.16 to 3155.76 MPa⁻¹ m a⁻¹ was chosen (Fig. c).

Due to the range of spatial scales in bed roughness that can affect basal drag, estimating Cmax from observations of bed topography is not straightforward. We therefore base our Cmax prior (Fig. 3d) on a combination of coarse-resolution bed topography beneath PIG retrieved from Bedmap2 data Figs. S7 and S8;, as well as high-resolution autonomous underwater vehicle (AUV) data collected downstream of Thwaites Glacier 1.5 m; and under the Thwaites Eastern Ice Shelf 2 m;Figs. S9 and S10. Although shear resistance is most likely built at spatial scales smaller than the resolution of Bedmap2, these data provide a conservative lower bound on Cmax (Sect. S6.2).

μ is a frequently used parameter and its prior (Fig. e) aims to capture the overall distribution within the glaciological community e.g.,. Note that although Cmax and μ serve similar roles in, e.g., the Schoof and Zoet-Iverson sliding law, they represent distinct physical properties and are thus assigned separate prior distributions (Sect. ).

As CB and CS are positive scaling coefficients that may vary over several orders of magnitude, even within the same glacial catchment , a log-uniform prior was chosen for these parameters (Fig. f).

Due to the computational cost of the grid search, we currently limit the model parameter space Θi to 4D. For example, we do not consider variations in the exponents m, q, and p (Sect. ). However, computationally more efficient methods, such as Monte Carlo algorithms, can be explored in future studies to simultaneously vary more than four parameters.

Based on a previous study examining the same acoustic impedance data and due to the smoothing effect of the inversion (1 km horizontal grid resolution), we do not expect to capture acoustic impedance variations for each individual data point but rather the general trend across the five data sites. Given this context, all sliding laws reasonably match the acoustic impedance observations when using the parameter values yielding the minimum misfit across all data sites (Fig. ). However, for some sliding laws, the minimum misfit parameter values are at the limits of the likely range (e.g., extremely small grain diameter (dg=0.003 mm) for the Budd sliding law). While the minimum misfit might correspond to a rather unlikely parameter value, a narrow band of similarly small misfits spans a more reasonable parameter range, indicating some indistinctness in the selected minimum misfit parameter values. This is a key characteristic of the misfit distribution in all of our experiments. As an example, Fig.  shows how the misfit varies with the three model parameters dg, ϕ, and μ when using a Coulomb sliding law. The same plots for all other sliding laws with a maximum 3D parameter space are shown in Figs. S11 to S22.

To consider the misfit distribution across the entire parameter range and any prior assessment of the probability of the parameter values used, we infer the best-candidate sliding law based on Bayesian model selection. The Coulomb sliding law has the highest posterior probability of all sliding laws tested (increase of 27.5 % relative to the prior; Fig. ). However, the Schoof and Zoet-Iverson sliding laws show a similarly strong increase, hindering the determination of a single-best sliding law. The Tsai-Budd sliding law exhibits the smallest increase (4.8 %) out of all the laws incorporating a Coulomb friction term of the form μN or CmaxN. Nonetheless, the increase in posterior probability for all sliding laws incorporating a Coulomb friction term suggests this is a desirable property of a sliding law. In comparison, the Budd sliding law, without the μN modification of the Tsai-Budd law, performs worse (0.8 % decrease). The fixed effective pressure endmember scenario that assumes N=pi everywhere performs worst of all, leading to the smallest posterior probability (83.4 % decrease). The endmember scenario with N=0 Pa everywhere yields the highest posterior probability of all fixed effective pressure experiments (4.1 % increase; see also Fig. S23).

The relatively high posterior probabilities of sliding laws incorporating a Coulomb friction term and the N=0 Pa endmember scenario are consistent with the widespread occurrence of deformable sediment under the fast-flowing tributaries of PIG . Furthermore, the high probabilities of these sliding laws align with previous studies identifying (quasi-)plastic deformation of the underlying sediment as the primary mode of sliding for PIG . While the sensitivity of grounding-line retreat patterns and mass loss projections to the choice of sliding law is high , determining the exact implications of using a (quasi-)plastic sliding law on glacier behaviour through prognostic simulations for all sliding laws and parameter values is out of the scope of this study. In general, sliding laws representing a (quasi-)plastic rheology lead to higher sea level rise contributions .

As for the minimum misfit model parameters, the predicted acoustic impedance under the model parameters with the highest posterior probability generally agrees with the observations within uncertainties for all sliding laws tested (Fig. S24). In the remainder of this paper, we refer to the model parameters with the highest posterior probability as the maximum a posteriori (MAP) parameters. When examining the MAP parameters in more detail (Fig. S24), the effect of the chosen prior distributions is evident. Although covering the full range within this size classification, the MAP grain diameter for all sliding laws is Silt-sized (highest prior probability; Fig. ). The MAP porosities (0.39 to 0.44) are at the upper end of the high-prior probability range (ϕ=[0.3,0.45]) for all sliding laws except the fixed effective pressure endmember scenario N=pi (ϕ=0.55; Fig. S24), indicating comparatively porous sediments beneath PIG. Similarly, the MAP values of the unique sliding law parameters without a log-uniform prior distribution (μ and Cmax) are in the vicinity of the highest prior probability.

Even when using log-uniform prior distributions for scaling coefficients and uniform priors for other parameters – thus making no use of the Bedmap2 or AUV data to constrain the Cmax prior – the sliding laws incorporating a Coulomb friction term still yield the highest probabilities, with the Coulomb and Schoof sliding law showing the greatest increase (26.3 % for both; Fig. S25). This demonstrates the robustness of our key result against variations in prior distributions.

Excluding the fixed effective pressure scenarios, the predicted effective pressure for the MAP model parameters is generally below 0.1 MPa (1 bar) for the 4 sites within fast-flowing tributaries (Fig. S26). The relatively high probability of the N=0 Pa endmember scenario (Figs.  and S23) further supports a low effective pressure. This is in agreement with previous effective pressure estimates derived from, e.g., shear wave velocities , borehole water level measurements , and the widespread presence of active subglacial lakes .

Site iSTARit, located between two tributaries, has higher effective pressures (0.1 to 1 MPa), with the effective pressure derived from the Coulomb sliding law being ∼0.1 MPa. We hypothesize that the higher effective pressure and resulting increased basal drag at this site hinder basal sliding.

Retrieving the effective pressure for the Coulomb sliding law with the MAP parameters across the whole Amundsen Sea Embayment indicates the effective pressure is generally below 0.5 MPa (Fig. b). Being closely related to the basal drag, this map represents the slipperiness of the bed, with areas of low effective pressure being susceptible to fast retreat. However, the effective pressure map is based on a spatially uniform μ obtained from five sites in PIG and does not capture (local) dynamic subglacial systems as, e.g., represented by a subglacial hydrology model. Furthermore, using only the Coulomb sliding law with the MAP parameters neglects the probabilities of other sliding laws and parameter values. Therefore, the provided effective pressure map should be used with caution. Following the Bayesian framework to determine the most probable effective pressure map by weighting the individual maps for all sliding laws and parameter values, incorporating spatially variable model parameters, as well as applying BASLI–VGS in regions characterized by higher basal heterogeneity (e.g., Thwaites Glacier), should be explored in future studies.