To test the relative influences of the background rock density and Bouguer disturbance uncertainties on bathymetry estimates, we ran three ensembles: one with variable background densities sampled from a Gaussian prior distribution centered at 2700 kg m⁻³ with a standard deviation of 80 kg m⁻³ and SGS interpolation of the Bouguer disturbance; one with constant background density and SGS interpolation of the Bouguer disturbance, and finally; one with constant background density and a constant Bouguer disturbance which is a kriging interpolation of the Bouguer disturbance (Table ). Each ensemble is composed of 100 inversions run in parallel on a 2021 Mac Studio M1 computer. Each SGS interpolation and MCMC inversion has a combined wall time of about 5 min. The inversions are completed using a random walk Metropolis MCMC approach that iteratively updates the bathymetry with block updates until the root mean squared error (RMSE) between the forward modeled gravity and the target terrain effect is less than 1.2 mGal. This stopping criterion was chosen because the inversions were able to reliably achieve this RMSE. The final iteration of each inversion, the misfit throughout optimization, and the accepted iterations are saved at the end of each inversion. Figure  describes the MCMC approach used for the inversions. The following sections describe how the Bouguer disturbance is stochastically interpolated with SGS, derive the random walk MCMC approach for this problem, explain the choice of block updates, describe the forward modeling of the gravity data, and finally describe the initialization of the bathymetry models.

SGS is a geostatistical interpolation technique that preserves the spatial statistics of conditioning data while sampling the uncertainty of the interpolation . In contrast to SGS, traditional interpolation methods such as kriging, splines, and Radial Basis Functions provide overly smooth interpolations that do not retain the spatial statistics of the conditioning data. SGS works by moving through a random sequential path in the grid. At each grid cell, if the Bouguer disturbance is unknown, a value is sampled from a Gaussian distribution defined by the kriging mean and variance and entered into the grid . In this way, as the interpolation moves through the grid there are more and more values from which to compute the kriging mean and variance, which is conditioned by a specified number of neighbors away from the grid cell. The kriging equations incorporate the geostatistics of the conditioning data through the (semi)variogram, which is a measure of the pairwise dissimilarity between data points with respect to the distance, or lag, between the pair of data points. We generate the experimental variogram and fit Gaussian, exponential, spherical and Matérn covariance models to the variogram using Scikit GStat . The Matérn covariance model fit the experimental variogram the best and thus was used for the interpolation. The experimental variogram and fit are shown in Fig. b. We use the Ordinary Kriging SGS interpolation function from the open-source Python package GStatSim where the Ordinary Kriging SGS methods are robust to trends in the Bouguer disturbance. We generate 100 realizations of the SGS interpolation for use in an ensemble that quantifies the uncertainty in the bathymetry due to the interpolation of the Bouguer disturbance. Finally, we interpolated the Bouguer disturbance deterministically with Ordinary Kriging using the same Matérn variogram (Fig. e) to create an ensemble without the uncertainty due to the interpolation.

We solve the gravity inverse problem using an MCMC approach where random blocks of the bed topography are iteratively perturbed until convergence is achieved. The approach works by iteratively proposing new bathymetric surfaces, forward modeling the gravity due to the new model, and accepting or rejecting the new model. After many iterations, the MCMC algorithm is able to converge to a bathymetry model that approximately matches the forward modeled gravity and the target terrain effect. An outline of the inversion process is shown in Fig. .

MCMC is a popular approach for generating samples from a posterior distribution of model parameters, often by proposing new models and accepting or rejecting moves to new models . Typically, MCMC is viewed in a Bayesian context in which prior beliefs about the model parameters are transformed into the posterior distribution by a likelihood function which calculates the probability of a dataset given the model parameters . However, our MCMC approach is best viewed as a stochastic optimization approach to generate a sample from the posterior within a larger Monte Carlo framework. The background density is sampled first (or remains constant as in ensembles 2 and 3) and then an SGS interpolation of the Bouguer disturbance is simulated. The MCMC approach is then used to perturb the bathymetry model until the forward modeled bathymetry approximately matches the target terrain effect, g^terrain. We take only the final iteration from each MCMC chain because we are primarily concerned with generating ensemble members that correspond to different SGS interpolations. Previous work has enforced geostatistical priors over model parameters in MCMC in the form of SGS e.g.,, however, we do not specify a geostatistical prior for the sake of simplicity and to fit the gravity data as closely as possible. We use a random walk Metropolis (RWM) approach, a simplification of the popular Metropolis-Hastings MCMC algorithm, in which new model proposals are the previous model with the addition of a zero-mean multivariate normal perturbation . In our case, these updates are 2D Gaussian random fields (see Sect. ).

We derive the RWM gravity inverse problem from Bayes' theorem where m represents a vector of seafloor depth values and d represents a vector of gravity disturbances: 4P(m|d)=P(d|m)⋅P(m)P(d) where P(m|d) is the posterior distribution of m given d, P(d|m) is the likelihood, P(m) is the prior distribution of m and P(d) is the evidence. We define the forward model as 5d=g(m) where d is the gravity data and g(m) is the forward gravity function evaluated on the bathymetry model. We assume independent and identically distributed Gaussian uncertainties represented by s, as in , making the likelihood : 6P(d|m)=exp⁡-S(m)s2 where 7S(m,d)=12∑i=1Ngi(m)-di2 is the sum of squared residuals divided by two and s is the uncertainty of the gravity data derived from crossover errors. The Metropolis-Hastings acceptance probability, α(mnew), is defined as 8αmnew=minPmnew|dqmnew,moldPmold|dqmold,mnew,1 where q(mnew,mold) is the probability of proposing the new model given the old model. Given a symmetric proposal generation mechanism, as is the case with a zero-mean multivariate update, the probability of proposing the forward and reverse jumps are the same and the proposal distributions cancel. Using Bayes' theorem we can expand the posterior: 9αmnew=minPd|mnewPmnewP(d)Pd|moldPmoldP(d),1. The prior probability distribution for the model space, P(m), is a uniform distribution and thus P(mnew) and P(mold) are equivalent and cancel. Thus, the acceptance probability simplifies to 10αmnew=minPd|mnewPd|mold,1 and the acceptance probability can then be put in terms of the misfit function as 11αmnew=minexp⁡-Smnew,d-Smold,ds2,1. This expression implies that if the new model improves the least squares misfit, the model is accepted and if it does not, there is still a non-zero probability that it is accepted. This simple approach is advantageous because we require no evaluation of gradients and only require the forward gravity model and a multivariate normal perturbation method.

The large number of bathymetry values to be modeled poses computational challenges in achieving convergence to the desired posterior distribution. In geostatistical inverse problems, using whole domain updates often results in very low acceptance rates and slow refinement of the model because one area may improve while another area worsens the fit with the data . Because of this, it is advantageous to limit the updates to smaller portions of the model where the bathymetry is expected to be correlated in space .

To address this optimization challenge, we used block updates to perturb the bathymetry models. For each MCMC iteration, a random location within the inversion domain is selected as the center of the block. The block perturbation is a Gaussian random field confined to the block which is added to the previous bed. The blocks are varied by their size in pixels, their correlation length (range) in pixels, and their amplitude in meters.

Our ensemble approach relies on taking a sample from the ends of many MCMC chains, motivating a RWM approach which efficiently converges to producing samples within the distribution. In order to have the RWM converge as efficiently as possible we follow an annealing schedule which specifies the block size, the range and amplitude of the Gaussian random field, and number of iterations that the block update parameters are used for. The block update schedule used for the inversion is shown in Table . As the inversion converges to a bathymetry model, smaller block sizes with shorter ranges and lower amplitudes become more efficient. We tested different annealing schedules and qualitatively compared their acceptance rates and efficiency in reducing the gravity data misfit. We targeted acceptance rates between 0.2 and 0.5, which is an acceptable range for most MCMC applications . The annealing schedule shown in Table was chosen because it kept acceptance rates within the desired range and the reduction in RMSE per iteration high as seen in Fig. .

The block update method can introduce edge artifacts into the bathymetry. Many of these edges are smoothed out by the continued updating of the model, but some edges remain visible in the final iteration. To eliminate these features, the final iteration is low-pass filtered with a Gaussian filter with a cutoff wavelength of 5 km.

The gravity response, d, is forward modeled g(m) using the gridded BedMachine v3 ice surface and ice thickness to generate rectangular prisms of constant density. We use an ice density of 917 kg m⁻³ and sea water density of 1027 kg m⁻³. We use the open-source Python package, Harmonica , to forward model the vertical gravitational acceleration of the prisms at the gridded gravity coordinates using the equations from . We then take the residual between the forward modeled data and the target terrain effect, calculate the Metropolis acceptance criterion (Eq. ), and accept or reject the block update. The prism gravity method is advantageous because of its simplicity, but it can be computationally challenging because the gravity of each prism must be added together to compute the gravity at a single observation location. In order to speed up the gravity calculation, we leveraged the additive nature of the prism gravity method and limited the calculation to the updated block. For each gravity observation point, the gravity due to the new and previous prisms of the block are computed and the new gravity is calculated by subtracting the previous gravity due to the block prisms and adding the gravity due to the new block prisms to the total gravity. This approach eliminates redundant gravity calculations and greatly speeds up the forward model calculation.

It is typical in MCMC to initialize chains with different initial conditions to avoid getting stuck in local modes . Each inversion is initialized with the BedMachine bed with the addition of a Gaussian random field conditioned on the margins of the inversion domain so that each MCMC chain has a different initial condition. The primary and secondary range, or correlation distance, of the random field were both randomly sampled from uniform distributions with a maximum of 30 and 50 km to add anisotropy to the random fields. The random field is additionally scaled by a random value between 1 and 300 m and finally is conditioned on the margins of the inversion domain using Radial Basis Functions. These Gaussian random fields are generated using the open-source Python package GSTools .

Gravity inversions provide a means to infer the bathymetry beneath ice shelves, which is critical for understanding how mCDW moves beneath ice shelves and drives melting. However, gravity inversions are fundamentally non-unique and rely on assumptions about the background density and the Bouguer disturbance, or the portion of the gravity disturbance which comes from crustal density variations and the thickness of the crust. Previously, the Bouguer disturbance has been interpolated using one-time, deterministic interpolations. Instead, our new approach uses SGS to sample the uncertainty of the Bouguer disturbance interpolation. Our annealed RWM approach is able to efficiently solve the inverse problem, enabling us to construct ensembles of bathymetry models. We produce three ensembles with 100 MCMC inversions each of the bathymetry beneath the Dotson, Crosson, and Thwaites Eastern Ice Shelf in the Amundsen Sea. In the ensembles including SGS interpolations, each inversion includes an independent realization of the interpolation of the Bouguer disturbance. Our MCMC approach to solving the inverse problem uses block updates such that each inversion samples the uncertainty of the gravity data and the nonuniqueness of the problem. The annealed block update schedule has a clear effect on the efficiency of the inversion (Fig. ), with increases in acceptance rate after changing the parameters.

Through our set of three ensembles shown in Fig. , we can see that the Bouguer interpolation is the largest contributor to the uncertainty but the background density is not. This is shown by the fact that the ensemble with both Bouguer SGS interpolation and random background densities produces little difference with the ensemble with Bouguer SGS interpolation but a fixed background density of 2670 kg m⁻³. The final ensemble that uses a fixed density and a fixed mean Bouguer interpolation shows much smaller uncertainty, reflecting only the uncertainty due to crossover errors in the gravity data. The effect of the SGS interpolation of the Bouguer disturbance on the uncertainty is clearly visible in the difference between Figs.  and . These results show that uncertainty due to the background density may have been over-represented while uncertainty due to unknown geologic factors was unaccounted for. The previous inversions of and acknowledge that unknown geologic factors are difficult to account for. Our approach improves this issue by producing different bathymetry models that sample the uncertainty in the geology by stochastically interpolating the Bouguer disturbance. As such, we believe our approach represents the most robust inference and uncertainty quantification of the bathymetry beneath the Crosson, Dotson, and Thwaites Eastern ice shelves to date.

Our ensembles show consistent differences between the BedMachine bathymetry, which may have impacts on ocean circulation between the Crosson and Dotson ice shelves. Figure g–i shows that each ensemble produces a mean bathymetry that is generally deeper than BedMachine in the strait between the Crosson and Dotson ice shelves. This is because a portion of the strait is constrained by seismic soundings from , which showed deeper seafloor than BedMachine with the exception of a notable rise in the center of the seismic sounding area. The deeper seafloor measurements from the seismic soundings then drives the gravity inversion to be similarly deeper than BedMachine around it due to the influence of the Bouguer interpolation. A deeper strait could mean that there is more transport of water between the eastern and western portions of the Amundsen Sea than previously thought. The ability to move more water through this strait could indicate that more mCDW is delivered to the grounding line in this area than previously thought. Conversely, the mean of the ensembles are generally shallower than BedMachine below the Crosson and Dotson ice shelves, which could impact the amount of mCDW able to circulate in the sub-ice-shelf cavities. The means of the ensembles show little difference with BedMachine beneath Thwaites Eastern Ice Shelf. Individual models seen in Figs.  or  have much more variable water-column thickness than the mean models, meaning the individual models could impede the movement of water more than the smoother representative models because of their more complex geometries. The variability seen in the SGS ensemble members is justified by the fact that the seismic constraints on Dotson Ice Shelf show large deviations from the BedMachine bathymetry (Fig. a). As such, our ensemble members capture the uncertainty in such bathymetric features below the ice shelves. It will be useful for ocean modelers to see to what extent the roughness of the sub-ice-shelf cavities influences circulation and how this differs from a smoother model like the mean of an ensemble.

Our ensembles of bathymetry models could be useful for exploring glaciation history. The variability of our models in the ensembles including SGS is such that they include bathymetric highs which could have previously pinned the ice shelves. These possible pinning points could help explain past advance and retreat behavior. Conversely, ice-sheet-model spin up could be used to narrow the ensemble of bathymetry model to only those that can reproduce present conditions. As such, the ensemble of bathymetry models are useful for testing hypotheses about past ice-sheet configurations as well as further constraining the bathymetry.

Our ensembles not only satisfy the goal of quantifying uncertainty across the inversion domain, but allow for easy propagation of the uncertainty in bathymetry to ocean and ice-sheet models. This could be accomplished by running ocean and ice-sheet models with different bathymetry realizations, thereby producing an ensemble of model outputs. In this way, the uncertainty in the sub-ice-shelf melt and grounding-line retreat due to bathymetry can be better assessed. Given the computational cost of these downstream models, ensemble approaches to uncertainty quantification are not always feasible. However, recently both and demonstrated the use of deep learning to emulate an ocean model, thereby immensely reducing the computational cost of producing ice-shelf melt rates. Emulators like those could be used with our bathymetry ensembles to rapidly propagate the uncertainty in bathymetry to ice-shelf melt rates and ice loss projections. As such, our stochastic approach to bathymetry inversion enables improved uncertainty quantification of bathymetry but also for future ice-sheet stability.

The nonuniqueness of the the gravity method is such that the sediment thickness cannot be estimated without sediment-thickness constraints from boreholes or seismic measurements. One advantage of our method is that the Bouguer disturbance contains the gravity disturbance from density contrasts due to sedimentary rock, unconsolidated sediments, or dense igneous bodies. Therefore, despite not modeling sediment thickness directly, the SGS interpolation of the Bouguer disturbance can be interpreted as a stochastic interpolation of crustal density anomalies. However, some of the Bouguer disturbance in the conditioning area could arise from errors in the ice thickness from BedMachine as well as the gridding used to coarsen BedMachine to a 2 km grid. In order to account for the uncertainty in the conditioning data, the bed data (e.g. radar and sonar data) could be sampled from a distribution at each iteration in the chain instead of being treated as a deterministic value. Our inversion approach could also be improved by refining the grid near the grounding lines and pinning points, both of which exhibit complex geometries . While the gravity inversion could not necessarily resolve the bathymetry at these finer resolutions, the complex structures that are important to the ice-shelf dynamics could be better resolved. Furthermore, the annealing schedule used for the block updates was chosen arbitrarily through testing, and more systematic approaches to choosing the block update parameters could be used. Alternatively, the block size, correlation distance, and amplitude could all be treated as random variables that are sampled from each iteration, giving the benefits of larger and smaller updates throughout the whole inversion. Many such improvements should be considered in the future but ultimately, the combination of conditional simulation and stochastic optimization to interpolate the Bouguer disturbance and produce ensembles of bathymetry models is a step forward in incorporating the uncertainty due to unknown geologic factors in the inversion.

Our inversion uses only the vertical component of the acceleration due to gravity, but would greatly benefit from having direct measurements of the gradient of the gravity field. A test study from demonstrated that the inversion of bathymetry benefits from the inclusion of gravity-gradient data. Additionally, gravity-gradient data could improve the inversion of subglacial geology because of increased sensitivity to lateral changes in the gravity field. For the same reasons, the petroleum industry has routinely used gravity-gradient data to improve the delineation of subsurface oil and gas reservoirs e.g.,. Because of this, it would be beneficial for the community to explore future opportunities to acquire full-tensor gravity-gradient data.

We chose to study the Dotson, Crosson, and Thwaites Eastern Ice shelves because of their sensitivity to mCWD moving up the continental shelf and the important role they play in buttressing the fast-flowing glaciers terminating in the Amundsen Sea e.g.,. However, our method is readily transferable to other ice shelves in Antarctica and Greenland. Our approach requires airborne gravity data to be present and benefits from dense RES data on grounded ice and MBES or seismic bathymetry measurements offshore. The Thwaites region has dense airborne gravity data relative to other ice shelves. Where airborne gravity data is sparse, satellite gravity data can be used to fill in gaps. The ANTGG2021 continent-wide gravity product uses a combination of airborne and satellite gravity data, and can be used at ice shelves where airborne data is sparse . At other ice shelves, it may be necessary to detrend the Bouguer disturbance before interpolating with SGS. In addition, directional variograms and anisotropic SGS interpolations could be used in cases where there are elongated features in the Bouguer disturbance conditioning data. The fewer bed measurements there are, the less conditioning data are available to calculate the variogram of the Bouguer disturbance and the greater the uncertainty will be in SGS interpolations. Our approach motivates the continued acquisition of airborne and seaborne geophysical measurements along the Antarctic coast to improve bathymetric estimates and uncertainty quantification. The uncertainty derived from our bathymetry ensembles could be instructive in planning future underwater autonomous and remotely operated vehicle missions, seismic soundings, or further airborne geophysics to areas of high uncertainty see e.g.,.

Our optimization approach is flexible and could be expanded to include other data such as magnetics, so long as the forward model is sufficiently fast. The Python package Harmonica includes prism-based magnetic forward modeling functions so joint inversion of gravity and magnetics would be possible . The interpolated Bouguer disturbances represent an ensemble of gravity due to crustal density anomalies, mantle density, and crustal thickness, and thus could be used to generate an ensemble of inverted density models. Such an inversion could be done with our block perturbation approach where the surface is the bottom of sediments or the top of an igneous body. Our approach could be extended beyond surface inversions to 3D density inversions using 3D Gaussian random fields.