Comment on gmd-2021-321

The article studies non‐Gaussian extensions of a recently discovered link between certain Gaussian random fields, expressed as solutions to stochastic partial differential equations (SPDEs), and Gaussian Markov random fields. The focus is on non‐Gaussian random fields with Matérn covariance functions, and in particular, we show how the SPDE formulation of a Laplace moving‐average model can be used to obtain an efficient simulation method as well as an accurate parameter estimation technique for the model. This should be seen as a demonstration of how these techniques can be used, and generalizations to more general SPDEs are readily available.

Abstract. The Antarctic Ice Sheet represents the largest source of uncertainty in future sea level rise projections, with a contribution to sea level by 2100 ranging from −5 to 43 cm of sea level equivalent under high carbon emission scenarios estimated by the recent Ice Sheet Model Intercomparison for CMIP6 (ISMIP6). ISMIP6 highlighted the different behaviors of the East and West Antarctic ice sheets, as well as the possible role of increased surface mass balance in offsetting the dynamic ice loss in response to changing oceanic conditions in ice shelf cavities. However, the detailed contribution of individual glaciers, as well as the partitioning of uncertainty associated with this ensemble, have not yet been investigated. Here, we analyze the ISMIP6 results for high carbon emission scenarios, focusing on key glaciers around the Antarctic Ice Sheet, and we quantify their projected dynamic mass loss, defined here as mass loss through increased ice discharge into the ocean in response to changing oceanic conditions. We highlight glaciers contributing the most to sea level rise, as well as their vulnerability to changes in oceanic conditions. We then investigate the different sources of uncertainty and their relative role in projections, for the entire continent and for key individual glaciers. We show that, in addition to Thwaites and Pine Island glaciers in West Antarctica, Totten and Moscow University glaciers in East Antarctica present comparable future dynamic mass loss and high sensitivity to ice shelf basal melt. The overall uncertainty in additional dynamic mass loss in response to changing oceanic conditions, compared to a scenario with constant oceanic conditions, is dominated by the choice of ice sheet model, accounting for 52 % of the total uncertainty of the Antarctic dynamic mass loss in 2100. Its relative role for the most dynamic glaciers varies between 14 % for MacAyeal and Whillans ice streams and 56 % for Pine Island Glacier at the end of the century. The uncertainty associated with the choice of climate model increases over time and reaches 13 % of the uncertainty by 2100 for the Antarctic Ice Sheet but varies between 4 % for Thwaites Glacier and 53 % for Whillans Ice Stream. The uncertainty associated with the ice–climate interaction, which captures different treatments of oceanic forcings such as the choice of melt parameterization, its calibration, and simulated ice shelf geometries, accounts for 22 % of the uncertainty at the ice sheet scale but reaches 36 % and 39 % for Institute Ice Stream and Thwaites Glacier, respectively, by 2100. Overall, this study helps inform future research by highlighting the sectors of the ice sheet most vulnerable to oceanic warming over the 21st century and by quantifying the main sources of uncertainty.

Circular data can be found across many areas of science, for instance meteorology (e.g., wind directions), ecology (e.g., animal movement directions), or medicine (e.g., seasonality in disease onset). The special nature of these data means that conventional methods for non-periodic data are no longer valid. In this paper, we consider wrapped Gaussian processes and introduce a spatial model for circular data that allow for non-stationarity in the mean and the covariance structure of Gaussian random fields. We use the empirical equivalence between Gaussian random fields and Gaussian Markov random fields which allows us to considerably reduce computational complexity by exploiting the sparseness of the precision matrix of the associated Gaussian Markov random field. Furthermore, we develop tunable priors, inspired by the penalized complexity prior framework, that shrink the model toward a less flexible base model with stationary mean and covariance function. Posterior estimation is done via Markov chain Monte Carlo simulation. The performance of the model is evaluated in a simulation study. Finally, the model is applied to analyzing wind directions in Germany.

Abstract. Computer models of ice sheet behavior are important tools for projecting future sea level rise. The simulated modern ice sheets generated by these models differ markedly as input parameters are varied. To ensure accurate ice sheet mass loss projections, these parameters must be constrained using observational data. Which model parameter combinations make sense, given observations? Our method assigns probabilities to parameter combinations based on how well the model reproduces the Greenland Ice Sheet profile. We improve on the previous state of the art by accounting for spatial information and by carefully sampling the full range of realistic parameter combinations, using statistically rigorous methods. Specifically, we estimate the joint posterior probability density function of model parameters using Gaussian process-based emulation and calibration. This method is an important step toward calibrated probabilistic projections of ice sheet contributions to sea level rise, in that it uses data–model fusion to learn about parameter values. This information can, in turn, be used to make projections while taking into account various sources of uncertainty, including parametric uncertainty, data–model discrepancy, and spatial correlation in the error structure. We demonstrate the utility of our method using a perfect model experiment, which shows that many different parameter combinations can generate similar modern ice sheet profiles. This result suggests that the large divergence of projections from different ice sheet models is partly due to parametric uncertainty. Moreover, our method enables insight into ice sheet processes represented by parameter interactions in the model.

Bayesian whole-brain functional magnetic resonance imaging (fMRI) analysis with three-dimensional spatial smoothing priors has been shown to produce state-of-the-art activity maps without pre-smoothing the data. The proposed inference algorithms are computationally demanding however, and the spatial priors used have several less appealing properties, such as being improper and having infinite spatial range. We propose a statistical inference framework for whole-brain fMRI analysis based on the class of Matérn covariance functions. The framework uses the Gaussian Markov random field (GMRF) representation of possibly anisotropic spatial Matérn fields via the stochastic partial differential equation (SPDE) approach of Lindgren et al. (2011). This allows for more flexible and interpretable spatial priors, while maintaining the sparsity required for fast inference in the high-dimensional whole-brain setting. We develop an accelerated stochastic gradient descent (SGD) optimization algorithm for empirical Bayes (EB) inference of the spatial hyperparameters. Conditionally on the inferred hyperparameters, we make a fully Bayesian treatment of the brain activity. The Matérn prior is applied to both simulated and experimental task-fMRI data and clearly demonstrates that it is a more reasonable choice than the previously used priors, using comparisons of activity maps, prior simulation and cross-validation.

Abstract. The Greenland Ice Sheet is a key contributor to sea level rise. By melting, the ice sheet thins, inducing higher surface melt due to lower surface elevations, accelerating the melt coming from global warming. This process is called the melt–elevation feedback and can be considered by using two types of models: either (1) atmospheric models, which can represent the surface mass balance (SMB), or SMB estimates resulting from simpler models such as positive degree day models or (2) ice sheet models representing the surface elevation evolution. The latter ones do not represent the surface mass balance explicitly as well as polar-oriented climate models. A new coupling between the MAR (Modèle Atmosphérique Régional) regional climate model and the PISM (Parallel Ice Sheet Model) ice sheet model is presented here following the CESM2 (Community Earth System Model; SSP5-8.5, Shared Socioeconomic Pathway) scenario until 2100 at the MAR lateral boundaries. The coupling is extended to 2200 with a stabilised climate (+7 ∘C compared to 1961–1990) by randomly sampling the last 10 years of CESM2 to force MAR and reaches a sea level rise contribution of 64 cm. The fully coupled simulation is compared to a one-way experiment where surface topography remains fixed in MAR. However, the surface mass balance is corrected for the melt–elevation feedback when interpolated on the PISM grid by using surface mass balance vertical gradients as a function of local elevation variations (offline correction). This method is often used to represent the melt–elevation feedback and prevents a coupling which is too expensive in computation time. In the fully coupled MAR simulation, the ice sheet morphology evolution (changing slope and reducing the orographic barrier) induces changes in local atmospheric patterns. More specifically, wind regimes are modified, as well as temperature lapse rates, influencing the melt rate through modification of sensible heat fluxes at the ice sheet margins. We highlight mitigation of the melt lapse rate on the margins by modifying the surface morphology. The lapse rates considered by the offline correction are no longer valid at the ice sheet margins. If used (one-way simulation), this correction implies an overestimation of the sea level rise contribution of 2.5 %. The mitigation of the melt lapse rate on the margins can only be corrected by using a full coupling between an ice sheet model and an atmospheric model.

Stochastic methods have been typically used for the design and operations of hydraulic infrastructure. They allow decision makers to evaluate existing or new infrastructure under different possible scenarios, giving them the flexibility and tools needed in decision making. In this paper, we present a novel stochastic streamflow simulation approach able to replicate both temporal and spatial dependencies from the original data in a multi‐site basin context. The proposed model is a multi‐site extension of the modified Fractional Gaussian Noise (mFGN) model which is well‐known to be efficient to maintain periodic correlation for several time lags, but presents shortcomings in preserving the spatial correlation. Our method, called Weighted‐mFGN (WmFGN), incorporates spatial dependency into streamflows simulated with mFGN by relying on the Cholesky decomposition of the spatial correlation matrix of the historical streamflow records. As the order in which the decomposition steps are performed (temporal then spatial, or vice‐versa) affects the performance in terms of preserving the temporal and spatial correlation, our method searches for an optimal convex combination of the resulting correlation matrices. The result is a Pareto‐curve that indicates the optimal weights of the convex combination depending on the importance given by the user to spatial and temporal correlations. The model is applied to a number of river basins in Chile, where the results show that the WmFGN approach maintains the qualities of the single‐site mFGN, while significantly improving spatial correlation.

This paper presents a computational framework for uncertainty quantification in prognostics in the context of condition-based monitoring of aerospace systems. The different sources of uncertainty and the various uncertainty quantification activities in condition-based prognostics are outlined in detail, and it is demonstrated that the Bayesian subjective approach is suitable for interpreting uncertainty in online monitoring. A state-space model-based framework for prognostics, that can rigorously account for the various sources of uncertainty, is presented. Prognostics consists of two important steps. First, the state of the system is estimated using Bayesian tracking, and then, the future states of the system are predicted until failure, thereby computing the remaining useful life of the system. The proposed framework is illustrated using the power system of a planetary rover test-bed, which is being developed and studied at NASA Ames Research Center.

This paper is one in a sequence of presentations that consider the problem of uncertainty quantification (UQ) in model verification and validation. The particular sub-topic considered here is uncertainty quantification in modeling. The objectives of the paper are to enumerate some of the various sources of uncertainty in models, as-built systems and experimental data, to define uncertainty and errors and differences between them, to introduce different types of comparative difference metrics that can be used for uncertainty quantification, and to present examples illustrating their uses. Particular emphasis is placed on the ability to quantify modeling uncertainty that includes model form uncertainty, i.e. an epistemic uncertainty that results from an incomplete understanding of the physics of the phenomenon being modeled, and/or undetected modeling error.

Abstract. Observations and models agree that the Greenland Ice Sheet (GrIS) surface mass balance (SMB) has decreased since the end of the 1990s due to an increase in meltwater runoff and that this trend will accelerate in the future. However, large uncertainties remain, partly due to different approaches for modelling the GrIS SMB, which have to weigh physical complexity or low computing time, different spatial and temporal resolutions, different forcing fields, and different ice sheet topographies and extents, which collectively make an inter-comparison difficult. Our GrIS SMB model intercomparison project (GrSMBMIP) aims to refine these uncertainties by intercomparing 13 models of four types which were forced with the same ERA-Interim reanalysis forcing fields, except for two global models. We interpolate all modelled SMB fields onto a common ice sheet mask at 1 km horizontal resolution for the period 1980–2012 and score the outputs against (1) SMB estimates from a combination of gravimetric remote sensing data from GRACE and measured ice discharge; (2) ice cores, snow pits and in situ SMB observations; and (3) remotely sensed bare ice extent from MODerate-resolution Imaging Spectroradiometer (MODIS). Spatially, the largest spread among models can be found around the margins of the ice sheet, highlighting model deficiencies in an accurate representation of the GrIS ablation zone extent and processes related to surface melt and runoff. Overall, polar regional climate models (RCMs) perform the best compared to observations, in particular for simulating precipitation patterns. However, other simpler and faster models have biases of the same order as RCMs compared with observations and therefore remain useful tools for long-term simulations or coupling with ice sheet models. Finally, it is interesting to note that the ensemble mean of the 13 models produces the best estimate of the present-day SMB relative to observations, suggesting that biases are not systematic among models and that this ensemble estimate can be used as a reference for current climate when carrying out future model developments. However, a higher density of in situ SMB observations is required, especially in the south-east accumulation zone, where the model spread can reach 2 m w.e. yr−1 due to large discrepancies in modelled snowfall accumulation.

Spatiotemporal variation of soil organic matter (SOM) contents was significant to research on global warming, greenhouse effect, and ecosystem health and quality. However, the spatiotemporal modeling for soil properties in most studies focused on a discrete-time calibration and validation, and faced the problems of missing observations. Integrated Nested Laplace Approximation with the Stochastic Partial Differential Equation (INLA-SPDE) model that is robust to missing data and unbalanced sampling design was proposed as a potential model for spatiotemporal soil modeling. This study presented an application of INLA-SPDE for spatiotemporal modeling of SOM (2006, 2010, 2018) using 924 samples and 8 environmental covariates in Lishu County, Northeast China. The results demonstrated that the INLA-SPDE model incorporating spatiotemporal information generally outperformed the two-phase methods based on Cubist and Random Forest, particularly in years requiring greater temporal extrapolation (2006 and 2018), while achieving comparable performance in 2010. This superiority can be attributed to its comprehensive consideration of various sources of uncertainty. Furthermore, the posterior distributions derived from the model provided valuable insights into the effects of environmental covariates on SOM spatiotemporal variation, with clay content showing the strongest positive influence and annual precipitation exhibiting a notable negative effect. The spatial pattern of SOM consistently exhibited higher values in the east and lower values in the west. After an overall decline from 2006 to 2010, mean SOM content increased from 17.97 g kg-1 to 20.85 g kg-1 between 2010 and 2018 (a total increase of 2.88 g kg-1 at an annual rate of 0.36 g kg-1 yr-1), with notable recovery in central and eastern areas, likely associated with the implementation of straw returning practices. In addition to prediction accuracy, computational complexity, and uncertainty analysis, the study evaluated the model from new perspectives, including covariate interpretability and flexibility. This research provides a promising spatiotemporal modeling framework for digital soil mapping.

The accurate prediction of shear wave velocity (Vs) is critical for earthquake engineering applications. However, the prediction is inevitably influenced by geotechnical variability and various sources of uncertainty. This paper investigates the effectiveness of integrating explainable machine learning (ML) model and Bayesian generalized linear model (GLM) to enhance both predictive accuracy and uncertainty quantification in Vs prediction. The study utilizes an Extreme Gradient Boosting (XGBoost) algorithm coupled with Shapley Additive Explanations (SHAPs) and partial dependency analysis to identify key geotechnical parameters influencing Vs predictions. Additionally, a Bayesian GLM is developed to explicitly account for uncertainties arising from geotechnical variability. The effectiveness and predictive performance of the proposed models were validated through comparison with real case scenarios. The results highlight the unique advantages of each model. The XGBoost model demonstrates good predictive performance, achieving high coefficient of determination (R2), index of agreement (IA), Kling–Gupta efficiency (KGE) values, and low error values while effectively explaining the impact of input parameters on Vs. In contrast, the Bayesian GLM provides probabilistic predictions with 95% credible intervals, capturing the uncertainty associated with the predictions. The integration of these two approaches creates a comprehensive framework that combines the strengths of high-accuracy ML predictions with the uncertainty quantification of Bayesian inference. This hybrid methodology offers a powerful and interpretable tool for Vs prediction, providing engineers with the confidence to make informed decisions.

Machine learning (ML) has been leveraged to tackle a diverse range of tasks in almost all branches of nuclear engineering. Many of the successes in ML applications can be attributed to the recent performance breakthroughs in deep learning and the growing availability of computational power, data, and easy-to-use ML libraries. However, these empirical successes have often outpaced our formal understanding of the ML algorithms. An important but underrated area is the uncertainty quantification (UQ) of ML. ML-based models are subject to approximation uncertainty when they are used to make predictions due to sources including but not limited to, data noise, data coverage, extrapolation, imperfect model architecture, and the stochastic training process. The goal of this paper is to clearly explain and illustrate the importance of the UQ of ML. We elucidate the differences in the basic concepts of UQ of physics-based models and data-driven ML models. Various sources of uncertainties in physical modeling and data-driven modeling are discussed, demonstrated, and compared. We also present and demonstrate a few techniques to quantify the ML prediction uncertainties, including Monte Carlo dropout, deep ensemble, Bayesian neural networks, Gaussian processes, and conformal prediction. Finally, we discuss the need for building a verification, validation, and UQ framework to establish ML credibility.

Spatial-temporal imputation remains a challenging problem in transportation, environment and healthcare, where the missing value is filled based on spatial, temporal, and cross correlations. Previous research mainly focused on feature-level correlation integration and comprehension with the hand-crafted enhancement strategy. Meanwhile, the recently prevalent large language models (LLMs) provide token-level understanding for language linguistics, and whether they could be applied for spatial-temporal correlation enhancement is under exploration. To this end, we proposed an LLM-native framework STOMA to fully utilize the intrinsic relevance. We designed semantic enhancing methods by converting the complex correlations, e.g. spatial correlation in network, temporal correlation with periodicity and cross correlation from human behavior, into the embedded tokens. Specifically, we reform dynamic time warping as an asymmetric correlation constructor for complex dynamics. We adapt the proposed backbone along with the spatial-temporal fine-tuning technique, and the empirical results demonstrate the effectiveness of our methods over recent LLM-inspired methods evaluating on real-world datasets.

This paper develops a simple, robust framework for the problem of finding the route with the least expected travel time from any node to any given destination in a stochastic and time-dependent network. Spatial and temporal link travel time correlations are both considered in the proposed solution, which is based on a dynamic programming approach. In particular, the spatial correlation is represented by a Markovian property of the link states, in which each link is assumed to experience congested or uncongested conditions. The temporal correlation is manifested through the time-dependent expected link travel time given the condition of the link traversed. The framework enables the use of a route guidance system, in which at any decision node within a network, a decision can be made on the basis of current traffic information about which node to take next to achieve the shortest expected travel time to the destination. Numerical examples are presented to illustrate the computational steps involved in the framework to make route choices and to demonstrate the effectiveness of the proposed solution.