Assessing the impact of uncertainties in ice-sheet models is a major and challenging issue that needs to be faced by the ice-sheet community to provide more robust and reliable model-based projections of ice-sheet mass balance. In recent years, uncertainty quantification (UQ) has been increasingly used to characterize and explore uncertainty in ice-sheet models and improve the robustness of their projections. A typical UQ analysis first involves the (probabilistic) characterization of the sources of uncertainty, followed by the propagation and sensitivity analysis of these sources of uncertainty. Previous studies concerned with UQ in ice-sheet models have generally focused on the last two steps but have paid relatively little attention to the preliminary and critical step of the characterization of uncertainty. Sources of uncertainty in ice-sheet models, like uncertainties in ice-sheet geometry or surface mass balance, typically vary in space and potentially in time. For that reason, they are more adequately described as spatio-(temporal) random fields, which account naturally for spatial (and temporal) correlation. As a means of improving the characterization of the sources of uncertainties for forward UQ analysis within the Ice-sheet and Sea-level System Model (ISSM), we present in this paper a stochastic sampler for Gaussian random fields with Matérn covariance function. The class of Matérn covariance functions provides a flexible model able to capture statistical dependence between locations with different degrees of spatial correlation or smoothness properties. The implementation of this stochastic sampler is based on a notable explicit link between Gaussian random fields with Matérn covariance function and a certain stochastic partial differential equation. Discretization of this stochastic partial differential equation by the finite-element method results in a sparse, scalable and computationally efficient representation known as a Gaussian Markov random field. In addition, spatio-temporal samples can be generated by combining an autoregressive temporal model and the Matérn field. The implementation is tested on a set of synthetic experiments to verify that it captures the desired spatial and temporal correlations well. Finally, we illustrate the interest of this stochastic sampler for forward UQ analysis in an application concerned with assessing the impact of various sources of uncertainties on the Pine Island Glacier, West Antarctica. We find that larger spatial and temporal correlations lengths will both likely result in increased uncertainty in the projections.
©2021. California Institute of Technology. Government sponsorship acknowledged.
Despite large improvements in ice-sheet modelling in recent years, model-based estimates of ice-sheet mass balance remain characterized by large uncertainty. The main sources of uncertainty are associated with limitations related to poorly modelled physical processes, the model resolution, poorly constrained initial conditions, uncertainties in external climate forcing (e.g. surface mass balance and ocean-induced melting), or uncertain input data such as the ice sheet geometry (e.g. bedrock topography and surface elevation) or boundary conditions (e.g. basal friction and geothermal heat flux). In order to provide more robust and reliable model-based estimates of ice-sheet mass balance, we therefore need to understand how model outputs are affected by or sensitive to input parameters.
To this aim, uncertainty quantification (UQ) methods have become a powerful and popular tool to deduce the impact of sources of uncertainty on
ice-sheet projections (propagation of uncertainty) or to ascertain and rank the impact of each source of uncertainty on the projection uncertainty
(sensitivity analysis)
In this paper, we propose characterizing uncertain spatially varying input parameters as spatial random fields, that is, an infinite set of random
variables indexed by the spatial coordinate. More specifically, we focus on the class of Gaussian random fields, which provides a popular statistical
model to represent stochastic phenomena in engineering, spatial analysis, and geostatistics (e.g. kriging interpolation)
The goal of this paper is to introduce the implementation of a stochastic sampler for Gaussian Matérn random fields for forward uncertainty
quantification within the Ice-sheet and Sea-level System Model (ISSM)
This paper is organized as follows. In Sect.
In this work, we model a spatially varying uncertain input parameter as a random field
A Gaussian random field
Among the families of covariance functions, the Matérn family is a popular choice to represent spatial correlation in geostatistics. Applications
include spatial modelling of greenhouse gas emissions
In this paper, interest lies in generating realizations of Gaussian random fields for uncertainty quantification in ice-sheet models. Gaussian random
fields have already been employed in glaciology in a number of studies including
An intuitive interpretation for the rationale behind the SPDE (Eq.
Equation (
The SPDE approach has several advantages compared to other sampling methods. First, Eq. (
The SPDE approach can also be used to define a proper choice of a prior distribution for inverse problems in infinite dimension
We assume at this stage that
As shown in
More generally, the covariance matrix
The bulk of the computational cost is in evaluating the square root of the matrix
Numerically, this approximation leads to a representation of the precision matrix, i.e. the inverse of the covariance matrix, that is sparse
Temporal correlation between samples (for transient simulation) is represented with a first-order autoregressive model (AR1 process)
At every time step
The spatio-temporal model combining Eqs. (
As a means of verifying and testing the implementation of our stochastic sampler, we carry out a set of numerical experiments on a synthetic ice
sheet or ice shelf that is 100
Estimated marginal and bivariate distributions of the samples generated by solving the SPDE (Eq.
The same as Fig.
As a first experiment, we test our implementation by checking that the generated samples are actually drawn from a Gaussian random field with the
desired Matérn covariance function. More specifically, the marginal distributions have to follow Gaussian distributions with zero mean and unit
variance, while the bivariate distributions have to follow bivariate Gaussian distributions with zero mean and covariance given by
Eq. (
Comparison of estimated marginal standard deviations with homogeneous Neumann boundary conditions
The second experiment investigates the influence of the boundary conditions on the marginal variance of the samples generated following the SPDE
approach. Figure
Estimated autocorrelation functions (dotted lines) for different values of the temporal correlation factor
As a third experiment, we test our transient implementation by checking that the generated samples follow an AR1 process in time. For different values
of the temporal correlation factor
Fluxgates used to compute mass fluxes on tributaries of the PIG. Each gate is numbered from 1 to 13 and corresponds to one tributary. Gate 1 coincides with the ice front, and gate 2 coincides with the 1996 grounding line. The gates are superimposed on an InSAR surface velocity map of the area, in logarithmic scale
We set up a test problem similar to the application in
The anisotropic mesh used in this application comprises 2085 elements and 1112 nodes, with higher spatial resolution near the shear margins and the grounding line. The number of nodes is sufficiently small to make both the sampling (tenths of a second per sample) and the solving of the ice-sheet model (a few seconds per run) computationally fast.
Following
Samples of the random perturbation
The same as Fig.
The same as Fig.
Here, we represent the random perturbation
We use ISSM to propagate uncertainty in ice thickness into a probabilistic characterization of the depth-averaged mass flux across fluxgates. We
implement the propagation of uncertainty using Monte Carlo sampling. To this end, we begin by generating an ensemble of
Normalized histograms for the mass flux across the fluxgates depicted in Fig.
Figure
Normalized histograms for mass flux across fluxgate 3 (see Fig.
Figure
We carry out a global sensitivity analysis to determine where changes in ice thickness most influence uncertainty in mass flux across fluxgates. For a
given fluxgate, we can build a sensitivity map which gives for each location a sensitivity measure (or index) between the random field at this
location and the mass flux across the fluxgate. Here, we consider the coefficient of correlation between the random field and
the mass flux
Sensitivity map (visual representation of the coefficient of correlation) for mass flux across fluxgates depicted in Fig.
Figure
Sensitivity map (visual representation of the coefficient of correlation) for mass flux across fluxgate 3 (see Fig.
Figure
We perform another sensitivity analysis by considering multiple sources of uncertainty, namely uncertainty in ice thickness, basal friction and ice
hardness. We assume that these three sources of uncertainty are statistically independent, and we choose a parameter range of 30
The Sobol index of a given uncertain input parameter represents the fraction of the variance of the projections explained as stemming from this sole
uncertain parameter. A value of 1 indicates that the entire variance of the projections is explained by this sole uncertain parameter, and a value of 0
indicates that the uncertain parameter has no impact on the projection uncertainty. We estimate these Sobol indices using Monte Carlo sampling
following the estimation method by
Sobol sensitivity indices for the mass flux across flux gates. The gap between the height of a bar and the unit value represents the interaction index.
Figure
Transient samples of random perturbation over Pine Island Glacier with a temporal correlation of 0.5.
The same as Fig.
Finally, we present a transient experiment of the Pine Island Glacier over a 10-year period. The Pine Island Glacier is forced by basal melting
underneath the floating portion of the domain and surface mass balance. We impose a basal melting rate of 25
Normalized histograms for mass flux across fluxgate 3 (see Fig.
Figure
Coefficient of variation at the end of the simulation as a function of the temporal correlation
Figure
Model-based projections of ice-sheet mass balance should all be ideally given in the form of probabilistic projections that provide an assessment of
the impact of uncertainties in boundary conditions, climate forcing, ice sheet geometry or initial conditions. In this context, uncertainty
quantification analysis with ice-sheet models should capture the spatial and temporal variability of these sources of uncertainties. To this end,
random fields are an essential statistical tool for the modelling and analysis of uncertain spatio-temporal processes. Here, we take advantage of the
explicit link between Gaussian random fields with Matérn covariance function and a stochastic partial differential equation to implement a random
field sampler within ISSM. The FEM-oriented implementation of ISSM and its fully object-oriented and highly parallelized architecture allow for a
straightforward and computationally efficient implementation of this SPDE. This stochastic sampler provides an alternative to the pre-existing UQ mesh
partitioning approach implemented within the ISSM-DAKOTA UQ framework. The partitioning approach can be interpreted as the representation of a random
field with perfect correlation for locations in a same partition and zero
The results presented here provide complementary insight into the impact of uncertainties on the PIG. First, our results are consistent with previous
results by
Finally, we discuss a few limitations of the SPDE approach. First, this approach relies on the assumption that uncertainty in spatially varying input
parameters can be appropriately characterized by a Gaussian random field. Though Gaussian random fields are a practical model for many stochastic
phenomena, this assumption may not be appropriate because input parameters can have an asymmetric distribution or heavy tails or can simply be positive or
bounded. While this may sound restrictive, non-Gaussian random fields can be obtained through non-linear transformations of Gaussian random fields
We presented and implemented a Gaussian random field sampler within ISSM for uncertainty quantification analysis of spatially (and temporally) varying uncertain input parameters in ice-sheet models. So far, sampling of spatially varying uncertain input parameters within ISSM has relied on a partitioning of the computational domain and sampling from the partitions. This approach may be limited in representing spatial and temporal correlations. To improve the probabilistic characterization of spatially (and temporally) correlated uncertain input parameters, we proposed to generate realizations of Gaussian random fields with Matérn covariance function. Our implementation relies on an explicit link between Gaussian random fields with Matérn covariance function and a stochastic partial differential equation. This SPDE sampling approach allows for a computationally efficient sampling of random fields using the FEM and provides a flexible mathematical framework to build probabilistic models of spatio-temporal uncertain processes. In addition, model parameters allow us to intrinsically control the spatial and temporal correlations of the realizations and their variance and smoothness.
We applied this stochastic sampler to investigate the impact of spatially (and temporally) varying sources of uncertainties, namely uncertainties in ice thickness, basal friction, ice hardness and surface mass balance, on the Pine Island Glacier. More specifically, we investigated the impact of spatial and temporal correlations on ice-sheet mass balance. We showed that the SSA model is stable to uncertainties. We also showed that larger correlation lengths lead to increased uncertainty in mass balance estimates because the random perturbations have a larger length scale of influence. We found that the most influential source of uncertainties for estimating mass balance is the uncertainty in ice thickness. We also showed in a transient experiment that uncertainty in mass balance estimates increases in time and with higher temporal correlation. Overall, our results demonstrate the need to better constrain the spatial and temporal variability of physical processes impacting ice-sheet dynamics through data assimilation and modelling efforts.
The ISSM code can be downloaded, compiled and executed following the instructions available on the ISSM website
KB and EL discussed the results presented in this paper. KB implemented the stochastic sampler into ISSM and carried out the simulations. KB wrote the bulk of the manuscript with relevant comments from EL.
The contact author has declared that neither they nor their co-author has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We would like to thank the two anonymous referees for their very helpful comments that helped improve the overall quality and readability of the manuscript. Kevin Bulthuis acknowledges support from the NASA Postdoctoral Program (NPP), administered by the Universities Space Research Association (USRA) under contract with the National Aeronautics and Space Administration (NASA). The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. Government sponsorship is acknowledged.
This research has been supported by the Jet Propulsion Laboratory (NASA Postdoctoral Program).
This paper was edited by Alexander Robel and reviewed by two anonymous referees.