1.2 Computational ice-sheet models and uncertainty quantification . . . . 8

Contents

I Modelling ice sheets in the presence of uncertainties 1

1 Introduction 3

1.1 Research Context: Antarctic ice sheet and future sea-level rise . . . . 3

1.2 Computational ice-sheet models and uncertainty quantification . . . . 8

1.3 Uncertainty quantification: Methods, challenges, and opportunities . . . . 9

1.4 Contributions of the thesis . . . . 10

1.5 Overview of the thesis . . . . 12

2 Physics of ice sheets 15 2.1 Physics of ice sheets: overview . . . . 15

2.2 Full-order model . . . . 17

2.2.1 Mechanical problem . . . . 17

2.2.2 Thermal problem . . . . 20

2.2.3 Closure of the system of equations . . . . 21

2.2.4 Friction laws . . . . 22

2.3 Reduced-order models . . . . 24

2.3.1 Shallow-ice approximation . . . . 27

2.3.2 Shallow-shelf approximation . . . . 30

2.4 Glacial isostasic adjustment . . . . 34

2.5 Instability mechanisms in marine ice sheets . . . . 37

2.5.1 Marine ice-sheet instability . . . . 37

2.5.2 Marine ice-cliff instability . . . . 44

2.6 Chapter summary . . . . 46

3 The Antarctic ice sheet: Present, future, and challenges 47 3.1 The Antarctic ice-sheet in a nutshell . . . . 47

3.2 Mass balance of the Antarctic ice sheet . . . . 49

3.3 The future of the Antarctic ice sheet under climate change . . . . 50

3.3.1 Risk of MISI . . . . 50

3.3.2 Risk of MICI . . . . 51

3.3.3 Ice-air interactions . . . . 52

3.3.4 Ice-ocean interactions . . . . 52

3.3.5 Bedrock topography and glacial isostatic adjustment . . . . 53

3.4 Computational ice-sheet models: limitations for sea-level rise projections . . . 54

3.4.1 Ice-sheet model initialisation . . . . 54

3.4.2 Coupling with other components of the Earth system . . . . 55

3.4.3 Uncertainties in computational ice-sheet models . . . . 56

3.5 Conclusion . . . . 58

II Uncertainty Quantification: Theory and Methods 61 4 Uncertainty quantification methods for computational models 63 4.1 Introduction . . . . 63

4.2 Characterisation of uncertainties . . . . 64

4.3 Propagation of uncertainties . . . . 67

4.3.1 Monte Carlo sampling method . . . . 69

4.3.2 Surrogate models . . . . 72

4.3.3 Probabilistic learning on manifolds . . . . 85

4.4 Global sensitivity analysis . . . . 86

4.4.1 Correlation coefficients . . . . 88

4.4.2 Variance-based sensitivity indices . . . . 88

4.4.3 Generalised sentivity indices . . . . 96

4.5 Numerical illustrations . . . . 98

4.5.1 Propagation of uncertainties: Branin function . . . . 98

4.5.2 Sensitivity analysis: Ishigami function . . . 105

4.6 Summary . . . 107

5 A multifidelity quantile-based approach for confidence sets 109 5.1 Context . . . 109

5.2 Literature review . . . 111

5.3 Notations . . . 112

5.4 Random excursion and contour sets . . . 113

5.4.1 Excursion and contour sets . . . 113

5.4.2 Capacity functional . . . 114

5.4.3 Containment and inclusion functionals . . . 114

5.4.4 Coverage function . . . 115

5.4.5 Vorob’ev quantiles . . . 115

5.5 Confidence sets for random excursion and contour sets . . . 115

5.5.1 Optimisation within a parametric family . . . 118

5.6 Spatial discretisation . . . 122

5.7 Discretisation of the stochastic dimension . . . 125

5.7.1 Membership function . . . 126

5.7.2 Quantile estimation: Monte Carlo method . . . 126

5.7.3 Quantile estimation: spectral method . . . 129

5.7.4 Quantile estimation: bifidelity method . . . 131

5.8 Numerical example . . . 134

5.8.1 Problem set-up . . . 134

5.8.2 Sources of uncertainties . . . 135

5.8.3 Membership function . . . 136

5.8.4 Random variable ‰

. . . 136

5.8.5 Quantile estimation . . . 136

5.8.6 Confidence sets . . . 141

5.8.7 Spatial discretisation . . . 141

5.9 Conclusion . . . 142

5.A Asymptotic properties in Monte Carlo estimation of quantiles . . . 143

5.A.1 Monte Carlo quantile estimation based on the distribution function . . . 143

5.A.2 Monte Carlo quantile estimator based on the mid-distribution function . 143 5.B Proof of Theorem 5.1 . . . 145

III Uncertainty Quantification: Application to ice-sheet modelling 147 6 Essential ice-sheet models as an efficient tool for large-scale and long-term simula- tions and uncertainty quantification 149 6.1 High-fidelity vs essential ice-sheet models . . . 149

6.2 Thermomechanical ice-sheet model . . . 151

6.2.1 Grounded domain . . . 152

6.2.2 Floating domain . . . 153

6.3 Interactions with the Earth system . . . 156

6.3.1 Ice-air interface . . . 156

6.3.2 Ice-bedrock interface . . . 156

6.3.3 Ice-ocean interface . . . 158

6.3.4 Calving . . . 159

6.4 Model initialisation . . . 159

6.4.1 Initial conditions . . . 159

6.4.2 Input data . . . 159

6.4.3 Inversion of the basal sliding coefficient . . . 162

6.5 Numerical implementation . . . 162

6.5.1 Implementation of grounding-line flux parameterisation . . . 163

6.6 Chapter summary . . . 165

7 Illustration and performance evaluation of uncertainty quantification methods on ice-sheet model problems 167 7.1 Probabilistic projections of future sea-level rise . . . 167

7.1.1 Model problem . . . 167

7.1.2 Quantity of interest . . . 168

7.1.3 Sources of uncertainty . . . 169

7.1.4 Propagation of uncertainties: Monte Carlo method . . . 170

7.1.5 Propagation of uncertainties: Surrogate models . . . 170

7.1.6 Sea-level rise projections . . . 175

7.1.7 Propagation of uncertainty: Discussion . . . 176

7.2 Probabilistic assessment of grounded-ice retreat . . . 177

7.2.1 Quantity of interest . . . 177

7.2.2 Sources of uncertainty . . . 177

7.2.3 Membership function . . . 180

7.2.4 Quantile estimation: Monte Carlo method . . . 180

7.2.5 Quantile estimation: bifidelity method . . . 182

7.2.6 Confidence sets . . . 185

7.3 Computation of Sobol indices by probabilistic learning on manifolds . . . 187

7.3.1 Model problem . . . 187

7.3.2 Sources of uncertainty . . . 187

7.3.3 Quantity of interest . . . 187

7.3.4 Global sensitivity analysis: Monte Carlo method . . . 188

7.3.5 Global sensitivity analysis: Spectral method . . . 188

7.3.6 Global sensitivity analysis: PLoM method . . . 189

7.3.7 Global sensitivity analysis: Comparison of the numerical methods . . . 191

7.4 Chapter summary and conclusion . . . 192

8 Uncertainty quantification of the multicentennial response of the Antarctic ice sheet to climate change 195 8.1 Introduction . . . 196

8.2 Model description and methods . . . 198

8.2.1 Ice-sheet model and simulations . . . 198

8.2.2 Sources of uncertainty . . . 202

8.2.3 Uncertainty quantification methods . . . 206

8.3 Results . . . 209

8.3.1 Nominal projections . . . 209

8.3.2 Parameters-to-projections relationship . . . 211

8.3.3 Sea-level rise projections . . . 214

8.3.4 Stochastic sensitivity analysis . . . 220

8.3.5 Projections of grounded-ice retreat . . . 220

8.3.6 Influence of the parameter probability density function . . . 224

8.3.7 Projections under a more plastic sliding law . . . 224

8.3.8 TGL parameterisation . . . 225

8.4 Discussion . . . 226

8.4.1 Comparison of the sea-level rise projections with previous work . . . . 226

8.4.2 Comparison of the impact of parametric uncertainty with previous work 228 8.4.3 Comparison of projections of grounded-ice retreat with previous work . 228 8.4.4 Projections of ice loss and grounding-line retreat under parametric un- certainty . . . 228

8.4.5 Structural uncertainty and limitations . . . 229

8.5 Conclusion . . . 230

8.A Polynomial chaos expansion . . . 231

8.B Sobol sensitivity indices . . . 234

8.C Confidence regions for grounded ice . . . 235

8.D Supplementary figures . . . 236

9 Multi-model comparison of sea-level rise projections 247

9.1 Model description . . . 247

9.2 Multi-model comparison . . . 250

9.2.1 Short-term sea-level rise projections (2100) . . . 250

9.2.2 Medium-term sea-level rise projections (2300) . . . 254

9.2.3 Multi-model projections in the SROCC report . . . 257

9.3 Chapter summary . . . 259

IV Conclusion and directions for future work 273 10 Conclusion and perspectives 275 10.1 Summary and general conclusions . . . 275

10.2 Directions for future research . . . 277

10.2.1 Directions in ice-sheet modelling . . . 277

10.2.2 Directions in uncertainty quantification . . . 278

V Appendices 281 A Elements of probability theory 283 A.1 Probability space . . . 283

A.1.1 Conditional probability . . . 284

A.2 Random variable . . . 284

A.3 Random vector . . . 285

A.3.1 Gaussian random vector . . . 286

A.4 Random field . . . 286

A.4.1 Gaussian process . . . 287

Bibliography 289

List of Symbols 327