Abstract
Abstract. We consider the problem of inferring the basal sliding coefficient field for an uncertain Stokes ice sheet forward model from synthetic surface velocity measurements. The uncertainty in the forward model stems from unknown (or uncertain) auxiliary parameters (e.g., rheology parameters). This inverse problem is posed within the Bayesian framework, which provides a systematic means of quantifying uncertainty in the solution. To account for the associated model uncertainty (error), we employ the Bayesian approximation error (BAE) approach to approximately premarginalize simultaneously over both the noise in measurements and uncertainty in the forward model. We also carry out approximative posterior uncertainty quantification based on a linearization of the parameter-to-observable map centered at the maximum a posteriori (MAP) basal sliding coefficient estimate, i.e., by taking the Laplace approximation. The MAP estimate is found by minimizing the negative log posterior using an inexact Newton conjugate gradient method. The gradient and Hessian actions to vectors are efficiently computed using adjoints. Sampling from the approximate covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. We study the performance of the BAE approach in the context of three numerical examples in two and three dimensions. For each example, the basal sliding coefficient field is the parameter of primary interest which we seek to infer, and the rheology parameters (e.g., the flow rate factor or the Glen's flow law exponent coefficient field) represent so-called nuisance (secondary uncertain) parameters. Our results indicate that accounting for model uncertainty stemming from the presence of nuisance parameters is crucial. Namely our findings suggest that using nominal values for these parameters, as is often done in practice, without taking into account the resulting modeling error, can lead to overconfident and heavily biased results. We also show that the BAE approach can be used to account for the additional model uncertainty at no additional cost at the online stage.
Highlights
Inferring the basal sliding coefficient field using both the linear and nonlinear Stokes ice sheet model from noisy surface velocity measurements has received considerable attention in recent years
Our results suggest that setting rheology parameters to nominal values can result in severely misleading estimates of the basal sliding coefficient field and associated posterior uncertainty if the additional uncertainty in the rheology parameters is not accounted for
We summarize the Bayesian inference framework, which will be used in combination with the Bayesian approximation error approach, to account for uncertainties in rheology parameters
Summary
Inferring the basal sliding coefficient field using both the linear and nonlinear Stokes ice sheet model from noisy surface velocity measurements has received considerable attention in recent years (see, for example, Truffer, 2004, Raymond and Gudmundsson, 2009, Pollard and DeConto, 2012, Isaac et al, 2015b, Morlighem et al, 2013, Zhao et al, 2018a, b, Giudici et al, 2014, Petra et al, 2012, 2014, and Isaac et al, 2015a). The standard approach to this problem invariably assumes that the other parameters of the ice, such as those controlling the rheology, are known precisely This is common, for example, in the case of the so-called flow rate factor and the Glen’s flow law exponent, in which nominal values such as A = 10−16 Pa−n a−1 and n = 3, respectively, are prescribed; we refer, e.g., to Isaac et al (2015a), Petra et al (2014), Raymond and Gudmundsson (2009), Truffer (2004), Morlighem et al (2013), Zhu et al (2016), Zhao et al (2018b), Giudici et al (2014), and Pollard and DeConto (2012). For most ice sheet inverse problems considered in the literature, the field of interest is the basal sliding parameter, which arguably presents the largest uncertainty in determining the ice flow rate
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