Derivative-based uncertainty quantification in climate modeling P. Heimbach 1, D. Goldberg 2, C. Hill 1, C. Jackson 3, N. Petra 3, S. Price 4, G. Stadler 5, J. Utke 6 MIT, EAPS, Cambridge, MA U. Edinburgh, UK UT Austin, TX LANL, Los Alamos, NM ANL, Chicago, IL

Example of science questions Past, present, future contribution of mass loss from polar ice sheets to global mean sea level rise Rate of present-day heat uptake by the ocean The ocean’s role in the global carbon cycle

Posing the “UQ” problem For each of the examples given, how are estimates affected by … –…observation uncertainty? –…observation sampling? –…prior information on input parameters? –…model uncertainties, including artificial drift? Need a framework that: –accounts for these uncertainties –takes optimal advantage of information content in models and observations –is computationally tractable and relevant

The uncertainty space is very high-dimensional 3D fields of: –initial conditions –spatially varying model parameters, e.g.: vertical or eddy-induced mixing (ocean) material properties of ice(Arrhenius param.) 2D fields of surface or basal boundary conditions, e.g.: –surface forcing (heat flux, precipitation) –basal sliding, geothermal fluxes, basal melt rates –bed topography/bathymetry –air-sea gas (CO 2 ) exhange & transfer coefficients Underlying most of these questions: how well constrained by observations?

Deterministic, gradient-based approaches sensitivity analysis –use adjoint to infer sensitivity of climate indices (e.g., ocean heat content; MOC; ice sheet volume; total carbon uptake; …)to input fields optimal state & parameter estimation –optimal state/reconstruction of climate state from sparse, heterogeneous observations –optimal & “drift-free” initial conditions for prediction inverse/predictive uncertainty propagation

Example: Sensitivity of carbon uptake to changes in vertical diffusivity MIT general circulation model (MITgcm) coupled to biogeochemical module Adjoint model generated via open-source algorithmic/automatic differentiation tool OpenAD (Argonne National Lab) C. Hill, O. Jahn, et al., in prep.

Adjoint model also gives linear sensitivities Sensitivities of Grounded Volume of marine ice sheet highlight role of ice shelf margins Sensitivity to m Sensitivity to warming (softening) Example: Marine ice sheet/shelf adjoint sensitivities Goldberg & Heimbach (2013)

Example: ice sheet model inversion & initialization  UQ-enabled predictions for sea level rise require initial conditions for large ice masses that are consistent with: surface flow velocities present day ice geometry accumulation data or output from Earth system models  Compute MAP estimates for the basal friction coefficient field and the bedrock topography (each has about 33,000 parameters)  Overall 5500 adjoint-based gradients required: 11,000 (non)linear PDE solves Left : Implied accumulation rate without taking into account Earth system model data; Middle : implied accumulation rate after taking into account Earth system model data, which is shown on the Right.  Forward problem has 350,000 parameters, implementation based on LiveV FEM package and Trilinos solvers Perego, Price, Stadler (2014)

Posterior covariance of controls x ≈ Inverse of Hessian matrix Inverse uncertainty propagation – Hessian method Model–data misfit function: Solution / posterior uncertainty?  curvature of misfit function  Described by Hessian matrix of J controls x observations y M Data uncertainty  Controls uncertainty Δy  Δx Small curvature Large uncertainty Large curvature Small uncertainty R Linear term Nonlinear term

Assimilation of observations uncertainty Reduction of prior controls uncertainty Forward uncertainty propagation Data uncertainty  Controls uncertainty  Target uncertainty P yy  P xx  UQ scheme UQ algorithm for Ocean State estimation