Derivative-based uncertainty quantification in climate modeling

P. Heimbach1, D. Goldberg2, C. Hill1, C. Jackson3, N. Petra3, S. Price4, G. Stadler5, J. Utke6

MIT, EAPS, Cambridge, MAU. Edinburgh, UKUT Austin, TXLANL, Los Alamos, NMANL, 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 (CO2) 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 toolOpenAD (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)

1ˆ( )P H x

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

Observation uncertaintyMisfit function

controls x

observations y

M

Data uncertainty Controls uncertainty Δy Δx

Small curvatureLarge uncertainty

Large curvatureSmall uncertainty

R

2

2

12

12

ˆ ˆ( )

( )

) (

ˆT

T T

T

J

R x y

H x xx

Rx x x

M M MM

Linear term Nonlinear term

• Assimilation of observations uncertainty

• Reduction of prior controls uncertainty

• Forward uncertainty propagation

1

0 0 and xx P H PPP

1

xxyy R P HP

20 or or T

z xx PP g gP P

2z

Data uncertainty Controls uncertainty Target uncertainty Pyy Pxx

2

11

0

2

0

0

Tz

Tz

g g

P

P

R PH g

P

Pg

1 2 Tz xx xxgP PR H g

UQ scheme

UQ algorithm for Ocean State estimation