NACLIM CT1/CT3 Meeting 22-23 April 2013 Hamburg

1. NACLIM CT1/CT3 Meeting • 22-23 April 2013 Hamburg • Parameter optimization in an atmospheric GCM using Simultaneous Perturbation Stochastic Approximation (SPSA) • Reema Agarwal, Armin Köhl and Detlef Stammer • Centrum fürErdsystemforschung und Nachhaltigkeit (CEN) • UniversitätHamburg, Hamburg, Germany

2. Objective • To reduce AGCM errors through parameter optimization Model used and Control parameters • AGCM Planet Simulator (PlaSim) with spectral resolution T21 and 10 vertical sigma levels is used • 14 control parameters used in the parameterization of long wave-short wave radiation, cloud parameters; vertical diffusion time scales are chosen

3. Method used • SPSA technique (Spall 1998) is based on “Simultaneous Perturbation” and gradient approximation • The problem of estimation is solved by minimizing the cost function Ywith respect to the parameters • where M is the model state, d is observations used (pseudo or ERA data) and model-data error covariance matrix

4. The cost function Y() is minimized using the iterative procedure • where is a stochastic approximation of the gradient of the cost function • The stochastic gradient in SPSA is calculated by • is chosen as Bernoulli ±1 distribution with probability of for each ±1 outcome. The choice of a,c,A,α,γ is case dependent γ

5. Data set for optimization • Pseudo data (produced by model itself in a run with default values of all control parameters ) • ERA-Interim Reanalysis data of Temperature (all 10 models levels), Total precipitation and net heat Flux Sensitivity experiments • Shows the nonlinear cost function behavior with respect to perturbation in each parameter • Shows the range of minimum cost function that can achieved if parameter Perturbation are not precisely zero

6. Identical twin experiments • To test the assimilation procedure • Model run of 1-year was performed with a set of randomly chosen values of 14 parameters • The variables, a=0.01, c=0.2, A=40, α=0.602 and γ=0.101 are considered • The cost function reaches the acceptable minimum value of ~20 as shown in the sensitivity experiments Cost function vs. iteration number for selected 14 parameters in the identical twin experiment. Cost function is computed for a period of 1 year model integration.

7. Experiments with ERA-Interim a = 1e-5 10% perturbation a= 5e-5 • Three different experiments , two using different values of “a” and another with slightly perturbed values of default parameters, were performed • In all three cases cost function reduced to identical value indicating that the technique is robust Cost function vs. iteration number: To test the robustness of technique several different setups, differing in values of “a” and also using perturbed model parameters , were used. In all three cases nearby minima were attained at around 400 iterations

8. Results continued… Total RMSE The reduction in total RMSE after optimization is ~ 16%. RMSE is computed using total cost function contributions of temperature, fluxes and precipitation as given by the formula where N represents number of observations RMSE of individual contributions in original and optimized model states. The reduction is ~20%, 5%, 11% and 18% for Temperature, precipitation, net heat flux and surface temperature RMSE of individual contributions Global mean Temperature Profile RMSE of global mean Temperature between ERA-interim and Original (black line) and optimized (blue line) model runs. Model Levels Original Optimized RMSE(K)

9. Results continued… Original RMSE surface net heat flux(Watts RMSE near surface Temperature(K) Original Optimized Optimized • Error reduction in surface temperature takes place in the equatorial West Pacific, North Atlantic and Southern Indian Ocean regions. • Net heat flux shows improvement almost in every region with large errors of > 50 Watts/m2 coming down within 10 Watts/m2 in North Pacific, south-east Africa and also in Atlantic oceans.

10. Summary • SPSA technique can efficiently optimize parameters of AGCM by finding minimum cost function • It is easy to implement and computationally efficient • SPSA can handle chaotic models • The technique is robust and works well with pseudo data and reanalysis data • Overall reduction in RMSE is ~ 16% with respect to ERA-interim observations while surface temperatures shows improvement up to 18%

11. Thank You

12. The research leading to these results has received funding from the European Union 7th Framework Programme (FP7 2007-2013), under grant agreement n.308299 NACLIM www.naclim.eu