GEOS-CHEM adjoint development using TAF Monika Kopacz, Daniel Jacob, Dylan Jones (UT), Parvadha Suntharalingam, Paul Palmer April 5, 2005

What is an adjoint model? X Y adjoint model Inverse map x y K K* For linear operator K, mapping from space X to space Y. Adjoint operator KT For nonlinear model K represents linearization of the original model

Why the need for adjoint model? constraints on emissions with high resolution can consider nonlinear processes computationally efficient for sensitivity studies

Adjoint of GEOS-CHEM 3 approaches to obtaining derivative code: analytic derive adjoint operator analytically and discretize discrete Lagrange function define L = J +lT (K x – yobs)  li = li+1 + (Ki+1 xi+1 – yobs,i+1) direct differentiation of the code: Automatic Differentiation Transformation of Algorithm in Fortran (TAF) source to source compiler Recipes for Adjoint Code Construction, Giering and Kaminski, 1998

What is TAF? Source to source compiler developed by Ralf Giering and Thomas Kaminski; a commercial, license-based product supplied by Fast Opt® Forward differentiation, Tangent Linear Model: TAF dynamic_loop.f dynamic_loop_tl.f Reverse differentiation, Adjoint Model: TAF dynamic_loop_ad.f dynamic_loop.f TAF dynamic_loop(x, fc) ad_dynamic_loop(x, adx, fc, adfc) where x can represent emissions field fc can represent a scalar ‘cost’ variable then adx stores dfc/dx adfc stores dfc/dfc = 1 http://fastopt.com

First step Tangent Linear Model challenges benefits Semi-automatically derived, stand-alone model Tangent Linear Model challenges benefits ? general TAF-GC interfacing ? resolving flow nonlinearities ? Fortran 90 support ? manual changes verification: comparison with green’s function approach Check linearity of the code Resolve TAF-GC compatibility

Next step Adjoint Model challenges benefits Semi-automatically derived, stand-alone model Adjoint Model challenges benefits ? all TLM Fortran issues ? storage/recomputation for reverse mode ? non-trivial model evaluation ? manual changes ? Interfacing with optimization package verification: comparison with previous CO inversions, (Heald et al. 2003, Palmer et al. 2003) efficient sensitivity studies J  inversions for large state vector and large data sets J  data assimilation

Minimizing the gradient 0 ° 2 1 3 x2 x1 x3 x0 Adjoint Transport GEOS- CHEM Measurement Sampling Estimated Fluxes Modeled Concentrations simulated Measurements “True” Assumed Errors Weighted Residuals Cost function J Flux Update a priori fluxes Adjoint Fluxes = J Minimum of cost function J Courtesy of David Baker

Preliminary results Finite Difference Tangent Linear Model 3D field Scalar perturbation

SEMI-AUTOMATIC CAPABILITY TO GENERATE TLM/ADM End product ADJOINT MODEL TANGENT LINEAR MODEL OPTIMIZATION PACKAGE SEMI-AUTOMATIC CAPABILITY TO GENERATE TLM/ADM

Automatic differentiation Z = X * SIN (Y**2) differential transpose ADY = ADY + ADZ*X *COS(Y**2)*2*Y ADX = ADX + ADZ*SIN(Y**2) ADZ = 0.0 from Recipes for Adjoint Code Construction by R. Giering and T. Kaminski, 1998

Analytic adjoint operator for advection Consider