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

2. 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

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

4. 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

5. 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

6. 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

7. 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

8. 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

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

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

11. 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

12. Analytic adjoint operator for advection
Consider