Adjoint Models as Analytical Tools

1. Adjoint Models as Analytical Tools Dr. Ronald M. Errico Goddard Earth Sciences and Technology Center (UMBC) Global Modeling and Assimilation Office (NASA)

2. Outline 1. Sensitivity analysis: The basis for adjoint model applications 2. Examples of adjoint-derived sensitivity 3. Code construction: an example 4. Nonlinear validation 5. Efficient solution of optimization problems 6. Adjoint models as paradigm changers 7. Singular vectors 8. Observation sensitivity 9. Other applications 10. Other considerations 11. Summary

3. Sensitivity Analysis: The basis for adjoint model applications Impacts vs. Sensitivity

6. xi xj

8. Adjoint Sensitivity Analysis Impacts vs. Sensitivities A single impact study yields exact response measures (J) for all forecast aspects with respect to the particular perturbation investigated. A single adjoint-derived sensitivity yields linearized estimates of the particular measure (J) investigated with respect to all possible perturbations.

12. Examples of Adjoint-Derived Sensitivities

13. Example Sensitivity Field Errico and Vukicevic 1992 MWR Contour interval 0.02 Pa/m M=0.1 Pa/m

14. Lewis et al. 2001

15. J=average surface pressure in a small box centered at P J=barotropic component of vorticity at point P From Errico and Vukicevic 1992

16. Development of Adjoint Model From Line by Line Analysis of Computer Code

17. Development of Adjoint Model From Line by Line Analysis of Computer Code

18. Development of Adjoint Model From Line by Line Analysis of Computer Code Y = X * (W**A) Z = Y * X Parent NLM : Ytlm = Xtlm * (W**A) + Wtlm *A* X *(W**(A-1)) Ztlm = Ytlm * X + Xtlm * Y TLM : Xadj = Xadj + Zadj * Y Yadj = Yadj + Zadj * X Xadj = Xadj + Yadj * (W**A) Wadj = Wadj + Yadj * X *(W**(A-1)) Adjoint :

19. Development of Adjoint Model From Line by Line Analysis of Computer Code Y = X * (W**A) Z = Y * X Parent NLM : Ytlm = Xtlm * (W**A) + Wtlm *A* X *(W**(A-1)) Ztlm = Ytlm * X + Xtlm * Y TLM : Xadj = Xadj + Zadj * Y Yadj = Yadj + Zadj * X Xadj = Xadj + Yadj * (W**A) Wadj = Wadj + Yadj * X *(W**(A-1)) Adjoint :

20. Development of Adjoint Model From Line by Line Analysis of Computer Code Automatic Differentiation TAMC Ralf Giering (superceded by TAF) TAF FastOpt.com ADIFOR Rice University TAPENADE INRIA, Nice OPENAD Argonne Others www.autodiff.org See http://imgi.uibk.ac.at/MEhrendorfer/ work_7/present/session3/giering.pdf

21. Development of Adjoint Model From Line by Line Analysis of Computer Code • TLM and Adjoint models are straight-forward to derive from • NLM code, and actually simpler to develop. • Intelligent approximations can be made to improve efficiency. • TLM and (especially) Adjoint codes are simple to test rigorously. • 4. Some outstanding errors and problems in the NLM are typically • revealed when the TLM and Adjoint are developed from it. • 5. It is best to start from clean NLM code. • 6. The TLM and Adjoint can be formally correct but useless!

22. Nonlinear Validation Does the TLM or Adjoint model tell us anything about the behavior of meaningful perturbations in the nonlinear model that may be of interest?

23. Linear vs. Nonlinear Results in Moist Model 24-hour SV1 from case W1 Initialized with T’=1K Final ps field shown Errico and Raeder 1999 QJRMS Contour interval 0.5 hPa

24. Linear vs. Nonlinear Results in Moist Model Non-Conv Precip. ci=0.5mm Non-Conv Precip. ci=0.5mm Convective Precip. ci=2mm Convective Precip. ci=2mm

25. Comparison of TLM and Nonlinearly Produced Precip Rates 12-Hour Forecasts with SV#1 Non- Convective Linear Nonlinear Convective. Linear Nonlinear Errico et al. QJRMS 2004 Contours: 0.1, 0.3, 1., 3., 10. mm/day

26. Linear vs. Nonlinear Results • In general, agreement between TLM and NLM results • will depend on: • Amplitude of perturbations • Stability properties of the reference state • Structure of perturbations • Physics involved • Time period over which perturbation evolves • Measure of agreement The agreement of the TLM and NLM is exactly that of the Adjoint and NLM if the Adjoint is exact with respect to the TLM.

27. Efficient solution of optimization problems

33. Contours of J in phase (x) space Gradient at point P P M

34. Adjoint models as paradigm changers

35. Sensitivity field for J=ps with respect to T for an idealized cyclone 100 hPa 500 hPa 1000 hPa From Langland and Errico 1996 MWR

36. Bao and Errico: MWR 1997 Adjoint of Nudging Fields Contour 0.1 unit Contour 1.0 unit

37. t=0 0.4 units t=48 hours 5.0 units Errico et al. Tellus 1993

38. What is the modeled precipitation rate sensitive to? Errico, Raeder, Fillion: 2003 Tellus Non-Convective Forecast Precipitation Rate Contour interval 2 cm/day

39. Vort optimized Impacts for adjoint- derived optimal perturbations for forecasts starting indicated hours in the past. Errico et al. 2003 Tellus Rn Optimized Rc optimized

40. Perturbations in Different Fields Can Produce the Same Result 12-hour v TLM forecasts Initial u, v, T, ps Perturbation Initial q Perturbation Errico et al. QJRMS 2004

41. Singular Vectors

42. Gelaro et al. MWR 2000

43. Bred Modes (LVs) And SVs Results for Leading 10 SVs Gelaro et al. QJRMS 2002

44. Balance of Singular Vectors Vertical modes of the 10-level MAMS model (from Errico, 2000 QJRMS)

45. Balance of Singular Vectors E=E_t, K=KE, A=APE, R=R mode E, G=G mode E A,G E,K,R t=0 t=24 hours Errico 2000

46. The Balance of Singular Vectors Contour 2 units Initial R mode Initial G mode Contour 1 unit Errico 2000

47. How Many SVs are Growing Ones? EM E-norm Moist Model ED E-norm Dry Model TM R-norm Moist Model TD R-norm Dry Model Singular Value Squared Errico et al. Tellus 2001 Mode Index

48. From Novakovskaia et al. 2007 and Errico et al. 2007

49. Sensitivity to Observations The use of an adjoint of a data analysis algorithm