1. Adjoint models: Theory ATM 569 Fovell Fall 2015 (See course notes, Chapter 15) 1

2. Motivation We often look at a forecast and wonder “how could this be changed”, especially if it involves model error Often our strategy involves posing a hypothesis that explains the error’s origin, changing the model initial conditions and/or model parameters to compensate, and running the model… usually over and over and over Ideally, we want an active, dynamic tracer that shows how the error came about in the model – which fields, where and when The adjoint model is a simplified version of the forward model that runs backwards The forward model predicts temperature, pressure, winds, etc., forward in time “The future evolves from the past” The adjoint model predicts sensitivity to temperature, pressure, winds, etc.., backwards in time “Traces error back to its roots” 2

3. Background: simple problem 1D wave equation: M real grid points, 1 prognostic variable, 2 time level scheme Rewrite in explicit form c > 0 3

4. Background: matrix form Write in matrix form (real points j = 1, 2, …, M) Mx1 MxMMxM 4 For this simple problem, same C every time step. Not true for more realistic problems. WRONG

5. Background: matrix form Write in matrix form (real points j = 1, 2, …, M) Mx1 MxMMxM 5 For this simple problem, same C every time step. Not true for more realistic problems. Matrix fixed

6. Background: integrate model Initial condition is u 0. Integrate for N time steps. We can relate the final forecast to the initial condition through the transition matrix P N 6

7. Background: generalize model x is p prognostic fields x M gridpoints p = 4+ (i.e., u, w,  ’,  ’,…) M = NX x NY x NZ 7

8. Tangent linear model (TLM) #1 A simple model equation Examples: u = prognostic variable  = model parameter 1D wave equation Exponential decay 8

9. Tangent linear model (TLM) #2 Run the model twice, using two different initial conditions and/or two different values for parameter . Control solution u C (x,z,t) Alternative solution u A (x,z,t) Control parameter  C Alternative parameter  A Difference between simulations and their parameters 9

10. Tangent linear model (TLM) #3 You can always subtract two simulations. The TLM is a model that attempts to estimate the difference between the control and alternative runs, based on the control run run calculate TLM run run estimate Instead of… …do this Why? We’ll see… 10

11. Tangent linear model (TLM) #4 TLM can be formed via perturbation analysis, and as usual presumes the perturbations are (& remain) small so higher order terms absent 1 Uses Taylor series to approximate u” for so the TLM is 11 1 See course notes for qualifications and disclaimers

12. Tangent linear model (TLM) #5 The perturbation model has been linearized (no u’’  ’’ term) and is constrained to (“tangent to”) the control run (u C,  C ). “Tangent linear model” Discretize TLM and write in matrix form 12 C n based on control model run - Run control simulation - Archive C n every time step - Initialize and run TLM Ignore for simplicity

13. Tangent linear model (TLM) #6 Integrate the TLM. Initial condition is u’’ 0 13 C n based on control model run - Run control simulation - Archive C n every time step - Initialize and run TLM

14. Tangent linear model (TLM) #7 Generic form. x” is p variables by M points. Initial condition is x’’ 0 14

15. Forecast aspect J The forecast aspect J is something about the control run we want to examine How did some feature appear? Why did some error occur? A J at time N is a scalar function of the control run at that time J N = J(x N ) J N can be changed by perturbing the control run (ignoring higher order terms) 15

16. Change of forecast aspect ∆J N 16 Change of J at time N Sensitivity of J to x l at time N Perturbation applied to variable/location x l at time N p variables M locations Perturb a variable/location It only changes J if J is sensitive to it

17. Change of forecast aspect ∆J N 17 Let J N be surface pressure at one point, say 30˚N 60˚W, at time N ∆J N is how surface pressure at that place and time can be changed At time N, ∆J N is sensitive to only one variable and location – pressure at that location Therefore, of the p x M terms in the sum, only one sensitivity is nonzero, and it is equal to 1 Thus ∆J N is KNOWN information and is TRIVIAL

18. Rewrite as an inner product Postulate the adjoint model, a prediction model for sensitivity x* Making this less trivial 18 Take the TLM model and (a)Replace x’’ by x* (b)Transpose C n (c)Operate it backwards Note C n T ≠ C n -1, so we are not running TLM backwards From control run

19. TLM vs. adjoint 19 The control run “information” used to step perturbations forward in time is transposed and used to step sensitivity backwards in time

20. TLM and adjoint Relating initial and final times for TLM and adjoint models Next, we will make use of the adjoint property This is how the adjoint model got its name… 20

21. The recipe 21 trivial by definition relate final to initial time invoke adjoint property relate final to initial time Note therefore that

22. The recipe 22 KNOWNTRIVIAL KNOWN NOT TRIVIAL! Tells me what my perturbations x’’ have to be at the initial time to get that change to J at time N

23. Integrating the adjoint 23 (1) Run the control model to time N and save C n every time step (2) Initialize adjoint at time N (3) Integrate adjoint backwards, reading in C n from archive You DON’T need to integrate the TLM

24. To summarize The control simulation is made by integrating a (likely nonlinear) model forward in time, producing forecasts of temperature, pressure, winds, etc.. The tangent linear model is a linearized version of the forward model, producing forecasts of perturbations (deviations) from the control forecast The inescapable assumption is the deviations are small The adjoint model is a transposed version of the TLM The adjoint model runs backwards in time The adjoint propagates sensitivity to temperature, pressure, winds, etc., backwards It represents an active, dynamical “tracer” It also must assume that deviations are small Our simple examples involved differentiating the model differential equation to create the TLM. In practice, we differentiate the model code. 24