1. Lijun Liu Seismo Lab, Caltech Dec. 18, 2006 Inferring Mantle Structure in the Past ---Adjoint method in mantle convection

2. Geoid Seismic Tomography Other Data Present Past Much less is known about the earth’s interior in the past ? Dynamic Topography Plate Tectonics

3. Motivation Data on earth surface (I) and that of the interior (II) are somehow independent of each other for mantle study. Without exact knowledge of rheology and dynamics, both I and II are not coupled. Plate motion record is not long enough for the “known” subducted slabs to regulate lowermost mantle structures. Forward modeling has no feedback to the initial state; crude backward integration suffers from accumulated artifacts at thermal boundary layers. The adjoint method which constrains the initial condition by the output can provide II in the past. Combination of adjoint method and data I is promising for study of dynamics of solid earth system.

4. Governing Equations: (continuity) (momentum) (energy) : velocity, P: dynamic pressure, : density, : dynamic viscosity, : thermal diffusivity, H: internal heat source.

5. Adjoint Equation: integration by part and let Lagrange function: where is the adjoint quantity : error in the initial; T p : prediction; T d : data The idea of adjoint: (cost function) (e.g. R. Errico, 1997)

6. Flow Chart Target T 0 Data T 1 Forward run Adjoint run J 1 st guess

7. 1D Model with Finite Element Method

8. CitcomS is a fully spherical FEM code, solving advection diffusion problems. I developed the adjoint version of CitcomS to realize the time inversion. Adjoint method with CitcomS CMB Surface  = 1.27  = 1.87  

9. Simple case Boundary Condition: velocity : free slip & non-penetrative temperature: isothermal at top/bot.; zero heat flux on sidewalls. Viscosity: no depth dependence; no temperature dependence Thermal BL: no Reference states: Blank 1 st guess

10. caseshaperadiusmagnitudemesh Target T 0 sphere0.150.3depend blanksphere0.150.00133x33x33 smallersphere0.070.533x33x33 biggersphere0.200.333x33x33 SBI(1)SBI 33x33x33 SBI(2)SBI 73x73x73 Note: column 2 to 4 describe the anomaly properties. SBI: simple backward integration from present to past List of the first initial guesses

11. Retrieved initial conditions from various first guesses

12. The main feature is always recovered; a better first guess leads to a better recovery. Recovery with SBI first guess has the smallest residual. Higher resolution mesh greatly reduces the residual.

13. More Complicated Earth Model Viscosity: reference visc. = 1.0e21 (Pa sec) Temperaturewith thermalboundarylayer:

14. Residual plot and retrieved initial conditions

15. Conclusions about the adjoint method The adjoint method works well for whole mantle convection models. The SBI first guess is optimal because it is unique and it leads to a good recovery. However… The adjoint method requires that both model parameters and the present day observation be perfectly known, neither of which is true in general. Present day observation = seismic tomography Model unknowns => mainly rheology (viscosity) !

16. Dynamic topography as additional constraint

17. Theoretically Numerically h: dynamic topography

18. What happens now …

19. Start with a trial viscosity and an estimated temperature from seismic tomography Adjoint calculation predicted dynamic topography compare with observation Update temperature and viscosity accordingly

20. Example 1

21. Example 2

22. Summary 1.The dynamic topography, as another constraint, makes the adjoint method practical in real geological problem. 2.More complicated models are to be tested, e.g. layered viscosity structure, real seismic tomography as observation, etc. 3.Incorporating plate tectonics history…