Lijun Liu

Seismo Lab, Caltech

Dec. 18, 2006

Inferring Mantle Structure in the Past---Adjoint method in mantle convection

Geoid

Seismic Tomography

Other Data

Present

PastMuch less is known about the

earth’s interior in the past

?

DynamicTopograph

y

Plate Tectonics

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.

Governing Equations:

(continuity)

(momentum)

(energy)

: velocity, P: dynamic pressure, : density,

: dynamic viscosity, : thermal diffusivity,

H: internal heat source.

0 u

HTTut

T

2

guP 2

u

Adjoint Equation: integration by part and let

HTTutTJL 2/

TkTutTJL 2/

Lagrange function:

0L

where is the adjoint quantity

: error in the initial; Tp: prediction; Td: dataad

The idea of adjoint:

(cost function) V

dp dVTTtJ 21 )()(

adaJdJ

)/(

2

utT

J

(e.g. R. Errico, 1997)

Flow Chart

Target T0Data T1

Forward run

Forward run

Adjoint run

J

TJ /

1st guess

n0

nT0nT1

J

J

n1

nnnn TT 001

0

1D Model with Finite Element Method

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

=

1.8

7

Simple caseBoundary 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 1st guess

case shape radius magnitude mesh

Target T0 sphere 0.15 0.3 depend

blank sphere 0.15 0.001 33x33x33

smaller sphere 0.07 0.5 33x33x33

bigger sphere 0.20 0.3 33x33x33

SBI(1) SBI SBI SBI 33x33x33

SBI(2) SBI SBI 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

Retrieved initial conditions from various first guesses

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.

More Complicated Earth ModelViscosity: reference visc. = 1.0e21 (Pa sec)

Temperaturewith thermal boundary layer:

Residual plot and retrieved initial conditions

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)

!

Dynamic topography as additional constraint

• Theoretically

• Numerically

T C1h

22 hC

h

h: dynamic topography

What happens now …

• 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

Example 1

Example 2

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…