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Lijun Liu
Seismo Lab, Caltech
Dec. 18, 2006
Inferring Mantle Structure in the Past---Adjoint method in mantle convection
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Geoid
Seismic Tomography
Other Data
Present
PastMuch less is known about the
earth’s interior in the past
?
DynamicTopograph
y
Plate Tectonics
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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.
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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
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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)
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Flow Chart
Target T0Data T1
Forward run
Forward run
Adjoint run
J
TJ /
1st guess
n0
nT0nT1
J
J
n1
nnnn TT 001
0
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1D Model with Finite Element Method
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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
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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
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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
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Retrieved initial conditions from various first guesses
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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.
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More Complicated Earth ModelViscosity: reference visc. = 1.0e21 (Pa sec)
Temperaturewith thermal boundary layer:
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Residual plot and retrieved initial conditions
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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)
!
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Dynamic topography as additional constraint
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• Theoretically
• Numerically
T C1h
22 hC
h
h: dynamic topography
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What happens now …
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• 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
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Example 1
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Example 2
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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…