Time reverse simulations in regional seismology
Daniel Stich
Instituto Andaluz de Geofísica
Universidad de Granada
stich@ugr.es
Colaborators (in time-reversed order): Antonio Molina, Peter Danecek, Rosa
Martín, Jeroen Tromp, Chris Bean, Gareth O’Brian, Ivan Lokmer, Andrea Morelli…
The direction of time
We’re quite sure:
The direction of time is forward
The arrow of time points to the future
Reasons:
Psychological arrow of time, fundamentally different characteristics of
- The past (memory, known)
- The present (perception, brief interaction of environment and brain)
- The future (plans, unknown)
Logical arrow of time
- A cause precedes its effects (try throwing the egg)
But take care
- Seems evident, but the causal relation is an interpretation (David Hume)
- Apparent contradiction between causal and psychological arrow
What about physics ?
Symmetries in the laws of physics
Mechanics (vertical throw)
Altitude vs. time: A parabola
1r 2 r
r
r
x (t)  g t  v 0  t  x 0
2
And what happened if time went backward ?
Symmetries in the laws of physics
Mechanics (vertical throw)
1r 2 r
r
r
x (t)  g t  v 0  t  x 0
2
Nothing changes !
T : t t
1r
r
r
r
2
x (t)  g (t)  (v 0 ) (t)  x 0
2
1r 2 r
r
r
x (t)  g t  v 0  t  x 0  x(t)
2
-Impossible distinguish a vertical throw from the future or the past !
-No preferred direction of time
-The laws of Newton's mechanics are time reversal invariant

Symmetries in the laws of physics
Another example: Electrodynamics (Maxwell's equations)
Nothing changes !
-Electrodynamics is time reversal invariant
-The same for Coulomb and Lorentz forces -> link to mechanics
-Do it yourself: General relativity, quantum field theory, etc…
Where’s the arrow of time in physics !?
Symmetries in the laws of physics
Back to mechanics (but more realistic)
Friction:
r
r
F  bv
Time asymmetry !

1r 2 r
b
r
r
x (t)  g t  v 0  t  x 0 
2
m
2b
r
r
x (t)  x (t) 
m
 vdt
r
 vdt  x(t)
2
- Dissipation of
energy (kinetic -> heat) is not time reversal invariant
- Time reversal: Friction -> mysterious push; unknown physical process
-> Blame it on thermodynamics

2
Symmetries in the laws of physics
The 2nd principle of thermodynamics
Suni  0
?
T=50ºC
-Spontaneously, processes happen in 
only
one direction
-All processes in nature are irreversible
-There’s an increase of entropy
-> Thermodynamic arrow of time !
T=75ºC
25ºC
Symmetries in the laws of physics
The arrow of time and seismology
Elastic wave equation:
r
u
r
r
 2  (  2)(( u))  (  (  u))
t
2
For negligible attenuation, seismic wave propagation is time reversal
invariant
Conclusion: There is time-symmetry in wave propagation, but asymmetry
in the state of the system (initial conditions: An earthquake does
radiate energy)
Symmetries in the laws of physics
Time reversal of the seismic wavefield
Numerical experiment
- Spherical waves
from a point source
- Time reversal
-> Focussing at
source location
Anderson, LANL
Time reversal works
Technical application for ultrasonic waves: non destructive testing,
medical imaging…
And earthquake seismology?
(Sparse point sampling at the free surface, attenuation, noise, bad
Earth models…)
Does it still work?
Earthquake source studies
An example: Time reversal of
global record sections
The 2004 Mw 9.3 Sumatra earthquake
Time reversal of long-period
seismograms from 165 stations
-> successful recovery of source
location and radiation pattern !
Larmat et al., 2006
Earthquake source studies
Application to more enigmatic sources:
Glacial earthquakes
Episodic rapid slip of large outlet glaciers,
proxy for climate change
Radiate seismic waves, but very weak and
emergent signals
Time reversal: location and radiation
pattern (single force)
Larmat et al., 2008
Earthquake source studies
Another application to a more enigmatic source
Volcanic tremor/ LP events
Synthetic tests for different source models,
complex topography, heterogeneous structure:
For densely instrumented volcanoes and
appropriate imaging field, focussing occurs in
the right location, indicating also the radiation
pattern.
Promising ! Pending: application to real data…
Lokmer, O’Brian, Stich, Bean, 2009
Earthquake source studies and… Earth structure?
Seismic sources: Time reversal ≠
inversion, no source model and
objective function involved
-> unprejudiced imaging of known
(shear faulting) and unknown (e.g.
volcanic, glacial) events
Conclusion: Blurry images, difficult
to interpret, but new insights
And Earth structure?
Scatterers (reflection, diffraction)
act as secondary sources, can we
locate those through time reversal?
Forward problem
Model m
Predictions
Inverse
problem
Observations
Let’s try time reversal of the seismogram coda
d
Evaluation
problem
Earth structure
A good example: very long (7 minutes !)
surface waves are observed at the
Gibraltar Arc for EQs in the Atlantic.
Panels show vertical, radial and transverse
waveforms (15-50s) for the 17/12/2009
Mw 5.5 EQ along a ~N-S profile.
Earth structure
Very long seismograms.
Is it due to 12km of
sediments in the Gulf of
Cadiz? Possible basin
resonance (continuous
secondary source)?
Earth structure
Modelling the 2009 EQ:
-Wave propagation is simulated with
the spectral-element code SPECFEM3D
(www.geodynamics.org)
- Lateral dimensions of the model volume 20°x 20°
- Conforming hexahedral mesh for periods down to ~8 s
- Time step of 0.05 s
-Simulations done at RES (Red Española de Supercomputación).
- ~10 GBytes of distributed memory
- 64-100 processors
- ~3 hours.
Earth structure
Modelling the 2009 EQ:
Snapshots of vertical velocity
Earth structure
Time reversal of coda waves:
-many stations (+IberArray)
-cut direct waves
-flip direction of time
-apply 3C recordings as
single force sources at
surface
Earth structure
Time reversal of coda waves from the
2009 EQ (snapshots of ~energy density)
Earth structure
Modelling the 2009 EQ:
Snapshots of vertical velocity
Earth structure
Another good example: Simple coda waves
Indirect waves reflected at some obstacle away
from the great circle path (multipathing)
Alpine area: Late arrival with characteristic
polarization (Love waves), reflection at Apennines?
Time reversal ->
imaging of
reflector?
Earth structure
Time reversal of the surface
wave coda
Apennines: Moho discontinuity -> change of
phase velocity -> reflection of surface waves
Time reversal ->
imaging of
lateral
heterogeneity?
Earth structure
Time reversal of the surface wave coda
and imaging:
Imaging: correlation between the backward
wavefield and the original forward wavefield
(Claerbout, 1985)
Known application: Reflection seismics
New: Do it in the horizontal direction.
The part of propagating a terminal wavefield
backward in time is the “adjoint problem”
Tarantola, 1988:
-Wave equation operator is symmetric,
-Displacement fields and source fields are dual
spaces when initial and final conditions are
interchanged
Earth structure
Waveform adjoint source for a least
squares waveform misfit function:
Synchronized, time-reversed, linear
differences between predicted and
observed displacement seismograms applied
at the receiver locations):
Coda waves are the residuals compared to
an earth where reflection/ scattering does
not occur
Earth structure
Time reversal links to tomographic
inversion:
Model
The interesting point: An adjoint
wavefield that incorporates data
residuals, and the regular forward
wavefield correlate where the sources ofInverse
problem
the residuals could be located
computing first derivatives of the misfit
function with respect to the model
parameters (sensitivity kernels):
Tromp et al., 2005, Fichtner et al., 2006
Forward problem
m
Observations
Predictions
d
Evaluation
problem
Tromp et al.,
2005, Fichtner
et al., 2006
Earth structure
Evolution of the time reversed coda wavefield (top) and regular
wavefield (bottom)
Earth structure
Correlation between the time reversed and regular coda wavefield
(adjoint approach, quantitative framework, transpose operators and
dual spaces require time reversal)
Reconstruction of the reflector (S-wave sensitivity kernels, synthetic
data)
Earth structure
Correlation between the time reversed and regular coda wavefield
(adjoint approach, quantitative framework, transpose operators and
dual spaces require time reversal)
Reconstruction of the reflector (S-wave sensitivity kernels, synthetic
data left, real coda right)
End.
Thank you !
Earth structure
Time reversal of coda waves