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 ?

Mechanics (vertical throw) Altitude vs. time: A parabola

And what happened if time went backward ?

Symmetries in the laws of physics

r x (t)

1

2

r g t2

r v 0 t

r x 0

Mechanics (vertical throw)

Nothing changes ! -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

r x (t)

1

2

r g t2

r v 0 t

r x 0

T : t t

r x (t)

1

2

r g (t)2 (

r v 0) (t)

r x 0

r x (t)

1

2

r g t 2

r v 0 t

r x 0 x(t)

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:

Time asymmetry !

- Dissipation of energy (kinetic -> heat) is not time reversal invariant - Time reversal: Friction -> mysterious push; unknown physical process

-> Blame it on thermodynamics

Symmetries in the laws of physics

r x (t)

1

2

r g t2

r v 0 t

r x 0

b

mvdt

2

r x (t)

r x (t)

2b

mvdt2

r x (t)

r F b

r v

The 2nd principle of thermodynamics -Spontaneously, processes happen in only one direction -All processes in nature are irreversible -There’s an increase of entropy -> Thermodynamic arrow of time !

Symmetries in the laws of physics

Suni 0

? 25ºC T=75ºC T=50ºC

The arrow of time and seismology Elastic wave equation:

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

2

r u

t 2 ( 2)((

r u )) ( (

r u ))

Time reversal of the seismic wavefield

Numerical experiment - Spherical waves from a point source - Time reversal -> Focussing at source location Anderson, LANL

Symmetries in the laws of physics

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?

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

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

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?

Earthquake source studies and… Earth structure?

Forward problem

Evaluation problem

Model m

Predictions Inverse problem

Observations d

Let’s try time reversal of the seismogram coda

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:

Earth structure

-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.

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:

Earth structure

Modelling the 2009 EQ:

Snapshots of vertical velocity

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

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

Time reversal links to tomographic inversion:

The interesting point: An adjoint wavefield that incorporates data residuals, and the regular forward wavefield correlate where the sources of 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

Evaluation problem

Model m

Predictions Inverse problem

Observations d

Earth structure

Tromp et al.,

2005, Fichtner

et al., 2006

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)

Earth structure

End. Thank you !

Time reversal of coda waves

Earth structure