LMU Geophysics/Seismology
Heiner Igel, Andreas Fichtner, Martin
Käser, Marcus Mohr, Karin Sigloch, a.o.
Spatial Scales and Memory
(back of the envelope)
Highest
Highestfrequency:
frequency:
Shortest
Shortestwavelength:
wavelength:
Shortest
Shortestwavelength:
wavelength:
Grid
Gridpoints
pointsper
perwavelength:
wavelength:
Grid
Gridspacing:
spacing:
Grid
Gridspacing:
spacing:
0.1
0.1 Hz
Hz
20
20km
km(crust)
(crust)
50
50km
km(mantle)
(mantle)
55
2000
2000mm(crust)
(crust)
5000
5000mm(mantle)
(mantle)
Required grid points: O(109)
Required memory:
O(100 GBytes)
T>20s
Data
Synthetics
The FORWARD Problem
Elastic wave equations
3D Model
ρ∂ t2ui = ∂ j (σ ij + M ij ) + f i
σ ij = cijkl ε kl
Grid
ε kl = 1 / 2(∂ k ul + ∂ l uk )
Parallelisation
Synthetic seismograms
Simulation
Elastic Wave Equations
theory
theoryof
oflinear
linearelasticity
elasticity
(stress-strain,
Hooke´s
(stress-strain, Hooke´slaw)
law)
++
Newton´s
Newton´slaw
law
(acceleration
through
(acceleration throughforces
forces
caused
causedby
bystress)
stress)
==
velocity-stress
velocity-stressformulation
formulation
(linear
hyperbolic
(linear hyperbolicsystem)
system)
In
Inaaheterogeneous
heterogeneousmedium:
medium:
space-dependent
material
space-dependent materialcoefficients
coefficients
Lamé
(x,y,z),
Laméconstants:
constants: λλ==λλ(x,y,z),
μμ==μμ(x,y,z),
(x,y,z),
density
ρρ==ρρ(x,y,z)
density
(x,y,z)
Structure of the Hyperbolic System
Compact vector-matrix notation gives
with the vector of unknowns and Jacobian matrices
Numerical
methods
••
Finite
FiniteDifferences
Differences(high
(highorder,
order,optimal
optimaloperators)
operators)
••
Pseudospectral
Pseudospectral methods
methods(Chebyshev,
(Chebyshev,Fourier)
Fourier)
••
Finite/spectral
Finite/spectral elements
elementson
onhexahedral
hexahedralgrids
grids
••
Unstructured
Unstructured (tetrahedral)
(tetrahedral) grids
grids (finite
(finitevolumes/elements,
volumes/elements,
natural
naturalneighbours,
neighbours,discontinuous
discontinuous Galerkin)
Galerkin)or
orcombinations
combinations
••
Parallelization
Parallelizationusing
usingMPI
MPI(message
(messagepassing
passinginterface)
interface)
Spectral element techniques
Discontinuous Galerkin Method
Slip map of an earthquake fault
Mesh spacing is proportional to P-wave velocity
Käser, Mai, Dumbser, 2007
The (structural) inverse problem
Data vector d:
Traveltimes of phases observed at
stations of the world wide
seismograph network, now we move
to complete waveforms
Model m:
3-D seismic velocity model in the
Earth’s mantle. Discretization using
splines, spherical harmonics,
Chebyshev polynomials or simply
blocks.
Sometimes 10000s of travel times (millions of seismogram samples)
and a large number of model blocks: underdetermined system
Model Uncertainties – Degrees of
Freedom
Decreasing misfit
Increasing model complexity
Increasing number of degrees of freedom
after L. Boschi (2007)
Scientific problems
Geodynamics
Courtesy: G. Jahnke
Courtesy: H.P. Bunge, B. Schuberth
scientific problems
Earthquake scenarios
••
Accurate
Accurateforecasting
forecastingof
ofhazard
hazardand
andrisk
riskscenarios
scenariosfor
for
regions
and
time
intervals
specific
specific regions and time intervals
••
Incorporation
Incorporationof
ofearthquake
earthquakescenario
scenariosimulations
simulationsinto
into
probabilistic
hazard
analysis
probabilistic hazard analysis
M5.9 Roermond 1992
Shaking
hazard
Amplification
… and now …
• Andreas Fichtner: seismic waveform
inversion as an adjoint problem
• Heiner Igel: probabilistic desccription of
inverse problems – Monte Carlo Methods
• Karin Fichtner: finite frequency
tomography and relevance for geodynamic
issues
… some key questions …
related to waveform inversion
• How can we properly quantify uncertainties of
the inverted model(s)
• How can we properly visualize uncertainties in
large-dimensional model spaces
• Can we quantitatively describe prior information
on Earth‘s structure?
• Should we find optimal parameterizations of
Earth models?