Structure from surface waves
Stewart FISHWICK – Maggy HEINTZ
RSES - ANU
Body versus surface waves
P and S body waves interfere constructively to generate waves guided by
the surface of the Earth: the surface waves.
Love waves
Love waves result from the constructive
interference of SH waves
(e.g. the trapping of SH waves in the low velocity
surface layers)
The simplest case in which this occurs is a low
velocity layer overlying a high velocity half space.
The particle motion is in the horizontal plane,
perpendicular to the propagation direction.
Rayleigh waves
Rayleigh waves result from the constructive
interference of P and SV waves.
They are polarised is in the vertical plane,
parallel to the propagation direction.
The particle motion is elliptical retrograde near
the surface, and elliptical prograde at depth.
Observation on a seismogram
Surface waves arrive after direct P and S waves and show a large amplitude
Rayleigh waves mainly observed on the Z component
Love waves observed on the E and N components
Dispersion
Example of very well dispersed wave trains
The difficulty is that dispersion is not
usually pronounced enough to give peaks
that are associated with a single period
(e.g. a phase).
Typically phase velocity is measured in the
frequency domain.
Measurement of phase velocity between two
nearby stations for which common cycles of a
given phase can be reliably identified and
differential travel time measured.
Modes and depth dependence (1)
The wave train generated from the surface waves can be
constructed by summing the contributions of a number of
different ‘modes’.
Analogy:
Mode
Oscillations of a fixed string
Mode 0 (fundamental): the string is fixed at the end points
Mode 1: one additional fixed node point
Mode 2: two additional fixed node points….
While the amplitude of all surface waves decays
exponentially with depth, the higher modes
sample deeper parts of the Earth.
In recording of shallow events the
fundamental mode will dominate the
seismogram, while deeper events will excite the
higher modes.
Modes and depth dependence (2)
Alongside the depth dependence of the varying modes, all frequencies are sensitive to
structure at different depths.
Dispersion with period
Longer period surface waves, which sample deeper in the mantle, will arrive
before the shorter period waves
50 s period
Depth (km)
80 s period
Mode 0
Mode 1
Mode 2
Mode 3
Mode 4
Available Data
Lack of instrumentation in southern
hemisphere and oceanic regions.
Surface waves allow the sampling
of vast areas of the globe that are
otherwise devoid of seismic
stations and sufficiently strong
earthquake sources.
Particularly well adapted for oceanic
domains
Structures imaged
Large scale structures of geodynamics interest
Depth extent of cratonic roots
•Dorman et al. 1960 – Different structure beneath continent and
oceans required to fit dispersion data
•Macdonald 1963 – Explains seismic / gravity / heat flow through
contrasting structure to 700km deep
•Jordan 1975 – Continental tectosphere – at least 400km depth,
lateral contrasts stabilised by compositional gradients
•Anderson 1979 – Base of continental roots 150-200, no resolvable
differences below 250km
Deep roots may act as a buffer between crust and hot mantle
Stabilised due to chemical variations?
Suggestion is that old mantle has remained in contact with
overlying crust
Structures imaged
Large scale structures of geodynamics interest
Depth extent of cratonic roots
Structures imaged
Depth (km)
Depth extent of cratonic roots
Latitude
Structures imaged
Large scale structures of geodynamics interest
What else???
Subduction Zones
Mantle Plumes
But what are the difficulties associated with this for surface
wave tomography?
Structures imaged
Subduction zones
Measurement of dispersion (1)
Surface waves are well suited to study the elastic structure of the crust and upper
mantle, which can be deduced from their group and/or phase dispersion properties.
Originally, group velocity dispersion was obtained by measuring the arrival times of peaks
and troughs of waves in a dispersed wave train.
The time between two successive peaks would give the half period (T/2 = S/Z) and the
group velocity U(Z) is computed as:
U(Z) = X / t(Z),
With X the epicentral distance and t(Z) measured with respect to the earthquake’s origin time.
Measurement of dispersion (2)
Group velocities were preliminarily estimated by Multiple Filtering Techniques (MFT, e.g.
Dziewonski et al., 1969)
Aim
To isolate a surface wave mode along its group velocity curve (method
later perfected under the FTAN method (Lander, 1989, Levshin et al.,
1994)
How 1) Computation of an ‘energy diagram’:
plot of the energy contained in the
seismogram vs period and time (i.e.,
group velocity)
2)
The maxima of the 2D diagram
delineate the group velocity curve of
the dispersed surface wave modes
3)
The time domain seismogram is
filtered using multiple filters
centered, at each point in time, on
the frequency corresponding to the
maximum of energy at that time
The period range of each dispersion curve depends on the magnitude of the
earthquake, path length…
Measurement of dispersion (3)
Phase Velocities
•Direct measurements of phase dispersion is harder to
perform
•It is rare for the peaks to be adequately dispersed to be able
to measure a particular frequency
•Instead of calculating phase dispersion in the time domain,
commonly use frequency domain
•Need to know the initial phase of the seismogram –
knowledge about the earthquake source mechanism
•Tended to be global studies: Large events, long paths
•Possibility of two station methods ...
From dispersion measurements to the maps…
Group velocity maps: some examples… (1)
Global Models of Surface Wave Group Velocity (Larson and Ekstrom, 2001)
Love, 80 s
Rayleigh, 80 s
Love, 150 s
Rayleigh, 150 s
Group velocity maps: some examples… (2)
Love, 70s
Rayleigh, 70 s
Vdovin et al., 1999
Group velocity maps: some examples… (3)
Love, 70s
Rayleigh, 70 s
Vdovin et al., 1999
Feng et al., 2004
Waveform inversion (1)
Cells
Splines
Continuous regionalisation
Waveform inversion (2)
Cara and Leveque (1989): choice of secondary observables
The waveform of the surface wave portion of a seismogram depends in a highly non-linear
way of the elastic parameters.
The non linearity in the inverse problem is reduced by using a set of secondary
observables.
Uses single-mode cross correlograms,
the secondary observables are the
envelope and the instantaneous phase
Cross correlograms are computed for
real data with a single mode synthetic,
and for synthetic model with the same
single mode synthetic
Waveform inversion (3)
Cara and Leveque (1989)
The inversion process will aim at finding a 1d model of the Earth which minimises the
difference observed between the 2nd observables of the synthetic cross correlogram
and 2nd observables of the actual cross correlogram.
The secondary observables are chosen to extend the domain of quasi-linear
dependence between the data and model parameters.
Debayle (1999) developed an automated procedure around this inversion scheme.
The period range for the inversion is 50-150 seconds (however this depends on the
signal-to-noise ration in the seismogram). The inversion is only succesful if certain
criteria are fulfilled:
•Data Misfit (variance reduction) – envelopes
•Converged to a unique model
•Fit to the actual seismogram
Waveform inversion (3)
Partitioned Waveform Inversion: Nolet (1990)
The PWI code is based on the work of Nolet, 1990. It is a more direct inversion of the
waveform – rather than using secondary observables.
This, unfortunately, means the starting model for the inversion needs to be quite
accurate. There is a smaller domain of quasi-linearity than for the Cara and
Leveque method (Hiyoshi, 2001).
An advantage of PWI is that the linear constraints on the inversion give uncorrelated
errors and the inversion is for a 3D model – so somewhat easier to assess vertical
resolution.
Lebedev et al, (2005) have produced an automated multimode inversion. This
appears to be an interesting package with a lot of focus on:
•Case by case selection of the seismograms (time-frequency)
•Elaborate time- and frequency- dependent weighting
•Aiming to get the most out of the seismogram
Cells
Splines
Continuous regionalisation
1D models
The 1D models are obtained as result of the
waveform inversion.
We retrieve the perturbations with respect to a
reference Earth model (PREM for instance)
required to generate the observed seismogram.
Normally only one reference model is used in the
calculation of the final 1D model.
However, due to the non-linearity of the problem,
this may lead to erroneous path average models.
Parameters controlling the final image (2)
Choice of the reference model for the calculation of the 1D path average models
Cells
Splines
Continuous regionalisation
Tomographic inversion
The 1D models representing a path average for each source-receiver pair are used within a
tomographic inversion to derive a model of the 3D velocity structure.
Various parameterisations can be used:
•
•
•
•
•
spherical harmonics
wavelets
spherical B-spline functions
cells
continuous regionalisation
Can these parameterisations be easily
adapted for non-uniform data?
Example of B-spline function
Parameters controlling the final image
Path coverage
Van der Lee et al., 2001
Heintz et al., 2005
Parameters controlling the final image
Choice of the reference velocity (reference model)
Parameters controlling the final image
Choice of the color scale
150 km
150 km
Parameters controlling the final image
Parameterisation (1) : Correlation length
200 km
400 km
800 km
Results from a compromise between the physics of
surface waves (wavelengths impose the minimum
size of structures we can recover) and the path
coverage.
Parameters controlling the final image (6)
Parameterisation (2) : Knot point spacing
Parameters controlling the final image (6)
Regularisation : Choice of the damping
Parameters controlling the final image
Crustal Correction (1)
•In the waveform inversion we are using periods that are
strongly sensitive to mantle structure – and it is this structure
that we want to image
•How do we take account of the influence of the crust –
surface wave sensitivities are not zero
•In the calculation of the path average velocities (e.g., in fitting
the waveform) we can take into account the crust using a
global model
•3SMAC
•Crust 2.0
•Does it matter if they are wrong?
•Do we need to make any further corrections? - The crustal
model becomes very important when performing local
inversions (going from phase/group maps to a 1d model)
Parameters controlling the final image
Crustal correction (2): choice of the crustal model
Three global crustal models exist, but they highlight substantial differences
We compare here for the Australian and South American continents, 3SMAC and
CRUST2.0, both offering the same resolution
Parameters controlling the final image
Crustal correction (3): choice of the crustal model
Parameters controlling the final image
Crustal smearing: choice of the crustal model
Debayle and Kennett (2000) showed the effect
produced by a 10 km variation in crustal thickness
on a path average model.
The deviation exceeds the error bars only in the
uppermost 100 km for the shear velocity.
A 200 km wide zone with a 10 km difference in
crustal thickness will only produce a difference in
2 km in the average crustal model for paths as
short as 2000 km.
However, the necessity of working with paths with mixed continental and
oceanic components means that there is inevitably some influence from
crustal structure.
Limitations in Surface Wave Tomography (1)
Finite Frequency Effects
• One of the present hot-topics in tomography in general
• Trying to move away from the ray-theoretical approach,
great-circle and narrow – where smoothing is fro the choice of
parameterisation
• Is the theory good enough to incorporate finite frequency
effects – can we compute which regions actually influence the
observed waveform. Not everyone agrees.
• And while some ideas may be a theoretical improvement, the
parameterisations and regularisations are still the dominant
smoother of the tomographic models.
• There is some way to go ...
…on the sensitivity of seismic waves…
Limitation of the technique (2)
Finite frequency effect: first observed on body waves
Seismic ray: where the ray samples the Earth
Red = sensitive
Yellow = insensitive
Most of the ray seems to travel in
insensitive terrain !
From Nolet’s webpage
Dependence on the period of the wave: the sensitivity
kernel is narrower at short periods
Limitation of the technique (3)
…on the sensitivity of seismic waves…
Finite frequency effect: surface waves
The levels of heterogeneities in the upper mantle may be too large for the path average approximation
to be applied directly to the 1D models for surface waves under the assumption of propagation along
the great circle between source and receiver.
We should take into account finite frequency effects on surface wave propagation
rather than assuming sensitivity just on the ray path.
Examples of the influence zone kernels for Rayleigh
waves at period 50 s (left) and 100 s (right).
The approximate influence zone can be represented
as roughly 1/3 of the width of the first Fresnel zone.
From Yoshizawa and Kennett
Limitation of the technique (4)
…on the sensitivity of seismic waves…
Finite frequency effect: surface waves
Surface wave tomography including the effects of finite frequency
We see improvements in short wavelength structures.
There are significant differences in the models where velocity gradients are large.
The method of inversion for finite frequency surface waves is useful for multi-mode surface
waves. It has still difficulty in the treatment of the effects from a very strong heterogeneity
outside the influence zone, which has to be considered for the use of short period (< 40 s)
surface waves.
From Yoshizawa and Kennett
Limitation of the technique (5)
Periods that can be used
For waveform inversion shortest period is 40s
Much of the energy in the surface wavetrain is at
higher frequencies
Higher Modes
How are the higher modes treated?
How many modes to use?
Upweighting of higher modes?
Where do we have deep earthquakes
Error estimation ...
Error estimation
A priori error = 0.05 km.s-1
The tomographic inversion allows to compute an a posteriori covariance matrix which provides
an evaluation of the quality of the inverted model. In the inversion formulation, the a posteriori
covariance matrix incorporates the covariance matrix on the data.
The a posteriori covariance matrix Cm is related to the a priori covariance matrix Cm0 by
Cm = (I-R)Cm0
with I the identity matrix and R the resolution one.
When the resolution is null, the a posteriori error will be equal to the a priori error. For
perfect resolution, the a posteriori error should be null.
Assessing the resolution of the tomographic model (1)
Resolution test: checkerboard
a
b
c
Simons et al., 2002
A popular way of assessing the resolution in tomographic models is to calculate the
recovery of a synthetic (checkerboard) input pattern.
The damping parameters can be chosen to reproduce the input pattern optimally, a luxury
which is not available when choosing the damping needed to model the actual data for
unknown Earth surface.
Checkerboard tests are mainly used in a qualitative way.
Assessing the resolution of the tomographic model (2)
Resolution test: PREM at 50 km depth
Instead of checkerboard,
some people prefer
considering a ‘realistic’
input structure such as
PREM at 50 km depth.
The principle is however
the same, and it is the
ability of the path coverage
to recover known input
structures that is tested.
Assessing the resolution of the tomographic model (3)
Resolution tests
The problem with most resolution tests (in my opinion!), is that for surface wave tomography
they are not really that realistic.
It is difficult for them to include the waveform inversion stage (and when they do it is very
idealised – synthetics calculated from a 1d average, using the same code that will be used
in the inversion.
Therefore they are really only testing the tomographic inversion: path coverage.
Furthermore, noise is rarely added
The only problem is to publish a paper it’s almost expected that it’s included ...
The robustness of tomographic models can be asserted through extensive series of test.
Simons et al (2002) for instance, performed inversions with up to 40% of the data
randomly removed.
They calculated the mean model obtained from 500 inversions with 5% of the data
randomly removed, as well as the mean of 500 inversions performed on a dataset to
which noise was added.
However ... The interpretations on these tests are difficult – can’t just look at
averages/variance; as where you have no data you will damp towards the a priori and
get minimal variance!!!
Applications of Surface Wave Tomography
Physical Properties of the Upper Mantle (1)
Seismic velocities are most sensitive to temperature, but are also affected by:
grain size, water, melt and composition
Composition is a bit controversial – most of the experimental data suggests that
composition will have a small effect, but there are groups who feel it must/does play a more
important role.
Goes et al (2000) include temperature and composition and note that anelasticity is crucial
as it allows reasonable variations in temperature to give large variations in wavespeed.
Faul and Jackson (2005) combine temperature and grain size, based on experimental work
on torsional oscillations at seismic frequencies (olivine). They are able to then model the
1D wavespeed profiles of a region and estimate temp and grain size. (see next figure)
Priestley and McKenzie (2006) have a different approach: They use a shear wavespeed
model of the Pacific, and the expected age temperature relationship for oceanic lithosphere
to compute a relationship between wavespeed and temp. This is then applied to continental
regions.
Lots of different methods – all indicate that there are large variations in temp across the
continents, do these remain in place for a long time?
Application of surface wave tomography
Physical properties of the upper mantle (2)
Faul and Jackson (2005)
Application of surface wave tomography
Physical properties of the upper mantle (3)
How do we explain the low velocities in central Australia?
• The modelling of Faul and Jackson (2005) and the work of Shapiro and Ritzwoller
(2004) suggests that you must have increasing wavespeeds as you get shallower –
closer to the moho (temperatures must be decreasing)
• How confident are we that the low velocities are real!
• Could be crustal influence??
•However it is seen in another data set (body wave travel times)
So are we seeing the influence of something other than temperature?
Applications of Surface Wave Tomography
Diamond Exploration (1)
Jaques and Milligan (2004) investigated the structural controls on the location of
diamondiferous kimberlites and lamproites within Australia.
Locally: a relationship with known fault zones, and gradients/discontinuities in
potential field data
Regionally: fast wavespeeds (deep lithosphere required to be within the diamond
stability field) – but perhaps particularly the edge of these regions
From a three year old
surface wave model
Application of surface wave tomography
Diamond exploration (2): Australia
200km
Application of surface wave tomography
Diamond exploration (3): South America, Heintz et al. (2005)
Application of surface wave tomography
Diamond exploration (4): South Africa, Fouch et al. (2004) + Shirey et al. (2002)
Application of surface wave tomography
Diamond exploration (5): What are the edge features
We’ve already noted that the tomographic image is dependent on the choice of reference
model, the colour scale, the damping, the regularisation, etc., etc.
So what do we mean by an ‘edge’ feature of a tomographic model. Can investigate
horizontal edges through gradient maps – relate gradients to absolute velocities so
independent of the reference model (Fishwick, 2006)