Download M. Koch

The seismic Travel Time Problem
as applied to Tomography of the Earth
I. History and Developments in the Solution of the
forward and inverse Problem
Dr. M. Koch
Department of Geotechnology
University of Kassel
Lecture 1 / 41st INSC (2016)
M. Koch: The seismic travel time problem as applied to tomography of the Earth
28.07. 2016
Table of contents
1. Introduction
1.1. What is the seismic travel time problem and tomography?
1.2. Pioneering work and history
1.2.1. X-ray computed tomography
1.2.2. Local studies of the crust and upper mantle
1.2.3. Regional and global tomography
1.3. Recent trends: ambient noise and finite frequency tomography
2. Representation of structure
2.1 General approaches
2.2. Comparison of block/ smoothed parameterizations
3. The data prediction (forward) problem
3.1. Ray-based methods /shooting / bending
3.2. Grid-based methods / Eikonal/ shortest path ray tracing
M. Koch: The seismic travel time problem as applied to tomography of the Earth
28.07. 2016
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Table of contents
4. Solving the inverse problem
4.1. Nonlinear optimization formulation
4.2. The linear inverse problem
4.3. Analysis of solution robustness
5. Tomography as a tool to understand Earth’s geodynamics
5.1. Overview of the geodynamics of the Earth
5.2 How to relate seismic structure to geodynamics?
5.3. Tomography in geodynamically active regions
6. Future developments
7. References
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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1. Introduction
1.1. What is the seismic travel time problem and tomography?
Seismic tomography is the 3D- extension of the classical seismic travel time problem
technique for imaging the 3D- velocity structure v(x,y,z) of the Earth from travel times T of
seismic waves (P-, S-, and Surface waves) produced by earthquakes or explosions:
𝑇=
=>
1
𝑣 𝑥,𝑦,𝑧
𝑑𝑠
 d = g(m)
v(x,y,z) = F-1 (T)  m = g-1(m)
(forward problem)
(inverse problem)
a) Classical seismic travel time problem
Determination of v(r) by the
Wiechert- Herglotz -method
/ tau-p method
=> velocity model of the earth
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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1. Introduction
1.1. What is the seismic travel time problem and tomography?
b) Seismic tomography
b1) (Teleseismic) sources (hypocenters) known: Determination of structure
(Aki-Christofferson-Husebye (ACH) method):
Aki, K., A. Christoffersson, and E. S. Husebye (1977). Determination of the three-dimensional
seismic structure of the lithosphere, J. Geophys. Res., 82, 277-296
b2) (Local) sources unknown: Simultaneous Inversion for Structure and
Hypocenters
(SSH- Method)
(Focus of author’s research)
Aki K and Lee WHK (1976) Determination of three-dimensional velocity anomalies under a
seismic array using first P-arrival times from local earthquakes, 1, a homogeneous initial model.
J. Geophys. Res., 81: 4381–4399
Koch, M., (1985) Nonlinear inversion of local seismic travel times for the simultaneous determination
of the 3D-velocity structure and hypocenters - application to the seismic zone Vrancea, J. Geophys.,
56, 160 – 173.
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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1. Introduction
1.2. Pioneering work and history
1.2.1. X-ray computed tomography
2D- X-ray Computed Tomography (CT) developed in the 1970’s is at origin
of seismic tomography
Mathematics based originally on the Radon Transform (1917) (Koch, 1977)
Computationally nowadays CT done with
Algebraic Reconstruction Technique (ART)
(similar to the ACH-method)
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1. Introduction
1.2. Pioneering work and history
1.2.2. Local studies of the crust and upper mantle
Tomography using local earthquakes (SSH)
Aki K and Lee WHK (1976) Determination of three-dimensional velocity anomalies under a seismic array using first
P-arrival times from local earthquakes, 1, a homogeneous initial model. J. Geophys. Res., 81: 4381–4399
Eberhart-Phillips, D., 1986. Three-dimensional velocity structure in northern California coast ranges from inversion
of local earthquake arrival times. Bull. Seismol. Soc. Am. 76, 1025–1052.
Hirahara, K., 1988. Detection of three-dimensional velocity anisotropy. Phys. Earth Planet. Inter. 51, 71–85
References by the author M. Koch
Teleseismic tomography
Aki, K., A. Christoffersson, and E. S. Husebye (1977). Determination of the three-dimensional seismic structure of the
lithosphere, J. Geophys. Res., 82, 277-296
Koch, M., (1985) Nonlinear inversion of local seismic travel times for the simultaneous determination of the 3Dvelocity structure and hypocenters - application to the seismic zone Vrancea, J. Geophys., 56, 160 – 173.
=> extension of the linear ACH- and SSH – method to a nonlinear iterative gradient
optimization procedure with exact (shooting) ray-tracing
References by the author M. Koch
Nowadays non-linear inversion state of the art.
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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1. Introduction
1.2. Pioneering work and history
1.2.2. Local studies of the crust and upper mantle
Cross-bore hole tomography, for small-scale studies using Backprojection (ART) techniques of first
arrivals initiated by
McMechan, G.A., 1983. Seismic tomography in boreholes. Geophys. J. Royal Astr. Soc., 74, 601–612.
Bregman, N.D., Bailey, R.C., Chapmans, C.H., 1989. Crosshole seismic tomography. Geophysics 54, 200–215.
Reflection tomography, using artificial sources, to constrain both velocity and interface depth by
Bishop, T.P., Bube, K.P., Cutler, R.T., Langan, R.T., Love, P.L., Resnick, J.R., Shuey, R.T.,Spindler, D.A., Wyld, H.W., 1985.
Tomographic determination of velocity and depth in laterally varying media. Geophysics 50, 903–923
Wide-angle (refraction and wide-angle reflection) tomography on the regional scale by
Kanasewich, E.R., Chiu, S.K.L., 1985. Least-squares inversion of spatial seismic refraction data.
Bull. Seismol. Soc. Am. 75, 865–880.
Bleibinhaus, F., Gebrande, H., 2006. Crustal structure of the Eastern Alps along the TRANSALP profile from wide-angle
seismic tomography. Tectonophysics 414, 51–69.
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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1. Introduction
1.2. Pioneering work and history
1.2.3. Regional and global tomography
In these applications use of permanent global seismic networks (GSN).
Dziewonski, A.M., Hager, B.H., O’Connell, R.J., 1977. Large-scale heterogeneities in the lower mantle.
J. Geophys. Res. 82, 239–255
=> Used 700,000 P wave residuals from ISC bulletins to imagine lateral structure of the Earth’s mantle.
Use of PcP- and PKP- phases by
Karason, H., van der Hilst, R.D., 2001. Improving global tomography models of P-wavespeed. I. Incorporation of
differential travel times for refracted and diffracted core phases (PKP, Pdiff). J. Geophys. Res. 106, 6569–6587.
S-Phases and mostly spherical harmonical, but also irregular cell representation of heterogeneities,
allowing determination of Vp/Vs- ratios.
Su, W.-J., Dziewonski, A.M., 1997. Simultaneous inversion for 3-D variations in shear and bulk velocity in the mantle.
Phys. Earth Planet. Inter. 100, 135–156
Use of surface waves allowing for good resolution by the oceanic lithosphere
Shapiro, N.M., Campillo, M., Stehly, L., Ritzwoller, M.H., 2005. High-resolution surface wave tomography from ambient
seismic noise. Science 307, 1615–1618
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1. Introduction
1.2. Pioneering work and history
1.2.3. Regional and global tomography
Use of normal modes (free oscillations) allow for resolution of deep structures, but at large scales
Resovsky, J.S., Ritzwoller, M.H., 1999. A degree 8 mantle shear velocity model from normal mode observations
below 3 mHz. J. Geophys. Res. 104, 9931014
Anisotropy of the mantle is becoming a focus of tomography either
- by direct inversion of the P-wave anisotropy tensor
Tanimoto, T., Anderson, D.L., 1985. Lateral heterogeneity and azimuthal anisotropy of the upper mantle—Love and
Rayleigh waves 100–250 sec. J. Geophys. Res. 90, 1842–1858
References by the author M. Koch
- by shear wave splitting
Zhang, H., Liu, Y., Thurber, C., Roecker, S., 2007. Three-dimensional shear-wave splitting tomography in the Parkfield,
California, region. Geophys. Res. Lett. 34, doi:10.1029/2007GL031951
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1. Introduction
1.3. Recent trends: ambient noise and finite frequency tomography
Approaches that go beyond classical travel-time analysis:
Use of cross-correlations of ambient noise and the Coda of small-scale scattering
Shapiro, N.M., Campillo, M., 2004. Emergence of broadband Rayleigh waves from correlations of the ambient seismic
noise. Geophys. Res. Lett. 31, doi:10.1029/2004GL019491.
To overcome the limitations of geometrical resolution by the finite wavelength of body waves
(high frequency approximation of seismic ray- theory) use of first-order perturbation theory (Born
theory) to account for scattering has been proposed
Snieder, R., 1988a. Large-scale waveform inversions of surface waves for lateral heterogeneity.
1. Theory and numerical examples. J. Geophys. Res. 93, 12055–12065,
2. Application to surface waves in Europe and the Mediterranean. J. Geophys. Res. 93, 12067–12080
Understanding the sensitivity kernels of the ray-inversion (banana-doughnuts) allowed for better
localization of mantle plumes
Marquering, H., Dahlen, F.A., Nolet, G., 1999. Three-dimensional sensitivity kernels for finite-frequency travel times:
the banana–doughnut paradox. Geophys. J. Int. 137, 805–815.
Finite frequency tomography, using phase information, improves resolution of ray-based tomography
Hung, S.H., Shen, Y., Chiao, L.Y., 2004. Imaging seismic velocity structure beneath the Iceland hotspot: a finite frequency
approach. J. Geophys. Res. 109, B08305
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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Table of Contents
1. Introduction
1.1. What is the seismic travel time problem and tomography?
1.2. Pioneering work and history
1.2.1. X-ray computed tomography
1.2.1. Local studies of the crust and upper mantle
1.2.3. Regional and global tomography
1.3. Recent trends: ambient noise and finite frequency tomography
2. Representation of structure
2.1 General approaches
2.2. Comparison of block/ smoothed parameterizations
3. The data prediction (forward) problem
3.1. Ray-based methods /shooting / bending
3.2. Grid-based methods / Eikonal/ shortest path ray tracing
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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2. Representation of structure
2.1 General approaches
Structural representation dependent on resolution purpose of the model, whether model contains
horizontal (Moho) or vertical (faults) boundaries, and of the data (station) coverage.
Parametrization possible with
a) blocks (no a priori smoothing)
(ACH-method, SSH- method of the author, see References by the author M. Koch )
smoothing may occur a posteriori during the regularization (damping) of the solution, in account of
the Backus-Gilbert resolution-covariance trade-off
Backus, G.E., Gilbert, J.F., 1968. The resolving power of gross earth data. Geophys. J. Royal Astr. Soc. 16, 169–205
b) with interpolation and/or B-Spline smoothing
Thurber, C.H., 1983. Earthquake locations and three-dimensional crustal structure in the Coyote Lake area, central
California. J. Geophys. Res. 88, 8226–8236.
Thomson, C.J., Gubbins, D., 1982. Three-dimensional lithospheric modelling at NORSAR: linearity of the method and
amplitude variations from the anomalies. Geophys. J. Royal Astr. Soc. 71, 1–36
c) adaptive irregular Delauney tetrahedra meshes
Sambridge, M., Gudmundsson, O., 1998. Tomographic systems of equations with irregular cells.
J . Geophys. Res. 103, 773–781.
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2. Representation of structure
2.2 Comparison of block/ smoothed parameterizations
Synthetic reconstruction test with direct block parametrization (a) and with B-Spline smoothing (b).
(Rawlinson et al., 2010)
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Table of Contents
1. Introduction
1.1. What is the seismic travel time problem and tomography?
1.2. Pioneering work and history
1.2.1. X-ray computed tomography
1.2.1. Local studies of the crust and upper mantle
1.2.2. Regional and global tomography
1.3. Recent trends: ambient noise and finite frequency tomography
2. Representation of structure
2.1 General approaches
2.2. Comparison of block/ smoothed parameterizations
3. The data prediction (forward) problem
3.1. Ray-based methods /shooting / bending
3.2. Grid-based methods / Eikonal/ shortest path ray tracing
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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3. The data prediction (forward) problem
3.1. Ray-based methods /shooting / bending
Seismic ray tracing involves the computation of
𝑇=
1
𝑣 𝑥,𝑦,𝑧
𝑑𝑠
From the elastic wave-equation under the high-frequency approximation results
the Eikonal equation
|∇T|= s
with, s=1/v, the slowness)
or for the characteristics, the trajectory r(l), the kinematic ray equation
Cerveny, V., 2001. Seismic Ray Theory. Cambridge University Press, Cambridge
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3. The data prediction (forward) problem
3.1. Ray-based methods /shooting /
bending
Solution of ray-equation by
a) shooting method (initial value problem)
(iterative application of Snell’s law)
Thurber, C.H., Ellsworth, W.L., 1980. Rapid solution of
ray tracing problems in heterogeneous media. Bull.
Seismol. Soc. Am. 70, 1137–1148
References by the author M. Koch
b) bending method (boundary value
problem, to be solved iteratively)
Julian, B.R., Gubbins, D., 1977. Three-dimensional
seismic ray tracing. J. Geophys. 43, 95–113
Comparison of shooting- (top) and bending methods
(bottom) (Rawlinson et al., 2010)
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3. The data prediction (forward) problem
3.2. Grid-based methods / Eikonal/ shortest path ray tracing
Grid-based methods
a) Eikonal –equation
b) Shortest path ray tracing (using Fermat’s principles)
compute the global travel-time field on a defined a grid of points, not only for one pair of
source-receiver, as previously.
Advantage:
Highly efficient and stable for very heterogeneous media
Disadvantage: Computationally demanding (CPU-time scales with n3 of grid size)
Vidale, J.E., 1988. Finite-difference calculations of traveltimes. Bull. Seismol. Soc. Am., 78, 2062–2076
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Table of contents
4. Solving the inverse problem
4.1. Nonlinear optimization formulation
4.2. The linear inverse problem
4.3. Analysis of solution robustness
5. Tomography as a tool to understand Earth’s geodynamics
5.1. Overview of the geodynamics of the Earth
5.2 How to relate seismic structure to geodynamics?
5.3. Tomography in geodynamically active regions
6. Future developments
7. References
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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4. Solving the inverse problem
4.1. Nonlinear optimization formulation
The (nonlinear) seismic forward (travel time) relationship between
model m and data d
d = g(m)
is formulated as a nonlinear least-squares optimization problem
for the objective function
||d - g(m) ||2 -> minimum
Solution methods are
Iterative inversion procedure, adapted from
Liu, Q. and Y.J. Gu (2012), Seismic imaging:
From classical to adjoint tomography,
provide only a local minimum, i.e. depend on an accurate initial models, but Tectonophysics, 566-567, 31–66
provide robust measures of model uncertainty, at least within the linear
sub-framework (see Koch, Lecture 2)
1) non-linear (iterative) gradient techniques (most common),
(Newton, Gauss-Newton, Levenberg-Marquardt)
2) fully nonlinear methods, i.e. Monte Carlo (MC) methods, i.e.
simulated annealing and genetic algorithms, are recent,
allow a global search of the “true” minimum, but very time consuming.
Sambridge, M., Mosegaard, K., 2001. Monte Carlo methods in geophysical
inverse problems. Rev. Geophys. 40,
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4. Solving the inverse problem
4.1. Nonlinear optimization solution
Non-linear (iterative) gradient technique (Gauss-Newton / Levenberg-Marquardt)
For the model update Δm (from a starting solution mo) in step n+1
Δmn+1 = (GTG + Cm−1) * GTCd−1(d – g(mn))
with
G = 𝝏g/𝝏mn, the Fréchet (Jacobian) matrix, computed either analytically, but mostly numerically
Cm , the covariance matrix of the model, related to the a priori uncertainty of the model
Cd, the covariance matrix of the data, related to the a priori uncertainty of the data
The equation above is also “the one-step” of the so-called damped least squares solution,
or ridge regression of the linear inverse problem (see Koch, Lecture 2).
Iterative procedure is stopped, whenever
|| Δmn+1 ||2 < εm
or
|| d-g(mn)||2 = || r ||2 < εd
No guarantee to reach the local minimum, let alone the global one.
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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4. Solving the inverse problem
4.2. The linear inverse problem
 One-step Gauss-Newton / Levenberg-Marquardt method
=>
Δm = (GTG + kI) * GTCd−1(d – Gm0)
with
k , the damping- or ridge parameter, to stabilize (regularize) the ill-posed (unstable)
inverse solution (see Koch, Lecture 2).
The proper choice of k ~ trace(Cm-1 ) is one of the most intriguing issues of
ill-posed inverse problems
Koch, M., 1992. The optimal regularization of the linear seismic inverse problem,
In: Geophysical Inversion, Bednar, J.B., L. Lines, R.H. Stolt and A.B. Weglein (eds.), Society for Industrial and
Applied Mathematics (SIAM), Philadelphia, PY, pp. 170 – 234.
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4. Solving the inverse problem
4.3. Analysis of solution robustness (error-resolution analysis)
This is one of the most intriguing issues in inverse theory (e.g. Koch, Lectures 2 and 3).
Frequently overlooked, when dealing with non-linear inversion, as no exact theory exists
Koch, M., 1992, Optimal models in seismic inversion of local travel times: unifying the different viewpoints
within both the linear and nonlinear inversion approaches,
19th International Conference on Mathematical Geophysics, Taxco, Mexico, June 21 - 26, 1992.
Two common techniques are
-
synthetic resolution checkerboard test
(simple procedure, but works also for nonlinear inversion)
-
covariance- resolution - “trade-off” analysis (Backus-Gilbert-methodology)
Backus, G.E., Gilbert, J.F., 1968. The resolving power of gross earth data.
Geophys. J. Royal Astr. Soc. 16, 169–205.
(only verified within the framework of linear theory)
(see Lecture 2)
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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4. Solving the inverse problem
4.4. Analysis of solution robustness (error – resolution analysis)
Synthetic test example (Rawlinson et al., 2010)
Input model (a) and final (nonlinear inversion) model (d)
Resolution checkerboard test (a) and covariance and
error estimates (b) for model left
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Table of contents
4. Solving the inverse problem
4.1. Nonlinear optimization formulation
4.2. The linear inverse problem
4.3. Analysis of solution robustness
5. Tomography as a tool to understand Earth’s geodynamics
5.1. Overview of the geodynamics of the Earth
5.2 How to relate seismic structure to geodynamics?
5.3. Tomography in geodynamically active regions
6. Future developments
7. References
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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5. Tomography as a tool to understand
Earth’s geodynamics
5.1. Overview of the geodynamics of the Earth
Geodynamical processes in the Earth mainly driven by
density differences due to
- mineral composition differences
(layering of the earth)
- thermally driven flow
(mantle convection => plate tectonics)
(mantle plumes
=> hot spots, rift zones, volcanism)
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5. Tomography as a tool to understand
Earth’s geodynamics
5.2. How to relate seismic structure to geodynamics?
Seismic velocities
;
depend on (decreasing order?)
-
temperature T,
composition (ρ),
presence of partial melt,
anisotropy.
Based on
Goes, S., Govers, R., Vacher, P., 2000. Shallow mantle temperatures under Europe from P and S wave tomography.
J. Geophys. Res. 105, 11153–11169.
a) when mantle is below the solidus, temperature T is dominant factor,
with a decrease of 0.5–2% in P- and 0.7–4.5% in S-velocity per 100 oC - T-change
b) compositional variations <1%
c) above the solidus, effect of partial melt is significant, but difficult to quantify
d) in lower mantle effects of temperature smaller (0.3-0.4%), depending on Fe/(Mg+ Fe)- ratio,
with a -0.5 % change  200 oC temperature change
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5. Tomography as a tool to understand
Earth’s geodynamics
5.3. Tomography in geodynamically active regions
The LLNL-G3Dv3 Model
(Simmons et al., 2012)
Note subduction and
spreading zones
and
small-scale anomalies
in the lower mantle
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5. Tomography as a tool to understand
Earth’s geodynamics
5.3. Tomography in geodynamically active regions
Surface wave- and S-wave- inversions combined
Comparison of three global shear velocity models at 150 km depth
(adapted from: Kustowski, B., Ekstrom, G., Dziewonski, A.M., 2008. Anisotropic shear-wave velocity structure of the
Earths mantle: a global model. J. Geophys. Res. 113, B06306)
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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5. Tomography as a tool to understand
Earth’s geodynamics
5.3. Tomography in geodynamically active regions
Subducting slab environments
P- and S- wave tomographic models under the
Japanese Island arc system.
(adapted from:
Widiyantoro, S., Gorbatov, A., Kennett, B.L.N.,
Fukao, Y., 2000. Improving global shear wave
traveltime tomography using three-dimensional
ray tracing and iterative inversion.
Geophys. J. Int. 141, 747–758)
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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5. Tomography as a tool to understand
Earth’s geodynamics
5.3. Tomography in geodynamically active regions
Mantle plume environments
P-wave tomographic “plume” images
under “thermally hot” (hotspots,
spreading and rift) zones.
adapted from
Zhao, D., 2004. Global tomographic
images of mantle plumes and subducting
slabs: insight into deep Earth dynamics.
Phys. Earth Planet. Inter. 146, 3–34
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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5. Tomography as a tool to understand
Earth’s geodynamics
5.3. Tomography in geodynamically active regions
Yellowstone hot spot
Upper P-wave mantle structure at a depth of 150 km across the US
(white line delineates the western region of high resolution)
(Source: earth scope)
Waite, G. P., R. B. Smith, and R. M. Allen (2006), VP and
VS - structure of the Yellowstone hot spot from
teleseismic tomography: Evidence for an upper mantle
plume, J. Geophys. Res., 111, B04303,
Husen, S. , R. B. Smith and G. P. Waite (2004). Evidence for gas
and magmatic sources beneath the Yellowstone volcanic feld
from seismic tomographic imaging.
J. Volcanology and Geothermal Res., 131, 397-410
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Table of contents
4. Solving the inverse problem
4.1. Nonlinear optimization formulation
4.2. The linear inverse problem
4.3. Analysis of solution robustness
5. Tomography as a tool to understand Earth’s geodynamics
5.1. Overview of the geodynamics of the Earth
5.2 How to relate seismic structure to geodynamics?
5.3. Tomography in geodynamically active regions
6. Conclusions and outlook
7. References
M. Koch: The seismic travel time problem as applied to tomography of the Earth
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6. Conclusions and outlook
-
Seismic tomography has experienced rapid advances on many fronts, since its inception in the
1970s, including improved techniques for solving the forward (ray-tracing) and inverse problems.
-
More efficient techniques of geometric ray tracing, to solve the two point (source–receiver path)
problem, such as grid-based methods, i.e. eikonal solvers and shortest-path ray tracing, as well as
multi-arrival schemes are coming to the fore.
-
The recently emerging ambient noise tomography, is having a significant impact, as ambient noise
information is independent of and complimentary to information from deterministic sources
-
Finite difference and even full waveform tomography is beginning to emerge as a powerful tool for
imaging the earth, as the computing power continues to increase (following Moore’s law)
-
At regional and global scales, one major main impediments to improve the resolution of
tomographic models is a lack of good data coverage, especially across ocean basins.
-
The steps from relating seismic tomographic properties to physical (thermal) and chemical
properties of the earth’s rock material for geodynamical inferences need further research
-
Quote (Nolet et al., 2007): “By all signs, seismic tomography is entering a Golden Age”. ??????
Further details in lectures 3 and 4 of the lecture series.
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7. References
General references (personal likings)
Iyer, H. and Hirahara, K., 1993. Seismic Tomography: Theory and Practice. Chapman & Hall, London, UK.
Nolet, G. (Ed), 1987. Seismic Tomography: With Applications in Global Seismology and Exploration Geophysics.
D. Reidel, Dordrecht, The Netherlands.
Nolet, G., Allen, T. and Zhao, D., 2007. Mantle plume tomography, Chemical Geology, 241, 248–263.
Nolet, G., 2008. A Breviary of Seismic Tomography: Imaging the Interior of the Earth and the Sun.
Cambridge University Press, Cambridge, UK.
Rawlinson, N., Pozgay, S. and Fishwick, S. 2010. Seismic tomography: A window into deep Earth. Physics of the
Earth and Planetary Interiors, 178, (3–4), 101–135.
Tarantola, A., 1987. Inverse Problem Theory. Elsevier, Amsterdam, The Netherlands.
References of the author (M. Koch)
Publications, Projects and Presentations of Research
>> Curriculum Vitae, Prof. Dr. M. Koch
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