1. Probabilistic tomography
Jeannot Trampert
Utrecht University

2. Is this the only model compatible with the data?
Project some chosen model onto the model null space of the forward operator
Then add to original model.
The same data fit is guaranteed!
Deal et al. JGR 1999

3. True model APPRAISAL Model uncertainty Data Estimated model
Forward problem
(exact theory)
APPRAISAL
Model uncertainty
Data
Estimated model
Inverse problem
(approximate theory, uneven data coverage, data errors)
Ill-posed (null space, continuity)
Ill-conditioned (error propagation)

4. In the presence of a model null space, the cost function has a flat valley and the standard least-squares analysis underestimates uncertainty.
Realistic uncertainty for the given data

5. Multiple minima are possible if the model does not continuously depend on the data, even for a linear problem.
Data
Model

6. Does it matter? Yes! dlnVs dlnVphi dlnrho
SPRD6 (Ishii and Tromp, 1999)
Input model is density from SPRD6
(Resovsky and Trampert, 2002)

7. The most general solution of an
inverse problem (Bayes, Tarantola)

8. Only a full model space search can estimate
Exhaustive search
Brute force Monte Carlo (Shapiro and Ritzwoller, 2002)
Simulated Annealing (global optimisation with convergence proof)
Genetic algorithms (global optimisation with no covergence proof)
Sample r(m) and apply Metropolis rule on L(m). This will result in importance sampling of s(m) (Mosegaard and Tarantola, 1995)
Neighbourhood algorithm (Sambridge, 1999)
Neural networks (Meier et al., 2007)

9. The model space is HUGE! Draw a 1000 models per second where m={0,1}
M=30  13 days
M=50  years
Seismic tomography M = O( )

10. The model space is EMPTY!
Tarantola, 2005

11. The curse of dimensionality  small problems M~30
Exhaustive search, IMPOSSIBLE
Brute force Monte Carlo, DIFFICULT
Simulated Annealing (global optimisation with convergence proof), DIFFICULT
Genetic algorithms (global optimisation with no covergence proof), ???
Sample r(m) and apply Metropolis rule on L(m). This will result in importance sampling of s(m) (Mosegaard and Tarantola, 1995), BETTER, BUT HOW TO GET A MARGINAL
Neighbourhood algorithm (Sambridge, 1999), THIS WORKS
Neural networks, PROMISING

12. The neighbourhood algorithm:
Sambridge 1999
Stage 1:
Guided sampling of the model space.
Samples concentrate in areas (neighbourhoods) of better fit.

13. The neighbourhood algorithm (NA):
Stage 2: importance sampling
Resampling so that sampling density reflects posterior
2D marginal
1D marginal

14. Advantages of NA Interpolation in model space with Voronoi cells
Relative ranking in both stages (less dependent on data uncertainty)
Marginals calculated by Monte Carlo integration  convergence check

15. NA and Least Squares consistency
As damping is reduced, LS solution converges towards most likely NA solution.
In the presence of a null space, LS solution will diverge, but NA solution remains unaffected.
NA-LS consistency

16. Finally some tomography!
We sampled all models compatible with the data using the Neighbourhood Algorithm (Sambridge, GJI 1999)
Data: 649 modes (NM, SW fundamentals and overtones)
He and Tromp, Laske and Masters, Masters and Laske, Resovsky and Ritzwoller, Resovsky and Pestena, Trampert and Woodhouse, Tromp and Zanzerkia, Woodhouse and Wong, van Heijst and Woodhouse, Widmer-Schnidrig
Parameterization: 5 layers
[ km] [ km] [ km]
[ km] [ km]
Seismic parameters
dlnVs, dlnVΦ, dlnρ, relative topography on CMB and 660

17. Back to density
Stage 1
Most likely model
SPRD6
Stage 2

18. Gravity filtered models over 15x15 degree equal area caps
Likelihoods in each cap are nearly Gaussian
Most likely model (above one standard deviation)
Uniform uncertainties
dlnVs=0.12 %
dlnVΦ=0.26 %
dlnρ=0.48 %

19. chemical heterogeneity
Likelihood of correlation
between
seismic
parameters
Evidence for
chemical heterogeneity
Vs-VF
Half-height vertical bar corresponds to SPRD6

20. chemical heterogeneity
Likelihood of
rms-amplitude
of seismic
parameters
Evidence for
chemical heterogeneity
RMS density
Half-height vertical bar corresponds to SPRD6

21. The neural network (NN) approach:
Bishop 1995, MacKay 2003
A neural network can be seen as a non-linear filter between some input and output
The NN is an approximation to any function g in the non-linear relation d=g(m)
A training set is used to calculate the coefficients of the NN by non-linear optimisation

22. The neural network (NN) approach:
Trained on forward relation
Trained on inverse relation

23. 1) Generate a training set: D={dn,mn}
2) Network training:
3) Forward propagating a new datum through the trained network (i.e. solving the inverse problem)
dobs
p(m|dobs,w*)
1) Generate a training set:
D={dn,mn}
Network Input:
Observed or synthetic
data for training dn
Neural Network
We first need to generate a so called training set. Consisting of a collection of random earth models and the corresponding dispersion curves. We then design a neural network which takes dispersion curves as input vector and it’s output are the various parameters of a guassian mixture model, which
Network Output:
Conditional probability
density
p(m|dn,w)

24. Model Parameterization
Topography
(Sediment)
Vs (Vp, r linearly scaled)
3 layers
Moho Depth
Vpv, Vph, Vsv, Vsh, h
7 layers
+- 10 % Prem
220 km Discontinuity
As already mentioned we consider local dispersion curves as the response of a 1-dimensional earth model parameterized as a stratigraphical earth model with various layers. The data we consider do not have above 400 km we allow the main discontinuites such as 220, Moho and Topography to vary as well as the indicated paramters in the layers. We have three crustal layers with a sedimentary layer of variable thickness on top. All the prior information is explicitly defined by the bounds of variations on all the model parameters. Moho depth for example is allowd to vary between km over contients and 0-40 km under oceans. We proceed by randomly generating a series of earth models where all the indicated parameters are allowed to vary and compute the corresponding synthetic dispersion curves. Surface waves in the period range considered can not resolve all three crustal layers. Instead of inverting for shear wave velocity in each crustal layer individually we invert for the vertically averaged crustal shear wave velocity. We actually do this as allready mentioned with an extenden neural network approach the mixture density network. The application of which can best be explained with a short flowchart.
Vp, Vs, r
8 layers
+- 5 % Prem
400 km Discontinuity

25. Example of a Vs realisation

26. Advantages of NN
The curse of dimensionality is not a problem because NN approximates a function not a data prediction!
Flexible: invert for any combination of parameters
(Single marginal output)

27. Advantages of NN
4. Information gain Ipost-Iprior allows to see which parameter is resolved
Information gain for
Moho depth (Meier et al, 2007)

28. Standard deviation s [km]
Mean Moho depth [km]
The aim of my project is to map discontinuities such as crustal thickness shown here within the Earth. Additionally we would like to obtain the corresponding uncertainties in our estimate as well. Over the last decades seismologist deduced images of the Earth interior from the information contained in 100 thousands of recorded seimograms. These images generally show fast and slow velocity anomalies of p and s wave velocity with respect to a reference model at various depth. Relatively little effort has been undertaken to actually map the topography of the various discontinuities within the Earth. Knowledge of which is crucial for various other fields in the Earth Sciences. This forms the main motivation of my Phd research, developing a new approach to invert surface wave data for discontinuities within the Earth. First I give a short introduction about the kind of data we consider, then I will explain the basic concepts of the method we use and then I will show an application of this approach to global crustal thickness.
Meier et al.
2007
Standard deviation s [km]

29. Mean Moho depth [km]
Meier et al.
2007
CRUST2.0 Moho depth [km]

30. NA versus NN NA: joint pdf  marginals by integration  small problem
NN: 1D marginals only (one network per parameter)  large problem

31. Advantage
of having a pdf
Quantitative
comparison

32. Mean models of Gaussian pdfs, compatible with observed gravity data, with almost uniform standard deviations: dlnVs=0.12 % dlnVΦ=0.26 % dlnρ=0.48 %

33. Thermo-chemical
Parameterization:
Temperature
Variation of Pv
Variation of total Fe

34. Sensitivities

35. Ricard et al. JGR 1993
Lithgow-Bertelloni and Richards Reviews of Geophys. 1998
Superplumes

36. Tackley, GGG, 2002