Post on 16-Jan-2016
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The good sides of Bayes
Jeannot Trampert
Utrecht University
Bayes gives us an answer!Example of inner core anisotropy
Normal mode splitting functions are linearly related to seismic anisotropy in the inner core
The kernels Kα, Kβ and Kγ are of different size, hence regularization affects the different models differently
Regularized inversion
Full model space search (NA, Sambridge 1999)
Resolves 20 year disagreement between body wave and normal mode data (Beghein and Trampert, 2003)
Bayes or not to Bayes?
We need proper uncertainty analysis to interpret seismic tomography
probability density functions for all model parameters
Do models agree?No knowledge of uncertainty
implies subjective comparisons.
Partial knowledge of uncertainty allows hypothesis testing
Deschamps and Tackley, 2009
Mean density model separated into its chemical and temperature contributions (full pdf obtained with NA)
Trampert et al, 2004)
Deschamps and Tackley, 2009
Full knowledge of uncertainty allows to evaluate the probability
of overlap or consistency between models
What is uncertainty?
Consider a linear problem where d are data, m the model, G partial derivatives and e the data uncertainty
where m0 is a starting model and L the linear inverse operator
The estimated solution is
What is uncertainty?
where (I-R) is the null-space operator
This can be rewritten as
Resulting in a formal statistical uncertainty expressed with covariance operators as
What is uncertainty?
How can we estimate uncertainty?
① Ignore it: should not be an option but is the common approach
② Try and estimate m: Regularized extremal bound analysis (Meju,
2009)Null-space shuttle (Deal and Nolet, 1996)
③ Probabilistic tomographyNeighbourghood algorithm (Sambridge, 1999)Metropolis (Mosegaard and Tarantola, 1995) Neural Networks (Meier et al., 2007)
The most general solution of an inverse problem (Bayes)
evidence
x
),(
),(),(),(
likelihoodpriorposterior
md
mdmdkmd
Tarantola, 2005
A full model space search should estimate )()()( mLmkm m
• 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)• Neighbourhood algorithm (Sambridge, 1999)• Sample(m) and apply Metropolis rule on L(m). This will
result in importance sampling of (m) (Mosegaard and Tarantola, 1995)
• Neural networks (Meier et al., 2007)
The neighbourhood algorithm (NA): Sambridge 1999
Stage 1:Guided sampling of the model space.Samples concentrate in areas (neighbourhoods) of better fit.
The neighbourhood algorithm (NA):
Stage 2: importance samplingResampling so that sampling density reflects posterior
2D marginal 1D marginal
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
• Marginals are a compact representation of the seismic data and prior rather than a model
Example: A global mantle model
•Using body wave arrival times, surface wave dispersion measurements and normal mode splitting functions
•Same mathematical formulation
Mosca et al., 2011
Mosca et al., 2011
What does it all mean?
Mineral physics willtell us!
Thermo-chemicalparameterization:
• Temperature• Fraction of Pv (pPv)• Fraction of total Fe
Example: Importance sampling using the Metropolis rule (Mosegaard and Tarantola, 1995)
Disadvantages of NA
and Metropolis
Works only on small linear and non-linear problems(less than ~50 parameters)
The neural network (NN) approach: Bishop 1995, MacKay 2003
•A neural network can be seen as a non-linear filter between any input and output•The NN is an approximation to a non-linear function g where d=g(m)•Works on forward or inverse function•A training set (contains the physics) is used to calculate the coefficients of the NN by non-linear optimisation
Properties of NN
1. Dimensionality is not a problem because NN approximates a function and not a data prediction!
2. Flexible: invert for any combination of parameters
3. 1D or 2D marginal only
Mantle transition zone discontinuities
Probabilistic tomography using Bayes’ theorem is possible but
challenges remain
• Control the prior and data uncertainty
• Full pdfs in high dimensions
• Interpret and visualize the information contained in the marginals