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