comparison of monte carlo methods for model probability distribution determination … ·...
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December 11, 2009 1
Comparison of Monte Carlo Methods for Model Probability Distribution Determination in SAR
Interferometry
Andy Hooper1 and Tim Wright2,
1Delft University of Technology2University of Leeds
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Interferogram
Model
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?
Non-linear Modelling
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(model) (data)
Non-linear Modelling
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Non-linear Modelling
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Non-linear Modelling
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Non-linear Modelling
• Optimal M, from e.g. simulated annealing, at minimum/maximum value of f(D).
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Non-linear Modelling
• Optimal M, from e.g. simulated annealing, at minimum/maximum value of f(D).
• Acknowledging we have measurement errors, aim is to find a probability distribution of model parameters.
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Simulated Noise Method
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Simulated Noise Method
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• Adding simulated noise changes f(D).
+noise
Simulated Noise Method
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• Adding simulated noise changes f(D).• Find optimal M again and repeat.
Wright et al, 1999
Simulated Noise Method
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• Adding simulated noise changes f(D).• Find optimal M again and repeat.
Wright et al, 1999
+noise
Simulated Noise Method
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• Adding simulated noise changes f(D).• Find optimal M again and repeat.
Model Parameters
Wright et al, 1999
P(M|D)
Markov-Chain Method
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• Set f(D) as probability density of data.
Includesnoise covariance
Markov-Chain Method
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• Set f(D) as probability density of data.• Bayes Theorem: P(M|D)=K.P(M).P(D|M)
Markov-Chain Method
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• Set f(D) as probability density of data. • Bayes Theorem: P(M|D)=K.P(M).P(D|M)• Build P(M|D) directly by repeated sampling from
M, mapping M to D, and keeping or rejecting sample based on p(D) e.g. Metropolis algorithm.
Mosegaard and Tarantola, 1995; Hooper et al, 2007
Markov-Chain Method
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• Set f(D) as probability density of data. • Bayes Theorem: P(M|D)=K.P(M).P(D|M)• Build P(M|D) directly by repeated sampling from
M, mapping M to D, and keeping or rejecting sample based on p(D) e.g. Metropolis algorithm.
Mosegaard and Tarantola, 1995; Hooper et al, 2007
Markov-Chain Method
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• Set f(D) as probability density of data. • Bayes Theorem: P(M|D)=K.P(M).P(D|M)• Build P(M|D) directly by repeated sampling from
M, mapping M to D, and keeping or rejecting sample based on p(D) e.g. Metropolis algorithm.
Mosegaard and Tarantola, 1995; Hooper et al, 2007
Markov-Chain Method
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Model Parameters
• Set f(D) as probability density of data. • Bayes Theorem: P(M|D)=K.P(M).P(D|M)• Build P(M|D) directly by repeated sampling from
M, mapping M to D, and keeping or rejecting sample based on p(D) e.g. Metropolis algorithm.
Mosegaard and Tarantola, 1995; Hooper et al, 2007
Comparison of Methods
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Markov-chain
Simulated noise
Comparison of Methods
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Markov-chain
Simulated noise
(Bayesian, so can also include prior model distribution)
Questions
• Do the results of the 2 methods agree
• What is the performance of each method?
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Synthetic Data
Deformation Atmosphere+Decorrelation Total phase
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Optimal Models
Simulated noise
True Model Residual
Markov-chain
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Simulated Noise Method
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Markov-Chain Method
95% Bounds comparison
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Normalised by true values
Generally, good agreement
Markov-chain method
Sim
ulat
ed n
oise
met
hod
Convergence Simulated Noise
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Number of iterations
Convergence Simulated Noise
~900 iterations needed
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Number of iterations
Convergence Simulated Noise
~900 iterations needed
BUT, 37 000 forward models per iteration ~3 x 107 runs.12/11/2009 31
Number of iterations
Convergence Markov-Chain
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Number of iterations
Convergence Markov-Chain
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~400,000 iterations needed
BUT, 1 forward model per iteration
Number of iterations
Real Data
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2002 Mw6.7 Nenana Mountain earthquake
(Wright et al, 2003) Simulated Noise Method
Real Data
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2002 Mw6.7 Nenana Mountain earthquake
(Wright et al, 2003) Simulated Noise Method
Markov-Chain Method
Agreement
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Agreement not as good, as simulated data, but reasonable
(only 100 runs using simulated noise)
Markov-chain method
Sim
ulat
ed n
oise
met
hod
Conclusions• Probability distributions from both methods agree.
• Optimal models differ (simulated noise gives non-weighted least-squares, Markov-chain gives maximum likelihood).
• Markov-chain method more efficient (2 orders of magnitude).
• Markov-chain method can include prior constraints.
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