Team 2: Final Report Uncertainty Quantification in Geophysical
Team 2: Final Report Uncertainty Quantification in Geophysical Inverse Problems Statement of the Tomography Problem
Observe the arrival time, given the first order physics Invert for the
slowness, or density (x,z)x,z))
Creating Synthetic Data Our unknown earth model is a layered model
Vertical/Horiz)ontal observations are important! The observations are
calculated as line integrals through the synthetic model.
Choose A Model There are many choices for model! Each choice leads to a different solution
Each solution can be evaluated for goodness of fit. Haar wavelets provide an easy way to describe a region.
Our Model Has Errors! The travel times do not allow us to reconstruct all the details of the layers.
We use covariance matrices are used to measure the uncertainty of the model. Prior covariance matrix is used to account for the model uncertainty without considering the
observed traveltimes. Posterior covariance is accounts for the observed traveltimes.
Solving the Inverse Problem for a single choice of model Partial Data Set
Full Data Set The Prior Distribution The natural choice for a prior
pdf is the distribution that allows for the greatest uncertainty while obeying the constraints imposed by the prior information, and it
can be shown that this least informative pdf is the pdf that has maximum entropy (x,z)Jaynes 1968, 1995, Papoulis 1984)
From Malinverno, 2000 Prior Mean
Prior Uncertainty Colormap To Code Mean and Uncertainty
Uncertainty Mean
Prior Composite Image The Data Prediction Matrix For
each observation, we calculate the same line integral through
our wavelet model. These are the columns. The better the model is, the closer these
integrals match up with our observations Data Prediction Matrix
Posterior Surfaces- Full Data Posterior Mean Full Data
Posterior Uncertainty Full Data Posterior Composite Full Data
Posterior Surfaces Limited Data Posterior Mean Limited Data
Posterior Uncertainty Limited Data
Posterior Composite Limited Data Solving the Inverse Problem for
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