Team 2: Final Report Uncertainty Quantification in Geophysical

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

many choices of model Partial Data Set

Posterior Surfaces Posterior Mean Surface

Posterior Uncertainty Posterior Composite Image

Decimation Histogram MCMC Statistics

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