Samples, Symmetries and Extensions for Travel Time Tomography and Source Location Problems

Abstract:

Nonlinear inverse problems are often solved using stochastic or deterministic sampling methods. Given some observed data, possible values of model parameters are estimated by testing many examples (samples) of parameter values against the data. We simulate data that would be recorded if each set of values was true, and compare the result to observations and prior information. Their consistency defines how likely the parameter values are to be true.

However, where we have neither samples nor prior information, we do not know how likely the values are to be true, and two mathematical theorems obstruct all uncertainty estimates. The Curse of Dimensionality (CoD) is the observation that the number of samples required to achieve similar levels of information per parameter (or dimension) rises exponentially with the number of parameters: real problems with only tens of parameters often make dense sampling intractable. The No Free Lunch (NFL) theorem states that any sampling algorithm applied to all problems is on average no better than any other. We appear to reach an impasse, requiring exponential computation for many parameter problems.

High-D parameter spaces can be explored despite the CoD if each simulation provides full information about sufficiently high-D subspaces of parameter values (since these then need not be explored). The NFL can be circumnavigated by developing bespoke algorithms for particular problems. This talk shows that physics-based symmetries of particular inverse problems provide infinitely many other sample simulations for free, dramatically increasing the information from each simulation. Define the extension of any sample to be all extra information available from the sample without additional computed simulations.

Travel time tomography is used to image the wave slowness structure of the interior of the Earth and other media. Such images allow sources to be located given arrival time measurements of waves from the source. This talk will show that all simulations from all source locations that lie on any ray to any receiver location, through any slowness structure in a high-dimensional extension, are already known from any single simulation. None of the wealth of information in this extension is currently used for source location, nor for travel time tomography: this talk will reveal its value.