Seismic tomography refers to a broad class of techniques that aim to estimate spatial variations in the propagation speed of seismic waves in Earth's interior using information gleaned from seismograms recorded at Earth's surface. The concept has many aspects in common with medical applications, such as CAT imaging. Over the past decades global tomography has made significant progress with the imaging of Earth's large-scale, deep-interior structure. However, currently available techniques do not do a very good job in constraining Earth's structure on a wide range of length scales, which is a prerequisite for understanding the relationship between near-surface and deeper-mantle processes. Indeed, uneven data coverage and heterogeneous data quality often render non-unique, fuzzy images, with substantial spatial variation in reliability. The approximate nature of the 'red and blue' images impedes quantitative interpretation and keeps tomography from reaching its full potential as a probe of Earth's deep interior. This is particularly important in the context of EarthScope, a nationwide, multi-year geosciences project funded by the National Science Foundation. Its seismology component, USArray, has begun to provide spectacular broad-band data from dense distributions of seismograph stations, but to make optimal use of such data, that is, to construct the best possible models of the crust and mantle beneath North America, one needs to use better tomographic imaging methods than are available today.
Traditional tomographic techniques use only a small part of the recorded data, for instance the arrival time or (filtered) waveform of a particular seismic wave. We need to improve our ability to interpret the broad-band wavefield excited by earthquakes (or other sources). From a geosciences point of view, we wish to have better images of mantle heterogeneity (including anisotropy) beneath North America and better understanding of the causative physical and chemical processes. From a physical point of view, this requires full consideration of how the elastic waves propagate in media with strong (and often non-smooth) heterogeneity. From a mathematical point of view this requires the use of more accurate wave-propagation theory (based on the wave equation) in order to capture this complexity, clever wavefield representation and model parameterization (for instance by means of curvelet frames) to allow faster computation in view of the massive size of modern (academic and industrial) data sets, and new statistics to estimate realistic uncertainties in the resulting depictions of Earth's sub-surface. Indeed, we must consider full wave dynamics in all steps on the trajectory from 'data' to 'image', including data representation, wave theory, parameterization, regularization, and uncertainty analysis.
