Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces.

In this article, we prove optimal convergence rates results for regularization methods for solving linear ill-posed operator equations in Hilbert spaces. The results generalizes existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions.

Seismic imaging of transition zone discontinuities suggests hot mantle west of Hawaii.

The Hawaiian hotspot is often attributed to hot material rising from depth in the mantle, but efforts to detect a thermal plume seismically have been inconclusive. To investigate pertinent thermal anomalies, we imaged with inverse scattering of SS waves the depths to seismic discontinuities below the Central Pacific, which we explain with olivine and garnet transitions in a pyrolitic mantle. The presence of an 800- to 2000-kilometer-wide thermal anomaly (ΔT(max) ~300 to 400 kelvin) deep in the transition zone west of Hawaii suggests that hot material does not rise from the lower mantle through a narrow vertical plume but accumulates near the base of the transition zone before being entrained in flow toward Hawaii and, perhaps, other islands. This implies that geochemical trends in Hawaiian lavas cannot constrain lower mantle domains directly.

Seismostratigraphy and thermal structure of Earth's core-mantle boundary region.

We used three-dimensional inverse scattering of core-reflected shear waves for large-scale, high-resolution exploration of Earth's deep interior (D'') and detected multiple, piecewise continuous interfaces in the lowermost layer (D'') beneath Central and North America. With thermodynamic properties of phase transitions in mantle silicates, we interpret the images and estimate in situ temperatures. A widespread wave-speed increase at 150 to 300 kilometers above the coremantle boundary is consistent with a transition from perovskite to postperovskite. Internal D'' stratification may be due to multiple phase-boundary crossings, and a deep wave-speed reduction may mark the base of a postperovskite lens about 2300 kilometers wide and 250 kilometers thick. The core-mantle boundary temperature is estimated at 3950 +/- 200 kelvin. Beneath Central America, a site of deep subduction, the D'' is relatively cold (DeltaT = 700 +/- 100 kelvin). Accounting for a factor-of-two uncertainty in thermal conductivity, core heat flux is 80 to 160 milliwatts per square meter (mW m(-2)) into the coldest D'' region and 35 to 70 mW m(-2) away from it. Combined with estimates from the central Pacific, this suggests a global average of 50 to 100 mW m(-2) and a total heat loss of 7.5 to 15 terawatts.