M. V. de Hoop

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 Earths 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 (ΔT = 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.

Seismic Imaging of Transition Zone Discontinuities Suggests Hot Mantle West of Hawaii

Hot material upwelling from deep below Hawaii may pool up far below and to the west of the islands. 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 (ΔTmax ~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.

Seismic imaging with the generalized Radon transform: a curvelet transform perspective*

A key challenge in the seismic imaging of reflectors using surface reflection data is the subsurface illumination produced by a given data set and for a given complexity of the background model (of wave speeds). The imaging is described here by the generalized Radon transform. To address the illumination challenge and enable (accurate) local parameter estimation, we develop a method for partial reconstruction. We make use of the curvelet transform, the structure of the associated matrix representation of the generalized Radon transform, which needs to be extended in the presence of caustics and phase linearization. We pair an image target with partial waveform reflection data, and develop a way to solve the matrix normal equations that connect their curvelet coefficients via diagonal approximation. Moreover, we develop an approximation, reminiscent of Gaussian beams, for the computation of the generalized Radon transform matrix elements only making use of multiplications and convolutions, given the underlying ray geometry; this leads to computational efficiency. Throughout, we exploit the (wave number) multi-scale features of the dyadic parabolic decomposition underlying the curvelet transform and establish approximations that are accurate for sufficiently fine scales. The analysis we develop here has its roots in and represents a unified framework for (double) beamforming and beam-stack imaging, parsimonious pre-stack Kirchhoff migration, pre-stack plane-wave (Kirchhoff) migration and delayed-shot pre-stack migration.

Data analysis tools for uncertainty quantification of inverse problems

We present exploratory data analysis methods to assess inversion estimates using examples based on l2- and l1-regularization. These methods can be used to reveal the presence of systematic errors such as bias and discretization effects, or to validate assumptions made on the statistical model used in the analysis. The methods include bounds on the performance of randomized estimators of a large matrix, confidence intervals and bounds for the bias, resampling methods for model validation and construction of training sets of functions with controlled local regularity.

Reciprocity and the optimization approach to migration/inversion

Conceptually, depth migration algorithms in seismic prospecting can be envisaged as a large-scale “remote sensing” problem where, through the use of judiciously chosen computational methods, the physical properties of a remote part of the configuration in which the probing waves propagate are to be reconstructed from the measured data elsewhere. Mathematically, such interaction is described by a reciprocity theorem. As A.T. de Hoop (1988) has pointed out, there are two reciprocity theorems applying to acoustic wave motion: a reciprocity theorem of the time-convolution type and one of the time-correlation type. The reciprocity theorem of the time-convolution type leads, via the introduction of pointsource solutions (Green’s functions) to the wave problem, to integral representations for the quantities that characterize the wave motion. These representations (usually in the Born approximation) form the basis for the “Kirchhoff” depth migration/inversion algorithms (Miller et al., 1987). In the present contribution, the properties of the reciprocity relation of the time-correlation type are investigated with a view of their relation to an optimization approach to migration. In its single scattering approximation, one can extract from it the algorithms that are known under the names “reversetime” and “wave-equation” (Berkhout, 1982) depth migration. (2)