Velocity inversion: A case study in infinite-dimensional optimization

The goal of seismic velocity inversion is the estimation of seismic wave velocities inside the earth by attempting to predict, in a least-error sense, seismic waveforms measured at its surface. We present velocity inversion as a case study in the various infinite-dimensional pathologies which may afflict practically important problems of distributed parameter identification, treated as optimization problems in function spaces. These features differentiate various problem formulations far beyond the degree one would expect for finite- (small-) dimensional problems. We illustrate this differentiation by comparing the characteristics of three different least-squares formulations of velocity inversion.     

On the principal axes of diffusion in difference schemes for 2d transport problems

Via Taylor series, we associate with a difference stencil L^h approximating Lu := au_x + bu_y its modified equation:

Full-size image (1K)

. By rotating axes to eliminate the 2Bu_xy term, the principle axes through which the diffusion in L^h acts is calculated. Interestingly, for many schemes proposed for 2D transport problems these axes have little to do with the “streamline” and ”crosswind” directions of the continuous problem. Several examples are considered from this point of view.     

Crystal and molecular structure of crenatine carbonate

The crystal and molecular structure of 1-ethyl-4-methoxy-9H-pyrido[3,4b]indole (crenatine) carbonate C₁₄H₁₄N₂O·H₂CO₃, (MS, m/z 226)M _R 288.3, a-carboline alkaloid, has been determined from X-ray diffraction data. The compound crystallizes in the space group Pbca with cell parameters:a=11.616(4),b=18.450(8),c=12.992(5)Å,V=2784(2)ų,Z=8,D _calc=1.375 g cm^–3, (MoK)=0.71069Å,(Mo K)=0.94 cm^–1,F(000)=1216,R/R _w =8.2/10.3% for 1099 reflections. The ring system of the-carboline nucleus is planar. The title compound shows a two center hydrogen bond between the indole N-H group and the oxygen atom of a carbonate group. The structure does not display hydrogen bonding between-carboline groups but rather a bonding network involving the carbonate group.