Stability considerations for one-dimensional inverse problems

Summary. We investigate the issues of stability and conditioning for the one-dimensional seismic inverse problem. We show that these issues are distinct; i.e. that numerically stable implementations of solutions to the inverse problem will not give accurate results if the problem is ill-conditioned. In addition, we identify the factors which determine the condition of the inverse problem. These are the total variation of the acoustic impedance profile being sought and the accuracy of the low-frequency content of the reflection data. We illustrate these results on implementations of two numerically stable algorithms for the inverse problem, one of which has a reputation for being unstable. A comparison shows nearly identical results for the two methods on noise-contaminated and frequency band limited reflection data. In fact, we conjecture that all of the well-known 'layer-stripping' inverse scattering methods share the same mathematical stability characteristics. On the other hand, we also show that ill-conditioning can lead to failure of such algorithms, through amplification of error due either to inaccurate data or to discretization or roundoff. Finally, we observe that appropriate smoothing of the seismic data for an ill-conditioned inverse problem (high-variation impedance profile) can cause the problem to become well-conditioned (lower-variation profile). As is typical with regularizations, the price paid for the newly-acquired ability to solve the problem is a loss of accuracy in the solution.     

Copper(II) interaction with carbonate species based on malachite solubility in perchlorate medium at the ionic strength of seawater

Equilibrium constants for copper(II)-carbonate and -bicarbonate species have been determined at 25°C from consideration of malachite, Cu₂(OH)₂CO_3(s), solubility in UV-photo-oxidized perchlorate solutions of 0.72 m ionic strength. The ratios of total dissolved copper, T(Cu), to free copper(II) ion, [Cu ²⁺], in 30 malachite saturated experimental solutions of 1–10 × 10^?3eq kg^?1 H₂O initial total alkalinity (TA_i in the pH range 5.0–9.3 were fitted to a copper(II)-ion speciation model. The experimental data indicate the existence of CuCO₃⁺, CuHCO₃⁺ and Cu(OH)CO₃^? in addition to the hydrolys and Cu(OH)CO₃^? in addition to the hydrolysis products in the range of conditions defined by this study. The stoichiometric equilibrium constants, applicable to seawater at 0.72 m ionic strength, 25°C and 1 atm are

A speciation model employing the equilibrium constants determined in this study and copper(II) hydrolysis constants from previous work suggests that the inorganic speciation in seawater (pH = 8.2, TA = 2.3 meq kg ^?1, 25°C) is dominated by the CuCO₃⁰ complex (82%) and that only 2.9% of the total inorganic copper exists as the free copper(II) ion. Hydrolysis products, CuOH⁺ and Cu(OH)₂⁰, account for 6.5% while CuHCO₃⁺ and Cu(OH)CO₃^? species comprise 1.0 and 6.3% of the total inorganic copper, respectively.     

Inversion of reflection seismograms by differential semblance analysis: algorithm structure and synthetic examples1

Seismograms predicted from acoustic or elastic earth models depend very non-linearly on the long wavelength components of velocity. This sensitive dependence demands the use of special variational principles in waveform-based inversion algorithms. The differential semblance variational principle is well-suited to velocity inversion by gradient methods, since its objective function is smooth and convex over a large range of velocity models. An extension of the adjoint state technique yields an accurate estimate of the differential semblance gradient. Non-linear conjugate gradient iteration is quite successful in locating the global differential semblance minimum, which is near the ordinary least-squares global minimum when coherent data noise is small. Several examples, based on the 2D primaries-only acoustic model, illustrate features of the method and its performance.     

Dispersion and cost analysis of some finite difference schemes in one-parameter acoustic wave modeling

Given a precision threshold to be imposed on the group velocity error and a finite difference scheme for the acoustic wave equation, it is possible to determine time-step and grid-spacing in an optimal manner, i.e., so as to minimize the computational cost. Using this optimal cost as a criterion, it becomes easy to compare schemes for efficiency in homogeneous media. Heterogeneous media with constant density can be accommodated to a certain extent by minimizing the cost over a range of Courant numbers. Such analysis shows that, amongst the second-order Taylor series schemes in time, higher-order schemes are generally more efficient than lower-order schemes. However, this result does not extend to very high order schemes.     

Tailored Blast Wave Production Pertaining to Supernova Remnants

We report on the first production of “tailored” blast waves in cluster media using a 1 ps laser pulse focused to 2×10¹⁶ W/cm². This new technique allows cylindrical blast waves to be produced with a strong axial modulation of variable spatial frequency, as a seed for instability growth. Energy deposition is modified by changing the cluster density whilst keeping the atomic density of the target constant. Electron density maps show the production of strongly modulated blast waves and the development of a thin shell structure in H at late times, and the trajectories show blast waves forming in H, and Ar. In Xe, a blast wave does not form on the timescales studied. Analysis of astrophysical similarity parameters suggests that a hydrodynamically similar situation is created in H, and that further evolution would create a regime where radiative effects may be influential in Ar and Xe.     

CALCIUM VARIATION IN OLIVINES OF THE MURCHISON AND VIGARANO METEORITES

Olivine grains were analysed by microprobe for Si, Ca, Mg, and Fe. Distribution of Mg and Fe is similar to that already established for Type II and III carbonaceous chondrites. Calcium concentration varies by a factor of about 4 and seems to be independent of fayalite content: its range (about 0.1 ? 0.45 weight percent) resembles that in the olivines of Murray (Fredriksson and Keil, 1964). Olivines of carbonaceous chondrites are the product of mechanical mixing of pre-existing crystals (Van Schmus, 1969). More than one fractionation process is necessary to account for Ca variation superimposed on Mg:Fe fractionation observed in Murray, Murchison and Vigarano.     

Properties of chondrules in EL3 chondrites,comparison with EH3 chondrites,and the implications for the formation of enstatite chondrites

Abstract— The study of chondrules provides information about processes occurring in the early solar system. In order to ascertain to what extent these processes played a role in determining the properties of the enstatite chondrites, the physical and chemical properties of chondrules from three EL3 chondrites and three EH3 chondrites have been examined by optical, cathodoluminescence (CL), and electron microprobe techniques. Properties examined include size, texture, CL, and composition of both individual phases and bulk chondrules. The textures, distribution of textures, and composition of silicates of the EL3 chondrules resemble those of EH3 chondrules. However, the chondrules from the two classes differ in that (1) the size distribution of the EL chondrules is skewed to larger values than EH chondrules, (2) the enstatite in EL chondrules displays varying shades of red CL due to the presence of fine‐grained sulfides and metal in the silicates, and (3) the mesostasis of EH chondrules is enriched in Na relative to that of EL chondrules. The similarities between the chondrules of the two classes suggest similar precursor materials, while the differences suggest that there was not a single reservoir of meteoritic chondrules, but that their origin was fairly local. The differences in the size distribution of chondrules in EH and EL chondrites may be explained by aerodynamic and gravitational sorting during accumulation of the meteoric material, while differences in CL and mesostasis properties may reflect differences in formation conditions and cooling rate following chondrule formation. We argue that our observations are consistent with the formation of enstatite chondrites in a thick dynamic regolith on their parent body.