Splitting Inversion Research Articles

Erratum: Signature splitting inversion and backbending in Rb80 [Phys. Rev. C 87 , 034320 (2013)

Erratum: Signature splitting inversion and backbending in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Rb</mml:mi><mml:mprescripts/><mml:none/><mml:mn>80</mml:mn></mml:mmultiscripts></mml:math> [Phys. Rev. C <b>87</b> , 034320 (2013)

The Rotational Shear Layer inside the Early Red-giant Star KIC 4448777

Abstract We present the asteroseismic study of the early red-giant star KIC 4448777, complementing and integrating a previous work, aimed at characterizing the dynamics of its interior by analyzing the overall set of data collected by the Kepler satellite during the four years of its first nominal mission. We adopted the Bayesian inference code diamonds for the peak bagging analysis and asteroseismic splitting inversion methods to derive the internal rotational profile of the star. The detection of new splittings of mixed modes, which are more concentrated in the very inner part of the helium core, allowed us to reconstruct the angular velocity profile deeper into the interior of the star and to disentangle the details better than in Paper I: the helium core rotates almost rigidly about 6 times faster than the convective envelope, while part of the hydrogen shell seems to rotate at a constant velocity about 1.15 times lower than the He core. In particular, we studied the internal shear layer between the fast-rotating radiative interior and the slow convective zone and we found that it lies partially inside the hydrogen shell above r ≃ 0.05R and extends across the core–envelope boundary. Finally, we theoretically explored the possibility for the future capabilty to sound the convective envelope in the red-giant stars and we concluded that the inversion of a set of splittings with only low-harmonic degree l ≤ 3, even supposing a very large number of modes, will not allow us to resolve the rotational profile of this region in detail.

Publisher's Note: Signature splitting inversion and backbending in80Rb [Phys. Rev. C87, 034320 (2013)

Publisher's Note: Signature splitting inversion and backbending in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mrow /><mml:mn>80</mml:mn></mml:msup></mml:math>Rb [Phys. Rev. C<b>87</b>, 034320 (2013)

Conics in the hyperbolic plane intrinsic to the collineation group

In the manner of Steiner’s interpretation of conics in the projective plane we consider a conic in a planar incidence geometry to be a pair consisting of a point and a collineation that does not fix that point. We say these loci are intrinsic to the collineation group because their construction does not depend on an imbedding into a larger space. Using an inversive model we classify the intrinsic conics in the hyperbolic plane in terms of invariants of the collineations that afford them and provide metric characterizations for each congruence class. By contrast, classifications that catalogue all projective conics intersecting a specified hyperbolic domain necessarily include curves which cannot be afforded by a hyperbolic collineation in the above sense. The metric properties we derive will distinguish the intrinsic classes in relation to these larger projective categories. Our classification emphasizes a natural duality among congruence classes induced by an involution based on complementary angles of parallelism relative to the focal axis of each conic, which we refer to as split inversion (Definition 5.3).

High-spin states and level structure inRb84

High-spin states in $^{84}\mathrm{Rb}$ have been studied by using the $^{70}\mathrm{Zn}$($^{18}\mathrm{O},p3n)$$^{84}\mathrm{Rb}$ reaction at beam energy of 75 MeV. The $\ensuremath{\gamma}$-$\ensuremath{\gamma}$ coincidence, excitation function, and ratios for directional correlation of oriented states were determined. A new level scheme was established in which the positive- and negative-parity bands have been extended up to ${17}^{+}$ and ${17}^{\ensuremath{-}}$ with an excitation energy of about 7 MeV. The signature splitting and signature inversion of the positive-parity yrast band were observed. To understand the microscopic origin of the signature inversion in the yrast positive-parity bands of doubly odd Rb nuclei, as an example, we performed calculations using the projected shell model to describe the energy spectra in $^{84}\mathrm{Rb}$. It can be seen that the main features are reproduced in the calculations. This analysis shows that the signature splitting, especially its inversion, can be reproduced by varying only the $\ensuremath{\gamma}$ deformation with increasing spin. To research the deformation of $^{84}\mathrm{Rb}$ carefully, we calculate the total Routhian surfaces of positive-parity yrast states by the cranking shell model formalism. In addition, the results of theoretical calculations about the negative-parity yrast band in $^{84}\mathrm{Rb}$ with configuration $\ensuremath{\pi}({p}_{3/2},{f}_{5/2})\ensuremath{\bigotimes}\ensuremath{\nu}{g}_{9/2}$ are compared with experimental data, and a band diagram calculated for this band is also shown to extract physics from the numerical results.

Switching slow relaxation in a MnIII3MnIV cluster: an example of grafting single-molecule magnets onto polyoxometalates

Binding an established single-molecule magnet cluster of distorted cubane type [Mn(III)(3)Mn(IV)O(4)] to the lacunary site of an {alpha-P(2)W(15)} polyoxotungstate scaffold results in a surprising zero-field splitting inversion and the subsequent loss of magnetization bistability.

The Inversion Potential of Ammonia: An Intrinsic Reaction Coordinate Calculation for Student Investigation

A computational–experimental project is described in which upper-level students construct the double minimum inversion potential for NH3 from intrinsic reaction coordinate (IRC) calculations using ab initio methods (MP2/cc-pVTZ). They use this potential to obtain the inversion eigenvalues from which they can predict the frequencies of the split inversion mode transitions and make comparisons with experimental results. In an experimental component they acquire the IR spectrum of NH3 vapor in the 900–1000 cm-1 region. The results obtained from the calculation are used to estimate the rates of inversion from both tunneling and thermal processes. Suggestions for extensions of this project to other compounds are provided.

On the effect of error correlation on linear inversions

We have examined the effect on linear helioseismic inversions of correlations in data errors, taking an example from one-dimensional rotational splitting inversion. Artificial data with correlated errors were generated and then inverted with or without using the proper covariance matrix. The effects of using incorrect covariance matrices, on solutions as well as on trade-offs, are discussed. It is found that improper account of the correlations can be deleterious to the faithfulness of the inversions, and yields incorrect error estimates, which under some circumstances can lead to misleading inferences.

Real time 2D inversion of induction logging data

Real time 2D inversion for an induction logging instrument may be achieved using a fast forward modeling and special inversion strategy. The fast forward modeling employs a low-frequency approximation of an induction response known as Doll's geometric factor. Modeling geometric factors is much faster than modeling the electromagnetic field in the frequency domain. To transform real data into the Doll's limit, multi-frequency skin-effect correction is applied. The correction technique involves an asymptotic theory of the integral equation for a 2D boundary value problem. The inversion is based on separating the parameter space into subspaces of lower dimension. Initially, adaptive overlapping windows split logging data into manageable portions. Each window consists of three subwindows: the predictor, corrector and upgrader. Further separation of parameters is introduced by Doll's approximation: the low-frequency response is linear with respect to formation conductivity. This allows us to split inversion for conductivity and geometric parameters. The next level of splitting inversion is achieved by independently determining parameters of the near borehole zone and remote formation areas. This is done by utilizing different subsets of sensors. The inversion does not require initial guess: layers are introduced dynamically, if necessary. The resolution is improved in sequential iterations by adding finer details to the previously obtained models. The final selection of parameters satisfies a variety of a priori constraints formulated as target resistivity distributions. The technique for imposing constraints is based on the analysis of data mapping into the model space. Interpretation of synthetic and real data confirms the viability of the method.

Traveltime inversion for the geometry of aquifer lithologies

Crosswell traveltime tomography can provide detailed descriptions of the geometry and seismic slowness of lithologic zones in aquifers and reservoirs. Traditional tomographic inversions that estimate a smooth slowness field to match traveltime data, provide limited information about the dominant scale of subsurface heterogeneity. We demonstrate an alternative method, called the multiple population inversion (MPI), that co‐inverts traveltimes between multiple well pairs to identify the spatial distribution of a small number of slowness populations. We also compare the MPI with the split inversion method (SIM) that was recently introduced to address the same problem. The lithologies and hydraulic parameters for these populations can then be determined from core data and hydraulic testing. The MPI iteratively assigns pixels to a small number of slowness populations based on the histogram of slowness residuals. By constraining the number of slowness values, this method is less susceptible to inversion artifacts, such as those related to slight variations in ray coverage, and can resolve finer scale sedimentary structures better than methods that smooth the slowness field. We demonstrate the MPI in two dimensions with a synthetic aquifer and in three dimensions with the Kesterson aquifer in the central valley of California. In both cases, the constrained inversion algorithm converges to an equal or smaller average traveltime residual than obtained with unconstrained‐value tomography. The MPI accurately images the dominant lithologies of the synthetic aquifer and provides a geologically reasonable image of the Kesterson aquifer.

Estimating Lithologic and Transport Properties in Three Dimensions Using Seismic and Tracer Data: The Kesterson aquifer

The identification of aquifer heterogeneities, particularly flow paths and barriers, has become a critical research topic in hydrology. Cross‐well seismic tomography may provide the needed resolution when used in conjunction with hydraulic head and tracer concentration measurements. We demonstrate a field application and sensitivity analysis of the split inversion method (SIM), which combines seismic, hydraulic, and tracer data to estimate the three‐dimensional zonation of aquifer properties along with the hydraulic properties for these zones. For the Kesterson aquifer in the San Joaquin Valley, California, we first invert seismic travel times measured between six well pairs to obtain seismic slowness (1/seismic velocity) cross sections, or tomograms. We then use conditional simulation to provide three‐dimensional seismic slowness realizations. Next, the SIM is used to split several realizations into three lithologic zones and assign hydraulic properties to the zones to best match six tracer concentration histories.

Coupled seismic and tracer test inversion for aquifer property characterization : D. W. Hyndman, J. M. Harris & S. M. Gorelick, Water Resources Research, 30(7), 1994, pp 1965–1977

Seismic and tracer test data can be combined to estimate the spatial patterns of aquifer properties. We present an algorithm that estimates the geometry of large-scale lithologic zones, the effective hydraulic conductivities and seismic velocities for these zones, and the effective small-scale dispersivity. The heart of this algorithm is our split inversion method, which extracts the geometry of lithologic zones from an estimated seismic velocity field. This method determines the zonation that best matches multiple types of data, such as seismic travel times and tracer concentrations. Although the current implementation of the algorithm uses only cross-well seismic travel times and tracer concentrations, the algorithm could incorporate other data types that are sensitive to changes in large-scale lithology. We demonstrate the approach for two synthetic sandy aquifers, one with interbedded clay lenses and another with interbedded silt and gravel lenses. These examples differ in the uniqueness of the relationship between seismic velocities and hydraulic conductivities. For these examples our algorithm successfully maps interwell heterogeneities and accurately estimates hydraulic and seismic parameters.

Coupled seismic and tracer test inversion for aquifer property characterization

Seismic and tracer test data can be combined to estimate the spatial patterns of aquifer properties. We present an algorithm that estimates the geometry of large‐scale lithologic zones, the effective hydraulic conductivities and seismic velocities for these zones, and the effective small‐scale dispersivity. The heart of this algorithm is our split inversion method, which extracts the geometry of lithologic zones from an estimated seismic velocity field. This method determines the zonation that best matches multiple types of data, such as seismic travel times and tracer concentrations. Although the current implementation of the algorithm uses only cross‐well seismic travel times and tracer concentrations, the algorithm could incorporate other data types that are sensitive to changes in large‐scale lithology. We demonstrate the approach for two synthetic sandy aquifers, one with interbedded clay lenses and another with interbedded silt and gravel lenses. These examples differ in the uniqueness of the relationship between seismic velocities and hydraulic conductivities. For these examples our algorithm successfully maps interwell heterogeneities and accurately estimates hydraulic and seismic parameters.

A general linear prediction approach to the complex split inversion algorithm

A general linear prediction framework is developed for the derivation of the complex split Levinson algorithm, which leads naturally to a structural interpretation of the redundancy present in the classical Levinson-Durbin algorithm. A novel order-recursive predictor structure is presented. It is conceptually simpler than a lattice realization since it propagates only one (generalized) prediction error. When the usual prediction error is desired, it can be recovered through a simple terminal stage. >