Checkerboard Models Research Articles

Acoustic wave simulation in strongly heterogeneous models using a discontinuous Galerkin method

In recent years, the discontinuous Galerkin method (DGM) has been rapidly developed for the numerical simulation of seismic waves. For wavefield propagation between two adjacent elements, it is common practice to apply a numerical flux to the boundary of each element to propagate waves between adjacent elements. Several fluxes, including the center, penalty, local Lax-Friedrich (LLF), upwind, and Rankine-Hugoniot jump condition-based (RH condition) fluxes, are widely used in numerical seismic wave simulation. However, some fluxes do not account for media differences between adjacent elements. Although different fluxes have been successfully used in DGM for many velocity models, it is unclear whether they can produce sufficiently accurate or stable results for strongly heterogeneous models, such as checkerboard models commonly used in tomographic studies. We test different fluxes using the acoustic wave equation. We analyze the accuracy of the penalty, LLF, upwind, and RH-condition fluxes based on the results of the numerical simulations of the homogeneous and two-layer models. We conduct simulations using checkerboard models, and the results indicate that the LLF, penalty, and upwind fluxes may have instability problems in heterogeneous models with long-time simulations. We observe instability issues in the LLF, penalty, and upwind fluxes when the wave-impedance contrast is high at the media interface. However, the results of the RH-condition flux remain consistently stable. The series of numerical examples presented in this work provide insights into the characteristics and application of fluxes for seismic wave modeling.

3D Multiresolution Velocity Model Fusion with Probability Graphical Models

ABSTRACT The variability in spatial resolution of seismic velocity models obtained via tomographic methodologies is attributed to many factors, including inversion strategies, ray-path coverage, and data integrity. Integration of such models, with distinct resolutions, is crucial during the refinement of community models, thereby enhancing the precision of ground-motion simulations. Toward this goal, we introduce the probability graphical model (PGM), combining velocity models with heterogeneous resolutions and nonuniform data point distributions. The PGM integrates data relations across varying resolution subdomains, enhancing detail within low-resolution (LR) domains by utilizing information and prior knowledge from high-resolution (HR) subdomains through a maximum posterior problem. Assessment of efficacy, utilizing both 2D and 3D velocity models—consisting of synthetic checkerboard models and a fault-zone model from Ridgecrest, California—demonstrates noteworthy improvements in accuracy, compared to state-of-the-art fusion techniques. Specifically, we find reductions of 30% and 44% in computed travel-time residuals for 2D and 3D models, respectively, as compared to conventional smoothing techniques. Unlike conventional methods, the PGM’s adaptive weight selection facilitates preserving and learning details from complex, nonuniform HR models and applies the enhancements to the LR background domain.

Comparison of P-Wave Tomography Model in Sunda-Banda Arc by Using Data from BMKG and ISC

The tectonic setting of our study area is located between the Island of Java and Timor Leste. The complexity of this region is started with two different plates, The Indo-Australian plate and the Eurasian plate that move with different orientations and convergence rate. This area also shows active seismic activity and has a series of active volcanoes as a product of subduction and collision. To deepen understand this area, we perform delay time tomography using FMTOMO package that includes 3-D finite-difference based ray tracing and sub-space inversion procedure. We used two different sets of data, the first one is 4 years data catalog from the Indonesian Agency of Meteorology, Climatology and Geophysics, and the second one is 47 years of data from the International Seismological Centre. Data from the local Indonesian show agency shows a fewer number of events but more focus clusters. Meanwhile, the data from ISC catalog has more events and evenly distributed data. However, we also noticed that data from ISC has cluster events located at the same depth that can be improved with events relocation for better depth estimation. The Checkerboard models from both data set show a comparable result, though data from ISC show a better recovered model at a deeper depth and shallow part in the eastern area. The checkerboard from the local Agency shows slightly better results in the shallow part. Next, we invert delay time for each data set using we optimized damping and smoothing parameters. Final tomogram models show that data from the local Agency show a more continuous fast velocity band representing a downgoing subducting slab and possible back-arc thrust while results from the ISC data show a more detached fast velocity band that could be contributed from fixed depth problem in the data set. However, we noticed that data from ISC show a higher amplitude low-velocity anomaly especially in the shallow depth

Initial Result of P Wave Tomography Model in Sunda-Banda Arc Transition using FMTOMO

Our study area is located near island Sumbawa, Sumba, Flores, West Timor, Indonesia and East Timor, popularly known as Sunda-Banda arc transition zone. The tectonic setting is mainly controlled by the movement of the oceanic lithosphere Indo-Australian plate subducting the Eurasian plate and Northward migration of Australian continental lithosphere into western Banda-arc in the region of Flores, Sumba and Timor island. We tried to image velocity structure beneath these regions using regional events and tomography inversion model. We collected 5 years of regional events from the Indonesian Agency of Meteorology, Climatology and Geophysics. In total, we reserved 3186 events recorded on 29 stations. For data processing, we used fast marching method as ray tracing between sources and receiver. We then employed subspace inversion as the tomography procedure to estimate the best velocity model representing the tectonic model in the region. Hypocenter data distribution is concentrated on shallow parts of the region and along the Benioff zone down to a maximum depth of 400 km. One of challenge of this study is that although events are abundance, the stations used are mostly located onshore and does not extend in the south-north direction that leads us to under determined problem in the inversion process. However, checker-board models show most our target area can be retrieved to its initial model with sign of smearing effects shown start from a depth of 50 km. After six iteration and optimized selection of damping and smoothing parameters, we observed low velocity anomaly under Bali, Lombok, Sumba, East Nusa Tenggara at shallow depth that may be related with volcanic activity. Deeper low anomaly can also be seen that may be related with partial melting process. A band of fast velocity is clearly seen that goes deepen to the north depicting subducting slabs own to a depth of 300 km. We also observed a possible of fast velocity in the northern part of our stations at shallow depth that we believe may represent the back arc thrust.

Magnetic excitations of diagonally coupled checkerboards* *Project supported by the National Key R&D Program of China (Grant Nos. 2018YFA0306001 and 2017YFA0206203), the National Natural Science Foundation of China (Grant No. 11974432), GBABRF-2019A1515011337, and Leading Talent Program of Guangdong Special Projects.

By using quantum Monte Carlo based stochastic analytic continuation (QMC-SAC) and spin wave theory, we study magnetic excitations of Heisenberg models with diagonally coupled checkerboard structures. We consider three kinds of checkerboard models (DC 2 × 2, DC 3 × 3, and CDC 3 × 3) consisting nearest-neighbor strong J 1 and weak J 2 antiferromagnetic interactions. When the coupling ratio g = J 2/J 1 approaches 1, all three diagonal checkerboards have the same long-range antiferromagnetic Néel order at T = 0. When g decreases, the quantum fluctuation can drive DC 2 × 2 model to quantum paramagnetic state, while DC 3 × 3 and CDC 3 × 3 models still have the long-range Néel order. By calculating the magnetic excitations at different coupling ratios, we find that the low-energy part of magnetic excitations calculated by QMC-SAC can be well explained by the spin wave theory. However, the high-energy parts even deep in the long-range antiferromagnetic phase are beyond the spin wave description. Compared to the g = 1 uniform square lattice, the high-energy excitations are more rich in our models. Our study may also draw the attention to the high-energy exctitaions beyond the spin wave theory.

Checkerboard: a Bayesian efficacy and toxicity interval design for phase I/II dose-finding trials

ABSTRACT The rise of targeted therapy and immunotherapy has challenged the conventional more-is-better phase I trial design paradigm that focuses on finding the MTD. In this article, we propose a novel model-assisted phase I/II design, called checkerboard design, that considers both toxicity and efficacy. As an extension of the keyboard design, the checkerboard design models the joint distribution of toxicity and efficacy, and divides toxicity and efficacy domain into a series of equal-width intervals or keys. In light of interim data, the checkerboard design continuously updates the posterior distribution of toxicity and efficacy, and adaptively determine the optimal dose for treating the next cohort of patients based on the posterior probability of toxicity and efficacy keys. As a model-assisted design, one important advantage of the checkerboard design is that its decision rule can be pretabulated, greatly simplifying its implementation. We also extend the checkerboard design to handle continuous efficacy endpoint. Simulations study shows that the checkerboard design yields competitive performance comparable to existing model-based phase I/II designs, but is simpler and easier to implement in real applications.

Fracton fusion and statistics

We introduce and develop a theory of fusion and statistical processes of gapped excitations in Abelian fracton phases. The key idea is to incorporate lattice translation symmetry via its action on superselection sectors, which results in a fusion theory endowed with information about the non-trivial mobility of fractons and sub-dimensional excitations. This results in a description of statistical processes in terms of local moves determined by the fusion theory. Our results can be understood as providing a characterization of translation-invariant fracton phases. We obtain simple descriptions of the fusion theory in the X-cube and checkerboard fracton models, as well as for gapped electric and magnetic excitations of some gapless U(1) tensor gauge theories. An alternate route to the X-cube model fusion theory is provided by starting with a system of decoupled two-dimensional toric code layers, and giving a description of the p-string condensation mechanism within our approach. We discuss examples of statistical processes of fractons and sub-dimensional excitations in the X-cube and checkerboard models. As an application of the ideas developed, we prove that the X-cube and semionic X-cube models realize distinct translation-invariant fracton phases, even when the translation symmetry is broken corresponding to an arbitrary but finite enlargement of the crystalline unit cell.

Criticality and scaling corrections for two-dimensional Heisenberg models in plaquette patterns with strong and weak couplings

We use the stochastic series expansion quantum Monte Carlo method to study the Heisenberg models on the square lattice with strong and weak couplings in the form of three different plaquette arrangements known as checkerboard models C$2\times2$, C$2\times4$ and C$4\times4$. The $a\times b$ here stands for the shape of plaquette consisting with spins connected by strong couplings. Through detailed analysis of finite-size scaling study, the critical point of C$2\times2$ model is improved as $g_{c}=0.548524(3)$ compared with previous studies with $g$ to be the ratio of weak and strong couplings in the models. For C$2\times4$ and C$4\times4$ we give $g_{c}=0.456978(2)$ and $0.314451(3)$. We also study the critical exponents $\nu$, $\eta$, and the universal property of Binder ratio to give further evidence that all quantum phase transitions in these three models are in the three-dimensional O(3) universality class. Furthermore, our fitting results show the importance of effective corrections in the scaling study of these models.

Spin excitation spectra of the two-dimensional S=12 Heisenberg model with a checkerboard structure

We study the spin excitation spectra of the two-dimensional spin-$1/2$ Heisenberg model with a checkerboard structures using stochastic analytic continuation of the imaginary-time correlation function obtained from a quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions $J_{1}$ and $J_{2}$, and the tuning parameter $g$ is defined as $J_{2}/J_{1}$. The dynamic spin structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, spin wave theory is employed to explain some features of numerical results, especially for the low-energy part. When $g$ is close to $1$, the features of the spin excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them is discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\times 3$ checkerboard model has distinctive excitation features, such as a gap between a low-energy gapless branch and a gapped high-energy part that exists when $g$ is small. The gapless branch in this case can be regarded as a spin wave in N$\mathrm{\acute{e}}$el order formed by a "block spin" in each $3\times 3$ plaquette with an effective exchange interaction originating from renormalization. One unexpected finding is that the continuum also appears in this low-energy branch.

Time-domain full-waveform inversion of Rayleigh and Love waves in presence of free-surface topography

Correct estimation of near-surface seismic-wave velocity when encountering lateral heterogeneity and free surface topography is one of the challenges to current shallow seismic. We propose to use time-domain full-waveform inversion (FWI) of surface waves, including both Rayleigh and Love waves, to solve this problem. We adopt a 2D time-domain finite-difference method with an improved vacuum formulation (IVF) to simulate shallow-seismic Rayleigh wave in presence of free-surface topography. We modify the IVF for SH-wave equation for the simulation of Love wave in presence of topographic free surface and prove its accuracy by benchmark tests. Checkboard model tests are performed in both cases when free-surface topography is included or neglected in FWI. Synthetic model containing a dipping planar free surface and lateral heterogeneity was then tested, in both cases of considering and neglecting free-surface topography. Both checkerboard and synthetic models show that Rayleigh- and Love-wave FWI have similar ability of reconstructing near-surface structures when free-surface topography is considered, while Love-wave FWI could reconstruct near-surface structures better than Rayleigh-wave when free-surface topography is neglected.

Modeling near-infrared reflectance spectra of clay and sulfate mixtures and implications for Mars

High-resolution mapping by visible and near-infrared orbital spectrometers has revealed a diversity of hydrated mineral deposits on the surface of Mars. Quantitative analysis of mineral abundances within these deposits has the potential to distinguish depositional and diagenetic processes. Such analysis can also provide important constraints on the nature of putative global and local-scale mineralogical transitions on Mars. However, the ability of models to extract quantitative mineral abundances from spectra of mixtures relevant to sedimentary rocks remains largely untested. This is particularly true for clay and sulfate minerals, which often occur as fine-grained components of terrestrial sedimentary rocks and are known to occur in a number of sedimentary deposits on Mars. This study examines the spectral properties of a suite of mixtures containing the Mg-sulfate epsomite mixed with varying proportions of smectitic clay (saponite, nontronite, and montmorrilonite). The goal of this work is to test the ability of checkerboard (linear) and intimate (non-linear) mixing models to obtain accurate estimates of mineral abundances under ideal and controlled laboratory conditions. The results of this work suggest that: (1) spectra of clay–sulfate mixtures can be reproduced by checkerboard and intimate mixing models to within 2% absolute reflectance or single scattering albedo, (2) clay and epsomite abundance can be modeled to within 5wt.% when particle diameter is optimized, and (3) the lower threshold for modeling clay in spectra of clay–epsomite mixtures is approximately 10wt.%, below which the models often fail to recognize the presence of clay.

Superconductivity in inhomogeneous Hubbard models

We present a controlled perturbative approach to the low-temperature phase diagram of highly inhomogeneous Hubbard models in the limit of small coupling, ${t}^{\ensuremath{'}}$, between clusters. We apply this to the dimerized and checkerboard models. The dimerized model is found to behave like a doped semiconductor, with a Fermi-liquid groundstate with parameters (e.g., the effective mass) which are smooth functions of the Hubbard interaction, $U$. By contrast, the checkerboard model has a Fermi liquid phase at large $U>{U}_{c}=4.58$, a $d$-wave superconducting state with a full gap for ${U}_{c}>U>0$, and a narrow strip of an intermediate $d$-wave superconducting phase with gapless ``nodal'' quasiparticles for $\ensuremath{\mid}U\ensuremath{-}{U}_{c}\ensuremath{\mid}<\mathcal{O}({t}^{\ensuremath{'}})$.

Multiple-spin exchange models on the checkerboard lattice with an exact two-fold degenerate Shastry-Sutherland ground state

Two quantum spin models with bilinear-biquadratic exchange interactions are constructed on the checkerboard lattice. It is proved that, under certain sufficient conditions on the exchange parameters, their ground states consist of two degenerate Shastry-Sutherland singlet configurations. The constructions are studied for arbitrary spin-S. The sufficient conditions for the existence of ferromagnetic ground state are also found exactly. The approximate quantum phase diagrams are presented using the exact results, together with a variational estimate for the N\'eel antiferromagnetic phase. A two-leg spin-1/2 ladder model, based on one of the above constructions, is considered which admits exact solution for a large number of eigenstates. The ladder model is shown to have exact level-crossing between the rung-singlet state and the AKLT state in the singlet ground state. Also introduced is the notion of perpendicularity for quantum spin vectors, which appears in the discussion on one of the two checkerboard models, and is discussed in the Appendix.

Lateral velocity resolution from three-dimensional seismic refraction data

SUMMARY A method for estimating the lateral velocity resolution from 3-D seismic refraction traveltimes is presented. The method is non-linear in that synthetic data calculated from ‘checkerboard’ models are inverted using an iterative tomographic approach incorporating smoothness constraints and updated ray paths at each iteration. Two applications are presented: (1) real data from the Faeroe Basin experiment using the ray coverage and noise level of approximately 50 000 picks, and (2) ideal data from the same experiment corresponding to 100 per cent data recovery, equivalent to about three times as many ray paths. The refraction data constrain the sedimentary and basement structure to approximately 12 km depth. A comparison of the results from the two data sets provides a qualitative check on the resolution method, illustrates the pitfalls of using ray coverage as an indicator of resolution, and demonstrates the potential resolution of currently feasible 3-D refraction experiments. 32 checkerboard models consisting of alternating positive and negative velocity anomalies superimposed on the preferred final model of the basin were tested, each with a diVerent cell size, position or orientation. The problem of ray bending associated with the 5 per cent checkerboard anomalies is shown to be small. Using models with four cell sizes allowed for resolution estimates from 3 to 15 km, this range being imposed by the frequency content of the data and the areal dimensions of the study area (44◊26 km). An operator centred on each model node measures the local semblance between the known and recovered checkerboard models. A determination of the smallest well-resolved cell size at each model node is then made using a 0.7 semblance threshold to provide a spatially dependent lateral resolution estimate. Resolution for the real data set varies from an average of just over 3 km at 2 km depth, to 10 km at 10 km depth, where there is a local maximum in ray coverage due to a concentration of ray-bottoming points. For the ideal data set, the average lateral resolution drops oV roughly linearly from better than 3 km at 2 km depth, to 6.3 km at 10 km depth. The largest diVerence between the resolution associated with the ideal and real data occurs at 7.2 km depth, where it averages 4.6 km for the former and 10.8 km for the latter due to the near absence of ray-bottoming points between 7 and 9 km. The results show that the final model for the Faeroe Basin obtained by regularized inversion with smoothness constraints is consistent with the resolution estimates in the sense that the model does not contain structure with a wavelength smaller than the resolution estimate at any point. The results for the ideal data show that good-quality 3-D refraction experiments can provide a lateral resolution of no worse than the receiver spacing (shot spacing for land data) throughout most of the sampled volume.

Critical Behavior of Transport in Lattice and Continuum Percolation Models

It has been observed that the critical exponents of transport in the continuum, such as in the Swiss cheese and random checkerboard models, can exhibit nonuniversal behavior, with values different than the lattice case. Nevertheless, it is shown here that the transport exponents for both lattice and continuum percolation models satisfy the standard scaling relations for phase transitions in statistical mechanics. The results are established through a direct, analytic correspondence between transport coefficients for two component random media and the magnetization of the Ising model, which is based on the observation we made previously that both problems share the Lee-Yang property.

Rigorous link between the conductivity and elastic moduli of fibre-reinforced composite materials

We derive rigorous cross-property relations linking the effective transverse electrical conductivity cr* and the effective transverse elastic moduli of any transversely isotropic, two-phase ‘fibre-reinforced’ composite whose phase boundaries are cylindrical surfaces with generators parallel to one axis. Specifically, upper and lower bounds are derived on the effective transverse bulk modulus k* in terms of cr* and on the effective transverse shear modulus //* in terms of cr*. These bounds enclose certain regions in the ct*-ac* and cr*-/r* planes, portions of which are attainable by certain microgeometries and thus optimal. Our bounds connecting the effective conductivity cr* to the effective bulk modulus ft* apply as well to anisotropic composites with square symmetry. The implications and utility of the bounds are explored for some general situations, as well as for specific microgeometries, including regular and random arrays of circular cylinders, hierarchical geometries corresponding to effective-medium theories, and checkerboard models. It is shown that knowledge of the effective conductivity can yield sharp estimates of the effective elastic moduli (and vice versa), even for infinite phase contrast.

Spontaneous magnetisations of the Ising model on the bathroom tile lattice

The authors evaluate the site spontaneous magnetisations on the general 4-8 or bathroom tile lattice using mappings to checkerboard, free-fermion and Union Jack Ising models. Their results verify the conjecture of Lin et al (1987), who proposed their formula from twelfth-order low-temperature series expansions.

Disorder solutions for Ising and Potts models with a field

The exact expressions of the disorder varieties and the corresponding values taken by the partition functions of the checkerboard Ising and Potts models with a magnetic field are obtained. Different specialisations and extensions of these exact results are examined.

Disorder points of the IRF and checkerboard Potts models

Conditions are given for a general 'interactions-round-a-face' (IRF) model in statistical mechanics to have a disorder point, and expressions obtained for the free energy and intra-low correlations. These are applied to the checkerboard Potts model, thereby verifying a recent conjecture by Jaeckel and Maillard (1984).