Ergodic Models Research Articles

Nonergodic Ground-Motion Models for Subduction Zone and Crustal Earthquakes in Japan

ABSTRACT We investigate the nonergodic behavior of ground motions from subduction zone earthquakes and crustal earthquakes in Japan using the National Research Institute for Earth Science and Disaster Resilience strong-motion flatfile together with various reference ergodic ground-motion models (GMMs). For the nonergodic path effects, our nonergodic GMM has two path effects terms: a cell-specific linear-distance scaling, denoted as δP2PQ, that mimics the effects of a 3D Q structure, and a source- and site-specific term related to the effects of the 3D velocity structure, denoted δP2PV. The resulting model provides spatially varying nonergodic source, site, and path terms. The GMM smoothly interpolates and extrapolates the nonergodic terms in space so that the GMM can be applied to any combination of source and site locations in the region. In regions where data are sparse, the median nonergodic terms will approach zero but with large epistemic uncertainty. Over the period range of 0–10 s, the ranges of the standard deviations of the nonergodic source, path, and site terms are 0.2–0.65, 0.25–0.40, and 0.15–0.40 natural log units, respectively. The fully nonergodic models lead to a 40%–55% decrease in the aleatory standard deviation compared to the reference ergodic GMMs. This large reduction in the aleatory standard deviation combined with the change in the median given by the nonergodic terms can exert a significant impact on the computed seismic hazard for the Japan region.

Small-strain site response of soft soils in the Sacramento-San Joaquin Delta region of California conditioned on VS30 and mHVSR

Sites located in the Sacramento-San Joaquin Delta region of California typically have peaty-organic soils near the ground surface, which are characteristically soft, with shear wave velocities as low as 30 m/s. These unusually soft geotechnical conditions, which are outside the range of applicability of existing ergodic site amplification models, can be anticipated to produce significant site effects during earthquake shaking. We evaluate site response for 36 seismic stations in the Delta region using non-ergodic methods with low-amplitude ground motion data. We model first-order site effects using a period-dependent relation conditioned on the 30 m time-averaged shear wave velocity ( VS30). Relative to extrapolations of global ergodic models, this Delta-specific model provides lower and higher levels of amplification for short and long periods, respectively. While the local model provides unbiased predictions for Delta sites as a whole, it smooths over site-specific effects such as resonant peaks. Microtremor-based horizontal-to-vertical spectral ratios (mHVSRs) were measured for 34 sites, from which additional site parameters such as peak frequency ( fp), peak amplitude ( ap), and average mHVSR amplitude over some frequency bandwidth ([Formula: see text]) are derived. Sites with prominent mHVSR peaks are found to often exhibit site resonance effects, while sites without prominent peak features generally do not. A modified Ricker wavelet pulse function conditioned on fp is used to model resonance effects, while a logistic function whose amplitude is correlated with [Formula: see text] is used to model general broadband amplification that transitions to zero at long periods. These mHVSR-informed models are implemented as additive components to the VS30-scaling model. Relative to a global ergodic model, the VS30-scaling model does not appreciably change the site-to-site aleatory variability ([Formula: see text]) for periods shorter than about 1.5 s but reduces [Formula: see text] by ~0.1 (natural log units) at long periods. When the mHVSR-informed model components are used, [Formula: see text] is reduced by ~0.05 to 0.1 for short-to-intermediate periods (up to ~2 s). A model for nonlinear effects, derived from nonlinear ground response analyses, will be presented in a separate paper.

SCALE-INVARIANT MODE IN COLLISIONLESS SPHERICAL STELLAR SYSTEMS

An analytical solution of the perturbed equations is obtained, which exists in all ergodic models of collisionless spherical stellar systems with a single length parameter. This solution corresponds to variations of this parameter, that is, the stretching/contraction of the sphere keeping the total mass constant. During the process, the system remains in an equilibrium state. The simplicity of the solution makes it possible to explicitly give expressions for the distribution function, potential, and density in all orders of perturbation theory. This, in turn, helps to clarify the concept of perturbation energy, which, being a second-order magnitude in amplitude, cannot be calculated in linear theory. The expression for the 2nd-order perturbation energy does not match the well-known quadratic form for perturbation energy derived from 1st-order perturbations in linear theory. However, both of these energies are integrals of motion and differ only by a constant. The obtained solution can be used to control the correctness of codes and the accuracy of calculations in the numerical study of collisionless stellar models.

Scale-Invariant Mode in Collisionless Spherical Stellar Systems

An analytical solution for the perturbed equations, applicable to all ergodic models of collisionless spherical stellar systems with a single length parameter, has been derived. This solution corresponds to variations in this parameter, i.e., the expansion or contraction of the sphere while conserving total mass. During this process, the system maintains an equilibrium state. The simplicity of the solution allows for the explicit expression of the distribution function, potential, and density across all orders of perturbation theory. This, in turn, aids in clarifying the concept of perturbation energy, which, being of second order in amplitude, cannot be calculated using linear theory. It is demonstrated that the correct expression for perturbation energy, accounting for second-order perturbations, does not align with the well-known expression for perturbation energy via a quadratic form, derived from first-order perturbations within linear theory. However, both these energies are integrals of motion and differ only by a constant. The derived solution can be utilized to verify the correctness of codes and the precision of calculations in the numerical study of collisionless stellar models.

Regional physics-based simulation of ground motion within the Rhȏne Valley, France, during the MW 4.9 2019 Le Teil earthquake

In this paper we introduce the 3D physics-based numerical simulations (PBS) of ground motion during the Nov 11, 2019, MW 4.9 Le Teil earthquake, which occurred in a low-to-moderate seismicity area in the South-East of France, within the Rhône river valley, which hosts several operating nuclear installations. The numerical code SPEED, developed at Politecnico di Milano, Italy, was used to produce the PBS. After introducing the criteria to construct the numerical model, based on the relatively limited data available, a numerical convergence test was made to identify the frequency range for accurate simulations. Furthermore, the performance of the numerical results against the available strong motion records was assessed quantitatively using Goodness-of-fit (GoF) measures. According to the GoF scores, a good-to-excellent agreement was found on the horizontal components up to 8 Hz, showing that, even without a very detailed 3D numerical model of the medium, that would imply detailed knowledge of the basin shape, of the bedrock-to-basin impedance ratio, as well as of the damping ratio in the basin and its dependence on frequency, the PBS may provide realistic broadband predictions of earthquake ground motion. Nevertheless, as shown by the poorer performance on the vertical component, the high-frequency limitations of PBS, also in relation to the energy radiated by the kinematic source model, is still an issue to be carefully addressed. In spite of these limitations, the results obtained in this work demonstrate that PBS, if suitably calibrated and validated, can be either an alternative or a useful complement to empirical ground motion models, especially in those cases where the region- and site-specific features of ground shaking, including near-source conditions, are typically not accounted for by ergodic empirical models, such as for the seismic risk evaluation of large urban areas and/or of strategic structures, infrastructures and industrial plants.

Strictly ergodic distal models and a new approach to the Host-Kra factors

Cocycles are a key object in Antolín Camarena and Szegedy's theory of nilspaces. We introduce variants of these cocycles, named nilcycles, enabling us to give conditions which guarantee that an ergodic group extension of a strictly ergodic distal system admits a strictly ergodic distal topological model. In particular we show that if the base space is a dynamical nilspace then a dynamical nilspace topological model may be chosen for the extension. This approach combined with a structure theorem of Gutman, Manners and Varjú applied to the ergodic group extensions between successive Host-Kra characteristic factors gives a new proof that these factors are inverse limit of nilsystems for Zk-actions.

Nonlinear Autoregression With Convergent Dynamics on Novel Computational Platforms

Nonlinear stochastic modeling is useful for describing complex engineering systems. Meanwhile, neuromorphic (brain-inspired) computing paradigms are being developed to tackle tasks that are challenging and resource-intensive on digital computers. An emerging scheme is reservoir computing, which exploits nonlinear dynamical systems for temporal information processing. This brief introduces reservoir computers with output feedback as stationary and ergodic infinite-order nonlinear autoregressive models. We highlight the versatility of this approach by employing classical and quantum reservoir computers to model synthetic and real datasets, further exploring their potential for control applications.

Site response of sedimentary basins and other geomorphic provinces in southern California

Ergodic site amplification models for active tectonic regions are conditioned on the time-averaged shear wave velocity in the upper 30 m ( VS30) and the depth to a shear wave velocity isosurface ( zx). The depth components of such models are derived using data from sites within many geomorphic domains. We provide a site amplification model utilizing VS30 and depth, with the depth component conditioned on type of geomorphic province: basins, valleys, and mountain/hills. As with current models, the depth component of our model is centered with respect to the VS30-scaling model using differential depth δzx, taken as the difference between a site-specific depth and a VS30 -conditioned average depth. Using data from southern California, we find that long-period site response for all sites combined exhibits relative de-amplification and amplification for negative and positive differential depths, respectively. Individual provinces exhibit broadly similar trends with depth, but amplification levels are on average stronger in basins such that little relative de-amplification occurs at negative differential depths. Valley and mountain/hill sites have, on average, weaker amplification levels but stronger scaling with δzx. Site-to-site standard deviations vary appreciably across geomorphic provinces, with basins having lower dispersions than mountain/hill sites and the reference ergodic model.

Estimation for change point of discretely observed ergodic diffusion processes

AbstractWe treat the change point problem in ergodic diffusion processes from discrete observations. Tonaki et al. (2021a) proposed adaptive tests for detecting changes in the diffusion and drift parameters in ergodic diffusion process models. When any change in the diffusion or drift parameter is detected by this or any other method, the next question to consider is where the change point is located. Therefore, we propose the method to estimate the change point of the parameter for two cases: the case where there is a change in the diffusion parameter, and the case where there is no change in the diffusion parameter but a change in the drift parameter. Furthermore, we present rates of convergence and distributional results of the change point estimators. Some examples and simulation results are also given.

Pure strictly uniform models of non-ergodic measure automorphisms

<p style='text-indent:20px;'>The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model.</p>

Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random models

We prove quantitative bounds on the dependence of the density of states on the potential function for discrete, deterministic Schrödinger operators on infinite graphs. While previous results were limited to random Schrödinger operators with independent, identically distributed potentials, this paper develops a deterministic framework, which is applicable to Schrödinger operators independent of the specific nature of the potential. Following ideas by Bourgain and Klein, we consider the density of states outer measure (DOSoM), which is well defined for all (deterministic) Schrödinger operators. We explicitly quantify the dependence of the DOSoM on the potential by proving a modulus of continuity in the ℓ∞-norm. The specific modulus of continuity so obtained reflects the geometry of the underlying graph at infinity. For the special case of Schrödinger operators on Zd, this implies the Lipschitz continuity of the DOSoM with respect to the potential. For Schrödinger operators on the Bethe lattice, we obtain log-Hölder dependence of the DOSoM on the potential. As an important consequence of our deterministic framework, we obtain a modulus of continuity for the density of states measure (DOSm) of ergodic Schrödinger operators in the underlying potential sampling function. Finally, we recover previous results for random Schrödinger operators on the dependence of the DOSm on the single-site probability measure by formulating this problem in the ergodic framework using the quantile function associated with the random potential.

Efficient Propagation of Epistemic Uncertainty for Probabilistic Seismic Hazard Analyses (PSHAs) Including Partial Correlation of Magnitude–Distance Scaling

ABSTRACT Probabilistic seismic hazard analysis (PSHA) is moving from ergodic ground-motion models (GMMs) to nonergodic GMMs that account for site-source-specific source, path, and site effects and which require a much larger number of GMM branches on the logic tree to capture the full epistemic uncertainty. An efficient method for computing PSHA with a large number of GMM branches was developed by Lacour and Abrahamson (2019) using polynomial chaos (PC) expansion with the key assumption that the epistemic uncertainty in the median ground motion is fully correlated. In the current study, we remove the assumption of full correlation using a multivariate PC expansion. The correlation structure of the available median GMMs across scenarios is computed empirically. The median ground motion is modeled as a Gaussian random process with the correlation structure of the GMMs across the range of relevant earthquake scenarios. This Gaussian random process is discretized using the Karhunen–Loeve expansion, which leads to multivariate PC expansions of uncertain hazard curves. The hazard fractiles can be reconstructed during an efficient postprocessing phase that includes the effects of partial correlation between the GMMs. Multivariate PC expansions require significantly more terms than for the fully correlated case, which increases the calculation time by about a factor of 5, but it is still much more efficient than direct sampling of the branches of the GMM logic for a large number of branches. An example hazard calculation shows that the effect of using partial correlation in place of full correlation of the GMMs is small for the Next Generation Attenuation-West2 (NGA-West2) set of GMMs, indicating that the fully correlated assumption may be adequate for many applications. The multivariate PC method can be used to evaluate the effects of the partial correlation of the GMMs for sets of GMMs that are different from the NGA-West2 GMMs.

Application of empirical and simulation-based site amplification models for Central and Eastern North America to selected sites

The Next Generation Attenuation Relationships for Central & Eastern North-America (NGA-East) Geotechnical Working Group (GWG) has presented models for site amplification in Central and Eastern North America that represent a significant change from past practice, which was based on models developed for active tectonic regions. The GWG models are ergodic in their formulation, meaning that they produce an average level of amplification conditional on VS30 and other the site parameters. We illustrate the application of these models to four sites in Texas, South Carolina, Mississippi, and New York City, and compare results with site-specific ground response analyses. The results indicate that substantial advantage is possible when ergodic models conditioned only on VS30 are supplemented with a modular term that produces a peak at one or more site natural periods ( Tnat). The article demonstrates features and limitations of the GWG models for sites in Central and Eastern North America and provides useful recommendations for coupling ergodic and non-ergodic (site-specific) modeling as part of seismic hazard studies.

Mean equicontinuity, almost automorphy and regularity

The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with (measurable) discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of a minimal topological dynamical system has trivial (one point) fibers. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibers have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.

Varied phenomenology of models displaying dynamical large-deviation singularities.

Singularities of dynamical large-deviation functions are often interpreted as the signal of a dynamical phase transition and the coexistence of distinct dynamical phases, by analogy with the correspondence between singularities of free energies and equilibrium phase behavior. Here we study models of driven random walkers on a lattice. These models display large-deviation singularities in the limit of large lattice size, but the extent to which each model's phenomenology resembles a phase transition depends on the details of the driving. We also compare the behavior of ergodic and nonergodic models that present large-deviation singularities. We argue that dynamical large-deviation singularities indicate the divergence of a model timescale, but not necessarily one associated with cooperative behavior or the existence of distinct phases.

Adaptive efficient analysis for big data ergodic diffusion models

We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete fixed time moments in the case when diffusion coefficients are unknown. To this end on the basis of sequential analysis methods we develop model selection procedures, for which we show non asymptotic sharp oracle inequalities. Through the obtained inequalities we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the model regularity is unknown. For the first time for such problem, we found in the explicit form the celebrated Pinsker constant which provides the sharp lower bound for the minimax squared accuracy normalized with the optimal convergence rate. Then we show that the asymptotic quadratic risk for the model selection procedure asymptotically coincides with the obtained lower bound, i.e this means that the constructed procedure is efficient. Finally, on the basis of the constructed model selection procedures in the framework of the big data models we provide the efficient estimation without using the parameter dimension or any sparse conditions.

The ambiguity of nestedness under soft and hard constraints

Many real networks feature the property of nestedness, i.e. the neighbours of nodes with a few connections are hierarchically nested within the neighbours of nodes with more connections. Despite the abstract simplicity of this notion, various mathematical definitions of nestedness have been proposed, sometimes giving contrasting results. Moreover, there is an ongoing debate on the statistical significance of nestedness, since random networks where the number of connections (degree) of each node is fixed to its empirical value are typically as nested as real ones. By using only ergodic and unbiased null models, we propose a clarification that exploits the recent finding that random networks where the degrees are enforced as hard constraints (microcanonical ensembles) are thermodynamically different from random networks where the degrees are enforced as soft constraints (canonical ensembles). Indeed, alternative definitions of nestedness can be negatively correlated in the microcanonical one, while being positively correlated in the canonical one. This result disentangles distinct notions of nestedness captured by different metrics and highlights the importance of making a principled choice between hard and soft constraints in null models of ecological networks.

Epistemic Uncertainty in Site Response as Derived from One-Dimensional Ground Response Analyses

One-dimensional ground response analyses (GRAs) are often used with an expectation that they provide unbiased estimates of site effects. Under that hypothesis, epistemic uncertainty in site response arises mainly from uncertain soil properties. This approach has dominated practice for projects where site-specific site response is estimated for use in probabilistic seismic hazard analyses. We extend the uncertainty framework to also consider modeling errors (i.e., inability of GRAs to model site response for some sites). We quantify this epistemic uncertainty using vertical array data in which the downhole motion is input to GRAs to predict ground motions at the surface. Residuals (i.e., difference between observed and predicted ground motion intensity measures in natural log units) are partitioned into between- and within-site components. After correcting for overprediction bias near the site period, we quantify epistemic uncertainty using between-site standard deviation, which ranges from 0.2 to 0.35 using California data. A Japan data set analyzed in the literature provides similar results. This dispersion is only modestly smaller than the site-to-site variability from ergodic models for active tectonic regions, which limits the apparent benefits of site-specific GRAs. Dispersion results are not appreciably affected by varying damping models, although a model informed by site-specific observations minimizes bias relative to alternative models based on geotechnical laboratory tests and seismological crustal attenuation studies.

Regional-Scale 3D Ground-Motion Simulations of Mw 7 Earthquakes on the Hayward Fault, Northern California Resolving Frequencies 0–10 Hz and Including Site-Response Corrections

ABSTRACTLarge earthquake ground-motion simulations in 3D Earth models provide constraints on site-specific shaking intensities but have suffered from limited frequency resolution and ignored site response in soft soils. We report new regional-scale 3D simulations for moment magnitude 7.0 scenario earthquakes on the Hayward Fault, northern California with SW4. Simulations resolved significantly broader band frequencies (0–10 Hz) than previous studies and represent the highest resolution simulations for any such earthquake to date. Seismic waves were excited by a kinematic rupture following Graves and Pitarka (2016) and obeyed wave propagation in a 3D Earth model with topography from the U.S. Geological Survey (USGS) assuming a minimum shear wavespeed, VSmin, of 500 m/s. We corrected motions for linear and nonlinear site response for the shear wavespeed, VS, from the USGS 3D model, using a recently developed ground-motion model (GMM) for Fourier amplitude spectra (Bayless and Abrahamson, 2018, 2019a). At soft soil locations subjected to strong shaking, the site-corrected intensities reflect the competing effects of linear amplification by low VS material, reduction of stiffness during nonlinear deformation, and damping of high frequencies. Sites with near-surface VS of 500 m/s or greater require no linear site correction but can experience amplitude reduction due to nonlinear response. Averaged over all sites, we obtained reasonable agreement with empirical ergodic median GMMs currently used for seismic hazard and design ground motions (epsilon less than 1), with marked improvement at soft sedimentary sites. At specific locations, the simulated shaking intensities show systematic differences from the GMMs that reveal path and site effects not captured in these ergodic models. Results suggest how next generation regional-scale earthquake simulations can provide higher spatial and frequency resolution while including effects of soft soils that are commonly ignored in scenario earthquake ground-motion simulations.

Schwartz‐type model selection for ergodic stochastic differential equation models

AbstractWe study theoretical foundation of model comparison for ergodic stochastic differential equation (SDE) models and an extension of the applicable scope of the conventional Bayesian information criterion. Different from previous studies, we suppose that the candidate models are possibly misspecified models, and we consider both Wiener and a pure‐jump Lévy noise‐driven SDE. Based on the asymptotic behavior of the marginal quasi‐log likelihood, the Schwarz‐type statistics and stepwise model selection procedure are proposed. We also prove the model selection consistency of the proposed statistics with respect to an optimal model. We conduct some numerical experiments and they support our theoretical findings.