Critical Sensitivity and Trans-scale Fluctuations in Catastrophic Rupture

-- Rupture in the heterogeneous crust appears to be a catastrophe transition. Catastrophic rupture sensitively depends on the details of heterogeneity and stress transfer on multiple scales. These are difficult to identify and deal with. As a result, the threshold of earthquake-like rupture presents uncertainty. This may be the root of the difficulty of earthquake prediction. Based on a coupled pattern mapping model, we represent critical sensitivity and trans-scale fluctuations associated with catastrophic rupture. Critical sensitivity means that a system may become significantly sensitive near catastrophe transition. Trans-scale fluctuations mean that the level of stress fluctuations increases strongly and the spatial scale of stress and damage fluctuations evolves from the mesoscopic heterogeneity scale to the macroscopic scale as the catastrophe regime is approached. The underlying mechanism behind critical sensitivity and trans-scale fluctuations is the coupling effect between heterogeneity and dynamical nonlinearity. Such features may provide clues for prediction of catastrophic rupture, like material failure and great earthquakes. Critical sensitivity may be the physical mechanism underlying a promising earthquake forecasting method, the load-unload response ratio (LURR).     

Damage Localization,Sensitivity of Energy Release and the Catastrophe Transition

-- Large earthquakes can be viewed as catastrophic ruptures in the earth's crust. There are two common features prior to the catastrophe transition in heterogeneous media. One is damage localization and the other is critical sensitivity; both of which are related to a cascade of damage coalescence. In this paper, in an attempt to reveal the physics underlying the catastrophe transition, analytic analysis based on mean-field approximation of a heterogeneous medium as well as numerical simulations using a network model are presented. Both the emergence of damage localization and the sensitivity of energy release are examined to explore the inherent statistical precursors prior to the eventual catastrophic rupture. Emergence of damage localization, as predicted by the mean-field analysis, is consistent with observations of the evolution of damage patterns. It is confirmed that precursors can be extracted from the time-series of energy release according to its sensitivity to increasing crustal stress. As a major result, present research indicates that the catastrophe transition and the critical point hypothesis (CPH) of earthquakes are interrelated. The results suggest there may be two cross-checking precursors of large earthquakes: damage localization and critical sensitivity.     

Experimental Evidence of Critical Sensitivity in Catastrophe

The paper presents an experimental study on critical sensitivity in rocks. Critical sensitivity means that the response of a system to external controlling variable may become significantly sensitive as the system approaches its catastrophic rupture point. It is found that the sensitivities measured by responses on three scales (sample scale, locally macroscopic scales and mesoscopic scale) display increase prior to catastrophic transition point. These experimental results do support the concept that critical sensitivity might be a common precursory feature of catastrophe. Furthermore, our previous theoretical model is extended to explore the fluctuations in critical sensitivity in the rock tests.     

强非线性杜芬系统随机振动的等效化研究

以相对穿阈率为评估指标，探讨了强非线性杜芬系统稳态随机振动的等效化问题。结果表明，在不同位移阈值条件下，随着系统振动非线性的增强，立方恢复力等效模型预测精度逐步提高。该等效模型对强非线性杜芬系统具有良好的适用性。     

Catastrophic Rupture Induced Damage Coalescence in Heterogeneous Brittle Media

In heterogeneous brittle media, the evolution of damage is strongly influenced by the multiscale coupling effect. To better understand this effect, we perform a detailed investigation of the damage evolution, with particular attention focused on the catastrophe transition. We use an adaptive multiscale finite-element model (MFEM) to simulate the damage evolution and the catastrophic failure of heterogeneous brittle media. Both plane stress and plane strain cases are investigated for a heterogeneous medium whose initial shear strength follows the Weibull distribution. Damage is induced through the application of the Coulomb failure criterion to each element, and the element mesh is refined where the failure criterion is met. We found that as damage accumulates, there is a stronger and stronger nonlinear increase in stress and the stress redistribution distance. The coupling of the dynamic stress redistribution and the heterogeneity at different scales result in an inverse cascade of damage cluster size, which represents rapid coalescence of damage at the catastrophe transition.     

The aftershock dynamics of the Sumatra-Andaman earthquake

The aftershocks of the catastrophic Sumatra-Andaman earthquake of December 26, 2004 (M = 9.0) are analyzed in the general context of the theory of critical phenomena. The analysis relies on the idea that, according to this theory, critical transitions have two key properties. The first is that the intensity of the fluctuations in a dynamical system monotonically increases with the approach of the bifurcation point, so that at a certain time instant, a sufficiently strong internal pulse initiates the catastrophe. This transition can be treated as spontaneous. The second property is that the reactance of the dynamical system drastically increases on the approach of the bifurcation. Even a weak external perturbation in the near-threshold interval can result in a catastrophe. In this case, it is reasonable to refer to the critical transition as an induced transition. The aftershocks of the Sumatra-Andaman earthquake are likely to demonstrate the typical features of induced seismicity. First, the strongest aftershock (M = 7.2) occurred 3 h 20 min after the main shock. It could have probably been induced by the round-trip seismic echo. Second, it was found that the spectral density of the aftershock sequence significantly increases at about ～0.3 mHz, which is close to the frequency of the spheroidal mode ₀S₂. This suggests that the spheroidal oscillations of the Earth, which are excited by the main seismic shock, modulate the aftershock activity. Both hypotheses are supported by the analysis of the aftershocks of the Tohoku earthquake of March 11, 2011 (M = 9.0).     

Study of discontinuities in hydrological data using catastrophe theory

Abstract

Modelling and prediction of hydrological processes (e.g. rainfall–runoff) can be influenced by discontinuities in observed data, and one particular case may arise when the time scale (i.e. resolution) is coarse (e.g. monthly). This study investigates the application of catastrophe theory to examine its suitability to identify possible discontinuities in the rainfall–runoff process. A stochastic cusp catastrophe model is used to study possible discontinuities in the monthly rainfall–runoff process at the Aji River basin in Azerbaijan, Iran. Monthly-averaged rainfall and flow data observed over a period of 20 years (1981–2000) are analysed using the Cuspfit program. In this model, rainfall serves as a control variable and runoff as a behavioural variable. The performance of this model is evaluated using four measures: correlation coefficient, log-likelihood, Akaike information criterion (AIC) and Bayesian information criterion (BIC). The results indicate the presence of discontinuities in the rainfall–runoff process, with a significant sudden jump in flow (cusp signal) when rainfall reaches a threshold value. The performance of the model is also found to be better than that of linear and logistic models. The present results, though preliminary, are promising in the sense that catastrophe theory can play a possible role in the study of hydrological systems and processes, especially when the data are noisy.

Citation Ghorbani, M. A., Khatibi, R., Sivakumar, B. & Cobb, L. (2010) Study of discontinuities in hydrological data using catastrophe theory. Hydrol. Sci. J. 55(7), 1137–1151.     

利用逾渗模型研究辽宁地区的地震活动

首先,我们利用现代非线性物理理论及震源理论建立了适合地震活动的逾渗模型。然后,我们利用这一模型,对辽宁地区１９７０年以来的地震活动进行了研究。通过对地震活动的逾渗相变过程中逾渗阈值附近的各种临界指数的计算,对破裂概率达到或超过逾渗阈值的子区域进行了讨论并对近期辽宁地区的地震活动趋势进行了探讨。     

Conditional nonlinear optimal perturbation: Applications to stability,sensitivity, and predictability

Conditional nonlinear optimal perturbation (CNOP) is a nonlinear generalization of linear singular vector (LSV) and features the largest nonlinear evolution at prediction time for the initial perturbations in a given constraint. It was proposed initially for predicting the limitation of predictability of weather or climate. Then CNOP has been applied to the studies of the problems related to predictability for weather and climate. In this paper, we focus on reviewing the recent advances of CNOP’s applications, which involves the ones of CNOP in problems of ENSO amplitude asymmetry, block onset, and the sensitivity analysis of ecosystem and ocean’s circulations, etc. Especially, CNOP has been primarily used to construct the initial perturbation fields of ensemble forecasting, and to determine the sensitive area of target observation for precipitations. These works extend CNOP’s applications to investigating the nonlinear dynamical behaviors of atmospheric or oceanic systems, even a coupled system, and studying the problem of the transition between the equilibrium states. These contributions not only attack the particular physical problems, but also show the superiority of CNOP to LSV in revealing the effect of nonlinear physical processes. Consequently, CNOP represents the optimal precursors for a weather or climate event; in predictability studies, CNOP stands for the initial error that has the largest negative effect on prediction; and in sensitivity analysis, CNOP is the most unstable (sensitive) mode. In multi-equilibrium state regime, CNOP is the initial perturbation that induces the transition between equilibriums most probably. Furthermore, CNOP has been used to construct ensemble perturbation fields in ensemble forecast studies and to identify sensitive area of target observation. CNOP theory has become more and more substantial. It is expected that CNOP also serves to improve the predictability of the realistic predictions for weather and climate events plays an increasingly important role in exploring the nonlinear dynamics of atmospheric, oceanic and coupled atmosphere-ocean system. Supported by National Basic Research Program of China (Grant Nos. 2006CB403606, 2007CB411800), National Natural Science Foundation of China (Grant Nos. 40830955, 40675030, 40505013), Institute of Atmospheric Physics, Chinese Academy of Sciences (Grant No. IAP07202), and LASG State Key Laboratory Special Fund     

A test of terminal Mesozoic “catastrophe”

Terminal Mesozoic “catastrophe”-type extinction models that advocate synchronous marine and terrestrial extinctions spanning short time intervals (a few days up to a few millennia) have a common foundation: the simultaneous terminations of geological ranges of some taxa of marine CaCO₃-producing microplankton (and possibly the dinosaurs) at the end of the Cretaceous. Gartner and McGuirk [1] propose a new catastrophe theory that at the end of the Cretaceous fresh-brackish water from the Arctic Ocean spread over the surface of the world's oceans, causing global cooling, aridity, and the extinctions. Like other catastrophe models, this one also fails to address the possibility of hiatus control of ranges at the end of the Cretaceous; a well documented, seemingly nearly universal hiatus of variable and unknown duration separates Cretaceous and Tertiary strata. Documented terminal Cretaceous marine regression (perhaps 10 times more rapid than a typical regression according to Cooper [8] would have caused terrestrial erosion and stripping away of the latest Cretaceous stratigraphic record, thus truncating geological ranges along a seemingly planar datum. The terminal Cretaceous marine CaCO₃ dissolution event would have had the same effect on ranges of marine planktonic CaCO₃-producing microplankton (the event was a shallow-water phenomenon). The simultaneous terminations of geological ranges is thus possibly the result of hiatus control, and the terminal Cretaceous “catastrophe” an illusion. Attempts to use Cretaceous-Tertiary transition floras to support global cooling at the time of the extinctions are not based on sound stratigraphic foundations; realistic paleobotanical-climatic inferences can only be based on the precise correlation of the Cretaceous-Tertiary contact in marine and terrestrial stratigraphic sections, and these correlations have not been made with sufficient precision to support catastrophe theory. The much used “across the Cretaceous-Tertiary boundary” glosses over ignorance of the true terminal Cretaceous scenario, lost forever in most places by the destruction of the terminal Cretaceous stratigraphic record. For now, stable isotope paleotemperature data from marine strata that can be dated radiometrically provide the most reliable estimates of the Cretaceous-Tertiary transition climate; Boersma et al. [5] indicate global warming of deep and shallow oceans “across” the contact (and not surficial cooling only as is required by the spillover model). Older much-cited climate inferences based on leaf physiognomy are suspect in light of Dolph and Dilcher's [23] work that shows little correlation between leaf physiognomy and climate.     

Comparison of automatic procedures for selecting flood peaks over threshold based on goodness‐of‐fit tests

In comparison with the traditional analysis of annual maximums, the peaks over threshold method provides many advantages when performing flood frequency analysis and trend analysis. However, the choice of the threshold remains an important question without definite answers and common visual diagnostic tools are difficult to reproduce on a large scale. This study investigates the behaviour of some automatic methods for threshold selection based on the generalized Pareto model for flood peak exceedances of the threshold and the Anderson–Darling test for fitting this model. In particular, the choice of a critical significance level to define an interval of acceptable values is addressed. First, automatic methods are investigated using a simulation study to assess fitting and prediction performance in a controlled environment. It is shown that P values approximated by an existing table of critical values can speed up computation without affecting the quality of the outcomes. Second, a case study compares automatically and manually selected thresholds for 285 sites across Canada by flood regime and super regions based on site characteristics. Correspondences are examined in terms of prediction of flood quantiles and trend analysis. Results show that trend detection is sensitive to the threshold selection method when studying the evolution of the number of peaks per year. Finally, a hybrid method is developed to combine automatic methods and is calibrated on the basis of super regions. The outcomes of the hybrid method are shown to more closely reproduce the estimates of the manually selected thresholds while reducing the model uncertainty.     

Bistable plant–soil dynamics and biogenic controls on the soil production function

Soil formation results from opposite processes of bedrock weathering and erosion, whose balance may be altered by natural events and human activities, resulting in reduced soil depth and function. The impacts of vegetation on soil production and erosion and the feedbacks between soil formation and vegetation growth are only beginning to be explored quantitatively. Since plants require suitable soil environments, disturbed soil states may support less vegetation, leading to a downward spiral of increased erosion and decline in ecosystem function. We explore these feedbacks with a minimal model of the soil–plant system described by two coupled nonlinear differential equations, which include key feedbacks, such as plant‐driven soil production and erosion inhibition. We show that sufficiently strong positive plant–soil feedback can lead to a ‘humped’ soil production function, a necessary condition for soil depth bistability when erosion is assumed to vary monotonically with vegetation biomass. In bistable plant–soil systems, the sustainable soil condition engineered by plants is only accessible above a threshold vegetation biomass and occurs in environments where the high potential rate of erosion exerts a strong control on soil production and erosion. Vegetation removal for agriculture reduces the stabilizing effect of vegetation and lowers the system resilience, thereby increasing the likelihood of transition to a degraded soil state. Copyright © 2016 John Wiley & Sons, Ltd.     

Coupled Process Models of Fluid Flow and Heat Transfer in Hydrothermal Systems in Three Dimensions

Geothermal fields and hydrothermal mineral deposits are manifestations of the interaction between heat transfer and fluid flow in the Earth’s crust. Understanding the factors that drive fluid flow is essential for managing geothermal energy production and for understanding the genesis of hydrothermal mineral systems. We provide an overview of fluid flow drivers with a focus on flow driven by heat and hydraulic head. We show how numerical simulations can be used to compare the effect of different flow drivers on hydrothermal mineralisation. We explore the concepts of laminar flow in porous media (Darcy’s law) and the non-dimensional Rayleigh number (Ra) for free thermal convection in the context of fluid flow in hydrothermal systems in three dimensions. We compare models of free thermal convection to hydraulic head driven flow in relation to hydrothermal copper mineralisation at Mount Isa, Australia. Free thermal convection occurs if the permeability of the fault system results in Ra above the critical threshold, whereas a vertical head gradient results in an upward flow field.     

Model structure exploration for index flood regionalization

Although the nonlinear power form model structure is widely accepted by practitioners in the flood regionalization modelling, there is a lack of studies on whether there is a room for further improvement, and if the answer is yes, what should be done to explore alternative model structures. A framework is proposed in this study towards investigating this issue by the following steps: (i) a universal data‐driven model is utilized to see if there is a room for improvement compared with the conventional model, and (ii) if improvement is achieved, this means that there should exist more effective model structures than the current form. However, because the universal data‐driven models are usually opaque, more explicit model structures should be explored, which are convenient for practical usage. In this study, the proposed framework is applied in a case study using the catchment characteristics from the Flood Estimation Handbook in conjunction with the gamma test, support vector machine (SVM) and genetic programming (GP). First, the gamma test is used for the purpose of input variables selection where no model structure needs to be defined as a priori, and therefore, the result can be applied to any model structures for model building. Second, an SVM, which is a powerful data‐driven nonlinear model capable of modelling a variety of nonlinear systems, is applied to the index flood model for the first time. Once the best model is determined using those two data‐driven tools, GP is employed to find an alternative model structure. As the SVM is not formulated for producing explicit model functional form, the GP offers an advantage at this point where it can infer an explicit mathematical model functional form. Copyright © 2012 John Wiley & Sons, Ltd.     

Global sensitivity analysis of hydrologic processes in major snow‐dominated mountainous river basins in Colorado

The performance of watershed models in simulating stream discharge depends on the adequate representation of important watershed processes. In snow‐dominated systems, snow, surface and subsurface hydrologic processes comprise a complex network of nonlinear interactions that influence the magnitude and timing of discharge. This study aims to identify critical processes and interactions that control discharge hydrographs in five major mountainous snow‐dominated river basins in Colorado, USA. A comprehensive watershed model (Soil and Water Assessment Tool) and a variance‐based global sensitivity analysis technique (Fourier Amplitude Sensitivity Test) were used in conjunction to identify critical models parameters and processes that they represent. Average monthly streamflow and streamflow root mean square error over a period of 20 years were used as two separate objective functions in this analysis. Examination of the sensitivity of monthly streamflow revealed the influence of parameters on flow volume, whereas the sensitivity of streamflow root mean square error also exposed the influence of parameters on the timing of the hydrographs. A stability analysis was performed to investigate the computational requirements for a robust sensitivity analysis. Results show that streamflow volume is mostly influenced by shallow subsurface processes, whereas interactions between groundwater and snow processes were the key in the timing of streamflows. A large majority of important parameters were common among all study watersheds, which underlies the prospect for regionalization of process‐based hydrologic modelling in headwater river basins in Colorado. Copyright © 2013 John Wiley & Sons, Ltd.     

Diversity of threshold phenomena in geophysical media

The sample analysis of threshold phenomena in the lithosphere, atmosphere, and magnetosphere is conducted. The phenomena due to the flow of electric current and pore fluid in the rocks are considered, the scenario of wind-driven generation of atmospheric electricity is suggested, and the model of the geomagnetic storm time Dst variation is analyzed. An important general conclusion consists in the fact that in the geophysical media there is a wide class of threshold phenomena that are affine with phase transitions of the second kind. These phenomena are also related to the critical transitions in self-oscillatory systems with soft self-excitation. The integral representation of bifurcation diagrams for threshold phenomena is suggested. This provides a simple way to take into account the influence of the fluctuations on the transition of a system through the threshold. Fluctuations remove singularity at the threshold point and, generally, lead to a certain shifting of the threshold. The question concerning the hard transition through the threshold and several aspects of modeling the blow-up instability which is presumed to occasionally develop in the geophysical media are discussed.     

Study on the Seismogeny and Occurrence of Earthquakes by Using Pattern Dynamics

The mesoscopic damage dynamics and damage evolution have been applied to probe theevolutional process induced catastrophe in the earthquake activity.It is a new method.In thispaper,a brief introduction of the basic principle about damage dynamics and evolution ismade.At the same time,using the theory of the pattern dynamics we studied all earthquakes(M_L≥5.0)which occurred along Zhangjiakou-Bohai Sea earthquake belt in the capitalregion.The result indicates that the preparation and happening of the real earthquakes andthe theory are consistent.There are two kinds of evolutional model according to the finalstate,namely,global stability model(GS)and evolution induced catastrophe model(EIC).The two models haven’t evident boundary.The transitional zone exhibits the indeterminacyof the seismic process and the effect of the random variation.This research proposed newapproaches for earthquake prediction.     

Statistical determination of significant curved I-girder bridge seismic response parameters

Curved steel bridges are commonly used at interchanges in transportation networks and more of these structures continue to be designed and built in the United States. Though the use of these bridges continues to increase in locations that experience high seismicity, the effects of curvature and other parameters on their seismic behaviors have been neglected in current risk assessment tools. These tools can evaluate the seismic vulnerability of a transportation network using fragility curves. One critical component of fragility curve development for curved steel bridges is the completion of sensitivity analyses that help identify influential parameters related to their seismic response. In this study, an accessible inventory of existing curved steel girder bridges located primarily in the Mid-Atlantic United States (MAUS) was used to establish statistical characteristics used as inputs for a seismic sensitivity study. Critical seismic response quantities were captured using 3D nonlinear finite element models. Influential parameters from these quantities were identified using statistical tools that incorporate experimental Plackett-Burman Design (PBD), which included Pareto optimal plots and prediction profiler techniques. The findings revealed that the potential variation in the influential parameters included number of spans, radius of curvature, maximum span length, girder spacing, and cross-frame spacing. These parameters showed varying levels of influence on the critical bridge response.     

自组织临界现象在地震空间演化特征上的应用研究

将自组织临界现象理论在地震预报领域作深入的应用研究,根据各量的物理含义重新讨论了在地震科学中的应用,并用重整化群方法对强震前中小地震强现象的临界点问题进行研究。