Spatial Correlation Length Research Articles

Spatial correlations constrain cellular lifespan and pattern formation in corneal epithelium homeostasis.

Homeostasis in adult tissues relies on the replication dynamics of stem cells, their progenitors and the spatial balance between them. This spatial and kinetic coordination is crucial to the successful maintenance of tissue size and its replenishment with new cells. However, our understanding of the role of cellular replicative lifespan and spatial correlation between cells in shaping tissue integrity is still lacking. We developed a mathematical model for the stochastic spatial dynamics that underlie the rejuvenation of corneal epithelium. Our model takes into account different spatial correlations between cell replication and cell removal. We derive the tradeoffs between replicative lifespan, spatial correlation length, and tissue rejuvenation dynamics. We determine the conditions that allow homeostasis and are consistent with biological timescales, pattern formation, and mutants phenotypes. Our results can be extended to any cellular system in which spatial homeostasis is maintained through cell replication.

Cluster and information entropy analysis of acoustic emission during rock failure process

This study provided a new research perspective for processing and analyzing AE data to evaluate rock failure. Cluster method and information entropy theory were introduced to investigate temporal and spatial correlation of acoustic emission (AE) events during the rock failure process. Laboratory experiments of granite subjected to compression were carried out, accompanied by real-time acoustic emission monitoring. The cumulative length and dip angle curves of single links were fitted by different distribution models and distribution functions of link length and directionality were determined. Spatial scale and directionality of AE event distribution, which are characterized by two parameters, i.e., spatial correlation length and spatial correlation directionality, were studied with the normalized applied stress. The entropies of link length and link directionality were also discussed. The results show that the distribution of accumulative link length and directionality obeys Weibull distribution. Spatial correlation length shows an upward trend preceding rock failure, while there are no remarkable upward or downward trends in spatial correlation directionality. There are obvious downward trends in entropies of link length and directionality. This research could enrich mathematical methods for processing AE data and facilitate the early-warning of rock failure-related geological disasters.

Research on the propagation of partially coherent cosh-Gaussian beams through an ABCD optical system in non-Kolmogorov turbulence by effective tensor approach

An efficient tensor approach is used to study the propagation of partially coherent cosh-Gaussian beams through an ABCD optical system in non-Kolmogorov turbulence. Analytical expressions for the average intensity of the beam propagation are derived. The properties of the average intensity are investigated with a numerical example. One finds that the propagation of the beam with larger spatial coherence length is less affected by distance when the propagation distance is long enough, and as the Ch-parameter increases, the beam propagation is less effected by turbulent atmosphere. It is also found that the average intensity distribution of the cosh-Gaussian beams with larger spatial correlation length is more affected by the structure constant of turbulence (i.e., turbulence level). By choosing a suitable Ch-parameter and spatial coherence length, the partially coherent cosh-Gaussian beams can be better transmitted in non-Kolmogorov turbulence. Our results will be useful in free-space communication.

A real-space renormalization-group calculation for the quantum gauge theory on a square lattice

We revisit Fradkin and Raby’s real-space renormalization-group method to study the quantum gauge theory defined on links forming a two-dimensional square lattice. Following an old suggestion of theirs, a systematic perturbation expansion developed by Hirsch and Mazenko is used to improve the algorithm to second order in an intercell coupling, thereby incorporating the effects of discarded higher energy states. A careful derivation of gauge-invariant effective operators is presented in the Hamiltonian formalism. Renormalization group equations are analyzed near the nontrivial fixed point, reaffirming old work by Hirsch on the dual transverse field Ising model. In addition to recovering Hirsch’s previous findings, critical exponents for the scaling of the spatial correlation length and energy gap in the electric free (deconfined) phase are compared. Unfortunately, their agreement is poor. The leading singular behavior of the ground state energy density is examined near the critical point: we compute both a critical exponent and estimate a critical amplitude ratio.

Probabilistic Settlement Analysis of Strip Footing on Spatially Random Soil

The prediction of settlement in a traditional design approach usually uses a deterministic value of modulus of elasticity (E), which is estimated as an average value by testing the soil at selected locations. However, these deterministic properties of soil may not represent the actual properties of soil and site condition. Due to numerous sources of uncertainty, the properties of soil mass are spatially varying and anisotropic in the natural field condition. In this study, a random finite element method (RFEM) is used to evaluate the reliability of settlement of the strip footing on spatially random soil. Modulus of elasticity is the only considered random parameter. For this purpose, 2000 spatially random realizations of E-field are generated using Monte Carlo Simulation. Each of these realizations of heterogeneous soil profile is passed to FEM to analyze the settlement of footing. The final settlement results measured in all these realizations are then statistically evaluated and compared. The results of analysis show that the mean and standard deviation of the footing settlement are increased with increasing spatial correlation length. The large value of isotropic correlation length led to an increase in the mean settlement value by more than 25% as compared with the deterministic settlement calculations. Also, it is concluded that the rate of increase of settlements for anisotropic correlation length is lower than the one under isotropic condition.

Assessment of soil spatial variability for linear infrastructure using cone penetration tests

Soil spatial variability has a significant impact on the reliability of geotechnical structures. In particular, the horizontal variability is important for linear infrastructure, which has only limited vertical height and width, but extensive length. Due to depositional and geological processes, the variability is often substantially different in the vertical and lateral directions. This variability can be characterised by a spatial correlation length, or scale of fluctuation, which is a measure of how significantly soil properties are correlated in space. An analysis of the reliability of such a measure has been undertaken using synthetic data, leading to a design chart which quantifies the statistical uncertainty in the scale of fluctuation for specific site investigation designs, which can be an important input for probabilistic analyses of the structure response. Moreover, practical guidance for site investigation design is proposed which can reduce the statistical uncertainty. The method has been applied to a real site investigation comprising a row of 29 closely spaced cone penetration tests (CPTs), within a larger site investigation of 100 CPTs, and applied to a simple design calculation for a long embankment to illustrate the impact on slope stability assessment. The site investigation data are made available to add to the limited amount of detailed data in this field.

Comparative study on fracture characteristics of coal and rock samples based on acoustic emission technology

Monitoring on the damage evolution process of the coal and rock mass in coal resource mining play a vital role in safe production and avoiding economic losses. This study aims at further improving effectiveness and accuracy of acoustic monitoring method in the process of coal resource mining. For this purpose, a rock mechanics testing system and an acoustic emission (AE) test system are employed to conduct AE event location experiments on the coal and rock samples under uniaxial loading. According to the basic theory of b-value and single-link cluster (SLC) method, changing trend of related parameters with stress is studied, including b-value, spatial correlation length ξ and information entropy H. Then, the reason for variation of these three parameters is revealed. The analysis results show that b-value has not obvious change at the initial stage of loading, but spatial correlation length ξ presents upward trend during this period. Compared with b-value and spatial correlation length ξ, information entropy H is not sensitive to damage state of the coal and rock sample during the most time of loading process. However, it is worth noting that that these three parameters show significant change trend before the buckling failure of the coal and rock samples. Based on three-dimensional crack growth theory and criterion of microcrack density, the research results show that the change of the parameters are the macro characterization of transformation process from small-scale microfracture to large-scale microfracture and from local damage to overall damage for the coal and rock samples subjected to uniaxial loading. This research provides a new method to evaluate the damage state of the coal and rock mass under high stress using AE source location technology, and it has important guiding significance and reference value for ensuring the safety of life and property in the process of coal resource mining.

Stochastic properties of static friction

The onset of frictional motion is mediated by rupture-like slip fronts, which nucleate locally and propagate eventually along the entire interface causing global sliding. The static friction coefficient is a macroscopic measure of the applied force at this particular instant when the frictional interface loses stability. However, experimental studies are known to present important scatter in the measurement of static friction; the origin of which remains unexplained. Here, we study the nucleation of local slip at interfaces with slip-weakening friction of random strength and analyze the resulting variability in the measured global strength. Using numerical simulations that solve the elastodynamic equations, we observe that multiple slip patches nucleate simultaneously, many of which are stable and grow only slowly, but one reaches a critical length and starts propagating dynamically. We show that a theoretical criterion based on a static equilibrium solution predicts quantitatively well the onset of frictional sliding. We develop an efficient Monte-Carlo model by adapting the theoretical criterion and study the variability in global strength distribution caused by the stochastic properties of local frictional strength. The results demonstrate that an increasing spatial correlation length on the interface, representing geometric imperfections and roughness, causes lower global static friction. Conversely, smaller correlation length increases the macroscopic strength while its variability decreases. We further show that randomness in local friction properties is insufficient for the existence of systematic precursory slip events. Random or systematic non-uniformity in the driving force, such as potential energy or stress drop, is required for arrested slip fronts. Our model and observations provide a necessary framework for efficient stochastic analysis of macroscopic frictional strength and establish a fundamental basis for probabilistic design criteria for static friction.

The role of the spatial heterogeneity and correlation length of surface wettability on two-phase flow in a CO2-water-rock system

This study characterized and modeled heterogeneous surface wettability in sandstone and investigated the role of spatial heterogeneity and correlation length of surface wettability on relative permeability in a supercritical CO2 (scCO2)-brine-rock system. Understanding the role of wettability heterogeneity on relative permeability is essential to geological CO2 sequestration, oil and gas recovery, and contaminated groundwater remediation. Although numerous studies have attempted to understand the influences of surface wettability, capillary number (Ca), and viscosity ratio, the role of the spatial variation and correlation length of surface wettability on two-phase flow in three-dimensional (3D) porous media has not been unraveled due to the challenges in the measurement and representation of realistic rock surface wettability. In this work, we conducted in-situ measurements of surface contact angle (CA) in a Bentheimer sandstone after CO2 flooding using micro-computed tomography (micro-CT), and found that the pore-scale CA distribution on rock surfaces followed a log-normal distribution associated with a spatial correlation length. Based on the statistical information from CT scanning, a Gaussian random field was used to model CA distributions that had desired standard deviations and spatial correlation lengths, which were then adjusted within a certain range of values for sensitivity analyses to study their combined effects on the two-phase flow in the porous medium using the lattice Boltzmann (LB) method. The LB two-phase flow simulation was accelerated using hybrid, multicore parallel computing to overcome the challenges in simulating multiphase flow in a large 3D domain having 800 × 800 × 600 nodes. The simulation results showed that the surface wettability heterogeneity (i.e., standard deviation of CA) had a lesser effect on the relative permeability of the wetting fluid (water) but a more significant impact on the relative permeability of the non-wetting fluid (scCO2). The Corey model was used to fit the LB-simulated relative permeability curves of water and scCO2 and showed that the variations in the relative permeability curves for both water and scCO2 increased as the standard deviation and spatial correlation length of CA increased. This study illustrated that the assumption of homogeneous surface wettability may cause errors in multiphase flow simulations. The impacts of both the standard deviation and spatial correlation length of CAs should be accounted for. This is the first study that explored the spatial correlation lengths associated with CA distributions on sandstone surfaces and comprehensively investigated the roles of both spatial variation and correlation length of CA on two-phase flow properties in 3D porous media. The optimized LB multiphase flow model was proved a powerful tool to study the interplays and combined effects of these statistical parameters, which had critical applications in numerous natural and engineering processes that involved multiphase flow in porous media.

An Analytical Random Field Solution for the Reliability of Axially Loaded Piles in the Ultimate Limit State Considering the Effect of Soil Sampling

The present work is concerned with the effect of soil spatial variability on estimating the ultimate soil resistance of floating axially loaded piles from point measurements of soil properties along the pile. The ultimate limit state is considered. In particular, closed form formulae for (i) determining the optimal sampling depth for minimizing statistical uncertainty and (ii) the optimal—minimum required—safety factor for a desired failure probability level are derived. A dimensionless parameter, the cohesion-to-friction parameter Λ, is introduced which quantifies the weight of soil’s cohesion contribution relative to that of soil’s friction in the linear trend of the ultimate soil strength. The analysis shows that the probability of failure profile with the sampling depth attains a minimum, designating the optimal sampling point. This depends on the scaled spatial correlation length of the soil Θ (i.e., the spatial correlation length of soil over the length of the pile) and the parameter Λ, but not on the coefficient of variance of the ultimate soil strength (covu) or the safety factor. Furthermore, it was found that the optimal depth is always at the lower half of the pile, approaching the mid-point or the bottom end of the pile for Λ>>1 or Λ<<1, respectively. In addition, it was found that for large Θ, the optimal depth tends to be closer to the mid-point of the pile, while for small Θ, the optimal sampling depth arises closer to the bottom end. The practical usefulness of the results is related to a suitable choice of the safety factor. Inverting the probability of failure formula at its minimum value, an optimal safety factor is obtained as a function of the desired (acceptable) probability of failure, and the parameters Θ, Λ and covu. The optimal safety factor is the minimum value required for a desired level of the probability of failure.

Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements.

The aim of this work is to efficiently and robustly solve the statistical inverse problem related to the identification of the elastic properties at both macroscopic and mesoscopic scales of heterogeneous anisotropic materials with a complex microstructure that usually cannot be properly described in terms of their mechanical constituents at microscale. Within the context of linear elasticity theory, the apparent elasticity tensor field at a given mesoscale is modeled by a prior non-Gaussian tensor-valued random field. A general methodology using multiscale displacement field measurements simultaneously made at both macroscale and mesoscale has been recently proposed for the identification the hyperparameters of such a prior stochastic model by solving a multiscale statistical inverse problem using a stochastic computational model and some information from displacement fields at both macroscale and mesoscale. This paper contributes to the improvement of the computational efficiency, accuracy and robustness of such a method by introducing (i) a mesoscopic numerical indicator related to the spatial correlation length(s) of kinematic fields, allowing the time-consuming global optimization algorithm (genetic algorithm) used in a previous work to be replaced with a more efficient algorithm and (ii) an ad hoc stochastic representation of the hyperparameters involved in the prior stochastic model in order to enhance both the robustness and the precision of the statistical inverse identification method. Finally, the proposed improved method is first validated on in silico materials within the framework of 2D plane stress and 3D linear elasticity (using multiscale simulated data obtained through numerical computations) and then exemplified on a real heterogeneous biological material (beef cortical bone) within the framework of 2D plane stress linear elasticity (using multiscale experimental data obtained through mechanical testing monitored by digital image correlation).

Spatial coherence of the thermal emission of a sphere

Analytical expressions for calculating the energy density and spatial correlation function of thermal emission by a homogeneous, isothermal sphere of arbitrary size and material are presented. The spectral distribution and the power law governing the distance-dependent energy density are investigated in the near-field and far-field regimes for silicon carbide, silicon and tungsten spheres of various size parameters ranging from X = 0.002 to 5. The spatial coherence of thermal field emitted by spheres is also studied in both radial and polar directions. The energy density follows a power law of d^-2 (d is the observation distance) in the far field for all sizes and materials. The power law in the near field is strongly dependent on the material, size parameter, and the ratio d/a (a is the sphere radius). In the near field, the energy density follows a power law of d^-6 when X<<1 and d/a>>1 (similar to an electric point dipole). With increasing X or decreasing d/a, the contribution of multipoles to the energy density increases resulting in an increase in the power of d until the power law converges to that for a semi-infinite medium. The spatial correlation length in the radial direction is in the orders of $\lambda$, 0.1$\lambda$, and 0.001$\lambda$ in the far field, intermediate near field, and extreme near field, respectively. The correlation angle in the extreme near field is strongly dependent on the sphere size parameter, such that it decreases by three orders of magnitude (from 0.5$\pi$ to 0.001$\pi$) when X increases from 0.002 to 5. In the intermediate near field and far field, the correlation angle retains the same order of magnitude (0.15$\pi$ - 0.7$\pi$) for all considered Xs. While the excitation of dipolar localized surface phonons (LSPhs) does not affect the correlation length and angle, the multipolar LSPhs reduce the spatial coherence in both directions.

Computational de-noising based on deep learning for phase data in digital holographic interferometry

This paper presents a deep-learning-based algorithm dedicated to the processing of speckle noise in phase measurements in digital holographic interferometry. The deep learning architecture is trained with phase fringe patterns including faithful speckle noise, having non-Gaussian statistics and non-stationary property, and exhibiting spatial correlation length. The performances of the speckle de-noiser are estimated with metrics, and the proposed approach exhibits state-of-the-art results. In order to train the network to de-noise phase fringe patterns, a database is constituted with a set of noise-free and speckled phase data. The algorithm is applied to de-noising experimental data from wide-field digital holographic vibrometry. Comparison with the state-of-the-art algorithm confirms the achieved performance.

Diffusion of Anisotropic Particles in Random Energy Landscapes—An Experimental Study

If a colloidal particle is exposed to an external field, its Brownian motion is modified. In the case of an anisotropic particle, the external potential might not only affect its translation but also its rotation. We experimentally investigate the dynamics of a trimer, which consists of three spherical particles, within a random potential energy landscape. This energy landscape has energy values drawn from a Gamma distribution, a spatial correlation length similar to the particle size and is realised by a random light field, that is a laser speckle pattern. The particle translation and rotation are quantified by the mean squared (angular) displacement, the van Hove function and other observable quantities. The translation shows an intermediate subdiffusive regime and a long-time diffusion that slows down upon increasing the modulation of the potential. In contrast, the mean squared angular displacement exhibits only small deviations from a linear time dependence but a more detailed analysis reveals discrete angular jumps reflecting the symmetry of the trimer. A coupling between the translation and rotation is observed and found to depend on the length scale.

3D Atomic Mapping of Interfacial Roughness and Its Spatial Correlation Length in Sub-10 nm Superlattices.

The interfacial abruptness and uniformity in heterostructures are critical to control their electronic and optical properties. With this perspective, this work demonstrates the three-dimensional (3D) atomic-level mapping of the roughness and uniformity of buried epitaxial interfaces in Si/SiGe superlattices with a layer thickness in the 1.5-7.5 nm range. Herein, 3D atom-by-atom maps were acquired and processed to generate isoconcentration surfaces highlighting local fluctuations in content at each interface. These generated surfaces were subsequently utilized to map the interfacial roughness and its spatial correlation length. The analysis revealed that the root-mean-squared roughness of the buried interfaces in the investigated superlattices is sensitive to the growth temperature with a value varying from 0.17 ± 0.02 to 0.26 ± 0.03 nm in the temperature range of 500-650 °C. The estimated horizontal correlation lengths were found to be 8.11 ± 0.5 nm at 650 °C and 10.09 ± 0.6 nm at 500 °C. Additionally, reducing the growth temperature was found to improve the interfacial abruptness, with a 30% smaller interfacial width is obtained at 500 °C. This behavior is attributed to the thermally activated atomic exchange at the surface during the heteroepitaxy. Finally, by testing different optical models with increasing levels of interfacial complexity, it is demonstrated that the observed atomic-level roughening at the interface must be accounted for to accurately describe the optical response of Si/SiGe heterostructures.

Effect of spatial variability of block-type cement-treated ground on the bearing capacity of foundation under inclined load

Deep mixing methods are widely used for stabilizing soft clayey soils and improving their bearing capacity. However, spatial variability in the shear strength of the cement-treated ground introduces uncertainties in estimating the bearing capacity for design. This paper evaluates the reliability of, block-type, cement-treated foundation under inclined load conditions using random field numerical limit analyses. The undrained shear strength is modelled as a random field which is characterized by a log-normal distribution and a spatial correlation length. Monte Carlo simulations are then used to interpret the stochastic bearing capacity factor and failure mechanisms for inclined concentric loading conditions at selected ratios of the shear strength ratio of cement-treated ground to original clay, the coefficient of variation in undrained shear strength and correlation length of the cement-treated zone. Variability of the undrained shear strength can reduce the expected bearing capacity of the cement-treated ground by 50–70% compared to homogeneously mixed clay.

Temporal and Spatial Scales of Correlation in Marine Phytoplankton Communities

AbstractOcean circulation shapes marine phytoplankton communities by setting environmental conditions and dispersing organisms. In addition, processes acting on the water column (e.g., heat fluxes and mixing) affect the community structure by modulating environmental variables that determine in situ growth and loss rates. Understanding the scales over which phytoplankton communities vary in time and space is key to elucidate the relative contributions of local processes and ocean circulation on phytoplankton distributions. Using a global ocean ecosystem model, we quantify temporal and spatial correlation scales for phytoplankton phenotypes with diverse functional traits and cell sizes. Through this analysis, we address these questions: (1) Over what timescales do perturbations in phytoplankton populations persist? and (2) over what distances are variations in phytoplankton populations synchronous? We find that correlation timescales are short in regions of strong currents, such as the Gulf Stream and Antarctic Circumpolar Current. Conversely, in the subtropical gyres, phytoplankton population anomalies persist for relatively long periods. Spatial correlation length scales are elongated near ocean fronts and narrow boundary currents, reflecting flow paths and frontal patterns. In contrast, we find nearly isotropic spatial correlation fields where current speeds are small, or where mixing acts roughly equally in all directions. Phytoplankton timescales and length scales also vary coherently with phytoplankton body size. In addition to aiding understanding of phytoplankton population dynamics, our results provide global insights to guide the design of biological ocean observing networks and to better interpret data collected at long‐term monitoring stations.

The SAPBC Method on local, non-Cluster updates algorithms of Monte Carlo simulation: A study on more convergence of spin correlation at critical temperature.

In this method, we propose computational technique of Screw-Antisymmetric Periodic Boundary Condition (SAPBC Method) on local, non-cluster update algorithms of Isotropic square lattice Ising model of Monte Carlo simulations, as well. The SAPBC Method, actually, is an extended mixed method of Screw (helical) and Antisymmetric periodic boundary conditions beyond connection from of nearest neighbor spin of the main lattice to even far away block of the outer (foreign) neighbor spin arrays. Here, meanwhile of description of geometry exact details of method and way of spin interaction, have applied to critical slowing down in order to achieve more convergence of spin correlation at critical temperature. Actually, in general, at critical temperature algorithms performed by using SAPBC Method have faster correlation and much shorter autocorrelation time than algorithms performed by using PBC Method. We will also see that autocorrelation function for the typewriter Metropolis algorithm was found to be zero at high temperatures. For low temperatures it fell to zero and stayed there. The SAPBC Method also confirms and consists with the law of the spatial correlation length with its dynamical critical exponent. Therefore, it can be used as a trenchant method applied to boundary conditions of Monte Carlo simulation problems extending on a variety of other models such as XY-Pots-Heisenberg model and also cluster algorithms such as Wolf, Swendsen-Wangas, Hoshen-Koppelman as well.

Distributed Environmental Monitoring With Finite Element Robots

We introduce a distributed finite element algorithm that allows swarms of mobile robots to persistently monitor environmental quantities such as temperature or salinity. The robots deploy themselves into the environment, covering the domain and dividing it into nonoverlapping regions. Each robot estimates the environment over its own region using local measurements and communication with nearby robots. The algorithm ensures that each robot's estimate constitutes a piece of a global estimate that spans the entire domain, fuses the whole swarm's measurements, and accounts for the spatial correlation between measurement and estimation locations. By incorporating spatial correlation without requiring the transmission of measurements or measurement locations, the algorithm decouples its communication requirements from the spatial statistics of the environment and enables robots with fixed capabilities to monitor environments with different spatial correlation lengths. Analysis and simulation demonstrate that, as the number of robots increases, the memory and communication requirements of each individual robot decrease until reaching a minimum, after which the resolution of the environmental model increases. Additional robots, therefore, add computational resources to the swarm rather than introducing extra computational burdens.

A radar reflectivity operator with ice-phase hydrometeors for variational data assimilation (version 1.0) and its evaluation with real radar data

Abstract. A reflectivity forward operator and its associated tangent linear and adjoint operators (together named RadarVar) were developed for variational data assimilation (DA). RadarVar can analyze both rainwater and ice-phase species (snow and graupel) by directly assimilating radar reflectivity observations. The results of three-dimensional variational (3D-Var) DA experiments with a 3 km grid mesh setting of the Weather Research and Forecasting (WRF) model showed that RadarVar was effective at producing an analysis of reflectivity pattern and intensity similar to the observed data. Two to three outer loops with 50–100 iterations in each loop were needed to obtain a converged 3-D analysis of reflectivity, rainwater, snow, and graupel, including the melting layers with mixed-phase hydrometeors. It is shown that the deficiencies in the analysis using this operator, caused by the poor quality of the background fields and the use of the static background error covariance, can be partially resolved by using radar-retrieved hydrometeors in a preprocessing step and tuning the spatial correlation length scales of the background errors. The direct radar reflectivity assimilation using RadarVar also improved the short-term (2–5 h) precipitation forecasts compared to those of the experiment without DA.