Velocity Spectra Research Articles

Experimental Study on Vibration Velocity of Piled Raft Supported Embankment and Foundation for Ballastless High Speed Railway

In recent years, the high development of high-speed railway lines cross through areas with poor geological conditions, such as soft soil, offshore and low-lying marsh areas, resulting geotechnical problems, such as large settlements and reduction of bearing capacity. As a new soil reinforcement method in high speed railway lines, the piled raft structure has been used to improve soil conditions and control excess settlement. In order to study the dynamic behavior of piled raft supported ballastless track system in soft soil, an experimental study on vibration velocities of piled raft supported embankment and foundations is presented in soft soil with different underground water levels. Vibration velocities at specified positions of the piled raft supported embankment and foundations are obtained and discussed. The vibration velocity curves on various testing locations of piled raft foundations are clearly visible and have sharp impulse and relaxation pattern, corresponding to loading from train wheels, bogies, and passages. Vibration velocity distribution in the horizontal direction at three train speeds clearly follows an exponential curves. Most of the power spectrums of vibration velocity at various locations are mainly concentrated at harmonic frequencies. The change in water level has slight impaction on the peak spectrum of vibration velocity at harmonic frequencies. The vibration power induced by train loads are transmitted, absorbed, and weakened to a certain extent through embankment and piled raft structure. The dynamic response character of embankments are affected by their self-vibration characteristics and the dynamic bearing capacity of the piled raft structure.

Using cloud radar to investigate the effect of rainfall on migratory insect flight

Abstract The fate of migrating insects that encounter rainfall in flight is a critical consideration when modelling insect movement, but few field observations of this common phenomenon have ever been collected due to the logistical challenges of witnessing these encounters. Operational cloud radars have been deployed around the world by meteorological agencies to study precipitation physics, and as a byproduct, provide a rich database of insect observations that is freely available to researchers. Although considered unwanted ‘clutter’ by the meteorologists who collect the data, the analysis method presented here enables ecologists to delineate co‐occurring signals from insects and raindrops. We present a method that uses image processing techniques on cloud radar velocity spectra to examine the fate of migrating insects when they encounter precipitation. By analysing velocity spectra, we can distinguish flying insects from falling rain and compare the relative density of insects in flight before, during and after the rainfall. We demonstrate the method on a case of insect migration in Oklahoma, USA. Using this method, we show the first reconstructed images of migrating insect layers in flight during rainfall. Our analysis shows that mild to moderate rainfall diminishes the number of insects aloft but does not cause full termination of migratory flight, as has previously been suggested. We hope this technique will spur further investigations of how changing weather conditions impact insect migration, and enable some of the first of such studies in regions of the world that are underrepresented in the literature.

Analysis of subsurface velocity using CMP gathers picked up by unshielded GPR system: results from an experimental NAPL contaminated test site

Adequate knowledge of velocity is required for accurate data imaging and depth conversion, as well as for quantifying the distribution of soil water content. Without complementary borehole information in the form of dielectric permittivity and/or porosity logs along the profile, it is currently impossible to reliably estimate the high-frequency electromagnetic velocity distribution in the probed subsurface region. Here, we present a new method for calculating the precise subsurface velocity structure from ground penetrating radar (GPR) reflection data that does not require boreholes or log data. This study investigates the ability of the pulse_EKKO PRO GPR system to predict a vertical profile for the possible velocity estimation of a layered and contaminated geophysical test site in Hangzhou, China. All data were acquired and saved on the GPR system in various files (projects) before analysis using GPR software to obtain approximated velocity modelling using common midpoint (CMP) gathers. Using the velocity spectrum analysis, a vertical profile of the interval velocities can be derived from each CMP gather. The findings of this study indicate that the proposed method is effective and sustainable. Furthermore, owing to the efficacy of the method in terms of field effort and computational complexity, it can easily be expanded to 3D GPR velocity exploration, increasing its importance in comparison to standard offset-based techniques for estimating velocity using GPR.

Origin of eclipsing time variations in post-common-envelope binaries: Role of the centrifugal force

Eclipsing time variations in post-common-envelope binaries were proposed to be due to the time-varying component of the stellar gravitational quadrupole moment. This is suggested to be produced by changes in the stellar structure due to an internal redistribution of angular momentum and the effect of the centrifugal force. We examined this hypothesis and present 3D simulations of compressible magnetohydrodynamics performed with the PENCIL CODE. We modeled the stellar dynamo for a solar-mass star with angular velocities of 20 and 30 times solar. We included and varied the strength of the centrifugal force and compared the results with reference simulations without the centrifugal force and with a simulation in which its effect is enhanced. The centrifugal force causes perturbations in the evolution of the numerical model, so that the outcome in the details becomes different as a result of nonlinear evolution. While the average density profile is unaffected by the centrifugal force, a relative change in the density difference between high latitudes and the equator of ∼10−4 is found. The power spectrum of the convective velocity is found to be more sensitive to the angular velocity than to the strength of the centrifugal force. The quadrupole moment of the stars includes a fluctuating and a time-independent component, which vary with the rotation rate. As very similar behavior is produced in absence of the centrifugal force, we conclude that it is not the main ingredient for producing the time-averaged and fluctuating quadrupole moment of the star. In a real physical system, we thus expect contributions from both components, that is, from the time-dependent gravitational force from the variation in the quadrupole term and from the spin-orbit coupling that is due to the persistent part of the quadrupole.

Estimation of input energy spectrum from pseudo-velocity response spectrum incorporating the influences of magnitude, distance, and site conditions

The input energy spectrum is useful information for determining the hysteretic energy demand in energy-based seismic design. The input energy spectrum commonly expressed in terms of the energy equivalent velocity spectrum, Veq, can be easily established from the code-specified response spectra (RS) based on the relationship between Veq and RS. Several formulations for the Veq-RS relationship have been developed; however, all of them ignore the influences of magnitude, distance, and site conditions. The aim of this study is to propose a practical formulation for the relationship between Veq and the pseudo-velocity response spectrum, RSpv, incorporating the influences of magnitude, distance, and site conditions. Firstly, to explore the influences of magnitude, distance, and site conditions on the Veq-RSpv relationship, an approach for estimating Veq/RSpv is proposed based upon random vibration theory and verified by comparison with the values calculated by traditional time-series analysis. It is found that the magnitude influences Veq/RSpv the most among these parameters by changing the frequency content of the ground motion, and the Veq/RSpv values at long oscillator periods decrease significantly with an increasing magnitude. Then, based on these conclusions, a practical Veq/RSpv formulation incorporating the influences of magnitude, distance, and site conditions was developed using 16,660 real seismic ground motions in Japan.

An Automatic Velocity Analysis Method for Seismic Data-Containing Multiples

Normal moveout (NMO)-based velocity analysis can provide macro velocity models for prestack data processing and seismic attribute inversion. Datasets with an increasing size require conventional velocity analysis to be transformed to a more automatic mode. The sensitivity to multiple reflections limits the wide application of automatic velocity analysis. Thus, we propose an automatic velocity analysis method for seismic data-containing multiples to overcome the limit of multiple interference. The core idea of the proposed algorithm is to utilize a multi-attribute analysis system to transform the multiple attenuation problem to a multiple identification problem. To solve the identification problem, we introduce the local similarity to attribute the predicted multiples and build a quantitative attribute called multiple similarity. Considering robustness and accuracy, we select two supplementary attributes based on velocity and amplitude difference, i.e., velocity variation with depth and amplitude level. Then we utilize the technique for order preference by similarity to ideal solution (TOPSIS) to balance weights for different attributes in automatic velocity analysis. An RGB system is adopted for multi-attributes fusion in velocity spectra for visualization and quality control. Using both synthetic and field examples to evaluate the effectiveness of the proposed method for data-containing multiples, the results demonstrate the excellent performance in the accuracy of the extracted velocity model.

Research on Initial Model Construction of Seismic Inversion Based on Velocity Spectrum and Siamese Network

The initial model plays an important role in seismic inversion. Generally, the initial model is constructed by lateral extrapolation of parameters under horizons constraints. However, without horizon data, initial modeling becomes a challenging task. Velocity spectrum is a 2D image that can reflect the characteristics of the formations. We regard the problem of establishing the initial model as the problem of similarity analysis of seismic lateral characteristics and propose a method of establishing the initial inversion model based on velocity spectrum and Siamese network. Firstly, the lateral variation of formation characteristics is tracked on velocity spectra generated by common depth point (CDP) gathers. Then, the target tracking results at different CDP positions are obtained with the triple Siamese network. Finally, the discrete inversion parameters are extrapolated along the tracking paths to obtain the initial inversion model. The Siamese network can quickly obtain the similarity of 2D images and does not need manual labels. The theoretical and practical results show that our method can efficiently generate the initial model that conforms to the seismic structure and stratigraphic characteristics without the constraint of interpreted horizon data.

Power Spectrum of Acceleration and Angular Velocity Signals as Indicators of Muscle Fatigue during Upper Limb Low-Load Repetitive Tasks.

Muscle fatigue is a risk factor for developing musculoskeletal disorders during low-load repetitive tasks. The objective of this study was to assess the effect of muscle fatigue on power spectrum changes of upper limb and trunk acceleration and angular velocity during a repetitive pointing task (RPT) and a work task. Twenty-four participants equipped with 11 inertial measurement units, that include acceleration and gyroscope sensors, performed a tea bag filling work task before and immediately after a fatiguing RPT. During the RPT, the power spectrum of acceleration and angular velocity increased in the movement and in 6–12 Hz frequency bands for sensors positioned on the head, sternum, and pelvis. Alternatively, for the sensor positioned on the hand, the power spectrum of acceleration and angular velocity decreased in the movement frequency band. During the work task, following the performance of the fatiguing RPT, the power spectrum of acceleration and angular velocity increased in the movement frequency band for sensors positioned on the head, sternum, pelvis, and arm. Interestingly, for both the RPT and work task, Cohens’ d effect sizes were systematically larger for results extracted from angular velocity than acceleration. Although fatigue-related changes were task-specific between the RPT and the work task, fatigue systematically increased the power spectrum in the movement frequency band for the head, sternum, pelvis, which highlights the relevance of this indicator for assessing fatigue. Angular velocity may be more efficient to assess fatigue than acceleration. The use of low cost, wearable, and uncalibrated sensors, such as acceleration and gyroscope, in industrial settings is promising to assess muscle fatigue in workers assigned to upper limb repetitive tasks.

Modal Identification of Structures by Eliminating the Effect of the High Ocean Wind

Tropical cyclone is a rapidly rotating storm system with severe excitation such as high wind. This severe excitation may change the performance of structures, such as bridges, wharves and wind turbine structures. It is very necessary to monitor this important change. Modal parameters are the ones to reflect the structural instinct behavior. However, many identification methods assume that the excitation is white noise, which is not the truth during high ocean wind excitation. Therefore, the modal identification method to deal with severe ocean wind excitation should be investigated. This paper proposes an innovative method to eliminate the effect of high ocean wind on modal identification. The formulation to generate an impulse response is described, where the effect of high wind is pointed. Then the elimination method is derived using the wind velocity spectrum and correlation function. After the wind field is simulated, the wind velocities and spectra at all accelerometer positions are obtained. The real impulse response form is obtained. Then, modal identification using the real impulse response is performed. Finally, a practical cable-stayed bridge is employed and modal identification is performed. The results show that the identified modes can reflect structural real behavior.

Fourier neural operator approach to large eddy simulation of three-dimensional turbulence

Fourier neural operator (FNO) model is developed for large eddy simulation (LES) of three-dimensional (3D) turbulence. Velocity fields of isotropic turbulence generated by direct numerical simulation (DNS) are used for training the FNO model to predict the filtered velocity field at a given time. The input of the FNO model is the filtered velocity fields at the previous several time-nodes with large time lag. In the a posteriori study of LES, the FNO model performs better than the dynamic Smagorinsky model (DSM) and the dynamic mixed model (DMM) in the prediction of the velocity spectrum, probability density functions (PDFs) of vorticity and velocity increments, and the instantaneous flow structures. Moreover, the proposed model can significantly reduce the computational cost, and can be well generalized to LES of turbulence at higher Taylor-Reynolds numbers.

Oscillations of the large-scale circulation in experimental liquid metal convection at aspect ratios 1.4–3

We investigate the scaling properties of the primary flow modes and their sensitivity to aspect ratio in a liquid gallium (Prandtl number $Pr = 0.02$ ) convection system through combined laboratory experiments and numerical simulations. We survey cylindrical aspect ratios $1.4 \le \varGamma \le 3$ and Rayleigh numbers $10^{4} \lesssim Ra \lesssim 10^{6}$ . In this range the flow is dominated by a large-scale circulation (LSC) subject to low-frequency oscillations. In line with previous studies, we show robust scaling of the Reynolds number $Re$ with $Ra$ and we confirm that the LSC flow is dominated by a jump-rope vortex (JRV) mode whose signature frequency is present in velocity and temperature measurements. We further show that both $Re$ and JRV frequency scaling trends are relatively insensitive to container geometry. The temperature and velocity spectra consistently show peaks at the JRV frequency, its harmonic and a secondary mode. The relative strength of these peaks changes and the presence of the secondary peak depend highly on aspect ratio, indicating that, despite having a minimal effect on typical velocities and frequencies, the aspect ratio has a significant effect on the underlying dynamics. Applying a bandpass filter at the secondary frequency to velocity measurements reveals that a clockwise twist in the upper half of the fluid layer coincides with a counterclockwise twist in the bottom half, indicating a torsional mode. For aspect ratio $\varGamma = 3$ , the unified LSC structure breaks down into multiple rolls in both simulation and experiment.

A niching particle swarm optimization strategy for the multimodal inversion of surface waves

SUMMARY In practice, near-surface structures with shear wave velocity inversions or strong shear wave velocity contrasts may cause the phase velocity spectra of surface waves to be complex. Hence, it is sometimes difficult to identify mode numbers in the phase velocity spectrum. To avoid numbering different modes, the determinant misfit function has been applied to invert multimodal dispersion curves with a very limited computational cost due to the absence of the root-seeking procedure. However, this function presents a complicated relation with modal parameters and thus has multiple minima, resulting in an increase in model ambiguity. Therefore, it is more appropriate to adopt a multimodal optimization algorithm to find multiple minima instead of obtaining one optimal solution. In this study, we use a niching particle swarm optimization to find multiple minima with an enhanced fine search ability. Subsequently, we performed cluster analysis to distinguish different clusters in the inverted solutions and find the best-fitting profiles from multiple minima based on the Euclidean distance between the measured and inverted dispersion curves. Moreover, a modified Thomson–Haskell transfer matrix method is used to calculate the determinant misfit function for a better constraint on inversion because it can only resolve the surface wave modes possessing energy at the free surface, where both the sources and geophones are commonly deployed for active and passive surface wave exploration. Tests of synthetic and field data demonstrate that our inversion method is both effective and robust and emphasize its great potential in urban subsurface exploration and geotechnical characterization applications.

Interactions between upstream-propagating guided jet waves and shear-layer instability waves near the nozzle of subsonic and nearly ideally expanded supersonic free jets with laminar boundary layers

The interactions between upstream-propagating guided jet waves and shear-layer instability waves near the nozzle of subsonic and nearly ideally expanded supersonic, isothermal free jets are investigated for jets at Mach numbers between 0.50 and 2 with fully laminar exit boundary layers of different thicknesses. The velocity spectra in the shear layers downstream of the nozzle exhibit strong narrow peaks for the first azimuthal modes, associated with growing Kelvin–Helmholtz instability waves. The frequencies of the predominant peaks are close, but not necessarily equal, to those of the most amplified instability waves predicted from the mean flow fields using linear stability analysis. They also fall in most cases within or very near the allowable frequency bands of the free-stream upstream-propagating guided jet waves obtained using a vortex-sheet model and jump from one band to another as the Mach number increases. At these frequencies, moreover, high levels organized into elongated stripes are found in the jet potential core and standing-wave patterns are visible at the edges of the shear layers in the power spectral densities of pressure and velocity fluctuations. Therefore, the free-stream upstream-propagating guided jet waves appear to interact with and excite the instability waves near the nozzle of the present jets, as in screeching and impinging jets. This explains the disparities of the frequencies and azimuthal modes of the instability waves dominating early on in the shear layers of initially laminar jets and their discontinuous changes and staging behaviours as the jet velocity varies.

The Mueller maneuver attenuates left atrial phasic volumes and myocardial strain in healthy younger adults

Abstract Background Negative intrathoracic pressure swings alter cardiac loading conditions, which may impact left atrial hemodynamic function and augment atrial stress. If such changes occur and are repeated chronically, they may contribute to the marked atrial remodeling seen in states associated with forceful inspiratory efforts such as endurance training or obstructive sleep apnea. Purpose We studied the acute effects of static inspiratory efforts on left atrial chamber function, myocardial mechanics, and diastolic function. Methods We included healthy adults (10M/7F, 24±4 years) and used Mueller maneuvers to reduce intrathoracic pressure to a −30 cmH2O target for 15 seconds. Over 6 repeated trials, we used echocardiography to acquire left atrial and ventricular focused 2D images, and mitral Doppler inflow and annular tissue velocity spectra. Images were analyzed offline to determine left ventricular volumes, tissue relaxation velocities, trans-mitral filling velocities, and phasic left atrial volumes and myocardial strain using speckle-tracking imaging. Repeated measures were made at Baseline, within the last 5 seconds of the Mueller maneuver, and within the first 5 seconds after release, with 3–5 cardiac cycles averaged per measurement. Results Compared to Baseline, hemodynamic changes occurred during the late phase of the Mueller maneuver and immediately after release. Left ventricular stroke volume decreased by −10±4 mL (p<0.05) then returned to baseline upon release; this was associated with a −11±9 mL (p<0.05) reduction in end-diastolic volume. Left ventricular early diastolic filling attenuated, reflected by decreases in mitral annular relaxation velocity (−2±2 cm/s, p<0.05), E-wave filling velocity (−13±14 cm/s, p<0.05), and left atrial passive emptying volume (−5±5 mL, p<0.05), each normalizing upon release. Left atrial maximal volume decreased (−5±5 mL, p<0.05) during the Mueller maneuver, but increased relative to baseline (+4±5 mL, p=0.049) following release, while left atrial peak positive longitudinal strain decreased (−6±6%, p<0.05) and normalized upon release. Conclusions Attenuated left atrial and in turn ventricular filling may contribute to acute stroke volume reduction during forceful inspiratory efforts. Rapid changes in left atrial chamber volumes and myocardial strain with the release of forceful inspiratory efforts may reflect acute left atrial stress. Funding Acknowledgement Type of funding sources: Public grant(s) – National budget only. Main funding source(s): Natural Sciences and Engineering Research Council Discovery Grant

Acoustic tomography measurement and large-eddy simulation of macroturbulence in high flow tidal channels

Velocity spectra from a pilot tomography experiment in Grand Passage, Nova Scotia, exhibit the structure at 10 m to several 100 m scales similar to that of the spectra of vertically integrated velocity registered by bottom-mounted acoustic Doppler profilers. The experiment was motivated in part by the need for flow and turbulence measurement at sites being targeted for in-stream tidal power development, and by the potential of acoustic tomography for providing this information from shore-based locations in near real time. The measurements were made using a single pair of acoustic transceivers operating at 7 kHz and separated by 1.5 km at an oblique angle to the channel axis. Limited to a single cross-channel path, the experiment provided no information on the spatial structure of the turbulence. In order to investigate this structure, virtual tomographic experiments were carried out by propagating pulses through with meter-scale resolution velocity fields from a non-hydrostatic large-eddy simulation (LES) model of flow in Grand Passage. Results from these virtual experiments are presented, and their implications are discussed in relation to the potential of multi-transceiver tomography for macro-turbulence measurement and the role of horizontal shear in macro-turbulence production in coastal environments.

Unsteady gas dynamics and local heat transfer of pulsating flows in profiled channels mainly to the intake system of a reciprocating engine

• The secondary vortex flows of pulsating flows in profiled channels were analysed. • The turbulence intensity of pulsating flows in profiled channels was determined. • The effects of profiled channels on heat transfer in an engine intake system have been established. • An assessment of the key indicators of a diesel engine with a profiled intake system was performed. This article presents the results of experimental research into the gas dynamics and heat transfer of pulsating air flows in an intake system with square and triangular channels, as well as data from bench tests of a diesel engine. The research was carried out on both a full-scale model of a piston engine and a test bench with a diesel engine. In the first stage, some detailed data on the gas-dynamic and heat exchange characteristics of pulsating flows in the intake system were obtained. It is shown that the gas-dynamic effects typical of square and triangular channels have a significant impact on the flow and heat-transfer characteristics. It is established that the intensity of the turbulence of pulsating flows increases to 22% in the intake system with profiled channels (increased small-scale turbulence), while the amplitudes of the spectrum of flow velocity functions decrease to 33% (suppression of large-scale oscillations). It is shown that a square or triangular channel in the intake system leads to an increase in air mass flow in the range of 5% to 12% compared to the base system. It was found that the increase in the heat transfer coefficient is from 3% to 25% when profiled channels are used. In the second stage, bench tests of a diesel engine with different intake system configurations were carried out to evaluate the technical and economic performance. It is established that the use of profiled channels in the intake system leads to an increase in diesel power of up to 15% compared to the basic modification (while the engine efficiency remains unchanged). The data obtained supplement existing information on the gas-dynamic and heat-exchange characteristics of pulsating flows in profiled channels and can be used in the development of new designs for piston engine intake systems.

Diagnosis of fetal congenitally unguarded tricuspid valve orifice by echocardiography.

To review the imaging characteristics and evaluate the diagnostic value of echocardiography for fetal congenitally unguarded tricuspid valve orifice (CUTVO). Doppler echocardiography was performed and the images were compared with operative and necropsy findings in ten fetuses with CUTVO. The aim of the study was to summarize thecharacteristics of fetal echocardiography and analyze the causes of missed diagnoses and misdiagnoses. There were six cases with complete absence (60%) and four cases with partial absence of the tricuspid leaflet (40%).Among all cases, six cases were consistent with the ultrasonic diagnosis. CUTVO ultrasonographic characteristics consist of the atrioventricular connection with normal arteries and the tricuspid valve device partially or completely absent. The annulus of the tricuspid valve can be described as "empty" in the apical four-chamber view,Doppler evaluation shows to-and-fro flow across the tricuspid orifice with low velocity and two-way spectrum. Fetal echocardiography can accurately diagnose CUTVO, but missed diagnosis and misdiagnosis should be reduced.

Application of Random Forest Algorithm on Tornado Detection

Tornadoes are highly destructive small-scale extreme weather processes in the troposphere. The weather radar is one of the most effective remote sensing devices for the monitoring and early warning of tornadoes. The existing tornado detection algorithms based on radar data are unsupervised and have strict multi-altitude constraints, such as the tornado detection algorithm based on tornado vortex signatures (TDA-TVS), which may lead to high false alarm rates, and the performance of the detection algorithm is greatly affected by the radar data quality control algorithm. A novel TDA-RF algorithm based on the random forest (RF) classification algorithm is proposed for real-time tornado identification of the S-band China new generation of Doppler weather radar (CINRAD-SA). The TDA-RF algorithm uses velocity features to identify tornadoes and adds features related to reflectivity and velocity spectrum width in radar level-II data. Historical CINRAD-SA tornado data from 2006–2015 are used to construct the tornado dataset and train the TDA-RF model. The performance of TDA-RF is evaluated using CINRAD-SA data from five tornadoes of 2016–2020 with enhanced Fujita(EF) scale ratings ranging from EF0 to EF4 and distances from 10 to 130 km to the radar. TDA-RF performs well overall with the probability of detection (POD), false alarm ratio (FAR), and critical success index (CSI) of 71%, 29%, and 55%, respectively. Moreover, the TDA-RF improves POD and CSI, and reduces FAR compared to the TDA-TVS. The maximum tornado early-warning time of TDA-RF is 17 min, and the average is 6 min; TDA-RF can provide classification probability according to the tornado generation and development process to facilitate tracking ability.

Sea states influence on the behaviour of a bottom mounted full-scale twin vertical axis tidal turbine

As only a few tidal turbine developers have reached full-scale installation, the community lacks clear feedback on the validity of development tools to predict in-situ behaviour. From 2019 to 2021, HydroQuest tested its first 1 MW capacity bottom mounted twin vertical axis tidal turbine (VATT) OceanQuest at the Paimpol-Bréhat site, off the coast of Brittany, France. Thus, two years of operational data are available to analyse the behaviour of the full-scale VATT at sea. After describing the environmental conditions and the average performance of the machine, the wave effect on the turbine response is studied. The presence of strong waves multiplies almost by a factor 3 the fluctuation intensities of the torque, the rotational speed of the rotors and the turbine drag, without affecting significantly the mean quantities. The strong coherence between the loads and velocity spectra in the wave frequencies range indicates that the increase of fluctuation intensities is directly due to the periodic loads of surface waves.

Elementary Cellular Automata as Invariant under Conjugation Transformation or Combination of Conjugation and Reflection Transformations, and Applications to Traffic Modeling

This paper develops the analysis of properties of the cellular automata class introduced by the authors. It is assumed that the set of automaton cells is finite and forms a closed lattice, and there are two states for each automaton cell. We consider a new concept. This concept is the average velocity of a cellular automaton, which characterizes the average intensity of changes in the states of the automaton’s cells for a given initial state. The automaton velocity is equal to 1 if the state of any cell changes at each step. The spectrum of average velocities of a cellular automaton is the set of average velocities for different initial states. Since the state space is finite, the automaton, starting from a certain moment of time, is in periodically repeating states of a cycle, and thus, the research of the velocity spectrum is related to the problem of studying the set of the automaton cycles. For elementary cellular automata, the introduced class consists of a subclass of automata such that the conjugation trasformation of an automaton is the automaton itself (Subclass A) or the reflection of the automaton (Subclass B). For this class, it is proved that the spectrum of the automaton contains the value v0 if and only if the spectrum of the complementary automaton contains the value 1−v0 (the sum of the index of elementary cellular automaton and the complementary automaton is 255). For automata of Subclasses A and B, the set of cycles and the velocity spectrum are studied. For Subclass A, a theorem has been proved such that in accordance with this theorem, if two automata complementary to each other start evolving in the same initial state, then the sum of their average velocities is equal to 1. This theorem for Subclass A is generalized to cellular automata, invariant under the conjugation transformation, of more general type than elementary automata. Generalizations of the theorem have been given for the class of one-dimensional cellular automata with a neighborhood containing 2r+1 cells (the next state of the cell depends on the present states of this cell, r cells on the left and r cells on the right) and for some traditionally considered classes of two-dimensional automata. Some elementary cellular automata belonging to the class considered in the paper can be interpreted as transport models. The properties of the spectra for these automata are studied and compared with the properties of elementary cellular automata not invariant under the considered transformations and can also be interpreted as transport models. The analytical results obtained for these simple models can be used to study the qualitative properties and limiting behavior of more complex transport models.