Fourier Acceleration Research Articles

Exact computation of input‐energy spectra from Fourier amplitude spectra

AbstractA theoretically exact relation is presented between the Fourier amplitude spectrum and the elastic input‐energy spectrum. This energy spectrum is computed from the Fourier acceleration amplitude spectrum and the real part of the relative velocity transfer function of the single‐degree‐of‐freedom elastic oscillator. Some accelerometric records collected worldwide are used to prove the solution. It is shown that the relation presented in this work opens the door for a better understanding of the relationship between the input‐energy spectrum and other seismological variables, such as magnitude and focal distance. Copyright © 2003 John Wiley & Sons, Ltd.

Long-period ground-motions for large European earthquakes, 1905–1992, and comparisons with stochastic predictions

Data from European earthquakes in themagnitude range 5 to 8 between 1905 and1992 have been collected and collated, fordistances between 200 and 3400 km. The datainclude both analog and digital records,with priority on wave paths traversingnorthern Europe. Historical analog recordsfrom Uppsala and De Bilt have beendigitised, and appropriate responsefunctions established. New estimates ofseismic moments and moment magnitudes havebeen obtained, which together with momentmagnitudes from other sources have beencompared to surface wave magnitudes. Thelocation of the largest north Europeanearthquakes substantiate earliersuggestions that rifted regions (passivemargins, rifts and grabens) may have thelargest seismic potentials. Arandom-vibration (stochastic) model forprediction of observed peak amplitude,period and Fourier acceleration spectra hasbeen developed and calibrated againstintermediate and long-period observations.Reasonably good correspondence betweenpredictions and observations are obtainedwhen using a simple Brune source spectrum,new values for seismic moments andmoment-magnitude relations, together withreasonable assumptions for stress drop,geometrical spreading and anelasticattenuation. The model is useful first ofall for predicting broad regional averages,but as such it is robust, and it also hasthe potential to be used in an engineeringcontext for predicting spectral responseand peak ground accelerations. Some of theempirical data have also been studied interms of pseudo-relative spectral velocityand compared to strong-motion responsespectral prediction models established fornorthwestern Europe, again for lowfrequencies. Irrespective of these prediction models we emphasize, however, thatthe establishment of the data base itselfhas been an independent and importantpurpose of this study.

Ground Response in Lotung: Total Stress Analyses and Parametric Studies

Previous papers have reported on the performance of a recently developed nonlinear ground response analysis code, SPEC- TRA, with reference to the prediction of the free-field response at a Large-Scale Seismic Test site in Lotung, Taiwan during the M6.5 earthquake of May 20, 1986. Two more major earthquakes of different characteristics shook this test site later that same year, a M6.2 earthquake that occurred on July 30 and a M7.0 earthquake that occurred on November 15. The present article analyzes the free-field responses recorded by a downhole array from these latter two events using the code SPECTRA and a widely used equivalent linear analysis code SHAKE. The studies focus on the relative accuracy and sensitivity of the two codes with respect to the variations of the input material parameters, using time histories, acceleration response spectra, Fourier acceleration amplitude spectra, and Arias intensities as criteria for the comparison. The two codes captured the general wave form of the acceleration histories well, but there was a general tendency for both codes ~particularly SHAKE! to underpredict the Arias intensities of the earthquakes.

Fourier acceleration of Langevin molecular dynamics

Fourier acceleration has been successfully applied to the simulation of lattice field theories for more than a decade. In this paper, we extend the method to the dynamics of discrete particles moving in a continuum. Although our method is based on a mapping of the particles' dynamics to a regular grid so that discrete Fourier transforms may be taken, it should be emphasized that the introduction of the grid is a purely algorithmic device and that no smoothing, coarse-graining, or mean-field approximations are made. The method thus can be applied to the equations of motion of molecular dynamics (MD) or its Langevin or Brownian variants. For example, in Langevin MD simulations our acceleration technique permits a straightforward spectral decomposition of forces so that the long-wavelength modes are integrated with a longer time step, thereby reducing the time required to reach equilibrium or to decorrelate the system in equilibrium. Speedup factors of up to 30 are observed relative to pure (unaccelerated) Langevin MD. As with acceleration of critical lattice models, even further gains relative to the unaccelerated method are expected for larger systems. Preliminary results for Fourier-accelerated molecular dynamics are presented in order to illustrate the basic concepts. Possible extensions of the method and further lines of research are discussed.

Fast acceleration-encoded magnetic resonance imaging.

Direct acceleration imaging with high spatial resolution was implemented and tested. The well-known principle of phase encoding motion components was applied. Suitable gradient switching provides a signal phase shift proportional to the acceleration perpendicular to the slice in the first scan of the sequences. An additional scan serving as a reference was recorded for compensation of phase effects due to magnetic field inhomogeneities. The first scan compensated for phase shifts from undesired first- and second-order motions; the second scan was completely insensitive to velocity and acceleration in all directions. Advantages of the proposed two-step technique compared to former approaches with Fourier acceleration encoding (with several phase encoding steps) are relatively short echo times and short total measuring times. On the other hand, the new approach does not allow us to assess the velocity or acceleration spectrum simultaneously. The capabilities of the sequences were tested on a modern 1.5 T whole body MR unit providing relatively high gradient amplitudes (25 mT/m) and short rise times (600 micros to maximum amplitude). The results from a mechanical acceleration phantom showed a standard deviation of 0.3 m/s2 in sequences with an acceleration range between -12 and 12 m/s2. This range covers the expected maximum acceleration in the human aorta of 10 m/s2. Further tests were performed on a stenosis phantom with a variable volume flow rate to assess the flow characteristics and possible displacement artifacts of the sequences. Preliminary examinations of volunteers demonstrate the potential applicability of the technique in vivo.

Estimation of pressure gradients in pulsatile flow from magnetic resonance acceleration measurements.

A method for estimating pressure gradients from MR images is demonstrated. Making the usual assumption that the flowing medium is a Newtonian fluid, and with appropriate boundary conditions, the inertial forces (or acceleration components of the flow) are proportional to the pressure gradients. The technique shown here is based on an evaluation of the inertial forces from Fourier acceleration encoding. This method provides a direct measurement of the total acceleration defined as the sum of the velocity derivative vs. time and the convective acceleration. The technique was experimentally validated by comparing MR and manometer pressure gradient measurements obtained in a pulsatile flow phantom. The results indicate that the MR determination of pressure gradients from an acceleration measurement is feasible with a good correlation with the true measurements (r = 0.97). The feasibility of the method is demonstrated in the aorta of a normal volunteer. Magn Reson Med 44:66-72, 2000.

Estimation of pressure gradients in pulsatile flow from magnetic resonance acceleration measurements

A method for estimating pressure gradients from MR images is demonstrated. Making the usual assumption that the flowing medium is a Newtonian fluid, and with appropriate boundary conditions, the inertial forces (or acceleration components of the flow) are proportional to the pressure gradients. The technique shown here is based on an evaluation of the inertial forces from Fourier acceleration encoding. This method provides a direct measurement of the total acceleration defined as the sum of the velocity derivative vs. time and the convective acceleration. The technique was experimentally validated by comparing MR and manometer pressure gradient measurements obtained in a pulsatile flow phantom. The results indicate that the MR determination of pressure gradients from an acceleration measurement is feasible with a good correlation with the true measurements (r = 0.97). The feasibility of the method is demonstrated in the aorta of a normal volunteer. Magn Reson Med 44:66–72, 2000. © 2000 Wiley-Liss, Inc.

Estimation of pressure gradients in pulsatile flow from magnetic resonance acceleration measurements

A method for estimating pressure gradients from MR images is demonstrated. Making the usual assumption that the flowing medium is a Newtonian fluid, and with appropriate boundary conditions, the inertial forces (or acceleration components of the flow) are proportional to the pressure gradients. The technique shown here is based on an evaluation of the inertial forces from Fourier acceleration encoding. This method provides a direct measurement of the total acceleration defined as the sum of the velocity derivative vs. time and the convective acceleration. The technique was experimentally validated by comparing MR and manometer pressure gradient measurements obtained in a pulsatile flow phantom. The results indicate that the MR determination of pressure gradients from an acceleration measurement is feasible with a good correlation with the true measurements (r = 0.97). The feasibility of the method is demonstrated in the aorta of a normal volunteer. Magn Reson Med 44:66-72, 2000.

Generic finite-fault model for ground-motion prediction in eastern North America

AbstractGround-motion models based on the Brune point-source approximation have an underlying ω2 spectrum, with a single corner frequency. These models over-predict observed spectral amplitudes at low to intermediate frequencies (∼0.1 to 2 Hz), for earthquakes with moment magnitudes M of 4 or greater. The empirical spectra of moderate to large events tend to sag at these frequencies, relative to the level suggested by the Brune point-source model.A model that accounts for the finite extent of the fault plane correctly describes the observed spectral shapes. The model represents seismic radiation as a sum of contributions from several subfaults. Each subfault may be represented as a point source, and each subevent has an ω2 spectrum. When contributions to ground motion at an observation point are summed over all subfaults, the resulting spectral shape has two corner frequencies and more closely matches observed spectra. The more realistic spectral shape obtained through finite-fault modeling reflects the underlying reality that the radiation from real faults is formed by ruptures of their smaller parts, whose corner frequencies are higher than those implied by the full fault dimension. The two corners appear naturally as a result of subevent summation.We use the stochastic finite-fault methodology to simulate the recorded ground-motion data from all significant earthquakes in eastern North America (ENA). These data include eight events of M > 4 recorded on modern digital instruments (regional seismographs and strong-motion instruments), and three historical events of M 5.8 to 7.3 recorded on analog instruments. The goodness of fit of synthetics to the data is defined as simulation bias, which is indicated by the difference between the logarithms of the observed and the simulated spectrum, averaged over all recordings of an earthquake. The finite-fault simulations provide an unbiased fit to the observational database over a broad frequency range (0.1 to 50 Hz), for all events.A surprising conclusion of these simulations is that the subfault size that best fits the observed spectral shape increases linearly with moment magnitude, in an apparently deterministic manner. This strongly suggests that the subfault size can be unambiguously defined by the magnitude of the simulated earthquake. In this case, the radiation-strength factor(s), which is proportional to the square root of the high-frequency Fourier acceleration level, remains the only free parameter of the model. Its value is related to the maximum slip velocity on the fault. The strength factors for all modeled ENA events are within the range of 1.0 to 1.6, with the exception of the Saguenay mainshock (s = 2.2). This suggests a remarkable uniformity in earthquake slip processes.

Correlation of seismic intensity with Fourier acceleration spectra

We present a method for estimating the seismic intensity in terms of MMI or MSK scale using Fourier amplitude spectra of ground acceleration. The method implies that severity of earthquake ground motion is determined by spectral amplitudes in relatively narrow frequency band: so-called “representative frequencies”, at decreasing frequencies (from 7–8 Hz for small intensities to 0.7 – 1.0 Hz for MMI(MSK) = VIII–IX) with increasing intensity level. It is examined through estimation of probable intensity at a site using recordings of recent earthquakes in several seismic regions and prediction of intensity distribution patterns for some earthquakes. Seismic hazard maps, in terms of intensity levels based upon the proposed approach, should describe regional features of seismic waves excitation and propagation, as well as local ground conditions.

1) The influence of Gribov copies on gluon and ghost propagators in Landau gauge and 2) A new implementation of the fourier acceleration method

We study the influence of Gribov copies on gluon and ghost propagators in lattice Landau gauge. For the gluon propagator, Gribov noise seems to be of the order of magnitude of the numerical accuracy. On the contrary, for the ghost propagator, Gribov noise is clearly observable, at least in the strong-coupling regime. We also observe, in the limit of large lattice volume, a gluon propagator decreasing as the momentum goes to zero. Finally, we introduce an implementation of the method of Fourier Acceleration which avoids the use of the fast Fourier transform, being well suited for parallel and vector machines. We apply it to the case of Landau gauge fixing, and study its performance on APE computers.

Multigrid implementation of the Fourier acceleration method for Landau gauge fixing

We present a new implementation of the Fourier acceleration method for Landau gauge fixing. By means of a multigrid inversion we are able to avoid the use of the fast Fourier transform. This makes the method more flexible, and well suited for vector and parallel machines. We study the performance of this algorithm on serial and on parallel (APE100) machines for the four-dimensional SU(2) case. We find that our method is equivalent to the standard implementation of Fourier acceleration already on a serial machine, and that it parallelizes very efficiently: its computational cost shows a linear speedup with the number of processors. We have also implemented, on the parallel machines, a version of the method using conjugate gradient instead of multigrid. This leads to an algorithm that is efficient at intermediate lattice volumes.

An empirical study of earthquake source spectra for California earthquakes

AbstractWe generalize source, path, and site effects for California earthquakes as functions of magnitude and distance, based on regression analysis of 1000 Fourier acceleration spectra from 43 California earthquakes in the moment magnitude range from 4.4 to 7.4, recorded at rupture distances from 1 to 200 km. Empirically derived source spectra for California earthquakes are generally inconsistent with the spectral shape implied by a Brune (“omega-squared”) point-source model. This is manifested by magnitude- and frequency-dependence of the Brune model parameters. For example, the Brune stress parameter that best matches the data at high frequencies decreases from a value of about 120 bars at M5.5 to a value near 50 bars at M7.5. At frequencies below 1 Hz, though, source spectra have much lower spectral amplitudes than predicted by the Brune point-source model for these values of stress; this discrepancy grows with increasing magnitude. Finite-fault simulations indicate that this is a consequence of the breakdown of the validity of the point-source approximation for large ruptures. A stochastic finite-fault model, in which the fault is discretized as a number of subfaults, each of which is represented by a Brune omegasquared point source, correctly matches the observed spectral shapes and amplitudes.The spectral decay parameter kappa, representing average near-surface attenuation of high-frequency motion at rock sites, increases with increasing magnitude, from values near 0.035 sec at M5.5 to 0.050 sec at M7.5. Magnitude dependence of kappa may be interpreted as evidence of nonlinearity for typical California sites subjected to strong ground motion.Comparisons of our empirical source spectra for California to corresponding spectra for eastern North America suggest that the spectral amplitudes are similar in the two regions for low-frequency motions (f < 2 Hz for M5.5, f < 0.5 Hz for M7.5), for equivalent crustal conditions. The eastern events appear to have enhanced high-frequency near-source amplitudes relative to the California events; this is particularly pronounced for large-magnitude earthquakes.

Critical slowing-down in SU(2) Landau gauge-fixing algorithms

We study the problem of critical slowing-down for gauge-fixing algorithms (Landau gauge) in SU(2) lattice gauge theory on a 2-dimensional lattice. We consider five such algorithms, and lattice sizes ranging from 8 2 to 36 2 (up to 64 2 in the case of Fourier acceleration). We measure four different observables and we find that for each given algorithm they all have the same relaxation time within error bars. We obtain that the so-called Los Alamos method has a dynamic critical exponent z ≈ 2, the overrelaxation method and the stochastic overrelaxation method have z ≈ 1, the so-called Cornell method has z slightly smaller than 1 and the Fourier acceleration method completely eliminates critical slowing-down. A detailed discussion and analysis of the tuning of these algorithms is also presented.

Langevin simulation of SU ( N) × SU ( N)/SU ( N) spin systems

We apply improved Langevin methods with Fourier acceleration to SU ( N) × SU ( N)/SU ( N) spin models in two dimensions for N = 3,4 and 6. We show that Fourier acceleration combined with operator improvement techniques does lead to a considerable reduction of errors and correlation times and helps to overcome critical slowing down. We examine scaling in these theories and measure the spectrum in the different channels available in the case N = 4 and 6. The ratios of ground state masses in these channels is found to be consistent with theoretical predictions. By making use of multi-parameter fits to the relevant correlation functions we are able to measure the first excited states with reasonable accuracy. We discuss the efficacy of our algorithm in relation to other methods.

Bayesian Attenuation Regressions: an Application to Mexico City

We describe the application of a bayesian linear regression technique to the problem of deriving strong-motion attenuation relations. This approach provides a conceptual framework for the formal incorporation of knowledge about the involved phenomena that comes from sources other than the observed data (prior information, according to the bayesian terminology). The procedure produces numerical solutions that are more stable and rational than those obtained from conventional regression schemes. We illustrate the use of the proposed technique with the derivation of attenuation laws for the Fourier acceleration spectrum, as a function of magnitude and distance, at a hill-zone station in Mexico City.

LANGEVIN METHODS FOR SU(N) CHIRAL MODELS IN TWO DIMENSIONS

We apply improved Langevin methods with Fourier acceleration to SU(N)×SU(N)/SU(N) spin models in two dimensions for N=3, 4 & 6. We discuss the effect of fourier acceleration and operator improvement techniques, measure the fundamental correlation length on lattices upto 1282 and observe asymptotic scaling in a physical scheme. Bound state masses are also measured and evidence is found for excited states in all channels. Our results are found to be in good agreement with theory.

An investigation into a wavelet accelerated gauge fixing algorithm

We introduce an acceleration algorithm for coulomb gauge fixing, using the compactly supported wavelets introduced by Daubechies. The algorithm is similar to Fourier acceleration. Our provisional numerical results for SU(3) on 8 4 lattices show that the acceleration based on the DAUB6 transform can reduce the number of iterations by a factor up to 3 over the unaccelerated algorithm. The reduction in iterations for Fourier acceleration is approximately a factor of 7.

Langevin methods for SU(N) chiral models in two dimensions

We apply improved Langevin methods with Fourier acceleration to SU ( N) × SU ( N)/ SU ( N) spin models in two dimensions for N = 3,4 & 6. We discuss the effect of Fourier acceleration and operator improvement techniques, measure the fundamental correlation length on lattices up to 128 2 and observed asymptotic scaling in a physical scheme. Bound state masses are also measured and evidence is found for excited states in all channels. Our results are found to be in good agreement with theory.

Future ground motions in Mexico City

Future ground motions at a soft site of the Valley of Mexico are estimated for a postulated M 8.2 earthquake with epicentral distance of 280 km. Three techniques were used: 1. (a) a semi-empirical estimation of Fourier acceleration spectra at a hard (reference) site plus empirical determination of site effects through empirical transfer functions; 2. (b) semi-empirical computation of Fourier spectra at the reference site, with theoretical calculation of site response using one-dimensional horizontal S wave propagation corrected to account for surface waves; 3. (c) use of an M 6.9 recording as the Green's function of the postulated event. The three techniques use competing approaches, and we illustrate a rational way for combining them to obtain a single estimate that includes information from the three approaches.