Long-period Spectra Research Articles

Adjoint sensitivity kernels for free oscillation spectra

SUMMARY We apply the adjoint method to efficiently calculate sensitivity kernels for long-period seismic spectra with respect to structural and source parameters. Our approach is built around the solution of the frequency-domain equations of motion using the direct solution method (DSM). The DSM is currently applied within large-scale mode coupling calculations and is also likely to be useful within finite-element type methods for modelling seismic spectra that are being actively developed. Using mode coupling theory as a framework for solving both the forward and adjoint equations, we present numerical examples that focus on the spectrum close to four eigenfrequencies (the low-frequency mode, 0S2, and higher frequency modes, namely 2S2, 0S7 and 0S10 for comparison). For each chosen observable, we plot sensitivity kernels with respect to 3-D perturbations in density and seismic wave speeds. We also use the adjoint method to calculate derivatives of observables with respect to the matrices occurring within mode coupling calculations. This latter approach points towards a generalization of the two-stage splitting function method for structural inversions that does not rely on inaccurate self-coupling or group-coupling approximations. Finally, we verify through direct calculation that our sensitivity kernels correctly predict the linear dependence of the chosen observables on model perturbations. In doing this, we highlight the importance of non-linearity within inversions of long-period spectra.

Long-period seismicity at Shishaldin volcano (Alaska) in 2003–2004: Indications of an upward migration of the source before a minor eruption

We have analyzed the long-period (LP) seismic activity at Shishaldin volcano (Aleutians Islands, Alaska) in the period October 2003–July 2004, during which a minor eruption took place in May 2004, with ash and steam emissions, thermal anomalies, volcanic tremor and small explosions. We have focused the attention on the time evolution of LP rate, size, spectra and polarization dip angle along the dataset.We find an evolution toward more shallow dip angles in the polarization of the waveforms during the sequence. The dip angle is a manifestation of the source location. Because the LP seismic sources are presumed to reflect the aggregation of gas slug or pockets within the melt, we use the polarization dip at the LP onset as a proxy for the nucleation depth of the seismic events within the conduit. We refer to this parameter as the nucleation dip and the position along the conduit of the gas aggregation as nucleation depth.The nucleation dip changes throughout the dataset. It shows a sharp decrease between the end of December 2003 and the end of January 2004, followed by a gradual increase until the onset of the eruption. At the same time, a general increase of the LP rate occurs. We have associated the dip evolution with a sinking and a subsequent decrease of the nucleation depth, which would quickly migrate up to about 8km below the crater rim, followed by a slow depth decrease which culminates in the eruption.The change in the nucleation depth reflects either a pressure variation within the plumbing system, which would affect the confining pressure experienced by the gas aggregations. We have imputed such a pressure change to the intrusion of batches of magma from a deeper magma chamber (<10km) toward a shallower one (>5km). For a cylindric conduit with rigid walls, this leads to a volume of the injected new magma of 105–107m3, compatible with estimates in other areas, suggesting that the LP process can be considered a good proxy of the thermodynamical conditions of the shallow plumbing system.

Enhancement of long period components of recorded and synthetic ground motions using InSAR

Tall buildings and flexible structures require a better characterization of long period ground motion spectra than the one provided by current seismic building codes. Motivated by that, a methodology is proposed and tested to improve recorded and synthetic ground motions which are consistent with the observed co-seismic displacement field obtained from interferometric synthetic aperture radar (InSAR) analysis of image data for the Tocopilla 2007 earthquake ( M w =7.7) in Northern Chile. A methodology is proposed to correct the observed motions such that, after double integration, they are coherent with the local value of the residual displacement. Synthetic records are generated by using a stochastic finite-fault model coupled with a long period pulse to capture the long period fling effect. It is observed that the proposed co-seismic correction yields records with more accurate long-period spectral components as compared with regular correction schemes such as acausal filtering. These signals provide an estimate for the velocity and displacement spectra, which are essential for tall-building design. Furthermore, hints are provided as to the shape of long-period spectra for seismic zones prone to large co-seismic displacements such as the Nazca-South American zone.

Analysis of sustained long-period activity at Etna Volcano, Italy

Following the installation of a broadband network on Mt. Etna, sustained Long-Period (LP) activity was recorded accompanying a period of total quiescence and the subsequent onset of the 2004–2005 effusive episode. From about 56000 events detected by an automatic classification procedure, we analyse a subset of about 3000 signals spanning the December 17th, 2003–September 25th, 2004, time interval. LP spectra are characterised by several, unevenly-spaced narrow peaks spanning the 0.5–10 Hz frequency band. These peaks are common to all the recording sites of the network, and different from those associated with tremor signals. Throughout the analysed time interval, LP spectra and waveforms maintain significant similarity, thus indicating the involvement of a non-destructive source process that we interpret in terms of the resonance of a fluid-filled buried cavity. Polarisation analysis indicates radiation from a non-isotropic source involving large amounts of shear. Concurrently with LP signals, recordings from the summit station also depict Very-Long-Period (VLP) pulses whose rectilinear motion points to a region located beneath the summit craters at depths ranging between 800 and 1100 m beneath the surface. Based on a refined repicking of similar waveforms, we obtain robust locations for a selected subset of the most energetic LP events from probabilistic inversion of travel-times calculated for a 3D heterogenous structure. LP sources cluster in a narrow volume located beneath the summit craters, and extending to a maximum depth of ≈ 800 m beneath the surface. No causal relationships are observed between LP, VLP and tremor activities and the onset of the 2004–2005 lava effusions, thus indicating that magmatic overpressure played a limited role in triggering this eruption. These data represent the very first observation of LP and VLP activity at Etna during non-eruptive periods, and open the way to the quantitative modelling of the geometry and dynamics of the shallow plumbing system.

Ground Motion in Anchorage, Alaska, from the 2002 Denali Fault Earthquake: Site Response and Displacement Pulses

Data from the 2002 Denali fault earthquake recorded at 26 sites in and near Anchorage, Alaska, show a number of systematic features important in studies of site response and in constructing long-period spectra for use in earthquake engi- neering. The data demonstrate that National Earthquake Hazards Reduction Program (NEHRP) site classes are a useful way of grouping stations according to site ampli- fication. In general, the sites underlain by lower shear-wave velocities have higher amplification. The amplification on NEHRP class D sites exceeds a factor of 2 relative to an average of motions on class C sites. The amplifications are period dependent. They are in rough agreement with those from previous studies, but the new data show that the amplifications extend to at least 10 sec, periods longer than considered in previous studies. At periods longer than about 14 sec, all sites have motion of similar amplitude, and the ground displacements are similar in shape, polarization, and am- plitude for all stations. The displacement ground motion is dominated by a series of four pulses, which are associated with the three subevents identified in inversion studies (the first pulse is composed of P waves from the first subevent). Most of the high-frequency ground motion is associated with the S waves from subevent 1. The pulses from subevents 1 and 2, with moment releases corresponding to M 7.1 and 7.0, are similar to the pulse of displacement radiated by the M 7.1 Hector Mine earthquake. The signature from the largest subevent (M 7.6) is more subdued than those from the first two subevents. The two largest pulses produce response spectra with peaks at a period of about 15 sec. The spectral shape at long periods is in good agreement with the recent 2003 NEHRP code spectra but is in poor agreement with the shape obtained from Eurocode 8.

Recovery of aspherical earth structure from observations of normal mode amplitudes

On an aspherical earth, the normal mode amplitude pattern is perturbed by along-branch coupling. In the geometrical optics approximation, the effect of aspherical earth structure on normal mode amplitudes through along-branch coupling is equivalent to a perturbation Θ in the spatial phase of the amplitude pattern. This observable quantity is related to earth structure through a WKBJ phase integral along the source-receiver minor arc. The amplitude pattern is strongly sensitive to odd-degree earth structure and complements the information contained in the frequency shift pattern, which is sensitive to only even-degree structure. We have developed a method to estimate the amplitudes of unresolvably split multiplets from long-period spectra and have applied it to a large number of IDA and GEOSCOPE recordings to obtain a global data set of normal mode amplitudes for spheroidal modes 0S26−0S43. Normal mode amplitude estimation requires that the first surface wave arrival R1 be available in the seismogram, and coherent amplitude patterns have been obtained for 172 data records. The analysis of normal mode amplitudes affords several advantages not shared by surface wave phase velocity analysis or time-domain waveform fitting. Our method allows the estimation of errors in the model and the ability to cull away modes in the frequency domain, and it makes use of the information contained in the entire seismogram. In addition, our method makes no assumptions concerning the spectral shape of isolated multiplets in the frequency domain. The small data set employed here is only a fraction of the size of previous global data sets, but has the advantage that fewer assumptions are built in to the theory on which the structure inversion is based. Several factors, including uncertainty in the moment tensor scaling and focusing and defocusing effects, prohibit recovery of Θ for individual multiplets and limit us to the recovery of the average Θ over the mode band l = 26−43, for each source-receiver pair. Through a preferential weighting of subsets of this mode band, we may obtain estimates of Θ within narrower mode bands centred, respectively, on l = 31, l = 34, and l = 37. Least squares inversion of Θ within the various mode bands, utilizing approximately 100 source-receiver pairs, yields the geographic distribution of local eigenfrequency perturbation δδlocal(δ, δ) in a truncated spherical harmonic expansion. Inversion is first performed for the best fitting degree-6 earth model using all available modes within the band l = 26-43 with equal weight. The degree-6 earth model is obtained by inverting for degrees 1, 3, and 5 with degrees 2, 4 and 6 fixed according to published results from the frequency studies. Narrower band inversions are performed within l= 26-36, l = 29-39, and l = 31-43. Comparison with model M84A shows that the odd degree-1 and 3 components of aspherical earth structure may be reproduced from the amplitude data set. The degree-5 structure differs significantly from that of model M84A.

Summation of the Born series for the normal modes of the Earth

SUMMARY We derive an explicit formula for the response of a laterally heterogeneous, self-gravitating, rotating, dissipative, and physically dispersive earth to an earthquake. The long-period spectrum at a single station is given by u(ω) =RH0 [V(ω) +ωW−ω2T]−1·S0(ω) where R0 and S0(ω) are the unperturbed source and receiver vectors and H denotes the Hermitian transpose. The quantities V(ω), W, and T are the potential energy, Coriolis, and relative kinetic energy matrices, and V(ω) +ωW−ω2T is the Lagrangian matrix whose eigenvalues are the perturbed complex eigenfrequencies. This simple result is obtained by formal summation of the Born series.

Short-period S-wave attenuation under the United States

A study of short-period S -wave spectra from deep events recorded at LRSM stations indicates severe attenuation of S waves under the Western United States. The average differential in ts * between the Eastern and Western United States was found to be ∼0.67 ± 0.19 sec (95 per cent confidence interval), which is several times greater than Δ tp * ∼ 0.2 - 0.25 for short-period P . This indicates that most, if not all, energy loss occurs in shear deformation. The numerical value of the ts * differential is also smaller than that derived from long-period spectra indicating a frequency dependence of Q .

Analysis of amplitude spectra ofPwaves from earthquakes and underground explosions

A step-by-step display of the theoretical P-wave amplitude spectrum of shallow (focal depth ≤ 40 km) earthquakes with mb values ranging from 5.9 to 6.3 and underground explosions of comparable magnitudes as the signal propagates from source to receiver indicates that the factors most influential in shaping the respective spectra are the source crust transfer function, which includes the focal depth effect, and the source function. The results, although not unequivocal, favor a peak in the amplitude spectrum of the teleseismic P coda of earthquakes rather than a flat spectrum at low frequencies. The effect of focal depth appears to have more influence than the effect of source parameters on the peak frequency f0 in the spectra of earthquakes, whereas the source function is the important parameter for shallow underground nuclear explosions. For shallow seismic events of comparable mb the f0 of earthquake spectra is smaller than the f0 of explosion spectra by approximately an order of magnitude. The pronounced minimum (hole) at about 0.1 Hz in the long-period spectrum of earthquakes is associated with the focal depth effect and can be identified for several events. However, theoretically predicted holes starting at about 0.5 Hz in the short-period spectrum of assumed unilateral faulting earthquakes are due to the extended source effect and are not identifiable on any of the experimental spectra. For short-period records of Longshot and Milrow, Haskell's analytic formulation of the Werth and Herbst displacement potential, in conjunction with the outlined theory, predicts fairly accurately the increase (approximately 0.15 sec) in the duration of the first half-cycle with increasing yield. For the corresponding long-period records of Milrow his formulation predicts a value smaller than a mean of the measured values by about 15% (0.32 sec). Haskell's equations, together with the rest of the theory, appear to predict fairly accurately the variation in mb with yield for these two events when amplitude measurements are restricted to the first half-cycle.

Errors in the spectral analysis of long-period seismic body waves

Recent theories of earthquake radiation differ in their predictions as to the behavior of the long-period part of the spectrum, of radiated waves. A numerical model study shows that it is difficult experimentally to distinguish between these theories because of a fundamental difficulty in windowing seismic arrivals that consist essentially of a single cycle (as is often the case for long-period body waves). The effect of noise in the record and of truncation due to later arrivals is to produce spuriously large amplitudes for the long-period part of the spectrum. These errors are amplified if corrections must be made for low instrumental magnifications at these periods. Analyses of records of many earthquakes (four examples are presented) by various techniques show that the radiated body waves have a displacement amplitude spectrum characterized by the existence of a dominant period. This disagrees with theories that predict that long periods have a constant amplitude, and thus studies of the long-period spectrum interpreted in terms of these theories may yield results that are grossly in error. Our experimental results are in agreement with Archambeau's theory, which predicts the existence of a dominant period.

Low Even Configurations in the First Spectrum of Rhenium (Re i)

The theory of intermediate coupling with configuration interaction is used to calculate energy levels and $g$ values in the $5{d}^{5}6{s}^{2}$ and $5{d}^{6}6s$ configurations of Re i. Fifty-one observed levels are fitted with a mean deviation of \ifmmode\pm\else\textpm\fi{}325 ${\mathrm{cm}}^{\ensuremath{-}1}$ by using eleven free parameters. The $L(L+1)$ correction is important, as the mean deviation is \ifmmode\pm\else\textpm\fi{}514 ${\mathrm{cm}}^{\ensuremath{-}1}$ without it; the parameter for this correction ($\ensuremath{\alpha}=66.8\ifmmode\pm\else\textpm\fi{}13$) has a value similar to that in first and second long-period spectra. Standard deviations of the parameters are calculated, and within the limits so defined, Slater's parameters are consistent with expectations based on known behavior in first and second long-period spectra. Calculated $g$ values agree with observed values for thirty-seven levels to about \ifmmode\pm\else\textpm\fi{}0.03. Levels not previously assigned from experiment are given designations; the eigenvectors are used to illustrate qualitative features of these designations.