Multiple-taper spectral analysis of terrestrial free oscillations: part II

In Part I of this paper, we derived a set of data tapers designed to minimize the spectral leakage of decaying sinusoids immersed in white noise. Multiplying a long-period seismic record by K of these tapers creates K time series. A decaying sinusoid is fit to these K time series in the frequency domain at a number of chosen frequencies by a least-squares procedure. The fit is tested at each frequency using a statistical F -test. In Part I, we demonstrated that the multiple-taper method is a more sensitive detector of decaying sinusoids than the conventional direct spectral-estimate.
In this paper, we present a number of extensions to the multiple-taper method. We explain how the technique can be modified to estimate the harmonic components of records containing gaps. We discuss how sinusoids at frequencies between FFT bin frequencies can be detected, and how this method can be combined with conventional multi-station stacking procedures.     

Methodologies for estimating t*(f) from short-period body waves and regional variations of t*(f) in the United States

Summary. In this paper we discuss some aspects of estimating t * from short-period body waves and present some limits on t* (f) models for the central and south-western United States (CUS and SWUS). We find that for short-period data, with frequencies above 1 or 2 Hz, while the average spectral shape is stable, the smaller details of the spectra are not; thus, only an average t *, and not a frequency-dependent t *, can be derived from such information. Also, amplitudes are extremely variable for short-period data, and thus a great deal of data from many stations and azimuths must be used when amplitudes are included in attenuation studies.
The predictions of three pairs of models for t* (f) in the central and south-western United States are compared with time domain observations of amplitudes and waveforms and frequency domain observations of spectral slopes to put bounds on the attenuation under the different parts of the country. A model with the t * values of the CUS and SWUS converging at low frequencies and differing slightly at high frequencies matches the spectral domain characteristics, but not the time domain amplitudes and waveforms of short-period body waves. A model with t * curves converging at low frequencies, but diverging strongly at high frequencies, matches the time domain observations, but not the spectral shapes. A model with nearly-parallel t* (f) curves for the central and south-western United States satisfies both the time and frequency domain observations.
We conclude that use of both time and frequency domain information is essential in determining t* (f) models. For the central and south-western United States, a model with nearly-parallel t* (f) curves, where Δ t *∼ 0.2 s, satisfies both kinds of data in the 0.3–2 Hz frequency range.     

A probabilistic approach to the inversion of data from a seismic array and its application to volcanic signals

Array techniques are particularly well‐suited for detecting and quantifying the complex seismic wavefields associated with volcanic activity such as volcanic tremor and long‐period events. The methods based on the analysis of the signal in the frequency domain, or spectral methods, have the main advantages of both resolving closely spaced sources and reducing the necessary computer time, but may severely fail in the analysis of monochromatic, non‐stationary signals. Conversely, the time‐domain methods, based on the maximization of a multichannel coherence estimate, can be applied even for short‐duration pulses. However, for both the time and the frequency domain approaches, an exhaustive definition of the errors associated with the slowness vector estimate is not yet available. Such a definition become crucial once the slowness vector estimates are used to infer source location and extent. In this work we develop a method based on a probabilistic formalism, which allows for a complete definition of the uncertainties associated with the estimate of frequency–slowness power spectra from measurement of the zero‐lag cross‐correlation. The method is based on the estimate of the theoretical frequency–slowness power spectrum, which is expressed as the convolution of the true signal slowness with the array response pattern. Using a Bayesian formalism, the a posteriori probability density function for signal slowness is expressed as the difference, in the least‐squares sense, between the model spectrum and that derived from application of the zero‐lag cross‐correlation technique. The method is tested using synthetic waveforms resembling the quasi‐monochromatic signals often associated with the volcanic activity. Examples of application to data from Stromboli volcano, Italy, allow for the estimate of source location and extent of the explosive activity.     

The correlation of microtremors: empirical limits and relations between results in frequency and time domains

The use of the correlation of microtremor records is on its way to develop into a common tool to estimate local shear wave velocity structure. For this reason, the establishment of the conditions for the correct use of this method and its limitations when applied to real data is becoming increasingly important. In addition to the use of frequency domain spatial correlation technique [the Spatial Autocorrelation (SPAC) method], the use of time domain correlation to obtain the Green's function of the medium is rapidly gaining presence. We explore the use of microtremor correlation techniques in the time domain to determine local velocity structure and compare with previous results obtained with the same data using SPAC. Our data come from three experiments carried out in Parkway and Wainuiomata valleys in New Zealand, using broad-band portable stations. Interstation distances range from 5 m to 2.1 km, and our results are useful in the frequency band from 0.1 to 7 Hz. Frequency domain correlation requires an isotropic microtremor field, a condition that need not be satisfied in the time domain. Two station correlations provide useful results due to the temporal stationarity and isotropy, in average, of the microtremor wavefield. This manifests itself in the symmetry of the temporal correlation functions with respect to zero time. Our results show that the local velocity structure and the interstation distance are the key factors conditioning the frequency range where surface wave dispersion can be correctly measured either in frequency or time domains. We confirm that, when the interstation distance becomes much larger than the dominant wavelengths, only the correlation in time domain is useful. All our results indicate that the signal obtained in the correlation of vertical component microtremors is due to the fundamental mode of Rayleigh waves, which appears as the most stable propagation mode, without any indication of body waves.     

Internal waves in a rotating stratified spherical shell: asymptotic solutions

Summary. Small amplitude oscillations of a rotating, density-stratified fluid bounded by a spherical shell are examined. No restrictions are placed on the thickness of the shell. The internal mode spectrum is examined in the complete rotation-stratification parameter range including the regime that is appropriate for a plausible stratification distribution in the Earth's fluid core. A mathematical model is derived in terms of an eigenvalue PDE of mixed type. The existence of oscillatory solutions is exhibited in the limits of no rotation and no stratification. The frequency spectrum is extended asymptotically away from these limiting cases. A reduction in the complexity of the PDE for modes oscillating at the inertial frequency is exploited. A variational formulation is constructed in which the stratification parameter is treated as an eigenvalue of the system for fixed wave frequency. The spectral information is again extended asymptotically away from these 'accessible' points. Although the PDE reduces to Laplace's tidal equations (LTE) only under stringent parameter restrictions, it is observed that aspects of the behaviour of low frequency LTE modes are reproduced in the general model.     

Improved resolution of complex eigenfrequencies in analytically continued seismic spectra

Summary. The response of the Earth to an earthquake is a transient that is effectively zero several days after the event. A recording of the event, of finite duration in time, has a Fourier spectrum that is an entire, or integral, analytic function of frequency. We present a very simple procedure for computing the Fourier spectrum as a function of complex frequency; the analytically continued spectrum. By investigating the properties of the analytically continued spectrum we show how to extract high- Q modes, how to estimate Q either from the amplitude or from the width of a resonance function, and how to improve the resolution of splitting to the theoretical maximum. Examples of these procedures, using observed data, are presented.     

Seismic attenuation values obtained from instantaneous-frequency matching and spectral ratios

In this paper, attenuation values are obtained from seismic data using instantaneous-frequency matching and spectral ratios. to obtain differential t * values using instantaneous-frequency matching, a near offset reference pulse is attenuated until the resulting instantaneous frequency matches the observed value at the receiver. Prior to matching, filtering can be applied to each trace in order to reduce the effects of noise on the calculated instantaneous frequencies. In the second method, the spectral ratio between a receiver pulse and a reference pulse is used to obtain differential t * values. to obtain an unbiased estimate, a variable spectral bandwidth is used depending on the noise level of the data. the two methods are tested using synthetic traces and then applied to crustal refraction data from the 1986 PASSCAL Ouachita experiment. Results show that the differential t * values obtained using filtered, instantaneous-frequency matching are consistent with and have less scatter than those obtained from spectral ratios with a variable bandwidth.     

Discrimination of small earthquakes and artificial explosions in the Korean Peninsula using Pg/Lg ratios

The purpose of this study is to develop a technique to discriminate artificial explosions from local small earthquakes ( M ≤ 4.0) in the time–frequency domain. In order to obtain spectral features of artificial explosions and earthquakes, 3-D spectrograms (frequency, time and amplitude) have been used. They represent a useful tool for studying the frequency content of entire seismic waveforms observed at local and regional distances (Kim, Simpson & Richards 1994). P and S(L g ) waves from quarry blasts show that the frequency content associated with the dominant amplitude appears above 10  Hz and Rg phases are observed at close distances. P and S(L g ) waves from the Tongosan earthquake have strong amplitudes below 10  Hz. For the Munkyong earthquake, however, a broader frequency content up to 20  Hz is found.
  For discrimination between small earthquakes and explosions, Pg/L g spectral ratios are used below 10  Hz, and through spectrogram analysis we can see different frequency contents of explosions and earthquakes. Unfortunately, because explosion data recorded at KSRS array are digitized at 20  sps, we cannot avoid analysing below 10  Hz because of the Nyquist frequency. In order to select time windows, the group velocity was computed using multiple-filter analysis (MFA), and free-surface effects have been removed from all three-component data in order to improve data quality. Using FFT, a log-average spectral amplitude is calculated over seven frequency bands: 0.5 to 3, 2 to 4, 3 to 5, 4 to 6, 5 to 7, 6 to 8 and 8 to 10  Hz. The best separation between explosions and earthquakes is observed from 6 to 8  Hz. In this frequency band we can separate explosions with log ( Pg/L g ) above −0.5, except EXP1 recorded at SIHY1-1, and earthquakes below −0.5, except the Munkyong earthquake record at station KMH.     

On the superparamagnetic—stable single domain transition for magnetite, and frequency dependence of susceptibility

Sediments and soils often contain superparamagnetic (SP) magnetite or maghemite grains that cause a frequency dependence of low-field susceptibility X _fd which does not exceed 15 per cent/decade of frequency. Present models predict very different volume distributions for samples with the largest observed frequency dependence of susceptibility. While Stephensons' (1971) power-law model predicts most grains to be smaller than the stable single domain (SSD) threshold, the phenomenological model of >Dearing et al . (1996) suggests that most grains are between 10 and 25 nm in diameter. Finally, the recent calculations of Eyre (1997) indicate very broad volume distributions. This study reviews the nature of the superparamagnetic–stable single domain (SP–SSD) transition. The change of AC susceptibilities with grain size (or temperature) at the SP–SSD boundary is more gradual than commonly assumed. When distributions of particle coercivities and volumes are also considered, X _fd values are much smaller than those calculated by Eyre (1997). Nonetheless, X _fd can be larger than 15 per cent, and a larger frequency dependence has indeed been measured for some samples. The question whether the observed limited X _fd of soils and sediments is a result of a broad distribution or of a bimodal distribution, where SP and SSD grains are restricted to a certain relative abundance, can potentially be answered by susceptibility determinations at more than two frequencies and by measurements of the temperature dependence of susceptibility.     

Effects of Flow Regulation in Flow Regime on the Murrumbidgee River,South Eastern Australia: an assessment using a daily estimation hydrological model

In this paper, modelled hydrological data are used to quantify the effects of regulation on the flow regime of the lower Murrumbidgee River in the period 1970–1998. Although other studies report historical changes in flood frequency and duration, this study uses modelled natural daily flow data rather than pre-regulation records or aggregated modelled monthly data. The comparison of modelled natural and regulated daily flows shows the magnitude of changes to mean and seasonal flows, flood peaks and flow duration. At gauges upstream of major irrigation off-takes, mean flows have been increased by approximately 10 per cent, flood peaks have been reduced by 21–46 per cent, and there has been a seasonal redistribution such that flows in summer and autumn have been increased at the expense of those in winter and spring. At gauges downstream of the major irrigation off-takes, mean flows have been reduced by 8–46 per cent, flood peaks have been reduced by 16–61 per cent, and flows have been decreased in all seasons.     

A Conditional Dependence Adjusted Weights of Evidence Model

One of the key assumptions in weights of evidence (WE) modelling is that the predictor patterns have to be conditionally independent. When this assumption is violated, WE posterior probability estimates are likely to be biased upwards. In this paper, a formal expression for the bias of the contrasts will be derived. It will be shown that this bias has an intuitive and convenient interpretation. A modified WE model will then be developed, where the bias is corrected using the correlation structure of the predictor patterns. The new model is termed the conditional dependence adjusted weights of evidence (CDAWE) model. It will be demonstrated via a simulation study that the CDAWE model significantly outperforms the existing WE model when conditional independence is violated, and it is on par with logistic regression, which does not assume conditional independence. Furthermore, it will be argued that, in the presence of conditional dependence between predictor patterns, weights variance estimates from WE are likely to understate the true level of uncertainty. It will be argued that weights variance estimates from CDAWE, which are also bias-corrected, can properly address this issue.     

Phase-array decomposition of a seismic section

A seismic re fraction/wide-angle reflection profile is analysed for the presence of correlated events ('phases'). The correlation problem is formulated in terms of temporally, spatially and frequency-local complex covariances. For robustness, the method concentrates on phase rather than amplitude information. This allows a computationally efficient algorithm that can make allowance for signal correlation length and can model curved wavefronts. A statistical test based on residual phase misfit across the analysed subarray is used to assess the probability that a detected event represents a real correlated signal.
With our chosen analysis parameters and confidence level (over 99.9 per cent). 1222 events were detected in the data. Using simple techniques based on 1-D earth models, detected events are associated with a small number of particular wave types. In this way, we have succeeded in classifying almost 95 per cent of the detected events. Those that remain describe those components of the data that are inconsistent with our simple ray paths in the 1-D assumption and with our prescribed tolerance. These include reverberations, near-surface guided waves and reflected waves from strongly laterally inhomogeneous structures. According to our modelling, about 25 per cent of the detected events are consistent with simple P -wave reflected energy, and these are to a very large extent (over 85 per cent) distinct from all the other wave-type models we have used. A direct mapping of the detected events into the offset-depth domain reveals dear internal and external consistencies among the detections for the various wave types. Estimated earth structure is consistent with models from previous analyses based on much larger data sets.
We have thus succeeded in extracting correlated events from the data and decomposing these, approximately but meaningfully, into distinct classes (ray paths)     

A method for the computation of finite frequency body wave synthetic seismograms in laterally varying media

Summary. Body wave synthetic siesmograms for laterally varying media are computed by means of a slowness implementation of the extended WKBJ (EWKBJ) theory of Frazer & Phinney. An EWKBJ seismogram is computed by first tracing rays through a particular model to obtain conventional ray information (travel time, ray end point, ray slowness) and then using these data in the finite frequency integral expression for the EWKBJ seismogram. The EWKBJ seismograms compare favourably to geometrical ray theory (GRT) seismograms but are significantly better because of the finite frequency nature of the EWKBJ calculation. More realistic behaviour is obtained with EWKBJ seismograms at normal seismic frequencies near caustics, where the GRT amplitude is infinite, and within geometrical shadow zones where GRT predicts zero amplitudes. In addition the EWKBJ calculation is more sensitive than GRT to focuses and defocuses in the ray field. The major disadvantage of the EWKBJ calculation is the additional computer time over that of GRT, necessary to calculate one seismogram although an EWKBJ seismogram costs much less to compute than a reflectivity seismogram. Another disadvantage of EWKBJ theory is the generation of spurious, non-geometrical phases that are associated with rapidly varying lateral inhomogeneities. Fortunately the amplitudes of these spurious phases are usually much lower than that of neighbouring geometrical phases so that the spurious phases can usually be ignored. When this observation is combined with the moderately increased computational time of the EWKBJ calculation then the gain in finite frequency character significantly outweighs any disadvantages.     

Seismic response characteristics of marine reflection profiling systems

Summary. Factors influencing the seismic response characteristics of marine profiling systems are reviewed. The single frequency case is used to illustrate the influence of different frequencies on the response, as well as the towing depths of the source and receiver, and the geometry of a linear receiving array. The more realistic case of band-limited source waveforms is considered, using frequency spectra calculated from theoretically derived airgun signals. The results show that the number and shape of sidelobes of the profiling system response, as well as the filtering characteristics for reflections arising from reflectors in the vertical plane perpendicular to the axis of the receiver array are determined by the depths of the source and receiver and the relative amplitudes of the frequencies in the source waveform. These factors, along with the configuration of the hydrophone elements in the receiver array, determine the frequency and amplitude attenuation of reflections in the vertical plane containing the receiver array.
The filtering characteristics of the system both in and out of the vertical plane containing the receiver array are discussed, with implications for discriminating between off-axis and in-plane reflections. A plan view of the response of the system is constructed in the time domain for various profiling configurations and sources of different frequency content at a given time. This example shows how useful the resulting pictures are for optimizing acquisition parameters in profiling experiments.     

Autoregressive estimation of complex eigenfrequencies in low frequency seismic spectra

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The fact that a seismogram can be represented as an autoregressive (AR) time series and the use of Prony's method enable us to obtain the complex frequencies from the poles of the corresponding AR process. A frequency-domain formulation which employs a time-domain tapering technique is devised so that spectral peaks can be analysed individually, or in small groups. Statistical analysis of the estimators and several examples from two recent earthquakes illustrate the method.     

Inversion of seismic attributes for velocity and attenuation structure

We have developed an inversion formuialion for velocity and attenuation structure using seismic attributes, including envelope amplitude, instantaneous frequency and arrival times of selected seismic phases. We refer to this approach as AFT inversion for amplitude, (instantaneous) frequency and time. Complex trace analysis is used to extract the different seismic attributes. The instantaneous frequency data are converted to t * using a matching procedure that approximately removes the effects of the source spectra. To invert for structure, ray-perturbation methods are used to compute the sensitivity of the seismic attributes to variations in the model. An iterative inversion procedure is then performed from smooth to less smooth models that progressively incorporates the shorter-wavelength components of the model. To illustrate the method, seismic attributes are extracted from seismic-refraction data of the Ouachita PASSCAL experiment and used to invert for shallow crustal velocity and attenuation structure. Although amplitude data are sensitive to model roughness, the inverted velocity and attenuation models were required by the data to maintain a relatively smooth character. The amplitude and t * data were needed, along with the traveltimes, at each step of the inversion in order to fit all the seismic attributes at the final iteration.     

Multiplet-Multiplet Coupling Due to Lateral Heterogeneity: Asymptotic Effects On the Amplitude and Frequency of the Earth's Normal Modes

Summary. Starting with the first-order formulation of quasi-degenerate splitting theory for the normal modes of a laterally heterogeneous earth, we have obtained an asymptotic expression for the coupling terms corresponding to neighbouring multiplets along the same dispersion branch as the mode considered, valid to order 1/ℓ, where ℓ is the angular order of this mode (ℓ≥ 1).
We show that, to order zero, these coupling terms introduce a small shift in epicentral distance into the expression for the long period seismogram obtained by normal mode summation. This shift depends on the difference between the great circle and the minor arc averages of the local frequency. the coupling terms thus permit us to reconcile results obtained by normal-mode summation and by a propagating wave approach, as far as the dependence on structure of the phase of surface waves is concerned.
To order 1/ℓ, the coupling terms result in a perturbation in the amplitude of the mode considered, which depends on spatial derivatives of the local frequency and thus on the structure in the vicinity of the source station great circle path. We show that this term is equivalent to that which is found using ray perturbation methods for propagating surface waves. We compare and discuss the assumptions underlying both approaches and illustrate, by an example, the potential of the asymptotic normal-mode formulation for improved modelling of lateral heterogeneity in the earth.     

Measurements and interpretation of normal mode attenuation

summary . Measurements of Q for modes of free oscillation provide the most accurate information about the anelastic properties of the whole Earth in the period range from 100 to 3000 s. We have obtained more than 230 Q measurements, by using two different techniques. Individual LaCoste—Romberg gravimeter recordings of three large earthquakes were used to observe the time rate of decay of spectral peaks corresponding to different modes. This method provided measurements of Q for 37 different modes. By stacking 211 WWSSN recordings of two deep earthquakes, we were able to measure Q for 197 modes, including many overtones which cannot be analysed using the spectra of individual recordings.     

On the frequency contents of local events: source or path effect?

Local events which occurred in the Timanfaya volcanic field are analysed and their spectral contents are interpreted as being due to the effect of a finely layered medium with large impedance contrast, the presence of which is well known from well-logging data. By considering the layered medium as a filter, its spectral response (amplitude and phase) is computed from different randomly generated values of its parameters (total thickness, mean thickness of the individual layers and impedance contrast), showing that this stratigraphic filter acts as a low-pass filter. This effect can be correctly modelled by means of an exponential correlation function. However, in some cases resonant peaks appear in the spectrum superimposed on the low-pass filter effect, which can only be attributed to resonant scattering. An analysis of the phase and group delay associated with the amplitudes is made and strongly suggests that it is necessary to incorporate the theory of resonant scattering for the retrieval of medium properties. From the point of view of the amplitudes, the superposition of both effects, low-pass filter and resonant peaks, allows us to reproduce the gross features that are usually attributed to source effects. The need for a careful analysis of the local site response previous to any interpretation of the spectral features in terms of source characteristics is thus emphasized.     

Could the energy near the FCN and the FICN be explained by luni-solar or atmospheric forcing?

The observed time-series of precession/nutation show residuals with respect to an empirical model based on the rigid Earth theoretical nutations and a frequency dependent transfer function with resonances to the Earth's normal modes. These residuals display energy mainly in the frequency domain around 430 and 500 days in the inertial frame. In this frequency band, the energy is possibly related to two normalmode frequencies: the free core nutation (FCN) and the free inner core nutation (FICN). In this paper, we examine the possibility of obtaining this energy from the resonance effect induced by a luni-solar (or planetary) forcing, or by an atmospheric forcing at a frequency very close to these Earth free nutations. The amplification factor due to the resonance is computed from an analytical formula expressed in the case of a simplified three-layer ellipsoidal rotating earth (with an elastic inner core, a liquid outer core and an elastic mantle), as well as the empirical formula based on the analysis of VLBI observations. For the tidal forcing, the theoretical results do not show any resonance at the level of precision we have examined but it is still possible to find one frequency near the FCN or FICN frequencies which could be excited. In contrast, for the atmospheric pressure the level of energy needed could be obtained from the diurnal pressure, depending on the noise level of the Earth's global pressure. We also show that the combination of three waves can explain the observed decrease of energy with time. While the tidal potential amplitudes are too small, a pressure noise level of 0.5 Pa would be sufficient to excite these waves.