THE ESTIMATION OF SIGNAL SPECTRA AND RELATED QUANTITIES BY MEANS OF THE MULTIPLE COHERENCE FUNCTION *

A seismic trace recorded with suitable gain control can be treated as a stationary time series. Each trace, χ_j(t), from a set of traces, can be broken down into two stationary components: a signal sequence, α_j(t) *s(t—τ_j), which correlates from trace to trace, and an incoherent noise sequence, n_j(t), which does not correlate from trace to trace. The model for a seismic trace used in this paper is thus χ_j(t) =α_j(t) * s(t—τ_j) +n_j(t) where the signal wavelet α_j(t), the lag (moveout) of the signal τ_j, and the noise sequence n_j(t) can vary in any manner from trace to trace. Given this model, a method for estimating the power spectra of the signal and incoherent noise components on each trace is presented. The method requires the calculation of the multiple coherence function γ_j(f) of each trace. γ_j(f) is the fraction of the power on traced at frequency f that can be predicted in a least-square error sense from all other traces. It is related to the signal-to-noise power ratio ρ_j(f) by where K_j(f) can be computed and is in general close to 1.0. The theory leading to this relation is given in an Appendix. Particular attention is paid to the statistical distributions of all estimated quantities. The statistical behaviour of cross-spectral and coherence estimates is complicated by the presence of bias as well as random deviations. Straightforward methods for removing this bias and setting up confidence limits, based on the principle of maximum likelihood and the Goodman distribution for the sample multiple coherence, are described. Actual field records differ from the assumed model mainly in having more than one correctable component, components other than the required sequence of reflections being lumped together as correlated noise. When more than one correlatable component is present, the estimate for the signal power spectrum obtained by the multiple coherence method is approximately the sum of the power spectra of the correlatable components. A further practical drawback to estimating spectra from seismic data is the limited number of degrees of freedom available. Usually at least one second of stationary data on each trace is needed to estimate the signal spectrum with an accuracy of about 10%. Examples using synthetic data are presented to illustrate the method.     

Klauder wavelet removal before vibroseis deconvolution

The spiking deconvolution of a field seismic trace requires that the seismic wavelet on the trace be minimum phase. On a dynamite trace, the component wavelets due to the effects of recording instruments, coupling, attenuation, ghosts, reverberations and other types of multiple reflection are minimum phase. The seismic wavelet is the convolution of the component wavelets. As a result, the seismic wavelet on a dynamite trace is minimum phase and thus can be removed by spiking deconvolution. However, on a correlated vibroseis trace, the seismic wavelet is the convolution of the zero-phase Klauder wavelet with the component minimum-phase wavelets. Thus the seismic wavelet occurring on a correlated vibroseis trace does not meet the minimum-phase requirement necessary for spiking deconvolution, and the final result of deconvolution is less than optimal. Over the years, this problem has been investigated and various methods of correction have been introduced. In essence, the existing methods of vibroseis deconvolution make use of a correction that converts (on the correlated trace) the Klauder wavelet into its minimum-phase counterpart. The seismic wavelet, which is the convolution of the minimum-phase counterpart with the component minimum-phase wavelets, is then removed by spiking deconvolution. This means that spiking deconvolution removes both the constructed minimum-phase Klauder counterpart and the component minimum-phase wavelets. Here, a new method is proposed: instead of being converted to minimum phase, the Klauder wavelet is removed directly. The spiking deconvolution can then proceed unimpeded as in the case of a dynamite record. These results also hold for gap predictive deconvolution because gap deconvolution is a special case of spiking deconvolution in which the deconvolved trace is smoothed by the front part of the minimum-phase wavelet that was removed.     

ON THE BREMMER SERIES DECOMPOSITION: EQUIVALENCE BETWEEN TWO DIFFERENT APPROACHES*

A Bremmer Series decomposition of the solution y(t) to the lossless wave equation in layered media is where the y_j(t) are physically meaningful constituents (i.e., y₁(t) are primaries, y₂(t) are secondaries, etc.). This paper reviews Mendel's state space models for generating the constituents; reviews Bremmer's integral equation models for generating the constituents; and demonstrates how Mendel's state space models can be obtained by a careful decomposition of Bremmer's integral equation models. It shows that Mendel's equations can be viewed as approximate numerical solutions of Bremmer's integral equations. In a lossless homogeneous medium, the approximations become exact.     

A shot noise model for the generation of soil moisture data

An attempt is made to estimate the expected contribution of rainfall to soil moisture during the irrigation season. Effective rainfall and evapotranspiration are the parameters considered in the water balance carried out in the root zone. Rainfall occurrence is simulated by a Poisson process whereas evapotranspiration is described by a simple deterministic function of potential evapotranspiration and soil moisture in the root zone. Using the theory of shot noise models a closed form solution is derived from the expected soil moisture in the root zone at the end of the time interval (0,t]. For illustration purposes the proposed model is applied to a series of data from Mikra meteorological station in Greece.List of symbols x change in water storage in the root zone during the time interval t - X infiltrated rainfall of thei th storm event - ET actual evapotranspiration during thej th day - Poisson rate - number of storm events in (0,t] - t _r duration of rainfall - t _b interarrival time - h _i rainfall depth of thei th storm event - i _m mean rainfall intensity - i(t) instantaneous rainfall intensity - x(0),x(t) available soil moisture in the root zone at time 0 andt, respectively - PET potential evapotranspiration rate - x _F available soil moisture in the root zone at field capacity - soil moisture depletion rate (=PET/x _F ) - w impulse shape of filtered Poisson processes - E[·] mean value - S _i time of thei th rainfall event - N(t) time of storm events in (0,t] - estimated standard deviation The following symbols were used in this paper     

A multivariate gamma distribution arising from a Markov model

A Markov chain{X _t }, which has been useful for modelling in hydrology, can be specified by the Laplace transform (LT) of the conditional p.d.f. ofX _t+1 givenX _t =x _t , which is assumed to be of the exponential formH()exp{-G()x _t }. For appropriate choice ofH andG the marginal distribution ofX _t is the (univariate) gamma distribution. In this case, the joint p.d.f. ofX _t +1,...,X _t+n and its LT, are obtained, and this is extended to a seasonal version of the chain. A simple method of generating observations from these multivariate gamma distributions is noted, and the joint LT is applied to the problem of determining moments of weighted sums of such variables.     

Geopotential Reconstruction,Decomposition, Fast Computation,and Noise Cancellation by Harmonic Wavelets

Harmonic wavelets are introduced within the framework of the Sobolev-like Hilbert space H of potentials with square-integrable restrictions to the Earth's (mean) sphere _R . Basic tool is the construction of H-product kernels in terms of an (outer harmonics) orthonormal basis in H. Scaling function and wavelet are defined by means of so-called H-product kernels. Harmonic wavelets are shown to be building blocks that decorrelate geopotential data. A pyramid scheme enables fast computations. Multiscale signal-to-noise thresholding provides suitable denoising. Multiscale modelling of the Earth's anomalous potential from EGM96-model data is illustrated by use of bandlimited harmonic wavelets, i.e. Shannon and CP-wavelets.     

Hydroxyl (6/2) airglow emission intensity ratios for rotational temperature determination

OH(6/2) Q1/P1 and R₁/P₁ airglow emission intensity ratios, for rotational states up to j = 4.5, are measured to be lower than implied by transition probabilities published by various authors including Mies, Langhoff et al. and Turnbull and Lowe. Experimentally determined relative values of j transitions yield OH(6/2) rotational temperatures 2 K lower than Langhoff et al., 7 K lower than Mies and 13 K lower than Turnbull and Lowe.     

Duration Magnitude Scale and Site Residuals for Northern Morocco

— The first empirical duration magnitude (M_D) formula is developed and tested for the Northern Morocco Seismic Network (NMSNET). This relationship is obtained by relating the IGN (Instituto Geografico National, Madrid) body-waves mbLg^IGN to the duration (), and the epicentral distance (), at 25 analogue stations of the NMSNET for 479 earthquakes with 2.5 mb 5.4, from March 1992 to February 2001. M_D estimates are significantly more precise while introducing a correction term for each of these stations, cSta_j. The magnitude for the i^th event (M_D)_i is the mean value of individual M_Dij=–0.14+1.63log₁₀(_ij)+0.031(_ij)+cSta_j. The cSta_j corrections reduce considerably the local site effects which influence the recorded durations and cause stations to either overestimate, or underestimate M_D up to 0.5 magnitude units. Average station M_D residuals (–cSta_j) are found to be independent of the distance from the epicenter to at least 10 degrees. It seems evident that regional geological features in the immediate behavior of stations have a systematic effect on the corresponding obtained residuals: older well-consolidated Precambrian crystalline rocks produce high negative residuals (shorter durations), younger unconsolidated sediments produce high positive residuals (longer durations), whereas, intermediate M_D site residuals appear to be the result of the effect of various factors, principally age and state of consolidation of the bedrock, combined with the local tectonic.     

A TIME-DOMAIN APPROACH TO THE NORMAL-INCIDENCE INVERSE PROBLEM*

This tutorial paper presents a time-domain derivation of the Normal equations which are associated with the normal-incidence inverse problem. Connections are made between our time-domain derivation and the more familiar z-transform transfer function derivation.     

Propagation of the radiofrequency ground wave transient over a finitely conducting plane earth

Summary The introduction of precision radio navigation systems employing pulse techniques and the ever increasing interest in spherics have stimulated considerable interest in the propagation of the ground wave transient over the surface of the earth. The theory of the propagation of a transient radio frequency ground wave over a finitely conducting plane earth is presented for the particular case of theNorton surface wave by a consideration of a wave, interrupted abruptly at one point in time (t=0), a wave interrupted abruptly at two points in time (t=0,T ₂) and a wave interrupted at one point in time followed by an exponential decay. The first case is illustrated by several numerical examples of a cosine current wave applied to a vertical electric dipole source. It is apparent that the method of the inverse aplace transform for the particular cases considered yields some simple mathematical formulas.     

Theoretical investigation of the time variation of drainage density

A theoretical equation was developed to express the time variation of drainage density in a basin or geomorphic surface: D_i(t, T) is the drainage density at time T on the i-th basin or geomorphic surface, which was formed at time t; β(τ) is a factor related to the erosional force causing the development of the rivers of the basin or surface at time τ; δ_i is the maximum drainage density; and D_i is the initial drainage density on the i-th geomorphic surface or basin. The equation is based on the assumption that the drainage density increases with time until it reaches a specific upper limit δ_i(t)), the maximum drainage density, which is related to certain physical properties of the basin. The equations for various dated basins or geomorphic surfaces can be combined into one modified equation if the same relative erosional forces have acted on those basins or surfaces (β(t) = β(t) and if the basins or surfaces have the same physical properties δ_i(t) = δ_i(t), (D_i = D₀). The application of this equation to coastal terraces and glacial tills shows that the model is compatible with observed drainage densities on various dated basins or surfaces.     

Computation of travel times and rays in a medium of two-dimensional velocity distribution

A ray-tracing computer program is described for a two-dimensional velocity distribution defined by the local wave velocitiesv _{i, j} in points at arbitrary depthv _{i, j} below the surface points with the horizontal coordinatesx _i . The velocity variation is assumed to be linear in the triangles formed by three neighbouring points. Travel times and rays are then calculated after the exact analytic formulae for any position of the source within the model. No assumptions other than of a piecewise linear velocity structure are made. A first-order discontinuity can be approximated by a thin layer with a strong velocity gradient and refracted waves or wide-angle reflections obtained in this way. As an example,P-wave rays were computed for section No. 05 of the Alpine Longitudinal Profile. The model includes a low-velocity channel which is cut off on the eastern side, first-order discontinuities and a sediment basin.Paper presented at the ESC-Workshop Meeting Seismic Waves in Laterally Inhomogeneous Media, Liblice, 1978.     

A daily rainfall occurrence process

A model for the periodic (annual cycle), discrete rainfall occurrence process is presented. Using this model the probabilistic properties of the process in -day intervals can be investigated. In such an interval the rainfall occurrence process is approximated by some stationary processa _t ,tIN. The processa _t ,tIN is described by the distributions of the lengths of wet and dry sequences. It is assumed that the lengths of successive wet and dry sequences are independent. For this process the distribution of the number of wet days in -day intervals is calculated. The model is fitted to 50-year rainfall data from Wroclaw, Poland. Rainfall amounts of 0.1, 1.0 and 2.0 mm are considered as thresholds defining a wet day. To estimate the distribution of the length of wet and dry sequences the family of Pascal distribution is chosen.     

Approximate expression for the hilbert transform of a certain class of functions and their application to the ray theory of seismic waves

Summary Approximate expressions for the Hilbert transform of the functionf(t)=exp(- ₀ ² t ²/²) cos( ₀ t+v) are determined. This function, given a suitable choice of the three parameters ₀, and v, approximates a wide class of seismic signals very well. The approximate expressions for the Hilbert transform enable very simple formulae to be given for the elementary seismograms of the individual seismic body waves (in the zero approximation of the ray theory). This accelerates the computation of ray theoretical seismograms considerably.     

Stress wave patterns and the size of the non-elastic zones of an explosive source

Summary The travel-time curves of characteristic moments of stress wave patterns were investigated. The arrival time of the onset of stress wavet ₀, of the maximum amplitudet ₁ and of the moment terminating the pressure part (the first half-wave) of this wavet ₂ were taken to be representative. These travel-time curves were used to determine the velocities of propagation of these moments (V ₀,V ₁ andV ₂) as functions of the distance from the source. According to their variations it is possible to appoint the size of the cavity created by the explosion and to determine the distance at which the elastic source surface of stress waves is to be found. The radius of the cavity is given by 1.5 times the distance at whichV ₁ separates fromV ₀. The elastic source surface of stress waves is defined by the distance from whichV ₀ is constant and at whichV ₁ andV ₂ have minimum values. These two distances determine the points at which the character of the stress wave pattern changes: from shock wave in gases, through shock wave in a solid medium to a seismic wave.Paper presented at the General Assembly of the European Seismological Commission, Luxembourg, Sept. 21–29, 1970.     

Peculiarities of Rayleigh wave displacement in a transversely isotropic half space

We consider a transversely isotropic medium with vertical axis of symmetry (VTI). Rayleigh wave displacement components in a homogeneous VTI medium contain exp(±kr_jz), where z is the vertical coordinate, k is the wave number, and j?=?1, 2; r_j may be considered as depth-decay factor. In a VTI medium, r_j can be a real or imaginary as in an isotropic medium, or it can be a complex depending on the elastic parameters of the VTI medium; if complex, r₁ and r₂ are complex conjugates. In a homogeneous VTI half space, Rayleigh wave displacement is significantly modified with a phase shift when r_j changes from real to complex with variation of VTI parameters; at the transition, the displacement becomes zero when r₁?=?r₂. In a liquid layer over a VTI half space, Rayleigh waves are dispersive. Here, also Rayleigh wave displacement significantly modified with a phase shift when r_j changes from real to complex with a decrease of period. At very low period, phase velocity of Rayleigh waves becomes less than P-wave velocity in the liquid layer giving rise to Scholte waves (interface waves). The amplitudes of Scholte waves with a VTI half space are found to be significantly larger than those with an isotropic half space.     

Limits of transversely isotropic elastic parameters for the existence of classical Rayleigh waves

Solutions of P-SV equations of motion in a homogeneous transversely isotropic elastic layer contain a factor exp(±ν _j z), where z is the vertical coordinate and j?=?1, 2. For computing Rayleigh wave dispersion in a multi-layered half space, ν _j is computed at each layer. For a given phase velocity (c), ν _j becomes complex depending on the transversely isotropic parameters. When ν _j is complex, classical Rayleigh waves do not exist and generalised Rayleigh waves propagate along a path inclined to the interface. We use transversely isotropic parameters as α _H , β _V , ξ, ? and η and find their limits beyond which ν _j becomes complex. It is seen that ν _j depends on ? and η, but does not depend on ξ. The complex ν _j occurs when ? is small and η is large. For a given c/β _V , the region of complex ν _j in a ? -η plane increases with the increase of α _H /β _V . Further, for a given α _H /β _V , the complex region of ν _j increases significantly with the decrease of c/β _V . This study is useful to compute dispersion parameters of Rayleigh waves in a layered medium.     

Aspetto elettrogeosmotico bidimensionale variabile col tempo

Riassunto Fino ad oggi sono apparse solo teorie elettrogeosmotiche transitorie unidimensionali. Quì risolviamo un problema del genere bidimensionale, con elettrodi verticali cilindrici. Le curve rappresentative delle portate d'acqua catodicaq _K a cui perveniamo, denotano diminuzioni abbastanza rapide fino ai tempi dell' ordine dei «tempi elettrodici », (t=), portate che poi diminuiscono in modo molto lento. Si riscontrano quì più estesi «pianerottoli»q _K che non quelli delle monodimensionalità (t=/9). Inoltre, col crescere dei rapporti distanze eteropolari-raggi catodici, o col diminuire dei raggi catodici, tali portate (che si compongono dei contributi dei singoli anodi) s'incrementano nel modo diagrammato.

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Summary Till now only mono-dimensional transient electro-geoosmotical theories have been published. In this paper a solution is given for a problem of bi-dimensional type, with vertical cylindrical electrodes. The plots which represent the amounts of cathodic waterq _K show quite rapid lowerings as far as to reach time durations of the order of «electrodic times » (t=) volumes which, after, diminish very slowly. In this case more extendedq _K «lobbies» are found than those in mono-dimensional case (t=/9). Furthermore increasing the ratios heteropolar spacings—cathodic radii, or diminishing the cathodic radii, water discharges (which result from the contribution of the single anods) increase as plotted in the diagrams.

     

Formulae for errors for initial displacement and velocity problems using the Newmark method

A standard test of the Newmark method in structural dynamics is its application to the determination of the response of a damped or undamped single-degree-of-freedom system to a prescribed initial displacement or velocity. In this paper formulae for the error, Δ_j, in the response, after applying the Newmark method for j time-steps each of duration Δt, are proposed and their acceptable accuracy is demonstrated.     

The elastic properties of single-crystal fayalite as determined by dynamical measurement techniques

We present new elasticity measurements on single-crystal fayalite and combine our results with other data from resonance, pulse superposition interferometry, and Brillouin scattering to provide a set of recommended values for the adiabatic elastic moduliC _ij and their temperature variations. We use a resonance method (RPR) with specimens that were previously investigated by pulse superposition experiments. The nineC _ij of fayalite are determined from three new sets of measurements. One set of our newC _ij data is over the range 300–500 K. We believe that the relatively large discrep ncies found in someC _ij are due in large part to specimen inhomogeneities (chemical and microstructural) coupled with differences in the way various techniques sample, rather than only systematic errors associated with experimental procedures or in the preparations of the specimens.Our recommendeaC _ij's (GPa) and (C _ij/T) _p (GPa/K) are: The resulting values for the isotropic bulk and shear moduli,K _s and , and their temperature derivatives are:K _s=134(4) GPa; =50.7(0.3) GPa; (K _s/T) _p =–0.024(0.005) GPa/K; and (/T) _p =–0.013(0.001) GPa/K. An important conclusion is thatK _s increases as the Fe/(Fe+Mg) ratio in olivine is increased.