SEISMIC WAVE PROPAGATION IN LAYERED MEDIA IN TERMS OF COMMUNICATION THEORY

Geophysics ◽  
1966 ◽  
Vol 31 (1) ◽  
pp. 17-32 ◽  
Author(s):  
S. Treitel ◽  
E. A. Robinson
Keyword(s):  
Impulse Response ◽  
Communication Theory ◽  
Amplitude Spectrum ◽  
P Wave ◽  
Earth Model ◽  
Phase Lag ◽  
Elastic Plates ◽  
Pass System ◽  
Maximum Delay ◽  
Minimum Delay

The problem of a normally incident plane P wave propagating in a system of horizontally layered homogeneous perfectly elastic plates is reformulated in terms of concepts drawn from communication theory. We show how both the reflected and transmitted responses of such a system can be expressed as a z transform which is the ratio of two polynomials in z. Since this response must be stable, the denominators of the z transforms describing the reflected and transmitted motion are minimum delay (i.e., minimum‐phase lag). If the layered medium is bounded at depth by a perfect reflector, then the reflected impulse response recorded at the surface is in the form of a dispersive all‐pass z transform. A dispersive all‐pass system is one whose z transform is the ratio of the z transform of a maximum‐delay wavelet to that of its corresponding minimum‐delay wavelet; hence, the amplitude spectrum of a dispersive all‐pass system is unity for all frequencies. This means that the amplitude spectrum of the reflected response is identical to the amplitude spectrum of the input wavelet used to excite the system. More specifically, all the energy put in is returned with the same frequency content, but is differentially delayed. The phase‐lag spectrum of the reflected response lies everywhere above the phase‐lag spectrum of the input wavelet. Thus, the all‐pass situation implies that the layered earth model considered here, while not able to alter the amplitude of the frequency components of the input wavelet, will introduce differential time delays with certain properties into each such component. Finally, since the reflected impulse response is an all‐pass wavelet, its autocorrelation is a spike of unit magnitude at τ=0, and zero for all other lags.

Retrieval of source-extent parameters and the interpretation of corner frequency

1983 ◽  
Vol 73 (6A) ◽  
pp. 1499-1511
Author(s):  
Paul Silver
Keyword(s):  
Body Wave ◽  
Amplitude Spectrum ◽  
Low Frequency ◽  
P Wave ◽  
Corner Frequency ◽  
S Wave ◽  
Dislocation Source ◽  
Statistical Measures ◽  
Dislocation Models ◽  
Planar Dislocation

Abstract A method is proposed for retrieving source-extent parameters from far-field body-wave data. At low frequency, the normalized P- or S-wave displacement amplitude spectrum can be approximated by |Ω^(r^,ω)| = 1 − τ2(r^)ω2/2 where r^ specifies a point on the focal sphere. For planar dislocation sources, τ2(r^) is linearly related to statistical measures of source dimension, source duration, and directivity. τ2(r^) can be measured as the curvature of |Ω^(r^,ω)| at ω = 0 or the variance of the pulse Ω^(r^,t). The quantity ωc=2τ−1(r^) is contrasted with the traditional corner frequency ω0, defined as the frequency at the intersection of the low- and high-frequency trends of |Ω^(r^,ω)|. For dislocation models without directivity, ωc(P) ≧ ωc(S) for any r^. A mean corner frequency defined by averaging τ2(r^) over the focal sphere, ω¯c=2<τ2(r^)>−1/2, satisfies ωc(P) > ωc(S) for any dislocation source. This behavior is not shared by ω0. It is shown that ω0 is most sensitive to critical times in the rupture history of the source, whereas ωc is determined by the basic parameters of source extent. Evidence is presented that ωc is the corner frequency measured on actual seismograms. Thus, the commonly observed corner frequency shift (P-wave corner greater than the S-wave corner), now viewed as a shift in ωc is simply a result of spatial finiteness and is expected to be a property of any dislocation source. As a result, the shift cannot be used as a criterion for rejecting particular dislocation models.

scholarly journals 2D Multitransient Electromagnetic Response Modeling of South China Shale Gas Earth Model Using an Approximation of Finite Difference Time Domain with Uniaxial Perfectly Matched Layer

2016 ◽  
Vol 2016 ◽  
pp. 1-20 ◽  
Author(s):  
Olalekan Fayemi ◽  
Qingyun Di
Keyword(s):  
Finite Difference ◽  
Impulse Response ◽  
South China ◽  
Shale Gas ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Perfectly Matched Layer ◽  
Electromagnetic Response ◽  
Earth Model ◽  
Difference Time

In this study, we introduced multitransient electromagnetic (MTEM) method as an effective tool for shale gas exploration. We combined the uniaxial perfectly matched layer (UPML) equation with the first derivative diffusion equation to solve for a finite difference time domain (FDTD) UPML equation, which was discretized to form an algorithm for 3D modeling of earth impulse response and used in modeling MTEM response over 2D South China shale gas model. We started with stepwise demonstration of the UPML and the FDTD algorithm as an effective tool. Subsequently, quantitative study on the convergence of MTEM earth impulse response was performed using different grid setup over a uniform earth material. This illustrates that accurate results can be obtained for specified range of offset. Furthermore, synthetic responses were generated for a set of geological scenarios. Lastly, the FDTD algorithm was used to model the MTEM response over a 2D shale gas earth model from South China using a PRBS source. The obtained apparent resistivity section from the MTEM response showed a similar geological setup with the modeled 2D South China shale gas section. This study confirmed the competence of MTEM method as an effective tool for unconventional shale gas prospecting and exploitation.

On: “Homomorphic deconvolution” by D. J. Jin and J. R. Rogers (GEOPHYSICS, 48, 1014–1016).

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1559-1560
Author(s):  
Mark Lane ◽  
Tad Ulrych
Keyword(s):  
Standard Deviation ◽  
Relative Magnitude ◽  
Phase Unwrapping ◽  
Inverse Transform ◽  
Maximum Delay ◽  
Complex Cepstrum ◽  
Minimum Delay ◽  
Goertzel Algorithm ◽  
Noise Realization ◽  
Do So

The recent note by Jin and Rogers (1983) presented examples of the failure of the homomorphic transform to invert properly. Since this transform is not only of interest in geophysics, but has also found applications in other fields (Oppenheim and Schafer, 1975), these results are of concern. We consequently attempted to reproduce Jin and Rogers’ results. We failed to do so. In fact, in our experience, the transform has always inverted successfully. Our results using the first example of Jin and Rogers are shown in Figure 1. We used the algorithm of Tribolet (1977) with a modified Goertzel algorithm (Bonzanigo, 1978) for phase unwrapping. The figure is arranged as in Jin and Rogers’ paper. Figure 1a shows the input: impulses separated by 20 samples, of magnitude 2000 and 1999. Figure 1b shows its complex cepstrum. We have set the zero‐quefrency point to zero since this represents a scale factor and can dominate the plotting. Note the minimum delay cepstrum with a small amount of aliasing. The sequence returned by the inverse transform is shown in Figure 1c, demonstrating a successful inversion. The effect of noise is also shown. Noise with a standard deviation of 5 was added to the sequence of Figure 1a. This is shown in Figure 1d. Note that our noise realization is undoubtedly different from that of Jin and Rogers. The noise has changed the relative magnitude of the original spikes such that they are maximum delay. This is reflected in the cepstrum (Figure 1e). Figure 1f shows the returned sequence, again demonstrating the successful inversion.

scholarly journals Imaging structure and geometry of slabs in the greater Alpine area – A P-wave traveltime tomography using AlpArray Seismic Network data

2021 ◽  
Author(s):  
Marcel Paffrath ◽  
Wolfgang Friederich ◽  
Keyword(s):  
Alpine Region ◽  
Seismic Network ◽  
P Wave ◽  
Fault System ◽  
Earth Model ◽  
Low Velocity ◽  
European Origin ◽  
Traveltime Tomography ◽  
Model Domain ◽  
Asthenospheric Upwelling

Abstract. We perform a teleseismic P-wave traveltime tomography to examine the geometry and structure of subducted lithosphere in the upper mantle beneath the Alpine orogen. The tomography is based on waveforms recorded at over 600 temporary and permanent broadband stations of the dense AlpArray Seismic Network deployed by 24 different European institutions in the greater Alpine region, reaching from the Massif Central to the Pannonian Basin and from the Po plain to the river Main. Teleseismic traveltimes and traveltime residuals of direct teleseismic P-waves from 331 teleseismic events of magnitude 5.5 and higher recorded between 2015 and 2019 by the AlpArray Seismic Network are extracted from the recorded waveforms using a combination of automatic picking, beamforming and cross-correlation. The resulting database contains over 162.000 highly accurate absolute P-wave traveltimes and traveltime residuals. For tomographic inversion, we define a model domain encompassing the entire Alpine region down to a depth of 600 km. Outside this domain, a laterally homogeneous standard earth model is assumed. Predictions of traveltimes are computed in a hybrid way applying a fast Tau-P method outside the model domain and continuing the wavefronts into the model domain using a fast marching method. For teleseismic inversion, we iteratively invert demeaned traveltime residuals for P-wave velocities in the model domain using a regular discretization with an average lateral spacing of about 25 km and a vertical spacing of 15 km. The inversion is regularized towards an initial model constructed from an a priori model of the crust and uppermost mantle and a standard earth model beneath. The resulting model provides a detailed image of slab configuration beneath the Alpine and Apenninic orogens. Major features are an overturned Adriatic slab beneath the Apennines reaching down to 400 km depth still attached in its northern part to the crust but exhibiting detachment towards the southeast. A fast anomaly beneath the western Alps indicates a short western Alpine slab that ends at about 100 km depth close to the Penninic front. Further to the east and following the arcuate shape of the western Periadriatic Fault System, a deep-reaching coherent fast anomaly with complex interior stucture generally dipping to the SE down to about 400 km suggests a slab of European origin extending eastward to the Giudicarie fault. This slab is detached from overlying lithosphere at its eastern end below a depth of about 100 km. Further to the east, well-separated from the slab beneath the western and central Alps, another deep-reaching, nearly vertically dipping high-velocity anomaly suggests the existence of a slab beneath the Eastern Alps of presumably European origin which is completely detached from the orogenic root. Our image of this slab does not require a polarity switch because of its nearly vertical dip and full detachment from the overlying lithosphere. Fast anomalies beneath the Dinarides are weak and concentrated to the northernmost part and shallow depths. Low-velocity regions surrounding the fast anomalies beneath the Alps to the west and northwest follow the same dipping trend as the overlying fast ones, indicating a kinematically coherent subducting tectosphere in this region. In contrast, low-velocity anomalies to the east suggest asthenospheric upwelling presumably driven by retreat of the Carpathian slab and extrusion of eastern Alpine lithosphere towards the east while low velocities to the south are presumably evidence of asthenospheric upwelling and mantle hydration due to the backarc position behind the European slab.

Focal mechanism of intermediate depth earthquakes in the Alboran Sea (Western Mediterranean)

2021 ◽  
Author(s):  
Carolina López-Sánchez ◽  
Elisa Buforn ◽  
Maurizio Mattesini ◽  
Simone Cesca ◽  
Juan Vicente Cantavella ◽  
...  
Keyword(s):  
Moment Tensor ◽  
Probabilistic Approach ◽  
Focal Mechanisms ◽  
P Wave ◽  
Earth Model ◽  
Depth Range ◽  
Alboran Sea ◽  
Intermediate Depth ◽  
Velocity Models ◽  
Alborán Sea

<p>One of the characteristics of the seismicity in the Ibero-Maghrebian region is the occurrence of intermediate depth earthquakes (50<h<100 km), their largest concentration located at the western part of the Alboran Sea, with epicenters following an NNE-SSW alignment. In this study, we have relocated over 200 intermediate depth earthquakes (M≥3) occurred in this region in the period 2000-2020, using a non-linear probabilistic approach (NonLinLoc algorithm) together with a recent regional 3D tomography lithospheric velocity model for the Alboran-Betic Rif Zone. Maximum likelihood hypocenters confirm the NNE-SSW distribution in a depth range between 50 and 100 km. We have determined the focal mechanisms of 26 of these earthquakes with magnitudes (mb) greater than 3.9. We first derived focal mechanisms using the P-wave first motion polarity method and then performed a moment tensor inversion, using a probabilistic inversion approach based on the simultaneous fit of waveforms and amplitude spectra of P and S phases. We performed an accurate resolution study, by repeating the inversion using different 1-D velocity models and testing different moment tensor (MT) constraints: a full moment tensor, a deviatoric moment tensor and a pure double couple (DC). Misfit values are similar for different MT constraints. Most solutions have a non-DC component larger than 30%. This may be due to the tectonic complexity of the region and the use on the inversion of 1-D Earth model. The DC components obtained from the inversion show different orientations of the nodal planes. A first group of events to the northern part with epicenters inland on south Spain have horizontal tension axes in NE-SW direction. A second group of earthquakes with epicenters off-shore, but close to the Spanish coast, presents near-vertical pressure axes. The third group, formed by deeper earthquakes, with epicenters on the center of the Alboran sea have dip slip focal mechanisms of either normal or reverse motion with planes either vertical or dipping 45º plane oriented in NNE-SSW direction, approximately the same orientation as the alignment of their epicenters. The distribution of these intermediate depth earthquakes and their focal mechanisms evidence the seismotectonic complexity of the region related with a possible subduction.</p>

Transient lightning response to grounding grids buried in horizontal multilayered earth model considering time domain quasi-static complex image method and soil ionization effect

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Zhong-Xin Li ◽  
Peng Li ◽  
Ke-Chao Wang
Keyword(s):  
Impulse Response ◽  
Current Distribution ◽  
Time Domain ◽  
Earth Model ◽  
Content Type ◽  
Lightning Current ◽  
Ionization Effect ◽  
Lightning Impulse ◽  
Grounding Grids ◽  
Soil Ionization

Purpose The purpose of this paper is to propose a fast, accurate and efficient algorithm for assessment of transient behavior of grounding grids buried in horizontal multilayered earth model considering soil ionization effect. Design/methodology/approach The purpose of this paper is to develop a numerical simulation method to calculate the lightning impulse response of the grounding grid buried in a horizontal multilayered earth model. The mathematical model about the hybrid method based on PI basic function belonging to time domain is proposed in the paper; the mode can precisely calculate the lightning current distribution and lightning impulse response to grounding grids buried in horizontal multilayered soil model considering soil ionization effect. To increase computing efficiency, quasi-static complex image method (QSCIM) and its time-domain Green’s function closed form are introduced in the model. Findings The hybrid model is rather stable, with the respect to the number of elements used and with excellent convergence rate. In addition, because this mathematical model belongs to the time domain algorithm, it is very powerful for the simulation of soil ionization caused by high amplitude lightning current. Research limitations/implications To increase computing efficiency, QSCIM and its time domain Green's function closed form are introduced in the model. Practical implications The mathematical model about the hybrid method based on PI basic function can precisely calculate the lightning current distribution and lightning impulse response to grounding grids buried in horizontal multilayered soil model considering the soil ionization effect. Social implications Considering the soil ionization effect, the simulation calculation of lightning impulse response of substation grounding grid buried in the actual horizontal multilayered earth can effectively support the scientific and efficient design of lightning protection performance of substation grounding grid. Originality/value The hybrid model in time domain is originally developed by the authors and used to precisely calculate the lightning current distribution and lightning impulse response to grounding grids buried in horizontal multilayered soil model considering soil ionization effect. It is simple and very efficient and can easily be extended to arbitrary grounding configurations.

Mixed‐phase deconvolution

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 637-647 ◽  
Author(s):  
Milton J. Porsani ◽  
Bjorn Ursin
Keyword(s):  
Unit Circle ◽  
Amplitude Spectrum ◽  
Real Data ◽  
Coefficient Matrix ◽  
Mixed Phase ◽  
Time Signal ◽  
Lp Norm ◽  
Optimal Value ◽  
Minimum Delay ◽  
Delay Component

We describe a new algorithm for mixed‐phase deconvolution. It is valid only for pulses whose Z-transform has no zeros on the unit circle. That is, the amplitude spectrum cannot be zero for any frequency. Using the Z-transform of a discrete‐time signal, and assuming that the signal has α zeros inside the unit circle, the inverse of its minimum‐delay component may be estimated by solving the extended Yule‐Walker (EYW) system of equations with the lag α of the autocorrelation function (ACF) on diagonal of the coefficient matrix. This property of the solution of EYW equations is exploited to derive mixed‐phase inverse filters and their corresponding mixed‐phase pulses. For different values of α, a suite of inverse filters is generated using the same ACF. To choose the best decomposition and its corresponding mixed‐phase inverse filter, we have used the value of α which gives the maximum value of the Lp norm of the filtered signal. The optimal value of α does not seem to be very sensitive to the choice of norm as long as p > 2. In the numerical examples, we have used p = 5. The mixed‐phase deconvolution filter performs better than minimum‐phase deconvolution on the synthetic and real data examples shown.

Two methods for determining geophone orientations from VSP data

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. V87-V97 ◽  
Author(s):  
Xiaoxian Zeng ◽  
George A. McMechan
Keyword(s):  
A Priori ◽  
Polarization Plane ◽  
Seismic Profile ◽  
P Wave ◽  
Earth Model ◽  
S Wave ◽  
Coherent Signals ◽  
Plane Method ◽  
Wave Separation ◽  
Vsp Data

Vertical seismic profile (VSP) data are usually acquired with three-component geophones of unknown azimuthal orientation. The geophone orientation must be estimated from the recorded data as a prerequisite to processing such as P- and S-wave separation, calculation of wave-incident directions, and 3D migration. We compare and combine two methods for estimating azimuthal orientation by least-squares fitting over a large number of shots. Combining the two methods can be done in an automated manner, which provides more accurate estimates of the geophone orientations than previous methods. In the polarization-plane method, we calculate the polarization plane of the first P-wave arrival. Then we subtract the source azimuth to determine the geophone orientation, independently for each geophone, with an angular uncertainty of [Formula: see text], and with no accumulated errors. In the relative-angle method, we obtain relative angles between adjacent geophone pairs using trace crosscorrelations, and operate on all coherent signals (even noise). Swapped geophone components can be detected automatically using the polarization-plane method. The main limitation of these (and all other known) methods is that uncertainties associated with path refraction are not estimated, unless some geophones have a priori known orientations, or we have a known earth model to correct for refraction.

INFORMATIONAL CAPACITY OF ACOUSTIC MEASUREMENTS

2001 ◽  
Vol 09 (04) ◽  
pp. 1395-1406 ◽  
Author(s):  
EUGENE LICHMAN
Keyword(s):  
Energy Spectrum ◽  
Impulse Response ◽  
Seismic Data ◽  
Computational Procedure ◽  
System Response ◽  
Phase Lag ◽  
One Dimensional ◽  
Short Period ◽  
Measurement Function ◽  
Measured System

Presented is the theoretical model for extracting the system response from measurements of the acoustic wave propagating through the linear system. Based on the results of this analysis, measurements are described as a convolution of the impulse response of the system with the mixed-phase-lag nonstationary forward wavelet (or source-generated wavefield). The source-generated wavefield includes all multiple terms generated within the system as well as the energy source signature and the detector characteristics. It is shown that the decay ratio of the source-generated wavefield can be used to separate the energy spectrum of the source-generated wavefield and the energy spectrum of the impulse response from the measurement function. The level of separability of energy spectrum of the source-generated wavefield and the impulse response reflects the amount of information about the measured system, which can be obtained from experimental data. In particular, if the source-generated wavefield does not decay during the propagation through the system, or, if the effective distance of the decay is comparable with the size of the measured system, the impulse response cannot be extracted from the result of measurements. Based on the theoretical conclusions, the computational procedure is proposed for one-dimensional deconvolution algorithm. The application of this algorithm is illustrated using seismic data as an example. The forward wavelet is extracted from seismic data itself. The deconvolution of data with the extracted wavelet provides surface-consistent scaling along with peg-leg and short-period multiples attenuation.

Vertical seismic profile sonde coupling

Geophysics ◽  
1988 ◽  
Vol 53 (2) ◽  
pp. 206-214 ◽  
Author(s):  
James E. Gaiser ◽  
Terrance J. Fulp ◽  
Steve G. Petermann ◽  
Gary M. Karner
Keyword(s):  
Resonant Frequency ◽  
Impulse Response ◽  
Complex Modulus ◽  
Viscoelastic Behavior ◽  
P Wave ◽  
Borehole Wall ◽  
Initial Amplitude ◽  
S Wave ◽  
Surface Source ◽  
Contact Width

P-wave and S-wave displacements occur at high angles of incidence in vertical seismic profiles (VSPs). Therefore, the coupling of a geophone sonde to the borehole wall must be rigid in all directions. A sonde that is well coupled should have no resonant frequency within the seismic band and should provide geophone outputs that accurately represent the earth’s ground motion. An in‐situ coupling response experiment conducted under normal VSP field conditions provides a measure of the sonde‐to‐borehole wall coupling. The sonde is locked in the borehole and a surface source is excited at different offsets and azimuths. An analysis of the P-wave direct arrivals enhances damped oscillations that represent an estimate of the coupling impulse response. This response is characterized by the viscoelastic behavior of a Kelvin model related to the complex compliance [Formula: see text], where κ is the elastic spring constant, η is the damping constant, and ω is the angular frequency. The complex modulus κ−iωη is proportional to the contact width of the sonde with the borehole wall. Increasing the width by a factor of 4.5 causes a similar increase in κ−iωη where the resonant frequency and initial amplitude of the coupling impulse response increase by a factor of two. Also, the initial amplitude of the coupling impulse response appears to be inversely proportional to the locking force of the sonde. For a constant contact width, increasing the locking force by a factor of 1.37 decreases the amplitude of the response by 3.5 dB.

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