The fractal nature of the inhomogeneities in the lithosphere evidenced from seismic wave scattering

In this paper we show evidences of the fractal nature of the 3-D inhomogeneities in the lithosphere from the study of seismic wave scattering and discuss the relation between the fractal dimension of the 3-D inhomogeneities and that of the fault surfaces. Two methods are introduced to measure the inhomogeneity spectrum of a random medium: 1. the coda excitation spectrum method, and 2. the method of measuring the frequency dependence of scattering attenuation. The fractal dimension can be obtained from the inhomogeneity spectrum of the medium. The coda excitation method is applied to the Hindu-Kush data. Based on the observed coda excitation spectra (for frequencies 1–25 Hz) and the past observations on the frequency dependence of scattering attenuation, we infer that the lithospheric inhomogeneities are multiple scaled and can be modeled as a bandlimited fractal random medium (BLFRM) with an outer scale of about 1 km. The fractal dimension of the 3-D inhomogeneities isD ₃=31/2–32/3, which corresponds to a scaling exponent (Hurst number)H=1/2–1/3. The corresponding 1-D inhomogeneity spectra obey the power law with a powerp=2H+1=2–5/3. The intersection between the earth surface and the isostrength surface of the 3-D inhomogeneities will have fractal dimensionD ₁=1.5–1.67. If we consider the earthquake fault surface as developed from the isosurface of the 3-D inhomogeneities and smoothed by the rupture dynamics, the fractal dimension of the fault trace on the surface must be smaller thanD ₁, in agreement with recent measurements of fractal dimension along the San Andreas fault.     

SPECTRAL STRUCTURE OF VELOCITY INHOMOGENEITY OF CRUST MEDIUM BELOW THE SOUTHEASTERN MARGIN OF TIBETAN PLATEAU AND ITS ADJACENT REGIONS

In this paper we present results of spectral structure of crustal velocity inhomogeneity beneath the southeastern margin of Tibetan plateau and its adjacent region based on the S wave envelope broadening algorithm. The spectral structure of 8~16Hz band is selected to analyze the special character of crustal inhomogeneity and discuss the correlation between strong earthquakes and inhomogeneities. The result shows that strong and complex inhomogeneities of crustal medium are found in the southeastern margin of Tibetan plateau and its adjacent region. In the upper part of upper crust, the strong and small scale inhomogeneities are imaged in the Longmengshan fault zone and the north of the Anninghe fault zone, the weak and large scale inhomogeneites are imaged in the section from Huolu to Daofu of Xianshuihe fault zone and the south of the Anninghe fault zone. In the lower part of upper crust, strong inhomogeneites are found in the Longmengshan fault zone, Lianfeng fault zone, the north of the Anninghe fault zone and the sections from Huolu to Daofu of the Xianshuihe fault zone, weak inhomogeneites are found in the section from Daofu to Kangding of Xianshuihe fault zone. In the middle crust, strong inhomogeneities are observed in the section of the Baoxing to Dujiangyan, the Baoxing to Kangding, and Kangding to Shimian, and weak inhomogeneities are observed in the northwestern section from Huolu to Kangding, and the Lianfeng fault zone. Comparing the medium inhomogeneities with the location of the strong earthquakes, our results suggest existence of high correlation between them. Strong earthquakes are often located in the transitionary zone between the strong and the weak inhomogeneities. The spatial distribution of the strong and the weak medium inhomogeneities may be related to the broken medium from the strong movement of geological tectonic and the heat flow upwelling along active faults induced by frequent tectonic and volcanic activity.     

长白山天池火山区介质速度非均匀性谱结构

观测和研究表明,非均匀介质RMS(Root Mean Square)速度微扰动ε,特征尺度相关距离a以及幂指数k的空间分布特征可描述介质非均匀性特征.本文选用长白山天池火山区的小震记录,运用S波包络展宽法对长白山天池火山区地壳浅部介质速度非均匀性谱结构进行了研究.结果发现长白山天池火山区地壳浅部介质呈现强烈的介质非均匀...     

Estimation of the rocks statistical parameters from traveltime measurements

In this paper the method for estimating the statistical parameters of the medium from traveltime measurements of refracted waves is applied to study the statistical characteristics of crystalline rocks at the Multifunctional Station Faido (Gotthard Base Tunnel, Switzerland). The method is based on the geometrical optics (GO) approximation. A covariance function for traveltime fluctuations has been obtained by considering quasihomogeneous fluctuations of sound velocity in a plain-stratified medium. Strongly anisometric (having unequal dimensions in different directions) random inhomogeneities were embedded in this medium. To estimate the statistical parameters around the tunnel, the traveltime fluctuations are calculated. It is assumed that each observation of traveltime-distance relation for a given shot-receiver group corresponds to a particular realization of a medium statistical ensemble. By calculating the variance and the zero cross intervals of the first derivative of traveltime fluctuations, the standard deviation of the velocity fluctuations and the characteristic horizontal scale of the inhomogeneities are estimated. Although the method allows to obtain the characteristic lengths of the inhomogeneities in vertical as well as in horizontal direction, the limited offset of the field data made it only possible to measure the latter. The estimated horizontal characteristic scale is about 13 m, which is reasonably close to the direct geological measurements in the studied region, where quartz lenses are dominant among the inhomogeneities. The standard deviation of the velocity is estimated as 4.5%, which might be caused by the fractured structure around the tunnel and also by the fault zone near the study area.     

Depth migration of shot records in heterogeneous, transversely isotropic media using optimum explicit operators

A space–frequency domain 2D depth-migration scheme is generalized for imaging in the presence of anisotropy. The anisotropy model used is that of a transversely isotropic (TI) medium with a symmetry axis that can be either vertical or tilted. In the proposed scheme the anisotropy is described in terms of Thomsen parameters; however, the scheme can accommodate a wide range of anisotropy rather than only weak anisotropy. Short spatial convolution operators are used to extrapolate the wavefields recursively in the space–frequency domain for both qP- and qSV-waves. The weighted least-squares method for designing isotropic optimum operators is extended to asymmetric optimum explicit extrapolation operators in the presence of TI media with a tilted symmetry axis. Additionally, an efficient weighted quadratic-programming design method is developed. The short spatial length of the derived operators makes it possible for the proposed scheme to handle lateral inhomogeneities. The performance of the operators, designed by combining the weighted least-squares and weighted quadratic-programming methods, is demonstrated by migration impulse responses of qP and qSV propagation modes for the weak and strong TI models with both vertical and tilted symmetry axes. Finally, a table-driven shot-record depth-migration scheme is proposed, which is illustrated for finite-difference modelled shot records in TI media.     

SIGNAL ET BRUIT EN MAGNETOTELLURIQUE *

Conventional processes of extracting magnetotelluric signals from noisy records are reviewed: instrument noises and noises that are generated close to the detectors can be eliminated by the usual auto- and crosscorrelation processes. Identification of coherent noises, such as pulses due to field sources that are not uniform over at least 100 km in oil exploration or 1000 km in crustal studies, is much more tedious. The 5 components H_x, H_y, H_z, E_x, E_y, of the magnetotelluric field have been recorded in many areas in France at different periods of the year, (a) in non-uniform field sources in the vicinity of electric railways and of 50 cycle power lines, and (b) in areas of strong inhomogeneity at depth on the flanks of steep structures and near the sea shore. Means for detecting non-uniformity are reviewed. Measuring the vertical component of magnetic pulses is a good way of estimating field uniformity: if H vertical/H horizontal <10%, the uniform field assumption is valid, and the classical restitution formulas can be used; if H vertical/H horizontal > 10%, uniformity can not be assumed and there is some difficulty in deciding whether non-uniformity is due to the field source or to anisotropy or inhomogeneities at depth. Several ways to solve this difficulty are described. The reliability of calculation of actual resistivity at various depths is examined as a function of the precision of apparent resistivity measurements.     

Exact solutions ofSH wave equation in transversely isotropic inhomogeneous elastic media

Summary TheSH wave equation in a transversely isotropic inhomogeneous elastic medium, where the elastic parameters and density are functions of vertical coordinate, is considered. A general procedure is given for finding the inhomogeneities for which the equation can be solved in terms of hypergeometric, Whittaker, Bessel and exponential functions. A few simple inhomogeneities and the corresponding solutions in terms of these transcendental functions are presented.     

Refraction traveltime and amplitude corrections for very near- surface inhomogeneities

The performance of refraction inversion methods that employ the principle of refraction migration, whereby traveltimes are laterally migrated by the offset distance (which is the horizontal separation between the point of refraction and the point of detection on the surface), can be adversely affected by very near‐surface inhomogeneities. Even inhomogeneities at single receivers can limit the lateral resolution of detailed seismic velocities in the refractor. The generalized reciprocal method ‘statics’ smoothing method (GRM SSM) is a smoothing rather than a deterministic method for correcting very near‐surface inhomogeneities of limited lateral extent. It is based on the observation that there are only relatively minor differences in the time‐depths to the target refractor computed for a range of XY distances, which is the separation between the reverse and forward traveltimes used to compute the time‐depth. However, any traveltime anomalies, which originate in the near‐surface, migrate laterally with increasing XY distance. Therefore, an average of the time‐depths over a range of XY values preserves the architecture of the refractor, but significantly minimizes the traveltime anomalies originating in the near‐surface. The GRM statics smoothing corrections are obtained by subtracting the average time‐depth values from those computed with a zero XY value. In turn, the corrections are subtracted from the traveltimes, and the GRM algorithms are then re‐applied to the corrected data. Although a single application is generally adequate for most sets of field data, model studies have indicated that several applications of the GRM SSM can be required with severe topographic features, such as escarpments. In addition, very near‐surface inhomogeneities produce anomalous head‐wave amplitudes. An analogous process, using geometric means, can largely correct amplitude anomalies. Furthermore, the coincidence of traveltime and amplitude anomalies indicates that variations in the near‐surface geology, rather than variations in the coupling of the receivers, are a more likely source of the anomalies. The application of the GRM SSM, together with the averaging of the refractor velocity analysis function over a range of XY values, significantly minimizes the generation of artefacts, and facilitates the computation of detailed seismic velocities in the refractor at each receiver. These detailed seismic velocities, together with the GRM SSM‐corrected amplitude products, can facilitate the computation of the ratio of the density in the bedrock to that in the weathered layer. The accuracy of the computed density ratio improves where lateral variations in the seismic velocities in the weathered layer are known.     

基于双频磁共振的未知拉莫尔频率地下水探测方法

在利用地面磁共振方法进行地下水探测时,要求发射频率与拉莫尔（Larmor）频率相同.由于地磁场的不均匀性和噪声干扰,实际测量时无法获得准确的Larmor频率,导致发生偏共振现象.如果假设磁共振或频率偏量未知时,反演得到的含水量和弛豫时间（T₂^*）将存在较大误差.为了解决未知Larmor频率情况下的准确探测问题,本文在频率环测量方案的基础上提出了双频磁共振探测方法,即只需知道Larmor频率的范围区间,通过区间外的两个频率进行偏共振激发,利用频率差值的一半作为频率偏量计算核函数,即可消除未知Larmor频率的影响.通过仿真实验,证明了在电阻率大于100 Ωm时,双频磁共振和偏共振的核函数偏差较小.对于双层含水层模型,双频磁共振和偏共振的信号偏差小于3 nV,因此得到双频磁共振信号可以用偏共振核函数进行反演.通过对假设磁共振、假设偏共振和双频磁共振反演结果的对比,可以得到在未知频率偏量和设定频率偏量改变时,双频磁共振均能得到准确的反演结果.但是,当环境噪声增加和电阻率降低时,反演结果的准确度降低.最后,通过长春市烧锅镇采集数据的反演结果与已知钻探资料进行对比,验证了双频磁共振探测方法的有效性和准确性.     

Modelling of GPR waves for lossy media obeying a complex power law of frequency for dielectric permittivity

The attenuation of ground‐penetrating radar (GPR) energy in the subsurface decreases and shifts the amplitude spectrum of the radar pulse to lower frequencies (absorption) with increasing traveltime and causes also a distortion of wavelet phase (dispersion). The attenuation is often expressed by the quality factor Q. For GPR studies, Q can be estimated from the ratio of the real part to the imaginary part of the dielectric permittivity. We consider a complex power function of frequency for the dielectric permittivity, and show that this dielectric response corresponds to a frequency‐independent‐Q or simply a constant‐Q model. The phase velocity (dispersion relationship) and the absorption coefficient of electromagnetic waves also obey a frequency power law. This approach is easy to use in the frequency domain and the wave propagation can be described by two parameters only, for example Q and the phase velocity at an arbitrary reference frequency. This simplicity makes it practical for any inversion technique. Furthermore, by using the Hilbert transform relating the velocity and the absorption coefficient (which obeys a frequency power law), we find the same dispersion relationship for the phase velocity. Both approaches are valid for a constant value of Q over a restricted frequency‐bandwidth, and are applicable in a material that is assumed to have no instantaneous dielectric response. Many GPR profiles acquired in a dry aeolian environment have shown a strong reflectivity inside dunes. Changes in water content are believed to be the origin of this reflectivity. We model the radar reflections from the bottom of a dry aeolian dune using the 1D wavelet modelling method. We discuss the choice of the reference wavelet in this modelling approach. A trial‐and‐error match of modelled and observed data was performed to estimate the optimum set of parameters characterizing the materials composing the site. Additionally, by combining the complex refractive index method (CRIM) and/or Topp equations for the bulk permittivity (dielectric constant) of moist sandy soils with a frequency power law for the dielectric response, we introduce them into the expression for the reflection coefficient. Using this method, we can estimate the water content and explain its effect on the reflection coefficient and on wavelet modelling.     

Inhomogeneities near the core-mantle boundary evidenced from scattered waves: A review

Statistical properties of small-scale inhomogeneities (wavelengths between 20 and 70 km) near the core-mantle boundary are inferred from scattered core waves. Observations of scattered core waves at large seismic arrays and worldwide networks indicate that the inhomogeneities have a global nature with similar characteristics. However, there may exist a few regions having markedly stronger or weaker strengths. Scattering by volumetric inhomogeneities of about 1% inP-wave velocity in the lower mantle or by about 300 m of topographic relief of the core-mantle boundary can explain the observations. At present it is not possible to rule out either of these two alternatives, or a combination of both.     

The origin of low-frequency maxima in the spectra of seismic noise

The evidence attesting to the possibility of the largest spectral maximum in seismic noise with a peak frequency of 0.14–0.22 Hz to form as a result of low-frequency dissipation of elastic wave energy in rocks is presented. According to this mechanism, elastic wave energy in rock, which can be considered as a two-component medium (solid matrix + pore water), is dissipated in the form of low-frequency pulses whose energy is the lower the smaller the coefficient of porosity. It is assumed that weak seismic events continuously occur in the frequency range above 6 Hz, and their dissipated energy is a source of the low-frequency noise.     

LIMITS OF STRUCTURAL INVERSION OF SEISMIC HORIZONS*

Time horizons can be depth-migrated when interval velocities are known; on the other hand, the velocity distribution can be found when traveltimes and NMO velocities at zero offset are known (wavefront curvatures; Shah 1973). Using these concepts, exact recursive inversion formulae for the calculation of interval velocities are given. The assumption of rectilinear raypath propagation within each layer is made; interval velocities and curvatures of the interfaces between layers can be found if traveltimes together with their gradients and curvatures and very precise V_NMO velocities at zero offset are known. However, the available stacking velocity is a numerical quantity which has no direct physical significance; its deviation from zero offset NMO velocity is examined in terms of horizon curvatures, cable length and lateral velocity inhomogeneities. A method has been derived to estimate the geological depth model by searching, iteratively, for the best solution that minimizes the difference between stacking velocities from the real data and from the structural model. Results show the limits and capabilities of the approach; perhaps, owing to the low resolution of conventional velocity analyses, a simplified version of the given formulae would be more robust.     

长白山天池火山区介质非均匀性

高频S波随着传播距离的增大其均方根(RMS)包络逐渐变宽,我们把这种现象称为S波包络展宽现象.S波传播路径上随机分布的非均匀体对S波的多次前向散射和绕射作用是导致S波包络展宽现象的主要原因,因此可用S波包络展宽现象来研究介质非均匀性.本文采用S波包络峰值延时来对S波包络展宽现象进行量化.S波包络峰值延时定义为S波初至与其均方根包络峰值最大值之间的时间差.本文选用长白山天池火山区的小震记录,运用S波包络峰值延时对长白山天池火山口地区的介质非均匀性进行了评价.结果发现长白山天池火山区呈现强烈的介质非均匀性,在0~2 km深度范围内介质非均匀性表现出南部强北部弱的空间分布特征;在2~5 km深度范围内介质非均匀性的空间分布特征与频率具有相关性;天池火山区介质非均匀性具有明显的多尺度特性;强弱非均匀性接触带,往往是地震频发地带.根据地震与非均匀体在空间分布的相对位置,我们认为火山区介质非均匀性可能反映了火山早期喷发堆积物介质结构的差异.     

On faulting in a medium containing an inhomogeneity inplane and antiplane strain models

Faulting in a medium with an inhomogeneity is analysed applying two-dimensional models consisting of a shear crack in a presence of a circular inclusion. The stress drop and the stress intensity factor are calculated for mode II and III cracks of various positions in relation to the inclusion. The results demonstrate that the effect of an inhomogeneity on a shear zone strongly depends on the location of a zone for either mode II or mode III shear zone. This effect is mostly due to the spatial distribution of external effective shear stress around an inhomogeneity. Depending on the position, an inhomogeneity may have either a destabilizing effect (the stress intensity factor becomes greater) or a stabilizing influence (the stress intensity factor is decreased or faulting is prohibited so the inhomogeneity acts as an asperity or a barrier). There is a substantial difference, however, between mode II and mode III shear zones approaching an inhomogeneity centrally. Namely, the effect of inhomogeneity on the mode III shear zone located in the immediate vicinity of the inhomogeneity is in this case considerably more pronounced than that for mode II shear zone and depends to a far greater extent on the rigidity contrast between the inhomogeneity and the surrounding medium. Another important conclusion is that the quantitative effect of an inhomogeneity on faulting depends essentially on the initial value of the stress drop of a shear zone approaching an inhomogeneity, being decidedly higher for a shear zone of small stress drop. It means that in specified areas in the proximity of medium inhomogeneities one should expect substantially greater faulting activity in which weak events prevail than in other regions surrounding inhomogeneities where such activity should be distinctly reduced. Such conclusions apply to both high rigidity inhomogeneities, which, in particular, may be associated with intrusions from the upper mantle, and to low rigidity inhomogeneities such as volcanos. The present model sets forth the plausible explanation regarding why earthquakes from the same region are occasionally characterized by various values of the stress drop. The model also presents the quantitative insight concerning how heterogeneity of the medium, in the sense of spatial variation of elastic constants, affects faulting.     

Particle energy fluxes in an unstable plasma with vortex structures in an inhomogeneous geomagnetic field in the topside ionosphere

Plasma inhomogeneities extending along geomagnetic field lines in the ionosphere and magnetosphere can have a vortex structure. Electromagnetic waves can propagate in plasma inhomogeneities in the waveguide channel mode. It has been indicated that energy and particle fluxes related to the development of small-scale electrostatic turbulence in a magnetized plasma with an unstable electron component promotes an increase in plasma density gradients in the walls of waveguide channels and an enhancement in plasma vortices. At low L shells in the region of the geomagnetic equator, the development of plasma electrostatic instability and the damping of drifting plasma vortices in the inhomogeneous geomagnetic field in the topside ionosphere can be the main mechanism by which large-scale (∼1000 km) regions with a decreased plasma density are formed.     

NEW FILTERING METHODS WITH “VIBROSEIS”*

In order to obtain high resolution correlograms, it is of importance amongst other things to get reflection signals with large bandwidth. An advantage of the VIBROSEIS *** *** Trade Mark and Service book of the Continental Oil Company.
method is that the frequencies radiated by the vibrators can be matched to the transmission response of the subsurface involved. By choosing the right frequency range, the highest possible amplitude and most favourable form may be given to the reflection signals. In a reflection correlogram, individual signals cannot be considered in isolation. Signals of different origin are interfering with one another. They very often have different amplitudes, so that it may be desirable in many cases to filter out events of certain apparent velocity. With the VIBROSEIS method this may be achieved quite simply. All frequencies of the noise signal are uniformly suppressed. The advantage is that noise signals, e.g. refraction signals, which cannot be sufficiently attenuated by wavelength filtering, may be completely eliminated by this velocity filtering without affecting the bandwith of the desired signal. The total dynamic range of the tape recording can be used for the registration of wanted events. To perform this kind of filtering several vibrators are necessary in the field; each of them is controlled by an individual signal. There is an unavoidable error of static and dynamic corrections which causes the results of reflection measurements to deteriorate when using multiple coverage. High frequency components especially are seriously affected by destructive interference. This difficulty can be avoided by using a VIBROSEIS signal with high frequency component amplitudes supported. For the probability of error of corrections a normal distribution is assumed. A smoothed amplitude characteristic may be achieved after stacking. The amplitude characteristic of seismic devices is commonly reduced to about 100 cps bandwidth. For further improvement of resolution of VIBROSEIS correlograms it is necessary to apply special filtering methods. This is of particular interest when any kind of gain control is used to display weak events more clearly. With increasing amplification the sidelobes of the strong signals may reach the size of the weak events. In order to eliminate this effect, the amplitude characteristic of the VIBROSEIS signal is adjusted for optimum suppression of side-lobes.     

NEW FILTERING METHODS WITH “VIBROSEIS”*

In order to obtain high resolution correlograms, it is of importance amongst other things to get reflection signals with large bandwidth. An advantage of the VIBROSEIS *** *** Trade Mark and Service book of the Continental Oil Company.
method is that the frequencies radiated by the vibrators can be matched to the transmission response of the subsurface involved. By choosing the right frequency range, the highest possible amplitude and most favourable form may be given to the reflection signals. In a reflection correlogram, individual signals cannot be considered in isolation. Signals of different origin are interfering with one another. They very often have different amplitudes, so that it may be desirable in many cases to filter out events of certain apparent velocity. With the VIBROSEIS method this may be achieved quite simply. All frequencies of the noise signal are uniformly suppressed. The advantage is that noise signals, e.g. refraction signals, which cannot be sufficiently attenuated by wavelength filtering, may be completely eliminated by this velocity filtering without affecting the bandwith of the desired signal. The total dynamic range of the tape recording can be used for the registration of wanted events. To perform this kind of filtering several vibrators are necessary in the field; each of them is controlled by an individual signal. There is an unavoidable error of static and dynamic corrections which causes the results of reflection measurements to deteriorate when using multiple coverage. High frequency components especially are seriously affected by destructive interference. This difficulty can be avoided by using a VIBROSEIS signal with high frequency component amplitudes supported. For the probability of error of corrections a normal distribution is assumed. A smoothed amplitude characteristic may be achieved after stacking. The amplitude characteristic of seismic devices is commonly reduced to about 100 cps bandwidth. For further improvement of resolution of VIBROSEIS correlograms it is necessary to apply special filtering methods. This is of particular interest when any kind of gain control is used to display weak events more clearly. With increasing amplification the sidelobes of the strong signals may reach the size of the weak events. In order to eliminate this effect, the amplitude characteristic of the VIBROSEIS signal is adjusted for optimum suppression of side-lobes.     

WAVE FIELD EXTRAPOLATION TECHNIQUES FOR INHOMOGENEOUS MEDIA WHICH INCLUDE CRITICAL ANGLE EVENTS. PART I: METHODS USING THE ONE-WAY WAVE EQUATIONS*

Part I of this series starts with a brief review of the fundamental principles underlying wave field extrapolation. Next, the total wave field is split into downgoing and upgoing waves, described by a set of coupled one-way wave equations. In cases of limited propagation angles and weak inhomogeneities these one-way wave equations can be decoupled, describing primary waves only. For large propagation angles (up to and including 90°) an alternative choice of sub-division into downgoing and upgoing waves is presented. It is shown that this approach is well suited for modeling as well as migration and inversion schemes for seismic data which include critical angle events.     

A local stability analysis of astronomical disks

Abstract

A fifth-order dispersion relation describing the local stability of a differentially rotating flow against small perturbations is derived. Finite viscosity and conductivity and both vertical (parallel to the rotation axis) and radial gradients in density, temperature and pressure are included. A general form is assumed for the equation of state, although this is not exploited in the paper. A number of special cases are studied: with negligible viscosity and conductivity, it is shown that modes can often be separated into two high frequency (modified acoustic), two intermediate frequency (combined inertial and internal waves) and a low frequency mode. In convectively unstable situations the intermediate frequency modes may be replaced by a damped/growing pair of instablities. Various criteria for mode excitation are given. It is shown that viscosity always inhibits instability at very short wavelengths, while non-zero conductivity may destabilize the flow. At intermediate wavelengths viscosity could also play a destabilizing role. A parameter study of the effects of fluctuations in the conductivity shows that it could cause mode excitation under certain circumstances.