SUMMARY

submitted to Geophys.J.Int.

Sharpness characterization of the upper mantle discontinuity by?xed scale singularity analysis of converted phases

Felix Herrmann,S′e bastien Chevrot and Colin Stark

SUMMARY

Parametric waveform,mono-and multiscale analyses methods are used to characterize the properties of the discontinuity.The characterization involves the estimation of the sharpness,smoothness and extent of this transition from converted phases.In our approach sharpness is no longer associated with the extent of the?rst-order disconti-nuity.Instead,the sharpness is related to the fractional order of a generalized transition, parametrized by this order and two additional parameters,expressing smoothing and ex-tent/thickness length scales.Estimates for the parameters are derived by complementary parametric and direct scale analysis methods.Discontinuity characterization is used to construct a generalized transition model.This model is shown to accurately capture de-tailed information on the transition,which is necessary to obtain useful information on mineralogical changes that may occur at the transition.

Key words:Mantle discontinuities,waveform analysis,mineralogy

1INTRODUCTION

Analysis of differential travel times between and phases provides important constraints on the transition zone thickness(Stammler et al.1992;Chevrot et al.1999).Relevant additional infor-mation can be extracted from waveforms of these phases.For example,the amplitudes of converted phases can be used to infer the magnitude of the velocity jumps at and depth(Paulssen

2Felix Herrmann,S′e bastien Chevrot and Colin Stark

1988).Unfortunately,information only on the magnitude of the discontinuities is not suf?cient to con-strain mineralogical changes that may take place at these discontinuities.

Details of the seismic discontinuity depend on the partition coef?cient between the high and low pressure phases,and perhaps the depth interval over which the phase transition occurs(Stixrude1997). Due to the inherent bandwidth limitation of the seismic data,revealing the?ne structure of the upper mantle discontinuities is dif?cult.Thus far,various studies based on the analysis of converted phases have not led to de?nite conclusions concerning the behavior of the discontintuities(Paulssen1988; Petersen et al.1993;Gaherty et al.1999;Castle&Creager2000).In these studies,the upper mantle discontinuities are assumed to be given by?rst order discontinuities,parametrized by their thickness, generally referred to as https://www.wendangku.net/doc/5d7367524.html,rmation on the transition zone thickness can be obtained by comparing the temporal frequency content of the waves converted at the upper mantle discontinuities with the reference pulse(Petersen et al.1993).In this paper,we follow another approach where the sharpness of the transition is related to a more robust characteristic of the transition.

Recent research by Holschneider(1995)and Mallat(1997)on the detection and characterization of edges demonstrates the usefulness of the continuous wavelet transform as a tool to quantify the local characteristics of transitions/edges.Contrary to Fourier analysis,wavelets are able to unravel the local(both in time and frequency)nature of the transitions by studying their behavior as a function of scale zooms.Mathematically,the scale exponents,parametrizations of the zooms,provide precise estimates on the regularity of the transition.This regularity appears to be a more robust quantity to characterize the sharpness of transitions.Examples of wavelet techniques applied to geophysical data can be found in Alexandrescu et al.(1995),where geomagnetic jerks are being studied,or in Herrmann (1997,1998);Herrmann&Stark(2000a)where both detailed sedimentary records(well-log data)and high frequency(exploration)seismic data are being analyzed.

In this paper we?rst introduce a generalized parametrization for the transitions.This generaliza-tion is partly inspired by the precise mathematical de?nition of sharpness within the wavelet frame-work,and by the shape of the yield functions as given by Stixrude(1997).We use three parameters, sharpness,smoothing and extent,to describe transitions.Smoothing refers to the length scale of an intrinsic smoothing of the discontinuity,whereas the extent is related to the thickness.Given this gen-eralized transition we apply a specially designed suite of analysis techniques to rigorously quantify and reconstruct the nature of the generalized transitions.Two complementary scale methods will be reviewed and utilized.

The?rst method involves a parametric waveform inversion based on the minimization of the -norm difference between the observed converted phase and a parametrized family of waveforms.

Fixed scale analysis3 In this inversion scheme a linear convolution model is assumed to relate the unknown generalized transition and reference pulse to the measured converted phase.

Since the convolution model involves a differentiation,the wave conversion process can be inter-preted as a special case of a continuous wavelet transform(Herrmann2000b).Therefore,the second series of scale analyses is based on a direct multi-and monoscale analysis of the data.Due to the inherent bandwidth limitation of the seismic data,the applicability of the multiscale analysis proves to be limited for the estimation of the sharpness.However,the multiscale method provides informa-tion on possible scale crossovers linked to smoothing and/or extent length scales.To overcome the fundamental bandwidth limitation problem,the monoscale analysis technique,introduced by(Stark& Weissel1998;Herrmann&Stark1999,2000b,a;Stark et al.2000;Herrmann2000a;Stark&Weissel 2000),is applied to the data as well.The monoscale technique has the distinct advantage of being able to estimate the order/sharpness of the generalized transitions from the converted phase,which is essentially limited to only one scale.

After describing the scale analysis tools,the converted phase of the phase under station GEO-SCOPE CAN(Canberra,Australia)is analyzed.During these analyses,scale crossovers,marking smoothing length scales,are identi?ed while sharpness is consistently estimated using the paramet-ric and direct methods.Given the sharpness estimates,a generalized pro?le for the transition is constructed.Finally,we conclude by discussing the relevance of the order/sharpness parameter with respect to the mineralogy.

2PARAMETRIZATION OF SEISMIC DISCONTINUITIES

Seismic waves are sensitive to velocity and density variations in the Earth’s interior.When signi?cant variations occur over length scales of the order of the seismic wavelength,waves are re?ected and converted.Thus far,?rst order discontinuities have been used to represent the transitions(Paulssen 1988;Petersen et al.1993;Gaherty et al.1999).These transitions in the compressional(),shear() wavespeed or density()are given by

(1) with

the integral of the Heaviside step function,

where(2) is the amplitude.

4Felix Herrmann,S′e bastien Chevrot and Colin Stark

is the spatial extent/thickness.

Seismic waves convert at these?rst order discontinuities.When the characteristic wavelength is small enough compared to the discontinuity thickness,the converted phase broadens with respect to the reference pulse of the wave.The transmitted wave is unaffected by the transition.When the incident pulse has suf?cient low frequency content,the broadening increases as a function of the discontinuity thickness.By comparing the frequency content of the broadened converted phase with respect to the unaffected reference pulse,the transition thickness can be estimated(Paulssen1988; Petersen et al.1993).As the thickness decreases the broadening gradually disappears,an effect also observed when reducing the low frequency content of the incident pulse.

Fig.1(left)shows a number of amplitude normalized shear velocity pro?les,de?ned according to Eq.1,for an extent/thickness.For convenience the depth has been rescaled with the characteristic wavelength while the time axes are rescaled by the characteristic period.The total length of the pro?les is,and the onset of the discontinuity is located at.For the overburden the compressional,shear wavespeed and density are taken to be,and .Across the transition the density is kept constant while the wavespeeds increase to and.Synthetics for the converted phases are computed with a re?ectivity code(Kosarev et al.1979)and are shown on the right of Fig.1.The characteristic period of the pulse is and the slowness is.As expected the converted phase increasingly broadens as the transition thickness increases.

Although thickness appears to be a useful transition characteristic,it is insuf?cient for unam-biguously explaining all observed distortions of the converted waveform.Moreover,methods based on a?rst-order transition appear to be sensitive to the very low frequency content of the incident pulse.This sensitivity,in conjunction with mineralogical?ndings for upper mantle phase transitions (Stixrude1997),calls for a generalization of the transition model.This generalization entails the in-troduction of a larger class of transitions,capable of explaining the waveform distortions.This class of transitions is de?ned as

extent

de-sharpening

with and(3)

smoothing

where

is now a fractional order integration,

(4)

Fixed scale analysis5

is a spatial convolution with a Gaussian bell-shape smoothing function,,with width.

is a windowing with a Gaussian,,centered around and with support.

In contrast to the?rst order discontinuity de?ned in Eq.1,the generalized transitions of Eq.3contain only one singularity,located at.The order of this singularity equals and is taken to be .Examples of these generalized transitions(Fig.2,left column)clearly show the relevance of the parameterization for both the transitions and the corresponding converted phases(Fig.2,right column).

As indicated by the examples in Fig.2the most prominent parameter is sharpness,which is related to the amount of fractional integration.In Fig.2the sharpness is varied between,with a transition zone length set of corresponding to a characteristic time of.As increases,the initially sharp zero order Heaviside jump discontinuity(Fig.2,?rst trace on the left)is de-sharpened by the fractional integration,into a continuous and one time differentiable?rst order()ramp function.With the fractional integration,,the transition’s regularity increases to an times differentiable function.As shown by the plots in the?rst row,varying has a strong effect on both the transition and the converted wave.Not only is the wave increasingly broadened during the conversion but also the distortion changes with.For the small ’s the converted phase is“spiky”while for increasing the causal tail increases.The reason for the limited length of the tail is the?nite extent of the transition.In the case of an“in?nite”extent the tail would be in?nitely long for,and the wave conversion would no longer have a local maximum.

The two other parameters of the generalized transition,and,are introduced to invoke scale crossovers,changes in the scale,for a small smoothing length scale and large extent length scale.The ?rst crossover renders a smoothing of the transition by a spatial convolution.This convolution removes variations for scales smaller then.The second parameter limits the extent of the transition by tapering off the onset to a?xed?nite value.The effects of increasing both length scales are shown in the second and third row of Fig.2.In the second row the order of the transition is?xed to,while the transition is increasingly smoothed by a Gaussian bell-shape function with a width

in the non-dimensionalized depth coordinate.In the Fourier domain this smoothing simply amounts to applying a Gaussian taper around the zero frequency and with a width proportional to the. In the third row the order of the transition is also?xed to,but now the extent is varied from by applying a Gaussian bell-shape function,but now around the location of the singularity,yielding smoothing in the Fourier domain with a Gaussian of width proportional to.

The plots in the lower two rows in Fig.2show that both of these crossover length scales have an effect.In particular,limiting the extent to length scales of the order of the characteristic wavelength reduces the distortions substantially.The transitions look more and more like a jump discontinuity,

6Felix Herrmann,S′e bastien Chevrot and Colin Stark

eventually yielding a“spiky”and symmetric converted phase(see Fig.2,bottom).Apparently,this type of behavior is consistent with the behavior of the conventional formulation(cf.Eq.1)for a thickness going to zero.

Of the three parameters,variations in the order and extent have the largest effect on the distortions of the converted phase.Increasing the smoothing crossover length scale to scales close to the charac-teristic wavelength would also have a drastic effect.However,this situation is not relevant because the magnitude of the converted phase becomes too small to be observable.

In the absence of smoothing and extent crossovers,fractional integrations of the Heaviside func-tion are scale-invariant.This scale-invariance is expressed by the property

(5)

where the effect of dilations/compressions are removed by simple rescalings.As a result differences in scale of observation are interchangeable with the strength of the transition,which makes it dif?cult to issue precise statements,based on amplitudes only.For only the rescalings do not have an effect,yielding a constant transition amplitude,irrespective of the seismic wave dominant wavelength.

Parameterization of the discontinuity by the thickness of a?rst order transition only may be too restrictive,especially when low frequencies are absent in the data.By de?ning the sharpness as the degree of differentiability()one has the distinct advantage of being scale-invariant and indepen-dent of the amplitude of the discontinuity.In addition,sharpness de?ned in this way is related to the mathematical concept of regularity,for which rigorous estimation techniques exist based on the con-tinuous wavelet transform(Holschneider1995;Mallat1997).As a?nal argument the sharpness has the additional advantage of being robust because the scale exponent entails an order of magnitude characterization of the amplitude variations.

A?nal aspect of the fractional order transitions is directionality,which is due to their inherent asymmetrical shape.Fig.3illustrates the effects of the directionality on the converted phases.Flipped (sign reversed)and anti-causal transitions(direction reversed)are used in the example of Fig.2.

3SCALE ANALYSIS AND RECONSTRUCTION

Detection and characterization of discontinuities by scale analyses are based on a combination of a smoothing and(de-)sharpening operation acting on the data,

(de-)sharpening

Fixed scale analysis7 Depending on the application,the data refers to either the unknown transition or to the measured converted and reference phases.The smoothing is obtained by convolution with a family of dilated smoothing functions,,with a width proportional to the scale,.This scale is given by the sec-ond moment(variance)of the smoothing function and is roughly proportional to the dominant spatial wavelength or temporal period.The smoothing reduces the resolution of,hiding details smaller than .Fractional-order differentiations()or integrations(),on the other hand,sharpen or de-sharpen the data,revealing or obscuring discontinuities as appearing or disappearing local maxima.

For,Eq.6has the form of the linear convolution model for the converted phase(Richards &Frasier1976).After deconvolution for the source excitation function,the converted phase reads

Pds

obs.(10) with obs.the measured phases.The direct scale analysis consists of two approaches where local prop-

Typically given by dilations of the Gaussian Bell shape(Mallat1997)with standard deviation one.

8Felix Herrmann,S′e bastien Chevrot and Colin Stark

erties of discontinuities are studied(Stark&Weissel1998;Herrmann&Stark1999,2000b,a;Stark et al.2000;Herrmann2000a;Stark&Weissel2000),

via the modulus maxima as a function of the scale,for a?xed and large enough differentiation .

via the onsets or disappearances of modulus maxima as a function of the order of differentiation ()or integration()for a?xed scale,.

Multiscale analysis by the continuous wavelet transform(Holschneider1995;Mallat1997)is based on the?rst approach,where singularities in the derivatives of reveal themselves as local maxima for the modulus when the order of differentiation exceeds the order of the discontinuity.For the linear convolution model underlying the converted phases(cf.Eq.7),the order of differentiation equals one, ,while the“smoothing”function is given by the incident pulse,which after deconvolution equals the residual seismic wavelet,.Consequently,converted phases are able to detect upper mantle discontinuities as local maxima in the data at the seismic wavelength scale.

Within multiscale analysis,the sharpness and scale of the discontinuities are characterized by inspecting the behavior of the wavelet coef?cients along the modulus maxima.The scale exponents measure the asymptotic decay or growth rate of the modulus maxima as the scale decreases.In addi-tion,signi?cant changes in the scaling,which break the asymptotic scaling of the wavelet coef?cients, provide information on possible scale crossovers.

To obtain unambiguous asymptotic sharpness estimates,a wide scale range in the data unaffected by scale crossovers is critical.Unfortunately,seismic data themselves are more or less con?ned to a single scale,as can readily be seen from Eq.7.This equation shows that the converted phases exactly correspond to the smoothed derivative,i.e.the wavelet transform for,of the time parametrized medium at the?xed seismic scale.To overcome limitation in scale,sharpness is estimated from the converted phase,using the property that,at a?xed scale,the local maxima of the converted phases disappear when the fractional integration in?nitesimally exceeds the order of the transition minus one.

3.1Parametrized waveform inversion

As shown in Fig.2,converted phases generated at generalized transitions,as de?ned in Eq.3,depend on the parameters of the generalized transition https://www.wendangku.net/doc/5d7367524.html,rmation on these parameters can be obtained by generating a family of waveforms according to Eq.6,with set to either Dirac’s delta distribution or to the reference pulse.Estimates for the parameters can be obtained by minimizing the-norm mismatch between the observed and modeled waveforms(cf.Eq.9)

In addition to the transition’s order and smoothness/scale,its location and direction(see Fig.3)

Fixed scale analysis9 are estimated.Fig.4summarizes the proposed inversion method for a synthetic example with the scale and order set to and.The transition is taken to be anti-causal()and?ipped(), i.e.given by.In this example the reference pulse,,is taken to be a fractional left-right differentiated Gaussian.The degree of differentiation is set to,a value close to the estimated value obtained from analyzing the measured reference pulse itself.The synthetic phase is computed with the convolution model.Parametric inversion of this synthetic phase is conducted by minimizing the difference between this phase and

Pds

(13)

where is a suf?ciently smooth(differentiable)smoothing function with width,and is an th order wavelet,generated by dilatations of

10Felix Herrmann,S′e bastien Chevrot and Colin Stark

order Taylor approximation of around the point,i.e.

(14)

with.Scale exponents are introduced as order of magnitude estimates for the remainder,,and can be measured by analyzing the wavelet coef?cients for decreasing scales and along curves where the modulus of the wavelet coef?cients has a local maximum,i.e.

as(15)

for lines where at abscissa.Following Holschneider(1995)the ac-curacy of?nding the location of the maxima is enhanced by analytical continuation of the wavelet transform,de?ned in Eq.13,into the complex plane,yielding

(16)

where is the Hilbert transform.

Curves connecting the modulus maxima across the different scales(see e.g.Fig.5)are called wavelet transform modulus maxima lines,WTMML’s.The scale exponents are obtained via the slope of a linear regression of the of the wavelet coef?cients versus the along a WTMML(Fig.5). These exponents characterize both the local sharpness(regularity)and scaling(cf.Eq.5)and are equal to the order of the transitions as de?ned in Eq.’s3and4.Fig.5depicts the multiscale analysis of idealized converted phases emerging at an in?nite extent,unsmoothed transition with, ,and at a smoothed,?nite extent transition with,and .Without loss of generality the second wave conversion can be interpreted as a low-pass?ltered converted wave.Evidently,the dashed line in the-plot for the modulus along the WTMML displays the expected powerlaw behavior for the converted wave at the unsmoothed and in?nite extent transition.The slope approximately equals the scale exponent.Because there is no maximum for the WTMML amplitudes,an intrinsic length scale is absent in the unsmoothed and in?nite extent case. This is one of the main characteristics of transitions given by pure fractional order onsets.

Unfortunately,this ideal behavior observed in Fig.5is absent in the simulated converted wave (solid lines)and the observed,as can be seen from Fig.10of section4.2.Evidently,this effect is caused by the bandwidth limitation of the seismic wave,which makes it dif?cult to estimate a unique scale exponent from the converted phases.

Fixed scale analysis11 3.3Scale regimes and crossovers

So far the multiscale analysis has focused mainly on obtaining information on the sharpness of transi-tion.The degree of smoothing and extent of the transition are also important.The bandwidth limitation of the residual wavelet,in conjunction with a possible intrinsic smoothing and extent of the transition, gives rise to a more intricate scaling behavior(cf.Eq.15)for the wavelet transform(),

for(17)

When is a generalized transition,three different scaling regimes are expected for the exponent:

as

(18)

as

These three regimes(small,intermediate and large)show that it is dif?cult to(i)assign a single unique sharpness exponent;(ii)estimate the sharpness when the smoothing and extent scale are not well separated;and(iii)obtain information on the extent at the?xed seismic scale of the converted phase. Both the parametric inversion and the monoscale analysis resolve the?rst issue.The second and third issues remain dif?cult to resolve because of the limitation in available scales for the converted phases. However,the identi?cation of the different scale regimes facilitates interpretation of the data.The scale crossovers are studied via the extrema in the derivative of the local slope of the wavelet coef?cients along the WTMML,i.e.

(19)

The is an“instantaneous”scale exponent,whose derivative has a minimum when the wavelet coef?cients change from the scaling regime dominated by smoothing to the regime dominated by discontinuity.For small scales the sharpness equals the order of the wavelet transform,which is a clear indication of smoothness.As the scale increases the exponent asymptotically decreases to either a value close to the order of the discontinuity or to=0.The latter is the case where the extent of the transition is small compared to the dominant wavelength.

In addition to localizing crossovers,the also has the advantage of being independent of the actual amplitudes of the wavelet coef?cients.These instantaneous exponents represent local orders of magnitude which allow for elaborate scaling comparisons between observed reference pulses and converted phases.Refer to section4.2for a detailed analysis of versus.

With the criteria given by Eq.18,we interpret the example depicted in Fig.5.As expected,for

12Felix Herrmann,S′e bastien Chevrot and Colin Stark

small scales the wavelet coef?cient scale with the number of vanishing moments.For very large scales the local scale exponent is close to,which is consistent with the behavior predicted by Eq.18.The value occurs because the wave conversion involves a single differentiation. For intermediate scales the local exponent crosses over from the scale range dominated by smoothing to the scale range dominated by the extent.Despite the quasi-linear behavior of the largest scales it remains dif?cult to assign a single scale exponent to the large scale-range.The slopes for the large scales appear more or less the same for these two examples since the axes are logarithmic,but small non-linear effects can be quite large.Identifying the extent via a crossover at the very large length scale is certainly a challenge.

3.4Monoscale analysis by fractional order“wavelets”

The lack of available scale ranges to conduct the linear regression within the asymptotic multiscale analysis prevents a sharpness characterization by a single scale exponent.As a result,asymptotic techniques,attempting to?t powerlaw dependence in the seismic data(Fig.5),are not applicable.

Observed waveform variations in the synthetic examples shown in Fig.2,however,suggest con-verted phases contain information on the transition sharpness.To obtain this information a monoscale analysis method is introduced,which overcomes the fundamental bandwidth limitation problem(S-tark&Weissel1998;Herrmann&Stark1999,2000b,a;Herrmann2000a;Stark&Weissel2000). The method is based on varying the amount of sharpening or de-sharpening instead of the scale.As a result,Eq.6yields a convolution with a series of“wavelets”with a varying number of fractional vanishing()or diverging moments.The transform reads

Fixed scale analysis13 if the order of fractional differentiation exceeds the order of the transition.Conversely,the second criterion in Eq.22uses the property that a local maximum disappears when the fractional integration exceeds the negative sharpness exponent.This exponent is negative because differentiation reduces the exponent by the order of differentiation.Consequently,the exponent of the transition reduces by one during the conversion,i.e.the order becomes.

The monoscale method is tested on a generalized transition and the corresponding modeled con-verted phase.The results are presented in Fig.’s6and7.Scale exponents for the sharpness are esti-mated,using the“on-off”criteria of Eq.’s21and22for a varying number of scales.The transition and converted phase are plotted on the left of Fig.6and7,respectively.Monoscale analysis results are depicted as colored balls superimposed on the increasingly smoothed transition and converted phase. The color and size of the balls refer to the estimated scale exponent and the derivative at the location of the singularity.Evidently,the estimates are not seriously affected by the smoothing and?niteness of the extent(and)of the transition.For the converted phase(Fig.7),we observe the same behavior at intermediate and small scales.

So far,the analysis was restricted to asymmetric right-handed positive transitions only.Transitions can,however,be left-handed or?ipped in sign.The onset-criteria of Eq.’s21and22are affected by this directivity.To circumvent this problem,the monoscale analysis is conducted using both causal and anti-causal fractional derivatives or integrations.Fig.8shows an example with smoothed causal and anti-causal transitions which are submitted to the analysis.From the third plot it is clear that the location and direction of the singularities are estimated correctly.

3.5Reconstruction

Reconstruction of the transition is possible,given the location,sharpness and gradient of the singu-larities.The derivative(read conversion amplitude)determines the amplitude of the discontinuity.For the converted phases the estimated exponents have to be corrected for both the differentiation during the conversion as well as for the sharpness of the reference pulse.Instead of the gradient,the corrected amplitude of the converted phase is used.This correction is obtained by multiplying the conversion coef?cient by the reciprocal of the slowness dependent transmission coef?cient(cf.Eq.7).The tran-sitions are reconstructed via

(23)

where the’s are the right-handed()or left-handed()transitions.The’s and’s are the estimated exponents and location of the singularities.

Reconstructions of the type yielded by Eq.23are based on information contained in the location

14Felix Herrmann,S′e bastien Chevrot and Colin Stark

and order of the https://www.wendangku.net/doc/5d7367524.html,rmation on the regular,or differentiable,part of the transitions is not taken into account.Therefore,the reconstruction is modulo a polynomial of order,which represents the trend of the background medium.This latter trend can be obtained by other means,for instance from a reference model like PREM(Anderson1989;Dziewonsky&Anderson1981).

The bottom plot of Fig.8displays the reconstructed pro?le from the synthetic trace depicted on the top.Estimates for the location,order,direction and relative magnitude are taken from both the second and third plot.Because the background trend is unknown,it is not possible to retrieve the absolute velocities.However,the amplitude variations are nicely recovered.The deviations between the?rst and second transitions are mainly due to a lack of information on the smoothing and extent of the transitions.

4SCALE CHARACTERIZATION OF THE DISCONTINUITY

Both the parametric inversion and multi-/monocale analysis techniques are applied to the reference pulse and converted phase recorded by the GEOSCOPE station CAN(Canberra,Australia). The data processing is described in detail by Chevrot et al.(1999).First,the parametric inversion of section3.1is applied,followed by a detailed study of the scaling properties via the direct multi-and monoscale analysis techniques presented in sections3.2and3.4.

4.1Parametric inversion of the

By generating a family of waveforms according to Eq.6,with set to the reference pulse and min-imizing the difference between the parametric waveforms and the observed,both the scale and sharpness can be determined.Figure9contains the results of the parametric inversion.The con-verted phase and reference pulse are depicted in the top of the?gure.The converted phase is obtained by windowing the data(dashed line)around with an effective window width of(). The time is measured with respect to the arrival and,for comparison,the reference pulse is shifted to the arrival of the converted phase.Moreover,the amplitudes of both phases are normalized to one. Compared to the reference pulse the converted phase is slightly broadened,as can be seen from the right plot in Fig.9(top),where the solid line refers to the reference pulse and the dashed-dotted line to reference pulse.The-norm differences are displayed in the second row with the position of the global minimum denoted by the.In the bottom row(left),comparison is made between the measured and inverted phases.Finally,on the bottom right the reconstructed transition is shown for both the unsmoothed,in?nite extent and smoothed?nite extent case.The estimated smoothing length

Fixed scale analysis15 scale,which corresponds to the time scale of the reference pulse,was used together with the applied taper window width for the extent.

The estimated values for the scale and sharpness are and.The estimate yields a discontinuity given by the generalized transition of,where.The degree of smoothing is found to be,which is large because the method estimates the relative scale with respect to the reference pulse.An explanation for this overestimation is the applied window size,which less penalizes large scales during the optimization.Consequently,the parametric waveform inversion does not allow for a determination of a possible intrinsic smoothing of the.Finally,notice that the directionality of the discontinuity has also been recovered accurately from the minimization.

Changing the window size corresponds to changing the extent.Inverting for this window size proves to be dif?cult because the inversion scheme tends to prefer very large window sizes.The win-dow size chosen above yields inversion results for the()parameter pair.Reducing the window size too much reduces both the order and smoothing scale,an observation consistent with scale crossover predicted by Eq.18.

4.2Multiscale analysis of the

To verify the?ndings of the parametric inversion,both the reference pulse and converted phase are submitted to a multiscale analysis by the continuous wavelet transform with one vanishing moment ().Fig.10summarizes the results.On the left the reference pulse and converted phase are depict-ed.The color scale plot in the middle displays the absolute value of the complex wavelet coef?cients (cf.Eq.16).From the right column one can see that there is a distinct scale crossover which takes place at for the converted phase and at for the reference pulse.The estimate for the scale crossover of the reference pulse is consistent with the low-pass period?ltering of the data at.For small time scales,the local scale exponent equals the number of vanishing moments indicating a smooth inner scale.For large scales the local exponent does not taper off to a value of, indicative of a?nite extent.Therefore,no evidence is found in the data suggesting a?nite extent length scale.Indeed,there is a small bump around a scale of.However,this bump is,as one can see from the deformation of the WTMML,caused by contributions from events arriving at later times.Reducing the window size removes the events,but in that case the extent crossover appears to be related to the applied window size and not to the data.

The band-limitation of the data precludes a unique estimation for the sharpness parameter using the multiscale wavelet analysis method.However,the analyses do indicate a broadening of the con-verted phase of.For a shear velocity of this yields a spatial smoothing length scale of410.

16Felix Herrmann,S′e bastien Chevrot and Colin Stark

4.3Monoscale analysis of the

A complementary estimate for the sharpness of the discontinuity is obtained by applying monoscale analysis to the phase.Fig.11summarizes our?ndings.The location,order and magnitude of the wavelet coef?cients are denoted by the position,color and size of the colored balls depicted on top of smoothings of the converted phase plotted on the left.The applied window size for the taper is taken to be.As the smoothing length scale increases,monoscale sharpness estimates change, an effect especially apparent for scales exceeding the scale crossover(Fig.12).For small scales,the scales unaffected by the additional smoothing,the sharpness estimates equal,yielding a transition sharpness of.To arrive at this estimate we used the property that the converted phase acts as the temporal derivative of the time parametrized medium?uctuations(cf.Eq.7).This differentiation reduces the sharpness by one.Hence,the transition’s sharpness is given by. Not only is the estimate of the monoscale analysis equal to the one obtained by the parametric inver-sion,the direction is also consistent with the parametric inversion?ndings(cf.Fig.9).

5DISCUSSION AND CONCLUSIONS

The CAN dataset that we have analyzed is exceptional in many respects.First,we were able to select 69high quality records corresponding to epicentral distances between33and88in a320-360 azimuthal window.The excellent distance distribution allows us to separate converted waves from other phases present in the wave coda.Additionally,by analyzing converted waves coming from a narrow azimuthal window,it is possible to focus on a small spot of the410and minimize the effects of discontinuity topography during the stacking process necessary to detect the converted phases. Second,this station is characterized by a high signal-to-noise ratio at high frequency which allows one to determine waveforms at periods around.The study by Petersen et al.(1993)is the only other study to our knowledge where detection of converted phases at such short periods was reported, to our knowledge.Interestingly,their results show a clear asymmetry of the pulses at stations BRF,CRF and NRE0,YKW similar to the one observed at station CAN.

The parametric inversion and multi-and monoscale analyis of the data under station CAN demonstrate that the observed waveform can be described by the proposed generalized transi-tion model.Sharpness,de?ned as the order of the transition,is consistently observed by the different analysis techniques applied to the data.By allowing a fractional order transition,the broadening and more importantly the waveform asymmetry are described.Fig.13summarizes the results of the anal-ysis.On the top of the?gure,comparison is made between the converted phase and the?t from the parametric inversion.Both the negative trough and the distortion with respect to the converted phase

Fixed scale analysis17 are captured.The negative trough is due to the shape of the reference pulse,which can be parametrized by a symmetric superposition of a th-order causal and anti-causal derivative of the Gaussian with width.The waveform distortion itself is caused by the fractional integration of the transition, which corresponds to a410transition.Monoscale analysis on the phase con?rms the sharpness estimate,obtained by the parametric https://www.wendangku.net/doc/5d7367524.html,pared to the sharpness,the extent and smoothing length scales are more dif?cult to estimate.For example,there is no evidence of a scale crossover associated with an extent length scale.However,differences in the smoothing scale crossover occur,revealing an intrinsic smoothing length scale of approximately410,which corresponds to410.Notice that this smoothing length scale is relatively small compared to the thickness estimates published by Petersen et al.(1993)and Gaherty et al.(1999).

Reconstructions for the are included in Fig.13(middle and bottom).All three reconstructions (Fig.13)are based on the estimated sharpness of410.The reconstructions differ in extent and smoothing length scales.The solid lines refer to the smoothed and?nite extent transitions with the smoothing410,set to the estimate from the multiscale analysis,and the extent,, set to the applied window size.Finally,the dotted and dashed-dotted lines refer to in?nite extent transitions which are smoothed and unsmoothed,respectively.On the bottom of Fig.13,the time parametrized transitions(middle)are converted to depth via the differential traveltime given by the PREM model.

Sharpness proves to be the most robustly constrained parameter describing the transition.The es-timated fractional value for the sharpness exponent is an indication that the elastic parameters contain a non-trivial singularity at.The order of this singularity describes the rate at which the elastic parameters decrease as the-discontinuity is approached from below.As opposed to?rst order singularities,which follow from the yield function(Stixrude1997),the non-linear shape and non-trivial singularity order found within our approach cannot easily be reconciled with a non-critical re-lationship between mineralogical composition and the elastic parameters.This discrepancy may result from a biased estimate of the pulse resulting from stacking many records,but we believe that our dataset puts us in ideal observational conditions.Another explanation may be that within a critical phenomenon,the elastic parameters undergo a sudden singular,pressure and/or temperature-induced change.Typically,these phase transitions occur over many length scales which may be supported by the lack of a?nite extent length scale.In addition,these transitions are characterized by fractional order onset functions,found in thermodynamical phase transitions of disordered systems(Stauffer& Aharony1994).

Changing the order of a polynomial description for the transition does not change the order of the singularity.Only the functional dependence between the broadening and the characteristic length scale

18Felix Herrmann,S′e bastien Chevrot and Colin Stark

(extent)changes.Consequently,there are indications that linear gradients and/or cubic polynomials, such as proposed by Gaherty et al.(1999),may not provide a complete and accurate description of the shape of the transitions.Contrary to conventional transition characterizations,sharpness as de?ned in this paper refers to a scale invariant description of the transition which could be related to a possible critical phenomenon.

REFERENCES

Alexandrescu,M.,Gibert,D.,Hulot,G.,Le Mou¨e l,J.-L.,&Saracco,G.,1995,Detection of geomagnetic jerks using wavelet analysis,J.geophys.Res.,,100(B7),12557–12572.

Anderson,D.,1989,Theory of the Earth,Blackwell Scienti?c Publications.

Castle,J.&Creager,K.,2000,Sharpness and shear velocity jump across the660-km discontinuity,J.geophys.

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Chevrot,S.,Vinnik,L.,&Montagner,J.P.,1999,Global scale analysis of the mantle phases,J.Geophys.

Res.,104,20203–20219.

Dziewonsky,A.&Anderson,D.,1981,Preliminary Reference Earh Model,Phys.Earth.Planet.Inter.. Gaherty,J.,Wang,Y.,Jordan,T.,&Weidner,D.,1999,Testing plausible upper-mantle compositions using ?ne-scale models of the410-km discontinuity,Geophys.Res.Lett.,,26(11),1641–1644.

Herrmann,F.,1997,A scaling medium representation,a discussion on well-logs,fractals and waves,Ph.D.

thesis,Delft University of Technology,Delft,the Netherlands.

Herrmann,F.,1998,Multiscale analysis of well and seismic data,in Mathematical Methods in Geophysical Imaging V,edited by S.Hassanzadeh,vol.3453,pp.180–208,SPIE.

Herrmann,F.,2000,Singularity characterization by monoscale analysis,Journal of Applied Harmonic Analy-sis,invited paper in special issue on applications of wavelets.

Herrmann,F.,2000,Global texture characterization of well-and seismic data,in prep.

Herrmann,F.&Stark,C.,1999,Monoscale analysis of edges/re?ectors using fractional differentiation-s/integrations,in Expanded Abstracts,Soc.Expl.Geophys.,Tulsa.

Herrmann,F.&Stark,C.,2000,Seismic facies characterization by monoscale analysis,submitted for publica-tion.

Herrmann,F.&Stark,C.,2000,A scale attribute for texture in well-and seismic data,in Expanded Abstracts, Soc.Expl.Geophys.,Tulsa.

Holschneider,M.,1995,Wavelets an analysis tool,Oxford Science Publications.

Kosarev,G.L.,Makeyeva,L.I.,Savarensky,E.F.,&Chesnokov,E.M.,1979,In?uence of anisotropy beneath seismograph station on body waves,Izv.Acad.Nauk,Fizika Zemli,2,26–37.

Mallat,S.G.,1997,A wavelet tour of signal processing,Academic Press.

Fixed scale analysis19 Paulssen,H.,1988,Evidence for a sharp670-km discontinuity as inferred from-to-converted waves,J. Geophys.Res.,93,10489–10500.

Petersen,N.,Vinnik,L.,Kosarev,G.,Kind,R.,Oreshin,S.,&Stammler,K.,1993,Sharpness of the mantle discontinuities,Geophys.Res.Lett.,20,859–862.

Richards,P.G.&Frasier,C.W.,1976,Scattering of elastic waves from depth-dependent inhomogeneities, Geophysics,41,441–458.

Stammler,K.,Kind,R.,Petersen,N.,Kosarev,G.,Vinnik,L.,&Qiyuan,L.,1992,The upper mantle disconti-nuities:correlated or anticorrelated?,Geophys.Res.Lett.,19,1563–1566.

Stark,C.&Weissel,J.,1998,Fixed scale characterization of singularities using complex fractional order wavelets,unpublished manuscript.

Stark,C.&Weissel,J.,2000,Fractional derivative wavelet analysis of edges,in prep.

Stark,C.,Herrmann,F.J.,&Weissel,J.,2000,Monoscale wavelet analysis and its application to stream?ow time series,in preparation.

Stauffer,D.&Aharony,A.,1994,Introduction to Percolation Theory,Taylor and Francis.

Stixrude,L.,1997,Structure and sharpness of phase transitions and mantle discontinuities,Journal of Geo-physical Research,102(B7),14835–14852.

20Felix Herrmann,S′e bastien Chevrot and Colin Stark

英文summary写作范例

Article Children Must be Taught to Tell Right from Wrong William Kilpatrick Many of today 's young people have a difficult time seeing any moral dimension （ 道德层 面 ） to their actions. There are a number of reasons why that 's true, but none more prominent than a failed system of education that eschews （ 回避 ） teaching children the traditional moral values that bind Americans together as a society and a culture. That failed approach, called “decision - making, ” was introduced in schools 25 years ago. It tells children to decide for themselves what is right and what is wrong. It replaced “character education. （ 品格教 育 ）” Character education didn 't ask children to reinvent the moral wheel （ 浪费时间重新发明早已存 在的道德标准）; instead, it encouraged them to practice habits of courage, justice and self-control. In the 1940s, when a character education approach prevailed, chewing gum; today they worry about robbery and rape. Decision-making curriculums pose thorny （ 棘手的 ） ethical dilemmas to students, with the impression that all morality is problematic and that all questions of right and wrong are in dispute. Youngsters are forced to question values and virtues they 've never acquired in the first place or upon which they have only a tenuous （ 薄弱的 ） hold. The assumption behind this method is that students will arrive at good moral conclusions if only they are given the chance. But the actual result is moral confusion. For example, a recent national study of 1,700 sixth- to ninth-graders revealed that a majority of boys considered rape to be acceptable under certain conditions. Astoundingly, many of the girls agreed. This kind of moral illiteracy is further encouraged by values-education （价值观教育 ） programs that are little more than courses in self-esteem （ 自尊 ）. These programs are based on the questionable assumption that a child who feels good about himself or herself won 't want to do anything wrong. But it is just as reasonable to make an opposite assumption: namely, that a child who has uncritical self-regard w ill conclude that he or she can 't do anything bad. Such naive self-acceptance results in large part from the non-directive （ 无指导性的 ）, non-judgmental （ 无是非观的 ）, as-long-as-you-feel-comfortable-with-your-choices mentality （ 思 想） that has pervaded （ 渗透） public education for the last two and one-half decades. Many of today 's drug education, sex education and values -education courses are based on the same 1960s philosophy that helped fuel the explosion in teen drug use and sexual activity in the first place. Meanwhile, while educators are still fiddling with （ 胡乱摆弄 ） outdated “feel - good ” approaches, New York, Washington, and Los Angeles are burning. Youngsters are leaving school believing that matters of right and wrong are always merely subjective. If you pass a stranger on the street and decide to murder him because you need money —if it feels right —you go with that feeling. Clearly, murder is not taught in our schools, but such a conclusion —just about any conclusion —can be reached and justified using the decision-making method. It is time to consign （ 寄出 ） the fads （风尚 ） of “decision - making ” and “non- judgmentalism ” to the ash heap of failed policies, and return to a proved method. Character education provides a much more realistic approach to moral formation. It is built on an understanding that we learn morality not by debating it but by practicing it. Sample teachers worried about students leaving them

summary 范文

Original: My neighbor's children love playing hide-and-seek as all children do, but no one imagine that a game they played last week would be reported in the local newspaper. One afternoon, they were playing in the vacant lot down the corner. Young Paul, who is only five years old, found the perfect place to hide. His sister, Natalie, had shut her eyes and was counting to ten when Paul noticed the storage mail box at the corner and saw that the metal door was standing open. The mailman had just taken out several sacks of mail and had carried them to his truck which was standing at the curb a few feet away. Paul climbed into the storage box and pulled the door closed so hard that it locked. Soon realizing what he had done, he became frightened and started crying. Meanwhile, Natalie was looking for him everywhere but could not find him. It was lucky that she happened to pause at the corner for a minute and heard her brother's cries. She immediately ran to tell the mailman who hurried back from his truck to unlock the metal door. Paul was now free, but he had had such a bad scare that he could not stop crying. The mailman, however, soon found a way of making him laugh again. He told him that the next time he wanted to hide in a mail box, he should remember to put a stamp on himself! Summary: The children were playing hide-and-seek in a vacant lot one afternoon. Finding that the storage mailbox had been left open, Paul hid and locked himself in it accidentally. His sister, Natalie, heard his cries and realized where he was hiding, so she immediately told the mailman to unlock the metal door. After letting him out, the mailman made him stop crying by telling him to put a stamp on himself the next he wanted to hide in a mailbox. Original: Why do some animals die out? In the past two hundred years people have caused many kinds of animals to die out--to become extinct. People keep building houses and factories in fields and woods. As they spread over the land, they destroy animals' homes. If the animals can't find a place to live, they die out. Sixteen kinds of Hawaiian birds have become extinct for this reason. Other animals, such as the Florida Key deer, may soon die out because they are losing their homes.Hunters have caused some animals to become extinct, too. In the last century, hunters killed all the passenger pigeons in North America and most of the buffalos. Today they are fast killing off hawks and wolves. Pollution is killing many animals today, too. As rivers become polluted, fish are poisoned. Many die. Birds that eat the poisoned fish can't lay strong, healthy eggs. New birds aren’t born. So far, no animals have become extinct because of pollution. But some, such as the bald eagle and the brown pelican, have become rare and may die out. Scientists think that some animals become extinct because of changes in climate. The places where they live become hotter or cooler, drier or wetter. The food that they eat cannot grow there any more. If the animals can't learn to eat something else, they die. Dinosaurs may have died out for this reason. Summary:

summary好例子

Steps to Writing a Summary 1.Read and understand the prompt or writing directions. What are you being asked to write about? Example: Summary of an Article Write a summary of the article. Your writing will be scored on how well you: ?state the main ideas of the article; ?identify the most important details that support the main ideas; ?write your summary in your own words, except for quotations; and ?express the underlying meaning of the article, not just the superficial details. 2. Read, think about, and understand the text. Review the material to make sure you know it well. Use a dictionary or context clues to figure out the meaning of any important words that you don’t know. 3. Take notes. Write down the main ideas and important details of the article. 4.Write a thesis statement. In a single sentence, state the main idea of the article. The thesis statement should mention the underlying meaning of the article, not just the superficial details. 5. Organize and outline ideas. Write down the important details you need to include in the summary. Put them in a logical order. Topic Sentence: Evidence: #1: #2: #3: 6.Write your essay. ?Your summary should be about one third of the length of the original article. ?Focus on the main point of the article and the most important details. ?Use your own words; avoid copying phrases and sentences from the article unless they’re direct quotations. 7.Revise. Have you indented all paragraphs? Have you captured the main point of the article? Have you included the most important details? Is there sentence variety? Have you avoided writing short, choppy sentences? Are there transitional words and phrases to connect ideas? 8. Proofread and edit. Check your spelling, grammar, and punctuation. Is the verb tense consistent? Are all names spelled correctly and capitalized? Have you avoided writing run-on sentences and sentence fragments? 9. Write your draft. Use blue or black ink. Skip lines. Write on one side of the paper only. Include a title on the top line. 10. Read your summary one last time before you turn it in. Look for careless spelling, punctuation, and grammar errors, especially omitted words or letters. Cross out errors neatly with a single line and write the correction above. Original Article: Bats In the distant past, many people thought bats had magical powers, but times have changed. Today, many people believe that bats are rodents, that they cannot see, and that they are more likely than other animals to carry rabies. All of these beliefs are mistaken. Bats are not rodents, are not blind, and are no more likely than dogs and cats to transmit rabies. Bats, in fact, are among the least understood and least appreciated of animals. Bats are not rodents with wings, contrary to popular belief. Like all rodents, bats are mammals, but they have a skeleton similar to the human skeleton. The bones in bat wings are much like those in arms and the human hand, with a thumb and four fingers. In bats, the bones of the arms and the four fingers of the hands are very long. This bone structure helps support the web of skin that stretches from the body to the ends of the fingers to form wings. Although bats cannot see colors, they have good vision in both dim and bright light. Since most bats stay in darkness during the day and do their feeding at night, they do not use their vision to maneuver in the dark but use a process called echolocation. This process enables bats to emit sounds from their mouths that bounce off objects and allow them to avoid the objects when flying. They use this system to locate flying insects to feed on as well. Typically, insect-eating bats emerge at dusk and fly to streams or ponds where they feed. They catch the insects on their wingtip or tail membrane and fling them into their mouths while flying. There are about 1,000 species of bat, ranging in size from the bumblebee bat, which is about an inch long, to the flying fox, which is sixteen inches long and has a wingspan of five feet. Each type of bat has a specialized diet. For seventy percent of bats, the diet is insects. Other types of bats feed on flowers, pollen, nectar, and fruit or on small animals such as birds, mice, lizards, and frogs. (continued on back)

有关summary的写作技巧

有关s u m m a r y的写作技 巧 Company number：【WTUT-WT88Y-W8BBGB-BWYTT-19998】

Summary的写法1 一、概括原文 （一）阅读 1.读懂文章 读文章的时候，要养成良好的阅读习惯，划划写写，英文阅读的时候，用铅笔轻轻划出重点词汇。 认真阅读给定的原文材料。如果一遍不能理解，就多读两遍。阅读次数越多，你对原文的理解就越深刻。 2.拆分文章 按照作者的思路，把文章分段，每个段落用几个词，几个短语概括。尽量简短，精炼。 段落中心句，在段落的开头或末尾。有时也会变态的在当中。 3.概括主旨 写出文章的thesis, 一句话概括文章的主旨。 （二）基本结构和技巧 1.重新拟定标题 给summary起一个标题。用那些能概括文章主题思想的单词、短语或短句子作为标题。也可以采用文中的主题句作为标题。 2.阐述观点 摘要应全部用自己的话完成。不要引用原文的句子。写概述的时候，如果能够明确是他人写作的文章，注意要把作者的名字放在第一句（或者是the

author…….）。接着写出要阐述的main ideas（主要观点）和supporting points （对主要观点的支持）。 3.词汇运用 注意概述的coherence（连贯性），运用好transition words（过渡词）, like however, furthermore, nonetheless, besides, therefore etc. 4.删除细节 只保留主要观点。 5.选择一至两个有代表性的例子 原文中可能包括5个或更多的例子，你只需从中筛选一至二个例子。 6.把长句变成短句，把长段的描述变成短小、简单的句子。 “ He was hard up for money and was being pressed by his creditor.” 可以概括为：“He was in financial difficulties.” “His courage in battle might without exag geration be called lion-like.” 可以概括为：”He was very brave in battle.” “He was hard up for money and was being pressed by his creditor.” 可以概括为：“He was in financial difficulties.” 6) 你还可以使用词组代替整句或者从句。请看下面的例子： “Beautiful mountains like Mount Tai, Lushan Mountain, and Mount Huang, were visited by only a few people in the past. Today, better wages, holidays with pay, new hotels on these mountains, and better train and bus services, have brought them within reach of many who never thought of visiting them ten years ago.” 可以概括为：”Beautiful mountains like Mount Tai, once visited by only a few people, are today accessible to many, thanks to better wages, paid holidays, new hotels and better transportation services.” 7) 使用概括性的名词代替具体的词，比如： “She brought home several Chinese and English novels, a few copies of Time and Newsweek and some textbooks. She intended to read all of them during the winter vocation.”

Summary_常用句式

Summary 常用句式 1.This article/ passage mainly tells (a story) about…… 2.This passage mainly deals with/discusses/explores/…… 3.In this passage (about ……), the author …… 4.In this passage about ……, the author …… 5.The author began the essay/ passage by telling/ presenting…… 6.First/Firstly/ In the beginning/In the first part, the author argues/ explains/ mentions/ states/ points out (that)…… 7.Secondly/ Next/ Further on/ Then/ In the next part/ In the main part, the author goes on with…… 8.Finally/ As a conclusion/, the author concludes/ adds/ stresses that…… 9.Finally, the author summarizes that …… 二、常见句型 1）This paper deals with.. 2）This article focuses on the topics of (that,having,etc). 3）This essay presents knowledge that... 4）This thesis discusses... 5）This thesis analyzes... 6）This paper provides an overview of...

一篇英语summary范文英语Summary写

一篇英语summary范文英语Summary写第一步：阅读 A.认真阅读给定的原文材料。如果一遍不能理解，就多读两遍。阅读次数越多，你对原文的理解就越深刻。 B.给摘要起一个标题。用那些能概括文章主题思想的单词、短语或短句子作为标题。也可以采用文中的主题句作为标题。主题句往往出现在文章的开头或结尾。一个好标题有助于确定文章的中心思想。 C.现在，就该决定原文中哪些部分重要，哪些部分次重要了。对重要部分的主要观点进行概括。 D.简要地记下主要观点——主题、标题、细节等你认为对概括摘要重要的东西。 第二步：动手写作 A. 摘要应该只有原文的三分之一或四分之一长。因此首先数一下原文的字数，然后除以三，得到一个数字。摘要的字数可以少于这个数字，但是千万不能超过这个数字。

B. 摘要应全部用自己的话完成。不要引用原文的句子。 C. 应该遵循原文的逻辑顺序。这样你就不必重新组织观点、事实。 D. 摘要必须全面、清晰地表明原文所载的信息，以便你的读者不需翻阅原文就可以完全掌握材料的原意。 E. 写摘要时可以采用下列几种小技巧： 1) 删除细节。只保留主要观点。 2) 选择一至两个例子。原文中可能包括5个或更多的例子，你只需从中筛选一至二个例子。 3) 把长段的描述变成短小、简单的句子。如果材料中描述某人或某事用了十个句子，那么你只要把它们变成一两句即可。 4) 避免重复。在原文中，为了强调某个主题，可能会重复论证说明。但是这在摘要中是不能使用的。应该删除那些突出强调的重述句。

5) 压缩长的句子。如下列两例： “His courage in battle might without exaggeration be called lion-like.” 可以概括为：”He was very brave in battle.” “He was hard up for money and was being pressed by his creditor.”可以概括为：“He was in financial difficulties.” 6) 你还可以使用词组代替整句或者从句。请看下面的例子： “Beautiful mountains like Mount Tai, Lushan Mountain, and Mount Huang, were visited by only a few people in the past. Today, better wages, holidays with pay, new hotels on these mountains, and better train and bus services, have brought them within reach of many who never thought of visiting them ten years ago.”

英文Summary写作方法、范例及常用句式

摘要是对一篇文章的主题思想的简单陈述。它用最简洁的语言概括了原文的主题。写摘要主要包括三个步骤：（1）阅读；（2）写作；（3）修改成文。 第一步：阅读 A．认真阅读给定的原文材料。如果一遍不能理解，就多读两遍。阅读次数越多，你对原文的理解就越深刻。 B．给摘要起一个标题。用那些能概括文章主题思想的单词、短语或短句子作为标题。也可以采用文中的主题句作为标题。主题句往往出现在文章的开头或结尾。一个好标题有助于确定文章的中心思想。C．现在，就该决定原文中哪些部分重要，哪些部分次重要了。对重要部分的主要观点进行概括。 D．简要地记下主要观点——主题、标题、细节等你认为对概括摘要重要的东西。 第二步：动手写作 A. 摘要应该只有原文的三分之一或四分之一长。因此首先数一下原文的字数，然后除以三，得到一个数字。摘要的字数可以少于这个数字，但是千万不能超过这个数字。 B. 摘要应全部用自己的话完成。不要引用原文的句子。 C. 应该遵循原文的逻辑顺序。这样你就不必重新组织观点、事实。 D. 摘要必须全面、清晰地表明原文所载的信息，以便你的读者不需翻阅原文就可以完全掌握材料的原意。 1 / 19

E. 写摘要时可以采用下列几种小技巧： 1) 删除细节。只保留主要观点。 2) 选择一至两个例子。原文中可能包括5个或更多的例子，你只需从中筛选一至二个例子。 3) 把长段的描述变成短小、简单的句子。如果材料中描述某人或某事用了十个句子，那么你只要把它们变成一两句即可。 4) 避免重复。在原文中，为了强调某个主题，可能会重复论证说明。但是这在摘要中是不能使用的。应该删除那些突出强调的重述句。 5) 压缩长的句子。如下列两例： “His courage in battle might without exaggeration be called lion-like.” 可以概括为：”He was very brave in battle.” “He was hard up for money and was being pressed by his creditor.” 可以概括为：“He was in financial difficulties.” 6) 你还可以使用词组代替整句或者从句。请看下面的例子：“Beautiful mountains like Mount Tai, Lushan Mountain, and Mount Huang, were visited by only a few people in the past. Today, better wages, holidays with pay, new hotels on these mountains, and better train and bus services, have brought them within reach of many who never thought of visiting them ten years ago.” 2 / 19

Summary的写作技巧和常见句型

S u m m a r y 的写作技巧和常见句型 、概括原文 一）阅读 1. 读懂文章读文章的时候，要养成良好的阅读习惯，划划写写，英文阅读的时候，用铅笔轻轻划出重点词汇。 认真阅读给定的原文材料。如果一遍不能理解，就多读两遍。阅读次数越多，你对原文的理解就越深刻。 2. 拆分文章按照作者的思路，把文章分段，每个段落用几个词，几个短语概括。尽量简短，精 炼。 段落中心句，在段落的开头或末尾。有时也会变态的在当中。 3. 概括主旨写出文章的thesis, 一句话概括文章的主旨。 二）基本结构和技巧 1.重新拟定标题给summary 起一个标题。用那些能概括文章主题思想的单词、短语或短句子作为标题。也可以采用文中的主题句作为标题。 2. 阐述观点 摘要应全部用自己的话完成。不要引用原文的句子。写概述的时候，如果能够明确是他人写作的文章，注意要把作者的名字放在第一句（或者是接着写出要阐述的main ideas （主要观点）和supporting points （对主要观点的the author .. ）。

支持）。 3. 词汇运用 注意概述的cohere nee （连贯性），运用好tran siti on words （过渡 词） , like however, furthermore, nonetheless, besides, therefore etc. 4. 删除细节 只保留主要观点。 5. 选择一至两个有代表性的例子 原文中可能包括5 个或更多的例子，你只需从中筛选一至二个例子。 6. 把长句变成短句，把长段的描述变成短小、简单的句子。 He was hard up for money and was being pressed by his creditor. 可以概括为：“ He was in finan cial difficulties. His courage in battle might without exaggeration be called lion- like. ” 可以概括为：” He was very brave in battle. He was hard up for money and was being pressed by his creditor. 可以概括为：“ He was in finan cial difficulties. 6）你还可以使用词组代替整句或者从句。请看下面的例子： Beautiful mountains like Mount Tai, Lushan Mountain, and Mount Huang, were visited by only a few people in the past. Today, better wages, holidays with pay, new hotels on these mountains, and better train and bus services, have brought them within reach of many who never thought of visiting them ten years ago. ” 可以概括为：” Beautiful mountains like Mount Tai, once visited by only a few people, are today accessible to many, thanks to better wages, paid holidays,

Summary的写作技巧和常见句型

Summary的写作技巧与常见句型 一、概括原文 (一)阅读 1、读懂文章 读文章的时候,要养成良好的阅读习惯,划划写写,英文阅读的时候,用铅笔轻轻划出重点词汇。认真阅读给定的原文材料。如果一遍不能理解,就多读两遍。阅读次数越多,您对原文的理解就越深刻。 2、拆分文章 按照作者的思路,把文章分段,每个段落用几个词,几个短语概括。尽量简短,精炼。 段落中心句,在段落的开头或末尾。有时也会变态的在当中。 3、概括主旨 写出文章的thesis, 一句话概括文章的主旨。 (二)基本结构与技巧 1、重新拟定标题 给summary起一个标题。用那些能概括文章主题思想的单词、短语或短句子作为标题。也可以采用文中的主题句作为标题。 2、阐述观点 摘要应全部用自己的话完成。不要引用原文的句子。写概述的时候,如果能够明确就是她人写作的文章,注意要把作者的名字放在第一句(或者就是the author……、)。接着写出要阐述的main ideas(主要观点)与supporting points(对主要观点的支持)。 3、词汇运用 注意概述的coherence(连贯性),运用好transition words(过渡词), like however, furthermore, nonetheless, besides, therefore etc、 4、删除细节 只保留主要观点。 5、选择一至两个有代表性的例子 原文中可能包括5个或更多的例子,您只需从中筛选一至二个例子。 6、把长句变成短句,把长段的描述变成短小、简单的句子。 “He was hard up for money and was being pressed by his creditor、” 可以概括为:“He was in financial difficulties、” “His courage in battle might without exaggeration be called lion-like、” 可以概括为:”He was very brave in battle、” “He was hard up for money and was being pressed by his creditor、” 可以概括为:“He was in financial difficulties、” 6) 您还可以使用词组代替整句或者从句。请瞧下面的例子: “Beautiful mountains like Mount Tai, Lushan Mountain, and Mount Huang, were visited by only a few people in the past、Today, better wages, holidays with pay, new hotels on these mountains, and better train and bus services, have brought them within reach of many who never thought of visiting them ten years ago、” 可以概括为:”Beautiful mountains like Mount Tai, once visited by only a few people, are today accessible to many, thanks to better wages, paid holidays, new hotels and better transportation services、” 7) 使用概括性的名词代替具体的词,比如: “She brought home several Chinese and English novels, a few copies of Time and Newsweek