Backward DVCS and Proton to Photon Transition Distribution Amplitudes

backward warping方法
反向变形（backward warping）是计算机视觉和图像处理中的一种技术。

它被广泛应用于实现图像的变形、图像纠正、跟踪与匹配等应用。

顾名思义，反向变形就是逆转了变形的方向，根据变形后的像素点坐标计算变形前的像素点的位置。

正向变形是指根据变形前的像素点坐标计算变形后的像素点位置的过程，例如图像的仿射变换、透视变换等。

这种变形方法对像素点坐标进行变换，从而实现图像的变形，但它不利于像素间的精确匹配。

反向变形的实现有很多方法，其中比较常见的包括三角剖分、逆距离加权和贝塞尔曲线等。

三角剖分（triangulation）是一种常用的反向变形方法。

对于一个图像，可以先将其用若干个三角形来划分，形成三角形网格。

对于变形后的图像，在三角形网格中找到其所在的三角形，然后再根据线性插值和重心坐标计算变形前的像素点位置。

逆距离加权（inverse distance weighting）则是一种较为简单的反向变形方法。

它的基本思想是根据变形后的像素点与其周围像素点的距离来计算变形前的像素点位置，距离越近的像素点权重越大。

该方法虽然简单，但在某些情况下可获得较好的变形效果。

贝塞尔曲线（Bezier curve）则是一种基于贝塞尔插值的反向变形方法。

该方法通过一系列控制点来确定几何形状，进而实现反向变形。

与三角剖分相比，贝塞尔曲线更加精确，但需要较为复杂的计算。

总之，反向变形在计算机视觉和图像处理的应用中具有广泛的应用前景。

无论是三角剖分、逆距离加权还是贝塞尔曲线，都是实现反向变形的有效方法。

a rXiv:h ep-ph/15121v114May21LAPTH-845/01LPT-Orsay 01-43May 2001Isolated prompt photon photoproduction at NLO M.Fontannaz a ,J.Ph.Guillet b ,G.Heinrich a a Laboratoire de Physique Th´e orique 1LPT,Universit´e de Paris XI,Bˆa timent 210,F-91405Orsay,France b Laboratoire d’Annecy-Le-Vieux de Physique Th´e orique 2LAPTH,Chemin de Bellevue,B.P.110,F-74941Annecy-le-Vieux,France Abstract We present a full next-to-leading order code to calculate the photoproduction of prompt photons.The code is a general purpose program of ”partonic event generator”type with large flexibility.We study the possibility to constrain the photon structure functions and comment on isolation issues.A comparison to ZEUS data is also shown.1IntroductionHigh energy electron-proton scattering at the DESY ep collider HERA is dominated by photopro-duction processes,where the electron is scattered at small angles,emitting a quasireal photon which scatters with the proton.These processes are of special interest since they are sensitive to both the partonic structure of the photon as well as of the proton.In particular,they offer the possibil-ity to constrain the(presently poorly known)gluon distributions in the photon,since in a certain kinematical region the subprocess qg→γq,where the gluon is stemming from a resolved photon, is dominating.Up to now,the experimental errors were too large to discriminate clearly between different sets of gluon distributions in the photon,but a high statistics analysis of the1996-2000 HERA data on prompt photon photoproduction announced by the ZEUS collaboration will shed new light on this issue.The calculation of higher order corrections to the Compton processγq→γq has been initiated some time ago[1]–[6].The most recent calculations for prompt photon photoproduction have been done by Gordon/Vogelsang[6]for isolated prompt photon production,Gordon[7]for photon plus jet production and by the group Krawczyk/Zembrzuski[8]for both the inclusive case and γ+jet.However,all of these calculations contain certain drawbacks.In[6],isolation is implemented by adding a subtraction term evaluated in the collinear approximation to the fully inclusive cross section.The programs of[7]and[8]do not contain the full set of NLO corrections.In[7],those parts where thefinal state photon comes from fragmentation of a hard parton were included only at leading order,arguing that isolation cuts will suppress the fragmentation component in any case to a large extent.Moreover,the box contribution has not been included.In[8],higher order corrections are included only for the case where initial andfinal state photons are both direct.So not only the contributions from fragmentation,but also the case where the initial photon is resolved are included at Born level only.However,the box contribution has been taken into account.The calculation presented in this paper takes into account the full NLO corrections to all four subparts.The corresponding matrix elements already have been calculated and tested in previous works[2,9,10].A major advantage of the present code is also given by the fact that it is constructed as a”partonic event generator”and as such is veryflexible.Various sorts of observables matching a particular experimental analysis can be defined and histogrammed for an event sample generated once and for all.This strategy already has been applied to construct NLO codes forγγproduction (DIPHOX)[11]and one or two jets photoproduction[12].The paper is organized as follows.In section2wefirst describe the theoretical framework and the treatment of the infrared singularities.Then we discuss the implementation of isolation cuts and outline the structure of the code.Section3is devoted to phenomenology.We study the effect of isolation,determine the kinematic region which is most sensitive to the gluon distribution in the photon and illustrate the sensitivity of the cross section to the energy of the incoming photon.We give results for inclusive isolated prompt photon production and compare with a recent analysis of ZEUS data[13],before we come to the conclusions in section4.2Theoretical formalism and description of the methodIn this section the general framework for prompt photon photoproduction will be outlined.We will review the contributing subprocesses,the treatment of infrared singularities and the implementation of isolation cuts.2.1The subprocesses contributing at NLOThe inclusive cross section for ep→γX can symbolically be written as a convolution of the parton densities of the incident particles(resp.fragmentation function for an outgoing parton fragmenting into a photon)with the partonic cross sectionˆσdσep→γX(P p,P e,Pγ)= a,b,c dx e dx p dz F a/e(x e,M)F b/p(x p,M p)dˆσab→cX(x p P p,x e P e,Pγ/z,µ,M,M p,M F)Dγ/c(z,M F)(1) where M,M p are the initial state factorization scales,M F thefinal state factorization scale andµthe renormalization scale.The subprocesses contributing to the partonic reaction ab→cX can be divided into four cat-egories which will be denoted by1.direct direct 2.direct fragmentation 3.resolved direct 4.resolved fragmentation.The cases”direct direct”and”resolved direct”correspond to c=γand Dγ/c(z,M F)=δcγδ(1−z)in(1),that is,the prompt3photon is produced directly in the hard subprocess.The cases with”direct”attributed to the initial state photon correspond to a=γ,with Fγ/e approximated by the Weizs¨a cker-Williams formula for the spectrum of the quasireal photonsf eγ(y)=αemylnQ2max(1−y)y .(2)The”resolved”contributions are characterized by a resolved photon in the initial state where a parton stemming from the photon instead of the photon itself participates in the hard subprocess. In these cases,F a/e(x e,M)is given by a convolution of the Weizs¨a cker-Williams spectrum with the parton distributions in the photon:F a/e(x e,M)= 10dy dxγf eγ(y)F a/γ(xγ,M)δ(xγy−x e)(3)Examples of diagrams contributing at Born level to the four categories above are shown in Figs.1 and2.In the case of the”direct direct”part,only the Compton processγq→γq contributes at leading order,at NLO the O(αs)corrections fromγq→γqg resp.γg→γq¯q and the corresponding virtual corrections contribute.We also included the box contribution(Fig.3)into the”direct direct”part since it is known to be sizeable[4],although it is formally a NNLO contribution.In the”direct fragmentation”part,thefinal state photon comes from the fragmentation of a hard parton participating in the short distance subprocess.From a technical point of view,afinal state quark-photon collinear singularity appears in the calculation of the subprocessγg→γq¯q.At higher orders,final state multiple collinear singularities appear in any subprocess where a high p T parton(quark or gluon)undergoes a cascade of successive collinear splittings ending up with a quark-photon splitting.These singularities are factorized to all orders inαs and absorbed,at some arbitrary fragmentation scale M F,into quark and gluon fragmentation functions to a photon,Dγ/c(z,M2F). When the fragmentation scale M F,chosen of the order of the hard scale of the subprocess,is large compared to any typical hadronic scale∼1GeV,these functions behave roughly asα/αs(M2F). Then a power counting argument tells that these fragmentation contributions are asymptotically of the same order inαs as the Born term.A consistent NLO calculation thus requires the inclusion of the O(αs)corrections to these contributions.Note that the singularity appearing in the processγg→γq¯q when thefinal state photon is emitted by the quark and becomes collinear,is subtracted and absorbed by the fragmentation function at the scale M F,as explained above.Therefore both the”direct direct”and the”direct fragmentation”parts separately depend strongly on M F and the attribution of thefinite terms to either of these parts is scheme dependent.Only in the sum of these parts the M F dependenceflattens as expected.The collinear singularities appearing at NLO if the incident photon splits into a collinear q¯q pair are absorbed into the functions F q/γ(xγ,M)at the factorization scale M.(Analogous for theorderFigure1:Examples of direct direct and direct fragmentation contributions at leadingFigure3:The box contributionproton distribution functions F b/p(x p,M p);we will set M p=M in the following.)Thus,by the same reasoning as above for thefinal state,the”initial direct”and”initial resolved”parts separately show a strong dependence on M which cancels out in the sum.Therefore it has to be stressed that only the sum over all four parts has a physical meaning.Figure5illustrates these cancellation mechanisms.The overall reduction of the scale dependence when going from leading to next-to-leading order can be seen in Fig.4.The scales M F and M have been set equal toµ,andµhas been varied between µ=pγT/2andµ=2pγT.One can see that the NLO cross section is much more stable against scale variations,it varies by less than10%in thisµrange.Figure4:Dependence of the total cross section on scale variations.µ=M=M F is varied between µ=pγT/2andµ=2pγT.Figure5:Cancellation of the leading dependence on the fragmentation scale M F between con-tributions from direct and fragmentationfinal states,and on the factorization scale M between parts with direct and resolved initial state.The results are normalized to the total cross section at M F=M=µ=pγT.2.2Treatment of infrared singularitiesThere are basically two methods to isolate the infrared singularities appearing in the calculation at NLO:The phase space slicing method[15]and the subtraction method[16].The method used here follows the approach of[11,14]which combines these two techniques.We will outline the strategy only shortly,for more details we refer to[11].For a generic reaction1+2→3+4+5,at least two particles of thefinal state,say3and4, have a high p T and are well separated in phase space,while the last one,say5,can be soft,and/or collinear to either of the four others.In order to extract these singularities,the phase space is cut into two regions:–part I where the norm p T5of the transverse momentum of particle5is required to be less than some arbitrary value p T m taken to be small compared to the other transverse momenta.This cylinder contains the infrared and the initial state collinear singularities.It also containsa small fraction of thefinal state collinear singularities.–parts II a(b)where the transverse momentum vector of the particle5is required to have a norm larger than p T m,and to belong to a cone C3(C4)about the direction of particle3(4), defined by(η5−ηi)2+(φ5−φi)2≤R2th(i=3,4),with R th some small arbitrary number.C3(C4)contains thefinal state collinear singularities appearing when5is collinear to3(4).–part II c where p T5is required to have a norm larger than p T m,and to belong to neither of the two cones C3,C4.This slice yields no divergence,and can thus be treated directly in4 dimensions.The contributions from regions I and IIa,b are calculated analytically in d=4−2ǫdimensions and then combined with the corresponding virtual corrections such that the infrared singularities cancel, except for the initial(resp.final)state collinear singularities,which are factorized and absorbed into the parton distribution(resp.fragmentation)functions.After the cancellation,thefinite remainders of the soft and collinear contributions in parts I and II a,b,c separately depend on large logarithms ln p T m,ln2p T m and ln R th.When combining the different parts,the following cancellations of the p T m and R th dependences occur:In part I,thefinite terms are approximated by collecting all the terms depending logarithmically on p T m and neglecting the terms proportional to powers of p T m.On the contrary,the R th dependence in the conical parts II a and II b,is kept exactly.This means that an exact cancellation of the dependence on the unphysical parameter R th between part II c and parts II a,b occurs,whereas the cancellation of the unphysical parameter p T m between parts II c,II a,b and part I is only approximate. The parameter p T m must be chosen small enough with respect to pγT in order that the neglected terms can be safely dropped out.On the other hand,it cannot be chosen too small since otherwise numerical instabilities occur.We have investigated the stability of the cross section by varying p T m and R th between0.005and0.1(see Figure6)and accordingly chosen the optimal values p T m=0.05GeV,R th=0.05.Figure6:Dependence of the total cross section on variations of the slicing parameter p T m.Rσdenotes the total(nonisolated)cross section normalized to the total cross section evaluated with p T m=0.005,Rσ(p T m)=σtot(p T m)/σtot(p T m=0.005).One can see that there is a plateau where the cross section is fairly insensitive to variations of p T m.The same study has been made for the dependence on R th,but there the cross section is completely stable within the numerical errors since the R th dependence has been kept exactly in all parts of the matrix element.2.3Implementation of isolation cutsIn order to single out the prompt photon events from the huge background of secondary photons produced by the decays ofπ0,η,ωmesons,isolation cuts have to be imposed on the photon signals in the experiment.A commonly used isolation criterion is the following4:A photon is isolated if, inside a cone centered around the photon direction in the rapidity and azimuthal angle plane,theamount of hadronic transverse energy E hadT deposited is smaller than some value E T maxfixed bythe experiment:(η−ηγ)2+(φ−φγ)2≤R2expE hadT≤E T max (4) Following the conventions of the ZEUS collaboration,we used E T max=ǫpγT withǫ=0.1and R exp=1.Isolation not only reduces the background from secondary photons,but also substantially reduces the fragmentation components,as will be illustrated in section3.1.Furthermore,it is important to note that the isolation parameters must be carefullyfixed in order to allow a comparison between data and perturbative QCD calculations.Indeed a part of the hadronic energy measured in the cone may come from the underlying event;therefore even the direct contribution can be cut by the isolation condition if the latter is too stringent.Let us estimate the importance of this effect and assume that the underlying event one-particle inclusive distribution is given bydn(1)2π4p T ,(5)n(1)being normalized to¯n particles per unit of rapidity.The probability that the isolation condition is fulfilled by a particle from an underlying event isn(1)isol= cone dφdη ∞E T max p T dp T dn(1)2 1+2E T max p T (6) With the ZEUS isolation parameters,E T max=0.5GeV for a photon of pγT=ing¯n=3 and p T ≈0.35GeV extracted from[18],one obtainsn(1)isol≈0.33This estimation is very rough and underestimates the true effect because there is also a non-negligible probability to fulfill the isolation condition with two underlying particles falling into the cone.Only a detailed Monte Carlo description of the underlying events can allow a reliable estimate of this non-perturbative effect.Here we just note that the cut put by ZEUS(E T max≈0.5GeV)is likely to be too low to eliminate any underlying event contamination and therefore makes a comparison between the partonic level QCD predictions and the(hadronic level)data difficult.2.4Features of the codeThe code consists of four subparts corresponding to each of the four categories of subprocesses. For each category,the functions corresponding to the parts I,II a,b,c described in section2.2are integrated separately with the numerical integration package BASES[23].Based on the grid produced by this integration,partonic events are generated with SPRING[23]and stored into an NTUPLE or histogrammed directly.It has to be emphasized that we generatefinal state partonic configurations.Hence this type of program does not provide an exclusive portrait offinal states as given by hadronic event generators like PYTHIA or HERWIG.On the other hand,the latter are only of some improved leading logarithmic accuracy.The information stored in the NTUPLE are the4-momenta of the outgoing particles,their types(i.e.quark,gluon or photon),the energy of the incident photon and, in the fragmentation cases,the longitudinal fragmentation variable associated with the photon from fragmentation.Furthermore a label is stored that allows to identify the origin of the event,e.g.if it comes from a2→2or a2→3process.Based on the information contained in these NTUPLES, suitable observables can be defined and different jet algorithms can be studied.The isolation cuts are included already at the integration level,but the user of the program can turn isolation on or offand vary the input parameters for the isolation cut at will.3Numerical results and comparison to ZEUS dataIn this section we present some numerical results for isolated prompt photon production.We restrict ourselves to the inclusive case,photon+jet production will be discussed in detail in a forthcoming publication.For the parton distributions in the proton we take the MRST2[19]parametrization.Our de-fault choice for the photon distribution functions is AFG[20],for comparisons we also used the GRV[21]distributions transformed to the(7)2E ewhere the sum is over all calorimeter cells,E is the energy deposited in the cell and p z=E cosθ.In order to obtain the”true”photon energy y,corrections for detector effects and energy calibration have to be applied to y JB.These corrections are assumed to be uniform over the whole y range and enter into the experimental systematic error.However,as the background varies with the photon energy y,these corrections may not be uniform.It has to be emphasized that the cross section is very sensitive to a variation of the energy range of the photon.(See Figure12and discussion below.)3.1Numerical results for inclusive prompt photon productionIf not stated otherwise,all plots showing the photon rapidity(ηγ)dependence are integrated over 5GeV<pγT<10GeV and0.2<y=Eγ/E e<0.9.Figure7shows a comparison of the NLO to the leading order result for the isolated cross section dσ/dηγThe importance of the box contribution is clearly visible.The higher order corrections enhance the isolated cross section by about40%.Fig.8shows the rapidity distribution of the full cross section before and after isolation.As already mentioned in section2.3,we used the isolation cuts E T max=ǫpγT withǫ=0.1and R exp =1to match those of the ZEUS collaboration.Fig.8also shows the effect of isolation on thefragmentation part5separately.Isolation reduces the fragmentation component to about6%of the total isolated cross section.In Fig.9the relative magnitude of all four components contributing to dσep→γX/dηγbefore and after isolation is shown.Note that isolation increases the contributions with a direct photon in the final state slightly since there the cut mainly acts on a negative term,which is the one where parton 5is collinear to the photon.It should be emphasized that Figure9has to be read with care since the individual parts have no physical meaning and are very sensitive to scale changes.Nevertheless the dominance of the resolved direct part remains if we choose e.g.µ=M=M F=pγT/2or2pγT.Figure10shows the relative magnitude of contributions from resolved and direct photons in the initial state to the isolated cross section.From the pγT distribution one can conclude that theresolved part dominates the cross section for small values of pγT such that it would be useful to lookat the photon rapidity distribution at pγT=5GeV in order to discriminate between different parton distribution functions in the photon.Since the gluon distribution in the photon is of particular interest,the sensitivity to the gluon in the photon is investigated in Fig.11.One can see that the gluon distribution in the photon starts to become sizeable only for photon rapiditiesηγ>1and dominates over the quark distribution for aboutηγ>2.5.Therefore the region of large photon rapidities and small photon p T is the one where the sensitivity to the gluon in the photon is largest.In order to test further the sensitivity to the gluon,we increased the gluon distribution in the photon uniformly by20%.As can be anticipated from Fig.11,the effect becomes sizeable only forηγ>2and leads to an increase of the cross section by about10%forηγ>2.5.We conclude that in the regionηγ<1,there is basically no sensitivity to the gluon in the photon.However,investigating the photon+jet cross section instead of the inclusive case offers larger possibilities to constrain the gluon in the photon since there one can vary the photon and the jet rapidities in order to single out a kinematic region where the sensitivity is large[24].Figure12shows the effect of a ten percent uncertainty in the”true”bounds of the photon energy y.One can see that a change of the lower bound on y has a large effect,in particular at large photon rapidities.This comes from the fact that the Weizs¨a cker-Williams distribution is large and steeply falling at small y.Increasing the lower bound on y therefore removes a large fraction of the direct events with lower energy initial photons.(y=x e for the direct events and largeηγcorrespond to small x e.)At large photon rapidities the spread due to the use of different parton distribution functions for the photon is smaller than the one caused by a10%variation of the lower bound on y.On the other hand,the region of large photon rapidities is of special interest since there the gluon in the photon is dominating.Therefore a small experimental error in the reconstruction of the ”true”photon energy is crucial in order to be able to discriminate between different sets of parton distribution functions in the photon.It has been tested that the effect of using different proton distribution functions–for example the CTEQ4M or the MRST1set of proton distribution functions–is of the order of3%at most. In all photon rapidity bins the spread is smaller than the one caused by different sets of photon distribution functions(which is about10%at small photon rapidities,see e.g.Fig.14).Thus a discrimination between different sets of photon distribution functions should be possible with the forthcoming full1996-2000data set analysis,where the errors on the data are expected to be small enough.Figure7:Comparison of NLO to LO result for the photon rapidity distribution.Figure8:Effect of isolation on the photon rapidity distribution dσep→γX/dηγfor the full cross section and for the fragmentation components separately.Isolation withǫ=0.1,R exp=1.Figure9:Relative magnitude of all four components contributing to dσep→γX/dηγfor the scale.choiceµ=M=M F=pγTFigure10:Comparison of contributions from resolved and direct photons in the initial state for the photon rapidity and transverse momentum distribution,with isolation.Figure11:Ratio of the contribution from quark resp.gluon distributions in the photon to the fullresolved part.Figure12:Photon rapidity distribution dσep→γX/dηγfor isolated prompt photons integrated over5GeV<pγT <10GeV and different lower bounds on y.Solid line:0.2<y<0.9with AFG photonstructure functions,dotted line:bounds on y changed by about10%,dashed line:0.2<y<0.9 with GRV photon structure functions3.2Comparison with ZEUS dataIn this section we compare our results to the ZEUS1996-97data on inclusive prompt photon photoproduction[13].Figures13and14show the photon p T and rapidity distributions with AFG resp.GRV sets of structure functions for the photon.For the p T distribution the agreement between data and theory is quite good.In the rapidity distribution(Fig.14)the datafluctuate a lot,such that the agreement is still satisfactory.However,it seems that theory underpredicts the data in the backward region,whereas the theoretical prediction tends to be higher at large photon rapidities. The curves of Gordon[7]and Krawczyk/Zembrzuski[8]given in[13]also show this trend.At high ηγthe reason for the difference could be that the isolation cut in the experiment removes more events than in the theoretical(parton level)simulation,as discussed in section2.3.Figure15shows that the discrepancy between theory and data at lowηγcomes mainly from the domain of small photon energies,whereas the discrepancy at largeηγis only present in the range of large photon energies.Note that at largeηγand large y the resolved part dominates and the underlying event could have a large multiplicity.Therefore the isolation criterion could also cut on the non-fragmentation contributions as discussed in section2.3.Figure13:Comparison to ZEUS data of photon p T distribution dσep→γX/dpγT for isolated prompt photons.Results for two different sets of parton distributions in the photon are shown.Figure14:Comparison to ZEUS data of photon rapidity distribution dσep→γX/dηγfor isolated prompt photons.Figure15:Photon rapidity distribution dσep→γX/dηγintegrated over5GeV<pγT<10GeV and different subdivisions of photon energies:(a)0.2<y<0.32,(b)0.32<y<0.5,(c)0.5<y<0.9.4ConclusionsWe have presented a program for prompt photon photoproduction which includes the full next-to-leading order corrections to all contributing subparts.It is a general purpose code of partonic event generator type and as such veryflexible.We used it to study the possibility to constrain the quark and gluon distributions in the photon. It turned out that the sensitivity to the gluon distribution in the photon is negligible in the rapidity range−0.7<ηγ<0.9studied by ZEUS.A discrimination between the AFG/GRV sets of parton distributions in the photon is not possible with the present experimental errors on the ZEUS1996/97data.However,a forthcoming analysis of all1996-2000data announced by the ZEUS collaboration will drastically improve this situation.We have shown that the cross section is very sensitive to small variations of the photon en-ergy range.Therefore a good control of the experimental error on the photon energy fraction y (reconstructed experimentally from the Jacquet-Blondel variable y JB)will be crucial for future comparisons.Despite the largefluctuations of the data,one can say that there is a trend that theory overpre-dicts the data in the forward region.The reason might be that the isolation cut imposed at partonic level in the perturbative QCD calculation does not have the same effect as the experimental one.If the experimental cut is too stringent,a large fraction of the hadronic energy in the isolation cone may come from underlying events,such that experimentally a larger number of events is rejected. We gave a rough estimate of the underlying events to be expected in the isolation cone.The possibilities offered by the study of photon+jet photoproduction will be investigated in a forthcoming publication[24].AcknowledgementsWe would like to thank P.Bussey from the ZEUS collaboration for helpful discussions.G.H. would like to thank the LAPTH for its continuous hospitality.This work was supported by the EU Fourth Training Programme”Training and Mobility of Researchers”,network”Quantum Chromo-dynamics and the Deep Structure of Elementary Particles”,contract FMRX–CT98–0194(DG12-MIHT).References[1]D.W.Duke and J.F.Owens,Phys.Rev.D26,1600(1982).[2]P.Aurenche,A.Douiri,R.Baier,M.Fontannaz and D.Schiff,Z.Phys.C24,309(1984).[3]A.C.Bawa,M.Krawczyk and W.J.Stirling,Z.Phys.C50,293(1991).[4]P.Aurenche,P.Chiappetta,M.Fontannaz,J.Ph.Guillet and E.Pilon,Z.Phys.C56,589(1992).[5]L.E.Gordon and J.K.Storrow,Z.Phys.C63,581(1994).[6]L.E.Gordon and W.Vogelsang,Phys.Rev.D52,58(1995).[7]L.E.Gordon,Phys.Rev.D57,235(1998).[8]M.Krawczyk and A.Zembrzuski,hep-ph/9810253.。

四步相移结构光的英文
English:
Four-step phase-shifting structured light is a method used in 3D scanning and measurement. It involves projecting a series of structured light patterns onto the subject being scanned, with each pattern shifted by a fraction of the pattern cycle. By phase-shifting the projected patterns, it is possible to extract depth information from the captured images, allowing for the reconstruction of a 3D model of the subject with high precision and resolution. This method is commonly used in industrial metrology, quality control, and reverse engineering applications.
中文翻译:
四步相移结构光是一种用于3D扫描和测量的方法。

它包括将一系列结构光模式投影到被扫描的对象上，每个模式都按照模式周期的一部分进行了偏移。

通过对投射的模式进行相位移，可以从捕获的图像中提取深度信息，从而可以重建对象的三维模型，实现高精度和高分辨率。

这种方法通常用于工业计量学、质量控制和逆向工程应用中。

Shade 着色Render 渲染Adjust 调节特效Brightness Conrtast亮度和对比度Channel Mixer通道混合Color Balance 颜色平衡Color StabilzerCurues曲线控制Hue/Saturation 色调饱和度Levels(Individual Controls)灰度级Posterize色调分离Threshold 阈值Channel通道特效Alpha Levels 调节图像Alpha通道Arithmetic算法Blend混合Cineon Lonverter转换Cineon帧文件Compound Arithmetic复合算法Invert转化Minimax扩亮扩暗Remove Color Mating删除蒙板颜色Set Channels设置通道Set Mattle设置蒙板Shift Channels转换通道Image Control图像控制特效Change Color颜色转变Color Balance(HLS)颜色平衡（HLS）Colorama彩光Equalize均衡Gamma(中介曲线)/pedestal(最低输出值)/Gain(最大输出值)调整每个通道的反应曲线Median中值PS Arbitary Map映像Tint 色彩Keying键控特效Color Difference Key对图像中含透明或半透明的素材键出Color Key对指定色键出Color Range对Lab,Yuv或RGB等不同颜色空间键出Difference Matte通过一个对比层与源层进行比较然后将源层中位置和颜色与对比层中相同的像素输出Extract通过指定一个亮度范围产生透明，键出图像中所有与指定键出亮度相近的像素，主要用于背景与保留对象明暗对比度强烈的素材Inner Outer Key指定两个遮罩路径，一个键出范围内边，一个键出范围外边，系统根据内外遮罩进行差异比较Linear Color Key通过指定RGB,HUE或Chroma键出，也可保留前边使用键控变为透明的颜色Luma Key键出与指定明度相似的区域适用，对比强烈图Paint艺术化特效Vector Paint模仿绘画，书写等过程性动画效果Render艺术化特效Audio Spectrum将指定的声音以其频谱形式图像化Beam 激光效果Audio Waveform以波形指定的音频图像化Ellipse依据给定的尺寸在图像上画一椭圆Fill以选定的颜色对目标遮罩进行填充Fractal纹理效果（万花筒）Fractal Noise产生闪电效果或其它的电子特技效果Lighthing产生闪电效果或其它的电子特技效果Radio Waves沿效果点中心向外扩展发射出无线电波的波纹Ramp在图像上创建一个彩色渐变斜面，可以将其原图融合Stroke沿指定的路径产生描边效果Vegas沿图像轮廓或指定的路径进行艺术化描边Stylize风格化效果（模仿各种画风模拟真实的艺术手法创作）Brush Strokes产生画笔描绘的粗糙外观效果Color Emboss产生彩色浮雕效果Emboss产生单色浮雕效果Find Edges强化颜色变化区域的过渡像素，模仿铅笔色边效果Glow搜索图像中明亮部分，然后对周像素明亮化，产生扩散的辉光效果Leave Color使指定颜色保持不变，而把其它部分转换成灰色显示Mosaic分割图像为许多正方形的方格，马赛克效果Motion Tile将多个源图像作为磁片复制到输出，屏幕分割为许多个正方形Noise在图像中加入细小的杂点，产生噪波效果Scatter在不改变每个独立象素色彩的前提下重新分配产生模糊的，涂抹的外观Strobe Light产生闪烁的效果Texturize指定层的纹理射到当层图像上Write On在指定层中产生笔书写效果Simulation仿真特效Card Dance根据指定层的特征分割画面，产生舞踏的效果Caustics模拟气泡，水珠等流体效果Shatter对图像进行粉碎爆炸处理，产生爆炸飞散的碎片Wave World创造液体波纹效果Particle Playground产生大量相似物体独立运动的动画效果Audio音效效果Backwards将声音从结束关键帧播放到开始关键帧，实现反向播放Base & Treble调整音频层音调Delay精确控制声音的延迟和调制，达到回声效果Flange & Chorus合成两种分离的音频特技效果High-Low Pass将低音和高音从声音中滤出Modulator通过变化频率和振幅给音频加颤音，比如逐渐消失Parametric EQ精确调整音频的声调Reverb表现宽阔的真实回声效果Stereo Mixe混合左右声道，产生一个声道到另一声道的完整音频Tone产生各种特技效果Blue & Sharpen模糊和锐化Channel Blur 对图像中的RGB和ALPHA通道进行单独的模糊Compound Blur沿指定的模糊层的的亮度为基准，对当层模糊Directional Blur沿指定方面产生模糊Fast Blur/Gaussian Blur高度模糊Radial Blur以效果点为基准，产生辐射模糊Sharpen通过相邻像素点之间的对比度进行图像清晰化Unsharp Mask通过增加定义边缘颜色的对比度产生边缘锐化效果Distort 扭曲特效Bezier Warp 在层的边界上沿一条封闭的Bezier曲线变形图像Bulge 以效果点为基准对图像进行变形处理使图像产生凹凸Corner Pin 通过改变图像四个边角的位置变形图像Displacement Map 以指定层的像素颜色值为基准变形产生变形效果Mesh Warp 在层上使用网格的Beizer切片控制图像的变形区域Mirror 沿分割线划分图像并反向一边图像到另一边Offset 根据设定的偏量对图像进行偏移对图像推向另一边Optics Compensation 产生摄像机透镜变形的效果Polar Coordinatess 将直角坐标转为极坐标或将极坐标转为直角坐标Reshape 产生涟漪效果，以圆心为轴向四周扩散Smear 在图像中定义一个区域内图像进行偏移延伸和变形Spherize 球面化效果，可以改变球形效果点位置Transform 产生二维几何变化Twirl 围绕指定点旋转图像，产生漩涡效果Wave Warp 在指定的参数范围内随机产生弯曲的波浪效果Perspective三维空间Basic 3D 建立一个虚拟的三维空间，在三维空间中对对象进行操作Bevel Alpha 在图像的Alpha通道区域出现导角外观Bevel Edges 在图像边缘产生导角外观Drop shadow 沿图像的Alpha通道边缘为图像制作阴影特效Text文本Basic Text 文本Numbers 产生随机和连续的数字效果Path Text 使文字沿路进行动画Time时间Echo 在层的不同点上合成关键帧，对前后帧进行混合，产生拖影或运动模糊Postering Time 为当前层指定一个新的帧速率产生特殊效果Time Displacement 通过按时转换像素以变形影像，产生各效Transitions两个镜头间如何进行连接BlockDissolve 以随机的方块对两个层的重叠部分进行切换Card Wipe 发和指定切换层进行卡片的反转擦拭Gradient Wipe 以指定层的亮值建立一个渐层Iris Wipe 指定顶点数产生多边形，对图像进行切换Radial Wipe 在指定的环绕方向上呈辐射擦拭层素材Venetian Blinds 在层素材或合成图像上产生百叶窗效果Linear Wipe 在层指定方向上显示擦拭效果，显示底层画面Video视频效果Brdcast Colors （广播级颜色）调整像素色彩的值Reduce Interlace FlickerTimecode 消除隔行扫描产生的闪烁的现象。

图像风格迁移算法流程There are various algorithms for image style transfer, which is a popular topic in the field of computer vision and image processing. 图像风格迁移是计算机视觉和图像处理领域一项热门话题，有各种算法可供选择。

One of the most well-known algorithms for image style transfer is the Gatys et al. algorithm, which utilizes neural networks to transfer the style of one image onto the content of another. 其中，最广为人知的算法之一是Gatys等人的算法，该算法利用神经网络将一个图像的风格转移到另一个图像的内容上。

The basic idea behind this algorithm is to use a pre-trained convolutional neural network (CNN) to extract features from both the style reference image and the content image. 该算法的基本思想是利用预训练的卷积神经网络（CNN）从风格参考图像和内容图像中提取特征。

Then, a new image is initialized with the content image and iteratively updated to minimize the difference between its contentfeatures and those of the content image, and its style features and those of the style reference image. 然后，使用内容图像初始化一个新图像，并迭代更新，以最小化它的内容特征与内容图像的差异，以及它的风格特征与风格参考图像的差异。

专利名称：Ortho projector to make photo maps from aerial photographs发明人：POLZLEITNER; FRANZ WOLFGANG申请号：US27877172申请日：19720808公开号：US3915569A公开日：19751028专利内容由知识产权出版社提供摘要：An autographic unit scans a model (1) representative of a terrain profile having profile zones, or planes and a displacement measuring device (21, 22) measures coordinate relative movement between one autograph optic and the aerial photo (2) to provide stepped digital position output signals of image coordinates representing line elements, at ground surface. A computer (38) calculates the average image coordinate (xon, yon) and the difference image coordinate value ( DELTA xon, DELTA yon) which are interpolated and applied to a differential distortion correction calculator (24) to calculate the re-establishment of the image and the required image enlargement and image rotation. A line aperture (9) exposes an ortho photo film (6) in accordance with the image portion, as re-established and modified by the calculated enlargement and rotation, the ortho photo film being moved in accordance with the plane movement of the scanning device of the autographic unit.申请人：POLZLEITNER; FRANZ WOLFGANG更多信息请下载全文后查看。

第30卷第2期2021年4月淮阴工学院学报Jonmat of Huaiyin Institute of Tech/obpyVoU33No.2Apo2021基于认知框架理论的谚语英译研究颜蓉(苏州大学外国语学院，江苏苏州215006)摘要：框架理论为人们认识世界提供了一种思路。

近年来，不少学者以框架理论为基础对文本进行解读和分析。

不同文化中的认知框架存在差异，这在翻译中有所体现。

从认知语言学的框架理论来看，翻译的过程可以理解为译者将原文中的框架转化为相应的译入语文化框架的过程，这一过程并非简单的框架对应,而需要进行一定的框架操作以使目标读者更准确地理解原文传递的信息。

以认知框架理论为基础，试图探讨谚语英译中的框架操作问题，以期丰富对翻译的认识,并为谚语英译提供借鉴和思考。

关键词：框架理论;谚语;英译中图分类号：H05文献标志码：A文章编号:1009-7761(2021)02-0044-07A Research on English Translation of Proverbs Basedon Cognitive Frame TheoryYANRogy(School of Foreign Languayos,Soochow Unmps/y,Suzhou Jiangsu215796,China)ASstroch:Frame Theo,,on which provi/os a perspective fhr people to perceive the woOU.In recent years,a nnmber of scholars have analyzef and intemotef texts basef on the0,,.Diberent calturos have different cognitive frameworfs,which are mflecUO in translation2From the perspective of Frame Theos m cognitive lin-yuistics,the process of translation can bo understooh as the process in which the translator transforms the frame of the source text into the covesponding caltural frame of the target lpgupo.The process is not a simple trans­formation of the frame,but repuiros ceOain opemtions on the iamo to enaPle the target reapers to understand the information conveyef by the source text more accnmtUy.Basef on Cognitive Frame Theos ,this paper attempts to explcue the ogerations on the frame in English translation of proverbs in order to enrich the understanding of translation and provi/o reference and thinking for the English translation of pmOs.Key worbt:Frame Theos；proverfs:English translation1774年,Minsky［在《表征知识的框架》一文中将框架定义为“一种特定情境的数据结构，例如身处某种类型的客厅,或者参加孩子的生日派对”。

Coupled equations for reverse time migration in transversely isotropic mediaPaul J.Fowler 1,Xiang Du 2,and Robin P .Fletcher 2ABSTRACTReverse time migration ͑RTM ͒images reflectors by using time-extrapolation modeling codes to synthesize source and receiver wavefields in the subsurface.Asymptotic analysis of wave propagation in transversely isotropic ͑TI ͒media yields a dispersion relation describing coupled P-and SV-wave modes.This dispersion relation can be converted into a fourth-order scalar partial differential equation ͑PDE ͒.In-creased computational efficiency can be achieved using equivalent coupled second-order PDEs.Analysis of the cor-responding dispersion relations as matrix eigenvalue systems allows one to characterize all possible coupled linear second-order systems equivalent to a given linear fourth-order PDE and to determine which ones yield optimally efficient finite-difference implementations.Setting the shear velocity along the axis of symmetry to zero yields a simpler approximate TI wave equation that is more efficient to implement.This sim-pler approximation,however,can become unstable for some plausible combinations of anisotropic parameters.The same eigensystem analysis can be applied using finite vertical shear velocity to obtain solutions that avoid these instability problems.INTRODUCTIONReverse time migration ͑RTM ͒propagates source wavefields for-ward in time and recorded receiver wavefields backward in time to enable the imaging of subsurface reflectors ͑Baysal et al.,1983;Mc-Mechan,1983;Whitmore,1983;de Faria et al.,1986;Etgen,1986;Hellmann et al.,1986;Chen,1987͒.It correctly handles multipath-ing and phase changes due to caustics.It has no intrinsic dip limita-tion and allows imaging of overturned and prismatic reflections.Re-verse time migration is thus a powerful tool for imaging complex structures,especially when implemented on prestack data.Most imaging of surface seismic data treats the wavefields as rep-resenting only primary energy and only P-wave modes.In isotropic media,it is common to use scalar acoustic wave equations to de-scribe time extrapolation of P-waves for RTM.However,anisotrop-ic media are inherently described by elastic-wave equations,with P-wave and S-wave modes usually intrinsically coupled.It could be possible to model propagation of anisotropic elastic waves incorpo-rating simultaneous separation of different wave modes in the sub-surface for imaging ͑Dellinger and Etgen,1990͒but the implemen-tation of such separation methods and corresponding anisotropic elastic imaging conditions remains a subject for ongoing research ͑Yan and Sava,2008͒.Moreover,modeling elastic waves is compu-tationally much more expensive than modeling acoustic waves.Rather than solving anisotropic elastic-wave equations,several authors have derived simpler two-way wave equations that can be efficiently solved to perform pseudo-acoustic anisotropic RTM.Alkhalifah ͑1998,2000͒introduces an approximate dispersion rela-tion for transversely isotropic ͑TI ͒media with a vertical symmetry axis ͑VTI media ͒based on setting the shear velocity along the axis of symmetry to zero.Direct implementation of Alkhalifah’s dispersion relation in space-time leads to a complicated fourth-order partial dif-ferential equation ͑PDE ͒.Other authors have implemented acoustic VTI modeling and migration based on various coupled second-order wave equations derived from Alkhalifah’s dispersion relation ͑Alkhalifah,2000;Klie and Toro,2001;Zhou et al.,2006a ;Hestholm,2007;Du et al.,2008͒.Alternatively,Duveneck et al.͑2008͒derive coupled first-order and second-order VTI wave equa-tions starting from Hooke’s law and the equations of motion with the vertical shear velocity again set to zero.In practice,one wants to be able to image data for tilted transverse isotropic ͑TTI ͒media not just VTI.Tilting the symmetry axis rela-tive to the coordinates does not add any new physics,just more alge-braic complexity.Extensions of coupled wave equations to TTI me-dia have been considered previously by several authors ͑Zhou et al.,2006b ;Fletcher et al.,2008;Zhang and Zhang,2008͒.The analysis and methods derived in the current paper can also be extended to TTI media;however,doing so introduces further computational com-Manuscript received by the Editor 30April 2009;revised manuscript received 22July 2009;published online 2February 2010.1WesternGeco,Denver,Colorado,U.S.A.E-mail:pfowler1@.2WesternGeco,Schlumberger House,West Sussex,UK.E-mail:xdu@crawley.oilfi;rfletcher1@.©2010Society of Exploration Geophysicists.All rights reserved.GEOPHYSICS,V OL.75,NO.1͑JANUARY-FEBRUARY 2010͒;P.S11–S22,7FIGS.10.1190/1.3294572S11D o w n l o a d e d 04/19/15 t o 121.251.254.73. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /plexity and some new stability issues.We address these aspects ofTTI RTM imaging in a companion paper ͑Fletcher et al.,2009͒.VTI MODELING EQUATIONSFor RTM,one needs accurate and efficient schemes for modeling P-waves in TI media.For simplicity,we develop the mathematical theory here for VTI media:The generalization to a tilted symmetry axis involves no additional physics but greatly complicates the alge-bra.Asymptotic analysis of free-space propagation of P-and SV-waves in VTI media ͑seeAppendix A ͒yields the dispersion relation0ס␻4p מ͓͑v px 2םv sz 2͒͑k x 2םk y 2͒ם͑v pz 2םv sz 2͒k z 2͔␻2pם͕v px 2v sz 2͑k x 2םk y 2͒2םv pz 2v sz 2k z 4ם͓v pz 2͑v px 2מv pn 2͒םv sz 2͑v pz 2םv pn 2͔͒͑k x 2םk y 2͒k z 2͖p ,͑1͒where ␻is temporal frequency;k x ,k y ,and k z are spatial wavenum-bers;v pz is the vertical P-wave velocity;v px סv pz ͱ1ם2␧is the hori-zontal P-wave velocity;v pn סv pz ͱ1ם2␦is the P-wave moveoutvelocity ͑relative to the vertical symmetry axis ͒;v sz is the vertical SV-wave velocity;and ␧and ␦are the anisotropic parameters de-fined by Thomsen ͑1986͒.Transforming equation 1using the rela-tions k x ←מi ץץx ,k x ←מi ץץy ,k x ←מi ץץz ,and ␻←i ץץt ͑Claerbout,1985͒yields the PDE0סץ4p ץt 4מ͑v px 2םv sz 2͒ͩץ4p ץx 2ץt 2םץ4p ץy 2ץt2ͪמ͑v pz 2םv sz 2͒ץ4p ץz 2ץt 2םv px 2v sz 2ͩץ4p ץx 4ם2ץ4p ץx 2ץy 2םץ4p ץy4ͪםv pz 2v sz 2ץ4p ץz4ם͓v pz 2͑v px 2מv pn 2͒םv sz 2͑v pz 2םv pn 2͔͒ϫͩץ4p ץx 2ץz 2םץ4pץy 2ץz 2ͪ.͑2͒The lower-case p as used in equations 1and 2represents a genericscalar wavefield variable and should not be confused with upper-case P used to designate P-waves.We return to the discussion of spe-cific physical interpretations of wavefield variables in a later section.Because most current RTM algorithms image only P-waves,it is common to use the simpler dispersion relation instead ͑Alkhalifah,2000͒:0ס␻4p מ͓v px 2͑k x 2םk y 2͒םv pz 2k z 2͔␻2pם͓v pz 2͑v px 2מv pn 2͒͑k x 2םk y 2͒k z 2͔p͑3͒and the corresponding PDE0סץ4p ץt 4מv px 2ͩץ4p ץx 2ץt 2םץ4p ץy 2ץt 2ͪמv pz 2ץ4p ץz 2ץt2םv pz 2͑v px 2מv pn 2͒ͩץ4p ץx 2ץz 2םץ4pץy 2ץz 2ͪ,͑4͒which are equivalent to the limit of equations 1and 2as v sz →0.Alkhalifah ͑2000͒refers to this as an “acoustic”approximation but we prefer to use the term “pseudo-acoustic”because modeling using equation 4still generates P-and SV-wave arrivals.Alkhalifah ͑2000͒treats the shear arrivals from equation 4as artifacts but theyare correctly modeled SV-waves for a VTI medium that has v sz ס0͑Grechka et al.,2004͒.We know of no laboratory or field measure-ments relevant to exploration seismology that demonstrate physical SV-waves propagating according to equation 4.However,it has been suggested that such a medium ͑VTI with v sz ס0͒might arise from the low-frequency averaging of finely layered liquids ͑Molot-kov and Khilo,1986͒or similar averaging of solid layers with per-fect-slip boundaries ͑Schoenberg,1983͒.Functionally,the impor-tance of equation 4in the context of RTM is that it is simpler in form and requires fewer parameters than equation 2but it still models P-wave arrivals that are nearly indistinguishable for practical imag-ing purposes from those of equation 2͑Alkhalifah,2000͒.Equations 2and 4are fourth-order PDEs in time and space and contain mixed space and time derivatives.To increase efficiency of finite-difference implementations,most published methods instead have used coupled PDEs derived from equation 4that are only sec-ond order in time and eliminate the mixed space-time derivatives.Alkhalifah ͑2000͒suggests using the coupled equationsץ2q ץt 2סv px 2ͩץ2q ץx 2םץ2q ץy 2ͪםv pz 2ץ2q ץz2מv pz 2͑v px 2מv pn 2͒ϫͩץ4r ץx 2ץz 2םץ4rץy 2ץz 2ͪץ2rץt 2סq .͑5͒In this coupled system,the q and r variables satisfy the single PDE in equation 4,as can be verified by simple substitution.The coupled system of equation 5,however,has only second-order time deriva-tives and isolates them explicitly on the left-hand side of the equa-tions with no mixing of time and space derivatives.This is easier and more efficient than equation 4to implement via finite-difference so-lutions.Zhou et al.͑2006a ͒suggest an alternative set of coupled equations given byץ2q ץt 2סv pn 2ͩץ2q ץx 2םץ2q ץy 2ͪםv pz 2ץ2q ץz 2םv pn 2ͩץ2r ץx 2םץ2r ץy 2ͪץ2r ץt 2ס͑v px 2מv pn 2͒ͩץ2q ץx 2םץ2q ץy2ͪם͑v px 2מv pn 2͒ϫͩץ2r ץx 2םץ2rץy 2ͪ.͑6͒As with equation 5,the q and r variables in equation 6satisfy the sin-gle PDE in equation 4;however,now only second-order space deriv-atives appear.These can be implemented by 1D convolutional finite-difference operators unlike the mixed fourth-order space derivatives in equation 5,which require more computationally expensive 2D convolutions.Du et al.͑2008͒and Zhang and Zhang ͑2008͒propose a third set of similar coupled equations:ץ2q ץt 2סv px 2ͩץ2q ץx 2םץ2q ץy 2ͪםv pz 2ץ2r ץz 2ץ2r ץt 2סv pn 2ͩץ2q ץx 2םץ2q ץy 2ͪםv pz 2ץ2r ץz2.͑7͒These are again equivalent in the sense that the coupled field vari-ables q and r will satisfy equation 4.Equation 7is even simpler inS12Fowler et al.D o w n l o a d e d 04/19/15 t o 121.251.254.73. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /form than equation 6and requires the evaluation of fewer distinct spatial derivatives.Yet another very similar system was presented by Duveneck et al.͑2008͒withץ2q ץt 2סv px 2ͩץ2q ץx 2םץ2q ץy 2ͪםv pz v pn ץ2r ץz 2ץ2r ץt 2סv pz v pn ͩץ2q ץx 2םץ2q ץy 2ͪםv pz 2ץ2r ץz2.͑8͒This is nearly identical in form to equation 7but with different coef-ficients on some of the derivative terms.However,it also can be shown to be equivalent to equation 4.These various coupled implementations all yield the same kine-matics but the amplitudes vary as can be seen from comparing im-pulse responses of each.Figure 1shows snapshots at time t ס0.4s for the q and r wavefields for each of these four methods ͑equations 3–6͒,using a causal point source in a homogeneous medium with v pz ס3km /s,␧ס0.24,and ␦ס0.1.In each of these,the P-wave ar-rival is the faster quasi-elliptical event and the SV-arrival is the star-shaped inner arrival.The shape of the SV arrival is discussed further by Grechka et al.͑2004͒.Note,however,in comparing amplitudes that the source function used for equations 5and 6was ͑q 0,r 0͒ס͑1,0͒whereas that for equations 7and 8was ͑q ,r ͒ס͑1,1͒.We discuss appropriate scaling of source functions in the next section;the source functions used here were chosen to be suitable for visual comparison of the wavefields corresponding to different coupled systems and to match figures shown in the original papers.Note also that the coupled systems of equations 6–8all have the same dimen-sional units for the two wavefields q and r whereas the dimensional units of the two wavefields in equation 5will be different.The coupled systems in equations 5–8all express the second time derivatives as various explicit linear combinations of space deriva-tives.The next sections examine how many such coupled systems are possible and which specific forms might be computationally op-timal.Before we examine these issues,however,we should point out that other approaches not of this general form have also been pro-posed in the literature.Klie and Toro ͑2001͒suggest a different ap-proximate PDE for P-waves than that of equation 2but it is generally less accurate ͑Fowler,2003͒.Hestholm ͑2007͒and Duveneck et al.͑2008͒suggest possible systems of first-order differential equations.Here,we focus principally on coupled second-order equations but we return to discuss relations between first-order and second-order systems in a later section examining the physical meanings of the wavefield parameters.Equivalent coupled second-order equationsJust how many such equivalent sets of coupled second-order equations exist and which ones are better in practice?Any system of linear differential equations with constant coefficients can be re-duced to a single linear differential equation by successive substitu-tion ͑Courant and Hilbert,1962͒.Going the other direction to find systems of coupled equations equivalent to a single higher-order dif-ferential equation is a bit trickier and the resulting systems will usu-ally not be unique.This problem can be analyzed algebraically by working with the dispersion relations from equations 1and 3.These dispersion relations are polynomial functions of the general form0סP ͑k x ,k y ,k z ,␻͒p ,͑9͒ס␻4p מf 1͑k x ,k y ,k z ͒␻2p םf 2͑k x ,k y ,k z ͒p .͑10͒A general form for explicit coupled linear second-order systems canbe written asa ͑k x ,k y ,k z ͒q םb ͑k x ,k y ,k z ͒r ס␻2qc ͑k x ,k y ,k z ͒q םd ͑k x ,k y ,k z ͒r ס␻2r ,͑11͒or equivalently as the matrix eigenvalue equation Aq ס␻2q ,whereq ס͑q r ͒and A ס͑a bc d ͒.Acoupled system of this form will be equiva-lent to the original fourth-order equation if new wavefield variables q and r are also solutions of equation 10.A necessary and sufficient condition for this to be true is that the characteristic polynomial of01000200010002000D e p t h (m )10002000100020001000200010002000Distance (m)D e p t h (m )1000200010002000Distance (m)01000200010002000D e p t h (m )100020001000200001000200010002000D e p t h (m )1000200010002000Equation 5Equation 6Equation 7Equation 8Equation 5Equation 6Equation 7Equation 8q rFigure 1.Wavefield snapshots at time t ס0.4s in a homogeneous VTI medium with v pz ס3000m /s,␧ס0.24,and ␦ס0.1.These show the q ͑left ͒and r ͑right ͒wavefields corresponding to equations 5–8,respectively,from top to bottom.The r wavefield for equation 5has been scaled by a factor of 1000for clarity.The top two examples,for equations 5and 6,used the source function ͑q 0,r 0͒ס͑1,0͒.The lower two examples for equations 7and 8used the source function ͑q 0,r 0͒ס͑1,1͒.Coupled equations for TI RTMS13D o w n l o a d e d 04/19/15 t o 121.251.254.73. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /the coupled system,considered as a matrix eigenvalue equation,willbe the same as the original single polynomial relation,so thatdet ͑A מ␻2I ͒ס␻4מ͑a םd ͒␻2ם͑ad מbc ͒סP .͑12͒Note that in general there will be many coupled systems equiva-lent to a given single polynomial relation.If a particular coupled sys-tem is defined by the coefficient matrix A ,then any invertible 2ϫ2matrix M induces a change of variables q ЈסM מ1q and the matrix B סM מ1AM will then define another equivalent coupled system Bq Јס␻2q Ј.This follows from recognizing that similarity transfor-mations preserve eigenvalues and so will have the same characteris-tic polynomial ͑Strang,1980͒.As a corollary,note that if one has two equivalent systems,Aq ס␻2q and Bq Јס␻2q Ј,then the two sets of variables will be related byq ЈסS A מ1S B q ,͑13͒where S A and S B represent the matrices whose columns are theeigenvectors of A and B ,respectively.We note here a few useful specific examples of generating equiva-lent coupled systems by similarity transforms.First,note that thechange of variables induced by the matrix M ס͑0110͒converts thesystem defined by A ס͑a bc d ͒into one defined by B סM מ1AMס͑d c b a ͒and ͑q Јr Ј͒סM מ1͑q r ͒ס͑0110͒͑q r ͒ס͑r q ͒.Thus,interchanging the coefficients a ↔d and b ↔c simply interchanges the meanings of the wavefield variables q and r .This symmetry implies that one needs to analyze only half the possible coefficient cases;the other half will follow from a trivial interchange of variables.Second,the interchange of variables induced by the matrixM ס͑␤00␣͒converts the system defined by A ס͑a bc d ͒into one de-fined by B סM מ1AM ס͑a ␣/␤b␤/␣c d ͒,with the implied change ofvariables ͑q Јr Ј͒סM מ1͑q r ͒ס͑1/␤001/␣͒͑q r ͒ס͑q /␤r /␣͒.Thus,for any cou-pled system,one can always scale one of the off-diagonal crosscou-pling coefficients upward if the other is scaled correspondingly downward.This scaling works for the corresponding space-time PDEs as well.As one simple example,if A represents the coeffi-cients of equation 7͑Du et al.,2008͒and B similarly represents the coefficients of equation 8͑Duveneck et al.,2008͒,the two coeffi-cient matrices are related by B סM מ1AM for the matrix Mס͑100v pn /v pz ͒.Thus,if ͑q ,r ͒represents a wavefield modeled using equation 7and ͑q Ј,r Ј͒represents a wavefield modeled using equa-tion 8,the wavefield variables will be related by ͑q Јr Ј͒סM מ1͑qr ͒ס͑100v pz /v pn ͒͑q r ͒ס͑q v pz r /v pn ͒ס͑qr /ͱ1ם2␦͒.Thus,one expects that the q wavefields for the two methods should be the same whereas the r wavefields should differ by the simple amplitude scalar 1/ͱ1ם2␦.Note that this scaling applies to the free-space propagat-ing wavefield but the amplitudes of modeled data depend also on the specific source functions used,which also must be scaled appropri-ately.Thus,for wavefields modeled using equations 7and 8,one willhave ͑q Јr Ј͒ס͑qr /ͱ1ם2␦͒,provided one also uses source functions scaled so that ͑q 0Јr 0Ј͒ס͑q 0r 0/ͱ1ם2␦͒.As a third example,the matrix M ס͑1110͒converts the coefficients of equation 7͑Du et al.,2008͒into those of equation 6͑Zhou et al.,2006b ͒via the similarity transform M מ1AM .One thus expects that if ͑q ,r ͒again represents the wavefields modeled using equation 7and ͑q Ј,r Ј͒represents wavefields modeled using equation 6,thenone will have͑qЈr Ј͒סM מ1͑qr ͒ס͑011מ1͒͑q r ͒ס͑rqמr ͒,provided the sources also satisfy the same condition,i.e.,͑q 0Јr 0Ј͒ס͑r 0q 0מr 0͒.Thiscan be verified numerically from the respective modeling responsesin Figure 1because the sources ͑q 0r 0͒ס͑11͒and ͑q 0Јr 0Ј͒ס͑10͒usedthere satisfy the necessary scaling condition.Minimal coupled second-order equationsThe VTI dispersion relations 1and 3that we consider here have coefficients of the particular formf 1סf 11͑k x 2םk y 2͒םf 12k z2f 2סf 21͑k x 2םk y 2͒2םf 22͑k x 2םk y 2͒k z 2םf 23k z 4.͑14͒To reduce computational cost of the resulting finite-difference schemes,we seek equivalent coupled second-order equations with no mixed fourth-order space derivatives:␻2q ס͓a 1͑k x 2םk y 2͒םa 2k z 2͔q ם͓b 1͑k x 2םk y 2͒םb 2k z 2͔r ␻2r ס͓c 1͑k x 2םk y 2͒םc 2k z 2͔q ם͓d 1͑k x 2םk y 2͒םd 2k z 2͔r .͑15͒The equivalence condition from equation 12then yields the follow-ing five equations in eight variables:f 11סמ͑a 1םd 1͒͑16͒f 12סמ͑a 2םd 2͒͑17͒f 21סa 1d 1מb 1c 1͑18͒f 22סa 1d 2םa 2d 1מb 1c 2מb 2c 1͑19͒f 23סa 2d 2מb 2c 2.͑20͒This leaves three free parameters;therefore,the solutions will be nonunique.To reduce computational costs further,an obvious approach is to try to set to zero as many coefficients as possible.For the dispersion relation in equation 1,one can show ͑Appendix B ͒that there are nine ways to set two coefficients in equation 15to zero ͑plus another sev-en that correspond to the trivial interchange of the variables q and r ͒but no combinations that allow one to set three or more coefficients to zero.All of these choices require six distinct derivative evalua-tions and therefore have equivalent nominal implementation cost.For Alkhalifah’s approximate dispersion relation in equation 3,there are four basic ways ͑see Appendix B ͒to set four of the coeffi-cients in equation 15to zero ͑plus four more from the interchange of the wavefield variables q and r ͒.These correspond to the four cou-pled systems of dispersion relations given byCase 1:␻2q סv px 2͑k x 2םk y 2͒q ם␣v pz v pn ͑k x 2םk y 2͒r␻2r ס1␣v pz v pn k z 2q םv pz 2k z 2r ͑21͒Case 2:␻2q ס͓v px 2͑k x 2םk y 2͒םv pz 2k z 2͔q ם␣͑v pn 2מv px 2͒ϫ͑k x 2םk y 2͒r␻2r ס1␣v pz 2k z 2q ͑22͒S14Fowler et al.D o w n l o a d e d 04/19/15 t o 121.251.254.73. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /Case 3:␻2q ס͓v px 2͑k x 2םk y 2͒םv pz 2k z 2͔q ם␣͑v pn 2מv px 2͒k z 2r␻2r ס1␣v pz 2͑k x 2םk y 2͒q ͑23͒Case 4:␻2q סv px 2͑k x 2םk y 2͒q ם␣v pz v pn k z 2r␻2r ס1␣v pz v pn ͑k x 2םk y 2͒q םv pz 2k z 2r .͑24͒Note that each of these has a free nonzero scaling parameter ␣.The coupled PDEs corresponding to these four cases then becomeCase 1:ץ2q ץt 2סv px 2ͩץ2q ץx 2םץ2q ץy 2ͪם␣v pz v pn ͩץ2r ץx 2םץ2r ץy 2ͪץ2r ץt 2ס1␣v pz v pn ץ2q ץz 2םv pz 2ץ2r ץz2͑25͒Case 2:ץ2q ץt 2סv px 2ͩץ2q ץx 2םץ2q ץy 2ͪםv pz 2ץ2q ץz2ם␣͑v pn 2מv px 2͒ϫͩץ2r ץx 2םץ2rץy 2ͪץ2r ץt 2ס1␣vpz 2ץ2qץz 2͑26͒Case 3:ץ2q ץt 2סv px 2ͩץ2q ץx 2םץ2q ץy 2ͪםv pz 2ץ2q ץz2ם␣͑v pn 2מv px 2͒ץ2r ץz2ץ2r ץt 2ס1␣v pz 2ͩץ2q ץx 2םץ2qץy 2ͪ͑27͒Case 4:ץ2q ץt 2סv px2ͩץ2q ץx 2םץ2q ץy 2ͪם␣v pz v pn ץ2r ץz 2ץ2r ץt 2ס1␣v pz v pn ͩץ2q ץx 2םץ2q ץy 2ͪםv pz 2ץ2r ץz2.͑28͒Note that case 1requires evaluation of six distinct derivatives,case 2requires five distinct derivatives,case 3requires four distinct deriva-tives,and case 4requires only three distinct derivatives.For imple-mentation,fewer distinct derivative evaluations implies lower com-putational cost,so one can expect that case 4should be optimally ef-ficient among methods of the type we have examined here.More-over,because even an isotropic acoustic wave equation requires three distinct spatial derivatives,the computational cost of case 4is essentially that of the isotropic case,albeit with the need to store two wavefields instead of one.Equation 7͑Du et al.,2008͒and equation 8͑Duveneck et al.,2008͒are particular examples of case 4͑equation 28͒,with the specific choices of the scaling parameter ␣סv pz /v pn or ␣ס1,respectively.Equation 6͑Zhou et al.,2006a ͒is not in the form of one of the four canonical cases given here because only three of the eight possible coefficients are set to zero rather than four but ithas the same nominal computational cost ͑five distinct derivatives ͒as case 2here.Stability requirementsModeling seismic waves always requires that one use material pa-rameter values for which the stiffness matrix is positive definite ͑see,e.g.,Slawinski,2003͒.For equation 2and coupled systems equiva-lent to it,this requires only the usual parameter restrictions needed by any elastic modeling,as described in Appendix A,although be-cause only P and SV modes are being modeled the actual value used for the stiffness component c 66has no effect.Violations of these lim-itations would represent physically unrealizable elastic media and so in practice would probably just indicate a problem with model con-struction.However,for the pseudo-acoustic equation 4and systems equivalent to it,the restrictions become more stringent and can present practical stability problems.Taking the limit as v sz →0͑and implicitly as c 66→0͒,the requirement that the stiffness matrix be positive definite yields the restriction that v px Ͼv pn or,equivalently,that ␧Ͼ␦.However,for the special case of ␧ס␦in a pseudo-acoustic medium,the SV-wave phase velocity is zero everywhere ͑Grechka et al.,2004͒and systems equivalent to equation 4will still remain formally stable.The pseudo-acoustic stability restriction in practice is thus that v px Նv pn or equivalently that ␧Ն␦.Where ␧Ͻ␦and as a result,coupled pseudo-acoustic systems such as equations 5–8can become unstable,instead one can use cou-pled systems equivalent to equation 2to restore stability.One such example of a coupled solution for finite v sz is given byץ2q ץt 2סv px 2ͩץ2q ץx 2םץ2q ץy 2ͪםv sz 2ץ2q ץz2ם␣ͱ͑v pz 2מv sz 2͒͑v pn 2מv sz 2͒ץ2rץz 2ץ2r ץt 2ס1␣ͱ͑v pz 2מv sz 2͒͑v pn 2מv sz 2͒ͩץ2q ץx 2םץ2q ץy 2ͪםv sz2ͩץ2r ץx 2םץ2r ץy 2ͪםv pz 2ץ2r ץz2.͑29͒Choosing,for example,␣סͱ͑v pz 2מv sz 2͒͑v pn 2מv sz 2͒yields the coupled systemץ2q ץt 2סv px 2ͩץ2q ץx 2םץ2q ץy 2ͪםv sz 2ץ2q ץz 2ם͑v pz 2מv sz 2͒ץ2r ץz 2ץ2r ץt 2ס͑v pn 2מv sz 2͒ͩץ2q ץx 2םץ2q ץy 2ͪםv sz 2ͩץ2r ץx 2םץ2r ץy 2ͪםv pz 2ץ2r ץz2,͑30͒which reduces in the limit as v sz →0to equation 7͑Du et al.,2008͒.Choosing instead ␣ס1gives a similar coupled system that reduces to equation 8͑Duveneck et al.,2008͒in the limit as v sz →0.For any choice of ␣,equation 29reduces to the general form of equation 28as v sz →0.Note,however,that equation 29requires computing all six distinct space derivatives instead of only the three derivatives needed for equation 28so it does have the drawback of being compu-tationally more expensive.Figure 2shows a comparison of model-ing for a case of v px Ͻv pn ,first with equation 7,which becomes un-Coupled equations for TI RTMS15D o w n l o a d e d 04/19/15 t o 121.251.254.73. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /stable,and then with equation 30,which reproduces physically real-istic SV kinematics and remains stable.In implementing systems such as equations 29or 30,one can choose any reasonable finite value for v sz that satisfies the general elastic-stability conditions ͑Appendix A ͒;the specific value used for v sz has little significant effect on the P arrival ͑Fowler,2003͒.In pro-cessing P-wave data,we usually do not know correct physical values to use for v sz .However,accurate modeling of the SV arrival is less important than insuring stability as long as we are most interested in accurate P-wave imaging.Note that SV-wave arrivals usually stack out after P-wave migration whether the SV velocities used are cor-rect or not just as long as the SV-wave velocities are substantially different from the P-wave velocities.Instability for pseudo-acoustic modeling ͑i.e.,v sz ס0͒is usually evidenced as exponential growth of the SV arrivals ͑Alkhalifah,2000;Grechka et al.,2004͒.No source-generated SV arrivals will be present for P-wave sources in isotropic or elliptically anisotropic media,as will ordinarily be the case for RTM of marine data.How-ever,mode conversion will still occur at interfaces in the parameter model;therefore,stability for pseudo-acoustic equations remains a potential problem unless one restricts parameter models by requir-ing v px Նv pn ͑or equivalently ␧Ն␦͒everywhere.This limitation is valid in many common sedimentary rocks but does not hold univer-sally ͑Thomsen,1986;Wang,2002͒.Note,finally,that the instability discussed in this section is a prop-erty of the physics underlying the particular differential equations and as such is distinct from any further numerical instability prob-lems that might arise from the incorrect design of specific finite-dif-ference schemes for subsequent implementation.Physical interpretations of the wavefield variablesWe have so far left indeterminate the physical meaning of the wavefield variables q and r used in the coupled PDE solutions here.Several interpretations are possible.The VTI P-SV dispersion rela-tion in equation 1is derived from the coupled eigensystem in the ͑x ,z ͒plane given by equation A-6.After substituting the velocity pa-rameters from equation A-8,equation A-6becomesͫv px 2k x 2םv sz 2k z 2מ␻2ͱ͑v pz 2מv sz 2͒͑v pn 2מv sz 2͒k x k zͱ͑v pz מv sz ͒͑v pn מv sz ͒k x k zv sz 2k x 2םv pz 2k z 2מ␻2ͬϫͫU 1U 3ͬס0.͑31͒Changing variables using the matrix M ס͑i k x 00␣ik z͒gives΄v px 2k x 2םv sz 2k z 2מ␻2␣ͱ͑v pz 2מv sz 2͒͑v pn 2מv sz 2͒k z21␣ͱ͑v pz 2מv sz 2͒͑v pn 2מv sz 2͒k x2v sz 2k x 2םv pz 2k z 2מ␻2΅ϫͫU 1ЈU 3Јͬס0͑32͒with͑U 1ЈU 3Ј͒סM מ1͑U 1U 3͒ס͑U 1/ik xU 3/i ␣k z͒.After extending to 3D by re-placing k x 2by k x 2םk y 2,equation 32fits the general form of equation 15with the specific case of coefficients b 1סc 2ס0.The meaning of these wavefield variables now will be that the first is proportional to the horizontal integral of the horizontal displacement measured in the direction of wavefield propagation,and the second is proportion-al to the vertical integral of the vertical displacement.Note that transforming equation 32to the space-time domain yields equation 29;therefore,variables for other coupled systems can be interpreted via the associated matrices defining the corresponding changes of variables.These are,however,not the only possible wavefield variables.One can also describe elastic-wave propagation in a VTI medium by the nine coupled first-order equations A-11.From asymptotic analy-sis ͑Appendix A ͒,we know that the value of c 66has no real effect on the propagation of P and SV modes so we can set it to zero if we are not considering SH modes.Doing so yields the seven simpler cou-pled first-order equations␳v˙1ס␴11,1ם␴13,3␳v˙2ס␴11,2ם␴23,3␳v˙3ס␴13,1ם␴23,2ם␴33,3␴˙11סc 11v 1,1םc 11v 2,2םc 13v 3,3␴˙33סc 13v 1,1םc 13v 2,2םc 33v 3,3␴˙23סc 44͑v 2,3םv 3,2͒␴˙13סc 44͑v 1,3םv 3,1͒͑33͒Distance (m)1000200010002000D e p t h (m)1000200010002000Distance(m)1000200010002000D e p t h (m )qr1000200010002000Figure 2.Wavefield snapshots at time t ס0.4s in a homogeneousVTI medium with v pz ס3000m /s,v sz ס1000m /s,␧ס0.1,and ␦ס0.2.These show the q ͑left ͒and r ͑right ͒wavefields correspond-ing to equation 7͑top ͒and equation 30͑bottom ͒.The black areas in the top plots are NaN values caused by the instability of equation 7when ␧Ͻ␦.All plots here used the same source function ͑q 0,r 0͒ס͑1,1͒.S16Fowler et al.D o w n l o a d e d 04/19/15 t o 121.251.254.73. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /。

全息照相的原理英语作文The principle of holographic photography is based on the interference pattern created by the interaction oflight waves. This pattern captures the three-dimensional information of an object, allowing us to reproduce a realistic and detailed image.Holographic photography uses a laser beam to illuminate the object and a photosensitive material to record the interference pattern. The laser light is coherent, meaning that all the light waves have the same frequency and phase, which is essential for creating a clear and sharp hologram.When the laser light reflects off the object, it combines with the reference beam to create an interference pattern on the photosensitive material. This pattern contains information about the object's shape, size, and texture, allowing us to reconstruct a lifelike image when the hologram is illuminated with laser light.Unlike traditional photography, holographic photography captures the complete wavefront of light, preserving both the intensity and phase information. This allows us to reproduce not only the appearance of the object but alsoits depth and spatial relationships, creating a truly realistic representation.One of the key advantages of holographic photography is its ability to capture and display three-dimensional images without the need for special glasses or viewing devices. This makes holograms an ideal tool for scientific research, medical imaging, and artistic expression, opening up new possibilities for visual communication and storytelling.In conclusion, holographic photography offers a unique and powerful way to capture and reproduce three-dimensional images with unparalleled realism and detail. By harnessing the principles of interference and wavefront reconstruction, holograms enable us to experience the world in a new and immersive way, pushing the boundaries of visual representation and storytelling.。