Characteristic Wino Signals in a Linear Collider from Anomaly Mediated Supersymmetry Breaki

a r X i v :h e p -p h /0610277v 1 21 O c t 2006SLAC-PUB-12168TU-778October,2006Possible Signals of Wino LSP at the Large Hadron Collider M.Ibe 1,Takeo Moroi 2and T.T.Yanagida 3,41Stanford Linear Accelerator Center,Stanford University,Stanford,CA943092Department of Physics,Tohoku University,Sendai,Japan 3Department of Physics,University of Tokyo,Tokyo 113-0033,Japan 4Research Center for the Early Universe,University of Tokyo,Tokyo 113-0033,JapanAbstractWe consider a class of anomaly-mediated supersymmetry breaking models wheregauginos acquire masses mostly from anomaly mediation while masses of other superparticles are from K¨a hler interactions,which are as large as gravitino mass ∼O (10−100)TeV.In this class of models,the neutral Wino becomes the lightest superparticle in a wide parameter region.The mass splitting between charged and neutral Winos are very small and experimental discovery of such Winos is highly non-trivial.We discuss how we should look for Wino-induced signals at Large Hadron Collider.1IntroductionIn the vacua of broken supersymmetry(SUSY)not only the gravitino but also squarks and sleptons acquire tree-level SUSY-breaking masses through supergravity(SUGRA) effects[1].On the other hand,if there is no singletfield in the hidden sector,quantum-loop corrections generate gaugino masses,which is often called as anomaly mediation[2,3]. This is the most economical mechanism for giving SUSY-breaking masses to all SUSY particles in the SUSY standard model(SSM)since it does not require any extrafield other than ones responsible for the SUSY breaking.In this model the gauginos are much lighter than the squarks and sleptons due to the loop effects.That is,there is a little disparity in the spectrum of SUSY particles.The masses of squarks and sleptons(and also that of the gravitino)are O(10−100)TeV for the gaugino masses less than O(1)TeV. Because of this disparity the anomaly mediation model is free from all problems in the SSM such asflavor-changing neutral current and CP problems[4].Furthermore,the mass of the lightest Higgs boson is naturally above the experimental lower bound m H>114.4 GeV[5].From the cosmological point of view,the gravitino-overproduction problem[6] in the early universe is much less serious in the above model due to the relatively large gravitino mass.It is also notable that a consistent scenario of thermal leptogenesis[7]is possible in the present scheme[8],and that the lightest superparticle(LSP),which is the neutral Wino,can be a good candidate for the dark matter in the universe[9,10].The purpose of this paper is to discuss possible tests of the above anomaly mediation model at the Large Hadron Collider(LHC)experiments.The crucial point is that the charged and neutral Winos are the most-motivated LSPs whose masses almost degenerate with each other,and the neutral Wino˜W0is slightly lighter than the charged one˜W±[11].In a cosmological scenario[10],a sizable amount of Winos are produced by the decay of gravitino originating from inflaton decay[12],and provided that such Wino can be the dark matter in the universe,the mass of the Wino is predicted as M2≃100GeV−2TeV.Therefore,it is highly possible that the Wino is within the reach of the LHC.1Unfortunately,however,it is also possible that superparticles other than Winos can be hardly produced at the LHC,depending on the model parameters.Indeed,as we will see,the gluino-Wino mass ratio M3/M2can be larger than the prediction of the pure anomaly mediation model,M3/M2≃8.Thus,it is difficult to produce the gluinos atthe LHC except for very optimistic cases as well as sfermions.Therefore it is now highly important to discuss signals of the Wino production at the LHC.2Properties of WinosWefirst discuss the mass spectrum of superparticles.We assume that the masses of squarks,sleptons and Higgs bosons are dominantly from SUGRA effects.With a generic K¨a hler potential,all the scalar bosons except the lightest Higgs boson acquire SUSY breaking masses of the order of the gravitino mass(m3/2).For the gaugino masses,on the contrary,tree-level SUGRA contributions are extremely suppressed if there is no singlet elementaryfield in the hidden sector.In this paper,we adopt the above setup and consider the situation that the scalar masses(as well as the gravitino mass)are of the order of O(10−100)TeV while gauginos are much lighter.In addition,we also assume that Higgsinos and heavy Higgs bosons have masses of O(10−100)TeV.Hereafter,we call such a scenario as“no singlet scenario”.If there is no singletfield in the hidden sector,effects of anomaly mediation becomes very important in generating gaugino masses[2,3].At Q∼m3/2(with Q being a renormalization scale),anomaly-mediation contributions to the gaugino masses are given byM(AMSB) a =−b a g2amass in the“no singlet scenario”.Indeed,we expect the following K¨a hler potential:K∋cH u H d+c′M2G,(4) where F X is the vacuum expectation value of the F-component of X.Thus,µ-and B-parameters are both of O(m3/2)and they are independent.For the successful electro-weak symmetry breaking,one linear combination of Higgs bosons should become light: with the so-calledβ-parameter,the(standard-model-like)light doublet is denoted as sinβH u−cosβH∗d,where H u and H d are up-and down-type Higgs bosons,respectively.Then,threshold corrections to the gaugino masses from the Higgs-Higgsino loop are given by[3,15]∆M(Higgs)1=316π2L,(5)∆M(Higgs)2=g22|µ|2−m2A ln|µ|22We assume that the vacuum expectation value of X is much smaller than M G.Figure1:Gaugino masses as functions of L/m3/2for a real value of L.The Wino mass is given in the solid line,while the dotted and dashed lines are for Bino and gluinomasses,respectively.Here we take m3/2=50TeV.The cancellation of the Wino mass for L/m3/2≃−1is a result of our specific choice of the phase,Arg(L)=π.For a generic phase of L,the cancellation becomes milder,though the Wino mass can be much lighterthan M(AMSB)2for|L/m3/2|≃1and Arg(L)∼π.Theµ-parameter(and L)is a complex variable and the Wino and Bino masses depend on the relative phase betweenµand m3/2.(Here,we use the bases where the gravitinomass is real and positive.)Importantly,with a distractive interference between M(AMSB)2and∆M(Higgs)2,the Wino mass can be even smaller than the purely anomaly mediated one.As an example,in Fig.1,we plot the gaugino masses for m3/2=50TeV as functions of L/m3/2,assuming that L is real for simplicity.In this case,we see that the threshold corrections drastically change the gaugino mass and even cancel the anomaly mediatedWino mass.In our analysis,gaugino masses are given by M a=M(AMSB)a +∆M(Higgs)aatQ=m3/2,and we take into account the effects of renormalization group evolutions below this scale.We have checked that the ratios of two gaugino masses are insensitive to the gravitino mass for a given L/m3/2.A wide variety of gaugino mass in the“no singlet scenario”has strong implications forcollider physics.In the pure anomaly mediation,all the gaugino masses are determined only by the gravitino mass and corresponding gauge coupling constants as shown in Eq.(1),and hence a negative search for gluino-induced signals results in lower bounds on other gaugino masses,in particular,that of Wino mass.In the“no singlet scenario”, on the contrary,gluino and Wino masses are independent.Thus,even if gluino-induced signals will not be found at the LHC,we will still have a strong motivation to look for Wino-induced signals for the parameter region where Wino is relatively light.In the pure anomaly mediation model,the lightest gaugino is always the Wino.How-ever,in the“no singlet scenario”,the Wino LSP is not a generic consequence any more since the Bino may become lighter than Wino.However,Fig.1shows that Wino becomes the LSP if|L/m3/2|is less than a few,which is the case when|µ|and m A are both fairlyclose to the gravitino mass.In addition,the Wino LSP scenario can never cease from being the most motivated scenario from the point of view of cosmology.This is because the Bino LSP results in the overclosure of the universe[3].Therefore,in this paper,we concentrate on the Wino LSP scenario,where the lightest neutralino and the lightest chargino are both(almost)purely Winos.Now,we consider detailed properties of the Wino.As discussed in Ref.[11],the dominant mass splitting between the charged and neutral Winos comes from one-loop gauge boson contributions to the gaugino masses whenµis large.The splitting is given by∆M=m˜W±−m˜W0=g22πcos2θc f2π∆M3 1−m2πprocesses are a few%or smaller for M2>∼88GeV.3We also estimate the lifetime of charged Wino,which is given by O(10−10)sec.Then,charged Winos produced at the LHC travel typically O(1−10)cm before they decay.As we discuss in the following,this fact has very important consequences in the study of our scenario at the LHC.3Wino at the LHCNow we are at the position to discuss how we can test the“no singlet scenario”at the LHC.In studying superparticles at the LHC,it is often assumed that the productions of superparticles are mostly via productions of colored superparticles.Even though the primary superparticles are scalar quarks and/or gluino,they decay into various lighter superparticles which may be scalar leptons,charginos,and/or neutralinos.Of course, those processes are important when scalar quarks and/or gluino are not too heavy.Since we are interested in the case where all the sfermions(as well as heavy Higgs multiplets)have masses of order100TeV,they are irrelevant for the LHC.Hereafter,we will not consider the production of those particles.On the contrary,masses of gauginos(in particular,that of gluino)are of order100 GeV–1TeV.As discussed in the previous section,the gluino-to-Wino mass ratio is a free parameter in the“no singlet scenario”,since the prediction of pure anomaly mediation, which gives M3/M2≃8,can be significantly altered.For a given Wino mass M2,the discovery of SUSY at the LHC depends on the ratio M3/M2.When M3/M2is relatively small,the gluino is light enough so that a significant amount of gluino is produced at the LHC.In this case,the dominant production processes of superparticles are gluino pair productions.The primary gluinos decay into lighter super-particles and,at the end of decay chain,the neutral Winos are produced.Consequently, we observe events with energetic jets and large missing E T.In this case,signals from the production of superparticles are basically the same as those in well-studied SUSY models.Hereafter,we concentrate on a rather pessimistic case where the gluino mass is so largethat a gluino production rate is suppressed at the LHC.In this case,the only possibility of detecting superparticles is direct productions of charged and neutral Winos.(Notice that the Bino production cross section is very small since the scalar quarks are extremely heavy.)In our case,the neutral Wino is the LSP and hence is stable.Thus,even if it is produced at the LHC,it does not leave any consequence on the detector.The charged Wino is heavier than the neutral one and hence is unstable.The mass difference∆M= mW±−m W is expected to be160–170MeV,and the charged Wino mainly decays as ˜W±→˜W0π±with lifetime of O(10−10)sec.Thus,most of charged Winos decay beforeit reaches a muon detector.In addition,the emitted pion has very tiny energy;its boost factor is typicallyγπ∼O(1).Thus,it is challenging to identify such a low-energy pion. These facts make the discover of Wino production events very difficult.Winos can be pair-produced at the LHC via Drell-Yang processes q¯q→˜W+˜W−and q¯q′→˜W±˜W0.However,these events cannot be recorded since there is no trigger relevant for them.Indeed,in order for the event to be recorded,thefinal state should contain high energy jet(s)and/or high energy electro-magnetic activity.In the Drell-Yang process,we do not expect such activities since the charged Winos decay before reaching the muon detector.However,events may be recorded if Winos are produced with high E T jet(s).In particular,ATLAS[18]and CMS[19]both plan to trigger on missing E T events with energetic jets;if the transverse energy of the jet and missing E T are both larger than50–100GeV,events will be recorded.(Hereafter,we call such a trigger as J+XE trigger.) With pp collision at the LHC,a Wino pair can be produced in association with a high E T jet via the following parton-level processes:q¯q→˜W+˜W−g,gq→˜W+˜W−q,g¯q→˜W+˜W−¯q,(11)q¯q′→˜W±˜W0g,gq→˜W±˜W0q′,g¯q→˜W±˜W0¯q′.(12) Gluon and(anti-)quark in thefinal state are hadronized and become energetic jets.Thus, at the trigger level,this class of events are characterized by a high E T mono-jet,missingFigure2:Total cross section of a Wino pair+mono-jet production eventσ(pp→˜W+˜W−j)+σ(pp→˜W±˜W0j)at√4We have cross checked our results by using the codes,FeynArts[22]and FormCalc[23].of Tevatron[11].If the charged Wino travels O(10)cm or longer before it decays,it hits some of the detectors and the track from charged Wino may be reconstructed.The track from charged Wino may be also distinguished from tracks from other standard-model charged particles by using the time-of-flight information and/or by measuring ionization energy loss rate(dE/dx).In addition,it is important to note that,with the lifetime of O(10−10)sec,most of the charged Winos are likely to decay inside the detector.Then,we may observe tracks which disappear somewhere in the detector.This will be regarded as a spectacular signal of new physics beyond the standard model.In particular,in the ATLAS detector,the transition radiation tracker(TRT)will be implemented,which is located at56–107cm from the beam axis[20].The TRT continuously follows charged tracks.Thus,if the charged Wino decays inside the TRT,the charged Wino track will be identified by off-line analysis.In addition,with the TRT,charged pions emitted by the decays of˜W±may be also seen. Such events have very rare backgrounds and hence they can be used for the discovery of the Wino production.We calculate the cross section of the processes pp→˜W+˜W−j and pp→˜W±˜W0j,with the requirement that at least one charged Wino travels transverse length L T longer than L(min)before it decays.At the trigger level,no hadronic nor electro-magnetic activities Twill be identified except for the mono-jet,so the missing E T is equal to the transverse energy of the mono-jet.In order to use the J+XE trigger,we impose a cut such that E T≥100GeV for thefinal-state jet.L T is sensitive to the decay width of charged Wino. Since we consider the case where gauginos are the only light superparticles,we use Eq.(9)for∆M.The decay width of charged Wino is given in Eq.(10).Cross sections for several values of L(min)are shown in Fig.3.Assuming L∼100fb−1Tfor high luminosity run of the LHC,for example,a sizable number of charged Winos with L T∼O(10)cm will be produced.Thus,if we canfind their tracks,it will provide an intriguing signal which cannot be explained in the standard model.Requiring10events with L T≥50cm(1m)with L=100fb−1,for example,the LHC can cover M2≤350 GeV(200GeV).Thus,the search for(short)tracks of heavy charged particles is strongly√Figure3:Cross section of the process pp→˜W˜W j atprecision.In order to perform a statistical analysis of mono-jet+missing E T event,it is crucial to understand the total background cross section.As well as the process pp→Zj,other processes may contribute to the background.One of such is the process pp→ZZj if both Z’s decay invisibly.Thus,we have also calculated the cross section of this process. Taking account of the branching ratio of the invisible mode of Z,we have found that the background cross section from pp→ZZj is29–0.23fb for E T=100−500GeV, which is smaller than the signal cross section when the Wino mass is smaller than∼500 GeV.In addition,we will obtain some information about this class of backgrounds using leptonic decay modes of Z.Thus,we consider that the process pp→ZZj is not a serious background.The process pp→W±j followed by leptonic decay of W±may also give backgrounds if the charged lepton goes into the beam pipe.However,the probability of charged lepton escaping into the beam pipe may not be so large.In addition,the number of this type of backgrounds will be also estimated by studying mono-jet events with a single charged lepton and missing E T.We consider that cross sections of other processes resulting in the mono-jet+missing E Tfinal state are also small enough to be neglected.If the standard-model cross section of the mono-jet+missing E T event is understood precisely,an excess beyond the statistical error may be regarded as a signal of Wino+ mono-jet events.That is,the expected number of background events is O(107−104)forE T>100−500GeV of the mono-jet at a high luminosity L=100fb−1,for example. Then,the number of the Wino production events with a mono-jet becomes statistically significant when M2<∼400GeV(see Fig.2).It may be the case that the background cross section cannot be determined with such a high precision because of some systematic errors.We will not go into the detailed study of systematic error.Instead,we estimate how well the systematic error should be controlled tofind an anomaly;in order to cover the Wino mass of100,200,300,400,and500GeV, systematic error in the determination of background cross section should be smaller than>100GeV 1.9,0.28,0.083,0.032,and0.015%(12,4.2,1.8,0.87,and0.45%)for E(min)T(500GeV).Finally,let us briefly discuss what happens if Higgsino mass(i.e.,theµ-parameter)is smaller than∼TeV.So far,we have considered the case where superparticles other than gauginos are as heavy as O(100)TeV,and hence the mass difference between charged and neutral Winos is determined mainly by the radiative corrections from gauge-boson loops.In this case,∆M is given by Eq.(9)and the lifetime of charged Wino is more or less predicted.If theµ-parameter is smaller,on the contrary,∆M is affected by the gaugino-Higgsino mixing.In this case,∆M can be enhanced or suppressed,depending onµ-and other parameters.In particular,∆M becomes smaller than the pion mass and the lifetime of charged Wino becomes longer in some parameter region.In this case,it is much easier tofind tracks of the charged Wino since˜W±does not decay inside the detector.It should be also noticed here that such a charged Wino with the lifetime of O(10−8)sec may be used as a probe of deep interior of heavy nuclei[24].AcknowledgmentThe authors would like thank S.Asai,J.Kanzaki and T.Kobayashi for valuable dis-cussion and useful comments.M.I.is grateful to T.Hahn and C.Schappacher for kind explanation of FormCalc.This work was supported in part by the U.S.Department of Energy under contract number DE-AC02-76SF00515(M.I.)and by the Grant-in-Aid for Scientific Research from the Ministry of Education,Science,Sports,and Culture of Japan, No.15540247(T.M.)and14102004(T.T.Y.).References[1]H.P.Nilles,Phys.Rept.110(1984)1;S.P.Martin,arXiv:hep-ph/9709356.[2]L.Randall and R.Sundrum,Nucl.Phys.B557,79(1999).[3]G.F.Giudice,M.A.Luty,H.Murayama and R.Rattazzi,JHEP9812,027(1998).[4]F.Gabbiani,E.Gabrielli,A.Masiero and L.Silvestrini,Nucl.Phys.B477,321(1996).[5]W.M.Yao et al.[Particle Data Group],J.Phys.G33,1(2006).[6]S.Weinberg,Phys.Rev.Lett.48(1982)1303;See,for a recent work,K.Kohri,T.Moroi and A.Yotsuyanagi,Phys.Rev.D73(2006)123511,and references therein.[7]M.Fukugita and T.Yanagida,Phys.Lett.B174,45(1986);For a recent review,W.Buchmuller,R.D.Peccei and T.Yanagida,Ann.Rev.Nucl.Part.Sci.55,311 (2005).[8]M.Ibe,R.Kitano,H.Murayama and T.Yanagida,Phys.Rev.D70,075012(2004);M.Ibe,R.Kitano and H.Murayama,Phys.Rev.D71,075003(2005).[9]T.Moroi and L.Randall,Nucl.Phys.B570,455(2000).[10]M.Ibe,Y.Shinbara and T.T.Yanagida,arXiv:hep-ph/0608127.[11]J.L.Feng,T.Moroi,L.Randall,M.Strassler and S.f.Su,Phys.Rev.Lett.83,1731(1999).[12]M.Kawasaki, F.Takahashi and T.T.Yanagida,Phys.Lett.B638,8(2006);M.Kawasaki,F.Takahashi and T.T.Yanagida,Phys.Rev.D74,043519(2006);M.Endo,M.Kawasaki,F.Takahashi and T.T.Yanagida,arXiv:hep-ph/0607170.[13]J.D.Wells,Phys.Rev.D71,015013(2005).[14]K.Inoue,M.Kawasaki,M.Yamaguchi and T.Yanagida,Phys.Rev.D45,328(1992).[15]T.Gherghetta,G.F.Giudice and J.D.Wells,Nucl.Phys.B559,27(1999).[16]A.Heister et al.[ALEPH Collaboration],Phys.Lett.B533,223(2002).[17]S.D.Thomas and J.D.Wells,Phys.Rev.Lett.81,34(1998).[18]ATLAS homepage,http://atlas.web.cern.ch.[19]CMS homepage,http://cms.cern.ch.[20]ATLAS Collaboration,CERN-LHCC-97-16;CERN-LHCC-97-17.[21]J.Pumplin,D.R.Stump,J.Huston,i,P.Nadolsky and W.K.Tung,JHEP0207,012(2002).[22]T.Hahn,mun.140,418(2001).[23]T.Hahn and M.Perez-Victoria,mun.118,153(1999).[24]K.Hamaguchi,T.Hatsuda and T.T.Yanagida,arXiv:hep-ph/0607256.。

信号用英语怎么说信号是表示消息的物理量，如电信号可以通过幅度、频率、相位的变化来表示不同的消息。

这种电信号有模拟信号和数字信号两类。

信号是运载消息的工具，是消息的载体。

那么你知道信号用英语怎么说吗?下面来学习一下吧。

信号英语说法1：signal信号英语说法2：semaphore信号的英语例句：信号响时，人人脱帽致敬。

Everyone uncovered when the signal sounded.红灯常常是危险的信号。

A red lamp is often a danger signal.没有视频信号。

There is no video signal.灯塔每一分钟发出两次信号。

The lighthouse flashes singals twice a minute.时钟信号输入端被设计用于提供时钟信号。

A clock signal input is designed to supply a clock signal.对随机信号和混沌信号的特性进行了分析比较。

The characteristics of stochastic and chaotic signals have been studied.相反，市场可能正在发出更深层次变化的信号。

Rather, markets may be signaling a deeper shift.通常，信号的主要用途是同步某个线程与其他线程的动作。

Usually, the main use of a semaphore is to synchronize a thread? s action with other threads.声音信号可以通过电缆传送而不失真。

Audio signals can be transmitted along cables without distortion.编码信号由碟形卫星天线接收。

The coded signal is received by satellite dish aerials.一根细电缆将信号传送给一台计算机。

LEOPOLD-FRANZENS UNIVERSITYChair of Engineering Mechanicso.Univ.-Prof.Dr.-Ing.habil.G.I.Schu¨e ller,Ph.D.G.I.Schueller@uibk.ac.at Technikerstrasse13,A-6020Innsbruck,Austria,EU Tel.:+435125076841Fax.:+435125072905 mechanik@uibk.ac.at,http://mechanik.uibk.ac.atIfM-Publication2-407G.I.Schu¨e ller.Developments in stochastic structural mechanics.Archive of Applied Mechanics,published online,2006.Archive of Applied Mechanics manuscript No.(will be inserted by the editor)Developments in Stochastic Structural MechanicsG.I.Schu¨e llerInstitute of Engineering Mechanics,Leopold-Franzens University,Innsbruck,Aus-tria,EUReceived:date/Revised version:dateAbstract Uncertainties are a central element in structural analysis and design.But even today they are frequently dealt with in an intuitive or qualitative way only.However,as already suggested80years ago,these uncertainties may be quantified by statistical and stochastic procedures. In this contribution it is attempted to shed light on some of the recent advances in the now establishedfield of stochastic structural mechanics and also solicit ideas on possible future developments.1IntroductionThe realistic modeling of structures and the expected loading conditions as well as the mechanisms of their possible deterioration with time are un-doubtedly one of the major goals of structural and engineering mechanics2G.I.Schu¨e ller respectively.It has been recognized that this should also include the quan-titative consideration of the statistical uncertainties of the models and the parameters involved[56].There is also a general agreement that probabilis-tic methods should be strongly rooted in the basic theories of structural en-gineering and engineering mechanics and hence represent the natural next step in the development of thesefields.It is well known that modern methods leading to a quantification of un-certainties of stochastic systems require computational procedures.The de-velopment of these procedures goes in line with the computational methods in current traditional(deterministic)analysis for the solution of problems required by the engineering practice,where certainly computational pro-cedures dominate.Hence,their further development within computational stochastic structural analysis is a most important requirement for dissemi-nation of stochastic concepts into engineering practice.Most naturally,pro-cedures to deal with stochastic systems are computationally considerably more involved than their deterministic counterparts,because the parameter set assumes a(finite or infinite)number of values in contrast to a single point in the parameter space.Hence,in order to be competitive and tractable in practical applications,the computational efficiency of procedures utilized is a crucial issue.Its significance should not be underestimated.Improvements on efficiency can be attributed to two main factors,i.e.by improved hard-ware in terms of ever faster computers and improved software,which means to improve the efficiency of computational algorithms,which also includesDevelopments in Stochastic Structural Mechanics3 utilizing parallel processing and computer farming respectively.For a con-tinuous increase of their efficiency by software developments,computational procedure of stochastic analysis should follow a similar way as it was gone in the seventieth and eighties developing the deterministic FE approach. One important aspect in this fast development was the focus on numerical methods adjusted to the strength and weakness of numerical computational algorithms.In other words,traditional ways of structural analysis devel-oped before the computer age have been dropped,redesigned and adjusted respectively to meet the new requirements posed by the computational fa-cilities.Two main streams of computational procedures in Stochastic Structural Analysis can be observed.Thefirst of this main class is the generation of sample functions by Monte Carlo simulation(MCS).These procedures might be categorized further according to their purpose:–Realizations of prescribed statistical information:samples must be com-patible with prescribed stochastic information such as spectral density, correlation,distribution,etc.,applications are:(1)Unconditional simula-tion of stochastic processes,fields and waves.(2)Conditional simulation compatible with observations and a priori statistical information.–Assessment of the stochastic response for a mathematical model with prescribed statistics(random loading/system parameters)of the param-eters,applications are:(1)Representative sample for the estimation of the overall distribution.4G.I.Schu¨e ller Indiscriminate(blind)generation of samples.Numerical integration of SDE’s.(2)Representative sample for the reliability assessment by gen-erating adverse rare events with positive probability,i.e.by:(a)variance reduction techniques controlling the realizations of RV’s,(b)controlling the evolution in time of sampling functions.The other main class provides numerical solutions to analytical proce-dures.Grouping again according to the respective purpose the following classification can be made:Numerical solutions of Kolmogorov equations(Galerkin’s method,Finite El-ement method,Path Integral method),Moment Closure Schemes,Compu-tation of the Evolution of Moments,Maximum Entropy Procedures,Asymp-totic Stability of Diffusion Processes.In the following,some of the outlined topics will be addressed stressing new developments.These topics are described within the next six subject areas,each focusing on a different issue,i.e.representation of stochastic processes andfields,structural response,stochastic FE methods and parallel processing,structural reliability and optimization,and stochastic dynamics. In this context it should be mentioned that aside from the MIT-Conference series the USNCCM,ECCM and WCCM’s do have a larger part of sessions addressing computational stochastic issues.Developments in Stochastic Structural Mechanics5 2Representation of Stochastic ProcessesMany quantities involving randomfluctuations in time and space might be adequately described by stochastic processes,fields and waves.Typical ex-amples of engineering interest are earthquake ground motion,sea waves, wind turbulence,road roughness,imperfection of shells,fluctuating prop-erties in random media,etc.For this setup,probabilistic characteristics of the process are known from various measurements and investigations in the past.In structural engineering,the available probabilistic characteristics of random quantities affecting the loading or the mechanical system can be often not utilized directly to account for the randomness of the structural response due to its complexity.For example,in the common case of strong earthquake motion,the structural response will be in general non-linear and it might be too difficult to compute the probabilistic characteristics of the response by other means than Monte Carlo simulation.For the purpose of Monte Carlo simulation sample functions of the involved stochastic pro-cess must be generated.These sample functions should represent accurately the characteristics of the underlying stochastic process orfields and might be stationary and non-stationary,homogeneous or non-homogeneous,one-dimensional or multi-dimensional,uni-variate or multi-variate,Gaussian or non-Gaussian,depending very much on the requirements of accuracy of re-alistic representation of the physical behavior and on the available statistical data.6G.I.Schu¨e ller The main requirement on the sample function is its accurate represen-tation of the available stochastic information of the process.The associ-ated mathematical model can be selected in any convenient manner as long it reproduces the required stochastic properties.Therefore,quite different representations have been developed and might be utilized for this purpose. The most common representations are e.g.:ARMA and AR models,Filtered White Noise(SDE),Shot Noise and Filtered Poisson White Noise,Covari-ance Decomposition,Karhunen-Lo`e ve and Polynomial Chaos Expansion, Spectral Representation,Wavelets Representation.Among the various methods listed above,the spectral representation methods appear to be most widely used(see e.g.[71,86]).According to this procedure,samples with specified power spectral density information are generated.For the stationary or homogeneous case the Fast Fourier Transform(FFT)techniques is utilized for a dramatic improvements of its computational efficiency(see e.g.[104,105]).Advances in thisfield provide efficient procedures for the generation of2D and3D homogeneous Gaus-sian stochasticfields using the FFT technique(see e.g.[87]).The spectral representation method generates ergodic sample functions of which each ful-fills exactly the requirements of a target power spectrum.These procedures can be extended to the non-stationary case,to the generation of stochastic waves and to incorporate non-Gaussian stochasticfields by a memoryless nonlinear transformation together with an iterative procedure to meet the target spectral density.Developments in Stochastic Structural Mechanics7 The above spectral representation procedures for an unconditional simula-tion of stochastic processes andfields can also be extended for Conditional simulations techniques for Gaussianfields(see e.g.[43,44])employing the conditional probability density method.The aim of this procedure is the generation of Gaussian random variates U n under the condition that(n−1) realizations u i of U i,i=1,2,...,(n−1)are specified and the a priori known covariances are satisfied.An alternative procedure is based on the so called Kriging method used in geostatistical application and applied also to con-ditional simulation problems in earthquake engineering(see e.g.[98]).The Kriging method has been improved significantly(see e.g.[36])that has made this method theoretically clearer and computationally more efficient.The differences and similarities of the conditional probability density methods and(modified)Kriging methods are discussed in[37]showing the equiva-lence of both procedures if the process is Gaussian with zero mean.A quite general spectral representation utilized for Gaussian random pro-cesses andfields is the Karhunen-Lo`e ve expansion of the covariance function (see e.g.[54,33]).This representation is applicable for stationary(homoge-neous)as well as for non-stationary(inhomogeneous)stochastic processes (fields).The expansion of a stochastic process(field)u(x,θ)takes the formu(x,θ)=¯u(x)+∞i=1ξ(θ) λiφi(x)(1)where the symbolθindicates the random nature of the corresponding quan-tity and where¯u(x)denotes the mean,φi(x)are the eigenfunctions andλi the eigenvalues of the covariance function.The set{ξi(θ)}forms a set of8G.I.Schu¨e ller orthogonal(uncorrelated)zero mean random variables with unit variance.The Karhunen-Lo`e ve expansion is mean square convergent irrespective of its probabilistic nature provided it possesses afinite variance.For the im-portant special case of a Gaussian process orfield the random variables{ξi(θ)}are independent standard normal random variables.In many prac-tical applications where the random quantities vary smoothly with respectto time or space,only few terms are necessary to capture the major part of the randomfluctuation of the process.Its major advantage is the reduction from a large number of correlated random variables to few most important uncorrelated ones.Hence this representation is especially suitable for band limited colored excitation and stochastic FE representation of random me-dia where random variables are usually strongly correlated.It might also be utilized to represent the correlated stochastic response of MDOF-systems by few most important variables and hence achieving a space reduction.A generalization of the above Karhunen-Lo`e ve expansion has been proposed for application where the covariance function is not known a priori(see[16, 33,32]).The stochastic process(field)u(x,θ)takes the formu(x,θ)=a0(x)Γ0+∞i1=1a i1(x)Γ1(ξi1(θ))+∞i1=1i1i2=1a i1i2(x)Γ2(ξi1(θ),ξi2(θ))+ (2)which is denoted as the Polynomial Chaos Expansion.Introducing a one-to-one mapping to a set with ordered indices{Ψi(θ)}and truncating eqn.2Developments in Stochastic Structural Mechanics9 after the p th term,the above representations reads,u(x,θ)=pj=ou j(x)Ψj(θ)(3)where the symbolΓn(ξi1,...,ξin)denotes the Polynomial Chaos of order nin the independent standard normal random variables.These polynomialsare orthogonal so that the expectation(or inner product)<ΨiΨj>=δij beingδij the Kronecker symbol.For the special case of a Gaussian random process the above representation coincides with the Karhunen-Lo`e ve expan-sion.The Polynomial Chaos expansion is adjustable in two ways:Increasingthe number of random variables{ξi}results in a refinement of the random fluctuations,while an increase of the maximum order of the polynomialcaptures non-linear(non-Gaussian)behavior of the process.However,the relation between accuracy and numerical efforts,still remains to be shown. The spectral representation by Fourier analysis is not well suited to describe local feature in the time or space domain.This disadvantage is overcome in wavelets analysis which provides an alternative of breaking a signal down into its constituent parts.For more details on this approach,it is referred to[24,60].In some cases of applications the physics or data might be inconsistent with the Gaussian distribution.For such cases,non-Gaussian models have been developed employing various concepts to meet the desired target dis-tribution as well as the target correlation structure(spectral density).Cer-tainly the most straight forward procedures is the above mentioned memo-ryless non-linear transformation of Gaussian processes utilizing the spectralrepresentation.An alternative approach utilizes linear and non-linearfil-ters to represent normal and non-Gaussian processes andfields excited by Gaussian white noise.Linearfilters excited by polynomial forms of Poisson white noise have been developed in[59]and[34].These procedures allow the evaluation of moments of arbitrary order without having to resort to closure techniques. Non-linearfilters are utilized to generate a stationary non-Gaussian stochas-tic process in agreement with a givenfirst-order probability density function and the spectral density[48,15].In the Kontorovich-Lyandres procedure as used in[48],the drift and diffusion coefficients are selected such that the solutionfits the target probability density,and the parameters in the solu-tion form are then adjusted to approximate the target spectral density.The approach by Cai and Lin[15]simplifies this procedure by matching the spec-tral density by adjusting only the drift coefficients,which is the followed by adjusting the diffusion coefficient to approximate the distribution of the pro-cess.The latter approach is especially suitable and computationally highly efficient for a long term simulation of stationary stochastic processes since the computational expense increases only linearly with the number n of dis-crete sample points while the spectral approach has a growth rate of n ln n when applying the efficient FFT technique.For generating samples of the non-linearfilter represented by a stochastic differential equations(SDE), well developed numerical procedures are available(see e.g.[47]).3Response of Stochastic SystemsThe assessment of the stochastic response is the main theme in stochastic mechanics.Contrary to the representation of of stochastic processes and fields designed tofit available statistical data and information,the output of the mathematical model is not prescribed and needs to be determined in some stochastic sense.Hence the mathematical model can not be selected freely but is specified a priori.The model involves for stochastic systems ei-ther random system parameters or/and random loading.Please note,due to space limitations,the question of model validation cannot be treated here. For the characterization of available numerical procedures some classifi-cations with regard to the structural model,loading and the description of the stochastic response is most instrumental.Concerning the structural model,a distinction between the properties,i.e.whether it is determinis-tic or stochastic,linear or non-linear,as well as the number of degrees of freedom(DOF)involved,is essential.As a criterion for the feasibility of a particular numerical procedure,the number of DOF’s of the structural system is one of the most crucial parameters.Therefore,a distinction be-tween dynamical-system-models and general FE-discretizations is suggested where dynamical systems are associated with a low state space dimension of the structural model.FE-discretization has no essential restriction re-garding its number of DOF’s.The stochastic loading can be grouped into static and dynamic loading.Stochastic dynamic loading might be charac-terized further by its distribution and correlation and its independence ordependence on the response,resulting in categorization such as Gaussian and non-Gaussian,stationary and non-stationary,white noise or colored, additive and multiplicative(parametric)excitation properties.Apart from the mathematical model,the required terms in which the stochastic re-sponse should be evaluated play an essential role ranging from assessing thefirst two moments of the response to reliability assessments and stabil-ity analysis.The large number of possibilities for evaluating the stochas-tic response as outlined above does not allow for a discussion of the en-tire subject.Therefore only some selected advances and new directions will be addressed.As already mentioned above,one could distinguish between two main categories of computational procedures treating the response of stochastic systems.Thefirst is based on Monte Carlo simulation and the second provides numerical solutions of analytical procedures for obtaining quantitative results.Regarding the numerical solutions of analytical proce-dures,a clear distinction between dynamical-system-models and FE-models should be made.Current research efforts in stochastic dynamics focus to a large extent on dynamical-system-models while there are few new numerical approaches concerning the evaluation of the stochastic dynamic response of e.g.FE-models.Numerical solutions of the Kolmogorov equations are typical examples of belonging to dynamical-system-models where available approaches are computationally feasible only for state space dimensions one to three and in exceptional cases for dimension four.Galerkin’s,Finite El-ement(FE)and Path Integral methods respectively are generally used tosolve numerically the forward(Fokker-Planck)and backward Kolmogorov equations.For example,in[8,92]the FE approach is employed for stationary and transient solutions respectively of the mentioned forward and backward equations for second order systems.First passage probabilities have been ob-tained employing a Petrov-Galerkin FE method to solve the backward and the related Pontryagin-Vitt equations.An instructive comparison between the computational efforts using Monte Carlo simulation and the FE-method is given e.g.in an earlier IASSAR report[85].The Path Integral method follows the evolution of the(transition)prob-ability function over short time intervals,exploiting the fact that short time transition probabilities for normal white noise excitations are locally Gaus-sian distributed.All existing path integration procedures utilize certain in-terpolation schemes where the probability density function(PDF)is rep-resented by values at discrete grid points.In a wider sense,cell mapping methods(see e.g.[38,39])can be regarded as special setups of the path integral procedure.As documented in[9],cumulant neglect closure described in section7.3 has been automated putational procedures for the automated generation and solutions of the closed set of moment equations have been developed.The method can be employed for an arbitrary number of states and closed at arbitrary levels.The approach,however,is limited by available computational resources,since the computational cost grows exponentially with respect to the number of states and the selected closurelevel.The above discussed developments of numerical procedures deal with low dimensional dynamical systems which are employed for investigating strong non-linear behavior subjected to(Gaussian)white noise excitation. Although dynamical system formulations are quite general and extendible to treat non-Gaussian and colored(filtered)excitation of larger systems,the computational expense is growing exponentially rendering most numerical approaches unfeasible for larger systems.This so called”curse of dimen-sionality”is not overcome yet and it is questionable whether it ever will be, despite the fast developing computational possibilities.For this reason,the alternative approach based on Monte Carlo simu-lation(MCS)gains importance.Several aspects favor procedures based on MCS in engineering applications:(1)Considerably smaller growth rate of the computational effort with dimensionality than analytical procedures.(2) Generally applicable,well suited for parallel processing(see section5.1)and computationally straight forward.(3)Non-linear complex behavior does not complicate the basic procedure.(4)Manageable for complex systems.Contrary to numerical solutions of analytical procedures,the employed structural model and the type of stochastic loading does for MCS not play a deceive role.For this reason,MCS procedures might be structured ac-cording to their purpose i.e.where sample functions are generated either for the estimation of the overall distribution or for generating rare adverse events for an efficient reliability assessment.In the former case,the prob-ability space is covered uniformly by an indiscriminate(blind)generationof sample functions representing the random quantities.Basically,at set of random variables will be generated by a pseudo random number generator followed by a deterministic structural analysis.Based on generated random numbers realizations of random processes,fields and waves addressed in section2,are constructed and utilized without any further modification in the following structural analysis.The situation may not be considered to be straight forward,however,in case of a discriminate MCS for the reliability estimation of structures,where rare events contributing considerably to the failure probability should be gener-ated.Since the effectiveness of direct indiscriminate MCS is not satisfactory for producing a statistically relevant number of low probability realizations in the failure domain,the generation of samples is restricted or guided in some way.The most important class are the variance reduction techniques which operate on the probability of realizations of random variables.The most widely used representative of this class in structural reliability assess-ment is Importance Sampling where a suitable sampling distribution con-trols the generation of realizations in the probability space.The challenge in Importance Sampling is the construction of a suitable sampling distribu-tion which depends in general on the specific structural system and on the failure domain(see e.g.[84]).Hence,the generation of sample functions is no longer independent from the structural system and failure criterion as for indiscriminate direct MCS.Due to these dependencies,computational procedures for an automated establishment of sampling distributions areurgently needed.Adaptive numerical strategies utilizing Importance Direc-tional sampling(e.g.[11])are steps in this direction.The effectiveness of the Importance sampling approach depends crucially on the complexity of the system response as well as an the number of random variables(see also section5.2).Static problems(linear and nonlinear)with few random vari-ables might be treated effectively by this approach.Linear systems where the randomness is represented by a large number of RVs can also be treated efficiently employingfirst order reliability methods(see e.g.[27]).This ap-proach,however,is questionable for the case of non-linear stochastic dynam-ics involving a large set of random variables,where the computational effort required for establishing a suitable sampling distribution might exceed the effort needed for indiscriminate direct MCS.Instead of controlling the realization of random variables,alternatively the evolution of the generated sampling can be controlled[68].This ap-proach is limited to stochastic processes andfields with Markovian prop-erties and utilizes an evolutionary programming technique for the genera-tion of more”important”realization in the low probability domain.This approach is especially suitable for white noise excitation and non-linear systems where Importance sampling is rather difficult to apply.Although the approach cannot deal with spectral representations of the stochastic processes,it is capable to make use of linearly and non-linearlyfiltered ex-citation.Again,this is just contrary to Importance sampling which can be applied to spectral representations but not to white noisefiltered excitation.4Stochastic Finite ElementsAs its name suggests,Stochastic Finite Elements are structural models rep-resented by Finite Elements the properties of which involve randomness.In static analysis,the stiffness matrix might be random due to unpredictable variation of some material properties,random coupling strength between structural components,uncertain boundary conditions,etc.For buckling analysis,shape imperfections of the structures have an additional impor-tant effect on the buckling load[76].Considering structural dynamics,in addition to the stiffness matrix,the damping properties and sometimes also the mass matrix might not be predictable with certainty.Discussing numerical Stochastic Finite Elements procedures,two cat-egories should be distinguished clearly.Thefirst is the representation of Stochastic Finite Elements and their global assemblage as random structural matrices.The second category addresses the evaluation of the stochastic re-sponse of the FE-model due to its randomness.Focusingfirst on the Stochastic FE representation,several representa-tions such as the midpoint method[35],the interpolation method[53],the local average method[97],as well as the Weighted-Integral-Method[94,25, 26]have been developed to describe spatial randomfluctuations within the element.As a tendency,the midpoint methods leads to an overestimation of the variance of the response,the local average method to an underestima-tion and the Weighted-Integral-Method leads to the most accurate results. Moreover,the so called mesh-size problem can be resolved utilizing thisrepresentation.After assembling all Finite Elements,the random structural stiffness matrix K,taken as representative example,assumes the form,K(α)=¯K+ni=1K Iiαi+ni=1nj=1K IIijαiαj+ (4)where¯K is the mean of the matrix,K I i and K II ij denote the determinis-ticfirst and second rate of change with respect to the zero mean random variablesαi andαj and n is the total number of random variables.For normally distributed sets of random variables{α},the correlated set can be represented advantageously by the Karhunen-Lo`e ve expansion[33]and for non-Gaussian distributed random variables by its Polynomial chaos ex-pansion[32],K(θ)=¯K+Mi=0ˆKiΨi(θ)(5)where M denotes the total number of chaos polynomials,ˆK i the associated deterministicfluctuation of the matrix andΨi(θ)a polynomial of standard normal random variablesξj(θ)whereθindicates the random nature of the associated variable.In a second step,the random response of the stochastic structural system is determined.The most widely used procedure for evaluating the stochastic response is the well established perturbation approach(see e.g.[53]).It is well adapted to the FE-formulation and capable to evaluatefirst and second moment properties of the response in an efficient manner.The approach, however,is justified only for small deviations from the center value.Since this assumption is satisfied in most practical applications,the obtainedfirst two moment properties are evaluated satisfactorily.However,the tails of the。

Eliminate Signal Noise With Discrete Wavelet TransformationThe wavelet transform is a mathematical tool that's becoming quite useful for analyzing many types of signals. It has been proven especially useful in data compression, as well as in adaptive equalizer and transmultiplexer applications.信号噪声消除和离散小波变换小波变换是一种在分析成为许多类型的信号很有用的数学工具。

它已被证明在数据压缩以及自适应均衡器和transmultiplexer应用中有特别的用途。

A wavelet is a small, localized wave of a particular shape and finite duration. Several families, or collections of similar types of wavelets, are in use today. A few go by the names of Haar, Daubechies, and Biorthogonal. Wavelets within each of these families share common properties. For instance, the Biorthogonal wavelet family exhibits linear phase, which is an important characteristic for signal and image reconstruction.小波是一种具有特定的形状与有限的时间的波。

nguage is a system of arbitrary vocal symbols used for human communication.2.Design features refer to the defining properties of human languagethat distinguish it from any animal system of communication.3.Arbitrariness refers that there is no logical connection between alinguistic symbol and what the symbol stands for(meaning and sounds).4.Duality means the property of having two levels of structures, suchthat units of the primary level are composed o f elements of the secondary level and each of the two levels has its own principles of organization. 5.Creativity(productivity)means language is resourceful because of its duality and recursiveness.6.Displacement means language can be used to refer to contexts removedfrom the immediate situations of the speaker.7.Cultural transmission: language is not biologically transmitted from generation to generation. The details of the linguistic system must belearned by each speaker.8.Interchangeability: means that any human b eing can be both a producer and a receiver of messages.9.Linguistics is the scientific study of language10.Descriptive means t he linguistic study aims to describe and analyze the language people actually use.Prescriptive means the linguistic study aims to lay down rules forcorrect and standard behavior in using language to tell people what they should say and not say.11.Synchronic共时in which languages are treated as self-containedsystems of communication at any particular time在那一刻、时、块的情况（当代、古代）Diachronic 历时 in which the changes to which languages are subjectin the course of time and treated historically.(在过程中都有什么变化、区别、有大时间变化)2个共时即为历时ngue and parole 找笔记重点。

Compressed Sensing of Monostaticand Multistatic SARIvana Stojanovic,Member,IEEE,MüjdatÇetin,Senior Member,IEEE,and W.Clem Karl,Senior Member,IEEEAbstract—In this letter,we study the impact of compressed data collections from a synthetic aperture radar(SAR)sensor on the reconstruction quality of a scene of interest.Different monostatic and multistatic SAR measurement configurations produce differ-ent Fourier sampling patterns.These patterns reflect different spectral and spatial diversity tradeoffs that must be made during task pressed sensing theory argues that the mutual coherence of the measurement probes is related to the reconstruc-tion performance of sparse domains.With this motivation,we propose a closely related t%-average mutual coherence parameter as a sensing configuration quality parameter and examine its relationship to the reconstruction behavior of various monostatic and ultranarrow-band multistatic configurations.We investigate how this easily computed metric is related to SAR reconstruction quality.Index Terms—Compressed sensing,multistatic SAR,mutual coherence,synthetic aperture radar(SAR),t%mutual coherence.I.I NTRODUCTIONS YNTHETIC aperture radar(SAR)is a remote sensing system capable of producing high-resolution imagery of target scenes independent of time of day,distance,and weather. Conventional SARs are monostatic,with collocated transmit and receive antenna elements.These SAR sensors coherently process multiple sequential observations of a scene under the assumption that the scene is static.Imaging resolution is de-termined by the bandwidth of the transmitted signals and the size of the synthesized antenna.Greater resolution requires wider bandwidths and larger aspect angles obtained from a longer baseline observation interval.An alternative approach is based on multistatic configurations,wherein spatially dispersed transmitters and receivers sense the scene.Such configurations provide the opportunity for spatial as well as frequency diver-sity and offer potential advantages inflexible sensor planning, sensing time reduction,and jamming robustness.To exploit the promise of multistatic sensing,robust methods of reconstructing imagery obtained from general multistatic configurations are needed,as well as tools to understand and evaluate the performance of various sensor configuration choices in a straightforward tractable manner.Recently,the area known as compressed sensing(CS)[1],[2]has received much attention in the signal processingfield.CS seeks to acquire as few measurements as possible about an unknown signal and,Manuscript received November9,2012;revised March24,2013;accepted April8,2013.I.Stojanovic is with Boston University,Boston,MA02215USA,and also with Scientific Systems Company,Inc.,Woburn,MA,01801USA.M.Çetin is with Sabanci University,80020Istanbul,Turkey.W.C.Karl is with Boston University,Boston,MA02215USA.Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/LGRS.2013.2259794given these measurements,reconstruct the signal either exactly or with provably small probability of error.The reconstruction methods used in CS are related to sparsity-constrained regular-ization.Signal reconstruction accuracy is shown to be related to the mutual coherence of the corresponding measurement operator[3],[4].Furthermore,the CS literature has demon-strated accurate signal reconstructions from measurement of extremely few but randomly chosen Fourier samples of a signal [2],[5].Since SAR can be viewed as obtaining samples of the spatial Fourier transform of the scatteringfield[6],these results suggest interesting opportunities for SAR sensing.The use of sparsity-constrained reconstructions for SAR im-age formation wasfirst presented in[7]although not within the CS framework.More recently,there have been several applica-tions of CS ideas to radar[8]–[10].The authors in[8]accurately reconstruct a small number of targets on a time–frequency plane by transmitting a sufficiently incoherent pulse and employing the techniques of CS.The authors in[10]propose the use of chirp pulses and pseudorandom sequences for CS with imaging radars.A CS technique for a SAR is also discussed in[9],where the authors obtain measurements by the random subsampling of a regular aspect–frequency grid in k-space.A SAR CS with reduced number of probes wasfirst discussed in[11]and[12]. In contrast to the previous work,we extend CS treatments to the multistatic scenario and propose a t%-mutual coherence parameter of the measurement operator as a simple measure of sensing configuration quality.Within this framework,we exam-ine different monostatic and ultranarrow-band multistatic con-figurations,which trade off frequency and geometric diversity. We show how the t%-mutual coherence of the measurement operator is affected by the number of transmitting probes as well as by the number of measurements,and we investigate and demonstrate how this simple metric is related to reconstruction quality.II.M ULTISTATIC SAR S IGNAL M ODELWe consider a general multistatic system with spatially dis-tributed transmit and receive antenna elements within a cone positioned at the center of a scene of interest.The scene of interest is modeled by a set of stationary point scatterers reflecting impinging electromagnetic waves isotropically to all receivers within the cone.We introduce a coordinate system with the origin in the center of the area of interest and,for simplicity,model the scene as2-D(Fig.1).The relative size of the scene is assumed to be small compared with the distances from the origin of the coordinate system to all transmitters and receivers such that transmit and receive angles would change negligibly if the coordinate origin moved to any point in the scene.Furthermore,we neglect signal propagation attenuation.1545-598X/$31.00©2013IEEEFig.1.Geometry of the kl th transmit–receive pair with respect to the sceneof interest.All transmit and receive pairs are restricted to lie within a cone of the angular extent Δθ.The complex signal received by the l th receiver,located at x l =[x l ,y l ]T ,for the narrow-band excitation from the k th transmitter,located at x k =[x k ,y k ]T ,reflected from a point scatterer at the spatial location x =[x,y ]T is given by r kl (t )=s (x )γk (t −τkl (x )),where s (x )is the reflectivity of the scat-terer,γk (t )is the transmitted waveform from the k th transmit-ter,and τkl (x )is the propagation delay from the transmitter to the scatterer and back from the scatterer to the receiver.The overall received signal from the entire ground patch with radius L is then modeled as a superposition of the returns from all the scattering centers.For narrow-band waveforms,defined by γk (t )=˜γk (t )e jωk t ,where ˜γk (t )is a low-pass slowly varying signal and ωk is the carrier frequency,the received signal is given byr kl (t )=x 2≤Ls (x )e jωk (t −τkl (x ))˜γk (t −τkl (x ))d x .(1)In the far-field case,when x 2 x k 2, x 2 x l 2,ωk /c x 22 x k 2,and ωk /c x 22 x l 2,we can use the first-order Taylor series expansion to approximate the propaga-tion delay τkl (x )asτkl (x )=1c ( x k −x 2+ x l −x 2)≈τkl (0)−1c x T e kl where τkl (0).=( x k 2+ x l 2)/c is the known transmitter–origin–receiver propagation delay and e kl .=e k +e l is the kl th transmit–receiver pair’s bistatic range vector.The vectorse k .=[cos φk ,sin φk ]T and e l .=[cos φl ,sin φl ]T are unit vec-tors in the direction of the k th transmitter and l th receiver,respectively.The chirp signal is the most commonly used spotlight SARpulse [6],given by γk (t )=e jαk t 2·e jωk t ,−τc /2≤t ≤τc /2,where ωk is the center frequency and 2αk is the so-called chirp rate of the k th transmit element.The narrow-band assumption is satisfied by choosing the chirp-signal parameters such that 2πB k /ωk 1.Ultranarrow-band waveforms are special cases of the chirp signal obtained by setting αk =0.We use a general transmitted chirp signal in (1),along with the far-field delay approximation,and apply typical demod-ulation and baseband processing [6]to obtain the following observed signal model:r kl (t )≈x 2<Ls (x )e j Ωkl (t )x Te kl d x (2)where Ωkl (t )=(1/c )[ωk −2αk (t −τkl (0))]depends on the frequency content of the transmitted waveform.The received signal model for the monostatic configura-tion corresponds to collocated transmit–receiver pairs,and for isotropic pointlike scattering,it is a special case of the multistatic model obtained by setting x k =x l .Note that in a general case,the nature of scattering is different for mono-static and multistatic collection scenarios.The radar collects energy backscattered from a target in the direction of incoming electromagnetic waves in the monostatic case,whereas the backscattered signal is measured in the direction of the receiver in the multistatic case.III.CS SARCS enables the reconstruction of sparse or compressible signals from a small set of linear nonadaptive measurements,much smaller in size than required by the Nyquist–Shannon theorem [1],[2].With high probability,accurate reconstructions can be obtained by sparsity-enforcing optimization techniques provided that the signal’s sparsifying basis and the random measurement basis are sufficiently incoherent.According to (2),SAR data represent Fourier k -space mea-surements of the underlying spatial reflectivity field.Different monostatic and multistatic SAR measurement configurations produce different Fourier sampling patterns.These patterns reflect different spectral and spatial tradeoffs that must be made during task planning.CS theory argues that random Fourier measurements represent good projections for the compressive sampling of pointlike signals [2].This suggests a natural application to the sparse aperture SAR sensing problem and opens a question of how different monostatic and multistatic SAR sensing configuration constraints influence reconstruction quality for a fixed number of measurements.Furthermore,we are interested in a simple goodness measure that can predict configuration quality before sensing even takes place.Such a quality predictor would allow better task planning and resource utilization.From CS,we know that the mutual coherence of the measurement probes is related to the worst-case reconstruction performance of sparse domains [3],[4].With this motivation,we examine the relationship of the sensing geometry and a closely related parameter we term the t %-average mutual coher-ence,as this parameter is expected to provide a better measure of the average reconstruction quality than the mutual coherence which is a pessimistic measure.In the following,we first describe reduced data collections with nonconventional SAR k -space sampling patterns.Next,we describe the sparsity-enforcing reconstruction and define the t %-average mutual coherence parameter used for the a priori evaluation of such sensing configurations.A.Sampling Configurations1)Monostatic SAR:One approach to reduced data collec-tion is to directly reduce the number of transmitted probes with regular or random interrupts in the synthetic aperture.STOJANOVIC et al.:COMPRESSED SENSING OF MONOSTATIC AND MULTISTATIC SAR3Fig.2.(a),(b)Monostatic SAR k-space sampling patterns for afixed k-space extent.(c),(d)Multistatic k-space sampling patterns for circular ultranarrow-band SAR operator.(a)Regular aspect–frequency sampling.(b)Random aspect,random frequency sampling.(c)Regular transmitter-receiver aspect positioning.(d)Random transmitter,random receiver aspect positioning.We consider regular and random observation sampling patterns within afixed observation extentΔΘcoupled with regular and random frequency sampling within a desired chirp-signal bandwidth B.We do not constrain random aspect/frequency samples to fall on a regular rectangular grid.Fig.2(a)illustrates the k-space sampling pattern when both aspect and frequency are sampled regularly,and Fig.2(b)illustrates a realization of a k-space sampling pattern when both aspect and frequency are sampled randomly.2)Multistatic SAR:Multistatic SAR offers the possibility of different k-space sampling patterns with tradeoffs in temporal frequency and spatial transmit/receiver location diversity.In the monostatic case,the chirp-signal bandwidth allowed for the extended coverage of the k-space in the range direction. However,in the multistatic case,extended k-space coverage can also be achieved with ultranarrow-band signals provided there is a spatial diversity of transmitter and receiver locations. Here,we consider a circular multistatic SAR with continuous-wave ultranarrow-band signal transmission.For the general multistatic SAR,the total number of measurements is calcu-lated as M=N tx N rx N f,where N tx is the number of trans-mitters/transmitted probes,N rx is the number of receivers, and N f is the number of frequency samples.In the case of the ultranarrow-band transmission,we have N f=1and different sampling patterns are achieved by varying transmitter and receiver angular locations.Fig.2(c)illustrates the k-space sampling pattern when both transmitter and receiver angular locations are sampled regularly,and Fig.2(d)illustrates a real-ization of k-space sampling when both transmitter and receiver locations are sampled randomly.B.Sparse ReconstructionWe consider the image formation of target scenes consisting of a sparse set of point reflectors.The spatial reflectivity is reconstructed on a rectangular grid,resulting in the linear discrete-data SAR modelr=˜Φs+n(3) where s is the unknown spatial reflectivity vector and˜Φis de-rived by discretizing(2).The vector r represents the observed, thus known,set of return signals at all receivers across time.Its elements are indexed by the tuple(k,l,t s),with t s being the sampling times associated with the kl th transmit–receive pair, and the spatial frequencyΩkl(t)and aspect-vector samples e kl are determined from the sampling configuration requirements. Both the observed SAR data r and the underlying scattering field s are complex valued.In SAR applications,reflectivity magnitudes are of primary interest.We apply sparsity-enforcing regularization directly on the magnitudes of the complex re-flectivityfield s.In particular,we define the reduced data SAR reconstruction problem ass=arg minss 11s.t. r−˜Φs 2≤σ(4)whereσrepresents the regularization parameter, s 11=i(Re{s i})2+(Im{s i})2,s i is the i th element of the vector s,and · 2represents the l2-norm.We solve the op-timization problem(4)using the software described in[13].C.t%-Average Mutual CoherenceThe mutual coherence of a measurement operator was dis-cussed as a simple but conservative measure of the ability of sparsity-enforcing reconstruction to accurately reconstruct a signal[3],[4].In the case of complex˜Φ,the mutual coherence of a sensing geometry is defined asμ(˜Φ)=maxi=jg ij,g ij=| φi,φj |φi 2 φj2,i=j(5)whereφi is the i th column of the matrix˜Φand the inner product is defined as φi,φj =φH iφj.The i th column vectorφi can be viewed as a range-aspect“steering vector”of a sensing geome-try or a contribution of a scatterer at a specific spatial location to the received phase-history signal.The mutual coherence measures the worst-case correlation between the responses of two distinct spatially distributed reflectors.A less conservative measure connected to average reconstruction performance was proposed in[14]for CS projection optimization.The t-average mutual coherence was defined as the average value of the set {g ij|g ij>t}.In contrast to the t-average mutual coherence,we propose to use the t%-average mutual coherence as a measure more closely related to the average reconstruction performance of(4).We define the t%-average mutual coherence,namely,μt%,as follows.Let E t%be the set of the t%percent of the largest column cross-correlations g ij.The t%-average mutual coherence is defined asμt%(˜Φ)=i=jg ij I ij(t%)i=jI ij(t%),I ij(t%)=1,g ij∈E t%0,otherwise.(6)In other words,μt%(˜Φ)measures an average cross-correlation value in a set of the t%most similar column pairs.The parameter t%should have a small value in order to accurately represent the tail of the column cross-correlation distribution. This measure is more robust to outliers,which can unfairly dominate the mutual coherence.A large value ofμt%(˜Φ)4IEEE GEOSCIENCE AND REMOTE SENSINGLETTERS Fig.3.Monostatic SAR when the ground scene consists of T=140scatter-ers.(a),(c)Performance versus sensing configuration for thefixed number ofmeasurements M=600.(b),(d)Performance versus number of measurementsM for sensing configurations with N tx=N f.(a),(b)t%-average mutualcoherenceμ0.5%.(c),(d)RMSE.indicates a large number of similar columns of˜Φthat canpotentially confuse the reconstruction algorithm.IV.S IMULATIONSIn this section,we present simulation results averaged over100Monte Carlo runs when n=0in(3).For regular sampling,we average over different ground-truth scene realizations.Incases that involve random sampling,we average over differentSAR operator and ground-scene realizations.For each real-ization of˜Φ,we measure the t%-average mutual coherenceμt%(˜Φ),for t%=0.5%,and display its average over all MonteCarlo runs.The ground-truth scene consists of T randomlydispersed scatterers each with unit magnitude and randomphase uniformly distributed in the range[0,2π].Reconstructionperformance is measured through the relative mean square error(RMSE),defined as RMSE=E[ s−s0 2/ s0 2],where s0 is the ground truth signal, s is its estimate from a reduced setof measurements,and E[·]stands for an empirical average overMonte Carlo runs.1)Simulation Results for Monostatic CS SAR:We consider the monostatic spotlight SAR imaging of a small ground patch of size(D x,D y)=(10,10)m when observed over a narrow-angle aspect cone ofΔθ=3.5◦.The transmitted waveforms are chirp signals with f o=10GHz and B=600MHz.The nominal range resolution isρx=c/(2B)=.25m,and the nominal cross-range resolution isρy=λ/(4sin(Δθ/2))=0.25m.Assuming that the pixel spacing matches the nominal resolution,we seek to reconstruct a40×40pixel reflectivity image.In Fig.3,we show the results when the ground scene consists of T=140randomly dispersed nonzero scatterers.The left column shows the results at thefixed number of measurements M=N tx N f=600as we vary the number of transmitted probes N tx.The right column shows the results as a function of the number of measurements M for sensing configurations with equal numbers of frequency and aspect samples,i.e., N tx=N paring theμt%curves to the RMSE curves,we see that as the t%-average mutual coherence is lowered,the reconstruction quality improves.Regular sampling introduces signal aliasing manifested as periodic and large-column cross-correlation peaks that confuse the reconstruction algorithm.The t%-average mutual coherence is the lowest when k-space sam-pling points cover the available k-space extent most uniformly. The most uniform coverage in the regular subsampling case is achieved when the ratio of the number of aspect angles to the number of frequency samples is approximately N tx/N f=ΔK x/ΔK y,whereΔK x(ΔK y)is the k-space extent in the cross-range(range)direction.On the other hand,in the random sampling case,the k-space coverage becomes more uniform as the number of transmitted probes increases.This is reflected in the lower values of theμt%.We see that increasing the number of transmitted probes after a certain value ofμt%is reached has only a small impact on the reconstruction performance.Finally, the monostatic random sensing enables high-quality reconstruc-tions with a smaller number of probes(N tx≥25)than required by the conventional SAR Nyquist sampling where N tx=40.2)Simulation Results for Multistatic CS SAR:The main advantage of CS in the monostatic scenario is the reduction of data storage and reduction in the number of transmitted probes. Data collection time cannot be reduced,as the monostatic SAR platform covers the whole aspect range sequentially in time. On the other hand,multistatic SAR has the potential to further reduce the data acquisition time through the use of a multitude of spatially dispersed transmitters and receivers.Theoretically, there exist many multistatic geometries with similar k-space coverage as in the monostatic case and,thus,similar reconstruc-tion results in the case of isotropic scattering.In an extreme case,we consider ultranarrow-band circular multistatic con-figurations with N f=1and transmitters and receivers placed around the scene in a full circle[15].In order to carry out simulations comparable with the mono-static case,the carrier wavelength is reduced such that spa-tial resolutions of the two configurations are approximately the same.In our simulations,each transmitter sends out an ultranarrow-band waveform signal with a frequency that satis-fiesρx=ρy=0.25=√2c/(4f o).The scene size is the same as in the monostatic simulation cases.We assume that the isotropic scattering assumption is valid for the circular multi-static configuration.When this assumption is violated,sparsity-driven imaging techniques that take the anisotropic nature of the scattering into account can be used(e.g.,[16]),and the reconstruction quality metrics may need to be modified.In Fig.4,we show the results as a function of the number of spatially dispersed transmitters N tx when the total number of measurements is heldfixed to M=600=N tx N rx N f and the signal support size is T=140.All sampling cases result in a k-space pattern that deviates significantly from a regular k-grid.This translates into a significantly reduced coherence of configurations with a few transmitted probes and higher reconstruction quality as compared with the monostatic case with the same number of transmitted probes.Furthermore, different multistatic sampling patterns achieve similar perfor-mance.While random sampling was the key to the improved performance in the monostatic case,circular multistatic config-uration is robust to transmit/receive sensor aspects.STOJANOVIC et al.:COMPRESSED SENSING OF MONOSTATIC AND MULTISTATIC SAR5Fig.4.Multistatic ultranarrow-band SAR performance versus sensing config-uration for fixed M =600and T =140.(a)μ0.5%.(b)RMSE.Fig. 5.GOTCHA SAR examples.(a)RMSE versus mutual coherence.(b)RMSE versus μ0.5%.(c)Sample reconstruction corresponding to the red ×point in (a)and (b)(i.e.,μ≈0.04and μ0.5%≈0.01).(d)Matched filter reconstruction corresponding to (c).3)Results Using GOTCHA SAR Imagery:In this section,we compare the utility of t %-average and conventional mutual coherence using the publicly released GOTCHA SAR data set [17].Starting with the complete phase history data of a 1024×1024-pixel image from this data set,we generate incomplete data by randomly deleting individual probes while keeping all frequency samples of retained probes.We use σ=0.1and set the maximum number of iterations to 200when solving (4).In Fig.5(a)and (b),we show the scatter plots of RMSE versus mutual coherence and RMSE versus μ0.5%-average mutual co-herence obtained using ten Monte Carlo realizations of incom-plete data when the total number of interrupts is 1+16i,i =1,...,31.We see that the RMSE variance at any particular value of μ0.5%is much smaller than the RMSE variance at any particular value of the mutual coherence,implying that μ0.5%is a better predictor of reconstruction quality than mutual coherence.A sample reconstruction from the data containing 240random synthetic aperture gaps is shown in Fig.5(c).The corresponding matched filter reconstruction is shown in Fig.5(d).We see that leakage artifacts due to reduced data are successfully removed with sparse reconstruction using (4).V .C ONCLUSIONIn this letter,we have studied different monostatic and multistatic SAR measurement configurations in the context of CS.We have shown that reconstructions of similar quality can be obtained using either the wide-band monostatic or ultranarrow-band multistatic configurations,effectively trading off frequency for geometric diversity.In both cases,configura-tions with sufficiently small values of the t %-average mutual co-herence achieve high-quality reconstruction performance.The t %-average mutual coherence is an easily computed parameter that can be used in the real-time design or evaluation of sensing configurations for,e.g.,the task planning of multimode radars.In the multistatic case,it is straightforward to obtain low coherence either by regular or random transmit/receive aspect sampling,whereas in the monostatic case,randomness in the sampling pattern leads to lower coherence.In the monostatic case,CS and sparsity-driven reconstruction allow for reduced onboard data storage and sensing with a reduced number of transmitted probes relative to what is conventionally required.In the multistatic case,CS and sparsity-driven reconstruction al-low for sensing with fewer transmitted probes but also reduced acquisition time when compared with 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