Azimuthal Correlations in the Target Fragmentation Region of High Energy Nuclear Collisions

Fourteen Easy Lessons in Density Functional TheoryJOHN P.PERDEW,ADRIENN RUZSINSZKYDepartment of Physics and Quantum Theory Group,Tulane University,New Orleans,LA70118Received1April2010;accepted5May2010Published online in Wiley InterScience().DOI10.1002/qua.22829ABSTRACT:Density functional theory(DFT)is now the most commonly usedmethod of electronic structure calculation in both condensed matter physics andquantum chemistry,thanks in part to the focus it has received over the first50years ofthe Sanibel Symposium.We present a short history,and review fourteen short and easybut important lessons about nonrelativistic DFT,with some partiality but with aminimum of technical complication.V C2010Wiley Periodicals,Inc.Int J Quantum Chem000:000–000,2010Key words:density functional theory;exchange–correlation energy;electronicstructure theory;Kohn–Sham theoryIntroduction and Short History of Density Functional TheoryT his article summarizes a talk given at the 50th anniversary Sanibel Symposium on Quantum Chemistry.Its original title,‘‘Some Things We have Learned About Exchange and Correlation in the Last Fifty Years,’’was some-what too restrictive in both subject matter and time frame.It presents14easy lessons in nonrela-tivistic density functional theory(DFT)at a quali-tative level.The selection of lessons is partial in both senses of the word:it is incomplete,and it reflects our own biases about what is most impor-tant and interesting(or at least most familiar). The same can be said of the short history we present.Our subject is appropriate to the occa-sion,since the Sanibel Symposium has played a major role in the development of modern elec-tronic structure theory,including but not limited to DFT.Possible companion pieces to this article are our short discussion[1]of‘‘perplexing’’issues in DFT and a detailed review[2]of the exact theory and its approximations.Often we need to predict the ground-state properties of an atom,molecule,solid,nanostruc-ture,or other system.We might need the equilib-rium geometry or structure,vibrational frequen-cies,electron density,total energy,and various total energy differences such as atomization and surface energies,as well as the linear and nonlin-ear responses to external static probes.One way to find the needed properties is to solve theCorrespondence to:J.P.Perdew;e-mail:*****************Contract grant sponsor:National Science Foundation.Contract grant number:DMR-0854769.International Journal of Quantum Chemistry,Vol.000,000–000(2010) V C2010Wiley Periodicals,Inc.N -electron Schroedinger equation for the N -elec-tron ground-state wavefunction.This approach is potentially accurate and complete (providing all the information that can be known),but it is com-putationally inefficient and impractical for large N .A second way is to solve for more limited in-formation,as provided by a Green’s function,density matrix,or electron density.This approach may be less accurate and complete,but it can achieve a useful computational efficiency even for large N .Kohn–Sham DFT [2,3]uses spin orbitals to predict the ground-state electron density,total energy,and related important properties.It pro-vides an often useful and improvable compromise between accuracy and computational efficiency.Thus,it is now the most widely used method of electronic structure calculation in both condensed matter physics and quantum chemistry,two sub-jects that have long been intertwined at the Sani-bel Symposia.However,other methods [4],including full wavefunction methods,continue to improve in accuracy and efficiency,providing for selected systems a benchmark of accuracy.Orbital-free DFT began in the 1920s with the Thomas–Fermi theory [5,6],which expresses the total energy E approximately in terms of the elec-tron density n ð~r Þ,where n ð~r Þd 3r is the average number of electrons in volume element d 3r at position ~r ,using the simplest density functional that makes sense.The density is then varied atfixed electron number N ¼R d 3rn ð~r Þto minimize the energy functional.This approach gives simple and useful estimates for the density and total energy of an atom (defined as minus the mini-mum work to strip all the electrons from the nu-cleus)but it is far too crude for chemistry.In fact,Teller [7]proved that in Thomas–Fermi theory,atoms do not bind together to form molecules and solids.Without the exchange–correlation energy,‘‘nature’s glue’’[8],chemical bonds are ei-ther absent or far too long and weak.However,orbital-free methods continue to improve;see recent work by Trickey and others [9–11].Spin orbitals or fictitious one-electron wave-functions w i ð~r ;r Þwere introduced in the 1930s by Hartree,Fock,and Slater [12].The energy (includ-ing sometimes exchange but not Coulomb correla-tion)was expressed in terms of these orbitals and their occupation numbers f i (1or 0,since electrons are fermions),and minimized with respect to them.This approach binds atoms into molecules and solids,although usually too weakly [8],andgenerally improves the total energy E and elec-tron density n ð~r Þ¼P i r f i r w i r ð~r Þj j 2over orbital-free methods.The Hartree and Hartree-Fock methods were not easy to implement on early computers and omitted important correlation effects.In the 1950s,Slater combined the orbital and density functional approaches by creating a local density approxima-tion (LDA)for the exchange–correlation energy and potential (which he called the X a approxima-tion [12]).He found that it could be easily imple-mented self-consistently on the computer and that it included a rough but useful estimate of correla-tion.After retiring from MIT in 1966,Slater joined the Quantum Theory Project at the University of Florida,Gainesville,which held the first Sanibel Symposium in 1961.In 1964–1965,the Hohenberg–Kohn [13]and Kohn–Sham [3]theorems,the twin pillars of mod-ern DFT,were published.These theorems showed that,given the right density functionals,one can find the exact ground-state density and energy of an N -electron system in an external scalar poten-tial using either the total density or the orbitals as variational objects.Kohn and Sham also proposed an LDA for the exchange–correlation energy,which,unlike Slater’s,is exact for an electron gas of uniform or slowly varying density.Kohn–Sham theory was not widely known until around 1970,when condensed matter physi-cists started to find that this theory in the LDA gives a remarkably realistic description of bulk solids and their surfaces.Since then,DFT has dominated electronic structure calculations for solids.For the surface energy [14,15]of a solid,density functionals proved to be more accurate than early correlated wavefunction calculations.The densities of many solids,especially simple metals,are sufficiently like those of uniform elec-tron gases for the LDA to work well.However,it did not work so well for atoms and molecules.In particular,the atomization energies of molecules were strongly overestimated (although they were still better than those of Hartree-Fock theory).In the period 1970–1986,DFT in chemistry had only a few prophets,including notably Parr [16],Jones and Gunnarsson [17],and Levy [18].This was however a time when theoretical work by Langreth and Per-dew [19,20],and by Gunnarsson and Lundqvist [21],explained why the local approximation worked as well as it did and suggested approaches to improve it.The new approaches included generalized gradient approximations and hybridPERDEW AND RUZSINSZKY2INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL.000,NO.000functionals,which greatly increased the accuracy of atomization energies and thus the relevance of the theory to chemistry.Since about1992,DFT has swept chemistry,as it earlier swept condensed matter physics.This development was led notably by Becke[22],who proposed many creative approaches to functional construction,and by John Pople and Nick Handy.Since the1920s,many important lessons have been learned about DFT.We summarize14of these lessons below.Lesson1:Density vs.Correlated WavefunctionThe electron density nð~rÞhas a lower dimen-sionality than the N-electron wavefunction Wð~r1;r1;…;~r N;r NÞ.Using n instead of W as thebasic variational object makes electronic structure calculations much faster but typically much less accurate.This lesson comes from Thomas[5]and Fermi [6].In his lecture for the1998Nobel Prize in Chemistry[23],Walter Kohn explained this fact in a clear if oversimplified way,which we para-phrase here:suppose we use a mesh of points in real space,with10points along each of the x,y, and z axes.Then we can compute and store the density on the mesh as103numbers.A one-elec-tron wavefunction can also be represented by 103numbers,but a10-electron correlated wave-function must be represented by(103)10¼1030 numbers.There are of course cleverer ways[4] to compute a10-electron wavefunction,but the fact remains that the effort to compute an N-electron wavefunction scales up very rapidly with N.Lesson2:Orbitals vs.Correlated WavefunctionWhile too much accuracy can be lost by using the density nð~rÞas the basic variational object, considerable accuracy can be restored by using also the occupied orbitals or fictitious one-electron wavefunctions w1ð~r;rÞ…w Nð~r;rÞ.This lesson comes from Hartree and Fock in the1920s and from Slater in the1950s[12].In our example from Lesson1,an orbital description of the10-electron system requires computing and storing only10Â103¼104numbers.Lesson3:Something Proved to Exist That Cannot be FoundIn principle,we can find the exact ground-state energy and density of N electrons in an external potential vð~rÞa.by solving an Euler equation for the densitynð~rÞas proved by Hohenberg and Kohn[13],orb.by solving self-consistent one-electronSchroedinger equations for the orbitals asproved by Kohn and Sham[3].In approach(b),the functional derivative d E xc=d nð~rÞserves as the exchange–correlation contribu-tion v xcð~rÞto the effective one-electron orKohn–Sham potential v KSð~rÞ.In practice,the exact density functional E v[n] for the total energy needed for approach(a),and the exact density functional E xc[n]for the exchange–correlation energy needed for approach (b),are not accessible in any practical way and must be approximated.Although the Hohenberg–Kohn–Sham theo-rems are only existence theorems,the knowledge that exact density functionals exist has strongly driven the quest for better and more accurate approximations.Lesson4:From Uniform ElectronGas to Atoms,Molecules,and SolidsThe local density approximation(LDA)E LDAxc½n ¼Zd3rnð~rÞe unifxcðnð~rÞÞ;(1)or better the local spin density approximation (LSDA)E LSDAxc½n";n# ¼Zd3rnð~rÞe unifxcðn"ð rÞ;n#ð~rÞÞ;(2)which start from e unifxc,the exchange–correlation energy per particle of a uniform electron gas,areFOURTEEN EASY LESSONS IN DENSITY FUNCTIONAL THEORY VOL.000,NO.000DOI10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY3accurate enough for useful calculations on solids and solid surfaces but not accurate enough for atoms and molecules.LDA comes from Kohn and Sham[3]and LSDA from von Barth and Hedin[24].The accu-racy of these local approximations for solids and surfaces was established by Lang and Kohn[25] and by Moruzzi et al.[26].Jones and Gunnarsson [17]found good structures but overestimated atomization energies for molecules.For open-shell or magnetic systems,in the ab-sence of an external magnetic field,one can use either the total density or the separate spin den-sities in principle,but in practice LSDA is more accurate than LDA because it inputs more infor-mation about the system[27].This was the first indication of a possible ladder of density func-tional approximations for the exchange–correla-tion energy to be discussed in Lessons9and10 below.All rungs of the ladder are actually imple-mented on the spin densities,but we will use the total density below to simplify the notation. Lesson5:The Electron Digs a HoleThe exact exchange–correlation energy is[19–21]the electrostatic interaction between the elec-tron density nð~rÞand the density n xcð~r;r0Þat r0of the exchange–correlation hole around an electronat~r:E xc¼ð1=2ÞZd3rnð~rÞZd3r0n xcð~r;~r0Þ=~r0À~rj j:(3)The exchange–correlation hole n xc¼n xþn c is the sum of the separate exchange and correlation holes.The exchange hole is the same as in Har-tree–Fock theory,with the Hartree–Fock orbitals replaced by Kohn–Sham orbitals,and the correla-tion hole is an average of an expectation value over the coupling constant for the electron–elec-tron interaction at fixed electron density.The exact holes satisfy the constraintsn x<0;Zd3r0n xð~r;~r0Þ¼À1;Zd3r0n cð~r;~r0Þ¼0:(4)These constraints are also satisfied by the LDA hole(that of a uniform gas)and this fact explains why LDA works as well as it does.The con-straints can also be imposed to develop other approximations that can improve over LDA by satisfying additional exact constraints.This lesson comes from Langreth and Perdew [19,20]and from Gunnarsson and Lundqvist [21].Lesson6:Searching Over Wavefunctions for theDensity FunctionalThe original proof of the Hohenberg–Kohn the-orem[13]was by reductio ad absurdum and thus not a constructive proof.Alternatively,the vari-ous density functionals,including E xc[n],can be defined by searches over N-electron wavefunc-tions constrained to yield a given electron density [18].Thus,the functionals can be intuitively understood and their exact properties can be derived.This lesson comes from Levy[18].It is an im-portant one because the derived exact properties (e.g.,Refs.[28,29])can be used to constrain the needed approximations.Lesson7:After the Local Density Comes its GradientSecond-order gradient expansionsE GEAxc½n ¼E LDAxc½n þZd3rC xcðnÞr nj j2=n4=3(5)improve on LDA for very slowly varying den-sities but are typically less accurate than LDA for realistic densities[30,31],because[31]the sec-ond-order gradient expansion of the hole violates the exact hole constraints of Lesson5.However, generalized gradient approximations or GGAs [30–35]E GGAxc½n ¼Zd3rn e GGAxcðn;r nÞ(6)can be derived with or without nonempirical parameters,and are more accurate than LDA for the separate exchange and correlation energies of atoms and for the atomization energies of mole-cules and solids.Nonempirical constructions often impose exact constraints on(1)the hole[33],or(2)the energyPERDEW AND RUZSINSZKY4INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI10.1002/qua VOL.000,NO.000functional[35],or(3)the coupling constant de-pendence[36].This lesson comes from Ma and Brueckner[30], Langreth and Perdew[31],Langreth and Mehl [32],Perdew and Wang[33],Becke[34,22],Per-dew et al.[35],and Seidl et al.[36].It was the main development that made DFT of interest to chemists.Lesson8:What is a Fractionof an Electron?The exact density functional E v[n]for the energy can be defined for an open system with noninteger average electron number N[37]by extending the constrained search of Lesson6from wavefunctions to ensembles.The energy E varies linearly between adjacent integer electron num-bers,with derivative discontinuities at the inte-gers.This explains why neutral molecules dissoci-ate to integer-charged atoms,and why the fundamental band gap[38,39]in the exact Kohn–Sham band structure is unphysical.This lesson comes from Perdew et al.[37], Perdew and Levy[38],and Sham and Schlueter [39].Lesson9:The More Information Input,the More Accuratethe OutputWe can make more accurate functionals by adding ingredients to the exchange–correlation energy density beyond n and!n:a.Meta-GGAs add the orbital kinetic energydensitysð~rÞ¼ð1=2ÞXi rf i r r w i rð~rÞj j2;(7)as proposed by Becke[40]with other non-empirical constructions by Perdew et al.[41–43].b.Hyper-GGAs or hybrid functionals addexact exchange information,such as the exact exchange energy or the exact exchange hole,as proposed by Becke[44]and refined by Savin[45],Vydrov and Scuseria[46],etc.Most hyper-GGAs do not properly scale to exact exchange under uniform density scal-ing to the high-density limit but some do[47].c.Random phase approximation(RPA)-likefunctionals add the unoccupied orbitals asdeveloped by Furche[48,49],Harl andKresse[50],etc.Lesson10:We are ClimbingJacob’s LadderThere is a five-rung Jacob’s ladder of common density functional approximations(LSDA,GGA, meta-GGA,hyper-GGA,and RPA-like functionals), as proposed by Perdew and Schmidt[51]and explained in Lesson9.All rungs except the hyper-GGA rung now have nonempirical constructions. Accuracy tends to increase up the puta-tional cost increases modestly from LSDA to GGA to meta-GGA(the three semilocal rungs)but can increase considerably on ascent to higher rungs.The Perdew-Burke-Ernzerhof(PBE)GGA[35] provides a moderately accurate description of atoms,molecules,and solids but is too simple to achieve high accuracy for all three kinds of sys-tems.The optimum GGA for solids has a weaker gradient dependence than the optimum GGA for atoms and molecules,as argued by Perdew and coworkers[52–54].However,a single nonempirical meta-GGA(revTPSS)can work well for the equi-librium properties of atoms,molecules,and solids, as shown by Perdew et al.[43].A meta-GGA can provide different GGA descriptions for solids(especially metals),which have important regions of strong orbital overlap where sð~rÞ)s Wð~rÞ r nj j2=ð8nÞ,and for atoms and mol-ecules,which have important regions where a sin-gle orbital shape dominates the density(as in an electron-pair bond),making sð~rÞ%s Wð~rÞ.Lesson11:Where Semilocal Functionals Fall DownSemilocal functionals(LSDA,GGA,and meta-GGA)necessarily fail(and thus full nonlocality is needed)when the exact exchange–correlation hole has a long-range tail,because semilocal function-als know nothing about the electron density far from an electron.This failure occurs when long-FOURTEEN EASY LESSONS IN DENSITY FUNCTIONAL THEORY VOL.000,NO.000DOI10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY5range van der Waals interactions between sepa-rating systems are important;the needed correc-tion is present in RPA-like functionals[48–50]but neither in semilocal functionals nor in hyper-GGAs.The failure also occurs when electrons are shared across stretched bonds between distant centers(as in certain transition states and dissoci-ation limits).Semilocal functionals cannot describe such stretched bonds,but hyper-GGAs and RPA-like functionals can do so by displaying more or less the right derivative discontinuities of Lesson8.Dramatic stretched-bond effects have been dis-cussed by Ruzsinszky et al.[55,56],Vydrov et al.[57],Tsuchimochi and Scuseria[58],and Yang and coworkers[59].Lesson12:Elaborate Functionals Need Corrections TooThe RPA,which uses the unoccupied orbitals in the simplest way that makes sense for all sys-tems,is not quite good enough for chemistry.It underestimates atomization energies of molecules by about the same amount that the PBE GGA overestimates them as discovered by Furche[48]. For this property,RPA is far less accurate than meta-and hyper-GGAs.RPA uses exact exchange but makes the correlation energy too negative by roughly0.02hartree/electron.Thus,a correction to RPA is needed.The needed correction to RPA is semilocal(de-scribable by LSDA or GGA)in atoms and at solid surfaces[60]but it is fully nonlocal in molecules [61].An accurate nonempirical nonlocal correction remains to be found.Lesson13:The Kohn–ShamPotential Shapes UpStarting from an orbital functional,the exact Kohn–Sham potential v KSð~rÞof Lesson3can be constructed by the optimized effective potential method of Talman and Shadwick[62,63].Starting from the correlated N-electron wave-function,we can directly construct the energy and electron density.However,it is also computation-ally practical to construct therefrom the exact Kohn–Sham potential and so to study its features. This can be done in two different ways:(1)The Zhao et al.[64]approach first constructs the den-sity and then numerically varies the Kohn–Shampotential to reproduce this density in a system of N noninteracting electrons.(2)Another approach due to Sahni[65]and others constructs the Kohn–Sham potential from expectation values computed from the correlated N-electron wavefunction.Lesson14:After Statics Comes DynamicsDFT can be extended(exactly in principle and approximately in practice)to time-dependent and excited states.The essential idea is to solve time-dependent one-electron Schroedinger equations in which the time-dependent exchange–correlation potential v xcð~r;tÞis a functional of the electron density nð~r0;t0Þ.Poles of the frequency-dependent linear density response function then yield the ex-citation energies.This lesson comes from Runge and Gross[66] and from others[67].ACKNOWLEDGMENTSThe authors thank Sam Trickey for the invita-tion to speak at the50th Sanibel Symposium,and Kieron Burke and Jianwei Sun for suggestions on the manuscript.References1.Perdew,J.P.;Ruzsinszky,A.;Constantin,L.A.;Sun,J.;Csonka,G.I.J Chem Theory Comput2009,5,902.2.Perdew,J.P.;Kurth,S.In A Primer in Density FunctionalTheory;Fiolhais,C.,Nogueira,F.,Marques,M.,Eds.;Lecture Notes in Physics;Springer:Berlin,2003;Vol.620,p1.3.Kohn,W.;Sham,L.J.Phys Rev1965,140,A1133.4.Shavitt,I.;Bartlett,R.J.Many Body Methods in Chemistryand Physics:MBPT and Coupled-Cluster Theory;Cambridge University Press:Cambridge,2009.5.Thomas,L.H.Proc Cambridge Philos Soc1926,23,542.6.Fermi,E.Zeit fuer Physik1928,48,73.7.Teller,E.Rev Mod Phys1962,34,627.8.Kurth,S.;Perdew,J.P.Int J Quantum Chem2000,77,819.9.Perdew,J.P.;Constantin,L.A.Phys Rev B2007,75,155109.10.Karasiev,V.V.;Jones,R.S.;Trickey,S.B.;Harris,F.E.Phys Rev B2009,80,245120.11.Huang,C.;Carter,E.A.Phys Rev B2010,81,045206.12.Slater,J. 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a r X i v :n u c l -e x /0209022v 1 25 S e p 20021High p T π0Production and Angular Correlations in 158A GeV p+A and Pb+Pb CollisionsS.Bathe a for the WA98-CollaborationaUniversity of M¨u nster,48149M¨u nster,GermanyRecent results of the WA98experiment with p and Pb induced reactions at 158A GeV are presented.Azimuthal γ-γcorrelations at high p T were studied in search for a signal of jet-like structures.A clear indication for back-to-back correlations can be seen in p+A with a strong dependence on the p T of the photons and the size of the system.In Pb+Pb collisions in plane elliptic flow has been observed.Results on transverse mass spectra of neutral pions measured at central rapidity are presented for impact parameter selected Pb+Pb collisions.In going from peripheral to medium central collisions there is a nuclear enhancement increasing with transverse mass similar to the Cronin effect,while for very central collisions this enhancement appears to be weaker than expected.1.IntroductionThe CERN experiment WA98[1,2]consisted of large acceptance photon and hadron spectrometers together with several other large acceptance devices which allow to measure various global variables on an event-by-event basis.The results presented here were obtained from an analysis of the data taken with p and Pb beams in 1995and 1996at 158A GeV.The Pb-induced reactions have been subdivided into samples of different centrality using the transverse energy E T measured in the MIRAC calorimeter.Photons are measured with the WA98lead-glass photon detector,LEDA,which consisted of 10,080individual modules with photomultiplier readout.The detector was located at a distance of 21.5m from the target and covered the pseudorapidity interval 2.2<η<2.9.Details about the photon measurement can be found in [2].2.Azimuthal γ-γ-CorrelationsWe attempt to use azimuthal correlations of photons,which mainly originate from de-cays of neutral hadrons,π0or η,to gain information on the relative fraction of produced particles which still carry memory of the primary production process.Particles produced in a primary two-body collision of incoming nucleons must be correlated in transverse momentum due to local momentum conservation.Dominantly,especially if their indi-vidual transverse momenta are large,the particles will be anti-correlated in azimuthal angle.This is of course reminiscent of jet structure in the particle emission in high-energy physics.In fact,there is expected to be a smooth transition of this purely kinematical effect to mini-jet or jet production.At high enough transverse momenta,such an analysis2∆φ o [ ]C n o r mM [ GeV]R101Figure 1.a)Exemplary C (∆φ)for p+C reactions.b)Ratio R of correlated to uncorrelatedpairs at ∆φ=180◦as function of the pseudo-mass M for p+C and p+Pb.should allow to study minijet or jet production and eventually allow to investigate effects such as jet quenching.If the particles undergo secondary and further rescatterings,as in an equilibrated system,the memory of the correlation discussed above will be lost.Of course,anti-correlation of particle pairs will also be generated by collective effects like hydrodynamical elliptic flow.One therefore has to check how much the measured effects might be altered by flow phenomena.The γ-pairs are characterized by the difference of their azimuthal angle ∆φand the so-called pseudo-mass M =p T 1+p T 2.The correlation function C(∆φ)is introduced:C (∆φ)=d 2Ndϕ1dNNdN31N N PartCollv 2Figure 2.a)v 2in Pb+Pb as function of the number of participants with and withoutconsideration of back-to-back effects.b)Strength of back-to-back correlations in p+A and Pb+Pb reactions.a function of different centralities in Pb+Pb with and without consideration of back-to-back effects.Especially in the more peripheral samples,elliptic flow does not describe the correlation completely.To compare the strength of an additional Gaussian-like correla-tion for different systems the ratio R of correlated to uncorrelated pairs at ∆φ=180◦is shown in Figure 2b as a function of the number of binary collisions.A parametrisation of the p+A data points with R (N Coll )=1/(aN Coll −1)is extrapolated to the Pb+Pb data,which represents scaling of the number of correlated pairs with N Coll .No deviation from this scaling is observed.3.Centrality Dependence of π0-ProductionAlready from the experimentally determined shape of transverse mass spectra of hadrons it is evident that heavy ion reactions are not merely a superposition of nucleon-nucleon collisions [5].In p+A collisions the flattening of the transverse mass spectra compared to p+p (Cronin effect [6])has been attributed to initial state multiple scattering of partons [7].In the analysis of central reactions of Pb+Pb at 158A GeV,however,it is seen that both predictions of perturbative QCD [8]and hydrodynamical parameterizations [9]can describe the measured neutral pion spectra reasonably well.The understanding of the relative contributions of the various soft and hard processes in particle production is espe-cially important in view of the recent interest in the energy loss of partons in dense matter [10,11],generally referred to as jet quenching ,as a possible probe for the quark gluon plasma.Since one of the suggested experimental hints of jet quenching is the suppression of particle production at high transverse momenta,it is important to understand other possible nuclear modifications of particle production in detail.More information in this respect may be gathered from the variation of the particle spectra for different reaction systems or different centralities [12].Here we study the variations in absolute multiplicities.Especially at high transverse momentum one na¨ıvely expects an increase of the multiplicity proportional to the number of collisions due to the4importance of hard scattering.In fact,it was already observed in p+A collisions at beam energies of200−400GeV[6]that the increase in cross section at high transverse momenta is even stronger than the increase in the target mass.The ratios of the measured pion multiplicity distributions for two different samples(labeled X and Y)normalized to the number of collisionsR XY(m T)≡ E d3N E d3N。

分子生物学名词解释大全AAbundance (mRNA 丰度)：指每个细胞中mRNA 分子的数目。

Abundant mRNA(高丰度mRNA)：由少量不同种类mRNA组成，每一种在细胞中出现大量拷贝。

Acceptor splicing site (受体剪切位点)：内含子右末端和相邻外显子左末端的边界。

Acentric fragment(无着丝粒片段)：(由打断产生的)染色体无着丝粒片段缺少中心粒，从而在细胞分化中被丢失。

Active site(活性位点)：蛋白质上一个底物结合的有限区域。

Allele(等位基因)：在染色体上占据给定位点基因的不同形式。

Allelic exclusion(等位基因排斥)：形容在特殊淋巴细胞中只有一个等位基因来表达编码的免疫球蛋白质。

Allosteric control(别构调控)：指蛋白质一个位点上的反应能够影响另一个位点活性的能力。

Alu-equivalent family(Alu 相当序列基因)：哺乳动物基因组上一组序列，它们与人类Alu家族相关。

Alu family (Alu家族)：人类基因组中一系列分散的相关序列，每个约300bp长。

每个成员其两端有Alu 切割位点(名字的由来)。

α-Amanitin(鹅膏覃碱)：是来自毒蘑菇Amanita phalloides 二环八肽，能抑制真核RNA聚合酶，特别是聚合酶II 转录。

Amber codon (琥珀MM子)：核苷酸三联体UAG，引起蛋白质合成终止的三个MM子之一。

Amber mutation (琥珀突变)：指代表蛋白质中氨基酸MM子占据的位点上突变成琥珀MM子的任何DNA 改变。

Amber suppressors (琥珀抑制子)：编码tRNA的基因突变使其反MM子被改变，从而能识别UAG MM子和之前的MM子。

Aminoacyl-tRNA (氨酰-tRNA)：是携带氨基酸的转运RNA，共价连接位在氨基酸的NH2基团和tRNA 终止碱基的3¢或者2¢－OH 基团上。

第47卷第3期2021年3月北京工业大学学报JOURNAL OF BEIJING UNIVERSITY OF TECHNOLOGYVol.47No.3Mar.2021基于压缩感知理论的二维DOA估计窦慧晶，梁霄，张文倩(北京工业大学信息学部，北京100124)摘要：二维波达方向(direction of arrival,DOA)估计在雷达探测、电子对抗、医学成像等领域有着广泛的应用.针对现有算法估计精度不足、计算量巨大的问题，在基于压缩感知理论的背景下提出一种二维均匀L型阵列信号的DOA估计算法.该算法首先对阵列信号的俯仰角和方位角构建空间合成角，并对空间合成角构建过完备冗余字典;再利用正交化高斯随机矩阵构造观测矩阵;最后通过改进RM-FOCUSS算法和求解三角函数的方法还原出方位角和俯仰角.理论研究表明，该方法在高信噪比、多快拍条件下比传统算法具有更高的估计精度和分辨力,且通过压缩采样降低了运算量.仿真实验验证了上述结论.关键词：DOA估计；压缩感知；过完备冗余字典；稀疏表示；压缩采样；测量矩阵中图分类号：TN911文献标志码：A文章编号：0254-0037(2021)03-0231-08doi：10.11936/bjutxb2019100002Two-dimensional DOA Estimation Based onCompressed Sensing TheoryDOU Huijing,LIANG Xiao,ZHANG Wenqian(Faculty of Information Technology，Beijing University of Technology，Beijing100124，China)Abstract：Two-dimensional direction of arrival(DOA)estimation has been widely used in radar detection，electronic reconnaissance，medical imaging and other fields.Aiming at the problems of inadequate estimation accuracy and enormous computational load of existing algorithms，a DOA estimation algorithm for two-dimensional uniform L-shaped array signals was presented in this paper based on compressed sensing theory.First，an over-complete redundant dictionary was established by using the space frequency of the azimuth angle and pitch angle.Then the orthogonal Gaussian random matrix was used to construct the measurement matrix.Finally,azimuth and elevation were restored by improving RM -FOCUSS algorithm and solving trigonometric function.The theoretical research shows that the proposed method has higher estimation accuracy and resolution than the traditional algorithm under the conditions of high SNR and multi-snapshot，and it reduces the computational complexity by compressing sampling.The simulation results verify the effectiveness and correctness.Key words：direction of arrival(DOA)estimation；compressed sensing(CS)；over-complete redundant dictionary；spare representation；compressed sampling；measurement matrix二维波达方向(direction of arrival,DOA)估计在阵列信号处理领域有着重要的研究意义，与一维收稿日期：2019-10-11基金项目：国家自然科学基金资助项目(61171137)；北京市教育委员会科研发展计划资助项目(KM201210005001)作者简介：窦慧晶(1969—),女，副教授，主要从事数字信号处理、信号参量估计阵列信号处理、语音信号处理方面的研究, E-mail:dhuijing@232北京工业大学学报2021年DOA估计相比,该估计算法能够更精确描述目标的空间特性，因此DOA估计在二维信号领域更具实际应用价值［1-2］.二维多重信号分类(two-dimensional multiple signal classification,2-D MUSIC)算法是目前已有的二维阵列信号DOA估计算法中最为经典的估计算法之一，该算法核心思想是将传统的一维MUSIC估计算法在二维空间进行直接推广，由于该算法需要二维谱峰搜索因而导致计算量巨大,且需要各信源的中心频率已知，因此很难满足实际应用⑶.为了解决上述缺陷，有学者提出一种无须谱峰搜索的二维旋转不变子空间(two-dimensional estimating signal parameter via rotational invariance techniques,2-D ESPRIT)算法以及二维传播算子(two-dimensional propagation method,2-D PM)算法⑷.这些算法的相继问世使阵列信号的处理性能得到一定的提高,但因其在小快拍数及低信噪比情况下估计性能严重下降而无法推广到实际应用中.在众多阵列结构中，由于L型阵列具有结构简单、实施容易、估计性能佳等优点而被广泛用于工程领域.为解决二维信号角度匹配精度不高且计算复杂的问题，文献［5］提出一种基于L型阵列的无须手动配对的二维DOA估计算法，通过引入新的合成角度计算出新的导向矢量，进而获得原信号的俯仰角和方位角.尽管该方法能够自动完成角度配对,但需要多次谱峰搜索及特征值分解导致计算复杂度过高.文献［6］提出一种新的二维DOA估计方法,该算法首先将方位角和俯仰角分别估计出来，再通过阵列输出的互相关和信号功率对2个角度进行匹配，由于需要大量的采样信号使得该方法不可有效避免大量的数值计算.为降低运算量有学者提出利用阵列数据的协方差矩阵进行二维角度估计的算法［7-8］.文献［9］提出一种利用多相干信号对方位角和俯仰角进行配对的方法,通过利用协方差矩阵最小化构造的代价函数从而实现角度配对,该算法存在的最大弊端是在构造协方差矩阵的过程中可能会引入外界噪声,从而影响其估计性能.压缩感知(comprehensive sensing,CS)理论的出现为现代信号处理带来一种更高效、更精确的方法,文献［10］提出基于该理论的£-SVD算法，该算法通过对接收信号进行奇异值分解(singular value decomposition,SVD)来降低算法复杂度和对噪声的敏感性，然后利用二阶锥规划的方法求解相应的优化问题，该算法在小快拍数和低信噪比时有很好的性能,并且可以直接用于相干信号［11］.该方法摆脱了传统奈奎斯特采样定理带来超大计算量的束缚.基于此，众多学者将压缩感知理论引入到DOA估计中来，从而达到降低计算量的目的.文献［12］提出一种基于协方差矩阵联合稀疏重构的降维波达方向估计算法，该算法充分利用阵列孔径,无须预先估计目标数目，参数估计性能在低信噪比及小快拍数据长度下优势明显，但在其他方面尚有改进余地.本文在基于压缩感知理论的背景下提出一种二维L 型阵列信号的DOA估计算法.该方法在高信噪比、多快拍条件下相较于传统算法具有更高的估计精度和分辨力，且具有较低的运算量.1信号模型本文试验采用L型均匀阵列，该模型中2个子阵互相垂直，成90。

3DLO 3D long offset seismic survey三维长偏移距野外资料采集3DHR-HR 3D High resolution –high ???seismic survey 三维高分辨－高密野外资料采集。

AAC = adjusted AC;ABI inclination at the drill bitAC acoustic 声波时差ACN =adjusted CN;ADN Azimuthal neotron densityAIT* Array Induction Imager T oolA&S admistration&serviceAHC Ascendant Hierarchical ClusteringARC Induction Resistivity GR annulus pressure ingrated toolARI Azimuth resistivity imager方位电阻率成像测井仪APD Elevation of Depth Reference (LMF) above Permanent DatumAPWD apparatus whle drillingASI Array seismic imager阵列地震成像仪AVG: AverageAVO Amplitude Versus Offset(Amplitude variation with offset calibration)振幅-炮检距关系AZI: Azimuth (deg)BBC Buy Back ContractBGG: Background Gas (%)BGP 物探局BHFP bottomhole flowing pressureBHS :Borehole StatusBHT :Bottom Hole TemperatureBHTA 声波幅度BHTT 声波返回时间BL WH Blue WhiteBML below mud lineBOP Blow out preventerBOP stack 防喷器组BS Bit SizeBSW basic?? saturation water(综合含水)CAL borehole diameter 井径CAST 声波扫描成像测井仪CBI Central Bank of IranCBIL 井周声波成像CBL Cement Bond LogCC correlation coefficientCCAL common core analysis常规岩心分析CCL Casing Collar LocatorCCM Contractors Committee MeetingCDF cumulative density functionCDF Calibrated Downhole ForceCDS condensatCF core faciesCF cut fluorescenceCFA Composition Fluid AnalyzerCGR Condensate-to-gas RatioCMC Crown mounted compensator安装在天车上的升沉补尝器CMR Combinable Magnetic Resonance 可组合的核磁共振测井仪CMR-A Timur/Coates PermeabilityCMR-B Timur/Coates PermeabilityCMR_DMRP Timur/Coates PermeabilityCMR_FLOW Timur/Coates PermeabilityCMR_SMRP Timur/Coates PermeabilityCMRT Timur/Coates PermeabilityCN neutron 中子COMP AMP Compessional amplitude纵波幅度COMP ATTN Compessional attunuation纵波衰减CON induction log 感应测井COPA Companhia de Petroleo de AngolaCP conductor pipeCQG Crystal Quartz GaugeCRA Corrosion resistant AlloyCRF cumulative recovery factor(采出程度)CSI 组合地震成像仪CT cut fluorescenceCt Conductivity of un-flushedCTRY COUNTRYcvgs cavings(井壁垮塌)Cxo Conductivity of flushed zonesDAC 阵列声波DC drilling collarDCCR Daily Cost Control ReportDDR Daily Drilling ReportDEN density 密度Dev deviation井斜DF direct fluorescenceDGR Daily Geological ReportDHI direct hydrocarbon indicatorDLS dog leg severityDPH Development PhaseDPPM Density Porosity Processing ModeDSC drill string compensatorDSI* Dipole Shear Sonic Imager偶极子横波声波成像仪DT1 Delta-T Shear - Lower DipoleDT1R Delta-T Shear, Receiver Array - Lower DipoleDT2 Delta-T Shear - Upper Dipole {F13.4}DT2R Delta-T Shear, Receiver Array - Upper DipoleDT3R Delta-T Stoneley, Receiver Array - Monopole StoneleyDT4P Delta-T Compressional - Monopole P&SDTCO Delta-T CompressionalDTRP Delta-T Compressional, Receiver Array - Monopole P&SDTSM Delta-T Shear {F13.4}DTTP Delta-T Compressional, Transmitter Array - Monopole P&SDVRT 数字垂直测井Dyn Clustering Dynamic ClusteringECD equivalent circulation densityEchoes(Echoescope) Integrated Resistivity neutron density tool标准回波数据EMW equivalent mud weightEFAC Electro-faciesEOS Equation of StateE&P exploration&productionFCM finantial committee meetingFFV free fluid voluneFG Formation GasFLD FIELDFL W: mud flow rate(gpm)FMI Formation Micro-scanner Image全井眼地层微电阻率成像仪EPD Elevation of Permanent Datum above Mean Sea LevelFRF final recovery factorFPSO 浮式采油轮FV: Funnel ViscosityGCSE Generalized Caliper SelectionGDAT GeoDetic DatumGDT "gas down to"G&G Geology&geophisicsGIIP gas initially in placegpm gallon per minuteGR natural gamma ray 自然伽马GRSE Generalized Mud Resistivity SelectionGTSE Generalized Temperature SelectionHDIP 六臂倾角HGNS Highly Integrated Gamma Ray Neutron SondeHRMS High-Resolution Mechanical SondeHRLA High-Resolution Azimuthal Laterolog SondeHWDP heavy wall drillpipelb/ft pounds/footIFD 低频偶极测井仪IGIP Initial Gas In placeIld deep investigate induction log 深探测感应测井Ilm medium investigate induction log 中探测感应测井Ils shallow investigate induction log 浅探测感应测井Image DIP 图像的倾角INAGEO Instituto Nacinal de GeologiaINC: Inclination (deg)IPL 孔隙度综合测井仪IRR Initial Remuneration Rate???JOA Joint Operating AggreementJMC Joint Management CommiteeK potassium 钾kips kilo poundsKRI Kernel Representative IndexKOP Kick Off PointKTH gamma ray without uranium 无铀伽马KTIM Timur/Coates Permeability, NMR_PermeabilityHALS 高分辨率方位侧向测井仪HGS high gravity solidsHRI 高分辨率感应测井仪LATI LATITUDELATD Latitude (N=+ S=- )L/D lay downLFA Live Fluid AnalyzerLGS low gravity solids?LMF Logging Measured From (Name of Logging Elevation Reference)LMRP lower marine riser packageLNG feed gas Liquified Natural Gas ---Sour-rich-wet gas at the outlet of the receiption facilities for hand over to NIOC LNG plantL/O lay outLOC LOCATIONLONG LONGITUDELOND Longitude (E=+ W=-)LOT leak of testLPG Liquified Petroleum GasL WD logging with drillingMAC 多极阵列声波成像MATR Rock Matrix for Neutron Porosity CorrectionsMBVI 束缚流体体积MBVM 可动流体体积MC motion compensator升沉不成器MCFL Micro-Cylindrically Focused LogMD: Measured Depth (m)MDP Master Development PlanMDRT Mesured Depth Rotary T ableMDT Modular Formation Dynamic T ester模块式地层动态测试器(组件式动态地层动力学测试仪)MER Maximum Efficient Ratemkwh milk whiteMIT 多道感应测井仪MMBBL metric million barelMMS Minerals Management ServiceMOA Memorandum of AgreementMOU Memorandum of UnderstandingMPERM 核磁共振渗透率MPHI 核磁共振有效孔隙度MRGC Multi-Resolution Graph-based ClusteringMRIL Magnetic Resonance Imager Log核磁共振成像测井MRIL WD magnetic resonance imaging logging while drillingMR_KLAMBDA_PERMS Timur/Coates PermeabilityMRP Magnetic Resonance PorosityMSIP Modular Sonic Imaging Platform 模块化声波成像平台msl mean sea levelMTSA Master T echnical Services AgreementM/U make up,mount upMUSD million United States DollarMW: Mud Weight (ppg)NGR neutron gamma ray 中子伽马NGS Natural Gamma SpectrumNIDC National Iranian Drilling CompanyNIOC National Iranian Oil CompanyNMR Nuclear Magnetic ResonanceNSP 防磨补心NTG Net to GrossOBM Oil Based MudOBMI Oil Base MicroImage toolOCM operating committee meetingODP overall development plan总体开发方案OGWC Original Gas Water ContactOOC Oil On CuttingsOOIP original oil in place 原始石油地质储量OPS operationsOWC oil water contactP90 which represents the realistic low sideP50 the Median value rangeP10 representing the realistic UPSIDE for the prospect.P16H_RT ARC Phase Shift Resistivity 16 inch Spacing at 2 MHz, Environmentally Corrected, Real-TimeP16L_RT ARC Phase Shift Resistivity 16 inch Spacing at 400 KHz, Environmentally Corrected,Real-TimePCG: Pipe connection Gas (%)接钻杆气Pd dew pointPD permanent datumPDAT Permanent DatumPEMA Pesquisas Minairas de AngolaPETRANGOL Companhia de Petroleos de AngolaPEX Platform Express (The Platform Express toolstring includes either the AIT* Array Induction Imager T ool or High-Resolution Azimuthal Laterolog Sonde (HALS) as the resistivity tool. The Three-Detector Lithology Density (TLD) tool and Micro-Cylindrically Focused Log (MCFL) are housed in the High-Resolution Mechanical Sonde (HRMS) powered caliper. Above the HRMS are a compensated thermal neutron and gamma ray in the Highly Integrated Gamma Ray Neutron Sonde (HGNS) and a single-axis accelerometer.)PHIE Effective PorosityPHIE Multimin Porositypl poorly??PML ParaMagnetic测井公司PMS positioning motor systemPNC 脉冲中子俘获测井POOH pull out of holePOOW pull out of wellPOS Possibility of SuccessPPG pond per gallonProdn productionPSA production sharing agreementPSIA pound per square inch apparentPSDM prestack depth migration叠前深度偏移PSTM prestack time migrationPTB interval Permian-Triassic boundary intervalP/U pick upPV: Plastic ViscosityRADOUTR 井眼的椭圆度Rd deep investigate double lateral resistivity log 深双侧向电阻率测井R/D rig downRHOB-CORR gas corrected Neutron DensityRIH run in holeRMR risked mean reserveRMLL micro lateral resistivity log 微侧向电阻率测井ROP: Rate of penetration (m/hr)ROR Rate of ReturnROV remotely operated vehicleRPM Rotations per minuteRs shallow investigate double lateral resistivity log 浅双侧向电阻率测井R/S/C Rotary/Slide/Circulation(Directional Drilling)Rt true formation resistivity. 地层真电阻率RT real timeRT Rotary T ableR/T running timeRTE Rotary T able ElevationR/U rig upRxo flushed zone formation resistivity 冲洗带地层电阻率S3 Surface Sensor Waveform 3SBM Oil Based Mud’ (OBM) include d synthetic oils . SBT 分区(扇段)水泥胶结测井仪,SC Service ContractSCAL special core analysis特殊岩心分析SDC Sonangol Data CentreSG specific gravityShear AMP Shear Amplitude横波幅度Shear ATTN ShearAttenuation横波衰减SHT Surface Hole TemperatureSICP Shut in casing presureSIDP Shut in drill pipe presure关井钻杆压力SOM Self-Organization Map ClusteringSONANGOL Sociodade Nacional Combustiveis AngolaSP spontaneous potential 自然电位SPP: Stand pipe Pressure(psi)SPD South Pars DrillingSRVC SERVICE COMPANYst stonelyST side trackStar Imager 微电阻率扫描成像STAR-II 声电组合成像测井STD stand 立柱STDS standsSTAT STATESTT 送入工具suc sucrosicSWAL 横波声波测井仪SWD Seismic While DrillingSWF shallow water flowS wb saturation of bond waterS wm saturation of movable waterS wirr saturation of irreducible waterT2 Dist T2分布数据TBRT thin bed resistivity tool薄层电阻率TCL Thorium Clay – GlobalTCM technical committee meetingTCRT 过套管电阻率测井仪TD: T otal Depth (m)TDEP HNG Coin Msg River DepthTENS Cable TensionTG Trip GasTG: T otal Gas (%)TH thorium 钍THP tubing head pressureTIM 公司TJVA T emporary Joint Venture Arrangement for Operating Purposes TLC T ough Logging Conditions pipe-conveyed systemTLD Three-Detector Lithology DensityTOC top of cementTP test pressureTPOR 总孔隙度TRQ: Rotary T orque KIb*ft)TVD: True Vertical Depth (m)TVDSS True Vertical Depth SubseaTVT true vertical thicknessT/W together withU uranium 铀U Photoelectric absorption cross-sectionUAE United Arab EmiratesUGC underground contourUMR unrisked mean reserveUSGS U.S. Geological SurveyUSI ultrasonic imager超声波成像仪UWI :UNIQUE WELL IDVDL Variable Density LogVPC vertical proportion curveVSI Versatile Seismic ImagerVSP-WD Vertical Seismic Profile While DrillingWBM Water Based MudWEC Well Evaluation ConferenceWH well headWHP well head pressureWOB: Weight on Bit(KIbs)WTHP well tubing head pressureX/O cross over 大小头, 转换接头XPT-A Xpress Perssure T ool-AXRD X-Ray diffractionYP: Yield PointΦR calculated porosity from resistivity with Archie formulaMRXabbr. <测井> Magnetic Resonance eXpert tool 核磁共振专家测井仪XPTabbr. <测井> Xpress Pressure ToolGPITabbr. <测井> General Purpose Inclinometer ToolFMIabbr. <测井> Fullbore Formation Micro Imager 全井眼地层微电阻率成像测井USIPabbr. <测井> UltraSonic Imaging PlatformHNGSabbr. <测井> Hostile Environment Natural Gamma Ray SondeVSIabbr. <测井> Versatile Seismic ImagerCBLabbr. <测井> Cement Bond Log 水泥胶结测井APSabbr. <测井> Accelerator Porosity SondeTLDabbr. <测井> Three-Detector Lithology DensityCCLabbr. <测井> Casing Collar Log 套管接箍测井.slb./modules/mnemonics/ChannelItem.aspx?codeLow fluid loss and gel strengths of pad mud spotted in the hole just prior to running casing.Riser pipe 隔水管Circulation circuit 循环管道5700系列的测井项目及曲线名称Retrievable tools, sometimes known as Slim Tools,Collar-mounted tools, also known as Fat Tools,a triple rig (able to trip 3 joints of pipe,*西方阿特拉斯的ECLIPS-5700成像测井系统*主要包括：声电组合成像测井（STAR-II）、核磁共振测井（MRIL-C）、多极子声波成像仪（M AC）和扇段水泥胶结评价（SBT）等*哈里伯顿的EXCELL-2000成像测井系统*主要包括：井周声波扫描成像测井（CAST），微电阻率扫描成像（EMI）*斯伦贝的MAXIS-500成像测井系统*主要包括：微电阻率扫描成像（FMI）、阵列感应（AIT）、方位电阻率（ARI）、偶极横波（DSI）、核磁共振（CMR）等成像测井*(Compressional)Slowness (time)(Δtc)*Shear Slowness (Δts)HSGR .GAPI :HNGS Standard Gamma Ray {F13.4}HCGR .GAPI :HNGS Computed Gamma Ray {F13.4}HFK .V/V :HNGS Formation Potassium Concentration {F13.4}HTHO .PPM :HNGS Formation Thorium Concentration {F13.4} HTPR . :HNGS Thorium/Potassium Ratio {F13.4} HTUR . :HNGS Thorium/Uranium Ratio {F13.4} HUPR . :HNGS Uranium/Potassium Ratio {F13.4} HURA .PPM :HNGS Formation Uranium Concentration {F13.4} GR_EDTC.GAPI :EDTC Gamma Ray {F13.4}CHR1 :Label Peak Coherence, Receiver Array - Lower Dipole {F13.4}CHR2 . :Label Peak Coherence, Receiver Array - Upper Dipole {F13.4} CHR3 . :Label Peak Coherence, Receiver Array - Monopole Stoneley {F13.4} CHRP . :Label Peak Coherence, Receiver Array, Compressional - Monopole P&S {F13.4} CHTP . :Label Peak Coherence, Transmitter Array, Compressional - Monopole P&S {F13.4} ITT .S :Integrated Transit Time {F13.4}HAZI .DEG :Hole Azimuth {F13.4}P1AZ .DEG :Pad 1 Azimuth {F13.4}RB .DEG :Relative Bearing {F13.4}SDEV .DEG :Sonde Deviation {F13.4}P1AZ_OBMT_2.DEG :Memorized Pad 1 Azimuth {F13.4}RB_OBMT_2.DEG :Memorized Rotated Relative Bearing {F13.4}PP_OBMT_2. :Pad Pressure {F13.4}DEVI .DEG :Hole Deviation {F13.4}RB_OBMT.DEG :Memorized Relative Bearing {F13.4}P1AZ_OBMT.DEG :Memorized Pad 1 Azimuth {F13.4}PP_OBMT. :Pad Pressure {F13.4}HAZI .DEG :Hole Azimuth {F13.4}P1AZ .DEG :Pad 1 Azimuth {F13.4}RB .DEG :Relative Bearing {F13.4} SDEV .DEG :Sonde Deviation {F13.4}OZA_2.OHMS :Injection Impedance Pad A {F13.4} OZB_2.OHMS :Injection Impedance Pad B {F13.4} OZC_2.OHMS :Injection Impedance Pad C {F13.4}OZD_2.OHMS :Injection Impedance Pad D {F13.4} OBRA_2[0].OHMM :Buttons resistivities, Pad A {AF13.4} OBRB_2[0].OHMM :Buttons resistivities Pad B {AF13.4}OBRC_2[0].OHMM :Buttons resistivities Pad C {AF13.4}OBRD_2[0].OHMM :Buttons resistivities Pad D {AF13.4}C1_OBMT_2.IN :OBMT2 Caliper 1 {F13.4}C2_OBMT_2.IN :OBMT2 Caliper 2 {F13.4} OZA .OHMS :Injection Impedance Pad A {F13.4}OZB .OHMS :Injection Impedance Pad B {F13.4}OZC .OHMS :Injection Impedance Pad C {F13.4} OZD .OHMS :Injection Impedance Pad D {F13.4}OBRA[0].OHMM :Buttons resistivities, Pad A {AF13.4}OBRB[0].OHMM :Buttons resistivities Pad B {AF13.4}OBRC[0].OHMM :Buttons resistivities Pad C {AF13.4}OBRD[0].OHMM :Buttons resistivities Pad D {AF13.4}C1_OBMT.IN :OBMT Caliper 1 {F13.4}C2_OBMT.IN :OBMT Caliper 2 {F13.4}。