《高等路面设计理论》课程授课教案（讲义）System Identification Method for Backcalculating Pavement Layer Properties

TRANSPORTATIONRESEARCHRECORD1384System Identification Method forBackcalculating Pavement Layer PropertiesFUMING WANG AND ROBERT L. LYTTONIn recent years pavement structural evaluation has relied increas-Avariety ofdifferent methods and computerprograms haveingly on determining material properties by nondestructive de-been developedfor backcalculation of layer moduli fromFWDflection testing and backcalculation procedures. The techniquetest results. Examples include the MODCOMP program de-used to achieve a convergence of the measured and predictedveloped by Irwin (2), the "sDEFseries of programsdeflection basins plays an important role in all backcalculationdescribed by Bush (3), and the MODULUS program devel-approaches.Aniterativemethodbasedonthesystemidentifi-oped by Uzan et al. (4). The MODCOMP program uses thecation (SID) scheme is developed, and the SID program is usedCHEVRON program for deflection calculations and is no-in conjunction witha multilayer elastic model (BISAR program)to backcalculate pavement layer properties.Numerical examplestable for its extensive controls on the seed moduli and theindicate that (a)the moduli backcalculated by the suggested SIDrange of acceptable moduli. The two programs reported bymethod comparewell with the results fromMODULUS,whichBush includetheCHEVDEFandBISDEFprograms,inwhichis a data base backcalculation program, and other developedthe deflection calculations are performed by the CHEVRONiterative backcalculation programs; (b)the SID is a quickly con-(5)andBISAR()programs,respectively,and thegradientverging procedure,and the influence of seed values,for a rela-search technique is used. MODULUS is a data base back-tively wide range, on the derived results is negligible; and (c) itis able tobackcalculate pavement layerthicknesses in additioncalculation program that departs from the usual microcom-to layermoduli.puter program pattern. Before the actual backcalculation pro-cess,MODULUScomputesa series of normalizeddeflectionbasinsusing theBISARprogram with layer moduli that coverNondestructive testing (NDT) has become an integral part ofthe range of anticipated values in the field. The deflectionpavementstructuralevaluationinrecentyears.Ofmanystatic.basins are stored ina data baseforsubsequentcomparisonvibration, impulse, and vehicular NDT devices, the fallingwithmeasureddeflectionbasins.Thepattern searchalgorithmweight deflectometer (FWD) has been most widely used fordeveloped by Hooke and Jeeves (7) and the three-point La-pavement evaluation (1). By dropping a mass from a pregrange interpolation technique (8) are used to find the layerdeterminedheight onto a baseplate resting on the pavementmoduli that minimize the error between measured and com-surface,the FWD can provide variable and large impulseputed basins.By replacing the direct computation of deflec-loadings to the pavement, which to some degree simulatetions with the interpolation scheme,MODULUS is distinctlyactual truck traffic.Pavement deflection is measured throughfaster than other iterative backcalculation programs for pro-a series of velocity transducers at various distances from theduction cases in which a large number of deflection basins inbase plate, and the data can be used to backcalculate the inthesamepavementgeometryaretobeevaluated.Whenpavesitu pavement properties, such as layer moduli.This infor.ment configuration changes, however, the time-consumingmation can in turn be used in pavement structural analysis totask of generating the deflection data base must be repeated.determine the bearing capacity, estimate the remaining life,Most current backcalculation procedures seek only to deand calculate an overlay requirement over a desired designtermine layer moduli and require the thickness of each pave-life.ment layer to be specified. The subgrade is assumed to beinfinitely thick, or a rigid layeris placed atan arbitrarydepth.As reported by other researchers,pavement deflections areFWDDATAREDUCTIONANDsensitive to layer thicknesses.Even modest errors in assumedBACKCALCULATIONMETHODSlayer thicknesses can lead to large errors in backcalculatedlayer moduli (9),and the existence of a rigid layer or bedrockThe analysis of the FWD test data is an inverse process.underlying the subgrade has a profound effect on the analysisInstead of predicting the pavement response,the deflectionof deflection data (10).The subgrade modulus may be sigis measured and the pavement properties are backcalculatednificantly overpredicted if a semi-infinite subgrade is falselyassumed when actual bedrock exists at a shallow depth, or itmay be underpredicted if a shallow rigid layer is arbitrarilyF.Wang,Materials,Pavements and ConstructionDivision,Texasintroduced when deepbedrock exists.Transportation Institute,TexasA&MUnivrsity,Collee StatinPavement thicknesses can be accurately measured throughTex.77843.Current affiliation:CAECenter,Zhengzhou Institute ofcoring, boring, ground-penetrating radar, or seismic testsTechogZhngzhouHenan0002ople'sRepublicoChHowever, pavements cover such a large area that it is im-R.L.Lyton,CivilEngineringDepartment,TexasA&MUniversitySystem, College Station, Tex.77843practical to use these techniques to determine the layer thick-

TRANSPORTATIONRESEARCHRECORD1384Wa2F.also be applied to determine properties of pavement struc-nesses at every point tested with deflection devices. Thustures in addition to layer moduli,even including the thicknessadvanced backcaiculationprocedures are clearlyneeded toof the layer as one of the unknown parameters.r:determine the layer thickness, especially the subgrade thick-ness, as well as moduli from the measured deflection infor-mation. In this paper an iterative procedure based on theIKsystem identification scheme is presented. It may be consid-ParameterAdjustmentAlgorithmered as an alternative approach to the subject.FThe system identification method requires the accurately mea-sured output data of the unknown system, a suitable modelF.SYSTEMIDENTIFICATIONMETHODto represent the behavior of the system, and an efficient pa-rameter adjustment algorithm that converges accurately andGeneral Procedurerapidly. If the data and the model are reliable, the success ofcsystem identification studies directly relies on the efficiencyThe objective of the system identification process is to esti-of the parameter adjustment algorithm.amate the system characteristics using only input and outputAn algorithm can be developed for adjusting model param-data from the system to be identified (11).The simplest andeters on the basis of the Taylor series expansion.Let the math-intellectually most satisfying method for representing the be-reematical model of some process be defined by n parameters:havior ofa physical process is to model it witha mathematical(1)representation. The model/process is identified when the errorrf = f(pi, P2,.-.,;x, I)between the model and the real process is minimized in some0sense; otherwise, the model must be modified until the desiredwherex and tare independent spatial and temporal variables.level of agreement is achieved.If anyfunction f(p.,P2,...,Pa; xx, t)isexpanded usingaFThere are three general strategies for error minimizationTaylor series and only first-order terms are kept, it can bein system identification procedures: forward approach, in-shown thatverse approach, and generalized approach. In the forwardt(2)approach, the model and the system to be identified are givenf (p + Ap) = fe(p) + VftApththe same known input, and the output error between the twosiis minimized. In the inverse approach, the outputs of thewheretheparametershaveall been collected intoavectorgmodel and the system are identical, and their input error isaminimized.The generalized approach is a combination of thep = [pu. P2, .+,p.JrFforward and inverse approaches.In all cases,theminimizationtlof the error between the model and the real process can beIf we equatef(p+Ap)with the actual outputof thesystemconducted with a model parameter adjustment.and f:(p)with the output of the model for the most recent setPThe forward approach is not as complicated as the inverseof parameters P,the error between thetwo outputs becomestor generalized approaches because,byusing a forward model,iit is easier to compute the output and to generate the param-ek=fr(p+Ap)-ft(p)neter adjustment algorithm.A system identification schemec= Vfk Apusing the forward approach and parameter adjustment al-tgorithm is shown in Figure 1.afk Api +afk Apa + :afkApacThe procedure shown in Figure 1 is exactly analogous to(3)rap,dp2apawhatisbeingdoneinbackcalculatingthemoduliofpavementsE(12,13).However,the systemidentification procedurecaniNote that e, represents the difference between the actual sys-Etem output and the model output when the independent vari-ables take on values Xk and g(1If the error is evaluated at m values (m n) of the inde-Apendent variables, m equations may be written:wuUNKNOWNX-SYSTEMafiafiafi: Api +feAp2+Apner=:apapap.!afaafafeAPeAp, +Ap2 +(4)MODELe2mYmapiap2apa.afmafu+PARAMETER ADJUSTMENTAp, +Ap2+e.mapiap2ALGORITHMEquation 4 can be conveniently nondimensionalized by di-System identification (forwardFIGURE1viding both sides by fu. Furthermore, if we define matrices r,approach)

1384Wang and Lytton3F,and α asruc-As soon as α is obtained, a new set of parameters is de-nesstermined asr[n2...m]pk+1 = P*(1 + α)(7)Sk=1,2,...,mfThe iteration process is continued until the desired conver-F= [Fa]gence is reached. In this paper the convergence criterion islea-set to 0.5 percent for α (i.e., the iterative procedure must beidelafu.prepeated until all parameter changes are not more than 0.5Fu=k=1,2,...,mi=1,2,.pa-.npercent).ap:fkandThe sensitivity matrix Fin Equation 6 is generated using asofmultilayer elastic model (BISAR program).The derivativesα= [αiα2.. .α,]Tncywheemrent the pavmApapi=1,2,..*,nα, =im-deflections at the sensor locations of FWD and p (i 1, 2,p:ath-.,n)thepavement layerpropertyparameters,arecom-'s:respectively,Equation 4may be rewritten asputed as the forward-derived differences.Thus the sensitivitymatrix F can be generated by n + I runs of BISAR.(1)r= Fα(5)The sensitivity matrix may be used for more than one it-eration.If theparameters have been changed"much,how-les.orever,it has to be regenerated because it only takes accountgaof the first-order Taylor series and the problem is highly non-beFTr=FTFα(6)linear, which means that the sensitivity values depend on theparameter values. Otherwise the iteration procedure mightThevector r is completelydetermined from the outputs ofnot converge, or,more often, it may converge very slowly.the model and the real system. The matrix Fis usually called(2)In this study the sensitivity matrix is updated when one of thethe sensitivity matrix, because its element Fa,reflects the sen-following conditions is encountered:sitivity of the output feto the parameter pi, and it can beorgenerated numerically if the analytical solution is not avail-1. One or more parameters have been increased by moreable.The technique used for generating the sensitivity matrixthan 100 percent during the past iterations;F and when it should be updated will be discussed later in2.One or more parameters have been decreasedmore thanthis sectionem50 percent during the past iterations; orThe unknown vector α reflects the relative changes of theset3.The sensitivity matrix has been used for three iterations,parameters.If the sensitivity matrix F or the system of equa-esbut the 0.5 convergence criterion has not been achieved.tions is wellbehaved, it can be obtained by using ageneralizedinverse procedure to solve Equation 5 (14,15).However, theremight be column degeneracies in the sensitivity matrix F.ThisBACKCALCULATIONOFLAYERMODULIconditionmaybe encountered when two ormoreparametershave similar effects, or any parameter has a negligible effect,On the basis of the procedure described above, the SID micro-on the behavior of the model f. In these cases Equation 5(3)may be ill conditioned from the mathematical point of view,computerprogramhas been developed.In thissection thepro-and the singularvalue decomposition (SVD)technique (16)gram is evaluated by comparing the backcalculated moduli withys-is one of the alternative approaches to give a stable solution.the results from MODULUS and other developed programs.tri-SVD diagnoses the sensitivity matrix by calculating its con-dition number, which is defined as the ratio of the largest ofde-the singular values to the smallest of the singular values. FisComparisonwithMODULUSsingular if its condition number is infinite, but the more com-mon situation is that some of the singular values are veryAn actual deflection basin is analyzed using the SID back-small but nonzero, thus Fis ill conditioned. Then SVD givescalculation program,and the results are compared with thosea solution by zeroing the small singular values,which corre-fromMODULUS.Deflectiondatawereobtainedusingthesponds to deleting some linear combinations of the set ofFWD (Dynatest Model 8000)on Section8at theTexasA&Mequations.The SVDsolution is very often better (in the sense(4)Research Annex (18). Section 8 consisted of 12.7-cm (5-in.)ofthe residual|Fx-r being smaller) than LU decompositionAC,30.48-cm(12-in.)crushed limestonebase,and30.48-cmsolution or Gaussian elimination solution.However, the SVD(12-in.)cement-stabilized subbase (very stiff layer) over clayuserhas to exercise some discretion in deciding at what thresh-subgrade.TheFWD geophones were located at 0, 30.48,old tozerothe smallsingularvalues.Inthisstudytheiteration60.96,91.44,121.92,152.4,and182.88cm(0,12,24,36,48,method developed by Han (17) is used to solve Equation 6.60, and72 in.)from the center of the load plate,whichhadHan's method not only gives the exact solution if Equation 6a radius of15cm (5.91 in.).is wellposed but alsogivesastablesolutionifEquation6isdi-By using the BISAR program to generate the deflectionillposed without deleting any equations.rdata base and assuming a 635-cm (250-in.)depth from pave-

TRANSPORTATIONRESEARCHRECORD1384Wment surface to bedrock,themodulibackcalculatedbyMOD-1.3ULUS for AC layer, base, subbase, and subgrade are E,1,2140740kg/cm2(2,000ksi),E,=3519kg/cm2(50ksi),E,=265 366 kg/cm2 (3,771 ksi), and E, = 915 kg/cm2 (13 ksi)1.1The SID backcalculation program is used to reduce thesame data for Section 8. As do other iterative approaches,the SID requires seed moduli values. Three sets of seed mod-0.9uli are selected to evaluate the effects of seed parameters onderived results.0.8First, the seed modulus values are assumed to be E, 0.7105555kg/cm2(1500ksi),E,=4222kg/cm2(60ksi),E,m140740 kg/cm2(2,000 ksi),and E, =704kg/cm2(10ksi),0.6which are relatively close to the results given by MODULUS.The 0.5 convergence criterion for α is reached after five it-0.523450erations, and the sensitivity matrix is regenerated after threeNO.OFITERATIONSiterations.Next,the seed moduli are changed toE,70370 kg/cm?(1,000ksi),Ez=7037kg/cm2(100ksi),E,=70370kg/cm2(1,000 ksi), and E, 2111 kg/cm2 (30 ksi). For this set ofE1E2—E3-0-E4seed moduli,onlythree iterations are needed,but thesen-sitivity matrix is regenerated after one iteration.FIGURE2Converging processLast, to verify the robustness of the SID approach, the seed(first set of seed moduli).moduliareassumedtobesignificantlydifferentfromthepre-vious values:E,=351851kg/cm2 (5,000ksi),Ez=35185kg/cm2(500ksi),E=351851kg/cm2(5,000ksi),andE,=2.53519 kg/cm2 (50 ksi).With these moduli the predicted deflections are approxi-matelyfour times less than the FWD data, which indicates2that very poor seed parameters have been entered. In prac-tice,anotherset of startingvaluesshouldbe selectedinthiscase. The SID procedure still converges, however. The sen-1.5sitivity matrix is updated four times, and altogether eightiterationsareperformedThe results for the preceding three cases are summarizedin Table 1,and the converging processfor each case is showninFigures2,3,and4,respectively.Theresultsbackcalculated0.5bytheSIDprogram agreevery well with those byMODULUS,0+1203TABLE1Backcalculated Moduli for Different SeedNO.OFITERATIONSValuesEsE2E4MODULIE,-E2- E3-E46--E1(kg/cm2)(3519′)(915)(140740′)(265366")FIGURE3Converging process(second set of seed moduli).4222704"Seed"1055551407403504262128906Backcalculated150451and the seed values have a negligible influence on the con-verged results, although they certainly affect the required"Seed"703707037703702111number of iterations.9061504513502262269BackcalculatedComparison with Other Iterative Backcalculation3519"Seed"35185135185351851Approaches9061514373478265507BackcalculatedThe SID program is compared with five other iterative back-calculation programs.Pavement data and deflection test data1kg/cm2-14.21psifor the comparison are obtained from a real pavement (19)The backcalculated moduli from the various programs are*moduli backcalculated by NODULUs (18)

1384Wang and Lyion111Backcalculation of Hypothetical Pavement Layer:10Moduli and Thicknesses9fThe SID program is evaluated by comparing the backcalcu-8lated layer moduli and thicknesses with hypothesized theo-7retical values. The comparison is done by assuming three6pavement sections with different thicknesses and moduli.Sur-7face deflections of the assumed pavement section arecalcu-lated using theBISARprogram and areused tobackcalculate4dthe layer thicknesses as well as the layer moduli.3The theoretical values and the backcalculated results forthe three pavement sections are presented in Table 3. TheSID program always converges toward the correct modulusand thickness for all layers.23456709NO.OFITERATIONSApplication Using Actual Deflection DataE1-E2E3-0E4The SIDprogram is applied to determine the subgrade thick-ness of Section 8 from the FWD data given in Table 1.Sinceincreasingthe number of unknownparameters requiresmoreFIGURE 4Converging processdata points to ensure the system overdeterminism, and be-(third set of seed moduli).cause of the likelihood of large measurement errors in realdata, backcalculating more than four parameters is not rec-ommended withoutperforming the dynamic analysis of FWDTABLE2Summary of Backcalculated Moduli (kg/cm)data.Therefore the process is divided into two steps:AC SurfaceSubgradeTest siteProgramAggregate Base1.The four layer moduli are backcalculated by assumingBISDEF136521776809the subgrade to be of infinite thickness. The results are com-788BOUSDEF114701809pared with those backcalculated previously by introducing a123711738851CHEVDEF635-cm(250-in.)depthfrom surface tobedrock.The subbase1661823and subgrade moduli are much more sensitive to the subgradeELSDEF14074thickness than the AC and base moduli. Thus the backcal-2350739MODCOMP211456809SID(BISAR)154741527TABLE3Summary of Backcalculated Layer Moduli and1084739BISDEF12223Layer Thicknesses1070697BOUSDEF11097MODULI (kg/cm*)THICKNESSES (cm)1168739CHEVDEF10605EE2E3hh2ha1070732ELSDEF1224492541907654MOOCOMP2"Seed"3518525.425.421111407635.0704114981217SID(BISAR)42222175930.530.5Backcalculated2815762.042222175930.530.5Theoretical2815762.01kg/cm14.21psi"Seed"3518525.42111140725.4635.0on-given in Table 2.The results from SID are close to those fromBackcalculated7037038.130.542212815889.0redthe other programs.Theoretical703704222281538.130.5889.0BACKCALCULATION OF LAYER MODULI AND"Seed"351852111140725.425.4LAYER THICKNESSES635.0Backcalculated70370105570415.238.1457.2By considering the layerthicknesses as unknown parameters,Theoretical70370105670438.115.2457.2ick-theSIDprogram canbe usedtobackcalculate thelayerthick-latanesses as well as layer moduli.This ability is illustrated by(9).1 cm = 0.3937 in.using hypothetical three-layer pavement structures and thearereal FWD data for Section 8.1kg/cm2。14.21psi

6TRANSPORTATIONRESEARCHRECORD1384WTABLE 4 Summary of Backcalculated Results for Section 8compared with those from other developed programs,andgood agreement is observed. The ability to backcalculateSubgrade Thickness (cm)ProgramBackcalculated Moduli (kg/cn)10-pavementlayer thicknesses is illustrated by using hypotheticalE,E2E.Eapavement sections and real FWD data.140740The backcalculated results for Section 8 indicate clearly thatInfinite (Assumed)MODULUS4081918331970the subgrade thickness should be carefully determined for the1SID(BISAR)14573639411010511900pavement under analysis.The backcalculated subgrade mod1561 (Assumed)MODULUS1407403519265365915ulus assuming infinite thickness may be twice that obtained3519from an analysis in which the depth to bedrock is arbitrarilySID(BISAR)150451262128.91513selected, such as 610 cm (20 ft).The SID program promises1082 (Backcalculated)SID(BISAR)14573639411518581407togivemore reliableresults by considering thesubgradethickness as one of the unknown parameters to be identified.141cm=0.3937inThe SID method is a very powerful and versatile analysistool and can be applied to a variety of backcalculations. As1 kg/cm2-14.21 psihas been successfully accomplished at Texas A&M Univer15sity,the parameters ofthe creep compliance of the AC layerculated AC and base moduli are fixed as known parameters,can be backcalculated from FWD data using the SID programand the derived subbase and subgrade moduli are taken asand the dynamic multilayer viscoelastic model UTFWIBM16the seed values for the next backcalculation step.(20)or SCALPOT(21),and the fracturepropertiesof asphalt2.The subbase and subgrade moduli together with theconcrete materials can also be backcalculated from fatiguesubgrade thickness are backcalculated.The seed subgradetest data using the SID program and the microcrack modelthickness is selected as 762 cm (300 in.),anda 1082-cm(426-MICROCR(22)in.) thickness is derived.The backcalculated results for Section 8, based on threeACKNOWLEDGMENTSdifferent subgrade thicknesses, are summarized in Table 4.The subgrade thickness significantly affects thebackcalculatedThe research described herein was sponsored in part by thesubbase and subgrade moduli.The subgrade modulus assum-Strategic HighwayResearchProgramandbytheExcellenting an infinite thickness is approximatelytwice the value back-Young Teacher Foundation of the State Education Commitcalculated assuming a subgrade thickness of 561 cm(221 in.)tee of China. The authors are pleased to acknowledge theirThebackcalculated subgrademoduluswiththebackcalculatedsupport.subgradethicknessof1082cm(426in.)fromtheSIDprogramis in between.This example clearly illustrates one substantial problem inmost current backcalculation procedures.If the subgrade isREFERENCESassumed to be infinitely thick, or a depth to bedrock is arbitrarilyselected,thebackcalculated subgrademodulusfrom1.R.E.Smith and R.L.Lytton.Synthesis Study of Nondestructivethese two assumptions may be quite different. Since the stiff-Testing Devices for Use in Overlay Thickness Design of FlexiblePavements.FHWA/RD-83/097.FHWA,U.S.Departmentofness of the supporting subgrade is a basic parameter in pave-Transportation,1984.ment structural analysis,over-or underprediction of subgrade2.L.H. Irwin.User's Guide to MODCOMPS, Version 2.I.Reportmodulus may lead to under-or overconservative resuits in83-8. 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