Pacific Earthquake Engineering Research Center

Transcription

1 Pacfc Earthquae Engneerng Research Center Uncertanty Specfcaton and Propagaton for Loss Estmaton Usng FOSM Method Jac W. Baer Stanford Unversty and C. Alln Cornell Stanford Unversty PEER 003/07 SEPT. 003

2 Uncertanty Specfcaton and Propagaton for Loss Estmaton Usng FOSM Methods Jac W. Baer and C. Alln Cornell Department of Cvl and Envronmental Engneerng Stanford Unversty PEER Report 003/07 Pacfc Earthquae Engneerng Research Center College of Engneerng Unversty of Calforna, Bereley September 003

3 ABSTRACT The estmaton of repar costs n future earthquaes s one component of loss estmaton currently beng developed for use n performance-based engneerng. An mportant component of ths calculaton s the estmaton of total uncertanty n the result, as a result of the many sources of uncertanty n the calculaton. Monte Carlo smulaton s a smple approach for estmaton of ths uncertanty, but t s computatonally expensve. The procedure proposed n ths report uses the frst-order second-moment (FOSM) method to collapse several condtonal random varables nto a sngle condtonal random varable, total repar cost gven (where s a measure of the ground moton ntensty). Numercal ntegraton s then used to ncorporate the ground moton hazard. The ground moton hazard s treated accurately because t s the domnant contrbutor to total uncertanty. Quanttes that can be computed are expected annual loss, varance n annual loss, and the mean annual rate (or probablty) of exceedng a gven loss. A general dscusson of element-based loss estmaton s presented, and a framewor for loss estmaton s outlned. The method wors wthn the framewor proposed by the Pacfc Earthquae Engneerng Research (PEER) Center The report maes suggestons for the representaton of correlaton among the random varables, such as repar costs, where data and nformaton are very lmted. Gudelnes for the estmaton of uncertanty n pea nterstory drft gven are also presented. Ths ncludes usng structural analyss to estmate aleatory uncertanty, and correlatons for an example structure. Several studes attemptng to characterze epstemc uncertanty are referenced as an ad. A smple numercal calculaton s presented to llustrate the mechancs of the procedure. The results of the example are also used to llustrate the effect of uncertanty on the rate of exceedng a gven total cost. Ths llustrates that uncertanty n total repar cost gven may or may not have a sgnfcant effect on the annual rate of exceedng a gven cost.

4 ACKNOWLEDGMENTS Ths wor was supported n part by the Pacfc Earthquae Engneerng Research Center through the Earthquae Engneerng Research Centers Program of the Natonal Scence Foundaton under award number EEC v

5 CONTENTS ABSTRACT... ACKNOWLEDGMENTS... v TABLE OF CONTENTS... v LIST OF FIGURES... v LIST OF TABLES... x INTRODUCTION... THE MODEL FRAMEWORK Total Repar Cost Element Damage Values Element Damage Measures Buldng Response Ste Sesmc Hazard Assumptons PROCEDURE Specfy ln EDP Specfy DM ln EDP and ln DVE DM, and collapse to ln DVE ln EDP Calculate ln DVE Swtch to the non-log form DVE Compute moments of TC Accountng for Collapse Cases Other Generalzatons of the Model Incorporate the ground moton hazard to determne E[TC] and Var[TC] Rate of Exceedance of a Gven TC Incorporaton of Epstemc Uncertanty Epstemc Uncertanty n TC Epstemc Uncertanty n the Ground Moton Hazard Expectaton and Varance of E[TC], Accountng for Epstemc Uncertanty Expectaton and Varance of the Annual Frequency of Collapse, Accountng for Epstemc Uncertanty Revsed Calculaton for E[TC], Accountng For Costs Due to Collapses Rate of Exceedance of a Gven TC, Accountng for Epstemc Uncertanty v

6 4 SPLE NUMERICAL EXAMPLE Specfy ln EDP Specfy Collapsed Functon ln DVE ln EDP Calculate ln DVE Swtch to the non-log form DVE Compute moments of TC Incorporate the ste hazard THE ROLE OF VARIANCE IN TC GIVEN GUIDELINES FOR UNCERTAINTY ESTATION Potental Sources of Structural Uncertanty Aleatory Uncertanty n EDP for The Van Nuys Testbed Expected value of EDPs Aleatory Uncertanty Correlatons Probablty of Collapse References for Estmaton of Epstemc Uncertanty n EDP CONCLUSION APPENDIX A: DERIVATION OF CORRELATION USING THE EQUI- CORRELATED MODEL APPENDIX B: DERIVATION OF MOMENTS OF CONDITIONAL RANDOM VARIABLES APPENDIX C: DERIVATION OF THE ANALYTICAL SOLUTION FOR λ (z) TC APPENDIX D: USE OF THE BOOTSTRAP TO COMPUTE SAMPLE VARIANCE AND CORRELATION APPENDIX E: ESTATING THE ROLE OF SUPPLEMENTARY VARIABLES IN UNCERTAINTY APPENDIX F: ESTATING THE VARIANCE AND COVARIANCE STRUCTURE OF THE GROUND MOTION HAZARD REFERENCES v

7 LIST OF FIGURES Fg. 3. Estmated medan of IDR ( exp( ln IDR ) ), condtoned on no collapse and Sa...8 Fg. 3. Collapsng out DM...9 Fg. 3.3 Element Fraglty Functons...0 Fg. 3.4 Element Repar Costs...0 Fg. 3.5 Collapsed dstrbuton DVE EDP... Fg. 3.6 Collapsng out EDP...3 Fg. 3.7 Sample Hazard Curve for the Van Nuys Ste...5 Fg. 4. E[TC ], plus/mnus one sgma, wth uncertanty only n EDP...44 Fg. 4. E[TC ], plus/mnus one sgma, wth uncertanty only n DVE EDP...44 Fg. 4.3 E[TC ], plus/mnus one sgma, wth all uncertantes...45 Fg. 4.4 Var [lntc ], denoted β TC...45 Fg. 4.5 Expected TC, for example soluton and.4.8 ft...47 Fg. 4.6 Fg. 4.7 Fg. 5. Fg. 5. Van Nuys ground moton hazard, and analytcal ft...47 λ TC (z): Comparson of numercal ntegraton and analytcal soluton...48 Effect of uncertanty, β TC, on frequency of exceedance of Total Cost, calculated usng numercal ntegraton...50 Effect of uncertanty examned usng the analytcal soluton for λ TC (z), when β =...5 Fg. 6. Estmated medan of IDR ( exp( ln IDR ) ), condtoned on no collapse and S a...55 Fg. 6. Estmated medan of ACC ( exp( ln ACC ) ), condtoned on no collapse and S a...56 Fg. 6.3 Estmated standard devaton n lnidr, condtoned on no collapse and S a...57 Fg. 6.4 Estmated standard devaton n lnacc, condtoned on no collapse and S a...57 Fg. 6.5 Fg. 6.6 Correlaton coeffcent vs. number of stores of separaton (for nterstory drft ratos) at 50/50 hazard level...59 Probablty of collapse, condtoned on...60 v

8 Fg. F. Example of aggregaton of nne hazard curves to obtan three curves (Fgure 6-4 taen from Pacfc Gas & Electrc, 988)...85

9 LIST OF TABLES Table 7. Sample Correlaton Coeffcents for lnedps...58 Table 7. Examples of β R and β U for NPP Structures, from Kennedy and Ravndra (984)...6 Table 7.3 Uncertanty Coeffcent β UD for Evaluaton of Steel Moment Frame Buldngs (FEMA, 000a)...6 Table 7.4 Example Uncertanty Coeffcents β U for Collapse Capacty n Dfferent Damage States (Bazzurro et al. 00)...63 x

10 Introducton The estmaton of annual losses n a buldng due to earthquae damage s a quantty of nterest to decson maers, and s a current topc of study n performance-based earthquae engneerng. Among the quanttes to be determned are the uncertanty n the annual losses and the contrbuton of each source of uncertanty to the total uncertanty n annual losses. Current efforts n ths feld consder the ground moton hazard, buldng response, damage to buldng elements, element repar costs, and total repar cost as ndvdual random varables, and then propagate uncertanty through each step to fnd a fnal result. One opton for propagatng ths uncertanty s through Monte Carlo smulaton (e.g. Porter and Kremdjan 00). Although straghtforward, t can be very expensve computatonally, especally when multple runs are requred to calculate senstvtes and/or when low probabltes are needed. In ths paper, we present an alternatve soluton employng a hybrd of smple numercal ntegraton supplemented by the frst-order second-moment, or FOSM, method (e.g., Melchers 999) appled to the hgher dmensonal response and loss varables. Regardless of the propagaton method used, t s necessary to represent and estmate what may be a complex varance and covarance structure. Ths report presents several models whch may be adopted for these estmates, and the models are equally useful when usng Monte Carlo smulaton. The report s broen nto several sectons, whch may be vewed somewhat ndependently. They are as lsted below:. Secton revews a framewor used n loss estmaton, and descrbes the elements of ths framewor.. Secton 3 presents a procedure for propagatng uncertanty through the varous random varables. Sectons 3. through 3.5, 3.8 and 3.9 present the propagaton procedure consderng aleatory uncertanty only. Secton 3.6 and 3.7 descrbe several

11 generalzatons of the model that have been ncluded for completeness, but are not needed for an understandng of the basc procedure. Sectons 3.0 through 3.4 ncorporate epstemc uncertanty and the model generalzatons nto the calculatons. Consderaton of epstemc uncertanty results n equatons of ncreased length and complexty, but f mplemented n a computer program, the computatons should not be overwhelmng. For a frst-tme reader, Sectons 3.6, 3.7 and 3.0 through 3.4 can be spped wthout losng the prmary methods of the procedure. 3. Secton 4 presents a numercal example to llustrate how the equatons of Secton 3 would be mplemented. It s ntended only as a smple llustraton, and so the model generalzatons are excluded from the example, as s consderaton of epstemc uncertanty. 4. Secton 5 uses the results from the example n Secton 4 to examne the effect of varance n TC gven on the fnal results. 5. Durng the course of ths procedure, there are many tmes when varances and covarances of random varables need to be estmated. Although the quantfcaton of these second moments s a relatvely undeveloped feld at present, several useful resources are descrbed n Secton 6. The report s concluded n Secton 7, whch summarzes the fndngs and features of ths report. Several Appendces are also ncluded to gve addtonal bacground nto deas presented n the body of the report. It s hoped that the modular format of ths report wll allow the reader to examne sectons of nterest, whle sppng sectons (e.g. Secton ) whch may not be new materal.

12 The Model Framewor The proposed procedure wors wthn the framewor proposed by the Pacfc Earthquae Engneerng Research (PEER) Center. Famlarty wth ths general framewor s presumed (Cornell and Krawnler 000; Krawnler 00a). Ths framewor s presented n the form of the followng equaton: λ ( z) G ( z, u) f ( u, v) f ( v, y) f ( y, x) dλ ( x) (.) = DVE DVE DM DM EDP EDP TC TC u v y x wth terms defned as follows: λ TC (z) s the annual rate of exceedng a total repar cost of z, where total repar cost, TC s the decson varable under study. GTC DVE(, z u) s the Complmentary Cumulatve Dstrbuton Functon (CCDF) of TC, condtoned on the vector of damage values of each element (DVE j s the damage value of element j) f DVE DM(u, v) s the PDF of the vector of damage values of each element, gven the vector of damage states of each element (DM j s the damage state of element j) f DM EDP(v, y) s the PDF of the vector of (the typcally dscrete) damage states, gven the vector of engneerng demand parameters f (y, x) s the PDF of the vector of engneerng demand parameters, gven the EDP dλ (x) ntensty measure s the absolute value of the dervatve of the annual rate of exceedng a gven value of the ntensty measure (the sesmc hazard curve). The absolute value s needed because the dervatve s negatve. Elements of ths framewor are developed further n the followng subsectons.

13 . TOTAL REPAIR COST The assumpton n ths study s that the total cost of repar to the structure s the sum of the repar costs of all elements n the structure, but ths can easly be generalzed. A modfcaton to ths assumpton accountng for collapse cases s presented n Secton ELEMENT DAMAGE VALUES Means and varances of repar costs for each possble damage state are needed for all element types under consderaton. Mean repar costs can be estmated from sources such as R.S. Means Company s publshed materals on constructon cost estmatng (R.S. Means Co. 00). Addtonal quantfcaton of repar costs s a topc of current research..3 ELEMENT DAMAGE MEASURES In current research, damage measures are typcally not contnuous, but a dscrete set of damage states (e.g., Porter and Kremdjan 00; Aslan and Mranda 00). We shall adopt ths format to facltate such efforts. Dscrete damage states are descrbed by fraglty functons, whch return the probablty of an element exceedng gven damage states at a gven Engneerng Demand Parameter (EDP) level. One fraglty functon s needed for each potental damage state of the element. Typcally, fraglty functons wth cumulatve lognormal functonal shapes are used. The development of fraglty functons for structural and nonstructural components s a topc of current research..4 BUILDING RESPONSE A probablstc model s needed for the dstrbuton of EDPs for the structure (e.g., the nterstory drfts and pea floor acceleratons for each floor), condtoned on a level of. Condtonal mean values and varances due to aleatory uncertanty can be determned from analyss technques such as Incremental Dynamc Analyss (Vamvatsos and Cornell 00)..5 SITE SEISMIC HAZARD It s necessary to determne the hazard curve for the predctor at the locaton of nterest, through ether PSHA or sesmc hazard maps. Ths topc has been developed n detal elsewhere (e.g., Kramer 995), and s not further covered n ths study. 4

14 .6 ASSUMPTIONS Several assumptons are made n ths framewor, and are descrbed below. They are beleved to be consstent wth the most advanced current sesmc loss estmaton efforts. Most can be reduced wthout formal dffculty. Marovan dependence s assumed for all dstrbutons n the framewor. For example, t s assumed that the dstrbuton of the DM vector can be condtoned solely on the EDP vector, and that nowledge of the provdes no addtonal nformaton. In ths way, prevous condtonng nformaton does not need to be carred forward through all future dstrbutons, reducng complexty. A condtonng varable that contans all necessary condtonal nformaton s deemed a suffcent descrptor (Luco 00). All damage s assumed to occur on an element level. The total cost of damage to the structure s then the sum of the damage cost of each element n the structure. Ths technque s explaned n more detal by Porter and Kremdjan (00). Ths assumpton can easly be generalzed, f desred. In ths paper, the excepton to ths assumpton s when collapse occurs, and repar costs wll be a functon of the collapse rather than ndvdual element responses. The collapse case s set asde n the ntal presentaton of the procedure, and then accounted for n a later secton. All relatons n the framewor are assumed to be scalar functons. For example, the condtonal dstrbuton of the Damage Measure of element j s a functon of only the th Engneerng Demand Parameter. Or alternatvely, f ( DM EDP v j, y) = f DM EDP ( v j, y ). Note also j j that the functon s not condtoned on varables from any prevous steps because of the Marovan process assumpton. To calculate total uncertanty n our decson varable, t wll be necessary to ncorporate both epstemc and aleatory uncertanty. These two sources of uncertanty are uncorrelated, allowng ther contrbutons to be calculated separately for smplcty. It s recommended that the varance of TC gven be calculated once for the varance due to epstemc uncertanty, and once for the varance due to aleatory uncertanty. 5

15 3 Procedure The procedure outlned maes use of FOSM approxmatons to calculate the mean and varance of TC gven. A dstrbuton F ( z, ) s then ft to these moments, and ntegrated TC x numercally or analytcally over the hazard curve,.e., dλ (x) to generate the mean annual rate of exceedng a gven repar cost: λ ( z) = F ( z, x) dλ ( x) (3.) TC TC The FOSM approxmatons used to obtan moments of TC from EDP, DM and DVE are justfed by the assumpton that the uncertanty n the hazard curve s the most sgnfcant contrbutor to varance of the total loss. Therefore, we are retanng the full dstrbuton for tself, but usng the FOSM approxmatons for all (frst and second) moments condtoned on. In addton, we lely do not have nformaton about the full dstrbutons of some varables (e.g., repar costs), and so usng only the frst two moments of these dstrbutons does not result n a sgnfcant loss of avalable nformaton. Note that we are worng wth natural logarthms of the varables descrbed prevously. Ths allows us to wor wth sums of terms, rather than products. We revert to a non-log form for the fnal result. 3. SPECIFY ln EDP The proposed model n ths study s EDP = H ( ) ε ( ), where H () s the (determnstc) mean value of EDP gven, and ε ( ) s a random varable wth mean of one, and condtonal varance adjusted to model the varance n EDP. Then when we use the log form of EDP, we have a random varable of the form ln( EDP ) ln( H ( )) + ln( ε ( )) =. We ntroduce the random varable notaton X Y, to denote that the model of X s condtoned on Y.

16 Note that the expected value of ln ( EDP ) s ln( ( )) ln ( EDP ) s equal to the varance of ln( ( )) H, and that the varance of ε. Both ln(h ()) and Var[ln( ε ( ))], as well as the correlatons between EDPs, can be determned from Incremental Dynamc Analyss. We wll need the followng nformaton for our calculatons : E[ln EDP ], denoted h () for all EDP (3.) Var[ln EDP ], denoted h* () for all EDP (3.3) ρ(ln EDP, ln EDP j ), denoted ĥ j () for all {EDP, EDP j } (3.4) As an example, usng nonlnear tme hstory analyss of the Van Nuys buldng (Lowes 00), the results needed for these equatons are presented n Secton 6. below. For llustraton, plots of the logs of floor nterstory drft ratos are shown n Fgure 3.. Note that these results have been condtoned on no collapse a procedure that wll be explaned n Secton 3.6. They are also presented as the exponental of the estmated natural logs of the results (denoted exp( ln IDR ) ), because we want the expected natural log of the drfts IDR IDR IDR3 IDR4 IDR5 IDR6 IDR = Sa at T = 0.85s Fg. 3.: Estmated medan of IDR ( exp( ln IDR ) ), condtoned on no collapse and Sa The data from ths fgure could be used n Equaton 3. above. Note that our model s lmted to the frst and second moments. The full dstrbuton model s not needed n what follows. 8

17 3. SPECIFY DM ln EDP AND ln DVE DM, AND COLLAPSE TO ln DVE ln EDP The dscrete states of the Damage Measure varable found n current loss estmaton (Porter and Kremdjan, 00) are not compatble wth the FOSM approach, whch requres contnuous functons for the moments. To deal wth the dscrete states, we tae advantage of the fact that we can always collapse the two dstrbutons DM ln EDP and ln DVE DM nto one contnuous dstrbuton ln DVE ln EDP by ntegratng over the approprate varable. Ths s llustrated n Fgure 3. below. λ ( x) (3.5) TC ( z) ftc DVE( z, u) fdve DM ( u, v) fdm EDP ( v, y) fedp ( y, x) dλ u v y x = because λ u ( y, x) dλ ( x) (3.6) TC ( z) ftc DVE( z, ) fdveedp ( u, y) fedp uyx = f u, y) = f ( u, v) ( v, y) (3.7) DVE EDP ( DVE DM fdm EDP v Fg. 3.: Collapsng out DM For a gven element wth n possble damage states, we use a set of element fraglty functons F, F F n, such that F ( y) = P( DM > d EDP y) (Fg. 3.3). These functons wll = have a correspondng set of dstrbutons c, c c n of element repar costs such that c (v) s a probablty dstrbuton of DVE, gven that the damage state equals d (Fg. 3.4). Wth ths nformaton, we can determne the frst two moments of the collapsed dstrbutons. Mranda et al. (00) documents the development of one set of these functons. 9

18 Fg. 3.3: Element Fraglty Functons Fg. 3.4: Element Repar Costs From the total probablty theorem, we now that n ths case, Equaton 3.7 can be wrtten n scalar form for each DVE as f = f DVE EDP DVE DM = d DM = d EDP (recall by assumpton that DamageStates each DVE s dependent on a sngle EDP). For our FOSM purposes, furthermore, t s suffcent to fnd smply the condtonal means, varances, and covarances of the DVEs gven the EDPs. Thus, tang the mean of ths PDF, we have the result: E [ DVE EDP] = E [ DVE DM = d ] P( DM = ) (3.8) Damage States P d µ ( F( EDP) F ( EDP) ) µ (3.9) + = Damage States Applyng the same thnng to E [ DVE EDP], and recognzng that [ X ] E[ X ] X Var = µ, we have the followng result (see Appendx B for dervaton): σ = EDM [ VarDVE[ DVE DM ]] + VarDM [ EDVE[ DVE DM ]] (3.0) DVE EDP = DVE Damage States + σ Damage States (( F ( EDP) F + ( EDP)) ( µ µ ) ( F ( EDP) F + ( EDP)) (3.) Fgure 3.5 shows an example of the mean and mean plus or mnus one sgma, as generated from the example dstrbutons shown n Fg. 3.3 and

19 Fg. 3.5: Collapsed dstrbuton DVE EDP Fnally, to complete ths collapsng exercse, we need to determne correlatons among the DVEs of all elements n the structure. Note that le the mean and varance, these correlatons are condtoned on the EDPs. Whle these calculatons are straghtforward, estmaton of the necessary correlaton nputs s a dffcult tas due to a lac of data. In the absence of addtonal nformaton, t may be helpful to use the followng characterzaton scheme. Let us assume for ths purpose a model of the form: where Struc ln DVE ln EDP = g (ln EDP ) + lnε + lnε + lnε, Struc ElClass m El ε represents uncertanty common to the entre structure, ε ElClass represents uncertanty m common only to elements of class m (e.g., drywall parttons, moment connectons, etc.), and ε El represents uncertanty unque to element. All of these ε s are assumed to be mutually uncorrelated. We then defne Struc Var[lnε ln ] = β, Struc EDP Var[lnε ln ] = β m ElClass EDP ElClass for all m, and Var[lnε El ln EDP ] = β El for all. Then the varance of ln DVE ln EDP s the sum of these varances. For ths specal case, a smple closed-form soluton exsts for the correlaton coeffcent (see Appendx A, Equatons A.9 and A.). Loosely speang, the correlaton coeffcent between two DVE s can be sad to be the rato of ther shared varances to ther total varance. The followng examples are llustratve:

20 Example : correlaton between two elements, DVE, DVE l, f both elements are of class ln DVE ln EDP = g (ln EDP ) + lnε + lnε + lnε (3.) Struc ElClass El ln DVE ln EDP = g (ln EDP ) + lnε + lnε + lnε (3.3) l Then (see Appendx A, Equaton A.9): ρ(ln DVE,ln DVE j l l ln j Struc ElClass βstruc + β ElClass Struc + β ElClass + EDP,ln EDPj ) = (3.4) β β El l El Example : correlaton between two elements, DVE, DVE l, f DVE s of element class, and DVE l s of element class ln DVE ln EDP = g (ln EDP ) + lnε + lnε + lnε (3.5) Struc ElClass El ln DVE ln EDP = g (ln EDP ) + lnε + lnε + lnε (3.6) l Then (see Appendx A, Equaton A.): ρ(ln DVE,ln DVE j l l ln j Struc Struc ElClass Struc ElClass β EDP,ln EDPj ) = (3.7) β + β + β El l El Note that ths formulaton requres β ElClass to be equal for all element classes, and be equal for all elements. If ths s excessvely lmtng, a closed-form soluton also exsts that allows β ElClass to vary by class, and β El β El to vary by element, and be functonally dependent on the EDP value. Ths soluton s also outlned n Appendx A. Note also that ths model can be expanded to more than three ε terms f desred. The use of more than two uncertan terms, and the use of β terms that vary by class or element are both generalzatons of the basc equcorrelated model, and thus we wll refer to a model ncorporatng any of these generalzatons as a generalzed equ-correlated model. The correlaton matrx for a generalzed equ-correlated model wll have off-dagonal terms that vary from term to term, as opposed to the strct equcorrelated model, where all off-dagonal terms are dentcal. We now have the condtonal mean and varance functons of ln DVE ln to EDP, obtaned by collapsng the dstrbutons provded (Equatons 3.9 and 3.), and correlaton coeffcents

21 determned usng the generalzed equ-correlated model (Equatons 3.4 and 3.7). We choose for future notatonal clarty to denote these results as: E[ln DVE ln EDP ], denoted g (ln EDP ) for all DVE (3.8) Var[ln DVE ln EDP ], denoted g* (ln EDP ) for all DVE (3.9) ρ(ln DVE, ln DVE l ln EDP, ln EDP j ), denoted ĝ l (ln EDP, ln EDP j ) for all {DVE, DVE l } (3.0) 3.3 CALCULATE ln DVE Usng nformaton from above, we can calculate the frst and second moments of ln DVE. Ths nvolves collapsng out the dependence onedp, as suggested n Fgure 3.6 below. λ ( z) f ( z, u) f ( u, y) f ( y, x) dλ ( x) (3.) = DVE DVE EDP EDP TC TC uyx because λ ( z) f ( z, u) f ( u, x) dλ ( x) (3.) = DVE DVE TC TC u x f u, x) = f ( u, y) f ( y, ) (3.3) DVE ( DVE EDP EDP x y Fg. 3.6: Collapsng out EDP To mantan tractablty, we shall do ths n an approxmate way referred to n structural relablty lterature as frst-order second-moment, or FOSM. To remove dependence on EDP, we tae the expectaton of ln DVE wth respect to ln EDP (gven ). We wrte ths as E[ln DVE ] = EEDP [ E[ ln DVE ln EDP ], where E [] denotes ths partcular condtonal expectaton operator. Substtutng our notaton from Equaton 3.8, we have E[ln DVE ] EEDP [ g ( ln EDP )] Tang a Taylor expanson of g ( ln EDP ) about E [ln EDP ] gves: E[ln DVE ] = EEDP [ g ( E[ln EDP ]) g ( ln EDP ) + ( ln EDP E[ln EDP ]) +K ] = (3.4) ln EDP E[ln EDP ] The frst-order approxmaton of ths seres s thus: EDP (3.5) 3

22 E[ln DVE ] E EDP + E EDP [ g ( E[ln EDP ])] ( ln EDP ) g ln EDP ( E[ln EDP ]) + 0 = g ( h ( )) E[ln EDP ] ( ln EDP E[ln EDP ]) = g (3.6) where the substtuton h( ) = E[ln EDP ] has been made, as defned n Equaton 3.. Usng a smlar approach to condtonal moments (Equaton B.7, Appendx B), t can be shown that: Var[ln DVE ] = E ln EDP + Var [ Var ln EDP ln DVE ln EDP [ E [ln ln DVE ln EDP DVE [ln DVE ln EDP ] ln EDP ] g g * ( h ( )) + h * ( ) ln EDP (3.7) h ( ) and smlarly, Cov[ln DVE,ln DVE l ] = E[ Cov[ln DVE + Cov[ E[ln DVE,ln DVE,ln DVE l EDP, EDP ] l EDP, EDP ] j j gˆ l ( h ( ), h j ( )) g * ( h ( )) g * l ( h j ( )) g + ln EDP h ( ) g l ln EDPj h ( ) j hˆ ( ) j h * ( ) h * j ( ) (3.8) Although not needed for the calculatons n ths framewor, correlaton coeffcents can be easly calculated usng the above nformaton, f they are of nterest: Cov[ln DVE,ln DVEl ] ρ (ln DVE,ln DVEl ) = (3.9) Var[ln DVE ] Var[ln DVE ] Subscrpts on the expectaton operators were ntally used on the expectaton and varance terms above to emphasze whch varable the expectaton was beng taen wth respect to. For the sae of concseness, these subscrpts are dropped n future equatons. l 3.4 SWITCH TO THE NON-LOG FORM DVE Usng frst-order second-moment methods, as n the prevous step, we have the followng results: E[ DVE ] ln DVE E[ln DVE ] g ( h ( )) = E[ e ] e e (3.30) 4

23 Var[ DVE ln DVE e ] ln DVE E[ln DVE ] Var[ln DVE ] g ( h ( )) = e Var[ln DVE ] (3.3) Cov[ DVE, DVE l ln DVE e ] ln DVE E[ln DVE ] ln DVEl e ln DVE l E[ln DVEl ] Cov[ln DVE,ln DVE l ] g ( h ( )) g ( h ( )) l j = e Cov[ln DVE,ln DVE ] (3.3) + l 3.5 COMPUTE MOMENTS OF TC We now aggregate the results from all ndvdual elements to compute an expectaton and varance for the total cost of damage to the entre buldng. Usng nformaton from prevous steps, we have the followng results: # elements # elements E[ TC ] = E DVE = E[ DVE ], denoted q() (3.33) = = # elements # elements Var[ TC ] = Cov[ DVE = l =, DVE ] l denoted q*() = # elements = # elements # elements Var[ DVE ] + Cov[ DVE, DVE ], (3.34) = l = + l 3.6 ACCOUNTING FOR COLLAPSE CASES At hgh levels, the potental exsts for a structure to experence collapse (defned here as extreme deflectons at one or more story levels). In ths buldng state, repar costs are more lely a functon of the collapse rather than ndvdual element damage. In fact, the structure s lely not to be repared at all. Thus, our predcted loss may not be accurate n these cases. In addton, the large deflectons predcted n a few cases wll sew our expected values of some EDPs such as nterstory drfts, although collapse s only occurrng n a fracton of cases. To account for the possblty of collapse, we would le to use the technque outlned above for no- 5

24 collapse cases, and allow for an alternate loss estmate when collapse occurs. The followng modfcaton s suggested. Note, n the followng calculatons, we are condtonng on a collapse ndcator varable. To communcate ths, we have denoted the collapse and no collapse condton as C and NC respectvely. At each level, compute the probablty of collapse. Ths probablty, PC ( ), s smply the fracton of analyss runs where collapse occurs. Calculate results usng the FOSM analyss as before, but usng only the runs that resulted n no collapse. We now denote these results E [ TC, NC] and Var [ TC, NC]. Defne an expected value and varance of total cost, gven that collapse has occurred, denoted E [ TC, C] and Var [ TC, C]. These values wll lely not be functons of, but the condtonng on s stll noted for consstency. The expected value of TC for a gven level s now the average of the collapse and no collapse TC, weghted by ther respectve probabltes of occurrng ( ) E[ TC ] = P( C ) E[ TC, NC] + P( C ) E[ TC, C] (3.35) The varance of TC for a gven level can be calculated usng the result from Appendx B: Var[ TC ] = E[ Var[ TC, NC or C]] + Var[ E[ TC, NC or C]] = ( PC ( ) ) VarTC [, NC] + PC ( VarTC ) [, C] ( PC ( ) )( ETC [ ] ETC [, NC] ) + + PC ( ) ( ETC [ ] ETC [, C] ) (3.36) The procedure can now be mplemented as before, usng these moments. Ths collapsecase modfcaton s probably necessary for any mplementaton of the model, as analyss of shang () levels suffcent to cause large fnancal loss are lely also to cause collapse n some representatve ground moton records. 3.7 OTHER GENERALIZATIONS OF THE MODEL Several other modfcatons to ths model can potentally be used to ncrease the accuracy of the estmate, wthout fundamentally changng the approach outlned above. One such modfcaton to the model s ncorporaton of demand surge (the ncrease n contractor costs after a major earthquae) usng the followng steps:. Determne the demand surge cost multpler as a functon of magntude, g(m) 6

25 . Determne the PDF p M (m m) from deaggregaton of the hazard 3. Demand surge as a functon of s h ( m) = g( m ) pm ( m m) 4. The new total cost as a functon of can be calculated as TC ( ) = h( ) E[ TC ], where E[ TC ] s the expected total cost gven, as calculated prevously. Another modfcaton to the model s a revson to allow the calculaton of an element damage measure based on a vector of EDPs, rather than just a scalar. A smple way to accommodate ths possblty s to create addtonal EDPs that descrbe the vector of nterest:. Create a new derved EDP that s a functon of the vector of basc EDPs of nterest (e.g., f DM s a functon of the average of EDP and EDP j, create a new EDP, EDP = (EDP + EDP j )/ ).. Compute mean, varance, and covarances of EDP usng frst-order approxmatons, and the second moment nformaton calculated for EDP and EDP j 3. The damage measure can then be a functon of the scalar EDP Ths method allows the smple scalar algebra to be used, at the expense of needng to trac an ncreased number of EDPs. If many addtonal EDPs are needed, t may be preferable to develop a more complex vector-based procedure. m 3.8 INCORPORATE THE GROUND MOTION HAZARD TO DETERMINE E[TC] AND VAR[TC] Usng the functons q() and q*(), and the dervatve of the ground moton hazard curve, dλ ( ), the mean and varance of TC per annum can be calculated by numercal ntegraton: E [ TC] = q( x) dλ ( x) (3.37) Var [TC] = E[ Var[ TC ]] + Var[ E[ TC ]] = q*( x) dλ ( x) + E[ q ( x)] E[ q( x)] = q * ( x) d ( x) + q ( x) dλ ( x) λ E[ TC] (3.38) Note that the frst term of Equaton 3.38 s the contrbuton from uncertanty n the cost functon gven, and that the second two terms are the contrbuton from uncertanty n the. 7

26 3.9 RATE OF EXCEEDANCE OF A GIVEN TC The frst- and second-moment nformaton for TC can also be combned wth a ste hazard to compute λ TC (z), the annual frequency of exceedng a gven Total Cost z. For ths calculaton, t s necessary to assume a probablty dstrbuton for TC that has a condtonal mean and varance equal to the values calculated prevously. The rate of exceedance of a gven TC s then gven by λ ( z) = G ( z, x) dλ ( x) (3.39) TC TC where G (, z x) = P( TC > z = x) s the Complementary Cumulatve Dstrbuton Functon TC of TC. By evaluatng the ntegral for several values of z, a plot can be generated relatng damage values to rates of exceedance. Generally, the ntegral above wll requre a numercal ntegraton. However, f the followng smplfyng assumptons are made, an analytc soluton s avalable:. E[TC =m] s approxmated by a functon of the form a (m) b, where a and b are constants. Note that ths s consstent wth fttng the medan of TC wth a(m) b, where β * TC a = a' e (3.40). The uncertanty TC s characterzed as follows: TC = E[ TC = m] ε TC, where ε TC s a lognormal random varable wth medan equal to and logarthmc standard devaton σ = β * ln ( ε ) TC (note that ths s constant for all ). TC 3. An approxmate functon of the form ˆ λ ( x ) x = 0 s ft to the true mean ste hazard curve. Note, ths form for the hazard curve has been proposed prevously by Kennedy and Short (984) and Luco and Cornell (998). Under these condtons, the annual rate of exceedng a gven Total Cost s gven by: b z λ TC ( z) = 0 exp β * TC (3.4) a' b b We note that f the a from Equaton 3.40 s substtuted nto Equaton 3.4, then the result becomes b z λ = TC ( z) 0 exp β * TC (3.4) a b 8

27 Equaton 3.4 s useful as a smple estmate of λ ( z ), but t s also very nformatve as a measure of the relatve mportance of uncertanty n the calculaton. The term b TC z 0 (3.43) a' n the equaton would be the result f β * TC were to equal zero that s, f we made all calculatons usng only expected values and neglected uncertanty. The term exp β * TC (3.44) b b s an amplfcaton factor that vares wth the uncertanty n TC present n the problem. Thus for ths specal case, t s smple to calculate the effect of uncertanty on the rate of exceedng a gven Total Cost. As we shall show below, t may not be unreasonable for ths factor to ncrease λ TC (z) by a factor of 0, so the effect of uncertanty may very well be sgnfcant. However, even for large values of β * TC, the annual rate of exceedance s stll domnated by the determnstc term. It s for ths reason that t has been proposed here that the FOSM approxmatons of β * TC performed above are suffcent to provde an accurate result. 3.0 INCORPORATION OF EPISTEMIC UNCERTAINTY Equatons 3.37, 3.38, and 3.4 are vald for the case when there s no epstemc uncertanty n the ground moton hazard curve or TC. However, we now need to extend our calculaton to account for ths uncertanty, whch s expected to be sgnfcant. To do ths, we frst ntroduce the effect of epstemc uncertanty n TC, and then ntroduce epstemc uncertanty n the ground moton hazard Epstemc Uncertanty n TC We have prevously assumed a model whch can be wrtten TC = E[ TC ] ε R, where ε R s a random varable representng aleatory uncertanty. Now, we extend that model to ncorporate epstemc uncertanty. We now assume a smplfed (frst-order) model of epstemc uncertanty n whch that uncertanty s attrbuted only to the central or mean value of a random varable and not for example, ts varance or dstrbuton shape. (In practce, one may nflate somewhat ths uncertanty n the mean to reflect these second-order elements of epstemc uncertanty.) Ths 9

28 model means that we represent the total uncertanty n TC as TC = E[ TC ] εε R U, where E[ TC ] s the best estmate of the (condtonal) mean and ε R and ε U are uncorrelated random varables representng aleatory uncertanty and epstemc uncertanty, respectvely. Note that ETC [ ε ] U s a random varable representng the (uncertan) estmate of the mean value TC, wth varance [ [ ]] of Var E TC. We have a total model of the form TC = q( ) εε R U. So tang logs gves us ln TC = ln q( ) + ln ε ( ) + ln ε ( ). The random varables ε R and R U ε U are uncorrelated, so we may deal wth them n separate steps. The above procedure, descrbed n Sectons 3. through 3.7, accounted for aleatory uncertanty and allowed us to fnd the varance due toε R. We must now repeat the procedure to calculate the varance due toε U. Note that we have swtched to logs agan to allow use of sums rather than products. The change can be made usng the followng relatonshp: VarR [ TC ] Var[ln ε R ( )] ln + ER[ TC ] (3.45) We denote Var[ln ε R ] and Var[ln ε U ] as β R as β U, respectvely. Note that n the prevous sectons, the uncertanty that we have denoted as β s now referred to as β R, to dstngush TC t from the β U that we are now addng. Representaton of Condtonal Varables n the Framewor Equaton To dstngush between aleatory and epstemc uncertantes of varous condtonal random varables, we ntroduce an addtonal notaton. For example, we denote the epstemc and aleatory uncertanty of ln EDP, as: Var EDP = β, for varance due to aleatory uncertanty n [ln ] REDP ; ln EDP (3.46) [ ] Var E[ln EDP ] = β U EDP, for varance due to epstemc uncertanty n the ; estmate of the mean of ln EDP (3.47) These values are equvalent to h* () n Equaton 3.3. Ths notaton s ntroduced smply to dstngush between aleatory and epstemc uncertanty. As a gudelne for estmatng 0

29 uncertanty, several references are ncluded n later sectons. Example results for β are R ; EDP presented n Secton 6., and gudelnes for estmatng β are presented n Secton 6.3. U ; EDP As a fnal note, one should be aware of the potental for double-countng of a source of uncertanty when constructng these models and classfyng sources of uncertanty as epstemc or aleatory. Any sngle source of uncertanty should be accounted for as ether aleatory or epstemc uncertanty, but t should not be ncluded n both. Accountng for Correlatons n EDP Estmates of correlatons need to be made at each step of the PEER equaton (.e., EDP, DVE EDP after DM has been collapsed out, and TC DVE ). In ths secton, we propose a model, and demonstrate ts use for correlatons n EDP. The same model s generally applcable to the other varables as well. Consder the followng model for ln EDP : ln( EDP ) = E[ln( EDP )] + εr EDP + ε U EDP (3.48) ; ; where E[ln( EDP )] s the mean estmate of E[ln( EDP )] and εr; EDP and ε U ; EDP are random varables representng aleatory and epstemc uncertanty, respectvely. Both random varables have an expected value of zero. Remember that we are usng boldface notaton because EDP s a vector of random varables. The aleatory uncertanty term ( ε REDP ; ) can be estmated drectly from data (see Secton 6..), so now we need to address the epstemc uncertanty term ( ε U; EDP ). Some of our epstemc uncertanty comes from model uncertanty (uncertanty about the accuracy of the structural model we are usng see Secton 6. and Appendx E). Another porton of our uncertanty comes from estmaton error we are estmatng the moments of ln( EDP ) by usng the sample averages of a fnte set of records. Ths estmaton uncertanty s most famously seen when estmatng a mean of a dstrbuton by the average of n samples, each wth varance σ. The varance of ths estmate s σ = σ / n. Ths ˆ µ σ / n s an epstemc uncertanty that we have referred to as estmaton uncertanty. So we now splt our epstemc uncertanty nto two terms: ε = ε + ε (3.49) U ; EDP U ; EDP U ; EDP model estmate

30 where ε s a random varable representng model uncertanty and ε U ; EDP s a Umod el ; EDP random varable representng estmaton uncertanty, and both random varables have a mean of zero. These two random varables are assumed to be uncorrelated, so they can be analyzed separately. When calculatng the epstemc uncertanty, we wll also need to calculate correlatons between estmates of means at dfferng levels (e.g., correlaton of estmates of the expected value of lnedp at = m and = m : ρ = = ). Although there s no E[ln EDP m ], E[ln EDP m ] correlaton between aleatory uncertantes, epstemc uncertanty (representng our uncertanty about the mean values) wll potentally be correlated. The modelng uncertanty, represented by ε U model ; EDP, may presumably, to a frst approxmaton, be assumed to have a perfect correlaton at two levels, because the models tend be common at least wthn the lnear and nonlnear ranges. The same perfect correlaton could be appled to two dfferent E[lnEDP] s at a sngle gven level. Our estmaton uncertanty, represented by ε U ; estmate EDP estmate, may also be correlated at two levels. For nstance, f we use the same set of ground moton records to estmate the E[lnEDP] s at more than one level by usng scalng, our estmates at the varyng levels wll be correlated. In order to measure ths correlaton, we can utlze the bootstrap. (Efron and Tbshran, 998). The use of bootstrappng to calculate the correlaton for a gven EDP at two levels s outlned n Appendx D. The varance of ε U ; EDP can also be calculated from the bootstrap, as mentoned n the Appendx. Once we have measured the varance and correlaton of ε U ; EDP and ε U ; EDP at two levels, we can combne them to fnd the correlaton of ε U; EDP at two levels. If the estmate model estmate varance of ε U ; model EDP, denoted β s equal at both levels, and the varance of U model ; EDP ε Uestmate ; EDP, denoted β s equal at both levels, then the correlaton of ε U; EDP at Uestmate ; EDP two levels, denoted ρ s: U ; EDP, ρ Umodel ; EDP Uestmate ; EDP ;, = βu ; EDP + βu ; EDP U EDP β model + ρ β estmate (3.50) where ρ s the correlaton between E[ln EDP ] at two levels due to estmaton uncertanty (the correlaton we measured from the bootstrap). We note, however, that f ρ s

31 expected to be near one, or f β U ; EDP s much greater than β U ; ρ U ; EDP, model estmate EDP, then wll be nearly one. Under these condtons, t s thus reasonable to smply assume a perfect correlaton, and thus sp the computatons of the bootstrap. Note that f the more complex model of Equaton 3.50 s used, t s not necessary that the varance of ε U ; EDP s equal at both levels as we have assumed above. An equaton followng the form of Equaton A.5 wll allow for the varance of ε U ; EDP to be dfferent at the two levels. estmate estmate Addtonal Correlatons In addton to correlatons between one E[ln EDP ] at two levels, t s also necessary to fnd correlatons between two E[ln EDP ] s at the same level (e.g., Corr[ E[ln EDP = m ], E[ln EDP = m ]] ). For aleatory uncertanty, t was possble to j mae a drect estmate from the data avalable, but t s slghtly more complcated for epstemc uncertanty. However, the model of Equaton 3.50 s sutable for ths stuaton as well. The correlaton coeffcent from estmaton uncertanty can be computed from the bootstrap, and for model uncertanty, a perfect correlaton could agan be assumed. Then the estmate of total correlaton can be calculated n a smlar manner to Equaton We also need an estmate of [ [ln ln ], [ln ln ]] Cov E DVE EDP E DVE EDP. The condtonal random varable E[ln DVE ln EDP ] wll have epstemc uncertanty. The model developed n the prevous secton for E[ln EDP ] could be appled n the same way to E[ln DVE ln EDP ]. Agan, t s worth consderng whether the smple assumpton of a perfect correlaton model may be approprate before usng the slghtly more complex model proposed here. Propagaton of Uncertanty, Accountng for Correlatons at Two Levels Earler, n Equaton 3.38, we dd not consder correlatons n TC at two levels. However, f there s correlaton n E[ln EDP ] at two levels, as we have ntroduced n ths secton, then ths correlaton wll propagate through to E[ TC ], and result n correlaton between E[ TC ] at two levels. We now show the FOSM approxmaton that accounts for ths correlaton. 3

32 Once we have specfed the correlaton between E[ln EDP ] and E[ln EDP ], and the correlaton between E[ln DVE ln EDP] and E[ln DVE ln EDP ], as descrbed n ths secton, and we have the expected values from Sectons 3. and 3., we can use FOSM to combne ths nformaton and compute the covarance between E[ln DVE ] and E[ln DVE ] as shown below: Cov[ E[ln DVE ], E[ln DVE ]] Cov[ E[ln EDP ], E[ln EDP ]] E[ln DVE ln EDP] E[ln DVE ln EDP] * ln EDP ln EDP E[ln EDP ] E[ln EDP ] + Cov[ E[ln DVE ln EDP], E[ln DVE ln EDP]] E[ln EDP ], E[ln EDP ] (3.5) In the same way, we can compute the covarance between E[ln DVE ] and E[ ln DVE ]: l Cov[ E[ln DVE ], E[ln DVE ]] l Cov[ E[ln EDP ], E[ln EDP ]] E[ln DVE ln EDP] * ln EDP j E[ln DVE ln EDP ] ln EDP E[ln EDP ] j E[ln EDPj ] + Cov[ E[ln DVE ln EDP], E[ln DVE ln EDP ]] l j l j E[ln EDP ], E[ln EDPj ] (3.5) As n Equaton 3.3, we must convert the covarance of the E[lnDVE] s to the covarance of the (non-log) E[DVE] s: Cov[ E[ DVE ], E[ DVE ]] l E[ln DVE ] E[ln DVEl ] e e (3.53) E[ln DVE ] E[ln DVE ] E[ln DVE ] l E[ln DVEl ] * Cov[ E[ln DVE ], E[ln DVE ]] l We sum the E[DVE ] random varables to get E[TC ] (e.g., ETC [ ] = EDVE [ ] ). So gven the values from the above equaton, we can compute all the covarance of E[TC ] at two levels: 4

33 CovETC [ [ ], ETC [ ]] = Cov EDVE [ ], EDVE [ l ] l = Cov[ E[ DVE ], E[ DVE ]] [ ] + Cov E[ DVE ], E[ DVE ] < l l (3.54) Ths covarance s merely a summaton of many covarance terms that we can now calculate. Note that we wll need to repeat ths calculaton for dfferent {, } pars. We wll use these values n the sectons below (e.g., n Equaton 3.6) Epstemc Uncertanty n the Ground Moton Hazard It s now necessary to account for epstemc uncertanty n the ground moton hazard. Ths uncertanty s often dsplayed qualtatvely as the fractle uncertanty bands about the mean estmate of the hazard curve, as shown n Fg Fg. 3.7: Sample Hazard Curve for the Van Nuys Ste Formally, we represent the ground moton hazard at a gven level as a random varable. λ ( m) = λ ( m) ε ( m) (3.55) U 5

34 where λ ( m ) s our best estmate or mean estmate of λ ( m ), and ε ( m) s a random varable wth a mean of. Consderng the entre range of levels mples that ε ( m ) s n U fact a random functon of. We wll agan need to consder correlatons, because the frst and second moment representaton of ths functon wll nvolve a covarance functon that s parameterzed by the two m levels consdered, Cov[ λ ( m), λ ( m)]. However, before we dscuss the random functon theory soluton to ths problem (e.g., Ngam, 983), let us realze that we wll be performng all of our calculatons usng numercal ntegraton. For example, the ntegral of Equaton 3.37 wll n practce be calculated as a dscrete summaton, whch we shall dscuss further n the next secton: E[ TC] = E[ TC = x] d ( x) λ E[ TC = x ] ( x ) λ ( x ) where 0 x < x <... < x n λ = x x (3.56) 0 Here t s only mportant to recognze that we are now dealng wth a dscrete set of λ ( x ). Therefore we defne a new random vector: λ ( x ) λ ( x ) λ ( x ) = x x n U (3.57) The mean and covarance of the array λ ( x ), =, K, n can be computed f we now the mean and covarance of the array of λ ( x ), =, K, n. We have prevously used the mean value of ths array, λ ( m ), whch we get from PSHA software. The varances can be estmated from the fractle uncertanty typcally dsplayed n a graph of the sesmc hazard curve (e.g., Fg. 3.7). The covarances of the array are potentally avalable from the output of PSHA software as well (see Appendx F). Usng ths formulaton, the random varable E[TC] can be represented as: n ETC [ ] = E ETC [ = x] λ( x) (3.58) = n whch we understand that there s now epstemc uncertanty n ETC [ = x ] and λ ( x ). Further, we assume that there s no stochastc dependence between the epstemc aspects of ETC [ = x ] and λ ( x ). 6

35 Now that we have quantfed the epstemc uncertanty n ETC [ = m] and n λ ( m ), we wll apply ths model to assessment of epstemc uncertanty n ETC [ ], λ collapse and λ ( z ). TC 3. EXPECTATION AND VARIANCE OF E[TC], ACCOUNTING FOR EPISTEMIC UNCERTAINTY Consder frst the effect of epstemc uncertanty on the mean estmate ( E[ TC ]) and epstemc varance ( Var[ E[ TC ]] ) of E[ TC ]. We calculate E[ TC ] by tang advantage of the ndependence of ETC [ = x ] and λ ( x ) n ETC [ ] = E E[ TC = x] λ ( x) [ ], and usng the lnearty of the expectaton operator: = n E x x ETC [ x ] = x x n λ ( x ) λ ( x ) ETC [ x ] = x x n = = = = = ETC = x λ ( x) = [ λ ( ) λ ( )] (3.59) Ths s the dscrete analog of Equaton 3.37 (where we used the mean hazard curve n the calculaton). So we see that our estmate of expected total cost s unchanged when we nclude epstemc uncertanty n the analyss, provded that we use the mean estmate of the ground moton hazard curve. However, because we are now uncertan about ETC [ = x ] and λ ( x ) (for all ), E[ TC ] s now uncertan. So we would le to calculate the epstemc varance n E[ TC ]. Ths calculaton nvolves a summaton of products of random varables. Consder equaton If we denote ETC [ = x ] as X, and λ ( x ) as Y, then ETC [ ] s of the form: n ETC [ ] = X Y (3.60) = where X, and Y are random arrays. There s no correlaton between X, and Y, but there s qute lely to be a correlaton between X and X j, and also between Y and Y j ( j), as dscussed above. We have calculated Cov[ X, X ] n Equaton 3.54 and Cov[ Y, Y ] s dscussed n Secton j j 7

36 3.0. and Appendx F. Gven that the needed covarance matrces have been calculated, from Dtlevsen (98) we have the followng result for a product of random arrays: Var X Y = E X X jcov[ Y, Yj] + Var X E[ Y] j Cov[ X, X j] Cov[ Y, Yj] = + EX [ ] EX [ j] CovYY [, j] j + EY [ ] EY [ j] CovX [, X j] (3.6) The above equaton becomes very long when X = ETC [ = x] and Y = λ ( x ) s substtuted bac n. It s left to the reader to mae ths change of notaton at the tme of mplementaton n a computer program. To match the other computatons n ths report, we would hope to have an analytcal soluton for the expectaton and varance of E[ TC ]. However, t can be shown that under smlar analytcal assumptons to those made elsewhere n ths report, an analytcal soluton does not exst 3. 3 An Analytcal Soluton? Equaton 3.6 should be easy to mplement n a smple computer program, although t s not feasble for bac-of-the-envelope calculatons. So we would hope to have a closed form soluton for E[ TC ] and Var[E[TC]]. We can compute E[ TC ] usng the followng ntegral: E[ TC] = z dλtc ( z) dz 0 We note however, that when we try to evaluate ths ntegral usng our analytc functonal forms, we have a problem. Substtutng our analytcal soluton for λ TC () z from Equaton 3.7 (developed n Secton 3.3 below), we have: d z b ETC [ ] = z 0 exp ( R U) dz dz a ' β + β b b 0 z b = z 0 exp ( βr + βu) dz b a' a' b b 0 where = 0 exp ( ) b βr βu z dz b a' + b b = K z 0 + b b 0 0 b K0 = 0 exp β + β b a ' b b ( R U ) 8