Required Number of Records for ASCE/SEI 7 GroundMotion Scaling Procedure 

By Juan C. Reyes and Erol Kalkan 

Open-File Report 2011–1083

U.S. Department of the Interior 

U.S. Geological Survey
ii 

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Required Number of Records for ASCE/SEI 7 GroundMotion Scaling Procedure 

By Juan C. Reyes and Erol Kalkan 

Open-File Report 2011–1083

U.S. Department of the Interior 

U.S. Geological Survey
U.S. Department of the Interior 

KEN SALAZAR, Secretary 

U.S. Geological Survey 

Marcia K. McNutt, Director 

U.S. Geological Survey, Reston, Virginia 2011 

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Suggested citation: Reyes, J.C. and Kalkan, E., 2011, Required number of records for ASCE/SEI 7 ground-motion 

scaling procedure: U.S. Geological Survey Open-File Report 2011-1083, 29 p. [http://pubs.usgs.gov/of/2011/1083/].

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v 

Contents 

 Page

Abstract ............................................................................................................................................................1 

Introduction.......................................................................................................................................................1 

Ground-Motions Selected.................................................................................................................................2 

Description of Inelastic SDF Systems ..............................................................................................................3 

Methodology .....................................................................................................................................................3 

Benchmark Inelastic Deformations...................................................................................................................4 

Evaluation of ASCE/SEI 7 Scaling Procedure: Less Than Seven Ground-Motions .........................................4 

Evaluation of ASCE/SEI 7 Scaling Procedure: Seven or More Ground-Motions..............................................5 

Modal-Pushover-Based Scaling Procedure: Alternative to the ASCE/SEI 7 Scaling Method ..........................7 

Conclusions ......................................................................................................................................................7 

Acknowledgments ............................................................................................................................................8 

References Cited..............................................................................................................................................8 

Notation ............................................................................................................................................................9 
1

Required Number of Records for ASCE/SEI 7 GroundMotion Scaling Procedure 

By Juan C. Reyes1 and Erol Kalkan2 

1 Universidad de los Andes, Bogota, Colombia, jureyes@uniandes.edu.co. 2 U.S. Geological Survey, Menlo Park, CA 94025, ekalkan@usgs.gov. 

Abstract

The procedures and criteria in 2006 IBC (International Council of Building Officials, 2006) and 2007 CBC 

(International Council of Building Officials, 2007) for the selection and scaling ground-motions for use in 

nonlinear response history analysis (RHA) of structures are based on ASCE/SEI 7 provisions (ASCE, 2005, 

2010). According to ASCE/SEI 7, earthquake records should be selected from events of magnitudes, fault 

distance, and source mechanisms that comply with the maximum considered earthquake, and then scaled 

so that the average value of the 5-percent-damped response spectra for the set of scaled records is not 

less than the design response spectrum over the period range from to sec (where is the 

fundamental vibration period of the structure). If at least seven ground-motions are analyzed, the design 

values of engineering demand parameters (EDPs) are taken as the average of the EDPs determined from 

the analyses. If fewer than seven ground-motions are analyzed, the design values of EDPs are taken as 

the maximum values of the EDPs. ASCE/SEI 7 requires a minimum of three ground-motions. These limits 

on the number of records in the ASCE/SEI 7 procedure are based on engineering experience, rather than 

on a comprehensive evaluation. This study statistically examines the required number of records for the 

ASCE/SEI 7 procedure, such that the scaled records provide accurate, efficient, and consistent estimates 

of “true” structural responses. Based on elastic-perfectly-plastic and bilinear single-degree-of-freedom 

systems, the ASCE/SEI 7 scaling procedure is applied to 480 sets of ground-motions. The number of 

records in these sets varies from three to ten. The records in each set were selected either (i) randomly, (ii) 

considering their spectral shapes, or (iii) considering their spectral shapes and design spectral-acceleration 

value, . As compared to benchmark (that is, “true”) responses from unscaled records using a larger 

catalog of ground-motions, it is demonstrated that the ASCE/SEI 7 scaling procedure is overly conservative 

if fewer than seven ground-motions are employed. Utilizing seven or more randomly selected records 

provides a more accurate estimate of the EDPs accompanied by reduced record-to-record variability of the 

responses. Consistency in accuracy and efficiency is achieved only if records are selected on the basis of 

their spectral shape and . 

Introduction

When nonlinear response history analysis (RHA) is 

required for design verification of building 

structures, the International Building Code (2006) 

and California Building Code (2007) refer to the 

ASCE/SEI 7 Section 16-21 (ASCE, 2005, 2010). 

 

1Ground motion selection and scaling procedure for two-dimensional 

analysis of structures are same in both ASCE/SEI 7-05 and 7-10 

According to these documents, earthquake records 

should be selected from events of magnitudes, fault 

distance, and source mechanisms that comply with 

the maximum considered earthquake. 

For two-dimensional analysis of symmetricplan buildings, ASCE/SEI 7 requires intensitybased scaling of ground-motion records using 

 

documents; therefore in the remaining of this report, we simply refer 

to this method as the ASCE/SEI 7 scaling procedure. 
2

appropriate scale factors, so that the average (that 

is, mean2) value of the 5-percent-damped response 

spectra for the set of scaled records is not less than 

the design response spectrum over the period 

range from to (where is the elastic 

first-“mode” vibration period of the structure). For 

three-dimensional analyses, ground-motions 

should consist of pairs of appropriate horizontal 

ground-motion acceleration components. For each 

pair of horizontal ground-motion components, a 

square root of the sum of the squares (SRSS) 

spectrum should be constructed by taking the 

SRSS of the 5-percent-damped response spectra 

of the unscaled components. Each pair of motions 

are then scaled with the same scale factor such 

that the mean of the SRSS spectra from all 

horizontal component pairs does not fall below the 

corresponding ordinate of the target spectrum in 

the period range from to . The design 

value of an engineering demand parameter 

(EDP)—member forces, member deformations or 

story drifts—is taken as the mean value of the EDP 

over seven or more ground-motions, or as its 

maximum value over all ground-motions, if the 

system is analyzed for fewer than seven groundmotions. This procedure requires a minimum of 

three records. The limits on the number of groundmotions in ASCE/SEI 7 were based on engineering 

experience rather than on a comprehensive 

evaluation (Charlie Kircher, oral commun.). 

This study, for the first time, statistically 

examines the required number of records for the 

ASCE/SEI 7 procedure such that the scaled 

records provide accurate, efficient, and consistent 

estimates of “true” median structural responses. 

The adjective “accurate” refers to the discrepancy 

between the “true” responses and those computed 

from the scaling procedure. The adjective “efficient” 

refers to the record-to-record (that is, intra-set) 

variability of responses, and the adjective 

“consistent” refers to the ground-motion set-to-set 

(that is, inter-set) variability of accuracy and 

efficiency. Smaller values of inter- and intra-set 

 

2In the remaining, “mean” is used in lieu of “arithmetic mean”. 

dispersion of responses indicate that the scaling 

procedure is more efficient and consistent. 

Based on elastic-perfectly-plastic and bilinear 

single-degree-of-freedom (SDF) systems, the 

accuracy, efficiency, and consistency of the 

ASCE/SEI 7 ground-motion scaling procedure are 

evaluated by applying it to 480 sets of groundmotions. The number of records in these sets 

varies from three to ten. The scaled records in each 

set were selected in three different ways: (i) 

randomly and (ii) minimizing discrepancy between 

scaled spectrum of a record and the target 

spectrum over the period range from to 

 (this approach will be referred to as “Best1”) 

and (iii) minimizing discrepancy between scaled 

spectrum of a record and the target spectrum over 

the period range from to , and then the 

final set of records was identified with spectralacceleration values at close to that of the design 

spectrum (this approach will be referred to as 

“Best2”). 

Ground-Motions Selected

The thirty records selected for this investigation 

(listed in table 1) were recorded from seven shallow 

crustal earthquakes compatible with the following 

scenario:

 Moment magnitude: Mw = 6.7 ± 0.2

 Joyner-Boore distance: km

 NEHRP3 soil type: C and D 

 Highest usable period≥4 sec 

Shown in figure 1 are the 5-percent-damped

geometric-mean response spectra for the xcomponent (identified as the maximum horizontal 

component) of the unscaled ground-motions. The 

geometric-mean spectrum of thirty records is taken 

as the design spectrum (that is, target spectrum) 

for purposes of this investigation. 

 

3National Earthquake Hazard Reduction Program
3

Description of Inelastic SDF Systems

The structures considered are 16 SDF systems 

with vibration periods equal to 0.2, 0.5, 1, and 

2.5 sec, and yield strength reduction factors R

equal to 1, 2, 4, and 8. The design base shear is

determined as the mass of the system (assumed to 

be 1 kip-sec2/in) times the geometric-mean 

pseudo-acceleration at reduced by R. The 

damping ratio of the selected SDF systems is 5-

percent. 

Two constitutive models, used for the inelastic 

SDF systems, are: (1) an elastic-perfectly-plastic 

model, and (2) a bilinear model with a strength 

hardening ratio of 10 percent. 

Methodology

According to the ASCE/SEI 7 procedure for twodimensional (or planar) analyses of “regular” 

structures, the ground-motions should be scaled 

such that the mean value of the 5-percent-damped

response spectra for the set of scaled motions is 

not less than the design spectrum over the period 

range from to sec. The ASCE/SEI 7 

scaling procedure does not insure a unique scaling 

factor for each record; various combinations of 

scaling factors can be defined to insure that the 

mean spectrum of scaled records remains above 

the target spectrum over the specified period 

range. To achieve the desirable goal of scaling 

each record with a minimum scale factor closest to 

unity, we implemented the ASCE/SEI 7 scaling 

procedure for randomly selected ground-motions 

as follows: 

1. For each of the thirty records listed in table 1, 

calculate the 5-percent-damped response 

spectrum and the vector of spectral 

values at 300 logarithmically spaced periods 

over the period range from to sec. 

2. Obtain a target (that is, “design”) pseudoacceleration spectrum as the geometricmean spectrum of thirty records. Define as 

a vector of target spectral values at periods 

 over the period range from to 

sec. 

3. Compute the scaling factor SF1 to minimize the 

difference between the target spectrum 

(Step 2) and the response spectrum 

(Step 1) by solving the following minimization 

problem for each ground-motion: 

min

SF1

log Aˆ − SF1 × log A ⇒ SF1

where is the Euclidean norm. Required for 

this purpose is a numerical method to minimize 

scalar functions of one variable; such methods 

are available in textbooks on numerical 

optimization (for example, Nocedal and 

Stephen, 2006). This minimization ensures that 

each scaled response spectrum is as close as 

possible to the target spectrum, as shown 

schematically in figure 2. 

4. Randomly select a set of m ground-motions to 

be used in nonlinear RHA of the systems 

described previously. No more than two 

records from the same event should be

included in a single set, so that no single event 

is dominant within a set. 

5. Determine the vector for the mean 

scaled spectrum defined as the mean of the 

scaled spectra ( ) of the set of m

records. The ordinates of this mean scaled 

spectrum could be smaller than the ordinates 

of the target spectrum at the same periods. 

6. Calculate the maximum normalized difference 

 (fig. 3a) between the target spectrum 

and the mean scaled spectrum over the 

period range from to ; that is,

, where 

and are the ordinates of the target and 

the mean scaled pseudo-acceleration spectra 
4

at vibration period , respectively. Define the 

scale factor SF2 = (1− ε ASCE )

−1

.

7. Determine the final scale factor 

SF = SF1 × SF2 for each ground-motion. 

Scaling ground-motions by the scaling factor 

 ensures that the mean value of the 

response spectra for the set of scaled motions 

is not less than the target spectrum over the 

period range from to (fig. 3b).

To select ground-motions using the approach 

“Best1”, where the discrepancy between scaled 

spectrum of a record and the target spectrum over 

the period range from to is minimized, 

Step 4 is modified as follows: 

4. Rank the scaled records based on their 

log Aˆ − SF1 × log A value; the record with 

the lowest value is ranked the highest. From 

the ranked list, select a set of m records to be 

used in nonlinear RHA of the systems 

described previously. 

Selection of ground-motions using the approach 

“Best2” requires that Steps 4-7 are iteratively 

implemented until the errors, defined as 

log Aˆ − SF1 × log A and A T( ) n − SF × A T( ) n , 

are minimized. At the end of Steps 1 to 7, the 

scaling factors for the sets of m ground-motions 

would have been determined. Nonlinear RHA is 

then used to obtain final EDP values. If at least 

seven ground-motions are analyzed ( ), the 

design values of EDPs are taken as the mean or 

median4 of the EDPs over the ground-motions 

used. If fewer than seven ground-motions are 

analyzed, the design values of EDPs are taken as 

the maximum values of the EDPs. 

 

4

Because the geometric mean and median of a lognormal 

distribution are the same, we decided to employ the term 

“median” instead of geometric mean, as is commonly done.

Benchmark Inelastic Deformations

Benchmark values of inelastic deformations ( ) 

were determined by using nonlinear RHA of the 

SDF systems described previously subjected to 

each of the 30 unscaled ground-motions, and 

computing the median and mean value of the data 

set. 

Hancock and others (2008) noted that the 

empirical ground-motion models that are used to 

derive the target spectrum assume the groundmotions to be lognormally distributed; therefore, the 

use of the median response spectrum of the 

records as a target spectrum is more consistent 

with the specification of the target spectrum. 

Similarly, it commonly is assumed that EDPs are 

lognormally distributed (Cornell and others, 2002); 

for this reason, it is more appropriate to represent 

the “mean” structural response by the median, a 

conclusion that is widely accepted. However, the 

ASCE/SEI 7 procedure states that the mean values 

of EDPs are used if at least seven ground-motions 

are considered. We decided to use both the 

median and the mean of the inelastic deformations 

as the benchmark values. In all cases of 

benchmark computations, the mean is larger than 

the median of inelastic deformations, indicating that 

the distribution of Dn is positively skewed. The 

percent differences between the two are in the 

ranges of 15 to 63, 23 to 39, 38 to 48, and 42 to 63 

for elastic-perfectly-plastic systems with R equal to 

1, 2, 4, and 8, respectively. For bilinear systems the 

percent differences are 11 to 26, 12 to 32, 20 to 45, 

and 27 to 56 for the same R values. Note that the 

difference between mean and median increases 

with increasing R value. 

Evaluation of ASCE/SEI 7 Scaling 

Procedure - Fewer Than Seven GroundMotions

The ASCE/SEI 7 scaling procedure was 

implemented for the inelastic SDF systems of this 

investigation subjected to one component of 

ground-motion (table 1). The accuracy of the 

ASCE/SEI 7 procedure was evaluated first by 
5

comparing the maximum value of the inelastic 

deformation due to seven sets of 3 to 6 scaled 

records against the benchmark value, defined as 

the median (or mean) value of due to the 30

unscaled ground-motions. These comparisons are 

shown in figures 4 and 5 for elastic-perfectly-plastic 

systems with Tn=0.2, 0.5, 1, and 2.5 sec, and R=1, 

2, 4, and 8 due to groups of 3, 4, 5, and 6 records

called as G3, G4, G5 and G6, respectively. Seven 

sets of records were considered in each of these 

groups. Among these seven sets, the first five sets 

of records were selected randomly out of 30 

records, and the two remaining sets of records 

were selected with the criteria explained previously; 

these are sets “Best1” and “Best2”. For each Tn, R, 

and constitutive model combinations, a total of 30 

sets of records are utilized. For the benchmark, the 

blue dot and the vertical line represent the median 

deformation value plus and minus one standard 

deviation (henceforth denoted as ), assuming a

lognormal distribution. For each set, the vertical line 

and the dot represent the range of the data set and 

maximum deformation value, respectively. Similar 

plots are presented in figures 6 and 7 for bilinear 

systems. Figures 4 and 5 permit the following 

observations: (1) Increasing the number of records 

from 3 to 6 has a minor effect in the accuracy of the 

procedure; overestimations range from 1 to 1500 

percent and from 35 to 820 percent for groups G3 

and G6, respectively. (2) The accuracy of the 

procedure decreases with increasing R value and 

decreasing period; the maximum error increases 

from 250 to 1,060 percent and from 620 to 1,500 

percent if R changes from 1 to 8, and Tn changes 

from 2.5 to 0.2 sec, respectively. (3) The 

improvement gained by the use of sets “Best1” and 

“Best2” is marginal. For R equals 8, the errors 

range from 10 to 190 percent and from 0.2 to 1,130 

percent for sets Best1 and Best2, respectively. For 

bilinear systems, the errors are slightly smaller than 

those for elastic-perfectly-plastic systems. 

The benchmark results shown in figures 4 

through 7 are based on the median deformation 

value; figure 8 compares the benchmark, 

calculated as the mean of 30 values, and the 

ASCE/SEI 7 deformation values for elasticperfectly-plastic and bilinear systems. Included also 

are the horizontal lines at 0.8 and 1.2 times the 

benchmark to indicate 20 percent error around 

the “true” value. It is apparent that the ASCE/SEI 7 

scaling procedure is not accurate and overly 

conservative, especially for elastic-perfectly-plastic 

systems. Insignificant improvement is gained by the 

use of sets “Best1” and “Best2”. 

Figures 9 and 10 show intra- and inter-set 

dispersion values to assess, respectively, the 

efficiency and consistency of the ASCE/SEI 7 

procedure. A lognormal distribution of values 

was assumed. As shown in figure 9, the intra-set 

dispersion increases with increasing period; 

implying that the procedure becomes less efficient. 

Similarly, inter-set dispersion increases with 

increasing R (fig. 10), indicating that the procedure 

becomes less consistent. Note that the inter-set 

dispersion tends to decrease with increasing 

number of records per set. According to the results 

presented in figures 4 through 10, the accuracy, 

efficiency, and consistency in the estimation of 

inelastic deformations are not achieved in the 

ASCE/SEI 7 procedure if fewer than seven records 

are employed. The procedure was found to be 

unacceptably conservative and unstable. 

Evaluation of ASCE/SEI 7 Scaling 

Procedure - Seven or More GroundMotions 

The accuracy of the ASCE/SEI 7 procedure was 

evaluated next by comparing the median (or mean)

value of the inelastic deformation due to seven 

sets of 7 to 10 scaled records against the 

benchmark value, defined as the median (or mean)

value of due to the 30 unscaled groundmotions.

Figures 11 and 12 show the range of inelastic 

deformation values for elastic-perfectly-plastic 

systems due to groups of 7, 8, 9, and 10 records

called G7, G8, G9, and G10, respectively. As 

explained previously, seven sets were considered 

in each of these groups (that is, a total of 28 sets 
6

plus sets Best1 and Best2 for each Tn, R and 

constitutive-model combinations). The dot and the 

vertical line represent the median deformation 

value assuming a lognormal distribution. 

Similar plots are presented in figures 13 and 14 for 

bilinear systems. Figures 15 and 16 compare the 

benchmark with the ASCE/SEI 7 deformation 

values for elastic-perfectly-plastic and bilinear 

systems for a range of R values. The ASCE/SEI 7 

results in figure 15 are the median of 7, 8, 9, and 

10 values from scaled records, and the 

benchmark is defined as the median of 30 

values from unscaled records, whereas in figure 

16, mean values are used for both the ASCE/SEI 7 

and the benchmark results. Included also in these 

figures are the lines at 0.8 and 1.2 times the 

benchmark to represent 20 percent error around 

the “true” value. By comparing figure 8 with figure 

16, it becomes obvious that the ASCE/SEI 7 

scaling procedure utilizing seven or more randomly 

selected records provides a more accurate 

estimate of inelastic deformations. However, the 

overestimations in median values of inelastic 

deformation generally are larger than 20 percent, 

especially for R=4 and 8, as shown in figure 17. 

Figures 11, 12, and 15 permit the following 

observations: (1) increasing the number of records 

from 7 to 10 has a minor effect in the accuracy of 

the procedure; overestimations range from 0.5 to 

140 percent and from 0.6 to 124 percent for groups 

G7 and G10, respectively. (2) The accuracy of the 

procedure decreases with increasing R value; the 

maximum error increases from 56 to 140 if the R

value changes from 1 to 8. (3) The fundamental 

period Tn does not affect significantly the accuracy 

of the ASCE/SEI 7 procedure if the records are 

selected randomly. The maximum error is 140 and 

134 percent for Tn=0.2 and 2.5 sec, respectively. 

Figures 13, 14, and 16 lead to similar conclusions. 

For short periods and large R values, the mean 

of randomly selected sets is not similar to the mean 

of the benchmark data set as demonstrated by the 

analysis of variance (ANOVA) values in figure 18. 

The ANOVA test returns the p value under the null 

hypothesis that both ASCE/SEI 7 and benchmark 

results are drawn from populations with the same 

mean. If p is near zero, it questions the null 

hypothesis and suggests that the ASCE/SEI 7 

mean is significantly different from the benchmark 

mean. This statistical test indicates that the random 

selection of records for the ASCE/SEI 7 procedure 

may lead to inconsistent results. 

For systems with Tn≥0.5 sec or small R values, 

set “Best2” is more accurate than set “Best1”, 

demonstrating that consideration of spectral shape 

and in selecting and scaling ground-motions 

improves the estimates significantly (Figs. 11-

18). For Tn=2.5 sec, the error of the procedure 

ranges from 9 to 60 percent and from 3 to 30 

percent for sets “Best1” and “Best2”, respectively. 

For systems with very short periods ( sec) 

and large R values ( and 8), both sets “Best 

1” and “Best 2” lead to inaccurate estimates of 

inelastic deformations (figs. 11-18); overestimations 

exceed 100 percent due to the high variability of 

spectral pseudo-accelerations and large 

discrepancies between elastic and inelastic spectra 

for periods in the acceleration sensitive region and 

large R values. 

The intra- and inter-set dispersion values are 

shown next in figures 19 and 20, where a 

lognormal distribution of values was assumed. 

The intra-set dispersion increases with increasing 

period, indicating larger variability of response 

values within a set. Similarly, the inter-set 

dispersion increases with increasing R values, 

implying that larger inelastic deformations would 

lead to increased set-to-set variability. As expected, 

the inter-set dispersion tends to decrease with 

increasing number of records per set. By 

comparing figures 10 and 20, it becomes evident 

that utilizing seven or more randomly selected 

records in the ASCE/SEI 7 reduces the inter-set 

dispersion significantly; the reduction is more 

pronounced for elastic-perfectly-plastic systems. 

The reduced inter-set variability indicates the 

consistency in the benchmark estimates of the 

ASCE/SEI 7 procedure using different sets. For 

systems with a fundamental period in the velocity 

or displacement sensitive region, accuracy, 

efficiency, and consistency are achieved only if 
7

records are selected on the basis of their spectral 

shape and as opposed to random selection. 

Modal-Pushover-Based Scaling 

Procedure - Alternative to the ASCE/SEI 

7 Scaling Procedure

Because the ASCE/SEI 7 ground-motion scaling 

method does not consider explicitly the inelastic 

behavior of the structure, it may not be appropriate 

for structures with short periods or for structures 

located in near-field sites where the inelastic 

deformation can be significantly larger than the 

deformation of the corresponding linear system. 

For such cases, scaling methods that are based on 

the inelastic deformation spectrum, or methods that 

consider the response of the first-“mode” inelastic 

SDF system, are more appropriate (Luco and 

Cornell 2007, Tothong and Cornell 2008, PEER 

2009). Kalkan and Chopra (2010a,b and 2011a,b) 

used these concepts to develop a modal pushoverbased scaling (MPS) procedure for selecting and 

scaling earthquake ground-motion records in a 

form convenient for evaluating existing structures 

and proposed designs of new structures. This 

procedure explicitly considers structural strength, 

determined from the first-“mode” pushover curve, 

and determines a scaling factor for each record to 

match a target value of the deformation of the first-

“mode” inelastic SDF system. If the MPS procedure

were applied to the systems of this investigation, it 

would lead to null error in the estimation of inelastic 

deformations and null intra- and inter-set 

dispersions. Therefore, the MPS procedure for SDF 

systems would be absolutely accurate, efficient and 

consistent. 

Conclusions

Based on elastic-perfectly-plastic and bilinear 

inelastic single-degree-of-freedom systems, the 

accuracy, efficiency, and consistency of the 

ASCE/SEI 7 ground-motion scaling procedure are 

examined by comparing the median and mean 

values of the inelastic deformation due to 480 sets 

of scaled records against benchmark values. The 

number of records in these sets varies from three 

to ten. The records in each set were selected either 

(i) randomly, (ii) considering their spectral shapes 

or (iii) considering the design spectral-acceleration 

value in addition to their spectral shapes. 

This evaluation of the ASCE/SEI procedure has led 

to the following conclusions: 

1. The ASCE/SEI 7 scaling procedure does 

not insure a unique scaling factor for each 

record; various combinations of scaling 

factors can be defined to insure that the 

mean spectrum of scaled records remains 

above the target spectrum over the 

specified period range. Utilizing a minimum 

scale factor closest to unity for each record 

may overcome this flaw in the ASCE/SEI 7 

scaling procedure. 

2. The ASCE/SEI 7 procedure is found to be 

conservative; it is neither efficient nor 

consistent if fewer than seven groundmotions are utilized. Only marginal 

improvements were gained by selecting 

and scaling records based on their spectral 

shape (as in sets Best1) and design 

spectral-acceleration value, (as in 

sets Best2). 

3. The ASCE/SEI 7 scaling procedure 

utilizing seven or more randomly selected 

records provides more accurate estimates 

of inelastic deformations. However, the 

overestimations in median values of 

inelastic deformation generally are larger 

than 20 percent. Increasing the number of 

records from 7 to 10 has a minor effect in 

the accuracy of the procedure. Thus, use 

of 7 records is found to be sufficient. 

4. In general, the accuracy of the procedure 

decreases with increasing R value. The 

fundamental period, Tn, (that is, long or 

short periods) does not significantly affects 

its accuracy if the records are selected 

randomly. 

5. For systems with Tn ≥ 0.5 sec or small R 

values (R < 4), consideration of spectral 
8

shape and in selecting and scaling 

ground-motions improves the displacement 

response ( ) estimates significantly. For 

systems with very short periods (

sec) and large R values ( and 8) 

however, both sets “Best 1” and “Best 2” 

lead to inaccurate estimates of inelastic 

deformations with overestimations 

exceeding 100 percent. This is due to the 

high variability of spectral pseudoacceleration values, large discrepancies 

between elastic and inelastic spectra for 

periods in the acceleration-sensitive region 

and large R values (that is, increased 

inelastic response). For such cases, 

scaling methods that are based on the 

inelastic deformation spectrum, or that 

consider the response of the first-“mode” 

inelastic SDF system, are more 

appropriate. 

This study has focused on the statistical 

examination of the required number of records for 

the ASCE/SEI ground-motion scaling method, 

which has been limited to elastic-perfectly-plastic 

and bilinear inelastic single-degree-of-freedom 

systems. 

Acknowledgments

Juan C. Reyes would like to acknowledge the 

generous support of Universidad de los Andes for 

providing him the financial support for his stay at 

the U.S. Geological Survey, Menlo Park. Special 

thanks are extended to Anil K. Chopra, Charlie 

Kircher and Nico Luco for providing their valuable 

comments and suggestions, which help improving 

the technical quality of this report. 

References Cited

ASCE, 2005, Minimum design loads for buildings,

ASCE/SEI 7-05, Reston, VA.

ASCE, 2010, Minimum design loads for buildings and 

other structures, ASCE/SEI 7-10, Reston, VA.

Cornell, C.A., Jalayer, F., Hamburger, R.O., and Foutch, 

D.A., 2002, Probabilistic basis for 2000 SAC federal 

emergency management agency steel moment 

frame guidelines, Journal of Structural Engineering 

(ASCE); v. 128, no. 4, p. 526–533.

Hancock, J., Bommer, J., and Stafford, J., 2008, 

Numbers of scaled and matched accelerograms 

required for inelastic dynamic analyses, Earthquake 

Engineering and Structural Dynamics, v. 37, p. 

1585-1607. 

International Conference of Building Officials (ICBO), 

2006, International Building Code, Whittier, CA.

International Conference of Building Officials (ICBO), 

2007, California Building Code, Whittier, CA.

Kalkan, E., and Chopra, A.K., 2010a, Practical 

Guidelines to Select and Scale Earthquake 

Records for Nonlinear Response History Analysis 

of Structures, USGS Open File Report 2010-1068, 

126 p. (http://pubs.usgs.gov/of/2010/1068/). 

Kalkan, E., and Chopra, A.K., 2010b, Evaluation of 

Modal pushover-based scaling of one component 

of ground-motion: Tall buildings, Earthquake 

Spectra (submitted for publication; 

http://nsmp.wr.usgs.gov/ekalkan/PDFs/Papers/J33

_Kalkan_Chopra.pdf). 

Kalkan, E., and Chopra, A.K., 2011a, Modal-pushoverbased ground-motion scaling procedure, Journal of 

Structural Engineering (ASCE), v. 138, no. 3, p. 

289-310 

(http://nsmp.wr.usgs.gov/ekalkan/PDFs/Papers/J25

_Kalkan_Chopra.pdf).

Kalkan, E., and Kwong, N.S., 2011b, Documentation for 

Assessment of Modal Pushover-based Scaling 

Procedure for Nonlinear Response History Analysis 

of “Ordinary Standard” Bridges, U.S. Geological 

Survey Open-File Report 2010-1328, 56 p. 

(http://pubs.usgs.gov/of/2010/1328/).

Luco, N., and Cornell, A.C., 2007, Structure-specific 

scalar intensity measures for near-source and 

ordinary earthquake ground-motions, Earthquake 

Spectra, v. 23, n. 2, p. 357–392.

Nocedal, J., and Stephen, J.W., 2006, Numerical 

optimization, 2nd ed.: Springer Series in Operations 

Research, Springer, USA.

PEER Ground-Motion Selection and Modification 

Working Group, 2009, Evaluation of Ground-Motion

Selection and Modification Methods: Predicting 

Median Interstory Drift Response of Buildings”, 
9

Haselton, C. B. (editor). PEER Report 2009/01, 

University of California, Berkeley, CA.

Tothong, P., and Cornell, A.C., 2008, Structural 

performance assessment under near-source pulselike ground-motions using advanced ground-motion

intensity measures, Earthquake Engineering and 

Structural Dynamics, v.37, no. 7, p. 1013-1037.

Notation

The following symbols are used in this report:

Pseudo-spectral-acceleration at period T

Vector of pseudo-spectral-acceleration values

Target value of pseudo-spectral-acceleration at period T

Vector of target pseudo-spectral-acceleration values

Mean scaled spectrum of m records

Maximum normalized difference between target and mean scaled spectrum

m Number of ground-motion records

Mw Moment magnitude

R Yield-strength reduction factor

RJB Joyner-Boore distance-perpendicular distance to surface projection of fault plane

SF Ground motion scaling factor

Tn Period of single-degree-of-freedom system; or elastic first-“mode” vibration period of structure

VS30 Average shear-wave velocity within 30 m depth from surface

Standard deviation
10

Table 1. Selected near-fault ground-motion records. [NEHRP, National Earthquake Hazard Reduction 

Program; AS, aftershock] 

Record 

sequence 

number

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Earthquake name Year

San Fernando, Calif. 1971

San Fernando, Calif. 1971

Imperial Valley (AS), Calif. 1979

Imperial Valley (AS), Calif. 1979

Imperial Valley (AS), Calif. 1979

Imperial Valley (AS), Calif. 1979

Imperial Valley (AS), Calif. 1979

Irpinia, Italy 1980

Superstition Hills (AS), Calif. 1987

Loma Prieta, Calif. 1989

Loma Prieta, Calif. 1989

Loma Prieta, Calif. 1989

Loma Prieta, Calif. 1989

Loma Prieta, Calif. 1989

Loma Prieta, Calif. 1989

Loma Prieta, Calif. 1989

Northridge, Calif. 1994

Northridge, Calif. 1994

Northridge, Calif. 1994

Northridge, Calif. 1994

Northridge, Calif. 1994

Northridge, Calif. 1994

Northridge, Calif. 1994

Northridge, Calif. 1994

Kobe, Japan 1995

Kobe, Japan 1995

Kobe, Japan 1995

Kobe, Japan 1995

Kobe, Japan 1995

Kobe, Japan 1995

Station Name

LA - Hollywood Stor FF

Santa Felita Dam (Outlet)

Calipatria Fire Station

Delta

El Centro Array #1

El Centro Array #13

Superstition Mtn Camera

Brienza

Wildlife Liquef. Array

Agnews State Hospital

Anderson Dam (Downstream)

Anderson Dam (L Abut)

Coyote Lake Dam (Downst)

Coyote Lake Dam (SW Abut)

Gilroy Array #7

Hollister - SAGO Vault

Castaic - Old Ridge Route

Glendale - Las Palmas

LA - Baldwin Hills

LA - Centinela St

LA - Cypress Ave

LA - Fletcher Dr

LA - N Westmoreland

LA - Pico & Sentous

Abeno

Kakogawa

Morigawachi

OSAJ

Sakai

Yae

Earthquake 

magnitude 

(Mw)

Joyner-Boore 

distance (km)

NEHRP site 

class

Highest 

usable 

period (sec.)

6.6 22.8 D 4

6.6 24.7 C 8

6.5 23.2 D 8

6.5 22.0 D 16

6.5 19.8 D 8

6.5 22.0 D 4

6.5 24.6 C 8

6.9 22.5 C 4

6.5 23.9 D 8

6.9 24.3 D 4

6.9 19.9 C 4

6.9 19.9 C 8

6.9 20.4 D 8

6.9 20.0 C 8

6.9 22.4 D 4

6.9 29.5 C 8

6.7 20.1 C 8

6.7 21.6 C 6

6.7 23.5 D 6

6.7 20.4 D 4

6.7 29.0 C 4

6.7 25.7 C 5

6.7 23.4 D 4

6.7 27.8 D 5

6.9 24.9 D 16

6.9 22.5 D 8

6.9 24.8 D 10

6.9 21.4 D 16

6.9 28.1 D 8

6.9 27.8 D 16
11

 

Figure 1. Response spectra of thirty near-fault ground-motions and their geometric-mean (that is, median) used 

as the target (that is, “design”) spectrum for evaluation of the ASCE/SEI 7 ground-motion scaling 

procedure. Damping ratio 5-percent. 
12

Figure 2. Schematic illustration of Step 3 of the evaluation methodology.

Figure 3. Schematic illustration of Step 6 of the evaluation methodology. 
13

Figure 4. Range of inelastic deformation values for elastic-perfectly-plastic systems with Tn=0.2 sec (top four 

panels) and Tn=0.5 sec (bottom four panels), and R=1, 2, 4, and 8 for sets of 3, 4, 5, and 6 records 

denoted respectively as G3, G4, G5, and G6. The blue dot and the vertical line represent the 

benchmark (denoted as B) median deformation value assuming a lognormal distribution. For each 

set, the vertical line and the dot represent the range of the data set and maximum deformation value, 

respectively. 
14

Figure 5. Range of inelastic deformation values for elastic-perfectly-plastic systems with Tn =1 sec (top four 

panels) and Tn=2.5 sec (bottom four panels), and R=1, 2, 4, and 8 for sets of 3, 4, 5, and 6 records 

denoted respectively as G3, G4, G5, and G6. The blue dot and the vertical line represent the 

benchmark (B) median deformation value assuming a lognormal distribution. For each set, the 

vertical line and the dot represent the range of the data set and maximum deformation value.
15

Figure 6. Range of inelastic deformation values for bilinear systems with Tn =0.2 sec (top four panels) and Tn=0.5 

sec (bottom four panels), and R=1, 2, 4, and 8 for sets of 3, 4, 5, and 6 records denoted respectively as 

G3, G4, G5, and G6. The blue dot and the vertical line represent the benchmark (B) median deformation 

value assuming a lognormal distribution. For each set, the vertical line and the dot represent the 

range of the data set and maximum deformation value, respectively. 
16

Figure 7. Range of inelastic deformation values for bilinear systems with Tn =1 sec (top four panels) and Tn =2.5 

sec (bottom four panels), and R=1, 2, 4, and 8 for sets of 3, 4, 5, and 6 ground-motion records denoted 

respectively as G3, G4, G5, and G6. The blue dot and the vertical line represent the benchmark (B) 

median deformation value assuming a lognormal distribution. For each set, the vertical line and the 

dot represent the range of the data set and maximum deformation value, respectively. 
17

(a) Elastic-perfectly-plastic systems

(b) Bilinear systems

Figure 8. Comparison of benchmark and ASCE/SEI 7 deformation values for (a) elastic-perfectly-plastic, and (b) 

bilinear systems. The benchmark engineering demand parameters (EDPs) correspond to the mean of 

30 deformation values. The deformation values for each set scaled by the ASCE/SEI 7 procedure are 

obtained as the maximum deformation values of 3, 4, 5 and 6 records in each of seven sets. Included 

also are sets Best1 and Best2.
18

(a) Elastic-perfectly-plastic systems

(b) Bilinear systems

Figure 9. Comparison of benchmark and ASCE/SEI 7 intra-set (that is, within set) dispersion values for (a) elasticperfectly-plastic, and (b) bilinear systems. G3, G4, G5, and G6 contain seven sets of 3, 4, 5 and 6 

ground-motion records, respectively. Lognormal distribution is assumed.
19

 (a) Elastic-perfectly-plastic systems

(b) Bilinear systems

Figure 10. ASCE/SEI 7 inter-set (that is, set-to-set) dispersion values for (a) elastic-perfectly-plastic, and (b) 

bilinear systems. G3, G4, G5, and G6 contain seven sets of 3, 4, 5 and 6 ground-motion records, 

respectively. Lognormal distribution is assumed.
20

Figure 11. Range of inelastic deformation values for elastic-perfectly-plastic systems with Tn=0.2 (top four panels) 

and 0.5 sec (bottom four panels), and R=1, 2, 4, and 8 for sets of 7, 8, 9, and 10 ground-motion records 

denoted respectively as G7, G8, G9, and G10. The blue dot and the vertical line represent the 

benchmark (B) median deformation value assuming a lognormal distribution. For each set, the 

vertical line and the dot represent the range of the data set and median deformation value, respectively. 
21

Figure 12. Range of inelastic deformation values for elastic-perfectly-plastic systems with Tn=1 (top four panels) 

and 2.5 sec (bottom four panels), and R=1, 2, 4, and 8 for sets of 7, 8, 9, and 10 ground-motion records 

denoted respectively as G7, G8, G9, and G10. The blue dot and the vertical line represent the 

benchmark (B) median deformation value assuming a lognormal distribution. For each set, the 

vertical line and the dot represent the range of the data set and median deformation value, respectively. 
22

Figure 13. Range of inelastic deformation values for bilinear systems with Tn=0.2 (top four panels) and 0.5 s 

(bottom four panels), and R=1, 2, 4, and 8 for sets of 7, 8, 9, and 10 ground-motion records denoted 

respectively as G7, G8, G9, and G10. The blue dot and the vertical line represent the benchmark (B) 

median deformation value assuming a lognormal distribution. For each set, the vertical line and the 

dot represent the range of the data set and median deformation value, respectively. 
23

Figure 14. Range of inelastic deformation values for bilinear systems with Tn=1 (top four panels) and 2.5 sec 

(bottom four panels), and R=1, 2, 4, and 8 for sets of 7, 8, 9, and 10 ground-motion records denoted 

respectively as G7, G8, G9, and G10. The blue dot and the vertical line represent the benchmark (B) 

median deformation value assuming a lognormal distribution. For each set, the vertical line and the 

dot represent the range of the data set and median deformation value, respectively. 
24

(a) Elastic-perfectly-plastic systems

(b) Bilinear systems

Figure 15. Benchmark and ASCE/SEI 7 deformation values for (a) elastic-perfectly-plastic, and (b) bilinear 

systems. The benchmark EDPs correspond to the median of 30 deformation values. The deformation 

values for each set scaled by the ASCE/SEI 7 procedure are obtained as the median of 7, 8, 9, and 10 

deformation values. Included also are sets Best1 and Best2.
25

(a) Elastic-perfectly-plastic systems

(b) Bilinear systems

Figure 16. Benchmark and ASCE/SEI 7 deformation values for (a) elastic-perfectly-plastic, and (b) bilinear 

systems. The benchmark EDPs correspond to the mean of 30 deformation values. The deformation 

values for each set scaled by the ASCE/SEI 7 procedure are obtained as the mean of 7, 8, 9, and 10 

deformation values. Included also are sets Best1 and Best2.
26

(a) Elastic-perfectly-plastic systems

(b) Bilinear systems

Figure 17. Error in the estimation of median deformation values for (a) elastic-perfectly-plastic and (b) bilinear 

systems. The benchmark EDPs correspond to the median of 30 deformation values. The deformation 

values for each set scaled by the ASCE/SEI 7 procedure are obtained as the median of 7, 8, 9, and 10 

deformation values. Included also are sets Best1 and Best2.
27

(a) Elastic-perfectly-plastic systems

(b) Bilinear systems

Figure 18. ANOVA values for (a) elastic-perfectly-plastic, and (b) bilinear systems. The ANOVA test compares 

deformation values from each set against the benchmark deformation values.
28

 (a) Elastic-perfectly-plastic systems

(b) Bilinear systems

Figure 19. Benchmark and ASCE/SEI 7 intra-set dispersion values for (a) elastic-perfectly-plastic, and (b) bilinear 

systems. Lognormal distribution is assumed.
29

(a) Elastic-perfectly-plastic systems

(b) Bilinear systems

Figure 20. ASCE/SEI 7 inter-set dispersion values for (a) elastic-perfectly-plastic, and (b) bilinear systems. 

Lognormal distribution is assumed.