Potential Differences between Time Series and Random Vibration Theory Site Response

Albert R. Kottke1 and Ellen M. Rathje2

1Pacific Earthquake Engineering Research Center, University of California, Berkeley; PH (612) 281-2259; Email: albert.kottke@gmail.com 2Department of Civil, Architectural, and Environmental Engineering, University of Texas, Austin; Email: e.rathje@mail.utexas.edu

ABSTRACT

Response spectra at the surface of two sites are computed with equivalent-linear site response methods using time series and random vibration theory (RVT) input motions. The results indicate a potential for significant differences between the expected response spectrum computed using RVT and time series methods. Even un-der linear-elastic conditions (i.e., constant properties), the RVT method computes higher peaks and lower troughs in the ratio between the surface response spectrum and the input response spectrum when compared to the time series analyses. The dif-ferences between the RVT and time series analysis are dependent on the site charac-teristics. The implication of this study is that the results of an RVT site response anal-ysis cannot be assumed to be representative of a time-series site response analysis.

INTRODUCTION

Site specific ground motions used in the assessment of seismic performance are computed using site response analyses that simulate the nonlinear response of the soil to earthquake ground shaking. In an analysis, the expected input motion is first estimated using a ground motion prediction equation and then characterized using either time series or random vibration theory (RVT). The use of RVT removes the time consuming process of selecting and scaling/modifying time series, and permits a statistically stable estimate of the expected surface response spectrum to be computed in one analysis – instead of the multiple time series analyses required due to motion-to-motion variability. While the RVT approach offers many advantages over the time series approach, there are few published validations of RVT based site response.

In the following study, linear and equivalent-linear site response simulations are conducted at three sites using simulated time series and RVT input motions. The relative differences in the spectral ratios are compared and factors governing the dif-ferences are identified.

RANDOM VIBRATION THEORY

Random vibration theory characterizes a motion by the amplitude of the Fou-rier amplitude spectrum (FAS) and the ground motion duration. No information re-garding the phase of the motion is known, thus the complete time series cannot be computed. Instead, RVT allows for calculation of the expected peak value in the time domain using a combination of Parseval’s theorem and extreme value statistics. Par-seval’s is used to relate the FAS to the root-mean-squared acceleration (arms) of the ground motion as follows:

544GeoRisk 2011 © ASCE 2011

Geo-Risk 2011

Dow

nloa

ded

from

asc

elib

rary

.org

by

RM

IT U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 08

/22/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

2| |

where Trms is the duration of the time series motion, and |X(f)| is the amplitude of the FAS at a frequency of f. A more simplified notion uses the zero-th moment (m0), where the n-th moment is defined as:

2 2 | |

The peak acceleration (amax) is related to arms by the peak factor (PF= amax / arms). Cartwright and Longuet-Higgins (1956) derived an integral expression for the ex-pected value of the peak factor in terms of the number of extrema (Ne) and the band-width (ξ) of the time series (Boore 2003):

√2 1 1 ξ

where Ne and ξ are based on the spectral moments (m0, m2,and m4) of the FAS and the ground motion duration (Tgm):

ξ

The duration used in the calculation of the arms differs from the duration used in the calculation of Ne. When computing an oscillator response, Boore and Joyner (1984) recommend that the Trms duration be increased to account for the extended duration.

ANALYTICAL MODELING

Site Profiles Analyzed. Site response simulations were performed for three sites: Turkey Flat (TF) near Parkfield, CA; Sylmar County Hospital (SCH) in Southern CA; and the Calvert Cliffs (CC) on the coast of Chesapeake Bay in MD. The TF site char-acteristics were based on data collected by Real et al. (2006). The velocity profile for the SCH site was based on the measurements made by Gibbs et al. (1999), but modi-fied slightly to reduce the potential for large strains in the low-velocity surface (i.e., the shear-wave velocity of the surface layer was increased from 200 to 250 m/s). The CC site profile was based on data presented in the UniStar Nuclear Services (2007) Final Safety Analysis Report. Equivalent-linear analyses were performed using the SCH and CC sites. For these analyses, nonlinear curves were computed using the Darendeli (2001) empirical model and the reported index properties. The three differ-ent velocity profiles, shown in Figure 1, have similar velocities but different depths to bedrock. The site frequency, defined by the quarter wavelength ( 4 / , where H is the height of the soil column and is the time-averaged shear-wave velocity), cor-responding to the TF, SCH, and CC sites are 4.8, 1.2 and 0.2 Hz, respectively.

Input Motions. The methods that are commonly used to define a ground motion for site response analysis include: scaled recorded motions, spectrally matched motions,

545GeoRisk 2011 © ASCE 2011

Geo-Risk 2011

Dow

nloa

ded

from

asc

elib

rary

.org

by

RM

IT U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 08

/22/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

and synthetically generated motions. To minimize differences between the time series and RVT input motions, stochastically simulated time series were selected for these comparisons. The stochastic method computes an FAS based on seismological mod-els that incorporate the effects of the source and the path. The RVT input motion is defined using the computed FAS and duration. The time series are simulated by using the program SMSIM which uses the same computed FAS to modify a windowed time series of white noise (Boore, 2003). Using these techniques, an RVT and 100 time series input motions were simulated for a magnitude 6.5 earthquake at a distance of 20 km. The peak ground acceleration (PGA) for this event was 0.17 g. Additionally, the motions were scaled up to PGA of 0.40 g by applying a scale factor of 2.35 so that a range of input intensities could be examined.

RESULTS

The stochastically simulated motions were propagated through the TF, SCH, and CC sites; first assuming the response of the soil is linear-elastic (LE) with 5% damping. The analyses were compared using the spectral ratio, which is defined as the ratio of the surface response spectrum to the input response spectrum. The relative difference between the time series and RVT analyses at a frequency f ( is de-fined as follows:

, ,

,

where , is the spectral ratio computed using RVT and , is the geometric mean of spectral ratios computed using the time series simulations.

Linear Elastic Analyses. The results of the TF site for the time series and RVT anal-yses are shown in Figure 2. The relative difference of the spectral ratio ( ) between the RVT and median time series motions is less than 5% at most frequencies, with the largest differences (~8%) at frequencies of strong amplification (7.7 and 14 Hz). Sim-ilar results are observed in the response calculated for the SCH site (Figure 3) with the maximum relative difference of approximately 6% occurring at the site frequency (1.7 Hz). In the response of the CC site the RVT analysis is again relatively high at the site frequency (0.3 Hz), but the relative difference of 25% is more significant than for the other two sites (Figure 3). In addition to being relatively high at the site fre-

Figure 1. Shear-wave velocity (Vs) profiles for the TF (left), SCH (center), andCC (right) sites.

546GeoRisk 2011 © ASCE 2011

Geo-Risk 2011

Dow

nloa

ded

from

asc

elib

rary

.org

by

RM

IT U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 08

/22/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

quency, the relative difference of RVT for the CC site is -12% at a frequency of 0.5 Hz. The spectral ratios for the CC site indicate a tendency for the RVT analysis to exaggerate the influence of the site and produce a response with higher peaks and lower troughs.

The results of the LE analyses indicate that the difference between RVT site response analysis relative to time-series analysis is dependent on the site characteris-tics. The site characteristics are reflected in the site transfer function used in the wave propagation analysis. The LE transfer functions for the three sites are shown in Figure 3 and these transfer functions were used for both the time series and the RVT anal-yses. Figure 3 shows that the transfer function of the TF site has a strong narrow peak at 14 Hz; the SCH site has a less strong, but wide, peak at 1.7 Hz; and the CC site has several strong, narrow peaks at frequencies below 1 Hz and its largest peak is at 0.2 Hz. Differences in the spectral ratios computed by RVT and time-series analysis are a result of differences in the shape of the FAS of the simulated time series and of the RVT input motion relative to these transfer functions.

Figure 4 shows the input FAS for the time series with the largest and smallest spectral ratios at a frequency of 1.7 Hz (the approximate site frequency) for the SCH

Figure 2. The spectral ratio (top row) computed with linear-elastic analyses forthe RVT and time series methods and the associated relative difference (bottomrow) for the TF (left column), SCH (center column), and CC (right column) sites.

Figure 3. Amplitude of the linear-elastic transfer functions for the TF (left), SCH(center), and CC (right) sites.

547GeoRisk 2011 © ASCE 2011

Geo-Risk 2011

Dow

nloa

ded

from

asc

elib

rary

.org

by

RM

IT U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 08

/22/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

site. Also shown is the input FAS for the RVT analysis. The time series with the smallest spectral ratio (SR(1.7 Hz)=1.72) shows a deficiency in motion at or near the frequency of the peak in the transfer function, while the time series with the largest spectral ratio (SR(1.7 Hz)=2.13) contains larger amplitudes around the same frequen-cy. Additionally, the time series FAS display peaks and valleys within the frequency range of the peak of the transfer function. On the other hand, the RVT input motion contains no peaks and valleys in its FAS and varies smoothly. Because RVT varies smoothly and does not have any valleys or peaks within the width of the peak of the transfer function, it propagates the full strength of the transfer function to the surface and predicts a larger spectral ratio. As all time-series motions will have some irregu-larities in the FAS across the peak in the transfer function, the median spectral ratio from a suite of time-series analyses will never be as large as that calculated by RVT.

The effect of the smooth RVT input motion is magnified when the peak in the transfer function is multiplied by only a small number of frequencies of the input time series. The frequency increment at which the FAS is defined is constant for a given time series, and therefore the effect is magnified when the natural frequency of the site is low. Additionally, the effect will be magnified when the width of the transfer function is very narrow or when there is significant amplification in the transfer func-tion.

For the CC site, the input FAS for the time series with the largest and smallest spectral ratios at a frequency of 0.3 Hz (the approximate site frequency) are shown in Figure 4 along with the LE transfer function and the input FAS for RVT. The first mode amplification for the CC site occurs at a lower frequency and over a narrower frequency range than the first mode amplification for the SCH site; thus, there is more difference between the spectral ratios from the time series analysis and RVT for the CC site (approximately 25% relative difference, see Figure 3). Additionally, there is more variability in the spectral ratios at the site frequency within the time series anal-

Figure 4. The input FAS for the time series with the largest and smallest spectralratios for the SCH (left) and CC (right) sites, along with the input FAS of theRVT analysis and the LE transfer function.

548GeoRisk 2011 © ASCE 2011

Geo-Risk 2011

Dow

nloa

ded

from

asc

elib

rary

.org

by

RM

IT U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 08

/22/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

ysis for CC site, with the maximum spectral ratio being 81% larger than the minimum spectral ratio. For SCH site, the maximum spectral ratio is only 23% larger than the minimum spectral ratio.

Equivalent-Linear Analyses. Equivalent-linear analyses were performed to investi-gate how the relative difference changes with input intensity and induced strain level. The input motions were propagated through the SCH site at two different intensity levels (0.17 and 0.40 g). Figure 5 shows the site response results for both the RVT and time-series analysis for a median input PGA of 0.17 g. In general, the results show good agreement between the time series and RVT analyses. However, the rela-tive difference in the RVT analysis has increased compared to the LE analysis (Figure 2), with the maximum relative difference at the site frequency now approaching 10%. At frequencies with no site amplification (i.e., troughs in the transfer function, such as at 2.3, 4.5, and 7.3 Hz), there is a consistent under prediction of the spectral ratio. These trends are even stronger when the intensity of the input motions increases to 0.40 g (Figure 5), where the relative difference at the site frequency exceeds 10%. These analyses show that as the intensity of the input motion increases, the relative difference at the site frequency increases and it shifts to lower frequencies due to the reduction in the site frequency associated with softening of the site.

To explain the increase in the relative difference with increase in intensity level, it is important to understand how the median response from the time series analyses are influenced by the different strain levels induced by each individual mo-tion. Consider the two stochastic motions shown in Figure 6. Motion #68 tends to strain the site less than the median, which makes the peak in the spectral ratio occur at a higher frequency (i.e., stiffer site). Motion #7 induces strains in the site larger than the median, causing the peak in the spectral ratio to be shifted towards lower frequen-cies. At the frequency of approximately 1.1 Hz (roughly the average degraded site

Figure 5. The spectral ratio (top row) and relative difference (bottom row) forthe SCH site computed with EQL site response at input PGA of 0.17 (left col-umn) and 0.40 g (right column).

549GeoRisk 2011 © ASCE 2011

Geo-Risk 2011

Dow

nloa

ded

from

asc

elib

rary

.org

by

RM

IT U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 08

/22/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

frequency), these two motions both have spectral ratios less than the median of all of the motions, and thus motions that strain a site more than or less than the median pro-duce smaller spectral ratios at the average site frequency. Motions that induce strains similar to the median response have the largest spectral ratio at the average site fre-quency. When all of the motions within the suite are averaged together, the effect of these differences is to pull down the median spectral response at the site frequency. At frequencies away from the peak spectral ratio, the motions #68 and #7 balance each other out (i.e., one is larger than the median and the other is smaller) such that the median of the time-series analyses is similar to RVT.

The relative difference of the EQL RVT analysis and the EQL time-series analysis for the CC site (Figure 7) shows slightly different trends than were observed for the SCH site. The low site frequency of approximately 0.3 Hz along with high bedrock shear-wave velocity of this Eastern US site (Figure 1) accentuates the rela-

Figure 6. The spectral ratio and strain profile for two time series that inducevarying levels of strain.

Figure 7. The spectral ratio (top row) and relative difference (bottom row) forthe CC site computed with EQL site response at input PGA of 0.17 (left column)and 0.40 g (right column).

550GeoRisk 2011 © ASCE 2011

Geo-Risk 2011

Dow

nloa

ded

from

asc

elib

rary

.org

by

RM

IT U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 08

/22/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

tive difference of the RVT spectral ratio at frequencies associated with peaks in the transfer function. This relative difference is as large as 25%. Additionally, the relative difference becomes increasingly negative at high frequencies and becomes increas-ingly negative as the input intensity increases.

SUMMARY

The analyses presented in this study demonstrate the time series and RVT site produce different estimates of the site response quantified by the spectral ratio. The difference in the site response computed using RVT analysis is affected by the char-acteristics of both the site and the input motions. The smooth shape of the RVT input FAS is more sensitive to the site transfer function than the irregular FAS of a time series, which results in larger amplification at the frequencies associated with peaks in the transfer function and less amplification at frequencies associated with troughs in the transfer function for RVT analyses. These differences are more significant for sites with low natural frequencies because the site amplification transfer function tends to be narrower and the number of points in the FAS affected by the amplifica-tion is smaller. Also important is the amplitude of the transfer function which increas-es for sites with large bedrock shear-wave velocity or other strong velocity contrasts. The relative difference of RVT at the site frequency increases with increasing intensi-ty because the RVT analysis does not take into account how individual motions strain a site differently.

Financial support for this research was provided by the Nuclear Regulatory Commission.

REFERENCES

Boore, D. (2003). “Simulation of ground motion using the stochastic method.” Pure and Applied Geophysics, 160(3-4), 635-676.

Boore, D. and Joyner, W. (1984). “A note on the use of random vibration theory to predict peak amplitudes of transient signals.” Bulletin of the Seismological Society of America, 74(5), 2035-2039.

Cartwright D. and Longuet-Higgins M. (1956). “The statistical distribution of the maxima of a random function.” In Proc. of the Royal Society of London. Se-ries A, Mathematical and Physical Sciences, 237(1209):212{232, 1956.

Gibbs J., Tinsley J., Boore D., and Joyner, W. (1999). Seismic velocities and geologi-cal conditions at twelve sites subjected to strong ground motion in the 1994 Northridge, California, earthquake: a revision of OFR 96-740. USGS OFR 99-446, U.S. Department of the Interior, U.S. Geological Survey, 1999.

Darendeli, M. (2001). Development of a new family of normalized modulus reduction and material damping curves. PhD thesis, The University of Texas, Austin, TX.

Real C., Shakal A., and Tucker, B. (2006). “Overview of the Turkey Flat ground mo-tion prediction experiment.” In Proc. SMIP06 Seminar on Utilization of Strong-Motion Data, 117-136.

UniStar Nuclear Services. (2007). Calvert Cliffs Nuclear Power Plant Unit 3, Com-bined License Application, Part 2: Final Safety Analysis Report, Rev. 6.

551GeoRisk 2011 © ASCE 2011

Geo-Risk 2011

Dow

nloa

ded

from

asc

elib

rary

.org

by

RM

IT U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 08

/22/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.