Bulletin of the Seismological Society of America, Vol. 104, No. 1, pp. 540–546, February 2014, doi: 10.1785/0120130176
Comment on “Comparison of Time Series and Random-Vibration Theory Site-Response Methods” by Albert R. Kottke and Ellen M. Rathje by Vladimir Graizer
Introduction to another? After reading this paper, one cannot make a decision about which method is more reliable and when it can be applied. From my point of view, this paper is lacking a critical part: the comparison with actually recorded earthquake data that can help in assessing the applicability and limitations of RVT. It is especially surprising because the authors are aware of existing data and publications in this area. I am referring to the strong-motion data recorded by the geotechnical arrays, which present an excellent opportunity to test site-response analysis methods. The California Geological Survey (CGS) has more than 20 such arrays with a number of well-recorded earthquake data, and these arrays were specifically designed to help testing site-response analysis methods (Shakal et al., 2004). The EQL analysis using the RVT approach was introduced in modeling strong ground motion by Vanmarcke (1975), Der Kiureghian (1980), Hanks and McGuire (1981), Boore (1983, 2003), Boore and Joyner (1984), and Herrmann (1985). It became even more popular in recent years, because it is much less labor intensive than classical TS, and because the RVT approach does not require choosing and matching strong-motion records (Silva and Lee, 1987; Silva et al., 1997; Deng and Ostadan, 2008; Kottke and Rathje, 2008; Kottke, 2010). In addition, the RVT approach uses Fourier, power, or response spectra as an input to the siteresponse analysis. In the meantime, it is important to remember that strong earthquake ground motions violate many of the assumptions on which RVT is based, including stationarity and randomness of the process. Despite these problems, studies conducted by Hanks and McGuire (1981), Boore (1983, 2003), Silva and Lee (1987), Silva et al. (1997), and Deng and Ostadan (2008) indicated that RVT can be used to provide reasonable estimates of mean response of earthquake ground motions. Unfortunately, none of these studies provided limitations of the RVT approach. There are also some publications pointing to the differences, which can be observed between the TS and RVT approaches (e.g., Kottke, 2010; Graizer, 2011; Renault and Hunfeld, 2011). Because the Kottke and Rathje (2013) paper is missing critical comparisons with recorded earthquake data, I will shortly describe results of my recently performed comparisons using the recordings from the TI downhole array located in San Francisco Bay between San Francisco and Oakland, California (Graizer, 2011). The TI downhole array was
Methods of site-response analysis require theoretical and empirical testing and validation before they can be used in seismic-hazard assessment. Comparison of site amplification functions (SAFs) calculated using earthquake data recorded by seismic arrays with SAF obtained using analytical approaches represent the most important tests of reliability of those methods. High-quality, low-amplitude earthquake data recorded by the Treasure Island (TI) downhole array between 1993 and 2010 were used for comparisons with the two versions of the equivalent linear (EQL) site-response methods implemented in the computer program STRATA (Kottke and Rathje, 2008; Rathje and Kottke, 2013): • time-series (TS) approach • random-vibration theory (RVT) approach. Use of the TS approach in STRATA matches well the empirically determined SAF between the bedrock and the surface. The STRATA version of the RVT approach can produce a significantly different SAF from the empirically determined and the TS approach. Further testing of different realizations of the RVT method is desirable to assess the method’s reliability and limitations. The paper of Kottke and Rathje (2013) raises a very important issue of comparison of the two site-response analysis approaches: classical TS SHAKE-type (Idriss and Seed, 1968; Idriss and Sun, 1992) and RVT (Vanmarcke, 1975; Der Kiureghian, 1980; Hanks and McGuire, 1981). These approaches represent two versions of EQL site-response analyses. I found this discussion timely, because more and more probabilistic seismic-hazard projects, including those for critical facilities, use the RVT approach. Kottke and Rathje (2013) properly pointed out that the fact that the RVT site-response analysis does not require input TS makes it an attractive alternative to the TS approach. Nevertheless, the main questions are: • Is this approach providing reliable results? • Are there limitations on applying this approach? • Are the comparisons provided in Kottke and Rathje (2013) paper applicable to all realizations of the RVT approach? Unfortunately, after reading the Kottke and Rathje (2013) paper these questions remain unanswered because the authors conduct comparisons of results provided by the two computational methods. What kind of conclusions can be made based only on comparing one computational approach 540
Comment
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Figure 1. P- and S-wave velocities and soil profile at Treasure Island array. Triangles show locations of the instruments. The color version of this figure is available only in the electronic edition. installed in 1992 by the CGS to study the response of softsoil-over-rock geologic structure to earthquake motion. The TI array has sensors located in the Franciscan bedrock (104 and 122 m depths), bay sediments (16, 31, and 44 m depths), artificial fill (7 m depth), and at the surface. High-quality low-amplitude earthquake data not exceeding 0:03g recorded by the array between 1993 and 2010 were used for comparisons with the TS and RVT versions of the EQL site-response methods implemented in the computer program STRATA (Kottke and Rathje, 2008; Rathje and Kottke, 2013).
Velocity and Geology Profile at Treasure Island Velocity and geology profiles at TI array are shown in Figure 1. Weathered Franciscan shale and sandstone are encountered at 88 m beneath the site with more-competent sandstone found at a depth of about 98 m (Darragh and Idriss, 1997). The TI array was installed in 1992 by the CGS with support of the National Science Foundation (NSF) to study the response of soft-over-rock geologic structure to earthquake motion (Darragh et al., 1993; de Alba et al., 1994). Original downhole S-wave velocity measurements were performed in the 104 m deep hole by the U.S. Geological Survey (Gibbs et al., 1992). I performed new S-wave velocity averaging based on the P–S suspension logging conducted in the more recently drilled 122 m deep downhole (Graizer and Shakal, 2004).
Earthquake Data Since 1993, more than a dozen low-amplitude earthquakes were recorded by the TI array. In this paper, records
of eight earthquakes (16 horizontal components) with 3:6 < M < 5:4 are used for site amplification studies (Table 1). The strongest shaking of 0:03g at the surface was recorded at an epicentral distance of 13.2 km from the Mw 4.0 Berkeley earthquake of 5 September 2003. Strong-motion records were processed using CGS standard processing procedure described in Shakal et al. (2003) with filtering using Butterworth filters of the 4th order on both low- and high-frequency ends. The frequency band of most processed records used in this study is 0.3–40 Hz. Figure 2 demonstrates an example of the north–south component of the array recordings (acceleration, velocity, and displacement) obtained during the 1999 M w 4.6 Bolinas earthquake. Acceleration, velocity, and displacement at both bedrock locations (104 and 122 m depths) are almost identical, and relatively simple. The motion becomes much more complex with longer duration and several times higher amplitudes in soft material (alluvium, bay mud, and artificial fill) and at the ground surface. Fourier transfer functions from the surface to rock (104 and 122 m instruments) are shown in Figure 3. Both horizontal components demonstrate amplification at the following six frequencies: 0.8, 1.9, 3.4, 4.4, 5.8, and 6.8 Hz. Peak amplifications at these frequencies are very stable. The lowest frequency (first mode) is associated with the alluvium-torock interface at a depth of 88 m. Using an average S-wave velocity of V S ∼ 267 m=s in the upper layer of thickness h 88 mm results in a resonance peak with fundamental frequency f V S =4h (e.g., Dobry et al., 2000) of 0.76 Hz, fairly close to the empirical value of 0.8 Hz. Figure 4 shows individual and median (thick line) ±1 standard error response spectral ratios of surface to
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Figure 2. Acceleration, velocity, and displacement recorded at Treasure Island at the surface and depths of 7, 16, 31, 44, 104, and 122 m during the Mw 4.6 Bolinas earthquake of 18 August 1999. downhole bedrock motions. Five percent dampedresponse spectral ratios demonstrate significant variations in amplitudes (Fig. 4) between different earthquakes with the first (0.8 Hz) and the second (1.9 Hz) amplification frequencies the same as at the Fourier amplification functions (Fig. 3), and the three other peaks also at close frequencies of 3.3, 4.5, and 6.25 Hz. Large variations in amplitudes of SAF are most likely due to the differences in wave-propagation paths through the anisotropic crust of Northern California with waves coming from different azimuths and angles. Significant variations of SAF are not uncommon in strong-motion observations. For example, large SAF variations during different earthquakes were also observed at the pair of Coyote Lake stations (Boore et al., 2004) and at Tarzana surface–downhole pairs of instruments (Graizer, 2009).
Site-Response Analysis Time Series Approach I used acceleration TS recorded at the bedrock (104 and 122 m depth) from the eight earthquakes shown in Table 1 (16 horizontal components) as an input to the STRATA TS version to calculate the surface-to-bedrock SAF. The STRATA TS version is a realization of the well-known
SHAKE EQL approach introduced by Idriss and Seed (1968) and implemented by Schnabel et al. (1972) in SHAKE and by Idriss and Sun (1992) in SHAKE91. Other inputs used are the shear-wave velocity profile shown in Figure 1 and the Electrical Power Research Institute (1993) shear-modulus reduction and damping ratio curves. Downhole bedrock recordings were treated as within the media motions. Figure 5 compares the empirically determined SAF (thick black line) and the SAF calculated using the recorded data. The SAF calculated using the STRATA TS method matches the empirically determined SAF well. Therefore, the STRATA TS method demonstrates stable results given the use of 16 horizontal components as inputs, and SAF matches the empirically determined SAF well. As a test of stability, I used 10 out of 16 available TS as an input, which is not all the data used in the empirical SAF determination. The differences between the calculations using the complete dataset of 16 TS and the abbreviated set of 10 TS are not significant. To further test the STRATA TS approach, I calculated the SAF using 10 generic (not recorded at the TI array) TS scaled to the same level of peak ground acceleration (PGA) as an average input motion. The SAF calculated using the generic set of TS data is also close to the empirically calculated series with peaks at the same
Table 1 Earthquakes Recorded by Treasure Island Array Number
Date (yyyy/mm/dd)
Time (UTC) hh:mm:ss.ss
M
M-type
North Latitude
West Longitude
Depth (km)
Epicentral Distance (km)
Surface PGA, g
1 2 3 4 5 6 7 8
1993/01/16 1994/06/26 1998/08/12 1998/12/04 1999/08/18 2003/09/05 2006/12/21 2007/10/31
06:29:35.01 08:42:50.31 14:10:25.14 12:16:07.77 01:06:18.94 01:39:53.68 03:12:28.76 03:04:54.81
4.8 4.0 5.1 3.9 4.6 4.0 3.6 5.4
Md Mw Mw Mw Mw Mw Mw Mw
37.018 37.915 36.753 37.920 37.907 37.843 37.857 37.434
121.463 122.285 121.462 122.289 122.686 122.222 122.245 121.774
7.7 6.3 9.1 6.5 7.9 11.1 8.8 10.0
120.2 12.6 143.8 13.0 29.0 13.2 12.6 68.4
0.014 0.021 0.006 0.014 0.017 0.027 0.020 0.013
Comment
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Figure 3. Average and individual earthquake Fourier spectral ratios of surface/rock-ground motion at Treasure Island. The color version of this figure is available only in the electronic edition.
Figure 4.4. Individual and median-response spectral ratios of surface/rock ground motion at Treasure Island. The color version of this figure is available only in the electronic edition.
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Figure 5. Comparison of empirical site amplification function with that of the STRATA time series and the RVT approaches using response or Fourier spectra as input. The color version of this figure is available only in the electronic edition. frequencies. There is underprediction of site amplification (exceeding ± one standard error range) in the frequency range of 0.1–0.3 and 4.5–12 Hz.
Random-Vibration Theory Approach Figure 5 also shows the SAF calculation using STRATA RVT based on the TI bedrock 5% damped-response spectrum as an input. The results show significant overprediction of the amplitude of the first mode of the site amplification (Fig. 5, dotted line). Use of Fourier spectrum as an input to the RVT analysis improves the result, but still demonstrates significant overprediction (Fig. 5, dashed line). The root mean square deviation (RMSD) can be used as a measure of the difference between values predicted by a model (Sest ) and the value actually observed (Si ): rP N 2 i Si − Sest ; RMSD N in which N is a number of measurements (number of frequencies of comparisons in our case). Comparisons show that in the frequency range of 0.3–25 Hz (for which lowamplitude earthquake data are reliably recorded) the RMSD of SAF prediction using TS approach RMSDTS 0:70, in which the RMSD of SAF prediction using RVT with response spectrum as an input RMSDRVT;RS 2:20, and using Fourier spectrum is RMSDRVT;F 1:73. Comparisons of RMSD demonstrate that the TS approach produces much better
agreement with recorded data than that of the RVT. Using Fourier instead of the response spectrum as an input to the RVT approach improves results by lowering RMSD. Figure 6 shows a comparison of peak ground velocity (PGV) recorded at different depths during the 21 December 2006 M w 3.6 Piedmont, California, earthquake with PGVs calculated using the STRATA TS and RVT approaches. Calculations performed using the TS approach demonstrate good agreement with the recorded data, with RVT significantly overpredicting PGV amplitudes. RVT calculations were performed twice with the Fourier and 5% damped-response spectrum as inputs. RVT (with Fourier spectrum input) and TS calculations produce the same results for the deep part of the profile up to the interface between the bedrock and alluvium at the depth of 88 m. In the upper part of the profile, RVT calculations differ significantly from those of the TS. Using Fourier input as opposed to response spectrum improves results, but still produces significant overprediction. These results clearly demonstrate the supremacy of the TS relative to the RVT approach using STRATA. I also performed a number of comparisons between SAF calculation using the STRATA TS and RVT methods for different types of profiles. These comparisons demonstrated that the largest differences between TS and RVT results occurred when the S-wave velocity profile had large impedance contrasts between layers (Graizer, 2011; Kottke and Rathje, 2013). These tests lead me to disagree with some other conclusions of Kottke and Rathje (2013). One correction,
Comment
545 typical for all RVT formulations? It is desirable to assess strengths and weaknesses of different implementations of the RVT method including potential limitations of the method by comparing results with recorded earthquake data.
Data and Resources Earthquake data used in this study were collected by the California Strong Motion Instrumentation Program (CSMIP) of the California Geological Survey http://www. strongmotioncenter.org/ (last accessed June 2011). The earthquake parameters were obtained from the Advanced National Seismic System (ANSS) catalog http://www. ncedc.org/ncedc/catalog-search.html (last accessed August 2011). Other data used in the paper came from published sources listed in the text or in references. Figure 6. Comparison of peak ground velocity recorded during the Mw 3.6 of 21 December 2006 Piedmont, California, earthquake with that of the STRATA TS and RVT approaches. The color version of this figure is available only in the electronic edition. for example, is that the duration of the signal improves RVT calculations. In my tests, the RVT method did not show sensitivity to the reasonable variations in the duration of the signal (assuming that duration was not grossly underestimated). For example, variation in durations of the RVT signal for TI data from 1 to 3, 5, or 10 s resulted in changes of predicted maximum at 0.8 Hz of less than 4%. Only unreasonable decrease of the signal’s duration to very short time of 0.5 s resulted in significant increase of 20% in predicted maximum.
Discussion The purpose of this short note is not to discredit the RVT approach, but to bring to the attention of researchers and practitioners the necessity of a comprehensive validation of this method against empirically recorded strong-motion data. Earthquake data recorded by downhole arrays at different depths and geologic settings present an excellent opportunity to test and calibrate analytical methods of site-response analysis. High-quality low-amplitude earthquake data recorded at the TI downhole array between 1993 and 2010 allow testing of these methods at low-strain levels. In contrast to the TS approach, realization of the equivalent-linear method in combination with the RVT approach in STRATA may produce results, which differ significantly from the empirically determined SAF and TS approach. It is important to further investigate the source of differences between the TS and RVT results, and to determine the limitations of the RVT approach using empirical observations as a benchmark. Because there are a number of different RVT codes developed independently, based on different approximation formulas, and apparently producing different results, the following question remains: are the above mentioned issues
Acknowledgments I wish to thank Tony Shakal and Robert Darragh for their help at different stages of data collection and analysis. Thanks also go to Albert Kottke for his support in testing the STRATA program. I would like to thank Jon Ake for initiating this study. Special thanks go to Diane Doser and to the anonymous reviewer for their reviews and suggestions helping to improve the manuscript. Any opinions, findings, and conclusions expressed in this paper are those of the author and do not necessarily reflect the views of the United States Nuclear Regulatory Commission.
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[email protected] Manuscript received 3 July 2013; Published Online 17 December 2013
