# An Empirical Model to Account for Spectral Amplification of Pulse-Like Ground Motion Records

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Near-Source Dataset

_{w}vs Joyner–Boore distance R

_{JB}distribution of NESS1 data grouped into pulse-like (230 records from which we excluded 29 waveforms, having pulses potentially related to geotechnical effects) and no-pulse data (430 records). Both pulse-like and ordinary data are uniformly distributed over the magnitude-distance range of the dataset. The best sampled magnitude range is around M

_{w}6.0, due to the preponderance of highly sampled Italian data recorded during the seismic sequences of Northern Italy (2012) and Central Italy (2016), where relevant near-fault effects, also related to the presence of pulses, are observed.

## 3. Pulses Identification

_{p}(i.e., the pseudo-period of the wavelet extracted from the velocity signal) and pulse azimuthal orientation α

_{pulse}.

_{p}versus magnitude of the causal events in NESS1 is reported in Figure 2a along with the corresponding least-squares linear regression model. As expected, we find that the logarithm of the mean of the pulse periods linearly depends on the event magnitude (M), according to the following empirical relationship:

_{p}is related to the rise time (i.e., duration of the slip at a single point on the fault) and the fault dimensions, tending to increase with magnitude; e.g., [5,6]. With reference to the pulse orientation (α

_{pulse}), we observe that it correlates with the normal direction (90°) to the strike of the fault (α

_{strike90°}) over all possible orientations of the strike angle, meaning that the pulses occur mainly on the fault normal (FN) direction, with no relevant differences for both Strike-Slip (SS) and dip-slip faults, that are Normal-Faulting (NF) and Thrust-Faulting (TF)—Figure 2b. On these data, we found the following relationship:

## 4. Effect of Pulses on Elastic Spectral Response

_{p}[21,22] and (ii) the pulse period on earthquake magnitude [2,5,6,23]. To investigate this dependency on spectral ordinates, in Figure 4 we show a sensitivity analysis of normalized SA, i.e., divided by the PGA value (Figure 4a), and SD (Figure 4b) to different bins of the pulse period T

_{p}, showing that spectral amplitudes become larger at longer periods as T

_{p}increases, starting to deviate from the ordinary mean trend in correspondence of the average value of T

_{p}in the bin. However, this effect is not visible for pulse periods shorter than 1 s. The SD (Figure 4b) are also markedly affected by T

_{p}, showing broadband amplifications at period pulses larger than 1 s.

_{p}bins. In these figures, the SA curves are plotted for pulse-like records with T

_{p}larger than 1 s in order to enhance the shape differences of the normalized SA. It can be noted that SS earthquakes show systematically higher spectral ordinates, both for SA and SD than the other styles of faulting, particularly above a period of 2 s. This effect may achieve about 40% of amplification for SD with respect to ground motions from dip-slip earthquakes. Among the latter, the TF earthquakes provide larger ordinates compared to NF and no-pulse ground motion.

_{p}with the style of faulting. Indeed, according to [23,24], the rise time for dip-slip earthquakes is, on average, about half the rise time for SS earthquakes. As a result, the pulse period for an SS earthquake is expected to be longer than that for a dip-slip earthquake, thus producing more significant spectral amplifications at longer periods, as evidenced by our analysis.

## 5. Calibration of the Empirical Corrective Factor

#### 5.1. Reference GMM of No-Pulse Records

_{JB}≤ 50 km.

_{M}, Equation (4)), the distance attenuation, also describing the magnitude-dependent geometrical spreading (F

_{D}, Equation (5)), the site (F

_{S}, Equation (6)), the style of faulting (F

_{SOF}, Equation (7)) and the error e (Equation (8)). The explanatory variables are the moment magnitude M

_{w}, the source-to-site distance R, the shear wave velocity in the uppermost 30 m, V

_{S,30}and the styles of faulting SOF

_{j}, which are dummy variables, introduced to specify SS (j = 1) and dip-slip, i.e., TF/NF (j = 2), fault types.

_{M}= b

_{1}(M

_{w}− M

_{h}) for M ≤ M

_{h}; F

_{M}= b

_{2}(M

_{w}− M

_{h}) for M > M

_{h},

_{D}= [c

_{1}(M

_{w}− M

_{ref}) + c

_{2}] log

_{10}(R),

_{S}= k log

_{10}(V

_{0}/800) with V

_{0}= V

_{S,30}if V

_{S,30}< 1500 m/s or V

_{0}= 1500 m/s if V

_{S,30}≥ 1500 m/s

_{SOF}= δ

_{j}SOF

_{j},

_{h}, the reference magnitude M

_{ref}and the pseudo-depth h are constants, which have been assumed as follows: M

_{h}= 6.75; M

_{ref}= 5.5 and h = 6 km according to the average values in the dataset.

_{1}, b

_{2}, c

_{1}, c

_{2}, k and f

_{j}(f

_{1}for SS, f

_{2}for TF and NF) are derived by a mixed-effect linear regression [25]. The predicted value by the model is a 10-base logarithm of the spectral acceleration SA (i.e., the peak ground acceleration, PGA and SA at 36 ordinates) at 5% damping in the period range 0.01–10 s. The model residuals r

_{es}are partitioned into between-event (δB

_{e}) and within-event (δWS

_{es}) terms of error (Equation (8), thus the total variability σ associated to the GMM (Equation (9) is obtained through the combination of between-event (τ) and within-event (ϕ) standard deviations [26]:

_{es}= δB

_{e}+ δWS

_{es}(e = event; s = station)

^{2}+ ϕ

^{2})

^{0.5}

_{JB}); rupture (R = R

_{rup}); rupture normalized to the fault length (R = R

_{norm}); line (R = R

_{line}) and nucleation-point (R = R

_{np}) distances (for a detailed description on these distance metrics definitions, see [15]). We find that the model prediction (Figure 6a) and the total standard deviation σ (Figure 6b) computed as mean over all the ordinary records are almost insensitive to the adopted distance metric. In light of this, we decide to adopt the R

_{JB}metric according to the majority of GMMs.

#### 5.2. Narrowband Amplification Factor

_{es}versus period T are plotted in Figure 8.

_{p}, as shown in Figure 9 for RotD50 (a) and FN (b) pulse-like waveforms. This approach to represent the residuals was also adopted by many authors [27,28,29,30,31].

_{p}leads evident that the mean curve of the residuals has the typical bell-curve shape with a “bump” centered close to ratio T/T

_{p}= 1. This shape of the mean amplification factor µ

_{Af}is well approximated by Equations (11) and (12), respectively for RotD50 and FN components, calibrated through the curve fitting toolbox provided by Mathworks-Matlab

^{®}environment:

_{lnAf}(obtained by Equation (11) or Equation (12)) can thus be used in conjunction with the reference GMM µ

_{lnSA,GMM}(converted in natural logarithm units) to compute the mean ground motion prediction corrected for the pulse-like effects (µ

_{lnSA,pulse}) as follows:

_{lnSA,pulse}); thus, for eventual applications we suggest it is conservatively assumed equal to the σ(T) of Equation (7). Indeed, according to the findings of Shashi and Baker [10], the σ

_{lnSA,pulse}is always lower than the standard deviation of the reference GMM, by a reduction factor dependent on the pulse-period.

#### 5.3. Application Example

_{lnAf,D50}(solid-colored lines) according to Equation (12) for two scenarios, compared to the observed spectral valuesfrom NESS1 dataset and divided by their PGA.

_{S,30}= 400 m/s and R

_{JB}10 km (Figure 11a); the second scenario is identical to the first one, except for the event that is M7.0 from NF rupture (Figure 11b).

_{JB}< 20 km and site properties V

_{S,30}= 400 ± 50 m/s. In these examples, the values of T

_{p}used to calculate µ

_{lnAf,FN}are estimated from Equation (1) calibrated on the pulse-like waveforms of NESS1 dataset.

_{p}.

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bolt, B.A.; Abrahamson, N.A. Estimation of strong seismic ground motions. In International Handbook of Earthquake and Engineering Seismology; Lee, W.H.K., Kanamori, H., Jennings, P.C., Kisslinger, C., Eds.; Academic Press: London, UK, 2003; pp. 983–1001. [Google Scholar]
- Mavroeidis, G.P.; Papageorgiou, A.S. A Mathematical Representation of Near-Fault Ground Motions. Bull. Seismol. Soc. Am.
**2003**, 93, 1099–1131. [Google Scholar] [CrossRef] - Kalkan, E.; Kunnath, S.K. Effects of fling step and forward directivity on seismic response of buildings. Earthq. Spectra
**2006**, 22, 367–390. [Google Scholar] [CrossRef] - D’Amico, M.; Felicetta, C.; Schiappapietra, E.; Pacor, F.; Gallovič, F.; Paolucci, R.; Sgobba, S.; Luzi, L. Fling Effects from Near-Source Strong-Motion Records: Insights from the 2016 M w 6.5 Norcia, Central Italy, Earthquake. Seismol. Res. Lett.
**2019**, 90, 659–671. [Google Scholar] [CrossRef] - Somerville, P.G.; Smith, N.F.; Graves, R.W.; Abrahamson, N.A. Modification of Empirical Strong Ground Motion Attenuation Relations to Include the Amplitude and Duration Effects of Rupture Directivity. Seismol. Res. Lett.
**1997**, 68, 199–222. [Google Scholar] [CrossRef] - Bray, J.D.; Rodriguez-Marek, A. Characterization of forward directivity ground motions in the near-fault region. Soil Dyn. Earthq. Eng.
**2004**, 24, 815–828. [Google Scholar] [CrossRef] - Baker, J.W. Identification of Near-Fault Velocity Pulses and Prediction of Resulting Response Spectra. Proc. Geotech. Earthq. Eng. Struct. Dyn. IV
**2009**, 40975, 1–10. [Google Scholar] - Tothong, P.; Cornell, C.A.; Baker, J.W. Explicit directivity-pulse inclusion in probabilistic seismic hazard analysis. Earthq. Spectra
**2007**, 23, 867–891. [Google Scholar] [CrossRef] - Chioccarelli, E.; Iervolino, I. Near-source seismic demand and pulse-like records: A discussion for L’ Aquila earthquake. Earthq. Eng. Struct. Dyn.
**2010**, 39, 1039–1062. [Google Scholar] [CrossRef] - Shahi, S.K.; Baker, J.W. An empirically calibrated framework for including the effects of near-fault directivity in probabilistic seismic hazard analysis. Bull. Seismol. Soc. Am.
**2011**, 101, 742–755. [Google Scholar] [CrossRef] - Chang, Z.; Sun, X.; Zhai, C.; Zhao, J.X.; Xie, L. An empirical approach of accounting for the amplification effects induced by near-fault directivity. Bull. Earthq. Eng.
**2018**, 16, 1871–1885. [Google Scholar] [CrossRef] - Akkar, S.; Moghimi, S.; Arıcı, Y. A study on major seismological and fault-site parameters affecting near-fault directivity ground-motion demands for strike-slip faulting for their possible inclusion in seismic design codes. Soil Dyn. Earthq. Eng.
**2018**, 104, 88–105. [Google Scholar] [CrossRef] - Spudich, P.; Chiou, B.S.J. Directivity in NGA earthquake ground motions: Analysis using isochrone theory. Earthq. Spectra
**2008**, 24, 279–298. [Google Scholar] [CrossRef] [Green Version] - Pacor, F.; Felicetta, C.; Lanzano, G.; Sgobba, S.; Puglia, R.; D’Amico, M.C.; Luzi, L. NEar-Source Strong-Motion Flatfile (NESS) (Version 1.0) [Data Set]; Istituto Nazionale di Geofisica e Vulcanologia (INGV): Rome, Italy, 2018. [Google Scholar] [CrossRef]
- Pacor, F.; Felicetta, C.; Lanzano, G.; Sgobba, S.; Puglia, R.; D’Amico, M.; Russo, E.; Baltzopoulos, G.; Iervolino, I. NESS1: A Worldwide Collection of Strong-Motion Data to Investigate Near-Source Effects. Seismol. Res. Lett.
**2018**, 89, 2299–2313. [Google Scholar] [CrossRef] - D’Amico, M.; Felicetta, C.; Russo, E.; Sgobba, S.; Lanzano, G.; Pacor, F.; Luzi, L. Italian Accelerometric Archive v 3.1; Istituto Nazionale di Geofisica e Vulcanologia, Dipartimento della Protezione Civile Nazionale: Rome, Italy, 2020. [Google Scholar] [CrossRef]
- Luzi, L.; Lanzano, G.; Felicetta, C.; D’Amico, M.C.; Russo, E.; Sgobba, S.; Pacor, F. ORFEUS Working Group 5. Engineering Strong Motion Database (ESM) (Version 2.0); Istituto Nazionale di Geofisica e Vulcanologia (INGV): Rome, Italy, 2020. [Google Scholar] [CrossRef]
- Puglia, R.; Russo, E.; Luzi, L.; D’Amico, M.; Felicetta, C.; Pacor, F.; Lanzano, G. Strong-motion processing service: A tool to access and analyse earthquakes strong-motion waveforms. Bull. Earthq. Eng.
**2018**, 16, 2641–2651. [Google Scholar] [CrossRef] - Baker, J.W. Quantitative classification of near-fault ground motions using wavelet analysis. Bull. Seismol. Soc. Am.
**2007**, 97, 1486–1501. [Google Scholar] [CrossRef] - Boore, D.M. Orientation-independent, nongeometric-mean measures of seismic intensityfrom two horizontal components of motion. Bull. Seismol. Soc. Am.
**2010**, 100, 1830–1835. [Google Scholar] [CrossRef] - Mavroeidis, G.P.; Dong, G.; Papageorgiou, A.S. Near-fault ground motions, and the response of elastic and inelastic singledegree-of-freedom (SDOF) systems. Earthq. Eng. Struct. Dyn.
**2004**, 33, 1023–1049. [Google Scholar] [CrossRef] - Alavi, B.; Krawinkler, H. Behavior of moment-resisting frame structures subjected to near-fault ground motions. Earthq. Eng. Struct. Dyn.
**2004**, 33, 687–606. [Google Scholar] [CrossRef] - Somerville, P.G. Magnitude scaling of the near fault rupture directivity pulse. Phys. Earth Planet
**2003**, 137, 201–212. [Google Scholar] [CrossRef] - Somerville, P.G. Development of an improved representation of near fault ground motions. In Proceedings of the SMIP98 Seminar on Utilization of Strong Ground Motion Data, Oakland, CA, USA, 15 September 1998; pp. 1–20. [Google Scholar]
- Bates, D.; Mächler, M.; Bolker, B.; Walker, S. Fitting linear mixed-effects models using lme4. J. Stat. Softw.
**2015**, 67, 1–48. [Google Scholar] [CrossRef] - Al-Atik, L.; Abrahamson, N.A.; Bommer, J.J.; Scherbaum, F.; Cotton, F.; Kuehn, N. The variability of ground-motion prediction models and its components. Seismol. Res. Lett.
**2010**, 81, 794–801. [Google Scholar] [CrossRef] [Green Version] - Hubbard, D.T.; Mavroeidis, G.P. Damping coefficients for near-fault ground motion response spectra. Soil Dyn. Earthq. Eng.
**2011**, 31, 401–417. [Google Scholar] [CrossRef] - Pu, W.; Kasai, K.; Kabando, E.K.; Huang, B. Evaluation of the damping modification factor for structures subjected to near-fault ground motions. Bull. Earthq. Eng.
**2016**, 14, 1519–1544. [Google Scholar] [CrossRef] - Alonso-Rodríguez, A.; Miranda, E. Assessment of building behavior under near-fault pulse-like ground motions through simplified models. Soil Dyn. Earthq. Eng.
**2015**, 79, 47–58. [Google Scholar] [CrossRef] - Liossatou, E.; Fardis, M. Near-fault effects on residual displacements of RC structures. Earthq. Eng. Struct. Dyn.
**2016**, 45, 1391–1409. [Google Scholar] [CrossRef] - Cao, Y.; Mavroeidis, G.; Meza Fajardo, K.C.; Papageorgiou, A. Accidental eccentricity in symmetric buildings due to wave passage effects arising from near-fault pulse-like ground motions. Earthq. Eng. Struct. Dyn.
**2017**, 46, 2185–2207. [Google Scholar] [CrossRef] - Baltzopoulos, G.; Iervolino, I.; Università degli Studi di Napoli, Federico II, Naples, Italy. Personal Communication, 2019.
- Cauzzi, C.; Faccioli, E.; Vanini, M.; Bianchini, A. Updated predictive equations for broadband (0.01–10 s) horizontal response spectra and peak ground motions, based on a global dataset of digital acceleration records. Bull. Earthq. Eng.
**2015**, 13, 1587–1612. [Google Scholar] - Boore, D.M.; Atkinson, G.M. Ground-motion prediction equations for the average horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between 0.01 s and 10.0 s. Earthq. Spectra
**2008**, 24, 99–138. [Google Scholar] [CrossRef] [Green Version] - Iervolino, I.; Cornell, C.A. Probability of Occurrence of Velocity Pulses in Near-Source Ground Motions. Bull. Seismol. Soc. Am.
**2008**, 98, 2262–2277. [Google Scholar] [CrossRef]

**Figure 1.**Magnitude-distance distribution of the NESS1 records grouped by pulse-like and no-pulse records.

**Figure 2.**Scatter plot of the pulse period T

_{p}versus Magnitude along with the fitting function Equation (1); the Root-Mean-Squared_error RMSE is equal to 0.6043 and the statistical coefficient of determination R-square is 48.56% (

**a**), relationship between pulse azimuth α

_{pulse}and strike normal angle α

_{strike90°}for Strike-Slip (SS), Thrust-Fault (TF) and Normal-Fault (NF)—RMSE 172.2 and R-square 31% (

**b**).

**Figure 3.**Average elastic 5%-damped acceleration response spectra acceleration (SA) as RotD50 (

**a**) and average elastic displacement spectra SD (

**b**) in log-log scale, for pulse-like and no-pulse records in NESS1. Ratio between RotD50 and fault normal (FN) spectral components for acceleration (

**c**) and displacement (

**d**) elastic response spectra. Thin gray lines indicate the individual spectra; solid-colored lines refer to the mean of pulse-like records; the dashed lines refer to no-pulse records.

**Figure 4.**Mean 5% damped acceleration response spectra SA divided by PGA (

**a**) and displacement spectra SD (

**b**) in log-log scale, grouped by 4 pulse period bins.

**Figure 5.**Mean 5% damped acceleration response spectra divided by PGA (

**a**) and displacement spectra (

**b**) in log-log scale, grouped by styles of faulting (NF—Normal Fault; TF—Thrust Fault; SS—Strike Slip).

**Figure 6.**Mean 5% damped acceleration spectra divided by PGA as predicted by the reference ground-motion model (GMM) (

**a**) and related mean standard deviation σ (

**b**) in log-log scale, for different distance metrics.

**Figure 7.**Mean 5% damped acceleration spectra divided by PGA according to the reference GMM for RotD50 (

**a**) and FN (

**b**) components in log-log scale, compared to the mean of pulse-like spectra in NESS1.

**Figure 8.**Individual and mean within-event residual values for pulse-like and no-pulse ground motions in NESS1.

**Figure 9.**Normalized residuals ε of acceleration response spectra in NESS1 for RotD50 (

**a**) and FN (

**b**) pulse-like components.

**Figure 11.**Observed and predicted SA according to the modified GMM for pulse-like ground motions (FN components), for two seismic scenarios: M6.5—SS type (

**a**) and M7.0—NF type (

**b**).

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**MDPI and ACS Style**

Sgobba, S.; Lanzano, G.; Pacor, F.; Felicetta, C.
An Empirical Model to Account for Spectral Amplification of Pulse-Like Ground Motion Records. *Geosciences* **2021**, *11*, 15.
https://doi.org/10.3390/geosciences11010015

**AMA Style**

Sgobba S, Lanzano G, Pacor F, Felicetta C.
An Empirical Model to Account for Spectral Amplification of Pulse-Like Ground Motion Records. *Geosciences*. 2021; 11(1):15.
https://doi.org/10.3390/geosciences11010015

**Chicago/Turabian Style**

Sgobba, Sara, Giovanni Lanzano, Francesca Pacor, and Chiara Felicetta.
2021. "An Empirical Model to Account for Spectral Amplification of Pulse-Like Ground Motion Records" *Geosciences* 11, no. 1: 15.
https://doi.org/10.3390/geosciences11010015