Prepared in cooperation with the U.S. Nuclear Regulatory Commission

Update of the Graizer-Kalkan Ground-Motion Prediction 

Equations for Shallow Crustal Continental Earthquakes

By Vladimir Graizer and Erol Kalkan

Open-File Report 2015–1009

U.S. Department of the Interior

U.S. Geological Survey
ii

U.S. Department of the Interior

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2015

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Suggested citation:

Graizer, V., and Kalkan, E., 2015, Update of the Graizer-Kalkan ground-motion prediction equations for shallow crustal 

continental earthquakes: U.S. Geological Survey Open-File Report 2015–1009, 79 p., 

http://dx.doi.org/10.3133/ofr20151009.

ISSN 2331-1258 (online)
iii

Acknowledgments

We wish to thank Martin Chapman, Jon Ake, and Dogan Seber for fruitful discussions on different 

aspects of ground-motion attenuation. Special thanks are extended to Stephen Harmsen and Nick Gregor 

for independent testing of our model. David Boore, Art Frankel, Kuo-Wan Lin, and Nilesh Shome have 

reviewed the material presented in this report; their valuable comments and suggestions helped to improve 

its technical quality and presentation. We also thank Paul Spudich for his help with mixed-effects 

residuals analysis, Jessica Dyke for her editing, and Brad Aagaard for his final review. 
iv

Contents

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

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

Dataset Selection ....................................................................................................................................................... 2

Functional Form of Ground-Motion Prediction Equation ............................................................................................. 5

GMPE for Peak Ground Acceleration ..................................................................................................................... 5

GMPE for Spectral Acceleration ............................................................................................................................. 6

Filter Functions ....................................................................................................................................................... 8

G1=Magnitude and Style of Faulting.................................................................................................................... 8

G2=Distance Attenuation..................................................................................................................................... 8

G3=Anelastic Attenuation .................................................................................................................................. 11

G4=Site Correction ............................................................................................................................................ 13

G5=Basin Effect................................................................................................................................................. 13

Mixed-Effects Residuals Analysis............................................................................................................................. 15

Intra-Event (Within-Event) Residuals Analysis of Path, Site, and Basin Depth Effects ......................................... 21

Analysis of Source Effects Using Inter-event (Between-Event) Residuals............................................................ 26

Terms of Standard Deviation .................................................................................................................................... 29

Model Results........................................................................................................................................................... 40

Comparisons With Graizer-Kalkan 2007–2009 Models ............................................................................................ 47

Comparisons With NGA-West2 Models.................................................................................................................... 50

Comparisons With Earthquake Data......................................................................................................................... 55

Example Calculations Using MatLAB Codes ............................................................................................................ 69

Range of Applicability ............................................................................................................................................... 69

Concluding Remarks ................................................................................................................................................ 70

Data and Resources................................................................................................................................................. 70

Disclaimer................................................................................................................................................................. 71

References Cited...................................................................................................................................................... 71

Appendix A. List of Earthquakes Used for Updating the Graizer-Kalkan Ground-motion Prediction Equation.......... 74

Appendix B. MatLAB Code for Graizer-Kalkan Ground-motion Prediction Equation (2015)..................................... 76

Appendix C. MatLAB Code to Generate Pseudo Spectral Acceleration Response Spectrum Using Graizer-Kalkan 

2015 (GK15) Ground-motion Prediction Equation .................................................................................................... 78

Figures

 1. Plots showing earthquake data distribution with respect to A, moment magnitude (M), and B, average shearwave velocity in the upper 30 m of the geological profile (VS30) against closest distance to fault rupture plane 

(R).. ............................................................................................................................................................... 3

 2. Plots showing earthquake data distribution with respect to closest distance to fault rupture plane (R) and 

basin depth (Bdepth). ....................................................................................................................................... 4

 3. Plots showing peak ground acceleration (PGA) distribution with respect to moment magnitude (M) and 

closest distance to fault rupture plane (R). .................................................................................................... 4

 4. Graph showing the generic spectral shape (SAnorm) model used and its controlling parameters (I, S,

μ,Tsp,0). .......................................................................................................................................................... 8

 5. Plots showing attenuation of maximum component ground motions during the 2004 magntiude (M) 6.0 

Parkfield, 1979 M6.5 Imperial Valley, and 2014 M6.0 South Napa earthquakes. ........................................ 10

 6. Graphs showing model results for anelastic attenuation with constant and variable Q0. ............................. 12
v

 7. Graphs showing A, dependence of amplitude on basin depth, and B, dependence of the response spectrum 

long period decay term (ζ) on basin depth (Bdepth)....................................................................................... 14

 8. Graphs showing comparison of pseudo spectral acceleration (SA) response computed using updated 

Graizer-Kalkan ground-motion prediction equation (GK15) for three cases: nonbasin, basin with 1.5-km 

depth, and basin with 3-km depth................................................................................................................ 15

 9. Plot showing overall mean bias of Graizer-Kalkan (GK15) and its standard deviation. ............................... 16

10. Plots showing distance dependence of Graizer-Kalkan (GK15) residuals................................................... 18

11. Plots showing magnitude dependence of Graizer-Kalkan (GK15) residuals................................................ 19

12. Plots showing VS30 dependence of Graizer-Kalkan (GK15) residuals.......................................................... 20

13. Plots showing depth to basin (Bdepth) dependence of Graizer-Kalkan (GK15) residuals. ............................. 21

14. Plots showing distribution of intra-event residuals in natural logarithmic units for peak ground acceleration 

(PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s with respect to closest fault distance to 

rupture plane (R). ........................................................................................................................................ 23

15. Plots showing distribution of intra-event residuals in natural logarithmic units for peak ground acceleration 

(PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s with respect to VS30............................ 24

16. Plots showing distribution of intra-event residuals in natural logarithmic units for peak ground acceleration 

(PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s with respect to Bdepth. ......................... 25

17. Plots showing distribution of event terms (ηi) in natural logarithmic units for peak ground acceleration (PGA) 

and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s with respect to moment magnitude (M)......... 27

18. Plots showing inter-event (between-event), intra-event (within-event), and total standard deviations of 

Graizer-Kalkan (GK15) in natural-logarithmic units computed based on mixed-effects residuals analysis.. 30

19. Plot showing total standard deviations (σ) of Graizer-Kalkan (GK15) and its approximation in natural

logarithmic units........................................................................................................................................... 32

20. Plots showing values of inter-event (between-event) standard deviations (τ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s in magnitude bins with equal 

number of data points.................................................................................................................................. 33

21. Plots showing values of intra-event (within-event) standard deviations (φ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0 and 3.0 s in magnitude bins with equal 

number of data points.................................................................................................................................. 35

22. Plots showing values of inter-event (between-event) standard deviations (τ) computed for peak ground 

acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0 and 3.0 s in distance bins with equal 

number of data points.................................................................................................................................. 36

23. Plots showing values of intra-event (within-event) standard deviations (φ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0 and 3.0 s in distance bins with equal 

number of data points.................................................................................................................................. 37

24. Plots showing values of inter-event (between-event) standard deviations (τ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0 and 3.0 s in VS30 bins with equal 

number of data points.................................................................................................................................. 38

25. Plots showing values of intra-event (within-event) standard deviations (φ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0 and 3.0 s in VS30 bins with equal 

number of data points.................................................................................................................................. 39

26. Graphs showing comparison of median pseudo spectral accelerations for strike-slip magnitude 5.0, 6.0, 7.0, 

and 8.0 earthquakes at R=1 and 30 km and VS30=270 and 760 m/s considering Q0=150 and Bdepth=0....... 41

27. Graphs showing comparison of distance scaling for strike-slip magnitude (M) 5.0, 6.0, 7.0, and 8.0 

earthquakes at median PGA and pseudo spectral accelerations at 0.2, 1.0, and 3.0 s considering VS30=760 

m/s, Q0=150 and Bdepth=0. ........................................................................................................................... 42
vi

28. Graphs showing comparison of magnitude scaling for strike-slip earthquakes at closest fault distance to 

rupture plane, R=1, 30, and 150 km for median peak ground acceleration (PGA) and pseudo spectral 

accelerations (SA) at 0.3, 1.0, and 3.0 s considering VS30=760 m/s, Q0=150 and Bdepth=0.......................... 43

29. Graph showing comparison of VS30 scaling on pseudo spectral accelerations (SA) for a strike-slip magntitude 

(M) 7.0 earthquake at R=30 km considering Q0=150 and Bdepth=0 .............................................................. 44

30. Graph showing comparison of Bdepth scaling on pseudo spectral accelerations (SA) for a strike-slip M7.0 

earthquake at R=30 km considering Q0=150 and VS30=270 m/s .................................................................. 44

31. Graph showing comparison of style of faulting scaling on pseudo spectral accelerations (SA) for a magnitude 

(M) 7.0 earthquake at R=30 km considering Q0=150, VS30=270 m/s, and Bdepth=0....................................... 45

32. Graphs showing comparison of Q0 scaling with closest fault distance to rupture plane (R) for a strike-slip 

magnitude (M) 7.0 earthquake for median peak ground acceleration (PGA) and pseudo spectral accelerations 

(SA) at 0.3, 1.0, and 3.0 s considering VS30=760 m/s and Bdepth=0.............................................................. 46

33. Graphs showing comparison of Q0 scaling on pseudo spectral accelerations (SA) for a strike-slip magnitude 

(M) 7.0 earthquake at R=30 and 120 km considering Q0=150, VS30=270 m/s and Bdepth=0 (R=closest fault 

distance to rupture plane ............................................................................................................................. 47

34. Graphs showing comparison of median pseudo spectral accelerations (SA) for strike-slip magnitude (M) 4.5, 

5.5, 6.5, and 7.5 earthquakes at closest fault distance to rupture plane R=1 and 30 km considering VS30=270 

and 760 m/s, Q0=150, and Bdepth=0 ............................................................................................................. 48

35. Graphs showing comparison of distance scaling for strike-slip magnitude (M) 5.0, 6.0, 7.0, and 8.0 

earthquakes at median peak ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, 

and 3.0 s considering VS30=760 m/s, Q0=150, and Bdepth=0......................................................................... 49

36. Plot showing comparison of total aleatory variability (σ) between Graizer-Kalkan (2007, 2009) (GK07–09) and 

Graizer-Kalkan (GK15) ground-motion prediction equations. ...................................................................... 50

37. Graphs showing comparison of distance scaling of Graizer-Kalkan (GK15) and Boore and others (2014) 

(BSSA14) for strike-slip magnitude (M) 6.0, 7.0, and 8.0 earthquakes at median peak ground acceleration 

(PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s considering VS30=760 m/s, Q0=150, and 

Bdepth=0........................................................................................................................................................ 51

38. Graphs showing comparison of distance scaling of Graizer-Kalkan (GK15) and Abrahamson and others 

(2014) (ASK14) for strike-slip magnitude (M) 6.0, 7.0, and 8.0 earthquakes at median peak ground 

acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s considering VS30=760 m/s, 

Q0=150, and Bdepth=0.................................................................................................................................... 52

39. Graphs showing comparison of spectra from Graizer-Kalkan (GK15) and Boore and others (2014) (BSSA14) 

for strike-slip magnitude (M) 6.0, 7.0, and 8.0 earthquakes at closest fault distance to rupture plane R =1 and 

30 km, considering VS30=270 and 760 m/s, Q0=150, and Bdepth=0................................................................. 53

40. Graphs showing comparison of spectra from Graizer-Kalkan (GK15) and Abrahamson and others (2014) 

(ASK14) for strike-slip magnitude (M) 6.0, 7.0, and 8.0 earthquakes at closest fault distance to rupture plane 

R =1 and 30 km, considering VS30=270 and 760 m/s, Q0=150, and Bdepth=0. ................................................ 54

41–53. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak ground 

acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from various earthquakes.

41. 1971 magntiude (M) 6.6 San Fernando earthquake......................................................................... 56

42. 1979 magntiude (M) 6.5 Imperial Valley earthquake........................................................................ 57

43. 1989 magntiude (M) 6.9 Loma Prieta earthquake............................................................................ 58

44. 1992 magntiude (M) 7.3 Landers earthquake .................................................................................. 59

45. 1994 magntiude (M) 6.4 Northridge earthquake............................................................................... 60

46. 1999 magntiude (M) 7.1 Hector Mine earthquake ............................................................................ 61

47. 1999 magntiude (M) 7.4 Kocaeli (Turkey) earthquake ..................................................................... 62

48. 1999 magnitude (M) 7.2 Düzce (Turkey) earthquake....................................................................... 63
vii

49. 1999 magnitude (M) 7.6 Chi-Chi (Taiwan) earthquake..................................................................... 64

50. 2002 magnitude (M) 6.5 San Simeon earthquake............................................................................ 65

51. 2002 magnitude (M) 7.9 Denali earthquake ..................................................................................... 66

52. 2004 magnitude (M) 6.0 Parkfield earthquake ................................................................................. 67

53. 2014 magnitude (M) 6.0 South Napa earthquake ............................................................................ 68

54. Site-specific 5-percent damped pseudo spectral acceleration (SA) response spectra generated using the 

MatLAB code in appendix C for a vertically dipping strike-slip magnitude (M) 6.0 earthquake at closest fault 

distance to rupture plane (R)=4.4 km considering VS30=350 m/s, Q0=150, and Bdepth=0 ............................. 69

Tables

 1. Estimator coefficients of Graizer-Kalkan (GK15) ground-motion prediction equation for peak ground 

acceleration. .................................................................................................................................................. 6

 2. Estimator coefficients for Graizer-Kalkan (GK15) spectral shape (SAnorm) model....................................... 7

 3. Event terms (ηi) using 35 events in the magnitude range of 4.9 to 7.9....................................................... 28

 4. Inter-event standard deviations (τ), intra-event standard deviations (φ), and total standard deviations (σ) in 

natural logarithmic units............................................................................................................................... 31

 5. Coefficients of inter-event (between-event) standard deviation term........................................................... 34
viii

Abbreviations

Bdepth basin depth under the site 

C constant term from the mixed-effects analysis

F style of faulting

G1 scaling function for magnitude and style of faulting

G2 path scaling function

G3 anelastic attenuation function

G4 site amplification function 

G5 basin scaling function

GK15 Graizer-Kalkan (2015)

GM ground motion

GMPE ground-motion prediction equation

I peak spectral intensity

M moment magnitude 

PGA peak ground acceleration

R closest distance to fault rupture plane

RRRii residual of j

th recording of ith earthquake

Q0 quality factor

S spectral wideness (area under the spectral shape)

SA pseudo spectral acceleration

T spectral period

Tss,0 predominant period of the spectrum

VS30 average shear-wave velocity in the upper 30 m of the geological profile

Y peak ground acceleration

Z1.5 depth to 1.5-km/s shear-wave velocity isosurface

σ total standard variability

μ function defining the predominant period of the spectrum 

ηi event term for event i

εii intra-event residual for recording j in event i

τ standard deviation of event term

φ standard deviation of intra-event term

ζ decaying function of the spectrum at long periods 
Update of the Graizer-Kalkan Ground-Motion Prediction 

Equation for Shallow Crustal Continental Earthquakes

By Vladimir Graizer1 and Erol Kalkan2

Abstract

A ground-motion prediction equation (GMPE) for computing medians and standard deviations

of peak ground acceleration and 5-percent damped pseudo spectral acceleration response ordinates of 

maximum horizontal component of randomly oriented ground motions was developed by Graizer and 

Kalkan (2007, 2009) to be used for seismic hazard analyses and engineering applications. This GMPE

was derived from the greatly expanded Next Generation of Attenuation (NGA)-West1 database. In this 

study, Graizer and Kalkan’s GMPE is revised to include (1) an anelastic attenuation term as a function 

of quality factor (Q0) in order to capture regional differences in large-distance attenuation and (2) a new 

frequency-dependent sedimentary-basin scaling term as a function of depth to the 1.5-km/s shear-wave 

velocity isosurface to improve ground-motion predictions for sites on deep sedimentary basins. The new 

model (GK15), developed to be simple, is applicable to the western United States and other regions with 

shallow continental crust in active tectonic environments and may be used for earthquakes with moment 

magnitudes 5.0–8.0, distances 0–250 km, average shear-wave velocities 200–1,300 m/s, and spectral 

periods 0.01–5 s. Directivity effects are not explicitly modeled but are included through the variability 

of the data. Our aleatory variability model captures inter-event variability, which decreases with 

magnitude and increases with distance. The mixed-effects residuals analysis shows that the GK15 

reveals no trend with respect to the independent parameters. The GK15 is a significant improvement 

over Graizer and Kalkan (2007, 2009), and provides a demonstrable, reliable description of groundmotion amplitudes recorded from shallow crustal earthquakes in active tectonic regions over a wide 

range of magnitudes, distances, and site conditions.

Introduction

The ground-motion prediction equation (GMPE) for for peak ground acceleration (PGA) and 5-

percent damped pseudo spectral acceleration (henceforth abbreviated as SA) response ordinates of 

maximum horizontal component of randomly oriented ground motions was developed by Graizer and 

Kalkan (2007, 2009) using the Next Generation of Attenuation (NGA)-West1 database (Chiou and 

others, 2008) along with many additional records from major California earthquakes, including the 2004 

Parkfield (M6.0, M=moment magnitude) and 2003 San Simeon (M6.5) earthquakes, and a number of 

smaller magnitude (5.0–5.7) earthquakes from Turkey, California, and other tectonically similar regions.

The Graizer-Kalkan GMPE is composed of two predictive equations: the first equation computes

PGA (Graizer and Kalkan, 2007), and the second equation obtains spectral shape (Graizer and Kalkan, 

 1

U.S. Nuclear Regulatory Commission

2

U.S. Geological Survey
2

2009). The term “spectral shape” refers to the SA response spectrum normalized by PGA. The SA 

response spectrum is constructred by anchoring the spectral shape to the PGA. In this model, the SA 

response spectrum is a continuous function of spectral period (T); all other GMPEs use a discrete 

functional form for predicting the SA response ordinates. The concept of continuous function de facto 

eliminates the structural difference between points in period and period intervals by making period

intervals infinitesimally short. As a consequence, the concept of continuous function allows spectral 

ordinates to be easily estimated 

Our predictive equations for PGA and spectral shape constitute a series of functions guided by 

empirical data and physical simulations. Each function represents a physical phenomenon affecting the 

seismic-wave radiation from the source. We call these functions filters because the seismic waves are 

filtered through a number of physical processes as they travel from source to location of measurement. 

The filter-based ground-motion prediction model is shown to provide accuracy (expected median 

prediction without significant bias with respect to the independent parameters) and efficiency (relatively 

small aleatory variabilty) (Graizer and Kalkan, 2009, 2011; Graizer and others, 2013). 

This report documents the recent improvements on the Graizer-Kalkan GMPE (denoted as

GK15), and provides a complete description of the basis for its functional form. The updates include (1) 

a new anelastic attenuation term as a function of quality factor to capture regional differences in largedistance attenuation and (2) a new frequency-dependent sedimentary-basin scaling term as a function of 

depth to the 1.5-km/s shear-wave velocity isosurface to improve ground-motion predictions for sites on 

deep sedimentary basins. We believe that these changes represent major improvements to our previous 

GMPE, and therefore justify the additional complexity in GK15. The analysis of mixed-effects residuals 

reveals that the revised GMPE is unbiased with respect to its independent parameters.

GK15 is applicable to earthquakes of moment magnitude 5.0 to 8.0 (except for M>7.0 normalslip events that lack constraint), at closest distances to fault rupture plane from 0 to 200 km, at sites 

having VS30 in the range from 200 to 1,300 m/s, and for spectral periods (T) of 0.01–5 s. We considered 

regional variability in source, path, and site effects but did not address hanging-wall effects.

In the following sections, we first describe the selection of data used in the update. We then 

present the changes made in this update, followed by evaluations of the updated model and comparisons 

to our 2008 model and the NGA-West2 relationships. Finally, we offer some guidance on model 

applicability. GK15 is coded into MatLAB (titled “GK15.m”) and provided in appendix B; appendix C 

presents an example MatLAB code (titled “runGK15.m”) showing how to easily use the model in 

engineering applications.

Dataset Selection

A total of 2,583 ground-motion recordings from 47 shallow crustal continental earthquakes with 

focal depths less than 20 km were selected. This dataset includes events gathered from the Pacific 

Earthquake Engineering Research Center database created under the NGA-West1 project3 and data from 

a number of additional events from additional stations in the NGA-West1 dataset. Specifically, data 

from the following earthquakes were included: 1994 Northridge, 1999 Hector Mine, 2002 Big Bear 

City, 2003 San Simeon, 2004 Parkfield, 2005 Anza and Yucaipa, 1976 Gazli (Uzbekistan), 1988 Spitak 

(Armenia), 1991 Racha (Georgia), 1999 Kocaeli, 1999 Düzce, and other Turkish earthquake data. This 

dataset is principally restricted to free-field motions from shallow crustal continental earthquakes

(except for one earthquake from the Gulf of California). Appendix A lists all the events in the dataset 

 3

NGA-West2 database was not used because it was not available to us at the time of this research. 
3

with relevant information on their moment magnitude, focal depth, epicenter coordinates, faulting

mechanism, and breakdown of record numbers from each event. A total of 47 earthquakes were selected

and can be summarized as follows: 32 earthquakes from California; 6 earthquakes from Turkey; 4 

earthquakes from Taiwan and Italy; 3 earthquakes from Armenia, Georgia, and Uzbekistan; and 2 

earthquakes from Alaska and Nevada. In general, approximately 70 percent of earthquakes in our

dataset are from California. 

Among 2,583 ground-motion recordings, 1,450 are from reverse fault events, 1,120 are from 

strike-slip fault events, and 13 are from normal fault events. The distributions of data with respect to 

moment magnitude and VS30 against the closest distance to fault rupture plane (R) are shown in figure 

1A and B, respectively. The current dataset includes data recorded within 0.2 to 250 km of the 

earthquake faults from events in the magnitude range of 4.9 to 7.9 (fig. 1A). The data used in the 

analysis represent main shocks only; hence, records from any aftershocks were excluded. 

 

Figure 1. Plots showing earthquake data distribution with respect to A, moment magnitude (M), and B, average 

shear-wave velocity in the upper 30 m of the geological profile (VS30) against closest distance to fault rupture plane

(R). National Earthquake Hazard Reduction Program (NEHRP) site categories SB, SC, and SD are shown.

Approximately half of the stations in our dataset have measured shear-wave velocity (VS30), and

the rest have inferred VS30 values. The VS30 ranges between 200 and 1,316 m/s. In figure 1B, groundmotion data is sorted according to the National Earthquake Hazard Reduction Program (NEHRP) site 

categories. The values of Bdepth (depth to 1.5-km/s shear-wave velocity isosurface) are plotted against 

the closest distance to fault rupture plane in figure 2. Bdepth is available for only 353 ground motion 

recordings in our dataset. 

A B
4

Figure 2. Plots showing earthquake data distribution with respect to closest distance to fault rupture plane (R) 

and basin depth (Bdepth).

The distribution of PGA values with respect to moment magnitude and closest distance to fault 

rupture are shown in figure 3; except for a handful of recordings, the values of PGA are less than about 

0.8 g (g=gravitational acceleration).

Figure 3. Plots showing peak ground acceleration (PGA) distribution with respect to moment magnitude (M) and 

closest distance to fault rupture plane (R).
5

PGA

G3

 

Anelastic 

attenuation

= × × × × 

G5

 

Basin effect 

G1

 

Magnitude and 

style of faulting 

G2 

Distance

attenuation 

G4

Site correction 

Functional Form of Ground-Motion Prediction Equation 

In the following, we first introduce the functional form of updated Graizer-Kalkan GMPE for 

PGA and spectral shape and then explain their updates in detail. 

GMPE for Peak Ground Acceleration

The revised ground-motion prediction model for PGA has 12 coefficients and 6 independent 

parameters. Its independent parameters are as follows:

• M=moment magnitude; 

• R=closest distance to fault rupture plane, in km (Rrup as in Campbell and Bozorgnia, 2008); 

• VS30=average shear-wave velocity in the upper 30 m of the geological profile, in m/s; 

• F=style of faulting; 

• Q0=regional quality factor; and

• Bdepth=basin depth under the site in km. 

The updates includes the following: 

i. a new anelastic attenuation term as a function of quality-factor, 

ii. a new frequency-dependent basin-scaling term as a function of depth to 1.5-km/s shear-wavevelocity isosurface (Z1.5), and 

iii. updated coefficients. 

The form of GMPE for PGA has a series of functions in a multiplication form: 

(1)

where G1 is a scaling function for magnitude and style of faulting, G2 models the ground-motion 

distance attenuation (path scaling), G3 adjusts the attenuation rate considering regional anelastic 

attenuation, G4 models the site amplification owing to shallow site conditions, and G5 is a basin scaling 

function. Equation 1 can be expressed in natural logarithmic space as

ln(Y) = ln(G1) + ln(G2) + ln(G3) + ln(G4) + ln(G5) + σln(PGA) (2)

where Y is PGA and σln(PGA) is the total aleatory variability. The functional forms for G1, G2, G3, G4, 

and G5 are given in equation 2.1. 

ln(G1) = ln[(c1 ∙ arctan(M + c2) + c3) ∙ F] (2.1)

where F denotes the style of faulting (F=1.0 for strike-slip and normal faulting, F=1.28 for reverse 

faulting, and F=1.14 for combination of strike-slip and reverse faulting). c1–3 are estimator coefficients.

ln(G2) = −0.5 ∙ ln[(1 − R/R0 )2 + 4 ∙ (D0)2 ∙ (R/R0 )] (2.2)
6

where R0 and D0 are

R0 = c4 ∙ M + c5 (2.2.1)

D0 = c6 ∙ cos[c7 ∙ (M + c8)] + c9 (2.2.2)

where c4–9 are estimator coefficients. In equation 2, ln(G3), ln(G4), and ln(G5) are

ln(G3) = −c10 ∙ R/Q0 (2.3)

ln(G4) = bv ∙ ln(VS30/ VA) (2.4)

ln(G5) = ln[1 + ABdist ∙ ABdepth] (2.5)

where c10, bv, and VA are the estimator coefficents. ABdepth and ABdist are given in equations 2.5.1 and 

2.5.2.

ABdepth = 1.077 /��1 − (1.5/(Bdepth + 0.1))2�

2

+ 4 ∙ 0.72 ∙ �1.5/(Bdepth + 0.1)�

2

 (2.5.1)

ABdist = 1 /�[1 − (40/(R + 0.1))2]2 + 4 ∙ 0.72 ∙ (40/(R + 0.1))2 (2.5.2)

The values of the estimator coefficients of the above equations are presented in table 1.

Table 1. Estimator coefficients of Graizer-Kalkan (GK15) ground-motion prediction equation for peak ground 

acceleration.

c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 bv VA

0.14 −6.25 0.37 2.237 −7.542 −0.125 1.19 −6.15 0.6 0.345 −0.24 484.5

GMPE for Spectral Acceleration

In GK15, the 5-percent damped SA response ordinates are constructed by anchoring the spectral 

shape to PGA. The revised spectral shape model has 15 coefficients and 4 independent parameters. Its 

independent parameters are M, R, VS30, and Bdepth.

Updates on the spectral shape model include the following:

i. a modified decay term for long periods as a function of basin depth, 

ii. a revised term for controlling the predominant period of the spectrum, and 

iii. updated coefficients.
7

 

Spectral 

Shape SA = PGA × 

The form of GMPE for SA is

 (3)

The spectral shape (S norm) is formulated as

S norm(T) = I ∙ exp[−0.5 ∙ [(ln(T) + μ)/S]2] + ��1 − � T

Tss,0

�

ζ

�

2

+ 4 ∙ �Dss�

2

∙ � T

Tss,0

�

ζ

�

−0.5

 (3.1)

where T is the spectral period, and Dss is the estimator coefficient.

μ = m1 ∙ R + m2 ∙ M + m3 ∙ VS30 + m4 (3.1.1)

I = (a1 ∙ M + a2) ∙ exp (a3 ∙ R) (3.1.2)

S = 1 ∙ R − ( 2 ∙ M + 3) (3.1.3)

Tss,0 = max � 0.3

|t1 ∙ R + t2 ∙ M + t3 ∙ VS30 + t4|

 (3.1.4)

ζ = 1.763 − 0.25 ∙ atan [1.4 ∙ �Bdepth − 1�] (3.1.5)

where m1–4, a1–3,t1–4, and 1–3 are estimator coefficients. 

As illustrated in figure 4, I defines the peak spectral intensity, μ and Tss,0 define the 

predominant period of the spectrum, S defines the spectral wideness (area under the spectral shape), and 

ζ controls the decay of the spectrum at long periods and depends upon basin depth. The estimation 

coefficients of equations 3.1 through 3.1.5 are presented in table 2. 

Table 2. Estimator coefficients for Graizer-Kalkan (GK15) spectral shape (S norm) model. 

m1 m2 m3 m4 a1 a2 a3 Dsp t1 t2 t3 t4

−0.0012 −0.38 0.0006 3.9 0.01686 1.2695 0.0001 0.75 0.001 0.59 −0.0005 −2.3

s1 s2 s3 

0.001 0.077 0.3251
8

Figure 4. Graph showing the generic spectral shape (S norm) model used and its controlling parameters (I, S,

μ,Tsp,0). Note that PGA = peak ground acceleration. 

Filter Functions

The physical aspects of each filter function in equation 2 are described below.

G1=Magnitude and Style of Faulting

The following scaling function models the ground-motion scaling owing to the magnitude and 

style of faulting.

G1 = (c1 ∙ arctan(M + c2) + c3) ∙ F (4)

where, c1, c2, and c3 are the estimator coefficients, and F is the style of faulting scaling term. This 

scaling function reflects the saturation of ground-motion amplitudes with increasing magnitudes. 

According to the results of Sadigh and others (1997), reverse fault events create ground motions 

approximately 28 percent higher than those from crustal strike-slip faults. Following this, we used 

F=1.0 for strike-slip and normal faults, F=1.28 for reverse faults, and F=1.14 for combination of strikeslip and reverse faulting. The G1 and its estimation coefficients are same as in Graizer and Kalkan 

(2007).

G2=Distance Attenuation 

One of the important features of our GMPE is the use of frequency-response function of a 

damped single-degree-of-freedom oscillator for modeling the distance attenuation of ground motion. 

This modeling approach is explained in detail in Graizer and Kalkan (2007). Following this approach, 

the G2 models the ground-motion distance attenuation as 

G2 = 1 ÷ �(1 − R/R0 )2 + 4 ∙ (D0)2 ∙ (R/R0 ) (5)
9

where R0 is the corner distance in the near-source of an earthquake defining the plateau where the 

ground motion does not attenuate noticeably. In other words, R0 defines the flat region of the 

attenuation curve. R0 is directly proportional to earthquake magnitude—the larger the magnitude, the 

wider the plateau. The ground-motion observations show that R0 varies from 4 km for magnitude 5.0 to 

10 km for magnitude 7.9 (Graizer and Kalkan, 2007). R0 is similar to the corner frequency of the

Brune’s model (1970, 1971) since both are related to the magnitude. 

In equation 5, D0 is the damping term that designates the amplitude of the bump; the term 

“bump” refers to the increase in amplitude of ground motion at a particular distance from the fault.

D0=0.7 results in no bump. The fact that the highest PGA may not be recorded at the closest distance 

but at some distance from the fault is evident during the 1979 M6.5 Imperial Valley, 2004 M6.0 

Parkfield, and most recently 2014 M6.0 South Napa earthquakes (fig. 5). The bump phenomenon was 

recently demonstrated through modeling geometrical spreading and relative amplitudes of ground

motions in eastern North America; the bump was attributed to radiation pattern effects combined with 

wave propagation through a one-dimensional layered earth model (Chapman and Godbee, 2012; 

Baumann and Dalguer, 2014). In the case of earthquakes, bump can be a result of one or many factors,

including the aforementioned radiation pattern, directivity, nonlinear behavior of media in the nearsource of an earthquake fault (low-velocity fault-zone-guided waves), and measuring distance as the 

closest distance to the rupture plane and not from the seismogenic (most energetic) part of the fault 

rupture.
10

Figure 5. Plots showing attenuation of maximum component ground motions during the 2004 magntiude (M) 6.0 

Parkfield, 1979 M6.5 Imperial Valley, and 2014 M6.0 South Napa earthquakes. Graphs show amplified peak 

ground acceleration (PGA) as a bump at near field (R<10 km); this phenomenon is captured well by Graizer-Kalkan 

(GK15). Note: solid line is for median, dashed lines are for 16th and 84th percentile predictions (R=closest fault 

distance to rupture plane; VS30=average shear-wave velocity in the upper 30 m of the geological profile).
11

G3=Anelastic Attenuation 

The G3 function in the Graizer and Kalkan (2007) GMPE for PGA was a simple scaling term to 

account for attenuation at large distances and basin effects. This term is now replaced with an anelastic 

attenuation term.

G3 = exp(−c10 ∙ R/Q0) (6)

where Q0 is the regional quality factor for propagation of seismic waves from source to site at 1 Hz, and 

c10 is the estimator coefficient. The value for Q0 is, on average, 150 for California and 640–1,000 for 

central and northeast United States (Singh and Herrmann, 1983; Mitchell and Hwang, 1987; Erickson 

and others, 2004). 

Figure 6A demonstrates the effects of G3 on PGA estimation; the PGA data are from the 1999 

M7.6 Chi-Chi earthquake. The earlier version of the G3 in Graizer and Kalkan (2007), denoted as GK07, 

results in a constant attenuation rate (R-1.5) at large distances (shown by grey line). The black, red, and 

blue lines are for GK15 with G3 in equation 6 using Q0=75, 150, and 300, respectively. It is clear that a 

lower crustal Q0 results in faster attenuation, and a higher Q0 yields slower attenuation at far distances. 

Q associated with strong motion is different from seismological measurements because the 

typical seismological Lg and Coda wave estimates of Q sample different volumes of the crust 

surrounding the station and different paths than typical propagation paths of strong-motion signals 

(Trifunac, 1994). Trifunac demonstrated that the strong-motion Q increases from very low values near 

the fault (Q=20 associated with the upper part of the soil profile with relatively low shear-wave 

velocity) to larger values at about 100–200 km away from the source associated with typical crustal 

attenuation. For the 2004 M6.0 Parkfield earthquake, frequency independent Q increased from 20 in the 

upper 300 m of the soil profile to higher values of 100–200 for depth range of 200 m to 5 km 

(Abercrombie, 2000). 

Although Q is distance dependent, Q0 in equation 6 is a constant. In Figure 6B, we made a 

simple assumption that Q increases with distance from a relatively low value of 10 in the vicinity of the 

fault to higher values of typical Lg-type crustal Q at far distances (R>100 km). Figure 6C compares the 

effects of constant Q0 with that of the distance-dependent Q. As expected, low Q0 in the near-source 

region produces slightly lower ground-motion intensity. However, this decrease does not exceed 3

percent at distances up to 50 km. Higher Q0 at far distances results in slower attenuation relative to the 

constant Q0. The effect of distance-dependent Q relative to the constant (distance-independent) Q0 is not 

significant. Considering other uncertainties, we concluded that it is reasonable to use a constant Q0

typical for a given region (usually that for Lg or Coda waves). 

In our updated GMPE for PGA, we assume a frequency-independent Q0. In equation 6, 

c10=0.345, based on average value of Q0=150 published for California to produce similar effects as our 

previous GMPE for distances of up to 200 km (shown in fig. 6A). We expect that our GMPE for PGA

can be adjusted to other active tectonic regions similar to California by using Q0 values typical for that 

region and determined using Lg or Coda waves.
12

Figure 6. Graphs showing model results for anelastic attenuation with constant and variable Q0. A, Comparison 

of Graizer and Kalkan (GK07-09) peak ground acceleration (PGA) predictions with Graizer-Kalkan (GK15) 

considering the 1999 magnitude (M) 7.6 Chi-Chi earthquake data. B, Constant and distance-dependent Q0.C,

Comparison of attenuation curves with constant and distance-dependent Q0 for an M7.0 event (VS30=400 m/s)

(R=closest fault distance to rupture plane).
13

G4=Site Correction

Based on published studies (a list of references is given in Graizer and Kalkan, 2007), a linear 

site-correction filter was adopted in GK07 because of the large variability in nonlinear site-correction 

models.

G4 = exp[bv ∙ In(VS30/VA)] (7)

Equation 7 is an equivalent form of the linear site-correction formula of Boore and others

(1997), where bv=−0.371, whereas our estimates yield bv=−0.24. Equation 7, with its parameters given 

in table 1, is similar to the equation of Field (2000) in exhibiting less amplification as VS30 decreases 

than that of Boore and others (1997). In our revised GMPE for PGA, there is no change in G4 from its 

earlier version in Graizer and Kalkan (2007).

G5=Basin Effect

A basin consists of alluvial deposits and sedimentary rocks that are geologically younger and 

have a significantly lower shear-wave velocity structure than the underlying rocks, which creates a 

strong interface. A number of publications show that the basin amplifies earthquake-induced body and 

surface waves (for example, Hanks, 1975; Lee and others, 1995; Campbell, 1997; Frankel and others, 

2001). Our new basin scaling function considers combined effects of amplification of both shear and 

surface waves owing to basin depth under the site according to Hruby and Beresnev (2003) and Day and 

others (2008). For simplicity, the basin shape and distance to the basin edge (Joyner, 2000; Semblat and 

others, 2002; Choi and others, 2005) are not accounted for. 

The mechanisms and results of shear and surface-wave amplifications in the basin are different. 

The basin amplification of S-waves affects mostly frequencies lower than ~10 Hz (Hruby and Beresnev, 

2003), and basin amplification of surface waves affects a range of spectral frequencies (from PGA to 

long spectral periods). During the 1992 M7.3 Landers, 1999 M7.1 Hector Mine, and 2010 M7.2 El

Mayor-Cucapah earthquakes, the PGA values observed in Los Angeles and San Bernardino basins were 

much higher than those measured at rock sites owing to amplified surface waves (Graizer and others, 

2002; Hatayama and Kalkan, 2012). 

In our previous GMPE, the spectral shape decayed at long periods with a slope of T-1.5

(T=spectral period), averaging basin and nonbasin effects (Graizer and Kalkan, 2009). We changed this 

by implementing the following basin scaling filter, which is a function of depth to 1.5-km/s shear-wave 

velocity isosurface Z1.5 (Bdepth), R, and T. 

G5 = 1 + ABdist ∙ ABdepth (8)

ABdepth = 1.077 /��1 − (1.5/(Bdepth + 0.1))2�

2

+ 4 ∙ 0.72 ∙ �1.5/(Bdepth + 0.1)�

2

 (8.1)

ABdist = 1 /�[1 − (40/(R + 0.1))2]2 + 4 ∙ 0.72 ∙ (40/(R + 0.1))2 (8.2)

Equations 8.1 and 8.2 (previously given as 2.5.1 and 2.5.2) are repeated here for convenience. 

ABdepth defines the amplitude of the basin effect depending upon Bdepth. The parameters of equations 
14

Equations 8.1 and 8.2 were constrained according to the 1999 M7.1 Hector Mine, M7.3 Landers, and 

1989 M6.9 Loma Prieta earthquakes. 

As shown in figure 7A, ABdepth varies from 0 for nonbasin to 1.077 for deep basin, and it 

saturates for basins deeper than 3 km. When Bdd dh is zero, ABdepthbecomes negligibly small, and the 

G5 does not have any effect (G5=1.0). It should be noted that our approach on modeling the basin effect 

is based on the three-dimensional simulations of Day and others (2008). They found that depth to the 

1.5-km/s S-wave velocity isosurface is a suitable parameter for use in GMPEs. Similar basin 

amplification was observed in the Northridge and Whittier Narrows earthquakes by Hruby and Beresnev 

(2003). Our dependence of period amplification on Z1.5 approximates the period dependence in table 3

of Hruby and Beresnev (2003). 

Figure 7. Graphs showing A, dependence of amplitude on basin depth, and B, dependence of the response 

spectrum long period decay term (ζ) on basin depth (Bdepth). 

The parameter controlling decay rate of spectrum at long periods (ζ in equation 3.1.5) varies in 

the range of 1.4 to 2. As shown in figure 7B, the spectral shape decays at long periods faster (T-2) for 

nonbasin sites and slower (T-1.4) for deep basin sites. Figure 8 compares the spectral accelerations for 

two different basin depths, Bdepth=1.5 and 3 km against the case without basin (Bdepth=0). The deeper 
15

basin produces a response spectrum with higher amplitudes at all periods with slower decay at long 

periods; it affects the long periods more than the short periods. Based on a three-dimensional modeling 

of ground motion, a possible explanation for the distance-dependent pattern was suggested by Olsen 

(2000). According to Olsen, the amplification factors are greater for events located farther from the 

basin edge. He suggested that the larger-amplitude surface waves generated for the distant events, in 

part at basin edges, are more prone to the amplification than are the predominant body waves impinging 

onto the basin sediments from nearby earthquakes. 

Figure 8. Graphs showing comparison of pseudo spectral acceleration (SA) response computed using updated 

Graizer-Kalkan ground-motion prediction equation (GK15) for three cases: nonbasin, basin with 1.5-km depth, and 

basin with 3-km depth. Background data is from the 1999 magnitude (M) 7.1 Hector Mine earthquake at about 80-

km closest fault distance to the fault rupture plane (R). Note that Bdepth =depth to basin. 

Mixed-Effects Residuals Analysis

We performed a mixed-effects residuals analysis to confirm that GK15 is not biased with respect 

to M, R, VS30, and Bdepth by examining trends of residuals against these independent parameters. The 

residuals at each spectral period are computed as follows:

RRRii = lnYii − μij(M, R, VS30, Bdepth) (9)
16

where i is the event and j is the recording index. RRRii is the residual of the j

th recording of the ith event. 

Yii is the intensity measure (PGA or 5-percent damped pseudo spectral acceleration ordinates) from j

th

recording of the i

th event. Term μij represents the GK15’s median (that is, geometric mean4

) estimate in 

natural logarithmic units. In order to check for overall bias, we used the maximum-likelihood method to 

recursively determine the mean of data points having the error structure of Joyner and Boore (1993), 

where the residuals correspond to an equation of 

RRRii = C + ηi + εii (10)

where C is a constant term (maximum-likelihood mean) from the mixed-effects analysis, which is a 

measure of the overall bias between the observations and the GMPE. The constant term (C) should be 

close to zero for unbiased estimates. In equation 10, ηi represents the event term for event i, and εii

represents the intra-event residual for recording j in event i. Event term ηi represents the approximate 

mean offset of the data for event i from the predictions provided by the median of the GMPE. Event 

terms (ηi) are used to evaluate the GMPE’s performance relative to source predictor variables. Both 

event and intra-event (within-event) terms are random Gaussian variables with zero mean. Their 

standard deviations are indicated by τ and φ, respectively.

For each spectral period, equation 10 is solved using the maximum-likelihood formalism given 

in the appendix of Spudich and others (1999). In Figure 9, maximum-likelihood mean values are plotted 

for each spectral period ranging from PGA to 5.0 s; figure 9 shows that the overall bias of GK15 is 

small. Some discrepancies are plausible because a continuous function of spectral period was forced to 

fit to all spectral acceleration data instead of a discrete data fitting at each spectral period as other 

GMPE developers have done. 

Figure 9. Plot showing overall mean bias of Graizer-Kalkan (GK15) and its standard deviation; maximumlikelihood mean C shows the overall bias between the observations and predictions. 

 4For a log-normal distribution of a random variable, the geometric mean (μ̂) and median (x50) are given 

by the same equation: x50 = μ̂= μ, where μ is the mean of a log-normal distributionTherefore, it is 

not misleading to use median instead of geometric mean. 
17

Close examination of figure 9 indicates that mean bias at PGA and spectral accelerations at 0.2 

and 1.0 s is near zero. Increased variability at long periods is consistent with other GMPEs (for example, 

Abrahamson and others, 2013; Boore and others, 2013) because spectral acceleration data at long 

periods demonstrate larger aleatory variations than at short periods. At long periods, longer than 2.5 s, 

GK15 overestimates the data on average by about 0.1 in natural logarithmic units, or about 10 percent. 

The underestimation between 0.2 and 1.0 s is similar, and it is on average by about 10 percent. The 

overall bias is negligible for PGA.

In order to separate inter-event disparities from intra-event variations, we performed a mixedeffects analysis with respect to M, R, VS30, and Bdepth, and we fit a slope a and intercept b to residuals 

according to the following formulation:

RRRii = a + b i + ηi + εii (11)

Both slope and intercept computed using equation 11 are plotted against the spectral period in 

order to check for systematic bias with respect to the independent parameters. Figure 10 shows distance

bias in the residuals; again both intercept and slope are near zero for all periods, indicating negligible

distance dependence of GK15. Figure 11 shows that there is no systematic magnitude bias in the 

residuals of GK15; both intercept and slope are essentially zero across the entire period band. In figures

12 and 13, the dependence of GK15 on VS30 and Bdepth are examined; these plots show that slope of fit is 

essentially zero for VS30 and Bdepth. Although varying degrees of VS30 and Bdepth dependence at certain 

periods (for example, 0.4 and 3 s) are noticeable, this small dependence is not surprising because VS30

and Bdepth are the two parameters with the least accuracy in the dataset. Overall, small a values indicate

that GK15 does not show overprediction or underprediction for the broad range of periods; thus, it is 

essentially unbiased. 
18

Figure 10. Plots showing distance dependence of Graizer-Kalkan (GK15) residuals. Intercept (a) and slope (b) of 

maximum-likelihood line fit through residuals as a function of closest fault distance to rupture plane (R). Note that yaxis scale is different in top and bottom panels.
19

Figure 11. Plots showing magnitude dependence of Graizer-Kalkan (GK15) residuals. Intercept (a) and slope (b)

of maximum-likelihood line fit through residuals as a function of magnitude (M). Note that y-axis scale is different in 

top and bottom panels.
20

Figure 12. Plots showing VS30 dependence of Graizer-Kalkan (GK15) residuals. Intercept (a) and slope (b) of 

maximum-likelihood line fit through residuals as a function of VS30. Note that y-axis scale is different in top and 

bottom panels (VS30=average shear-wave velocity in the upper 30 m of the geological profile).
21

Figure 13. Plots showing depth to basin (Bdepth) dependence of Graizer-Kalkan (GK15) residuals. Intercept (a) and 

slope (b) of maximum-likelihood line fit through residuals as a function of Bdepth. Note that y-axis scale is different in 

top and bottom panels.

Intra-Event (Within-Event) Residuals Analysis of Path, Site, and Basin Depth Effects

The intra-event residuals (εii) are used to test the GK15 with respect to distance and site effects. 

The residuals are shown in natural logarithmic units for PGA and spectral periods at 0.2, 1.0, and 3.0 s,

similar to Chio and Youngs (2013). In figure 14, we plot the intra-event residuals against R (0 to 150 

km) using the full dataset, with means and standard errors shown within bins. The bin sizes were 

adjusted so that each bin has approximately the same number of data points. The maximum-likelihood 

line is dashed, and its slope and intercept are provided on top of each plot. Although data are slightly 

underpredicted at 1.0 and 3.0 s for distances greater than 110 km, the results generally show no 

perceptible trend within the body of a predictor, indicating that the path-scaling functions in GK15

reasonably represent the data trends. 

The linear site response analyses consider trends of residuals ij with VS30. The intra-event 

residuals are plotted against VS30 (200 to 1,200 m/s) in figure 15. At 1.0 s, we note a slight 

overestimation for VS30 in the range 470 to 560 m/s; we believe that this is in part caused by the 

sparseness of data within this VS30 range. This overestimation trend is negligible for PGA and spectral 
22

acceleration at 0.2 s. In each figure, the flatness of the trends (dashed line) for VS30 indicates that our 

linear site response function (applicable for VS30>200 m/s) is a reasonable average for shallow crustal

continental regions. 

To examine possible sediment depth effects, we plot intra-event residuals in figure 16 against 

Bdepth in the range 0 to 2,500 m. In general, these residuals do not exhibit notable trends since they are 

near zero; there is little dependence on Bdepth between 1,200 and 1,400 m at 0.2 and 3.0 s; this is again 

attributed to the scarcity of the data within this range. It should be noted that number of data points 

contributing to these plots are less than those provided in previous figures because approximately 15

percent of data in our dataset have known Bdepth. 
23

Figure 14. Plots showing distribution of intra-event residuals in natural logarithmic units for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s with respect to closest fault 

distance to rupture plane (R). Black horizontal lines indicate size of the bins and their mean value; their standard 

deviation is shown by vertical solid red lines; dashed red line denotes a maximum-likelihood fit to residuals;

different symbols indicate different events. Note that a and b are intercept and slope of maximum likelihood fit. 
24

Figure 15. Plots showing distribution of intra-event residuals in natural logarithmic units for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s with respect to VS30. Black 

horizontal lines indicate size of the bins and their mean value; their standard deviation is shown by vertical solid red 

lines; dashed red line denotes a maximum-likelihood fit to residuals; different symbols indicate different events.

Note that a and b are intercept and slope of maximum likelihood fit. (VS30=average shear-wave velocity in the upper 

30 m of the geological profile). 
25

Figure 16. Plots showing distribution of intra-event residuals in natural logarithmic units for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s with respect to Bdepth. Black 

horizontal lines indicate size of the bins and their mean value; their standard deviation is shown by vertical solid red 

lines; dashed red line denotes a maximum-likelihood fit to residuals; different symbols indicate different events.

Note that a and b are intercept and slope of maximum likelihood fit (Bdepth= depth to basin).
26

Analysis of Source Effects Using Inter-event (Between-Event) Residuals

In figure 17, we show event terms (ηi) plotted against magnitude for PGA and SA at 0.2, 1.0,

and 3.0 s using 35 events in the range 4.9≤M≤7.9; the list of these events is given in table 3. The 

majority of the events, especially at small magnitudes, are from California. The events with less than 

five data points were excluded; this reduced the number of events from 47 to 35. In figure 17, the 

maximum-likelihood line is dashed, and its slope and intercept are provided on top of each plot. We see 

that our magnitude-scaling function (G1) captures the trends from various events as evident by near-zero 

intercept and near-zero slope of the maximum-likelihood fit, indicating that there is no significant trend 

with magnitude or a notable offset from zero. Except for the 2003 M6.5 San Simeon and 2004 Parkfield 

earthquakes, shown by  and ∗, respectively, all other 33 events exhibit an event term less than 1.0 

(ηi<1.0). <The event terms are provided in table 3 for PGA and SA at 0.2, 1.0, and 3.0 s.
27

Figure 17. Plots showing distribution of event terms (ηi) in natural logarithmic units for peak ground acceleration 

(PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s with respect to moment magnitude (M); in each 

plot dashed red line indicates a maximum-likelihood fit to event terms, its slope and intercept are provided on top of 

each plot. Note that a and b are intercept and slope of maximum likelihood fit.
28

Table 3. Event terms (ηi) using 35 events in the magnitude range of 4.9 to 7.9. [See appendix A for details of earthquakes. No, number of event listed in alphabetical order; s, second; PGA, peak ground acceleration]

No Event Event Term (ηi)

PGA 0.2 s 1.0 s 3.0 s

1 Anza 0.45 0.58 0.02 −0.76

2 Big Bear City −0.22 0.06 0.70 0.73

3 Borrego Mnt. 0.22 0.18 0.44 0.80

4 Chalfant Valley −0.33 −0.20 −0.25 0.15

5 Chi-Chi (Taiwan) −0.31 −0.37 0.08 0.29

6 Coalinga-01 −0.07 −0.34 0.40 0.33

7 Coalinga-05 0.29 0.18 0.00 0.10

8 Coyote Lake 0.04 −0.02 0.33 0.21

9 Denali (Alaska) −0.23 −0.11 −0.51 −0.34

10 Düzce (Turkey) −0.84 −0.72 −0.87 −0.71

11 Friuli (Italy) −0.12 −0.13 −0.31 −0.34

12 Gulf of California −0.12 −0.32 0.21 0.60

13 Hector Mine 0.12 −0.09 −0.39 −0.83

14 Imperial Valley −0.31 −0.22 −0.26 0.30

15 Kocaeli (Turkey) −0.05 0.04 0.06 0.31

16 Landers −0.23 −0.16 0.02 0.20

17 Lazio-Abruzzo (Italy) −0.10 0.04 −0.01 −0.50

18 Little Skul Mtn. (Nevada) −0.28 −0.20 −0.77 −0.63

19 Livermore −0.06 −0.02 0.44 0.78

20 Loma Prieta −0.02 0.05 0.26 0.30

21 Mammoth Lakes-06 0.39 0.28 0.08 −0.09

22 Manjil (Iran) 0.52 0.56 0.47 0.94

23 Morgan Hill −0.39 −0.45 −0.24 −0.33

24 Northridge −0.06 0.09 −0.03 −0.33

25 North Palm Springs 0.00 0.06 −0.37 −0.57

26 Parkfield −0.21 −0.10 −0.35 0.09

27 Parkfield (2004) −0.23 −0.99 −1.99 −2.33

28 San Fernando −0.42 −0.26 −0.55 −0.31

29 San Simeon 1.71 1.43 2.17 2.47

30 Sierra Madre 0.61 0.58 0.33 0.09

31 Superstition Hills-02 0.03 −0.08 0.11 0.47

32 Taiwan, Smart(5) 0.15 0.27 0.29 0.29

33 Whittier Narrows 0.04 0.23 −0.05 −0.55

34 Yountville −0.35 −0.35 0.60 0.27

35 Yucaipa 0.37 0.53 −0.05 −1.13
29

Terms of Standard Deviation

In GMPEs, total residuals are composed of intra-event residuals and event terms. The standard 

deviation (σ) of total residuals (that is, total aleatory variability) is defined as

σ = �τ2 + φ2 (12)

Figure 18 plots the inter-event standard deviations (τ), intra-event standard deviations (φ), and 

total standard deviations (σ) in natural logarithmic units (their values are tabulated in table 4). Figure 18 

shows that at long periods, there is likely energy content in the ground motions affecting the singledegree-of-freedom oscillator response, thus increasing the response variability and associated standard 

deviations. σ increases with period similar to the NGA-West2 GMPEs (for example, Abrahamson and 

others, 2014; Boore and others, 2014; Chiou and Youngs, 2014). For short periods (from 0.01 s to 0.3

s), σ is almost constant.

Recall that our GMPE for spectral shape is a continuous function of spectral period. To be 

consistent with this continuous form, we model the total aleatory variability (σ) with a continuous 

function of spectral period (T) as

σ(T) = max �

0.668 + 0.0047 ∙ log(T)

0.8 + 0.13 ∙ log(T) � (13)

Figure 19 compares the approximated σ using equation 13 with the actual σ. The approximation 

function matches well with the actual σ at all periods. 

The first stage of the variance analysis is to examine the magnitude dependence of inter-event

standard deviations (τ) and intra-event standard deviations (φ). Figures 20 and 21 show the values of 

τ and φ in natural logarithmic units computed from the residuals, considering eight magnitude bins. 

These figures show binned values of τ and φ for PGA and spectral acceleration ordinates at 0.2, 1.0, and 

3.0 s, considering residuals for all magnitudes in the range 4.9≤M≤7.9. In these figures, each bin has an 

equal number of data points; the ranges of bins are plotted with dashed black lines. The results shown in

figure 20 indicate lower values of τ for larger magnitudes at all periods. The fitted values of τ

demonstrate some level of magnitude dependence at most periods; therefore, a trilinear form is applied. 

The appropriate magnitude breakpoint is at 7.1 because most of the change occurs between magnitudes 

of approximately 7.0 and 7.5. The following magnitude-dependent model represents the inter-event

standard deviations (τ):

τ(M) = �

1 for M ≤ 7.1

1 + ( 2− 1) ∙ (M − 7.1) for 7.1 < M < 7.5

2 for M ≥ 7.5

 (14)

Terms 1 and 2 change slightly with spectral period; their values are provided in table 5. 
30

Figure 18. Plots showing inter-event (between-event), intra-event (within-event), and total standard deviations of 

Graizer-Kalkan (GK15) in natural-logarithmic units computed based on mixed-effects residuals analysis.
31

Table 4. Inter-event standard deviations (τ), intra-event standard deviations (φ), and total standard deviations

(σ) in natural logarithmic units.

Period (s) τ φ σ Period (s) τ φ σ

PGA 0.435 0.508 0.669 0.350 0.415 0.567 0.702

0.010 0.416 0.510 0.658 0.360 0.417 0.568 0.704

0.020 0.422 0.510 0.662 0.380 0.421 0.569 0.708

0.022 0.428 0.512 0.667 0.400 0.425 0.570 0.711

0.025 0.432 0.514 0.671 0.420 0.428 0.569 0.712

0.029 0.436 0.516 0.675 0.440 0.432 0.569 0.714

0.030 0.440 0.518 0.680 0.450 0.432 0.570 0.715

0.032 0.442 0.520 0.682 0.460 0.433 0.571 0.716

0.035 0.444 0.522 0.685 0.480 0.435 0.571 0.718

0.036 0.445 0.524 0.687 0.500 0.437 0.572 0.720

0.040 0.446 0.525 0.689 0.550 0.443 0.572 0.724

0.042 0.447 0.526 0.691 0.600 0.450 0.573 0.729

0.044 0.448 0.527 0.692 0.650 0.457 0.576 0.735

0.045 0.448 0.528 0.692 0.667 0.460 0.580 0.740

0.046 0.449 0.528 0.693 0.700 0.468 0.585 0.749

0.048 0.448 0.528 0.693 0.750 0.480 0.589 0.760

0.050 0.450 0.528 0.693 0.800 0.490 0.591 0.768

0.055 0.451 0.528 0.694 0.850 0.502 0.592 0.776

0.060 0.452 0.527 0.694 0.900 0.515 0.593 0.786

0.065 0.452 0.527 0.695 0.950 0.530 0.596 0.797

0.067 0.453 0.528 0.695 1.000 0.543 0.597 0.807

0.070 0.453 0.528 0.696 1.100 0.562 0.598 0.820

0.075 0.451 0.528 0.695 1.200 0.579 0.598 0.833

0.080 0.449 0.528 0.693 1.300 0.595 0.598 0.844

0.085 0.446 0.528 0.691 1.400 0.609 0.597 0.853

0.090 0.443 0.528 0.689 1.500 0.620 0.599 0.862

0.095 0.440 0.527 0.687 1.600 0.626 0.603 0.869

0.100 0.438 0.528 0.686 1.700 0.632 0.606 0.876

0.110 0.435 0.528 0.685 1.800 0.633 0.610 0.879

0.120 0.431 0.529 0.683 1.900 0.635 0.616 0.885

0.130 0.429 0.530 0.682 2.000 0.635 0.624 0.890

0.133 0.426 0.531 0.680 2.200 0.639 0.634 0.900

0.140 0.423 0.532 0.680 2.400 0.640 0.648 0.911

0.150 0.422 0.534 0.681 2.500 0.644 0.671 0.930

0.160 0.420 0.536 0.681 2.600 0.650 0.693 0.950

0.170 0.419 0.536 0.680 2.800 0.656 0.710 0.967

0.180 0.414 0.536 0.678 3.000 0.660 0.718 0.975

0.190 0.410 0.539 0.677 3.200 0.665 0.719 0.979

0.200 0.407 0.541 0.677 3.400 0.673 0.719 0.984
32

Period (s) τ φ σ Period (s) τ φ σ

0.220 0.407 0.544 0.679 3.500 0.680 0.723 0.992

0.240 0.409 0.547 0.683 3.600 0.683 0.722 0.994

0.250 0.408 0.550 0.685 3.800 0.687 0.727 1.000

0.260 0.406 0.554 0.687 4.000 0.682 0.721 0.992

0.280 0.407 0.558 0.690 4.200 0.680 0.715 0.986

0.290 0.405 0.561 0.692 4.400 0.676 0.717 0.985

0.300 0.406 0.564 0.694 4.600 0.670 0.718 0.982

0.320 0.410 0.565 0.698 4.800 0.667 0.718 0.980

0.340 0.414 0.566 0.701 5.000 0.699 0.745 1.022

Figure 19. Plot showing total standard deviations (σ) of Graizer-Kalkan (GK15) and its approximation in natural 

logarithmic units.

The φ values shown in figure 21 demonstrate that the intra-event standard deviations are stable 

with small variations over a wide range of magnitudes. In these figures, colored lines denote the 

population mean, which is close to the mean values of the bins. A complex relationship of intra-event 

standard deviations with magnitude at 3 s is apparent. However, for PGA and spectral acceleration 

ordinates at periods of 0.2 and 1 s, intra-event standard deviations seem to be magnitude independent.
33

Figure 20. Plots showing values of inter-event (between-event) standard deviations (τ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0, and 3.0 s in magnitude bins with equal 

number of data points; magnitude (M) range for each bin is shown by a horizontal black dashed line; colored lines 

denote the simplified τ model as a function of M.
34

The distance dependence of τ and φ is examined next in figures 22 and 23, respectively. The 

standard deviation data is divided into ten distance bins and plotted against R. Each bin has an equal 

number of data points. The inter-event standard deviations (τ) rise for the distances larger than 100 km.

This is attributed to the regional anelastic attenuation effects that are not fully captured by our model. 

Thus, we expect that this increase is strongly influenced by epistemic uncertainty in regional attenuation 

rates, not random site-to-site variability. To identify this variation, we adjust the model in equation 14 to 

include an additive term that is applicable for R>100 km as follows:

τ(M, R) = �τ(M)

τ(M) + r1 for R ≤ 100

+(r1 − r2) ∙ (R − 100) for 100 < R < 130

τ(M) + r2 for R ≥ 130

 (15)

Terms r1 and r2 are selected by visual inspection; they represent the maximum distance to which 

the complete GMPE terms are considered applicable (r1, generally up to 100 km) and the distance 

beyond which we capped increases with distance (r2 , generally up to 150 km). Figure 23 indicates that 

φ is overall independent of R; there are slight variations at large distances. 

In figures 24 and 25, we plot the values of τ and φ within VS30 bins. Each bin has an equal 

number of data points. We observe no change in τ for PGA and short periods. There is a slight variation 

of τ with VS30 at long periods above 500 m/s. For φ, there is no significant variation with VS30. This is

attributed to the smaller dataset that we used as compared to GMPEs of Abrahamson and others (2013) 

and Boore and others (2013), which have a VS30 dependence on τ because the NGA-West2 dataset has 

only 22 percent of VS30 measured, and the remaining 78 percent of NGA-West2 dataset are from 

estimations based on correlation of VS30 with local surface geology. Approximately half of the stations 

in our database have measured VS30, whereas the rest have inferred VS30 values. Overall, τ is magnitude 

and distance dependent. For a site with given M and R, τ can be computed using equation 15 with the 

parameters 1, 2, r1, and r2 provided in table 5.

Table 5. Coefficients of inter-event (between-event) standard deviation term. [SA = spectral acceleration]

 Peak ground 

acceleration SA (0.2) SA (1.0) SA (3.0)

s1 0.28 0.26 0.33 0.51

s2 0.04 0.07 0.06 0.06

r1 0.30 0.35 0.31 0.57

r2 0.52 0.50 0.58 0.90
35

Figure 21. Plots showing values of intra-event (within-event) standard deviations (φ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0 and 3.0 s in magnitude bins with equal 

number of data points; magnitude (M) range for each bin is depicted by a horizontal black dash line; color lines 

denote the simplified φ model as a function of M.
36

Figure 22. Plots showing values of inter-event (between-event) standard deviations (τ) computed for peak ground 

acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0 and 3.0 s in distance bins with equal number 

of data points; distance range for each bin is shown by a horizontal black dash line; color lines denote the simplified 

τ model as a function of closest fault distance to rupture plane (R).
37

Figure 23. Plots showing values of intra-event (within-event) standard deviations (φ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0 and 3.0 s in distance bins with equal number 

of data points; distance range for each bin is shown by a horizontal black dash line; color lines denote the simplified 

φ model as a function of closest fault distance to rupture plane (R).
38

Figure 24. Plots showing values of inter-event (between-event) standard deviations (τ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0 and 3.0 s in VS30 bins with equal number of 

data points; VS30 range for each bin is shown by a horizontal black dash line; color lines denote the simplified τ

model as a function of average shear-wave velocity in the upper 30 m of the geological profile (VS30).
39

Figure 25. Plots showing values of intra-event (within-event) standard deviations (φ) computed for peak ground 

acceleration (PGA) and pseudo spectral acceleration (SA) at 0.2, 1.0 and 3.0 s in VS30 bins with equal number of 

data points; VS30 range for each bin is shown by a horizontal black dash line; color lines denote the simplified φ

model as a function of average shear-wave velocity in the upper 30 m of the geological profile (VS30).
40

Model Results

The median SA response spectra for the GK15 model are shown in figure 26 for a vertical strikeslip earthquake scenario with M=5.0, 6.0, 7.0, and 8.0, at distances R=1 and 30 km, and VS30=760 and 

270 m/s, similar to comparisons given in Abrahamson and others (2013). Note that increase in 

magnitude shifts the predominant period of spectrum to larger values. The predominant period in our 

spectral shape model is controlled by μ and Tsp,0 as shown in figure 4, and both of them are magnitude 

dependent. A wider spectrum generated by larger magnitudes implies that energy at different periods is 

enriched by the complex wave propagation.

The path scaling is shown next in figure 27 for PGA and spectral periods at 0.3, 1.0, and 3.0 s. In 

this figure, the median ground motion from strike-slip earthquakes on rock site condition (VS30=760 

m/s) is shown for four different magnitudes, M=5.0, 6.0, 7.0, and 8.0. At intermediate distance range (5 

to 20 km from the fault), the GK15 model produces higher acceleration values. These high accelerations 

look like a bump on the attenuation curves. The reason for this bump was explained in section 

“G2=Distance Attenuation.” 

The magnitude scaling of the current model is shown in figure 28 for vertical strike-slip 

earthquakes on rock site conditions (VS30=760 m/s) for T=0.2, 1.0, and 3.0 s at distances R=1, 30, and 

150 km. Note that the break in the magnitude scaling at M5.5 is driven by consistency in response 

spectra of recorded data. The weak scaling of the short-period motion at short distances reflects the 

saturation with magnitude.

The site response scaling for an M7.0 vertical strike-slip earthquake at a closest rupture distance 

of 30 km is demonstrated in figure 29. The decrease in shear-wave velocity amplifies ground motion

and shifts the predominant period to higher spectral periods. Based on the same rupture scenario, figure 

30 displays the dependence of the spectra on Bdepth for a soil site with VS30=270 m/s.

The style of faulting scaling on the SA response spectra is shown in figure 31 for an M7.0

vertical strike-slip earthquake at a rupture distance of 30 km for a soil site with VS30=270 m/s. Lastly, Q0

is shown in figure 32 for PGA and spectral periods at 0.3, 1.0, and 3.0 s. The median ground motion 

from an M7.0 strike-slip earthquake on rock site condition (VS30=760 m/s) is shown for four different 

Q0. Figure 33 demonstrates the effects of Q0 on SA response spectra. The higher Q0 results in higher 

acceleration values. As Q0 increases, its effect on SA diminishes for all periods. 
41

Figure 26. Graphs showing comparison of median pseudo spectral accelerations (SA) for strike-slip magnitude

(M) 5.0, 6.0, 7.0, and 8.0 earthquakes at R=1 and 30 km and VS30=270 and 760 m/s considering Q0=150 and 

Bdepth=0 (R=closest fault distance to rupture plane; VS30=average shear-wave velocity in the upper 30 m of the 

geological profile; Q0=quality factor; Bdepth= depth to basin).
42

Figure 27. Graphs showing comparison of distance scaling for strike-slip magnitude (M) 5.0, 6.0, 7.0, and 8.0

earthquakes at median PGA and pseudo spectral accelerations at 0.2, 1.0, and 3.0 s considering VS30=760 m/s,

Q0=150 and Bdepth=0 (R=closest fault distance to rupture plane; VS30=average shear-wave velocity in the upper 30 

m of the geological profile; Q0=quality factor; Bdepth= depth to basin).
43

Figure 28. Graphs showing comparison of magnitude scaling for strike-slip earthquakes at closest fault distance 

to rupture plane, R=1, 30, and 150 km for median peak ground acceleration (PGA) and pseudo spectral

accelerations (SA) at 0.3, 1.0, and 3.0 s considering VS30=760 m/s, Q0=150 and Bdepth=0 (R=closest fault distance 

to rupture plane; VS30=average shear-wave velocity in the upper 30 m of the geological profile; Q0=quality factor; 

Bdepth= depth to basin).
44

Figure 29. Graph showing comparison of VS30 scaling on pseudo spectral accelerations (SA) for a strike-slip 

magntitude (M) 7.0 earthquake at R=30 km considering Q0=150 and Bdepth=0 (VS30=average shear-wave velocity in 

the upper 30 m of the geological profile; Q0=quality factor; Bdepth= depth to basin).

Figure 30. Graph showing comparison of Bdepth scaling on pseudo spectral accelerations (SA) for a strike-slip 

M7.0 earthquake at R=30 km considering Q0=150 and VS30=270 m/s (R=closest fault distance to rupture plane; 

VS30=average shear-wave velocity in the upper 30 m of the geological profile; Q0=quality factor; Bdepth= depth to 

basin).
45

Figure 31. Graph showing comparison of style of faulting scaling on pseudo spectral accelerations (SA) for a 

magnitude (M) 7.0 earthquake at R=30 km considering Q0=150, VS30=270 m/s, and Bdepth=0 (R=closest fault 

distance to rupture plane; VS30=average shear-wave velocity in the upper 30 m of the geological profile; Q0=quality 

factor; Bdepth= depth to basin).
46

Figure 32. Graphs showing comparison of Q0 scaling with closest fault distance to rupture plane (R) for a strikeslip magnitude (M) 7.0 earthquake for median peak ground acceleration (PGA) and pseudo spectral accelerations

(SA) at 0.3, 1.0, and 3.0 s considering VS30=760 m/s and Bdepth=0 (VS30=average shear-wave velocity in the upper 

30 m of the geological profile; Q0=quality factor; Bdepth= depth to basin).
47

Figure 33. Graphs showing comparison of Q0 scaling on pseudo spectral accelerations (SA) for a strike-slip 

magnitude (M) 7.0 earthquake at R=30 and 120 km considering Q0=150, VS30=270 m/s and Bdepth=0 (R=closest 

fault distance to rupture plane; VS30=average shear-wave velocity in the upper 30 m of the geological profile; 

Q0=quality factor; Bdepth= depth to basin).

Comparisons With Graizer-Kalkan 2007–2009 Models

Figure 34 compares median predictions of Graizer and Kalkan (2007, 2009) (denoted as GK07–

09) to GK15 for strike-slip (F=1.0), M5, 6.0, 7.0 ,and 8.0 earthquakes at R=1 and 30 km, considering 

two different soil conditions (VS30=270 and 760 m/s). For these comparisons, Q0 and Bdepth are taken as 

150 and 0 km (nonbasin), respectively. The notable differences between the predictions from GK15 and 

those from GK07–09 are for intermediate and long periods. The spectral ordinates predicted from GK15

are larger than those from GK07–09 at T<0.2 s, and they are smaller at longer periods. This adjustment 

was done based on data from California earthquakes such as Landers, Northridge, and Hector Mine.

This difference is more pronounced at R=30 km as compared to R=1 km. Overall, the PGA values are 

very similar in GK15 and GK07-09.

Figure 35 compares path scaling between GK07–09 and GK15 for conditions that are often 

significant for seismic hazard in the western United States. The difference in attenuation of PGA is 

minimum; GK15 predicts slightly lower PGA values only at close distances. The differences are most 

notable at long spectral periods (1.0 and 3.0 s), where the models provide lower estimates of spectral 

accelerations at all distances. 
48

Figure 34. Graphs showing comparison of median pseudo spectral accelerations (SA) for strike-slip magnitude 

(M) 4.5, 5.5, 6.5, and 7.5 earthquakes at closest fault distance to rupture plane R=1 and 30 km considering 

VS30=270 and 760 m/s, Q0=150, and Bdepth=0 (VS30=average shear-wave velocity in the upper 30 m of the geological 

profile; Q0=quality factor; Bdepth= depth to basin).
49

Figure 35. Graphs showing comparison of distance scaling for strike-slip magnitude (M) 5.0, 6.0, 7.0, and 8.0

earthquakes at median peak ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 

3.0 s considering VS30=760 m/s, Q0=150, and Bdepth=0 (R=closest fault distance to rupture plane; VS30=average 

shear-wave velocity in the upper 30 m of the geological profile; Q0=quality factor; Bdepth= depth to basin).

Figure 36 compares total aleatory variability in the present study to those of GK07–09. The total 

aleatory variability from the present study is generally higher at all periods than of GK07–09. We 

attribute the higher σ to the way that we computed total aleatory variability in GK15, by the method 

provided in Joyner and Boore (1993), which uses two random variables (εr and εe) replaced by a single 
50

random variable ε as in GK07–09. The higher σ could result from the triangle inequality and the lack of 

correlation between random variables εr and εe (P. Spudich, U.S. Geological Survey, oral commun., 

2014).

Figure 36. Plot showing comparison of total aleatory variability (σ) between Graizer-Kalkan (2007, 2009) (GK07–

09) and Graizer-Kalkan (GK15) ground-motion prediction equations.

Comparisons With NGA-West2 Models

We compare the model predictions of GK15 with those of Abrahamson and others (2014) 

(abbreviated as ASK14) and Boore and others (2014) (abbreviated as BSSA14). Figures 37 and 38

compare the median attenuation of PGA and spectral accelerations at 0.2, 1.0, and 3.0 s for horizontal 

motions for a vertically dipping strike-slip earthquake as a function of closest distance to fault rupture.

These results are for VS30=760 m/s. Visual comparison of the predictions of our model with those of the 

two NGA-West2 relationships shows that our predictions are in good agreement with the predictions of 

ASK14 and BSSA14 for a range of magnitudes and distances. For PGA and SA at 0.2 s, GK15 produces 

similar or slightly lower results as compared to the BSSA14 and ASK14 at close distances (0–5 km). At 

intermediate distances (5–20 km), the GK15 produces higher values because of the bump. 

The response spectra from magnitude 6.0, 7.0, and 8.0 earthquakes at R=1 and 30 km from a 

vertically dipping strike-slip fault and VS30=270 and 760 m/s are plotted in figures 39 and 40. The same 

parameters used in figures 37 and 38 were repeated. There is similarity among the three models for the 

M6–8 cases. The difference between models increases for the M8 case, especially at the long spectral 

periods.
51

Figure 37. Graphs showing comparison of distance scaling of Graizer-Kalkan (GK15) and Boore and others 

(2014) (BSSA14) for strike-slip magnitude (M) 6.0, 7.0, and 8.0 earthquakes at median peak ground acceleration 

(PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s considering VS30=760 m/s, Q0=150, and

Bdepth=0 (VS30=average shear-wave velocity in the upper 30 m of the geological profile; Q0=quality factor; Bdepth= 

depth to basin).
52

Figure 38. Graphs showing comparison of distance scaling of Graizer-Kalkan (GK15) and Abrahamson and 

others (2014) (ASK14) for strike-slip magnitude (M) 6.0, 7.0, and 8.0 earthquakes at median peak ground 

acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s considering VS30=760 m/s,

Q0=150, and Bdepth=0 (VS30=average shear-wave velocity in the upper 30 m of the geological profile; Q0=quality 

factor; Bdepth= depth to basin).
53

Figure 39. Graphs showing comparison of spectra from Graizer-Kalkan (GK15) and Boore and others (2014) 

(BSSA14) for strike-slip magnitude (M) 6.0, 7.0, and 8.0 earthquakes at closest fault distance to rupture plane R =1 

and 30 km, considering VS30=270 and 760 m/s, Q0=150, and Bdepth=0 (VS30=average shear-wave velocity in the 

upper 30 m of the geological profile; Q0=quality factor; Bdepth= depth to basin).
54

Figure 40. Graphs showing comparison of spectra from Graizer-Kalkan (GK15) and Abrahamson and others 

(2014) (ASK14) for strike-slip magnitude (M) 6.0, 7.0, and 8.0 earthquakes at closest fault distance to rupture plane 

R =1 and 30 km, considering VS30=270 and 760 m/s, Q0=150, and Bdepth=0 (VS30=average shear-wave velocity in the 

upper 30 m of the geological profile; Q0=quality factor; Bdepth= depth to basin).
55

Comparisons With Earthquake Data 

In figures 41–53, distance attenuation curves by GK15 for PGA and spectral accelerations at 0.2, 

1.0, and 3.0 s are plotted against the ground-motion data from 13 major events in California, Alaska, 

Taiwan, and Turkey. The results are based on randomly oriented maximum horizontal component of 

ground motions; average VS30 of dataset was used for each event. Figures are presented in chronological 

order of earthquakes. Visual comparison of predictions derived from the GK15 with observations reveal 

a reasonable to excellent performance of the revised model for a range of magnitudes and distances. 
56

Figure 41. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 1971 magntiude (M) 6.6 San Fernando earthquake (σ =aleatory deviation, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
57

Figure 42. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 1979 magntiude (M) 6.5 Imperial Valley earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
58

Figure 43. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 1989 magntiude (M) 6.9 Loma Prieta earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
59

Figure 44. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 1992 magntiude (M) 7.3 Landers earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
60

Figure 45. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 1994 magntiude (M) 6.4 Northridge earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
61

Figure 46. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 1999 magntiude (M) 7.1 Hector Mine earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile) 
62

Figure 47. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 1999 magntiude (M) 7.4 Kocaeli (Turkey) earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
63

Figure 48. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 1999 magnitude (M) 7.2 Düzce (Turkey) earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
64

Figure 49. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 1999 magnitude (M) 7.6 Chi-Chi (Taiwan) earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
65

Figure 50. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 2002 magnitude (M) 6.5 San Simeon earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
66

Figure 51. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 2002 magnitude (M) 7.9 Denali earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
67

Figure 52. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 2004 magnitude (M) 6.0 Parkfield earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
68

Figure 53. Comparisons of Graizer-Kalkan (GK15) median, 16th, and 84th percentile distance attenuation of peak 

ground acceleration (PGA) and pseudo spectral accelerations (SA) at 0.2, 1.0, and 3.0 s with ground motion data 

from the 2014 magnitude (M) 6.0 South Napa earthquake (σ = total aleatory variability, GM=ground motion, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile).
69

Example Calculations Using MatLAB Codes

The GK15 GMPE is coded as a MatLAB function (titled “GK15.m”) in appendix B. An example 

MatLAB code (“runGK15.m”) to generate a 5-percent damped pseudo spectral acceleration response 

spectrum for a given hazard condition is provided in appendix C. Running this code generates the 

spectrum shown in figure 54 for a vertically dipping strike-slip M6.0 event at a site with VS30=350 m/s 

and 4.4 km away from the rupture plane. The user can change the input parameters to construct a sitespecific response spectrum considering different hazard conditions. 

Figure 54. Site-specific 5-percent damped pseudo spectral acceleration (SA) response spectra generated using 

the MatLAB code in appendix C for a vertically dipping strike-slip magnitude (M) 6.0 earthquake at closest fault 

distance to rupture plane (R)=4.4 km considering VS30=350 m/s, Q0=150, and Bdepth=0 (σ = total aleatory variability, 

VS30=average shear-wave velocity in the upper 30 m of the geological profile; Q0=quality factor; Bdepth= depth to 

basin). 

Range of Applicability

The GMPE update developed in this study is considered to be applicable for estimating pseudo 

spectral accelerations (5 percent of critical damping) in the period range of 0.01 to 5 s and peak motions 

for earthquakes in which the following conditions apply:
70

• magnitude (M), 5.0–8.0;

• distance (R), 0–250 km;

• time-averaged shear-wave velocity (VS30), 200–1,300 m/s; 

• Q0, up to 250, typical for active tectonic environment; and

• basin depth (Bdepth), 0–10 km.

These limits are based on the distributions of the data used to develop the GK15 GMPE. The 

GMPE presented in this report is intended for application in western United States or in similar 

tectonically active shallow crustal regions. The data controlling the equations are derived principally 

from California; we have also used ground-motion data from Nevada, Alaska, Turkey, Taiwan, Italy, 

Armenia, Georgia, and Uzbekistan. Implementation of the GK15 to other active crustal regions may 

result in an additional degree of epistemic uncertainty.

The site and basin effects incorporated into the GK15 GMPE provide an average representation 

on specified values of VS30 and Bdepth. Some sites may show significant differences from these average 

site and basin effects. These differences are particularly likely when the geologic structure of the site is 

appreciably different than the average of the dataset. 

For application in other active tectonic regions where earthquakes at distances greater than about 

50 km are a major contributor to seismic hazard, there is a need for the user to obtain estimates of Q0 for 

that region for accurate ground-motion predictions. 

Concluding Remarks

In this study, the Graizer-Kalkan ground-motion prediction equation (GMPE) for peak ground 

acceleration and 5-percent damped pseudo spectral acceleration response ordinates is revised to account 

for differences in ground-motion scaling in terms of regional far-source distance attenuation and basin

effects. The new GMPE is controlled by a small number of measurable parameters including moment 

magnitude, closest distance to the fault rupture, style of faulting, shear-wave velocity in the upper 30 m

of site geological formation, regional quality factor, and basin depth (defined as the depth to 1.5-km/s

shear-wave velocity isosurface). 

The Graizer-Kalkan (GK15) offers a much simpler functional form than the NGA-West2 models

(for example, Abrahamson and others, 2014 (ASK14); Boore and others, 2014 (BSSA14); Campbell 

and Bozorgnia, 2014; Chio and Youngs, 2014), and it has either less or comparable standard deviations 

depending upon the spectral period of interest. The GK15 models spectral acceleration as a continuous 

function of spectral period, whereas the NGA-West2 models use a discrete functional form to compute 

SA ordinates at certain periods only (a total of 21 periods from 0.01 to 10 s). Comparisons of the GK15

model with ASK14 and BSSA14 demonstrate that GK15 produces similar or slightly lower results at 

very close distances to the fault (up to about 5 km) and at distances of more than 20 km from the fault 

for earthquakes with magnitude larger than 6.0. At intermediate distances (5 to 20 km from the fault), 

the GK15 produces higher estimates of ground motion than ASK14 and BSSA14. 

In conclusion, the GK15 GMPE is a significant improvement over GK07–09, and provides a 

demonstrable, reliable description of ground-motion amplitudes recorded from shallow crustal 

earthquakes in active tectonic regions over a wide range of magnitudes, distances and site conditions.

Data and Resources

Data used in this study came from published sources listed in the text or in references. The 

Graizer-Kalkan (GK15) ground-motion prediction equation is available from the authors upon request in 
71

Fortran and Excel. The MatLAB functions for GK15 are provided in appendix B. An example MatLAB 

routine to generate a site-specific response spectrum for given hazard conditions is provided in appendix 

C. 

Disclaimer

Any opinions, findings, and conclusions expressed in this report are those of the authors and do 

not necessarily reflect the views of the U.S. Nuclear Regulatory Commission.

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and around the Los Angeles Basin during the 2010 M7.2 El Mayor-Cucapah earthquake: Procedings

of the 15th World Conference on Earthquake Engineering, Lisbon, Portugal, 2012, 10 p. 

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attenuation of peak of strong ground-motion, in terms of magnitude, distance, path effects and site 

conditions: Department of Civil Engineering, University of Southern California, Report No. CE 95–

02, 268 p.

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United States: Bulletin of the Seismological Society of America, v. 77, no. 4, p. 1197–1210.

Olsen, K.B., 2000, Site amplification in the Los Angeles basin from three-dimensional modeling of 

ground-motion: Bulletin of the Seismological Society of America, v. 90, no. 6B, p. S77–S94.

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for shallow crustal earthquakes based on California strong motion data: Seismological Research

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alluvial basins; in-depth motion and focusing effect: Soil Dynamics and Earthquake Engineering, v. 

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Engineering, v. 13, p. 149–161.
74

Appendix A. List of Earthquakes Used for Updating the Graizer-Kalkan Groundmotion Prediction Equation

No Event Date Style of

faulting

Moment 

magnitude

Depth

(km)

Epicenter coordinates #

of Data

Distance

La Range (km) titude Longitude

1 Adana-Ceyhan (Turkey) 1998 Strike-slip 6.3 18.0 36.850 35.550 4 28.0 - 96.0

2 Anza 2005 Strike-slip 5.2 14.2 33.529 −116.573 279 4.8 - 197.6

3 Big Bear City 2003 Strike-Slip 4.9 6.3 34.310 −116.848 178 8.6 - 166.7

4 Bingol (Turkey) 2003 Strike-Slip 6.4 6.0 38.940 40.510 1 6.1

5 Bishop (Rnd. Val.) 1984 Strike-Slip 5.8 9.0 37.460 −118.590 1 21.9

6 Borrego Mnt. 1968 Strike-Slip 6.6 8.0 33.190 −116.142 5 45.7 - 222.4

7 Chalfant Valley 1986 Strike-Slip 5.8 6.7 37.577 −118.449 5 6.4 - 24.5

8 Chi-Chi (Taiwan) 1999 Reverse 7.6 16.0 23.860 120.800 420 0.3 - 172.2

9 Coalinga-01 1983 Reverse 6.4 4.6 36.233 −120.310 46 8.4 - 55.8

10 Coalinga-05 1983 Reverse 5.8 7.4 36.241 77.191 11 4.6 - 16.2

11 Coyote Lake 1979 Strike-Slip 5.7 9.6 37.085 −121.505 10 3.1 - 33.8

12 Denali (Alaska) 2002 Strike-Slip 7.9 4.9 63.538 −147.444 24 2.7 - 275.9

13 Dinar (Turkey) 1995 Normal 6.4 5.0 38.110 30.050 2 3.0 - 39.6

14 Düzce (Turkey) 1999 Strike-Slip 7.2 10.0 40.740 31.210 23 0.2 - 188.7

15 Erzincan (Turkey) 1992 Strike-Slip 6.9 9.0 39.720 39.630 2 5.0 - 65.0

16 Friuli (Italy) 1976 Reverse 6.5 5.1 46.345 13.240 5 15.8 - 102.2

17 Gazli (Uzbekistan) 1976 Reverse 6.8 10.0 40.381 63.472 1 5.0

18 Racha (Georgia, Ussr) 1991 Reverse 6.2 9.0 42.461 44.009 8 37.0 - 155.0

19 Gulf of California 2001 Strike-Slip 5.7 10.0 32.037 −114.906 12 76.7 - 134.1

20 Hector Mine 1999 Strike-Slip 7.1 5.0 34.574 −116.291 213 10.7 - 259.3

21 Imperial Valley 1979 Strike-slip 6.5 10.0 32.644 −115.309 33 0.1 - 50.1

22 Kocaeli (Turkey) 1999 Strike-Slip 7.4 15.0 40.727 29.990 31 3.2 - 349.6

23 Landers 1992 Strike-Slip 7.3 7.0 34.200 −116.430 69 2.2 - 190.1

24 Lazio-Abruzzo (Italy) 1984 Normal 5.8 14.0 41.710 13.902 5 18.9 - 51.3

25 Little Skul Mtn. (Nevada) 1992 Normal 5.7 12.0 36.720 −116.286 8 16.1 - 100.2

26 Livermore 1980 Strike-Slip 5.8 12.0 37.855 −121.816 7 16.7 - 56.1

27 Loma Prieta 1989 Reverse/Strike 6.9 17.5 37.041 −121.883 82 3.9 - 117.1

28 Mammoth Lakes-02 1980 Strike-Slip 5.7 14.0 37.628 −118.927 3 9.1 - 16.9

29 Mammoth Lakes-03 1980 Strike-Slip 5.9 16.0 37.561 −118.831 4 5.9 - 11.5

30 Mammoth Lakes-04 1980 Strike-Slip 5.7 5.0 37.625 −118.859 4 2.8 - 14.2

31 Mammoth Lakes-06 1980 Strike-Slip 5.9 14.0 37.506 −118.856 5 12.0 - 46.5

32 Manjil (Iran) 1990 Strike-Slip 7.4 19.0 36.810 49.353 7 12.6 - 174.6

33 Morgan Hill 1984 Strike-Slip 6.2 8.5 37.306 −121.695 28 0.5 - 70.9

34 Northridge 1994 Reverse 6.7 17.5 34.206 −118.554 174 4.0 - 78.1

35 North Palm Springs 1986 StrikeSlip/Thrust 6.1 11.0 34.000 −116.612 32 8.6 - 268.0

36 Parkfield 1966 Strike-slip 6.2 10.0 35.955 −120.498 6 6.3 - 63.3

37 Parkfield 2004 Strike-Slip 6.0 8.8 35.819 −120.364 94 0.3 - 169.6

38 San Fernando 1971 Reverse 6.6 13.0 34.440 −118.410 44 1.8 - 218.8

39 San Simeon 2002 Reverse 6.5 7.1 35.702 −121.108 138 12.4 - 317.8

40 Santa Barbara 1978 Thrust 5.9 12.7 34.399 −119.681 2 12.2 - 27.4

41 Sierra Madre 1991 Reverse 5.6 12.0 34.259 −118.001 9 10.4 - 48.2
75

No Event Date Style of

faulting

Moment 

magnitude

Depth

(km)

Epicenter coordinates #

of Data

Distance

La Range (km) titude Longitude

42 Spitak (Armenia) 1988 Reverse 6.8 5.0 40.987 44.185 1 25.0

43 Superstition Hills-02 1987 Strike-Slip 6.5 9.0 33.022 −115.831 11 1.0 - 27.0

44 Taiwan, Smart(5) 1981 Reverse 5.9 11.1 24.429 121.896 7 28.7 - 32.0

45 Whittier Narrows 1987 Reverse 6.0 14.6 34.049 −118.081 116 14.5 - 103.9

46 Yountville 2000 Strike-Slip 5.0 10.1 38.379 −122.413 25 9.9 - 95.7

47 Yucaipa 2005 Reverse 4.9 11.6 34.058 −117.011 388 2.6 - 160.1

Total = 2,583
76

Appendix B. MatLAB Code for Graizer-Kalkan Ground-motion Prediction 

Equation (2015)

%% 2015 GRAIZER-KALKAN GMPE

%

% Graizer & Kalkan 2015 updated GMPE for horizontal component ground motion

%

% Syntax:

% out = GK15(M, R, VS30, F, Q_0, Bdepth,flag)

%

% Input:

% M = Moment magnitude

% R = Closest distance to fault rupture in km (i.e., Rrup)

% VS30 = Average shear-wave velocity in the upper 30 m in m/s

% Bdepth = Depth to 1,500 m/s shear-wave velocity (i.e., Z1.5)

% Bdepth = 0 for non-basin sites

% F = Style of faulting

% F = 1.0 for strike-slip and normal faulting,

% F = 1.28 for reverse faulting, and

% F = 1.14 for combination of strike-slip and reverse faulting

% Q_0 = Regional Q_0 value (e.g., 150 for California)

% Bdepth = Depth to 1,500 m/s shear-wave velocity to define basin (i.e., Z1.5)

% Bdepth = 0 for non-basin sites 

% flag = 'm' for maximum horizontal component; 'a' for average 

% horizontal component

% Output:

% out = [t,SA,sigma]

% t = Spectral period vector (PGA to 5 s)

% SA = Median pseudo spectral acceleration ordinates (PGA to SA at 5 s)

% Sigma = Total standard deviations

%

% $Revision: 1.0 $ $Date: 01/18/2015 $

% Written by Dr. Erol Kalkan, P.E. (ekalkan@usgs.gov)

%

% Comment blocks and equation references in this function correspond to the

% following publication:

%

% Graizer, V. and Kalkan, E. (2015). Update of the Graizer_Kalkan ground-motion

% prediction equation for shallow crustal continental earthquakes,

% U.S. Geological Survey Open-File Report 2015-1009, p. 98.

function [out] = GK15(M,R,VS30,F,Q_0,Bdepth,flag)

if flag == 'm'; amp = 1; else

 amp = 1/1.12; % average to maximum component conversion

end

%% GMPE for PGA 

c1 = 0.14;

c2 = -6.25;

c3 = 0.37;

c4 = 2.237;

c5 = -7.542;

c6 = -0.125;

c7 = 1.19;

c8 = -6.15;

c9 = 0.6;

c10 = 0.345;

bv = -0.24;

VA = 484.5;

%% InPGA = G1 + G2 + G3 + G4 + G5;
77

G1 = log(( c1 * atan (M + c2) + c3) * F);

Ro = c4*M + c5;

Do = c6 * cos(c7 * (M + c8)) + c9;

G2 = -0.5 * log((1-R/Ro)^2 + 4 * (Do^2) * (R/Ro));

G3 = -c10 * R / Q_0;

G4 = bv * log(VS30 / VA);

A_Bdepth = 1.077/sqrt((1-(1.5/(Bdepth+0.1))^2)^2+4*0.7^2*(1.5/(Bdepth+0.1))^2);

A_Bdist = 1/sqrt((1-(40/(R+0.1))^2)^2+4*0.7^2*(40/(R+0.1))^2);

G5 = log(1 + A_Bdepth * A_Bdist);

InPGA = G1 + G2 + G3 + G4 + G5;

PGA = exp(InPGA) * amp;

%% GMPE for Spectral Shape (GK15SS)

t = [0.01 0.02 0.022 0.025 0.029 0.03 0.032 0.035 ...

 0.036 0.04 0.042 0.044 0.045 0.046 0.048 0.05 ...

 0.055 0.06 0.065 0.067 0.07 0.075 0.08 0.085 ...

 0.09 0.095 0.1 0.11 0.12 0.13 0.133 0.14 0.15 ...

 0.16 0.17 0.18 0.19 0.2 0.22 0.24 0.25 0.26 ...

 0.28 0.29 0.3 0.32 0.34 0.35 0.36 0.38 0.4 ...

 0.42 0.44 0.45 0.46 0.48 0.5 0.55 0.6 0.65 ...

 0.667 0.7 0.75 0.8 0.85 0.9 0.95 1 1.1 1.2 1.3 1.4 ...

 1.5 1.6 1.7 1.8 1.9 2 2.2 2.4 2.5 2.6 2.8 3 3.2 3.4 3.5 3.6 ...

 3.8 4 4.2 4.4 4.6 4.8 5];

m1 = -0.0012;

m2 = -0.38;

m3 = 0.0006;

m4 = 3.9;

a1 = 0.01686;

a2 = 1.2695;

a3 = 0.0001;

Dsp = 0.75;

t1 = 0.001;

t2 = 0.59;

t3 = -0.0005;

t4 = -2.3;

s1 = 0.001;

s2 = 0.077;

s3 = 0.3251;

if (M<5.5); M=5.5; end

for j = 1:length(t);

 I = (a1*M+a2)*exp(a3*R);

 mu = m1*R + m2*M + m3*VS30 + m4;

 S = s1*R - (s2*M + s3);

 Tsp_o = max(0.3,abs(t1*R + t2*M + t3*VS30 + t4));

 zay = 1.763-0.25*atan(1.4*(Bdepth-1));

 

 F1(j) = I*exp(-0.5*((log(t(j))+mu)/S)^2);

 F2(j) = 1/sqrt((1-(t(j)/Tsp_o)^zay)^2 + 4*Dsp^2*(t(j)/Tsp_o)^zay);

 Y(j) = F1(j) + F2(j);

end

Sigma = max(0.668 + 0.0047 * log(t),0.8 + 0.13 * log(t));

%% SA = GK15SS x exp(GK15PGA)

SA = [PGA Y * exp(InPGA) * amp];

t = [0 t];

Sigma = [0.66 Sigma];

%% Output [Spectral Period, SA in "g"]

out = [t' SA' Sigma'];

return
78

Appendix C. MatLAB Code to Generate Pseudo Spectral Acceleration Response 

Spectrum Using Graizer-Kalkan 2015 (GK15) Ground-motion Prediction 

Equation 

% RUN GRAIZER-KALKAN (GK15) GMPE TO GENERATE 5% DAMPED PSEUDO SPECTRAL 

% ACCELERATION RESPONSE SPECTRUM

%

% Syntax:

% runGK15

%

% Input:

% M = Moment magnitude

% R = Closest distance to fault rupture in km (i.e., Rrup)

% VS30 = Average shear-wave velocity in the upper 30 m in m/s

% Bdepth = Depth to 1,500 m/s shear-wave velocity isosurface (i.e., Z1.5)

% Bdepth = 0 for non-basin sites

%

% F = Style of faulting

% F = 1.0 for strike-slip and normal faulting,

% F = 1.28 for reverse faulting, and

% F = 1.14 for combination of strike-slip and reverse faulting

% Output:

% SA = Median pseudo spectral acceleration (SA) ordinates (PGA to 5 s)

% SA_16 = 16th percentile SA

% SA_84 = 84th percentile SA

%

% $Revision: 1.0 $ $Date: 01/18/2015 $

% Written by Dr. Erol Kalkan, P.E. (ekalkan@usgs.gov)

%

% Comment blocks and equation references in this function correspond to the

% following publication:

%

% Graizer, V. and Kalkan, E. (2015). Update of the Graizer-Kalkan ground-motion

% prediction equation for shallow crustal continental earthquakes,

% U.S. Geological Survey Open-File Report 2015-1009, p. 98.

clear all; clc; close all;

flag = input('maximum or average component (enter m or a): ', 's');

% Example Input Parameters

M = 6.0;

R = 4.4;

VS30 = 350;

F = 1.0; % Strike-slip fault

Bdepth = 0; % No basin

Q_0 = 50; % Regional quality factor for Napa, California

%% Call GK15

[out] = GK15(M,R,VS30,F,Q_0,Bdepth,flag);

t = out(:,1);

SA = out(:,2);

Sigma = out(:,3);

%% SA plus/minus sigma (16th and 84th percentile values)

SA_16 = SA./exp(Sigma);

SA_84 = SA.*exp(Sigma);

PeakSaAmp = max(SA);

for j = 1:length(t);
79

 if (SA(j) == PeakSaAmp); Tmax = t(j); break

 end

end

%% Plot spectrum

set(gcf,'position',[300 283 600 350]);

set(gca,'TickLength',[.0025 .0025]);

loglog(t,SA,'r-','LineWidth',2.5); hold;

loglog(t,SA_16,'k--','LineWidth',1.0);

legend('SA','+/- \sigma');

loglog(t,SA_84,'k--','LineWidth',1.0);

loglog(0.01,SA(1),'o','MarkerSize',12,...

 'MarkerEdgeColor','b','MarkerFaceColor','r');

loglog(Tmax,0.001,'o','MarkerSize',12,...

 'MarkerEdgeColor','b','MarkerFaceColor','r');

loglog(Tmax,PeakSaAmp,'o','MarkerSize',12,...

 'MarkerEdgeColor','b','MarkerFaceColor','r');

axis([0.01 5 0.01 4]);

set(gca,'XTickLabel',{'0.01','0.1','1'},'fontsize',12);

set(gca,'YTickLabel',{'0.01','0.1','1','4'},'fontsize',12);

xlabel('Period (s)','FontSize',[13],'FontWeight','bold','FontAngle','italic');

ylabel('Spectral Acceleration (g)','FontSize',[13],'FontWeight','bold','FontAngle','italic');

tit1 = strcat('{M = }',num2str(M,'%5.1f'));

tit2 = strcat('{R = }',num2str(R),' km');

tit3 = strcat('{V_S_3_0 = }',num2str(VS30),' m/s');

text('pos',[0.015 0.045],'str',tit1,'Color','k','FontSize',[12],'FontWeight','bold');

text('pos',[0.015 0.03],'str',tit2,'Color','k','FontSize',[12],'FontWeight','bold');

text('pos',[0.015 0.018],'str',tit3,'Color','k','FontSize',[12],'FontWeight','bold');

text('pos',[0.012 

SA(1)*0.8],'str',num2str(SA(1),'%5.2f'),'Color','k','FontSize',[12],'FontWeight','bold');

text('pos',[Tmax 

PeakSaAmp*1.4],'str',num2str(PeakSaAmp,'%5.2f'),'Color','k','FontSize',[12],'FontWeight','bol

d');

text('pos',[Tmax*1.2 

0.0015],'str',num2str(Tmax),'Color','k','FontSize',[12],'FontWeight','bold');

title('GK15 (5% damped) Pseudo Spectral Acc. Response 

Spectrum','FontSize',[13],'FontWeight','bold','FontAngle','italic');

hold on; grid on;