A Single Station Station Sigma Model Adrian Rodriguez Marek Virginia Tech with contributions from Fabrice Cotton, Norm Abrahamson, Sinan Akkar, Linda Al Atik, John Anderson, Fabian Bonilla, Julian Bommer, Hilmar Bungum, John Douglas, Stephane Drouet, Ben Edwards, Donat Fäh, Gonzalo Montalva, Haitham Dawood, Philippe Renault, Frank Scherbaum and Fleur Strasser Outline Single station sigma: overview Requirements for use of single station sigma Proposed model: Databases Proposed Phi (intra event) Model Conclusions 2 1
Single station sigma: overview A Ground Motion Prediction Equation (GMPE) ln(y) = f(m, R, Site, etc ) + ln(y) : median prediction : residual mean() = stdev( = tot 3 Breakdown of total sigma Residual component Standard deviation B e : event term (aleatoric) (aleatoric) W es : within event residual Required Data Global: Multiple recordings (surface) at different sites from EQ in multiple source regions 2
From Al Atik et al. (21) 5 Further breakdown Residual component S2S s : site term Standard deviation S2S (epistemic) Required Data Site specific: Multiple recordings (surface) at different WS es : site and sites from EQ in eventcorrected (aleatoric) SS residual multiple source regions Multiple recordings at individual sites 3
Example: Japanese KiKnet data, Sa(T=) 2 =51 W es -2 Station Number 2 S2S =28 - S2S s 2 =79 S2S s -2 Station Number WS es = W es -2 Station Number From Rodriguez-Marek et al. (21) 7 Note: S2S : site to site variability bl (after (f correction for site class in GMPE): spatial : single station within event variability: temporal Assumption: Standard deviation across spatial extent (across sites) is weighted equally to standard deviation across time: ergodic assumption * 8 4
Single station standard deviation: At a single site, S2S s is predictable (deterministic) SS : single station standard deviation Since ergodic assumption is removed: Partially non ergodic 9 MOTIVATION FOR USE OF SINGLE STATION SIGMA 12 5
Motivation for partially ergodic PSHA Single station within event standard deviation ( ) is less variable ibl across regions than its ergodic conterpart () Use of global datasets? 13 Data from various tectonic regions.8.8 PGA.1 1. 3. Period (Sec) California Swiss Taiwan Turkey Japan Model1 All 6 From Rodriguez-Marek et al. (21) PGA.1 1. 3. Period (Sec) 14 6
Motivation for partially ergodic PSHA Single station within event standard deviation () is less variable ibl across regions than Use of global datasets? Site response analyses: An exercise in estimating S2S s (and its uncertainty) If S2S is not removed: double counting uncertainty 15 Cost of partially ergodic PSHA Any term for which ergodicity is removed must be estimated t To do this: We must estimate S2S s (implies also estimate of its epistemic uncertainty) 16 7
PROPOSED MODEL 2 Project background Funded by the PEGASOS PRP project Task: compute estimatesof singlestationstation sigma Data provided by Resource Experts (GMPE developers) Data selection criteria (based on preliminary analyses) At least 5 records per station At least 5 stations per earthquake Residuals were computed by individual GMPE developers Single station sigma computed only from subsets of the data used for GMPEs and distance range of interest: M >= 4.5, R <= 2 km Not a constraint on data for GMPE development 21 8
Preliminary observations Decomposition into within event and between event (intra and inter event) event) residuals is possible No correlation between B e and W es estimates from regional dataset are more stable (e.g., no restriction on records per station) Develop only the intra event component ( ) 22 PROPOSED MODELS: DATABASE USED 23 9
Residuals were provided by California: N. Abrahamson Chiou et al. California SMME database (ShakeMap) Abrahamson and Silva NGA Japan: Rodriguez Marek et al. (211) KiKnet data as processed by Pousse et al. (24) Switzerland: Ben Edwards and Linda Al Atik Taiwan: N. Abrahamson Lin et al. (211) Turkey: S. Akkar 24 8 California 8 Switz. 7 7 6 5 4 6 5 4 3 3 2 1 2 3 8 7 6 5 4 3 Taiwan 2 1 2 3 2 1 2 3 8 7 6 5 4 3 Turkey 2 1 2 3 8 Japan 8 All Sites 7 7 6 5 4 6 5 4 3 3 2 1 2 3 2 1 2 3 25 1
Number of records used in analysis (Table 3.1) T (s) California Switzerland Taiwan Turkey Japan All Regions All M,R Sel. M, R All M,R Sel. M, R All M,R Sel. M, R All M,R Sel. M, R All M,R Sel. M, R All M,R Sel. M, R PGA 15295 1635 832 19 4756 462 145 145 3234 1834 24262 7695.1 3148 28 4756 462 145 145 3234 1834 11283 669 3514 28 145 145 3234 1834 6893 27 15295 1635 4756 462 145 145 3234 1834 2343 7676 3145 28 4756 462 145 145 3234 1834 1128 669 1 15287 1627 218 28 4753 459 145 145 3234 1834 25527 7693 3 432 3733 1 1 442 3833 26 Number of records used in analysis (Table 3.1) T (s) California Switzerland Taiwan Turkey Japan All Regions All M,R Sel. M, R All M,R Sel. M, R All M,R Sel. M, R All M,R Sel. M, R All M,R Sel. M, R All M,R Sel. M, R PGA 15295 1635 832 19 4756 462 145 145 3234 1834 24262 7695.1 3148 28 4756 462 145 145 3234 1834 11283 669 3514 28 145 145 3234 1834 6893 27 15295 1635 4756 462 145 145 3234 1834 2343 7676 3145 28 4756 462 145 145 3234 1834 1128 669 1 15287 1627 218 28 4753 459 145 145 3234 1834 25527 7693 3 432 3733 1 1 442 3833 27 11
T (s) Database bias: mean of within event residuals (natural log units) California Switz. Taiwan Turkey Japan All Regions Select. M,R Select. M,R Select. M,R Select. M,R Select. M,R Select. M,R PGA.11.31.9.111..4.3.37.22.26.1.12.1.1 24..6.24.3.12.1.4.1. 31.52.59.23.23.12.15.5.33..4.53.63.28.28.8.15.1.13...38.43.38.48.12.13 1.7.41.1.127..2.36.39.36.53.9.2 3...19.9..6 28 PROPOSED MODEL 29 12
Lack of regional dependence for.8 21,ss.8 PGA.1 1. 3. Period (Sec) California Swiss Taiwan Turkey Japan Model1 All 21 PGA.1 1. 3. Period (Sec) Comparison of single stations within-event standard deviation ( ) and within event standard deviations (). Shown are data for M<=Mc1 and Rc21 <=R<=2 km 31 No apparent Vs dependency for 1.9.8 s California France Alps France Pyr Japan Taiwan Turkey.1 1 2 1 3 Vs3 (m/s) 32 13
,T=. Observed Trends on Distance dependence for low magnitudes 8.8.1 M=[4.5 5.5] 1 1 2,T=. 8.8.1 M=[5.5 7.] 1 1 2,T=. 8.8.1 M=[7. 8.] CA TW JP 1 1 2 33 Observed Trends on dependence R=[ 16] R=[ 16 4] R=[ 4 2].8.8.8 CA TW JP,T=.,T=.,T=..1.1.1 5 6 7 8 5 6 7 8 5 6 7 8 34 14
Summary Join together all within event residuals from all regions No Vs3 dependency (more later) dependency important Distance dependency at small magnitudes 35 Proposed Phi Models Phi Model 1: Constant 36 15
Proposed Phi Models Phi Model 1: Constant Phi Model 2: Distance dependent ( R SS rup 1 R ) 1 2 1 R 2 rup c2 R R c1 c1 for R forr c1 forr rup R R rup R c1 R c2 rup c2 8 1 6 R c1 4 2 8 6 R c2 2 4 5 1 15 2 25 3 35 4 45 375 Proposed Phi Models Phi Model 3: Distance and magnitude dependent d t ( M, R SS rup ) C1 ( R rup ) C ( R 2 C ( R rup 1 ) C ( R 2 rup 1 C ( R rup ) ) rup M M c1 ) M c2 M c1 for M form M c1 M M form M c1 c2 c2 C 11 forr R Rrup Rc 11 11 21 11 Rc21 Rc 11 forr rup R c11 R c11 rup 21 forr rup Rc21 1 ( Rrup ) c21 5 5 5 5 5 8 7.5 7 6.5 6 M 5.5 5 4.5 4 1 2 R 3 4 38 16
Methodology to determine values a) A Random Effects regression was run using the entire database ofwithin event residuals for stations with 5 or more recordings. A constant model was assumed. The Random Effects give the site terms (S2S s ). S2S s computed with records for M < 4.5 b) The within event residuals were corrected with the S2S s term obtained in Step a c) Phi models were fitted to site corrected withinevent residuals (WS es ) Tabulated values of models A, B and C R dependent Model Period Rc1 Rc2.1 6 6 5 16 32.1 5 5 4 16 32 8 2 7 16 32 3 48 8 2 7 16 32 6 8 5 16 32 1 5 4 4 16 32 3 1 3 16 36 44 17
96 85 1/2/211 Methodology to determine values: M R dependent model Corner magnitudes were determined first using a similar approach to that described for the distance dependent model. Data favored a sharp drop in at M=6. A linear trend was forced by imposing the higher corner Mc2 to be equal to 7 5 M c1 5 M c1 5 M c2 5 M c2 4 5 6 7 8 4 5 6 7 8 Contour plots of Likelihood function, C, and C 2 T=3. L, T= x1, T= C2, T= 7 7 7 6.9-491 -49-49 -489.5 6.9 8 6 4 2 98 92 6.9 7 68 6.8 68 6.8 494 68 6.8 75 6.7-492 -491.5 6.7 6.7 8 6.6 6.6 6.6 85 Mc2 6.5-491 -49-49 -489.5 Mc2 6.5 4 6 8 2 98 Mc2 6.5 95 9 6.4 6.4 94 6.4 6.3 6.2 6.1-491.5-491 -49-49 -489.5-489 -488.5-489.5-49 49 91-49 -4-491.5 6 5 5.2 5.4 5.6 5.8 Mc1-492 6.3 6.2 6.1 8 1 6 4 2 96 98 6 5 5.2 5.4 5.6 5.8 Mc1 6.3 6.2 6.1 5 1 15 9 395 6 5 5.2 5.4 5.6 5.8 Mc1 9 Optimum Point Chosen Point 46 18
Methodology to determine values: M R dependent model Corner magnitudes were determined first using a similar approach to that described for the distance dependent model. Mc2 was forced to be equal to 7. even when data indicated that a sharper drop from Mc1 to Mc2. This resulted in a change of 12 and 22 values. Once the values of Mc1 and Mc2 were set, the distancedependence was obtained using the same approach as for the distance dependent model but using only data for M<Mc1. Once Mc1, Mc2, Rc1, and Rc2 were fixed, a maximum likelihood regression was run to obtain the values of the remaining parameters. Tabulated values of Model 3 M and R dependent Model Period C2 Mc1 Mc2 Rc11 Rc21.1 8 7 4 5.2 7. 16 36.1 4 4 3 5.2 7. 16 36 9 7 5.2 7. 16 36 3 6 5.2 7. 16 36 9 8 6 5.2 7. 16 36 1 54 4 45 5 7* 53 5.3 7 7. 16 36 3 4* 7* 7* 5.5 7. 16 36 * Values from standard deviation of residuals within corresponding bins are used to replace the Maximum Likelihood values (to avoid larger standard deviations for longer periods) 48 19
Model Fit 2.5 2 1.5 1 std( WS es within a distance bin 65 5 std( WS es ) std of selected bin 5 WS es - 5-1 -1.5-2 5-2.5 1 1 1 1 2 R (km) 1 1 1 1 2 R (km) Site-corrected within event residuals and their standard deviations for PGA. The plot on the left shows all the residuals, the red lines indicated the standard deviation of residuals within certain distance bins. The plot on the right shows only the standard deviation of residuals with the statistical error band. The blue line is the standard deviation for all the data. 49 Model fit: PGA All Data All Data 5 1 15 2 4.<M<5.2 4.5 5 5.5 6 6.5 7 7.5 8 4<M<52 7.<M<8. 5 1 15 2 <R< 16 5 1 15 2 32<R<2 6 6 4.5 5 5.5 6 6.5 7 7.5 8 Model 1 Model 2 Model 3 4.5 5 5.5 6 6.5 7 7.5 8 5 2
Model fit: T=s All Data All Data 5 1 15 2 4.5 5 5.5 6 6.5 7 7.5 8 4<M<52 4.<M<5.2 7<M<8 7.<M<8. 5 1 15 2 5 1 15 2 <R< 16 32<R<2 4.5 5 5.5 6 6.5 7 7.5 8 Model 1 Model 2 Model 3 4.5 5 5.5 6 6.5 7 7.5 8 51 Model fit: T=1.s All Data All Data 5 1 15 2 4.5 5 5.5 6 6.5 7 7.5 8 4.<M<5.3 7.<M<8. 5 1 15 2 5 1 15 2 <R< 16 32<R<2 4.5 4.5 5 5.5 6 6.5 7 7.5 8 5 5.5 6 6.5 7 7.5 8 Model 1 Model 2 Model 3 52 21
Comparison with ergodic values (magnitude independent model) Proposed Model 2 Based on within event residuals of AS8 Within event residuals in PRP data Period (s) 1 2 1 2 1 2.1 6 5 9 7.1 5 4 3 4 7 2 2 7 5 2.8 3 62 2 47 7 54 4 54 4 7 61 1 8 5 3 5 9 8 1 4 4 3 9 4 3 3 6 4 1 6 53 54 Al Atik (pers. Comm) 22
Comparison with ergodic values (magnitude dependent model) Proposed Model 3 (single station) Based on within event residuals of AS8 (ergodic) Within event residuals in PRP data (ergodic) Period (s) 11 21 C 2 11 21 C 2 11 21 C 2.1 8 7 4 9 4 4 7 8 1.1 4 4 3 1 1 9 4 3 5 9 7 2.9 2 7 3 6 1 4 2 3 4 9 8 6 5 3 2 7 1 1 4 5 7 4 4 2 3 2 3 4 7 7 8 5 6 6 3 8 55 56 Al Atik (pers. Comm) 23
VARIATION OF FROM STATION TO STATION 57 Epistemic uncertainty on Both the site term (S2S s ) and the singletti standard ddeviation ( ) have epistemic it i station uncertainties. Different stations sample different sources/paths 2D and 3D effects imply azimuthal dependency Degree of nonlinearity 58 24
Azimuthal dependency (KiK net data) T=1 s Stations with N>=15 59 Nmin = 1 7 =7 T=.,All SitesM>=4.5, R<=2, Nmin=1 Goodn. of fit test (1=reject) Normal Log Normal 6 =6 ss JB=1 JB= max =.86 ss,s 5 min = ss,s mean =3 ss,s 4 std =.1 ss,s 3 KS= Lillie=1 KS= Lillie= 2 1.1.8.9 1,s 6 25
Nmin = 15 3 =7 =6 ss 25 max =9 ss,s min =3 ss,s 2 mean =4 ss,s std =.9 ss,s 15 T=.,All SitesM>=4.5, R<=2, Nmin=15 Goodn. of fit test (1=reject) Normal Log Normal JB= KS= Lillie= JB= KS= Lillie= 1 5.1.8.9 1,s 61 Nmin = 2 T=.,All SitesM>=4.5, R<=2, Nmin=2 1 Goodn. of fit test (1=reject) 9 =7 Normal Log Normal 8 =6 ss 7 max =3 ss,s 6 min =8 ss,s 5 mean =4 ss,s 4 std =.8 ss,s JB= KS= Lillie= JB= KS= Lillie= 3 2 1.1.8.9 1,s 62 26
Variation in,s Period (s) Mean Std. Dev. Stations Mean Std. Dev. Stations Mean Std. Dev. Stations.1 3.1 326 4.9 133 4.8 32.1 5.12 316 5.1 133 5.8 32 7.12 5 2.1 13 6.11 5 6.11 326 7.1 133 7.9 32 46 6 11.11 316 47 7 9.9 133 46 6 8.8 32 1 4.1 326 3.8 133 3.7 32 3 1.1 245 2.8 89 1.7 16 63 Variation in range of,s Perio d (s) Mean Min Max Mean Min Max Mean Min Max.1 3.86 4 3 9 4 8 3.1 5 1.85 5 2 7 5 5 7 7 5 9 2 7 5 6 7 5 6.19.87 7 7 3 7 1 3 6.17.98 7 5 6 1 1 1 4.19.93 3 7 6 3 7 3 1.17.89 2 5 8 1 4 64 27
Note: possible Vs3 dependency? (KiK net data) 65 Epistemic Uncertainty on Summary There is variability of from station to station: tti Epistemic i uncertainty tit The standard deviation of,s can only be estimated for a constant Phi Model (Model 1) Standard deviation of,s reduces as N increases Range of,s reduces as N increases Can t reject normality for distribution of,s 66 28
Epistemic Uncertainty on Summary The standard deviation of the single station phi measured at each station tti [td( [std(,s )] can be used as a basis to assign epistemic uncertainty on Use well recorded stations (N>=2) to estimate std(,s ) Strictly applicable only to constant phi model Applicable to other phi models? [some of the variability may be due to M,R variability] 67 Notes: Consistency with other studies 29
Notes: Consistency across regions ss CA Taiwan Japan All PGA.51.51.5.51 T=.54.51.5.52 T=1.s.47.47.44.46 Note Consistency of ss across regions S2S is more variable across regions S2S CA Taiwan Japan All PGA.41.25.48.4 T=.44.28.46.43 T=1.s.44.37.37.41 Notes: dependency below M4.5? 2 < M < 4 4 < M < 7 T.,California. 2 M 4, R 2, Nmin 8 6 6 5 4 4 2 3 2 1 5 1 5 1,s,s Data for PGA, California 7 3
SUMMARY 71 Proposed models Model Epistemic Uncertainty on Phi Model 1 (Constant Phi) Phi Model 2 (Distance Dependent) Overpredicts at large magnitudes Phi Model 3 (Distance and dependent) Consider standard deviation in,s 73 31
Conclusions must be added Proponent position: consider tau estimates independently Conditions for application of single station sigma must be met Estimate of site term (S2S s ) and its epistemic uncertainty If not, must use total sigma Proponent position: estimates of S2S must also be given 74 Thank you 75 32
California data The California dataset used in this study consists of ground motion data from the Abrahamson and Silva (28) NGA dataset and small to moderate magnitude California data used for the small magnitude extension of the Chiou and Youngs (28) NGA model (Chiou et al., 21) at peak ground acceleration (PGA) and at periods of and 1 second. The Abrahamson and Silva (28) data are part of the NGA West dataset. The small to moderate magnitude data used in Chiou et al. (21) were obtained from ShakeMap. Between event residuals and within event residuals of these two California datasets were obtained by separately fitting the large magnitude and small to moderate magnitude datasets with the Abrahamson and Silva (28) GMPE and with the Chiou et al. (21) GMPE, respectively. 76 Switzerland The Swiss dataset used in this study consists of filtered and site corrected acceleration time series of Swiss Foreland events used in developing the stochastic ground motion model for Switzerland (Edwards et al., 21). The acceleration time series were filtered using a variable corner frequency acausal Butterworth filter and site corrected to correspond to the Swiss reference rock condition. A detailed description of the Swiss dataset is given in Edwards et al. (21). Recordings from co located stations were removed from the dataset and response spectra of the remaining recordings were computed using the Nigam and Jennings (1969) algorithm. Only recordings from events with more than 5 recordings and at stations with more than 5 recordings were used in this analysis. The form of the GMPE used to fit the Swiss data is discussed in Douglas (29) and is described as: 2 ln 1 2 3 4log( ) 5 y b b M b M b Rhypo b Rhypo The random effects algorithm developed by Abrahamson and Youngs (1992) was used to calculate the parameters of the GMPE fit to the Swiss dataset and to obtain the between event and within event residuals. 77 33
Japan The Japanese data used in this study is data downloaded from the KiK net network website and is described in Pousse et al. (25), Pousse et al. (26), and Cotton et al. (28). Only records between 1996 and October 24 with were used. In addition, only records with hypocenter depth less than 25 km were used in order to avoid subduction related events. The M JMA was converted to seismic moment magnitude using the Fukushima (1996) relationship (Cotton et al. 28). Data processing is described in Pousse et al. (25) and Cotton et al. (28). Cotton et al. (28) states that the longest usable period for the data set is 3. s. However, some of the spectral accelerations at long periods are lower than the number of decimals in the data set. For that reason, only spectral periods up to 1. s are used in this work. Closest distance to the rupture was assumed to correspond to hypocentral distance for small to moderate earthquakes or when the source dimensions remain unknown. For larger earthquakes, a closest distance to the fault was computed. A peculiarity in the development of the GMPE for the KiK net data is that both records from the surface and borehole stations were used (with appropriate site terms). This implies that the event terms and magnitude scaling was constrained both by surface and borehole data (Rodriguez Marek et al. 211). The site term uses Vs3 as the site parameter and includes only a linear amplification term. Parameters of the GMPE equation were obtained using the Random Effects regression of Abrahamson and Youngs (1992) 78 Taiwan The Taiwan dataset used in this study consists of ground motion data from shallow earthquakes that occurred in and near Taiwan from 1992 to 23. This dataset was assembled by Lin (29) and the ground motion recordings were baseline corrected and limited to distances of less than 2 km. The 1999 Chi Chi earthquake was excluded from the dataset. A more detailed description of the Taiwan data is given in Lin et al. (211). Within event and betweenevent residuals of the Taiwan dataset were computed using a mixed effects effects algorithm with respect to a revised version of the Chiou and Youngs (28) model. Only stations with at least 1 recordings were used to calculate the site terms. 79 34
Turkey The Turkish data used in this study is compiled within the framework of the project entitled Compilation of Turkish strong motion network according to the international standards. The procedures followed to assemble the database are described in Akkar et al. (21) and Sandıkkaya et al. (21). The dataset comprises of events with depths less than 3 km. Each event in the dataset has at least 5 recordings. The distance measure is Joyner Boore distance metric for all recordings. For small events (i.e., M w 5.5) epicentral distance is assumed to approximate Joyner Boore distance. The dataset is processed (band pass acausal filtering) by the method described in Akkar and Bommer (26). The same article also describes a set of criteria for computing the usable spectral period range that is used to define the maximum usable period for each processed record. The database consists of very few recordings with V S3 > 76 m/s (where Vs3 is the average shear wave velocity over the upper 3 m of a profile) and almost all events are either strike slip or normal. The database contains 239 recordings but this number is reduced to 26 at T = 3. s due usable period range criteria in Akkar and Bommer (26). Regression analysis is conducted using the above dataset to derive a set of GMPEs for the analysis of within event residuals. A one stage maximum likelihood regression method (Joyner and Boore, 1993) is employed while developing the GMPEs. The model estimates the geometric mean of the ground motion. The functional form is the same one that is presented in Akkar and Çağnan (21). It is capable of capturing magnitude saturation and magnitudedependent geometrical spreading. Site effects are considered using a function of Vs3 that also includes nonlinearity. 8 35