Generliztion o 2-Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville in SMSIM hve two orner requenies, ut only one o these models hs the option o vrying the high-requeny spetrl level, s the other 2- orner models re ompletely determined y speiied reltions etween the orner requenies nd mgnitude (see Tles 2 nd 3 in Boore, 23, or onise nd onvenient summry o the vrious models). In this note I provide equtions or generlizing two-orner models to llow the high-requeny soure spetrl level to e determined y the stress prmeter (the si ide eing tht the 2-orner model will hve the sme high-requeny soure spetrl level s single-orner soure model with the speiied ). The irst model is lredy in SMSIM (soure model 11); the soure spetrum is the multiplition o two untion o requeny. The seond model (soure model 12) is new to these notes; it is the summtion o two untions o requeny, nd s suh, it is generliztion o the soure spetr used y Atkinson nd Boore (1995) nd Atkinson nd Silv (2). I irst disuss the multiplitive spetrum model, nd this is ollowed y disussion o the dditive spetrum model. Generlized Multiplitive Soure Spetrum Let the elertion soure spetrum e proportionl to: A 2 1 1 p pd p 1 1 pd (1) Where p nd pd stnd or power o requeny nd power o denomintor. For high requenies, this eomes 2 ppd ppd AHF ( p pd p pd) (2) C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-1
For n 2 ω model, this onstrint must e stisied: p pd p pd 2 (3) I this onstrint is stisied, then the powers p nd pd n e relted to n equivlent stress prmeter nd single orner requeny model, s ollows. For single orner requeny model with orner requeny, the high-requeny spetrl level goes s A HF (4) 2 Equting (2) nd (4), with the onstrint (3) gives (5) 2 ppd ppd The proedure then is to use the reltion 13 6 4.96 1 β Δσ M (6) to otin given Δσ nd M, where M omes rom moment mgnitude M using the reltion Assuming tht is speiied y the user, in the ollowing wy: log M 1.5M 16.5. (7) log 1 2 M M, (8) then eqution (5) n e used to ind : 2 C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-
1 p 2 pd p pd (9) I illustrte this model or two sets o the powers, oth stisying the onstrint in eqution (3). In the irst exmple, p p 2 nd pd pd.5. Figure 1 shows the soure spetr or this model, ssuming M 6 nd 1 rs, or series o. Figure 1. Note tht with these hoies o the powers, the two-orner soures merge into the single orner model when.36 Hz, s expeted rom the ormultion ove. In ontrst, Figure 2 C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-3
shows the soure spetr or p p pd pd 1., nd or this se, the 2-orner model never pprohes the one-orner model. Figure 2. 4 C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-
Generlized Additive Soure Spetrum Let the elertion soure spetrum e proportionl to: A ε ε p pd p 1 1 2 1 pd (1) where p nd pd stnd or power o requeny nd power o denomintor. I proposed this soure model to Gil Atkinson in 1992 personl ommunition, nd she used it to derive soure spetrl model or ENA erthqukes (Atkinson, 1993), nd this orm o the soure model ws susequently used in other ppers y Gil nd her ollegues. For high requenies, this eomes 2 2 ppd ppd AHF (1 ε) p ε pd ppd (11) For n 2 ω model, the ollowing onstrint must e stisied: p pd p pd 2 (12) nd the high-requeny level is: A (1 ε) ε (13) 2 2 HF I the onstrint in eqution (12) is stisied, then given M nd M, M determined y equting the high-requeny soure spetrl level to the level or single orner 5 C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2- n e
requeny model, s ollows. This then generlizes the dditive two-orner model y letting the high-requeny level y determined y stress prmeter (lthough there my e some onstrints on M nd M in order or the resulting soure spetrum to mke sense, suh s eing rel numer---i need to do some explortion o this). For single orner requeny model with orner requeny, the high-requeny spetrl level goes s A HF (14) 2 Equting (13) nd (14) gives 2 2 (1 ) (15) The proedure now is to use the reltion to otin given Δσ nd M. 13 6 4.96 1 Δ (eqution 6, repeted) β σ M Assuming tht nd re speiied y the user, suh s in the ollowing wys nd log 1 2 M M (16) log ε 1ε 2ε ε M M (17) C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-6
then eqution (15) n e used to ind. Note tht or the nd 1 or given M tht there will e vlue o elow whih is not deined. This ours when the numertor under the rdil in eqution (15) equls.. From equtions (6) nd (15), the lower limit or is 1 3 M (18) where 6 4.96 1 (19) I hve revised the SMSIM progrms to inlude soure model 12 (soure model 11 ws lredy in the progrms). This required hnge to the prms iles, euse the oeiients o eqution (17) must e inluded in the prms ile. I tested the revision y using Atkinson nd Silv s (2) equtions or nd. Figure 3 shows the soure spetr or the dditive model or severl vlues o M nd, ompred to the single-orner soure spetr (to hek tht the 2-orner spetr hve the sme high-requeny levels s the single-orner spetr). I used p p 2. nd pd pd 1. or the exmple in this nd susequent igures. C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-7
Figure 3. Soure spetr Figure 4 shows the soure spetr or suite o vlues diering y tor o 2, nd rnging rom.1 to.64, or speiied vlue o. 8 C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-
Figure 4. I simulted the motions t 1 km or M 6 erthquke, with n eetive single-orner stress prmeter o 4 rs. Figure 5 shows the Fourier spetrum or the 1- nd 2-orner models or this se, omputed rom the si equtions or the spetr (wht is lled model in the igure), rom n verge o 1 time-domin simultions, nd the spetrum rom the lst o those simultions. The steep dey o the Fourier spetrum t low requeny is due to the inlusion o 8 low-ut ilter with orner requeny o.4 Hz, deying s t low requeny. C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-9
Figure 5. Fourier elertion spetr t R=1 km, M 6,.4 s, nd generi western rok rustl mpliitions nd Q. The response spetr re shown in Figure 6, omputed or the 1- nd 2-orner soure models, using oth rndom-virtion nd time-domin (with 1 simultions) omputtions. 1 C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-
Figure 6. Response spetr or ses disussed in Figure 5 (the jitter in the urves or low requenies is due to the requeny in the output iles not hving enough resolution). Note tht the TD nd RV simultions re in lose greement, exept or requenies etween out.4 nd.1 Hz. I m not sure o the reson or this disgreement, ut it might hve to do with the osilltor djustments eing used in the RV simultions (I m using those o Boore nd Thompson, 212). As usul, it is lwys good ide to hek RV results with TD omputtions. The dierene in mplitudes o the 1- nd 2-orner (soure 1 nd soure 12) PSA or requenies on either side o the region o the soure 12 sg (out.4 to 3 Hz, s shown in Figure 3) is primrily result o how the soure durtions re omputed. For soure 1 the soure durtion equls 1, while or the 2-orner soure model (soure 12), the soure durtion is given y C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-11
.5.5. The result is tht the soure durtion equls 1.77 nd 4.24 or soures 1 nd 12, respetively, nd this will result in dierene in the rms elertion o 4.24 1.77 1.5, with the soure 12 rms elertion eing smller thn the soure 1 rms elertion. The PSA in Figure 6 t high requeny, whih re equl to pek elertion, dier y tor o 1.5, s expeted rom the rtio o durtions. As n side, the eqution ove or the two-orner soure durtion diers rom tht used in Atkinson nd Boore (1995) nd Atkinson nd Silv (2):.5.. The prolem with this soure durtion is tht it leds to disontinuity t the mgnitude or whih the two orner requenies eome equl. I preer using equl weights o.5 or oth inverse orner requenies, s this voids the disontinuity. In ddition, s mgnitude inreses generlly inreses muh more rpidly thn, nd s result the durtion is primrily ontrolled y the term.5, whih is the eqution used y Atkinson nd Boore (1995) nd Atkinson nd Silv (2). C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-12
Disussion The two generlized 2-orner soure models re le to pproximte wide rnge o spetrl shpes, s indited in Figures 1, 2, nd 4. This lexiility omes t the expense o numer o oeiients tht must e speiied or ny pplition. I hve not worked with the models (nd dt to whih the models n e pplied) enough to mke strong reommendtions or these oeiients, ut t this time, I suggest the ollowing: or soure 11 (the multiplitive model), p p 2. nd pd pd.5, with to e determined y itting dt (or GMPEs, or inite-ult simultions); or soure 12 (the dditive model), p p 2. nd pd pd 1., rom Atkinson nd Boore (1995) or Atkinson nd Silv (2), nd to e determined y itting dt (or GMPEs, or inite-ult simultions). Reerenes Atkinson, G. M. (1993). Erthquke soure spetr in estern North Ameri, Bull. Seismol. So. Am. 83, 1778 1798. Atkinson, G. M. nd D. M. Boore (1995). Ground motion reltions or estern North Ameri, Bull. Seismol. So. Am. 85, 17--3. Atkinson, G. M. nd W. Silv (2). Stohsti modeling o Cliorni ground motions, Bull. Seismol. So. Am. 9, 255 274. Boore, D. M. (23). Predition o ground motion using the stohsti method, Pure nd Applied Geophysis 16, 635 676. Boore, D. M. nd E. M. Thompson (212). Empiril improvements or estimting erthquke response spetr with rndom-virtion theory, Bull. Seismol. So. Am. 12, 761 772. C:\smsim\generlize_2-orner_soure_model\smsim_generliztion_o_2-13