- Ground-motion model is:
_{jb}is the distance to the surface projection of rupture and δ is the dip of the fault; for uncorrected horizontal PGA: c_{1}= -2.896, c_{2}= 0.812, c_{3}= 0.0, c_{4}= -1.318, c_{5}= 0.187, c_{6}= -0.029, c_{7}= -0.064, c_{8}= 0.616, c_{9}= 0, c_{10}= 0.179, c_{11}= 0.307, c_{12}= -0.062, c_{13}= -0.195, c_{14}= -0.320, c_{15}= 0.370 and σ = c_{16}- 0.07M_{w}for M_{w}< 7.4 and σ = c_{16}- 0.518 for M_{w}≥ 7.4 where c_{16}= 0.964 or σ = c_{17}+ 0.351 for PGA ≤ 0.07g, σ = c_{17}- 0.132ln(PGA) for 0.07g < PGA < 0.25g and σ = c_{17}+ 0.183 for PGA ≥ 0.25g where c_{17}= 0.263; for corrected horizontal PGA: c_{1}= -4.033, c_{2}= 0.812, c_{3}= 0.036, c_{4}= -1.061, c_{5}= 0.041, c_{6}= -0.005, c_{7}= -0.018, c_{8}= 0.766, c_{9}= 0.034, c_{10}= 0.343, c_{11}= 0.351, c_{12}= -0.123, c_{13}= -0.138, c_{14}= -0.289, c_{15}= 0.370 and σ = c_{16}- 0.07M_{w}for M_{w}< 7.4 and σ = c_{16}- 0.518 for M_{w}≥ 7.4 where c_{16}= 0.920 or σ = c_{17}+ 0.351 for PGA ≤ 0.07g, σ = c_{17}- 0.132ln(PGA) for 0.07g < PGA < 0.25g and σ = c_{17}+ 0.183 for PGA ≥ 0.25g where c_{17}= 0.219; for uncorrected vertical PGA: c_{1}= -2.807, c_{2}= 0.756, c_{3}= 0, c_{4}= -1.391, c_{5}= 0.191, c_{6}= 0.044, c_{7}= -0.014, c_{8}= 0.544, c_{9}= 0, c_{10}= 0.091, c_{11}= 0.223, c_{12}= -0.096, c_{13}= -0.212, c_{14}= -0.199, c_{15}= 0.630 and σ = c_{16}- 0.07M_{w}for M_{w}< 7.4 and σ = c_{16}- 0.518 for M_{w}≥ 7.4 where c_{16}= 1.003 or σ = c_{17}+ 0.351 for PGA ≤ 0.07g, σ = c_{17}-0.132ln(PGA) for 0.07g < PGA < 0.25g and σ = c_{17}+0.183 for PGA ≥ 0.25g where c_{17}= 0.302; and for corrected vertical PGA: c_{1}= -3.108, c_{2}= 0.756, c_{3}= 0, c_{4}= -1.287, c_{5}= 0.142, c_{6}= 0.046, c_{7}= -0.040, c_{8}= 0.587, c_{9}= 0, c_{10}= 0.253, c_{11}= 0.173, c_{12}= -0.135, c_{13}= -0.138, c_{14}= -0.256, c_{15}= 0.630 and σ = c_{16}- 0.07M_{w}for M_{w}< 7.4 and σ = c_{16}- 0.518 for M_{w}≥ 7.4 where c_{16}= 0.975 or σ = c_{17}+ 0.351 for PGA ≤ 0.07g, σ = c_{17}- 0.132ln(PGA) for 0.07g < PGA < 0.25g and σ = c_{17}+ 0.183 for PGA ≥ 0.25g where c_{17}= 0.274. - Use four site categories:
- Firm soil
- Generally includes soil deposits of Holocene age (less than 11,000 years old) described on geological
maps as recent alluvium, alluvial fans, or undifferentiated Quaternary deposits. Approximately
corresponds to V
_{s,30}= 298 ± 92m∕s and NEHRP soil class D. Uncorrected PGA: 534 horizontal records and 525 vertical records and corrected PGA: 241 horizontal records and 240 vertical records. S_{V FS}= 0, S_{SR}= 0 and S_{FR}= 0. - Very firm soil
- Generally includes soil deposits of Pleistocene age (11,000 to 1.5 million years old) described
on geological maps as older alluvium or terrace deposits. Approximately corresponds to V
_{s,30}= 368 ± 80m∕s and NEHRP soil class CD. Uncorrected PGA: 168 horizontal records and 166 vertical records and corrected PGA: 84 horizontal records and 83 vertical records. S_{V FS}= 1, S_{SR}= 0 and S_{FR}= 0. - Soft rock
- Generally includes sedimentary rock and soft volcanic deposits of Tertiary age (1.5 to 100 million
years old) as well as ‘softer’ units of the Franciscan Complex and other low-grade metamorphic
rocks generally described as melange, serpentine and schist. Approximately corresponds to V
_{s,30}= 421 ± 109m∕s and NEHRP soil class CD. Uncorrected PGA: 126 horizontal records and 124 vertical records and corrected PGA: 63 horizontal records and 62 vertical records. S_{SR}= 1, S_{V FS}= 0 and S_{FR}= 0. - Firm rock
- Generally include older sedimentary rocks and hard volcanic deposits, high-grade metamorphic rock,
crystalline rock and the ‘harder’ units of the Franciscan Complex generally described as sandstone,
greywacke, shale, chert and greenstone. Approximately corresponds to V
_{s,30}= 830 ± 339m∕s and NEHRP soil class BC. Uncorrected PGA: 132 horizontal records and 126 vertical records and corrected PGA: 55 horizontal records and 54 vertical records. S_{FR}= 1, S_{V FS}= 0 and S_{SR}= 0.

Note that for generic soil (approximately corresponding to V

_{s,30}= 310m∕s and NEHRP site class D) use S_{V FS}= 0.25, S_{SR}= 0, S_{FR}= 0 and for generic rock (approximately corresponding to V_{s,30}= 620m∕s and NEHRP site class C) use S_{SR}= 0.50, S_{FR}= 0.50 and S_{V FS}= 0. - Use four fault types but only model differences between strike-slip, reverse and thrust:
- Normal
- Earthquakes with rake angles between 202.5
^{∘}and 337.5^{∘}. 4 records from 1 earthquake. - Strike-slip
- Includes earthquakes on vertical or near-vertical faults with rake angles within 22.5
^{∘}of the strike of the fault. Also include 4 records from 1975 Oroville normal faulting earthquake. Uncorrected PGA: 404 horizontal records and 395 vertical records and corrected PGA: 127 horizontal and vertical records. F_{RV }= 0 and F_{TH}= 0 - Reverse
- Steeply dipping earthquakes with rake angles between 22.5
^{∘}and 157.5^{∘}. Uncorrected PGA: 186 horizontal records and 183 vertical records and corrected PGA: 58 horizontal records and 57 vertical records. F_{RV }= 1 and F_{TH}= 0. - Thrust
- Shallow dipping earthquakes with rake angles between 22.5
^{∘}and 157.5^{∘}. Includes some blind thrust earthquakes. Uncorrected PGA: 370 horizontal records and 363 vertical records and corrected PGA: 258 horizontal records and 255 vertical records. F_{TH}= 1 and F_{RV }= 0.

Note that for generic (unknown) fault type use F

_{RV }= 0.25 and F_{TH}= 0.25. - Most records from 5.5 ≤ M
_{w}≤ 7.0. - Note that equations are an update to equations in Campbell (1997) because they used a somewhat awkward and complicated set of Ground-motion models because there used a mixture of functional forms. Consider that the new equations supersede their previous studies.
- Uncorrected PGA refers to the standard level of accelerogram processing known as Phase 1. Uncorrected PGAs are either scaled directly from the recorded accelerogram or if the accelerogram was processed, from the baseline and instrument-corrected Phase 1 acceleration time-history.
- Corrected PGA measured from the Phase 1 acceleration time-history after it had been band-pass filtered and decimated to a uniform time interval.
- Restrict data to within 60km of seismogenic rupture zone (r
_{seis}≤ 60km) of shallow crustal earthquakes in active tectonic regions which have source and near-source attenuation similar to California. Most data from California with some from Alaska, Armenia, Canada, Hawaii, India, Iran, Japan, Mexico, Nicaragua, Turkey and Uzbekistan. Note some controversy whether this is true for all earthquakes (e.g. Gazli and Nahanni). Exclude subduction-interface earthquakes. - Restrict earthquakes to those with focal depths < 25km.
- Exclude data from subduction-interface earthquakes, since such events occur in an entirely different tectonic environment that the other shallow crustal earthquakes, and it has not been clearly shown that their near-source ground motions are similar to those from shallow crustal earthquakes.
- Restrict to r
_{seis}≤ 60km to avoid complications related to the arrival of multiple reflections from the lower crust. Think that this distance range includes most ground-motion amplitudes of engineering interest. - All records from free-field, which define as instrument shelters or non-embedded buildings < 3 storeys high and < 7 storeys high if located on firm rock. Include records from dam abutments to enhance the rock records even though there could be some interaction between dam and recording site. Exclude records from toe or base of dam because of soil-structure interaction.
- Do preliminary analysis, find coefficients in f
_{3}need to be constrained in order to make Y independent on M_{w}at r_{seis}= 0, otherwise Y exhibits ‘oversaturation’ and decreases with magnitude at close distances. Therefore set c_{8}= -c_{2}∕c_{4}and c_{9}= -c_{3}∕c_{4}. - Functional form permits nonlinear soil behaviour.
- Do not include sediment depth (depth to basement rock) as a parameter even though analysis of residuals indicated that it is an important parameter especially at long periods. Do not think its exclusion is a serious practical limitation because sediment depth is generally not used in engineering analyses and not included in any other widely used attenuation relation.
- Do not apply weights during regression analysis because of the relatively uniform distribution of records w.r.t. magnitude and distance.
- To make regression analysis of corrected PGA more stable set c
_{2}equal to value from better-constrained regression of uncorrected PGAs. - Examine normalised residuals δ
_{i}= (lnY_{i}- ln)∕σ_{ln(Unc.PGA}where lnY_{i}is the measured acceleration, is the predicted acceleration and σ_{ln(Unc.PGA}is the standard deviation of the uncorrected PGA equation. Plot δ_{i}against magnitude and distance and find models are unbiased. - Consider equations valid for M
_{w}≥ 5.0 and r_{seis}≤ 60km. Probably can be extrapolated to a distance of 100km without serious compromise. - Note that should use equations for uncorrected PGA if only an estimate of PGA is required because of its statistical robustness. If want response spectra and PGA then should use corrected PGA equation because the estimates are then consistent.
- Note that should include ground motions from Kocaeli (17/8/1999, M
_{w}= 7.4), Chi-Chi (21/9/1999, M_{w}= 7.6), Hector Mine (16/10/1999, M_{w}= 7.1) and Duzce (12/11/1999, M_{w}= 7.1) earthquakes but because short-period motions from these earthquakes was significantly lower than expected their inclusion could lead to unconservative estimated ground motions for high magnitudes. - Prefer the relationship for σ in terms of PGA because statistically more robust. Note that very few records to constrain value of σ for large earthquakes but many records to constrain σ for PGA ≥ 0.25g.
- Find that Monte Carlo simulation indicates that all regression coefficients statistically significant at 10% level.