- Ground-motion model is:
_{1}(x) is the exponential integral function, α_{1}= 2.4862, α_{2}= 0.9392, α_{3}= 0.5061, α_{4}= 0.0150, b = -0.0181, σ = 0.7500 (total), σ_{e}= 0.4654 (inter-event) and σ_{r}= 0.5882 (intra-event). - All data from rock (NEHRP B) sites. Data from stations with known, significant site amplification and those located in volcanic belt are excluded. Use H/V ratios to verify that stations are all on generic rock. Data from 56 different stations.
- Focal depths between 10 and 29km.
- Functional form is based on the analytical solution of a circular finite-source model and body waves, which
also defines expression for r
_{0}(the radius of the circular fault based on Brune’s model) using a stress drop of 100bar in order to keep functional form as simple as possible. Note that functional form allows for oversaturation, whose existence is questionable. - Select data of interplate, thrust-faulting events (interface) from permanent networks between 1985 and
2004 on the Pacific coast between Colima and Oaxaca (majority of data from Guerrero but some data
from other regions, especially Oaxaca). Data from near-trench earthquakes whose high-frequency radiation
is anomalously low are excluded. To focus on ground motions of engineering interest, exclude data from
small (M
_{w}≤ 5.5) with few records that are only from distant stations (R > 100km). Exclude data from > 400km (use a larger distance than usual because of previously observed slow decay). To reduce potential variability of data, select only one record from two stations recording the same earthquake at less than 5km (based on visual inspection of data). - Data from 12–19bit digital accelerographs (66% of data), which have flat response down to less than 0.1Hz,
and 24bit broadband seismographs (34% of data), which have flat response for velocities between 0.01
and 30Hz. Broadband data mainly from M
_{w}< 6 and distances > 100km. Sampling rates between 80 and 250Hz. Instrumental responses and sampling rates mean data reliable up to 30Hz. - Roughly 45% of records from 20–100km. Only 16 records from < 25km and only 5 from 3 earthquakes with
M
_{w}> 7 and, therefore, note that any anomalous records will strongly influence results in this distance range. State that more near-source data from large Mexican interplate earthquakes needed. - Use Bayesian regression that accounts, for linear functions, for these correlations: 1) intra-event, 2) between
coefficients and 3) between different periods. To linearize function perform regression as: for a given period
and value of α
_{4}, compute coefficients α_{1}, α_{2}and α_{3}through Bayesian analysis and iterate for different values of α_{4}to find the value that gives best fit to data. This is repeated for each period. Note that this means the regression is not fully Bayesian. To obtain prior information on coefficients α_{1}, α_{2}and α_{3}use random vibration theory and theoretical expression for Fourier amplitude spectrum. Define other required prior parameters (covariances etc.) using previous studies. Smooth α_{4}w.r.t. period. Discuss differences between prior and posterior values and not that final results not over-constrained to mean prior values. - Find that model systematically overestimates in whole period range but since less than 5% consider bias acceptable.
- Plot residuals w.r.t. M
_{w}, distance and depth and find no significant trend. Note that even though focal depth is not included in model there is no significant dependence on it. - Adjust observed near-source PGAs to a common distance of 16km and include data from M
_{w}2.5–4.9 from r_{hypo}between 16 and 37km. Compare to predictions. Note the large scatter (more than an order of magnitude) so note that statistical significance is low. Note that model matches observations reasonably well.