where R is r_{epi}, v = 0 for horizontal and v = 1 for vertical components (believe both components should
have identical dependency on magnitude, distance and site conditions), S_{0} is correlation radius of source, β
is shear-wave velocity at earthquake source (assume as 3.5km∕s for NE India and 3.3km∕s in W Himalaya),
S is fault size in km and A_{0} is frequency-dependent attenuation function from previous study.
Model only valid for M_{min}≤ M ≤ M_{max} where M_{min}= -b_{2}∕(2b_{5}(T)) and M_{max}= -(1+b_{2}(T))∕(2b_{5}(T)).
For M < M_{min} use M = M_{min} in C_{2}M + C_{3}M^{2} terms. For M > M_{max} use M = M_{max} everywhere.
Consider three geological site conditions (thick strata of order of km):
s = 0
Sediment: 193 records.
s = 1
Intermediate sites or complex geological environments that cannot be categorized unambiguously: 26
records.
s = 2
Basement rock: 146 records.
Consider three local site categories (top 100–200m):
s_{L}= 0
Rock (V_{s}< 800m∕s only in top 10m): 35 records.
s_{L}= 1
Stiff soil (V_{s}< 800m∕s in top roughly 75m): 72 records.
s_{L}= 2
Deep soil (V_{s}< 800m∕s in top 150–200m): 258 records.
Response parameter is pseudo-velocity for 0, 2, 5, 10 and 20% damping.
Develop model only for 0.04 to 3s to minimize the effect of noise in the data. Extend model to shorter and
longer periods by theoretical methods.
217 (of which 5 are analogue and rest digital) records from 47 earthquakes and 80 stations in western
Himalaya and 148 (of which 5 are analogue and rest digital) records from 36 earthquakes and 56 stations
in northeastern India. Analogue data from local networks (of 40-50 stations) of IIT Roorkee in Kangra,
Garhwal-Kumaon areas and Shillong Plateau (1986–1999), which correct for instrument response and filter.
Digital data from National Strong Motion Instrumentation Network of 00 stations (since 2005), which
filter.
Select data from M ≤ 4.0 and r_{hypo}≤ 350km because weaker data contaminated by noise, lower magnitude
estimates are unreliable and more distant attenuation difficult to constrain.
Most data from W Himalaya from 4.5 ≤ M_{w}≤ 5.5 and 10 ≤ r_{hypo}≤ 185km (mean recording distance is
100km) and most data from NE India from 5.5 ≤ M_{w}≤ 6.0 and 35 ≤ r_{hypo}≤ 205km (mean recording
distance is 150km).
Focal depths (H) in W Himalaya between roughly 5 and 65km with most ≥ 30km, and in NE India
between roughly 5 and 55km with most ≤ 20km.
Assume common model for both region but different attenuation functions A_{0}, which find consistent with
the data.
Use decimation scheme before regress to eliminate possible bias due to non-uniform data distribution.
All 1095 spectral amplitudes for each T are grouped into magnitudes 4.0–4.9, 5.0–5.9 and 6.0–6.9.
Amplitudes in each group are separately sequentially per component and geology and soil categories.
Arrange amplitudes in each of the 54 sub-divisions are arranged in increasing order. Select maximum of
33 values from each group for regression corresponding to serial numbers closest to 3rd, 6th, …, 96th and
99th percentiles. Number of data points varies with T due to cut-off periods for each accelerogram.
Regress using a two-step weighted method.
Smooth coefficients to obtained predicted spectra that are physically realistic.
Fit models to residuals to obtain probability distributions for variability rather than σ.
Compare predicted and observed spectra for some example records and find a good match.