Site mixture of an analytic rock mixture by Gauss-Hermite quadrature
integrateSiteMixture.RdTransforms a rock spectrum represented at each oscillator period as a finite
lognormal mixture into the corresponding site spectrum, integrating the
site factor of evaluateSiteFactor() over the rock PGA of each component
with Gauss-Hermite quadrature. Every quadrature node becomes one component
of the returned mixture, so the result is again an analytic mixture:
summarizeMixture() reduces it to quantiles and mean, and
refineSiteMixture() chooses the order.
Arguments
- mixture
A table with one row per
(Tn, k)and columnsTn(oscillator period, s),k(component label),muLn,sigmaLn(natural-log location and scale,sigmaLnnon-negative) andw(positive weights summing to one at every period). It must hold the PGA rows (Tn == 0), and every component used at another period must have a PGA row of the same label.- VS30
One target velocity in m/s.
- Vref
One reference velocity in m/s; 760 or 3000 unless it equals
VS30.- model
Site factor model of
evaluateSiteFactor(),"ST20"or"ST17".- order
One integer, at least 2: Gauss-Hermite nodes per component.
Value
A data.table with Tn (s), k, muLn, sigmaLn and w,
ordered by Tn and k: the site spectrum as a lognormal mixture at
every period, conditional on the target velocity and the model, its
population being that of the rock mixture. mixture is not modified.
Mathematics
At period \(T_n\) the rock component \(k\) gives
\(\ln S_a \sim N(\mu_k, \sigma_k)\) with weight \(w_k\), and the
component of the same label at \(T_n = 0\) gives
\(\ln PGA \sim N(\mu_{0k}, \sigma_{0k})\). With the Gauss-Hermite rule
of order nodes \(x_m\) and weights \(\omega_m\) normalized to one
(Golub-Welsch, weight \(e^{-x^2}\)), the PGA nodes are
\(\ln PGA_{km} = \mu_{0k} + \sqrt{2} \sigma_{0k} x_m\). With
\(\rho = \rho_{BJ}(\max(T_n, 0.01), 0.01)\) of correlateResiduals(),
\(\rho_k = \rho\) and \(r_k = \sigma_k / \sigma_{0k}\) when
\(\sigma_{0k} > 0\) and both zero otherwise, the ordinate conditional on
the PGA node is
$$\ln S_a \mid PGA_{km} \sim N(\mu_k + \rho_k r_k (\ln PGA_{km} -
\mu_{0k}), \sqrt{1 - \rho_k^2}\, \sigma_k),$$
and with \((\mu_F, \sigma_F)\) of the factor at \(PGA_{km}\) the site
component \((k, m)\), labelled "k:m", is
$$\mu_{km} = \mu_k + \rho_k r_k (\ln PGA_{km} - \mu_{0k}) + \mu_F,
\quad \sigma_{km} = \sqrt{(1 - \rho_k^2) \sigma_k^2 + \sigma_F^2},
\quad w_{km} \propto w_k \omega_m,$$
the weights renormalized to one at every period. The site residual is
taken independent of the rock residual given PGA. When VS30 equals
Vref the site mixture is the rock mixture. No random numbers are drawn;
the error of the order is not measured here (see refineSiteMixture()).
Examples
Mixture <- data.table::CJ(Tn = c(0, 0.2, 1), k = c("a", "b"))
Mixture[, `:=`(muLn = log(0.4) + 0.1 * (k == "b") - 0.5 * Tn, sigmaLn = 0.5, w = 0.5)]
#> Key: <Tn, k>
#> Tn k muLn sigmaLn w
#> <num> <char> <num> <num> <num>
#> 1: 0.0 a -0.9162907 0.5 0.5
#> 2: 0.0 b -0.8162907 0.5 0.5
#> 3: 0.2 a -1.0162907 0.5 0.5
#> 4: 0.2 b -0.9162907 0.5 0.5
#> 5: 1.0 a -1.4162907 0.5 0.5
#> 6: 1.0 b -1.3162907 0.5 0.5
integrateSiteMixture(Mixture, VS30 = 360, Vref = 760, model = "ST20", order = 16)
#> Tn k muLn sigmaLn w
#> <num> <char> <num> <num> <num>
#> 1: 0.0 a:1 -4.07227009 0.3001148 7.489074e-11
#> 2: 0.0 a:10 -0.45403220 0.3436244 7.916919e-02
#> 3: 0.0 a:11 -0.11870193 0.3611579 2.364238e-02
#> 4: 0.0 a:12 0.22174094 0.3825849 3.633469e-03
#> 5: 0.0 a:13 0.57309755 0.4081712 2.629925e-04
#> 6: 0.0 a:14 0.94382934 0.4384487 7.650016e-06
#> 7: 0.0 a:15 1.34910175 0.4746981 6.547366e-08
#> 8: 0.0 a:16 1.82717329 0.5207557 7.489074e-11
#> 9: 0.0 a:2 -3.50358254 0.3003439 6.547366e-08
#> 10: 0.0 a:3 -3.02819994 0.3008427 7.650016e-06
#> 11: 0.0 a:4 -2.60101580 0.3018383 2.629925e-04
#> 12: 0.0 a:5 -2.20495637 0.3036751 3.633469e-03
#> 13: 0.0 a:6 -1.83102926 0.3068213 2.364238e-02
#> 14: 0.0 a:7 -1.47334911 0.3118378 7.916919e-02
#> 15: 0.0 a:8 -1.12732332 0.3193112 1.432843e-01
#> 16: 0.0 a:9 -0.78886995 0.3297702 1.432843e-01
#> 17: 0.0 b:1 -3.97373306 0.3001390 7.489074e-11
#> 18: 0.0 b:10 -0.36857139 0.3477454 7.916919e-02
#> 19: 0.0 b:11 -0.03417474 0.3661471 2.364238e-02
#> 20: 0.0 b:12 0.30555261 0.3883849 3.633469e-03
#> 21: 0.0 b:13 0.65637637 0.4147003 2.629925e-04
#> 22: 0.0 b:14 1.02672026 0.4456183 7.650016e-06
#> 23: 0.0 b:15 1.43171486 0.4824275 6.547366e-08
#> 24: 0.0 b:16 1.90958764 0.5289893 7.489074e-11
#> 25: 0.0 b:2 -3.40603575 0.3004142 6.547366e-08
#> 26: 0.0 b:3 -2.93188287 0.3010072 7.650016e-06
#> 27: 0.0 b:4 -2.50617067 0.3021766 2.629925e-04
#> 28: 0.0 b:5 -2.11177498 0.3043029 3.633469e-03
#> 29: 0.0 b:6 -1.73961030 0.3078852 2.364238e-02
#> 30: 0.0 b:7 -1.38367598 0.3134972 7.916919e-02
#> 31: 0.0 b:8 -1.03927050 0.3217098 1.432843e-01
#> 32: 0.0 b:9 -0.70223368 0.3330086 1.432843e-01
#> 33: 0.2 a:1 -3.58828671 0.3450321 7.489074e-11
#> 34: 0.2 a:10 -0.46681244 0.3673727 7.916919e-02
#> 35: 0.2 a:11 -0.19685655 0.3787511 2.364238e-02
#> 36: 0.2 a:12 0.07287302 0.3938000 3.633469e-03
#> 37: 0.2 a:13 0.34747420 0.4130450 2.629925e-04
#> 38: 0.2 a:14 0.63403632 0.4371810 7.650016e-06
#> 39: 0.2 a:15 0.94463103 0.4675207 6.547366e-08
#> 40: 0.2 a:16 1.30866511 0.5076983 7.489074e-11
#> 41: 0.2 a:2 -3.08754701 0.3451057 6.547366e-08
#> 42: 0.2 a:3 -2.66940417 0.3452724 7.650016e-06
#> 43: 0.2 a:4 -2.29450167 0.3456234 2.629925e-04
#> 44: 0.2 a:5 -1.94836816 0.3463151 3.633469e-03
#> 45: 0.2 a:6 -1.62383262 0.3475966 2.364238e-02
#> 46: 0.2 a:7 -1.31656173 0.3498302 7.916919e-02
#> 47: 0.2 a:8 -1.02331791 0.3534919 1.432843e-01
#> 48: 0.2 a:9 -0.74111888 0.3591422 1.432843e-01
#> 49: 0.2 b:1 -3.48958897 0.3450398 7.489074e-11
#> 50: 0.2 b:10 -0.38566132 0.3699590 7.916919e-02
#> 51: 0.2 b:11 -0.11759127 0.3821534 2.364238e-02
#> 52: 0.2 b:12 0.15060473 0.3980531 3.633469e-03
#> 53: 0.2 b:13 0.42400998 0.4181382 2.629925e-04
#> 54: 0.2 b:14 0.70967072 0.4430713 7.650016e-06
#> 55: 0.2 b:15 1.01960323 0.4741517 6.547366e-08
#> 56: 0.2 b:16 1.38315470 0.5150296 7.489074e-11
#> 57: 0.2 b:2 -2.98978386 0.3451287 6.547366e-08
#> 58: 0.2 b:3 -2.57286688 0.3453290 7.650016e-06
#> 59: 0.2 b:4 -2.19953592 0.3457469 2.629925e-04
#> 60: 0.2 b:5 -1.85533185 0.3465620 3.633469e-03
#> 61: 0.2 b:6 -1.53304251 0.3480526 2.364238e-02
#> 62: 0.2 b:7 -1.22823411 0.3506118 7.916919e-02
#> 63: 0.2 b:8 -0.93752186 0.3547381 1.432843e-01
#> 64: 0.2 b:9 -0.65776130 0.3609965 1.432843e-01
#> 65: 1.0 a:1 -2.55586619 0.4830057 7.489074e-11
#> 66: 1.0 a:10 -0.54196296 0.4947013 7.916919e-02
#> 67: 1.0 a:11 -0.33906813 0.4989103 2.364238e-02
#> 68: 1.0 a:12 -0.13105602 0.5040429 3.633469e-03
#> 69: 1.0 a:13 0.08523912 0.5102441 2.629925e-04
#> 70: 1.0 a:14 0.31473814 0.5177549 7.650016e-06
#> 71: 1.0 a:15 0.56664019 0.5270498 6.547366e-08
#> 72: 1.0 a:16 0.86466261 0.5393789 7.489074e-11
#> 73: 1.0 a:2 -2.25556568 0.4831000 6.547366e-08
#> 74: 1.0 a:3 -2.00195930 0.4832934 7.650016e-06
#> 75: 1.0 a:4 -1.77113922 0.4836532 2.629925e-04
#> 76: 1.0 a:5 -1.55383916 0.4842672 3.633469e-03
#> 77: 1.0 a:6 -1.34510261 0.4852373 2.364238e-02
#> 78: 1.0 a:7 -1.14173939 0.4866696 7.916919e-02
#> 79: 1.0 a:8 -0.94139007 0.4886650 1.432843e-01
#> 80: 1.0 a:9 -0.74208270 0.4913144 1.432843e-01
#> 81: 1.0 b:1 -2.45592985 0.4830159 7.489074e-11
#> 82: 1.0 b:10 -0.44238490 0.4956949 7.916919e-02
#> 83: 1.0 b:11 -0.23950674 0.5001036 2.364238e-02
#> 84: 1.0 b:12 -0.03150689 0.5054394 3.633469e-03
#> 85: 1.0 b:13 0.18477940 0.5118461 2.629925e-04
#> 86: 1.0 b:14 0.41427212 0.5195659 7.650016e-06
#> 87: 1.0 b:15 0.66616973 0.5290776 6.547366e-08
#> 88: 1.0 b:16 0.96418900 0.5416456 7.489074e-11
#> 89: 1.0 b:2 -2.15566854 0.4831281 6.547366e-08
#> 90: 1.0 b:3 -1.90210703 0.4833548 7.650016e-06
#> 91: 1.0 b:4 -1.67133581 0.4837702 2.629925e-04
#> 92: 1.0 b:5 -1.45408548 0.4844673 3.633469e-03
#> 93: 1.0 b:6 -1.24539620 0.4855504 2.364238e-02
#> 94: 1.0 b:7 -1.04207511 0.4871238 7.916919e-02
#> 95: 1.0 b:8 -0.84176130 0.4892837 1.432843e-01
#> 96: 1.0 b:9 -0.64248251 0.4921149 1.432843e-01
#> Tn k muLn sigmaLn w
#> <num> <char> <num> <num> <num>