Transform an analytic spectral mixture through site response

Composes a finite lognormal mixture at a reference condition with the lognormal dispersion of the selected site-response model. Conditioning on each mixture component and its same-realization rock PGA reduces the transformation to one-dimensional Gauss–Hermite quadrature. Every quadrature node becomes one component of the returned analytic mixture.

Usage

fitSaFMixture(gmm, vs30, vref = 760, models = "ST20", nq = 32L)

Arguments

gmm: A data.table with exactly one analytic component row per (Tn, k) and columns Tn, k, muLn, sigmaLn, and w. Tn == 0 must be present, every component used at another period must have a matching PGA component at Tn == 0, weights must sum to one at every period, and sigmaLn must be non-negative. Component participation and normalized weights may vary by period.
vs30: Numeric scalar target Vs30, in m/s.
vref: Numeric scalar reference Vs30, in m/s. Default 760.
models: Site-response model identifier: "ST20" (default) or "ST17".
nq: Positive integer number of Gauss–Hermite nodes per input component. Default 32.

Value

A data.table with exactly Tn, k, muLn, sigmaLn, and w. It is an analytic finite mixture; it contains no probability-level rows or report quantiles. Output weights sum to one at every period.

Details

If vs30 == vref, site response is the identity and the validated analytic input is returned without changing its parameters. No quantiles are materialized and no random numbers are drawn.

Examples

GMM <- data.table::CJ(Tn = c(0, 0.2, 1), k = c("a", "b"))
GMM[, `:=`(
  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
fitSaFMixture(gmm = GMM, vs30 = 360, nq = 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>