Distribution of permanent seismic displacement in a model ensemble

Permanent seismic displacement $D_n$ measures the accumulated movement of a sliding mass during an earthquake. In the rigid-block idealization, relative sliding begins when the driving acceleration exceeds the yield acceleration and continues until the relative velocity returns to zero. The yield coefficient $k_y=a_y/g$ expresses the yield acceleration $a_y$ relative to gravitational acceleration $g$. Rigid-block regressions relate displacement to measures of the base excitation; flexible-block regressions also represent the dynamic response of the deformable sliding mass through its initial fundamental period $T_s$ and spectral acceleration evaluated at a degraded period proportional to $T_s$.

For fixed model inputs, the positive displacement is represented by a lognormal distribution. Its parameters $\mu_{\ln D}$ and $\sigma_{\ln D}$ are the mean and standard deviation of the natural logarithm of displacement, with displacement expressed in $\mathrm{cm}$. evaluateRigidDisplacement() and evaluateFlexibleDisplacement() return these parameters as muLnD and sigmaLnD; they describe a distribution rather than one sampled displacement. For model $m$ and realization $s$, the displacement realization is

$$D_{n,s}^{(m)}(k_y)=\exp\left[\mu_{\ln D}^{(m)}(k_y,IM_s)+\sigma_{\ln D}^{(m)}(k_y,IM_s)\epsilon_s\right].$$

Here $IM_s$ denotes the intensity measures of realization $s$, and $\epsilon_s$ is its standard-normal residual; the remaining model inputs are held fixed for the scenario. In evaluateDisplacements(), the intensity measures and residual are shared across the requested models and yield coefficients. Fixing them makes each realization a displacement curve $D_{n,s}^{(m)}(k_y)$. The standardized residual is identical across these curves, while each model scales it by its own dispersion. The function receives the residuals through eps and performs a deterministic evaluation. Its output identifies the model, yield coefficient, realization and displacement by IDn, ky, s and Dn.

The flexible models BT07, BM17 and BM19 use the positive lognormal branch while omitting the probability of zero or negligible displacement. At fixed intensity measures, the branch describes displacement conditional on a nonzero displacement. When intensities vary, the propagated sample combines those positive branches under the supplied demand distribution. Pooling models with their original weights does not condition the full population on nonzero displacement: that conditioning would also reweight the models and the demand according to their probabilities of nonzero displacement.

At a fixed $k_y$, an ensemble pools the displacement realizations of its models. With $M$ models, their positive weights $w_m$ are normalized as

$$\omega_m=\frac{w_m}{\sum_{l=1}^{M}w_l}.$$

If each model contributes the same $N$ realizations, its empirical cumulative distribution function at displacement $x$, in $\mathrm{cm}$, is

$$\hat F_m(x)=\frac{1}{N}\sum_{s=1}^{N}\mathbf 1\left\{D_{n,s}^{(m)}(k_y)\le x\right\}.$$

The indicator $\mathbf 1$ equals one when the enclosed condition holds and zero otherwise. The pooled empirical distribution is the mixture

$$\hat F(x)=\sum_{m=1}^{M}\omega_m\hat F_m(x).$$

summarizeEnsemble() assigns the normalized model weight to each pooled realization at that yield coefficient. With equal realization counts, the ensemble mean is consequently the weighted combination of the model means. Ensemble quantiles are estimated from the pooled values and weights, rather than by averaging model quantiles. The estimator interpolates order statistics selected by cumulative weight; it is not the generalized inverse of the stepwise empirical distribution. With unequal realization counts, each model’s total contribution also depends on its number of rows, so the mixture weights are no longer the normalized model weights alone.

summarizeDisplacements() instead summarizes each model separately at each yield coefficient, returning the arithmetic mean or quantiles estimated by linear interpolation between order statistics. Both summary functions report level, the requested probability or mean, and n, the number of realizations summarized. These are marginal statistics at each $k_y$: adjacent points on a quantile curve do not identify one paired realization. The realization table retains the pairing required to follow an individual displacement curve across yield coefficients or across models.