Electronic Supplement to
Late Holocene Liquefaction at Sites of Contemporary Liquefaction during the 2010–2011 Canterbury Earthquake Sequence, New Zealand

by Sarah H. Bastin, Kari Bassett, Mark C. Quigley, Brett Maurer, Russell A. Green, Brendon Bradley, and David Jacobson

This electronic supplement provides an overview and further discussion of the probabilistic magnitude-bound methodology framework and derivative curves, a description of sediment units, and a table of peak ground accelerations (PGAs).

Overview of Probabilistic Magnitude-Bound Methodology

Probabilistic magnitude-bound curves are made possible by the development of probabilistic liquefaction triggering procedures (e.g., Cetin et al., 2004; Moss et al., 2006; Boulanger and Idriss, 2012, 2014). These procedures, based on the simplified liquefaction evaluation framework (Seed and Idriss, 1982; Whitman, 1971), recognize that there is a probability of liquefaction associated with every combination of earthquake-induced cyclic stress and the cyclic resistance of a soil to liquefaction. However, cyclic stress and cyclic resistance are also uncertain due to the variability of the inputs that define them. Collectively, these uncertainties highlight the need for a probabilistic assessment of liquefaction potential.

The methodology adopted herein does not attempt to quantify every uncertainty entering a paleoliquefaction study, but it incorporates the most significant parameter variability and demonstrates how quantifiable uncertainties can be accounted for in the analysis. Using the total probability theorem to integrate over select uncertainties, the probability that a site liquefies in an earthquake of magnitude M_w at site-to-source distance R can be expressed as

in which the conditional probability of liquefaction P(Site Liquefies|EQK:M, R) is given by a simplified probabilistic liquefaction triggering procedure; the conditional probability density function f(a_max|M, R) is given by a ground-motion prediction equation (GMPE); and f_rd(r_d) is a probability density function for r_d, a parameter in the simplified framework accounting for the nonrigid response of the soil column. In addition to the potential for liquefaction triggering, as defined by the simplified cyclic-stress framework (which is the current state-of-practice methodology), there exists a threshold shear strain γ_t below which excess pore pressures are not expected to develop, irrespective of the number of loading cycles (Dobry et al., 1980, 1982).

From this criterion, it follows that there is PGA below which liquefaction will not occur, regardless of shaking duration (in which earthquake magnitude is often used as a proxy for shaking duration), but this is not explicitly considered by the framework in equation (S1). The solution obtained from equation (S1) is therefore contingent on this requirement, which can affect the probability of liquefaction in large-magnitude earthquakes. Again utilizing the total probability theorem, the probability of an induced strain exceeding γ_t in an earthquake of magnitude M_w at site-to-source distance R can be expressed as

in which the conditional probability density function f(a_max|M, R) is given by a GMPE, whereas f_rd(r_d) and $f_{\frac{G}{G_{\max }}}\left(\frac{G}{G_{\max }}\right)$ are probability density functions for r_d and $\frac{G}{G_{\max }}$, respectively. Throughout equation (S2), $\frac{G}{G_{\max }}$ refers to the shear-modulus reduction coefficient at γ_t. From equations (S1) and (S2), and adopting Christchurch-appropriate inputs for liquefaction susceptibility, site response, and ground-motion prediction, a suite of probabilistic magnitude-bound curves are developed. For further discussion of this framework, and of the derivative curves, the reader is referred to Maurer et al. (2015).

In the probabilistic framework adopted herein (equations S1 and S2), the Joyner–Boore distance (R_JB) is needed between sites of paleoliquefaction and fault ruptures. However, because seismological coverage was insufficient to study New Zealand earthquakes until ∼1915 (Doser and Robinson, 2002), the locations, slip models, and magnitudes of historic ruptures are generally uncertain. Accordingly, we rely on a combination of paleoseismic field studies (e.g., Howard et al., 2005), proposed source models for large historic earthquakes (e.g., Doser et al., 1999), and empirical site-to-source distance conversions (Scherbaum et al., 2004) to estimate rupture magnitudes and site-to-source distances.

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Tables

Table S1. Sedimentological descriptions of sediment units. This table provides detailed sedimentological descriptions of the sediment units identified within the trenches and summarized in the main article.

Table S2. Modeled peak ground acceleration (PGA) and magnitude normalized PGA (PGA_7.5) of known active faults within the wider Canterbury area. The table presents the modeled PGA and PGA_7.5 for all the active faults within the wider Canterbury region. The maximum magnitude and R_rup to the study site are also listed for each active fault.

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