Liquefaction and Landslide models
Liquefaction models
Two liquefaction models were are implemented in the OQ-MBTK. The first is the method developed for the HAZUS software by the US Federal Emergency Management Agency. This model involves categorization of sites into liquefaction susceptibility classes based on geotechnical characteristics, and a quanitative probability model for each susceptibility class. The second model is an academic model developed by Zhu and others (2015). It is statistical model incorporating only DEM-derived quantities for site characterization.
HAZUS
The HAZUS model classifies each site into a liquefaction susceptibility class (LSC) based on the geologic and geotechnical characteristics of the site, such as the sedimentological type and the deposition age of the unit. In addition to the LSC and the local ground acceleration at each site, the depth to groundwater at the site and the magnitude of the causative earthquake will affect the probability that a given site will experience liquefaction.
The equation that describes this probability is:
\(P(L|PGA=a)\) is the conditional probability that a site will fail based on the PGA and the LSC. \(P_{ml}\) is the fraction of the total mapped area that will experience liquefaction if \(P(L|PGA=a)\) reaches 1. These terms both have LSC-specific coefficients; these are shown in Table 1.
\(K_m\) is a magnitude-correction factor that scales \(P(L)\) for earthquake magnitudes other than M=7.5, potentially to account for the duration of shaking (longer shaking increases liquefaction probability). \(K_w\) is a groundwater depth correction factor (shallower groundwater increases liquefaction probability).
LSC |
PGA min |
PGA slope |
PGA int |
\(P_{ml}\) |
|---|---|---|---|---|
very high |
0.09 |
9.09 |
0.82 |
0.25 |
high |
0.12 |
7.67 |
0.92 |
0.2 |
med |
0.15 |
6.67 |
1.0 |
0.1 |
low |
0.21 |
5.57 |
1.18 |
0.05 |
very low |
0.26 |
4.16 |
1.08 |
0.02 |
none |
\(\infty\) |
0.0 |
0.0 |
0.0 |
Table 1: Liquefaction values for different liquefaction susceptibility categories (LSC). PGA min is the minimum ground acceleration required to initiate liquefaction. PGA slope is the slope of the liquefaction probability curve as a function of PGA, and PGA int is the y-intercept of that curve. \(P_{ml}\) is the Map Area Proportion, which gives the area of liquefaction within each map unit conditional on liquefaction occurring in the map unit.
Zhu et al (2015)
The model by Zhu et al. (2015) is a logistic regression model requiring specification of the Vs30, the Compound Topographic Index (CTI), a proxy for soil wetness or groundwater depth, the PGA experienced at a site, and the magnitude of the causative earthquake.
The model is quite simple. An explanatory variable \(X\) is calculated as:
and the final probability is the logistic function
The term \(PGA_{M,SM}\) is the PGA times a nonlinear scaling factor for the magnitude.
Both the CTI and the Vs30 may be derived from digital elevation data. The Vs30 may be estimated from the topographic slope through the equations of Wald and Allen (2007), which uses a very low resolution DEM compared to modern offerings. As topographic slope tends to increase with increased DEM resolution, the estimated Vs30 does too; therefore a low-resolution DEM (i.e., a 1 km resolution) must be used to calculate Vs30, rather than the 30 m DEM that is the current standard. This results in a more accurate Vs30 for a given slope measurement, but it also means that in an urban setting, sub-km-scale variations in slope are not accounted for.
The CTI (Moore et al., 1991) is a proxy for soil wetness that relates the topographic slope of a point to the upstream drainage area of that point, through the relation
where \(d_a\) is the upstream drainage area per unit width through the flow direction (i.e. relating to the DEM resolution). It was developed for hillslopes, and is not meaningful in certain very flat areas such as the valley floors of major low-gradient rivers, where the upstream drainage areas are very large. Unfortunately, this is exactly where liquefaction is most expected away from coastal settings.