Coefficient variability

For both events the sign attached to each of the coefficients (i.e.: whether the coefficient is positive or negative and thus whether the variable has the effect of increasing or decreasing landslide probability) is the same. The values of coefficients vary between the two event models, as shown in Figure 4-22. Since the effect of individual predictors is co-dependent on the effect of all other predictors, it is difficult to attribute variability of individual coefficients to particular factors. This variability is

124 in part due to differences in the overall magnitude of the earthquakes. The higher intercept value for 1929 indicates the higher overall failure probability produced by the higher magnitude earthquake and the larger negative 𝐹𝑃𝐷 coefficient indicates a more gradual decrease in probability per unit distance from the seismic source. Larger positive 1929 coefficients for 𝑆𝑅, 𝐸𝑆 and 𝐷𝑆 are also likely to result from the higher overall failure probability for the 1929 event. Coefficients for 𝑁𝐷𝑆 are similar, suggesting that the increase in failure probability due to topographic amplification is reasonably consistent. The larger negative 𝐹𝐷𝑃 ∙ 𝑁𝐷𝑆 coefficient for 1968 suggests that the strength of the topographic amplification effect decreases more quickly with distance in the lower magnitude 1968 earthquake. 𝑆𝐿 coefficients are similar for both events, although the relative strength of the coefficient for Intrusives unit is slightly lower for 1929 than 1968.

125

Figure 4-22 Comparison plots of predictor coefficients for the 1929 and 1968 probability models. Note that the axes have different scales as the coefficients are not normalised and vary over a large scale between predictor variables

126 4.2.7 Predicted and observed probability comparison

Figure 4-23 shows a comparison of predicted and observed probability calculated across 100 equal frequency (1%) bins. These values have been plotted on logit (log-odds) scales, reflecting the mechanics of logistic regression: fitting linear correlation in log-odds space. At high values both models exhibit a close fit to the line of equality. At low values the 1929 model produces slightly over-predicted probabilities, while the 1968 model exhibits departure from equality with under-predicted probabilities. It is likely that these errors at the lower limit of the distribution of probabilities are at least in part statistical artefacts of low data frequency and near-zero probability values.

Figure 4-24 and Figure 4-25 show predicted probability values projected spatially, alongside the landslide source binary grids used to fit the models. The spatial projection of predicted failure probability closely reflects the spatial distributions of landslide source areas. For the 1929 earthquake, some of the large coastal landslides in the northwest of the study area occur outside of the region of higher failure probabilities. This may be due to the influence of coastal erosion on hillslope stability in these areas, which is not accounted for in the model.

127

Figure 4-23 Plotted comparison of predicted and observed probability for the 1929 Buller and 1968 Inangahua earthquakes, sampled within 100 equal frequency (1%) bins widths. The dotted line indicates the line of equality. The data are plotted on logit (log-odds) scales.

128

Figure 4-24 Map projected predicted probabilities (A) and observed landslide source locations (B) for the 1929 earthquake.

129

Figure 4-25 Map projected predicted probabilities (A) and observed landslide source locations (B) for the 1968 earthquake.

130 4.2.8 Model fit breakdown

Each predictor variable accounts for some part of the improvement in fit, although not all variables contribute equally. It is important to establish the relative contribution of different predictor variables to the fit of the models and thus the relative importance of different factors in determining hillslope failure probability. In order to achieve this, beginning with the full models presented above, the variable contributing the least increase in the R² value was repeatedly removed from the model and the model refitted. This process was carried out for both the 1929 and 1968 models and the R² values were recorded for each refinement. Figure 4-26 shows the ranked order of variable combinations and evolving values of R² resulting from this experiment. For both events, 𝐹𝑃𝐷, 𝑆𝐿 and 𝐺𝑇 are primary variables accounting for the majority of the model fit. Whilst the rank order of 𝐹𝑃𝐷 ∙ 𝑁𝐷𝑆, 𝐷𝑆, 𝑆𝑅, 𝑁𝐷𝑆 and 𝐸𝑆 differs, all these variables appear to be secondary in defining the spatial distribution of 𝑃 (𝐴). For 1929, the initial increase in R² produced by 𝐹𝑃𝐷 is much lower than in 1968. This suggests that less variability is accounted for by 𝐹𝑃𝐷 in 1929. Given the large scale and less well-constrained rupture in this event, the lower level of fit may be due to the effect of unresolved rupture complexity.

131

Figure 4-26 Plots showing the relative contribution of predictor variables to the fit of the 1929 and 1968 hillslope failure probability models. 1929 (top) and 1968 (bottom) model fit breakdown: sequence of model input predictors and resulting pseudo-R² goodness of fit values, produced by sequetially removing the least contributing predictor variable.

Maximum R² of full model

Maximum R² of full model

132 4.2.9 Discussion of model components and implications

The spatial probability distributions of landslides triggered by the 1929 and 1968 earthquakes can both be described using the logistic regression model given in Equation 4-8. The major influences upon the model behaviour are the distance of sites from the seismic source and the local hillslope gradient, where the underlying influence of hillslope gradient is dependent on geology. Secondary are the effects of topographic amplification, solar radiation, hillslope structural domains and local relief, which modify failure probability to a lesser extent. The influence of the model variables corresponds with that expected, based on their physical relationships to seismic and hillslope processes. The findings from this analysis have a number of similarities with the results of previous investigations but also notable differences.

The negative correlation between landslide probability and distance from the seismic source is commensurate with the attenuation of seismic ground accelerations (Campbell and Bozorgnia, 2008), and fits with the results of other ETL studies (e.g.:

Meunier et al., 2007, Dai et al., 2011, Lee et al., 2008a). While the decision of how to characterise the seismic source is somewhat arbitrary, the use of the fault plane provides the most physically accurate fault characterisation and the best fitting model, without the need to include an additional variable to account for differences between the hanging wall and footwall. The positive correlation between failure probability and 𝑁𝐷𝑆 corresponds with expected valley-scale patterns of topographic amplification and damping. The decay in the amplification effect with distance from the seismic source can be related to both the general lowering of relief and a change in the incidence angle of seismic waves. The change in wave incidence angle with increasing horizontal distance from coseismic faults results in a shift of topographic amplification peaks away from ridge crests (Meunier et al., 2008). The expected result of this is that failures will occur lower on hillslopes, resulting in the reduced influence of 𝑁𝐷𝑆 with increasing fault distance. While the 𝑁𝐷𝑆 variable accounts for the position of cells in the stream-ridge profile, it does not account for the local relief and therefore the severity of the topographic amplification effect (Benites et al., 1994). This corresponds with the greater influence of 𝑁𝐷𝑆 in high relief areas close to the fault, than in lower relief, coastal areas that occur at greater distances from the source.

133 Positive correlation between failure probability and hillslope gradient fits with the associated increase in local shear stresses (Section 2.1.4). A major difference in this model to similar published models of ETL susceptibility (e.g.: Lee et al., 2008a, Xu et al., 2012, Li et al., 2012) is that the influence of geology is not to modify landslide probability directly, but rather by changing the relationship between landslide probability and hillslope gradient. As the topography and lithology of landscapes are not independent, owing to the relationship between the strength of hillslope material and the maximum gradient and relief that hillslopes can sustain (e.g.: Hoek, 2000, Schmidt and Montgomery, 1995), it is logical that the influence of these variables should be coupled. The increase in failure probability with the local relief between drainage divides can be associated with the increase in gravitationally-induced shear stresses with the local height of hillslopes and the associated decrease in rock mass strength at increasing spatial scale (Culmann, 1866, Schmidt and Montgomery, 1995).

Although hillslope aspect is a common variable used in landslide susceptibility models, its influence is generally attributed to a combination of topographic effects on seismic waves, structural influences on hillslope stability and local climatic differences affecting the physical breakdown of hillslope material (Chen et al., 2012b, Dai et al., 2011, Lee et al., 2008a, Lee and Min, 2001). As a result, statistical relationships between aspect and landslide probability will be specific to particular earthquakes and regions. By separating the individual seismic, structural and climatic components that influence the aspect of landslides, model relationships derived here are more likely to be representative of other earthquakes and regions. The positive correlation between solar radiation and hillslope failure probability, and higher failure probability for slopes with out-of-slope dipping structures agree with the expected influence of these variables (Mcfadden et al., 2005, Selby, 2005, Moore et al., 2009, Hoek et al., 2002).

When these effects are controlled for in analysis, the orientation of hillslopes relative to seismic sources, associated with topographic effects on ground accelerations (e.g.:

Meunier et al., 2008), was not found to exhibit a significant influence on landslide probability. This may indicate that, when seismic waves interact with complex surface topography, this site effect may be weaker than suggested by results of seismic waves field experiments using simplified topography (Meunier et al., 2008).

134 Spatially distributed precipitation variables do not exhibit a significant influence on failure probability. This may suggest that rainfall and associated variability in pore-pressures have little influence on ETL activity, which may be true for well-drained sections of hillslope positioned high on mountainsides. As localised pore-pressure generation is highly dependent on surface and sub-surface hydrology (Montgomery et al., 1997, Wilson and Dietrich, 1987, Fannin and Jaakkola, 1999), and rainfall data represent long-term mean conditions rather than data at the time of the earthquakes, these variables may not accurately characterise the influence of rainfall on hillslope stability. In the absence of distributed pore-pressure data with which to compare the landslide distribution, understanding hydrological influences on ETL activity remains an area of uncertainty. Similarly, no significant relationship was observed between landslide probability and the distance of sites from mapped faults. While zones of weakened and damaged material occur around fault zones (e.g.: Mooney et al., 2007, Lockner et al., 2009), local hillslopes may have already adjusted to these conditions.

Mapped faults generally track along valleys, suggesting that erosional processes have already removed material weakened by faulting, which therefore exerts little influence on present day hillslope processes.

The consistency with which the model describes the spatial distribution of ETLs for both events suggests that this combination of underlying relationships can be applied generally to earthquakes in the region. In other words, landslides triggered by earthquakes in this area are sampled from this same spatial distribution of hillslope failure probability. The variables exhibiting a more minor influence in the model have greater potential to describe characteristics of the landslide distribution specific to these events and to this region. By removing these variables and identifying the major influences on failure probability, the model can be made less event-specific and more transferrable between different earthquakes and regions. The combination of the distance of sites from the seismic source and the local hillslope gradient, with influence dependent on geology, therefore provide a candidate variable subset for a generalised ETL probability model.

135