Reply to the discussion by J . M . Duncan , M . Navin , and T . F . Wolff on “ Probabilistic slope stability analysis for practice ” 1

Since the performance of earth slopes is controlled, in most cases, by the average material properties and pore pressures over the area of the slip surface, it is logical to consider the variability of the average values (rather than the variability between discrete points) in any probabilistic slope assessment. As demonstrated in the paper (El-Ramly et al. 2002), the variance of the spatial average of a property over an area or volume can be significantly less than that of the measured data. While this concept has been recognized for some time (Anderson et al. 1984; Li and Lumb 1987; Baecher 1987), for simplicity, several studies have used the variance of measured data without any reduction. The underlying assumption is that the operational shear strength, for example, controlling the stability of the slope is constant at all locations within the domain of the problem and can assume any value within the measured minimum–maximum range. There are two main consequences of this assumption. First, the probability of unsatisfactory performance can be grossly overestimated and second, the probability of unsatisfactory performance becomes detached from the scale at which failure takes place. To illustrate the above points, El-Ramly (2001) analysed a hypothetical slope configuration, Fig. D1, using soil data from an actual failure of the cut slope behind housing block No. 36 of the Shek Kip Mei Estate in Hong Kong (FMSW 2000). Three slip surfaces, corresponding to failure of the entire slope (slip surface 1), failure of the upper 2/3 of the slope (slip surface 2), and failure of the top 1/4 of slope height (slip surface 3), were analysed probabilistically. The analyses were performed using our probabilistic slope analysis methodology, as well as the simplified probabilistic approach with no variance reduction due to spatial averaging. The Spencer method of slices (Spencer 1967) was used in the stability calculations. The results of the analyses are summarized Table D1. As in the James Bay dyke study, the simplified analysis overestimates the probability of unsatisfactory performance. The difference in the estimated probabilities arises from ignoring the reduction in the variances of the input variables due to spatial averaging in the simplified analysis. For slip surface 3, the spatial averaging domain (the slip surface) is comparable in size to the assumed scale of fluctuation (δ ≈ 2ro = 10 m), and no variance reduction was introduced. Hence, the estimated probabilities of unsatisfactory performance were identical. For slip surface 1, the averaging domain is significantly larger than the scale of fluctuation, and the reduction in the variances of the input variables is substantial. The probability of unsatisfactory performance from the simplified analysis is 38 times that of the analysis accounting for spatial averaging. Such a large difference should have a significant impact on any decision making. For the analyses based on our methodology, the probabilities of unsatisfactory performance of larger slip surfaces are lower than those of the smaller surfaces, even though the latter have high factors of safety. This might seem unusual from a conventional point of view, but not from a probabilistic perspective. Because of the spatial variability of soil properties, encountering a sufficiently low strength to induce failure in localized areas of the slope is more likely than such an encounter over the entire slope height. Similarly, the probability of occurrence of small local zones of high pore pressure is much higher than that of large zones. In other words, the uncertainty in the average properties and pore pressure along the slip surface, and consequently the probability of unsatisfactory performance, are much higher for a small failure than for a large failure. This is shown clearly in the database of landslide incidents in Hong Kong, Table D2. The frequency of occurrence of minor failures is almost