June 29, 2010
One of the primary drawbacks to a Monte Carlo analysis is the large number of model runs required in order to determine the relative confidence of each input variable on impacting the factor of safety. The Alternate Point Estimation Method (APEM) allows a dramatic reduction in the number of model runs required to do a statistical analysis. The following table indicates the number of model runs recommended for the Monte Carlo analysis based on an 85% confidence expectation as related to the number of input variables.
|Input Variables||Number of Model Runs|
For a Monte Carlo analysis, it can be seen that when there are greater than 3 input variables it is problematic to apply the methodology with reasonable confidence.
APEM can calculate probability with a function of 2N + 1 model runs where N is the number of model input variables. Therefore, the methodology has significant application in settings where there are a number of input variables in which there is significant variance (the situation in many geotechnical projects).
The APEM method is a First Order Second Moment (FOSM) type of analysis and its use has been also documented in USACE slope stability papers. The use of FOSM methods is therefore not new, but the specific implementation of the APEM method in SVSlope® is unique in the industry. Its use as a reasonable statistical methodology has been confirmed through a peer-reviewed process on the Molycorp Questa Rock Pile Study.
The APEM analysis allows the results to be presented in terms of the probability of failure, Pf, and reliability index. It is also possible to display the results in terms of a tornado diagram, shown below, which shows the relative influence of each input variable. Therefore, the user can not only determine the overall probability failure but can determine the input variable which is most critical in influencing the factor of safety. This analysis can then be used as the basis for a risk analysis in which the project manager may decide to obtain more samples to further quantify the desired input variable.
Tornado Diagram from an APEM Analysis
In the Molycorp project, the method was successfully applied to calculate stability probabilities with up to 12 varying input fields. This methodology proved successful and reasonable in a peer-reviewed analysis setting.
The application of this theory is relevant for any slope stability analysis where there are a significant number of regions which have variance in the model input parameters (i.e., cohesion and/or friction angles). The methodology can be applied to calculate probabilities of failure. It can also be applied to determine the relative influences of input variables on the resulting factor of safety.
It should be noted that a complete theoretical description of all the probability methods may be found in the theory manual. The theory manual is provided with fully licensed versions of the SVSlope® software.