# Probabilistic Analysis

In *Slide2*, you can perform a Probabilistic slope stability analysis to determine the effect of uncertainty or variability of input parameters, on the results of the slope stability analysis.

For an overview of the Probabilistic Analysis procedure in *Slide2*, see the Probabilistic Analysis Overview topic.

The Probabilistic Analysis options in the Project Settings dialog, are described below.

## Probabilistic Analysis

To enable a Probabilistic Analysis with *Slide2*, the first thing you must do is select the **Probabilistic Analysis** checkbox on the **Statistics** page of the Project Settings dialog. When you select the Probabilistic Analysis checkbox:

- The Probabilistic Analysis options will be enabled in the Project Settings dialog (see below).
- A Statistics menu will be available in the
*Slide2*main menu. The Statistics menu contains all of the options which allow you to define the input data random variables for your probabilistic analysis (e.g. material properties, support properties, loading properties etc).

### SAMPLING METHOD

The **Sampling Method** determines how the statistical input distributions for the random variables you have defined for a Probabilistic Analysis, will be sampled. Two Sampling Methods are available in *Slide2* – Monte Carlo or Latin Hypercube sampling.

### MONTE CARLO METHOD

The **Monte Carlo** sampling technique uses random numbers to sample from the input data probability distributions. Monte Carlo techniques are commonly applied to a wide variety of problems involving random behavior, in geotechnical engineering.

Monte Carlo sampling of Normal distribution (1000 samples)

### LATIN HYPERCUBE METHOD

The **Latin Hypercube** sampling technique gives comparable results to the Monte Carlo technique, but with fewer samples. The method is based upon "stratified" sampling with random selection within each stratum. This results in a smoother sampling of the probability distributions. Typically, an analysis using 1000 samples obtained by the Latin Hypercube technique will produce comparable results to an analysis of 5000 samples using the Monte Carlo method.

Latin Hypercube sampling of Normal distribution (1000 samples)

**RESPONSE SURFACE**

The Response Surface sampling method uses a small number of strategically selected computations to create a response surface of factor of safety (FS) values for various combinations of input parameters. It then *predicts* the factor of safety values for any combination of samples and provides an estimated probability of failure. Since an Overall Slope probabilistic analysis can take extensive hours in 2D, this method is advantageous in significantly cutting down computation time.

Although many verification examples have proven it to agree well with Latin-Hypercube results, it cannot always guarantee a result that is identical. However, it will be able to give you a ballpark PF value. It is always recommended to run at least one Latin-Hypercube Overall Slope analysis overnight.

A probabilistic analysis with response surface is set up in the same way as any other probabilistic analysis:

Important Notes:

- Response Surface cannot be used in conjunction with a beta distributed input variable.
- The required initial computations used by response surface are a function of the number of random variables defined in the model. If the user's defined number of samples is smaller than the number of samples needed by Response Surface, the user's number of samples is computed with Latin-Hybercube only.
- After a Response Surface Overall Slope computation, the “Show All Surfaces” button will only show the surfaces for the samples that were computed in order to train the model.

## Stochastic Response Surface Theory

How it works:

1. Convert all random variables to standard normal random variables, N(0,1).

2. Represent resulting FS in polynomial chaos expansion form. In Slide2, a 3^{rd} order Hermite expansion is used.

3. Use a small number of computations to determine the coefficients of the polynomial in Step 2.

4. Generate Latin-Hypercube samples and plug them into the polynomial to estimate FS.

### STEP 1

The initial random variables are converted to standard normal random variables using transformation equations (Li et al., 2011).

### STEP 2

The Hermite polynomial expansion looks like this:

In the above, *F* is the factor of safety and *U _{i}* is the particular combination of standard normal random variables in a simulation. The coefficients

*a*must be determined.

In Slide2, a 3^{rd} order Hermite expansion is used.

### STEP 3

Let n=number of random variables defined, and N*=*number of computations required. The number of computations required is determined as follows:

If 1 random variable is defined (n=1), N=15 simulations are computed.

If n random variables are defined, the number of simulations computed (N) is:

If the N determined from the equation above is larger than the number of samples (S) defined by the user in the Statistics section of the Project Settings dialog, then only the user-defined number of samples are computed using the Latin-Hypercube sampling method.

These N computations are generated using Latin-Hypercube sampling (Choi et al., 2004) to ensure that the solution space is well-covered. They are then used to determine the polynomial coefficients associated with each variable or variable combination.

### STEP 4

The number of samples defined by the user (S) in the Statistics section of the Project Settings dialog, is generated using Latin-Hypercube sampling method. The samples are matched with the corresponding coefficients and the resulting FS is predicted. The PF is estimated from these predicted FS values.

**References for response surface:**

Choi, S. K., Grandhi, R. V., Canfield, R. A., & Pettit, C. L. (2004). Polynomial chaos expansion with latin hypercube sampling for estimating response variability. *AIAA journal*, *42*(6), 1191-1198.

Isukapalli, S. S. (1999). Uncertainty analysis of transport-transformation models.

Li, D., Chen, Y., Lu, W., & Zhou, C. (2011). Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables. *Computers and Geotechnics*, *38*(1), 58-68.

## Number of Samples

The **Number of Samples** which will be generated for each random variable, for the Probabilistic Analysis. For example, if Number of Samples = 1000, then 1000 values of each random variable (for example, Cohesion of Material 1) will be generated, according to the Sampling Method and statistical distribution for each random variable. The analysis will then be run 1000 times, and a safety factor calculated for each sample. This results in a distribution of safety factors, from which the Probability of Failure is calculated.

**TIP**: how many samples is "enough" ? You can get a good idea of how many samples is appropriate for your analysis, by viewing a Samples Convergence Plot in the *Slide2* Interpret program (select Statistics > Convergence Plot in the *Slide2* Interpret program). This plot will indicate the minimum number of samples which is necessary to converge to a final answer (i.e. mean safety factor, probability of failure).

## Spatial Variability Analysis

This option allows you to define spatial variability of material properties. See the Spatial Variability topic.

## Probabilistic Analysis Type

There are two different methods of performing a Probabilistic Analysis with *Slide2*:

- the
**Global Minimum**method - the
**Overall Slope**method.

These are described below.

### GLOBAL MINIMUM METHOD

The **Global Minimum** Probabilistic Analysis Type in *Slide2*, is commonly used in slope stability analysis. With this method:

- The regular (DETERMINISTIC) slope stability analysis is first carried out, in order to determine the slip surface with the overall (Global Minimum) factor of safety, of all the slip surfaces analyzed.
- The probabilistic analysis is then carried out on the Global Minimum slip surface, using the samples generated for each random variable. This means that the slope stability calculation is repeated N times (where N = Number of Samples) for the Global Minimum Slip surface. This results in N calculated safety factors. The Probability of Failure is then simply the number of analyses which result in a safety factor less than 1, divided by the total Number of Samples.
- NOTE: each Analysis Method in
*Slide2*(e.g. Bishop, Janbu etc) can result in a different Global Minimum slip surface. The Probabilistic Analysis is carried out independently on each Global Minimum slip surface which may result from each Analysis Method.

The **Global Minimum** Probabilistic Analysis Type, assumes that the Probability of Failure calculated for the (Deterministic) Global Minimum slip surface, is representative of the Probability of Failure for the entire slope. In many cases, this may be a valid or reasonable assumption. An alternative method which does not rely on this assumption (the Overall Slope Method) is described below.

NOTE: if you are using the Spatial Variability Analysis option, then the Global Minimum analysis type is NOT available; you MUST use the Overall Slope analysis type.

### OVERALL SLOPE METHOD

The **Overall Slope** Probabilistic Analysis Type in *Slide2* represents a different approach to the probabilistic analysis of slope stability.

- With the Overall Slope method in
*Slide2*, the ENTIRE SEARCH for a Global Minimum slip surface, is repeated N times (where N = Number of Samples). For each search iteration, a new set of random variable samples is first loaded, and the search is carried out. - A Global Minimum slip surface, FOR EACH SEARCH iteration, is then determined. This will generally result in the location of SEVERAL different Global Minimum slip surfaces, corresponding to different values of the sampled input data random variables.
- A typical Overall Slope probabilistic analysis, might result in the location of 10 to 50 different Global Minimum slip surfaces. However, the actual number will depend on your model, search method, and the distributions of your random variables.
- The definition of the Probability of Failure, for the
**Overall Slope**method, is the same as for the**Global Minimum**method. That is, the Probability of Failure is the number of analyses which result in a safety factor less than 1, divided by the total Number of Samples.

The advantage of the Overall Slope method, compared to the Global Minimum method, is that the Overall Slope method does NOT assume that the Probability of Failure for the slope, is equal to the Probability of Failure of the Deterministic Global Minimum slip surface.

Instead, the ENTIRE SEARCH is repeated, using different values (samples) of the input data random variables, for each search iteration. This is perhaps a more rational approach to probabilistic slope stability analysis, since it does not assume a fixed location of the Global Minimum slip surface. This approach would not have been feasible only a few years ago, due to the extensive computation time which would have been involved. With the current speed of the latest personal computers, such an analysis is now practical.

However, it must be pointed out, that the Overall Slope method will involve a substantially greater computation time, than the Global Minimum method. Depending on the Number of Samples, and the complexity of your model, the Overall Slope Probabilistic Analysis in *Slide2*, may take SEVERAL HOURS to complete. In general, you may wish to run an Overall Slope probabilistic analysis, at the end of a day, as an overnight run. Remember that the *Slide2* Compute Engine can run multiple files in succession, so you can set up several files for a Overall Slope Probabilistic Analysis, and run the analyses overnight.

If you have selected multiple analysis methods (e.g. Bishop, Janbu etc) the Overall Slope Probabilistic Analysis is carried out independently for each Analysis Method.

NOTE: if you are using the Spatial Variability Analysis option, then you MUST use the Overall Slope analysis type.

## Critical Probabilistic Slip Surface

In addition to the Overall Slope reliability, the **Critical Probabilistic Slip Surface** is also calculated, when the Probabilistic Analysis Type = Overall Slope. The Critical Probabilistic Surface is the individual slip surface which has the maximum Probability of Failure (and also the Minimum Reliability Index). This slip surface WILL NOT NECESSARILY BE THE SAME AS THE CRITICAL DETERMINISTIC SLIP SURFACE. See the Critical Probabilistic Surface topic for more information.