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# Statistics

Probabilistic Analysis can be applied to RocSlope models to apply statistics to the following:

• Material properties
• Joint properties
• Synthetic Joint definitions

To perform a Probabilistic Analysis in RocSlope:

1. Select Analysis > Project Settings in the menu and go to the Statistics tab. Tick the checkbox to turn on Probabilistic Analysis.
2. Add random variables to Material Properties (Statistics > Define Material Statistics or by selecting the Statistics button in the Define Materials dialog) and/or Joint Properties (Statistics > Define Material Statistics or by selecting the Statistics button in the Define Joint Properties dialog).

## Sampling Method

The Sampling Method determines how the statistical input distributions will be sampled for the random variables you have defined for a Probabilistic Analysis. Two Sampling Methods are available in RocSlope:

• Monte Carlo
• Latin Hypercube
For both sampling methods, sequences of random numbers are utilized to generate the random samples. The generation of random numbers in RocSlope is set as Pseudo-Random with a constant seed value (this ensures that runs with identical statistical parameters are sampled the same way for every run)

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 behaviour in geotechnical engineering.

LATIN HYPERCUBE METHOD

The Latin Hypercube sampling technique gives comparable results to the Monte Carlo technique, but with fewer samples [Iman et.al. (1980), Startzman et.al. (1985)]. 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 [Hoek et.al. (1995)].

## Number of Samples

The Number of Samples determines the number of samples which will be generated for each random variable, for a Probabilistic Analysis. For example, if Number of Samples = 1000, then 1000 values of each input data random variable (e.g. Friction Angle of Joint Property 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 set of factors of safety is calculated for each set of input data samples. This results in a distribution of factors of safety for each block, from which the Probability of Failure (PF) is calculated.

## Synthetic Joint Sampling

Statistics can be applied to Synthetic Joint Set Sampling by setting Dip, Dip Direction, Radius, and/or Spacing as random variables from the Define Synthetic Joint Properties dialog. Each Synthetic Joint from a Synthetic Joint Set is only sampled once to determine the orientation, size and location of that Joint. The number of samples is equal to the number of Synthetic Joints in a Synthetic Joint Set, which depends on the Spacing and how many Joints can fit on the Traverse, and not the Number of Samples set in Project Settings.

Once the Synthetic Joint Sets are created, Compute Block runs only once to determine the Deterministic geometries of the blocks formed (regardless whether Probabilistic Analysis is ON or OFF).

## Material Properties and Joint Properties Sampling

Any Material Property or Joint Property parameters which are modeled as random variables are sampled a number of times equal to the Number of Samples set in Project Settings. For each sample, Compute Kinematics is run to determine the set of kinematic results for all blocks. For example, if 50 valid blocks are formed after Compute Blocks and Number of Samples = 1000, then a total of 50,000 blocks are analyzed with Compute Kinematics.