Probabilistic Block Research Articles

Probabilistic Block Term Decomposition for the Modelling of Higher-order Arrays

Probabilistic Block Term Decomposition for the Modelling of Higher-order Arrays

A Probabilistic Block Economic Value Calculation Method for Use in Stope Designs under Uncertainty

Uncertainty is intrinsic to mine design and planning and introduces risk into the process. Nonetheless, most mine design and planning processes have historically been undertaken as deterministic processes, often resulting in unrealistic mine designs and plans which potentially lead to the destruction of shareholder value. This paper presents a probabilistic block economic value (BEV) calculation approach to minimise the shortcoming of using deterministic BEVs, and evaluates the impact of uncertainty on stope designs. The probabilistic BEV calculation approach was applied to a synthetic geological block model of a gold mineral deposit. The uncertainty associated with BEV input parameters was simulated using Monte Carlo simulation to create equally probable economic orebody models which were then used to create stope designs at different levels of risk. The probabilistic approach generated 20% to 53% higher net present values (NPVs) compared to the deterministic approach within 30% to 70% probability range. This indicates that, for the case study deposit, blocks with approximately 30% to 70% probability of having positive BEVs are the ones that should be used for mine design and planning. The results demonstrate that incorporating uncertainty early in underground mine design and planning potentially creates higher-value stopes.

Spatial Modelling of Concentration in Topsoil Using Random and Systematic Uncertainty Components: Comparison against Established Techniques

The assessment of contaminated land often requires the collection and analysis of soil samples and the discrete or continuous modeling of the contamination. The continuous modeling involves interpolating analyte concentrations between sampled positions. The contamination can be expressed either as a map showing the spatial distribution across the site, or as a frequency distribution, both of which can be compared with a threshold. The characterization of land contamination is affected by random and systematic uncertainty components of the sampling, chemical analysis, sampling positioning and modeling. This work describes the application and comparison of three techniques for developing continuous models of the contamination of a site supported by the quantification of none, some or all relevant uncertainty components. The 7.3 ha site was characterized using 100 soil samples and the lead contamination modeled by “inverse distance weighting” (IDW), “ordinary kriging”(OK) and a new Monte Carlo simulation method (MCM). The IDW only uses the positions and concentrations of the samples without their uncertainty. The OK also requires the “measurement error” and other parameters to select the variogram. The MCM uses the measurement uncertainty (including random and systematic effects arising from both sampling and analysis) and ‘GPS coordinates uncertainty’. The measurements of Pb concentration across the site were log-normally distributed, and therefore log-transformed prior to modeling. The output of the models was compared against a discrete model by ‘probabilistic block mapping’ (PBM) that also considers measurement uncertainty. The OK produced a smoothed spatial variation of Pb concentration that appears more realistic. However, IDW and OK underestimate the land contamination, while MCM prediction most closely matches that of the measured concentrations including the impact of uncertainty components. The MCM allowed a metrologically sound modeling of the contamination and the linear interpolation of data reduced assessment subjectivity. Both MCM and PBM made realistic estimates of the proportion of the site that was contaminated over the threshold and included all relevant uncertainty components in the continuous or discrete modeling, respectively. The PBM identifies areas where the true value of the contaminant concentration could exceed the threshold value, even though the single measured value did not.

Probabilistic Block Cipher for distributed systems

In this paper we would like to present a fully probabilistic symmetric block ciphering algorithm that can work on a distributed computer system. This fact allows to multiply the amount of processed information, so the algorithm becomes applicable in BigData.

A chaos-based probabilistic block cipher for image encryption

Traditional encryption is based on secrecy provided by secret-key. But this leads to generation of same cipher text when the encryption scheme is applied to same plaintext with same key. Thus, replay of messages can be effortlessly identified by an adversary which can be a weak link in any communication. Probabilistic encryption is an approach to overcome this weakness where different cipher texts are generated each time same plaintext is encrypted using the same key.Extending the probabilistic approach, which is generally employed in asymmetric encryption, this paper proposes a new chaos-based probabilistic symmetric encryption scheme with customizable block-size suitable for image encryption. It employs a Random Bits Insertion phase followed by four rounds of two-staged diffusion involving simple XOR (exclusive-OR) operation making it computationally efficient. Random Bits Insertion makes the scheme probabilistic. This phase also helps in increasing entropy and making intensity distribution more uniform in cipher. The generated cipher text is twice the size of plain text. An increase in cipher text space is inevitable for probabilistic encryption and it provides an advantage as the apparent message space for the attacker is increased. The observations show that the scheme offers high strength to resist statistical and cryptanalytic attacks.

Chaotic particle swarm optimization in optimal active control of shear buildings

The applications of active control is being more popular nowadays. Several control algorithms have been developed to determine optimum control force. In this paper, a Chaotic Particle Swarm Optimization (CPSO) technique, based on Logistic map, is used to compute the optimum control force of active tendon system. A chaotic exploration is used to search the solution space for optimum control force. The response control of Multi-Degree of Freedom (MDOF) shear buildings, equipped with active tendons, is introduced as an optimization problem, based on Instantaneous Optimal Active Control algorithm. Three MDOFs are simulated in this paper. Two examples out of three, which have been previously controlled using Lattice type Probabilistic Neural Network (LPNN) and Block Pulse Functions (BPFs), are taken from prior works in order to compare the efficiency of the current method. In the present study, a maximum allowable value of control force is added to the original problem. Later, a twenty-story shear building, as the third and more realistic example, is considered and controlled. Besides, the required Central Processing Unit (CPU) time of CPSO control algorithm is investigated. Although the CPU time of LPNN and BPFs methods of prior works is not available, the results show that a full state measurement is necessary, especially when there are more than three control devices. The results show that CPSO algorithm has a good performance, especially in the presence of the cut-off limit of tendon force; therefore, can widely be used in the field of optimum active control of actual buildings.

Development of a probabilistic block theory analysis procedure and its application to a rock slope at a hydropower station in China

The paper deals with a development of a new procedure to perform probabilistic block theory analysis using the vector method. The procedure includes the following three main features: (a) the joint orientation is considered as a bivariate distribution including the correlation that exists between the joint dip angle and dip direction angle; (b) the variability of discontinuity shear strength is incorporated in the analysis; (c) it performs deterministic and probabilistic block theory analysis including the mode of failure (MOF), maximum safe slope angle (MSSA), factor of safety (FS) and the instability probabilities for different cut slope dip angles (CSDA) and different MOFs. A rock slope at a hydropower station in China is used as a case study to illustrate the use of the developed procedure. The slope has six sets of discontinuities. Therefore, both deterministic and probabilistic block theory analyses were conducted for 4-plane, 5-plane, 6-plane and 7-plane blocks to study the stability of the rock slope under different MOFs and CSDA. For most MOFs, the MSSA obtained through deterministic analysis were found to be significantly higher than the CSDA obtained corresponding to zero instability cumulative probability. The differences obtained between the deterministic and probabilistic analyses resulted mainly due to the variability of discontinuity orientation. The case study illustrated the importance and superiority of the probabilistic analysis over the deterministic analysis. For the investigated rock slope by selecting the CSDA of 40° and 50°, the total instability cumulative probability can be limited to less than 5% and 10%, respectively.

Probabilistic block theory analysis for a rock slope at an open pit mine in USA

A new formulation is given to conduct a probabilistic block theory analysis. A new computer code (PBTAC) is developed to perform both deterministic and probabilistic block theory analysis. The variability of the discontinuity orientation and shear strength is incorporated in the probabilistic block theory analysis. Discontinuity orientation is treated as a bivariate random variable including the correlation that exists between the dip angle and dip direction. PBTAC code was applied to perform both deterministic and probabilistic block theory analyses for a part of an open pit mine in USA. Needed geological and geotechnical data for the analyses were obtained from field and laboratory investigations. The variability of the discontinuity orientations resulted in important differences between the probabilistic and deterministic block theory analyses results. The results confirmed that the design value selected for the maximum safe slope angle (MSSA) for a particular region in the open pit mine based on the deterministic block theory analysis can be on the unsafe side. In summary, the results showed clearly the superiority of probabilistic block theory analysis over the deterministic block theory analysis in obtaining additional important information with respect to designing rock slopes. The calculated values agree very well with the existing almost stable bench face angles reported by the mining company.

Improving Rebound Models in 3D Rockfall Simulation Codes Used for the Design of Protection Embankments

Rockfall propagation simulation models are widely used for assessing rockfall hazards as well as for the design of rockfall protection structures. This research work investigates the relevance of rockfall propagation models to assess rockfall hazard in the vicinity of embankments. In this article, particular focus is placed on the development of engineering oriented rebound models taking into account the shape of the block. Two different block impact modelling approaches are considered and compared. Small scale experiments involving blocks with different shapes impacting a small-scale embankment were conducted to provide data for calibrating the models. The trajectory of the blocks was tracked using a high speed camera (1000 fps) and an image processing algorithm was developed to extract the experimental trajectories. The two different approaches considered for modelling the impact of the block in the embankment vicinity were a probabilistic block impact model and a deterministic approach accounting for the block shape. In this latter case, the impact was calculated using an elasto-plastic contact model between the block and the slope surface. Both approaches succeed in simulating experimental results. Nevertheless the probabilistic model is limited by the introduction of the block shape parameter in the calculation while the deterministic approach seems to be limited in terms of computational efficiency.

Probabilistic key block analysis of a mine ventilation shaft stability – a case study

In this study, the probabilistic key block analysis was applied to evaluate the stability of a mine ventilation shaft developed in a rock mass of granite. The key blocks were identified based on the block theory. The variations of discontinuity orientations were fitted with the Beta distribution and taken into consideration. The key block forming probabilities were analyzed. For simplification of calculations the first-order second-moment (FOSM) approximation was employed for probability estimation. With the considerations of the rock properties as random variables and applications of several statistical analysis tools, the key block failure probabilities, the probabilistic distribution of safety factors, and the probabilistic distribution of potential maximum key block volumes were analyzed. The analysis indicated that although the safety factor calculated based on the mean values of the variables was above 1.0 for the stability of the most critical key block, the block had a considerable probability of failure with a significant rock volume due to variations in discontinuity orientations and rock properties. Without promptly applying supports to the rock excavation, the shaft had a significant likelihood of failure.

Improvement of methodology for block identification using mesh gridding technique

This paper revisits a methodology for block identification in rock masses using a mesh gridding technique recently published by the authors (Zhang et al., 2010. A new methodology for block identification and its application in a large scale underground cavern complex. Tunnelling and Underground Space Technology 25, 168–180). An extension to the originally proposed methodology is introduced in this paper. The improvement involves consideration of the finite nature of the geological fractures when defined in a mesh grid. In accordance with this consideration, modifications to the block identification algorithm are made, and as a result, the number of nodes in the reconstructed model can be decreased considerably. This, in turn, allows the inclusion of a significant larger number of geological fractures into the model and the implementation of probabilistic block analyses schemes for a large number of stochastic fractures in a model. The improved methodology is applied to the block identification in other projects, including an underground powerhouse at Three Gorges Project in China and a hydropower plant in Zambia. Results with the proposed extended methodology are compared with those obtained with other techniques, for example, a computer code (GeneralBlock) and also with field observations of unstable blocks at the site. Results (location, geometry and degree of stability of blocks) with the different methods are seen to agree, suggesting that the extended proposed methodology can be applied to get valuable results in the process of identifying unstable blocks in rock masses around engineering structures.

Stability analysis of tunnel keyblock: a case study

The probabilistic block theory which was suggested by Hatzor and Goodman (1992) was applied to an telecommunication tunnel by taking into consideration the individual joints obtained from face mapping during tunnel construction. Using the actual unrolled joint trace that was developed in the Banpo telecommunication tunnel during tunnelling, a statistical analysis of observed discontinuity data was performed. The deterministic analysis and probabilistic keyblock concept were applied and their results were compared with the observed failure modes. The result obtained from the deterministic analysis resulted in a large difference compared with the observed failure pattern; the analysis method which considers the joint combination probability gave more similar results to the observed data than the deterministic analysis. Furthemore, individual keyblock analysis was performed by means of the newly developed program. While the positions of the keyblocks were predicted properly by the individual keyblock analysis, the predicted sizes of the keyblocks were found to be different from the observed data. It was acknowledged from the individual keyblock analysis that most keyblocks are composed of steeply inclined joints in the crown or in the side wall fo a tunnel. But, in the fault-zone, any method based upon the block theory could not predict the failure pattern properly. It can be concluded that the continuous careful investigations of geological conditions were very important in tunnelling through a rock mass that experiences mainly structure-induced failures.