Deterministic generation

Uncertainties assessed at Tier 2 (deterministic) generate alternative point estimates for exposure and may be communicated in various ways, depending on the particular methods used for sensitivity analysis. As a minimum, this should identify which sources of uncertainty have been treated at Tier 2, state and justify the alternative quantitative estimates used for each one (e.g. minimum, maximum and most likely values), present exposure estimates for those combinations of alternative estimates that are considered plausible and state and justify any combinations of estimates that are considered implausible. In addition, it will be useful (especially if upper estimates exceed levels of concern) to show which of the quantified uncertainties have most influence on the outcome. 

A related algorithm can be written also for the Brownian trajectory [10]. However, the essential difference between an algorithm for a Brownian trajectory and equation (4) is that the Brownian algorithm is not deterministic. Due to the existence of the random force, we cannot be satisfied with a single trajectory, even with pre-specified coordinates (and velocities, if relevant). It is necessary to generate an ensemble of trajectories (sampled with different values of the random force) to obtain a complete picture. Instead of working with an ensemble of trajectories we prefer to work with the conditional probability. I.e., we ask what is the probability that a trajectory being at... 

Other methods which are applied to conformational analysis and to generating multiple conformations and which can be regarded as random or stochastic techniques, since they explore the conformational space in a non-deterministic fashion, arc genetic algorithms (GA) [137, 1381 simulation methods, such as molecular dynamics (MD) and Monte Carlo (MC) simulations 1139], as well as simulated annealing [140], All of those approaches and their application to generate ensembles of conformations arc discussed in Chapter II, Section 7.2 in the Handbook. 

An algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

The definitions above are an abbreviated version of those used in a veiy complex and financially significant exercise with the ultimate goal of estimating resei ves and generating production forecasts in the petroleum industry. Deterministic estimates are derived largely from pore volume calculations to determine volumes of either oil nr gas in-place (OIP, GIP). This volume when multiplied by a recovery factor gives a recoverable quantity of oil or natural gas liquids—commonly oil in standard barrels or natural gas in standard cubic feet at surface conditions. Many prefer to use barrels of oil equivalency (BOE) or total hydrocarbons tor the sum of natural gas, natural gas liquids (NGL), and oil. For comparison purposes 6,000 cubic feet of gas is considered to be equivalent to one standard barrel on a British thermal unit (Btu) basis (42 U.S. gallons). 

For the present purposes the deterministic function generator yields a constant signal x - 0, which means the summation output is identical with that of the noise generator. 

Purpose Generate data sets using mixed deterministic/stochastic models with N = 1. .. 1000. These data sets can be used to test programs or to do Monte Carlo studies. Five different models are predefined sine wave, saw tooth, base line, GC-peaks, and step functions. Data file SIMl.dat was... 

Deterministic trend models are based on the assumption that the trend of a time series can be approximated closely by simple mathematical functions of time over the entire span of the series. The most common representation of a deterministic trend is by means of polynomials or of transcendental functions. The time series from which the trend is to be identified is assumed to be generated by a nonstationary process where the nonstationarity results from a deterministic trend. A classical model is the regression or error model (Anderson, 1971) where the observed series is treated as the sum of a systematic part or trend and a random part or irregular. This model can be written as... 

Figure 2. (a) A deterministic self-similar fractal, i.e., the triadic Koch curve, generated by the similarity transformation with the scaling ratio r = 1/3 and (b) a deterministic self-affine fractal generated by the affine transformation with the scaling ratio vector r = (1/4, 1/2). 

A special kind of random noise, pseudo random noise, has the special property of not being really random. After a certain time interval, a sequence, the same pattern is repeated. The most suitable random input function used in CC is the Pseudo Random Binary Sequence (PRBS). The PRBS is a logical function, that has the combined properties of a true binary random signal and those of a reproducible deterministic signal. The PRBS generator is controlled by an internal clock a PRBS is considered with a sequence length N and a clock period t. It is very important to note that the estimation of the ACF, if computed over an integral number of sequences, is exactly equal to the ACF determined over an infinite time. 

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