Monte Carlo simulation propagator

Figure 4.11 Monte-Carlo simulation (100 trials) of error propagation for La/Yb fractionation in residual melts by clinopyroxene-garnet removal from a basaltic parent magma (see text for parameter description and distributions used). Top mineral-liquid partition coefficients for La and Yb. Bottom variations of the La/Yb ratio as a function of the fraction F of residual melt.

Pollock et al.(12) have also exploited the fact that poly dispersity index is a function of C2 only in a study utilizing a Monte-Carlo simulation technique to compare error propagation in the method of Balke and Hamielec to a revised method (GPCV2) proposed by Yau et al. (13) which incorporated correction for axial dispersion. 

In this method, each assessment factor is considered uncertain and characterized as a random variable with a lognormal distribution with a GM and a GSD. Propagation of the uncertainty can then be evaluated using Monte Carlo simulation (a repeated random sampling from the distribution of values for each of the parameters in a calculation to derive a distribution of estimates in the population), yielding a distribution of the overall assessment factor. This method requires characterization of the distribution of each assessment factor and of possible correlations between them. As a first approach, it can be assumed that all factors are independent, which in fact is not correct. 

MT Bowser, DDY Chen. Monte Carlo simulation of error propagation in the... 

MT Bowser, DDY Chen. Monte Carlo simulation of error propagation in the determination of binding constants from rectangular hyperbolae. 2. Effect of the maximum response range. J Phys Chem 103 197-202, 1999. 

Fig. 5 Schematic of Monte Carlo simulation of IR radiation propagating through a HWG without (top) and with (bottom) 15° acceptance angle...

Sensitivity analysis methods can be used in combination with methods for variance propagation. For example, Cullen Frey (1999) describe how variance in the sum of random numbers can be apportioned among the inputs to the sum. All of the statistical sensitivity methods mentioned above can be applied to the results of Monte Carlo simulation, in which... 

All of the considerations discussed lead naturally to the question of what price the analyst pays for this less-than-ideal spike/sample ratio. In most cases, error in the measurement of Rm makes the largest contribution to analytical uncertainty the isotopic compositions of sample and spike are usually well known in comparison to Rm. The matter of error propagation in isotope dilution analyses has been extensively treated by Adriaens et al., [13], and Patterson et al. used Monte Carlo simulation to study the problem [14]. Using propagation of error laws, Heumann derived the following relationship with which to calculate tfopt, the optimum spike-to-sample ratio (neglecting cost and availability) [8] ... 

Analytic meaiis do not exist to solve the problem of propagation of errors through nonlinear systems. Monte Carlo simulations can be used to eissess the magnitude and distribution of propagated errors. 

Figure 3.8 Distribution for the oxygen diffusion coefficient obtained by use of Monte Carlo simulations to assess propagation of errors for Example 3.2. The mean value for the diffusivity was 1.751 X 10 cm s" and the standard deviation was 1.3 x 10 cm s . ...

Uncertainties inherent to the risk assessment process can be quantitatively described using, for example, statistical distributions, fuzzy numbers, or intervals. Corresponding methods are available for propagating these kinds of uncertainties through the process of risk estimation, including Monte Carlo simulation, fuzzy arithmetic, and interval analysis. Computationally intensive methods (e.g., the bootstrap) that work directly from the data to characterize and propagate uncertainties can also be applied in ERA. Implementation of these methods for incorporating uncertainty can lead to risk estimates that are consistent with a probabilistic definition of risk. 

Figure 27. Plot of ln(KEg) -F PAVg/RT vs. reciprocal temperature ror the reaction YAG + OH-apatite + 25/4 quartz = 5/4 grossular + 5/4 anorthite + 3 YPO4 monazite + 1/2 H2O. Solid squares = xenotime-bearing assemblages, open squares = xenotime-absent assemblages. Least squares regression line is fit to all data points. Horizontal error bars represent temperature uncertainty of 30°C. Vertical error bars ate la (In Kgq + PAV/RT), derived from propagation of uncertainties in P ( 1000 bars), T ( 30°C), AVrxn (1%), compositional parameters (0.001 mole fraction YAG, 0.01 mole fraction all others), and /(H2O) ( 7.5 1000 trial Monte Carlo simulation). Labels on graph indicate sample numbers. From Pyle et al. (2001).

Patterson, K.Y., Veillon, C. and O Haver, T.C., Error propagation in isotope-dilution analysis as determined by Monte Carlo simulation. Anal. Chem., 66, 2829-2834 (1994). 

Figure 14-11. Images of the distribution of propagated charges in a pol)uner membrane with increasing redox-center concentrations as visualized by a Monte Carlo simulation. The left side is the electrode and the right side the solid/liquid interface [51].

In order to propagate modelling uncertainties and uncertainties stemming from the stochastic character of variables or insufficient knowledge of variables through the calculations and account for them in the final results the Monte Carlo simulation is used (cf. Example 4.5 and [14]). 

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