Random geochemical variables

When the geochemical variable is not uniquely determined but is a random variable, we would like to be able to assess how the parameters of the population change 

This equality holds true for any arbitrarily small segment dc and any c0, so the following equality is identically true 

In a case where F would contain a stochastic term (e.g., Brownian motion, noise), this equation would lead to the celebrated Fokker-Plank equation with a diffusion (second-order) term. 

This equation is a partial differential equation whose order depends on the exact form of/ and F. Its solution is usually not straightforward and integral transform methods (Laplace or Fourier) are necessary. The method of separation of variables rarely works. Nevertheless, useful information of practical geological importance is apparent in the form taken by this equation. The only density distributions that are time independent must obey 

If the process under investigation is radioactivity, for instance, then 

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In most natural situations, physical and chemical parameters are not defined by a unique deterministic value. Due to our limited comprehension of the natural processes and imperfect analytical procedures (notwithstanding the interaction of the measurement itself with the process investigated), measurements of concentrations, isotopic ratios and other geochemical parameters must be considered as samples taken from an infinite reservoir or population of attainable values. Defining random variables in a rigorous way would require a rather lengthy development of probability spaces and the measure theory which is beyond the scope of this book. For that purpose, the reader is referred to any of the many excellent standard textbooks on probability and statistics (e.g., Hamilton, 1964 Hoel et al., 1971 Lloyd, 1980 Papoulis, 1984 Dudewicz and Mishra, 1988). For most practical purposes, the statistical analysis of geochemical parameters will be restricted to the field of continuous random variables. 

A second reason to fit data to a function is to test whether the data are consistent with a model or to use a theoretical function to extrapolate experimental results to conditions otherwise not attainable. The goodness of fit of the data to a theoretical function can be gauged by the coefficient of determination EP), which is the correlation coefficient squared. B is often interpreted as the fraction of the variability of the response variable that is explained by its functional relationship to the independent variable. For example, if R = 0.8 then 80% of the variation in the response variable is explained by the model and 20% of the variation is the result of factors, including random error, that are not part of the model. Some geochemical processes occur under conditions or over time intervals that are difficult to simulate by experiments, making it necessary to use a theoretical model to predict their behavior. This approach uses data from experiments performed under easily attainable conditions to quantify parameters that are used to calibrate a theoretical model. These parameters are then used to predict the value of the variable at... 

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