Comparative study of simulation algorithms in mapping spaces of uncertainty

A viable method (such as stochastic or geostatistical simulation) is required for characterising and quantifying the uncertainty associated with predictions generated from any non-linear function of spatially varying parameters. Geostatistical simulation algorithms generate realisations of a random field with specified geostatistical properties, namely the reproduction of existing data, histogram and variogram or covariance of the data. A transfer function is a term used to describe a generally non-linear function, mathematical model or algorithm used to describe a process and predict its behaviour (responses). A transfer function may be an algorithm used to optimise the design of a open pit mine requiring as input the spatially varying properties of an orebody together with other parameters, or a three-phase reservoir flow simulator requiring as the spatially varying rock properties of the reservoir in addition to flow characteristics and engineering specifications or, similarly, a simulator of contaminant flow and so on. The predictions from transfer functions can be evaluated over each realisation of input parameters generated with stochastic simulations so as to obtain an uncertainty distribution of the response parameters) that reflect the spatial variability and uncertainty in the parameters) of interest. Examples of parameters of interest may be the production schedules in a mine over the life of the mine, or the production curved on a petroleum reservoir or the cash flows from oil or mineral production, or the parts of a contaminated site that needs to be remediated. It is important to recall that transfer functions are generally non-linear. Consequently, (i) an average type map of the complete spatial distribution of an attribute does not provide an average expected map of the spaces of response uncertainty; and (ii) a criterion for generating deposit or reservoir descriptions is defined: the approach selected must be evaluated in terms of the map of the uncertainty of the response, not the maps of the description of the field. In summary, the ability to map the uncertainty in response parameters of various transfer functions is critical. Thus it is important to assess the performance of various stochastic simulation methods in mapping uncertainty in response parameters.Stochastic simulation approaches address the issue of modelling the uncertainty about the values of continuous geological attributes at any particular un-sampled location (local uncertainty) as well as jointly over several locations (spatial uncertainty). The common past approach used in interpolating geological attributes of interest at un-sampled locations is estimation and amounts to minimising a local estimation error variance, which results in smoothing spatial distributions. Contrary to this, stochastic simulation aims to reproduce global statistics such as the histograms and variograms thus allowing the accounting of the effects of in-situ spatial variability on the outputs of the transfer functions used, as well as, considering the non-linearity of these transfer functions, simulations provide both an accurate average assessment in the predictions and the ability to map the uncertainty of these predictions. As mentioned above, the distribution of stochastically generated predictions are termed ‘space of uncertainty’ and cannot be defined analytically because of the complexity (non-linearity) of the related transfer functions. In this study, the focus is on two categories of stochastic simulation algorithms: (i) the wide class of stochastic simulation algorithms is known under the generic name of sequential simulations. In this category, instead of modelling the N-point conditional cumulative distribution function (ccdf), a one point ccdf is modelled and then sampled at each of the N locations visited along a random path. To ensure reproduction of the covariance model each point ccdf is made conditional not only to original data but also to all values simulated at previously visited locations. Sequential simulation algorithms namely sequential Gaussian, sequential indicator and joint sequential Gaussian simulation algorithms are used in this study; and (ii) the probability field approach which also trades the sampling of N-point ccdf for sampling of N successive one point ccdfs. Unlike the sequential approach, all one point ccdfs are conditional only to the original n data. Sampling techniques namely simple random sampling, stratified random sampling and systematic sampling are used to choose nine sample sets from two variables of the exhaustive data set. Transfer functions namely mean of geometric means, threshold proportion, minimum cost path from upper boundary to lower boundary and minimum cost path from upper left comer to lower right comer are used over each realisation to obtain uncertainty distributions of response that reflects the spatial variability and uncertainty in the parameter. First part of the study compares the above-mentioned stochastic simulation algorithms in a designed experiment using nine sample sets of different sizes chosen from an exhaustive data set. For joint sequential Gaussian simulation algorithm, nine sample sets are chosen from collocated variable as second data sets. For each of the sample set, a number of realisations are generated using each simulation algorithm. The realisations are used with each of the transfer functions used to produce a cumulative uncertainty distribution function of a response. The uncertainty distributions are then compared to the single value obtained from the exhaustive data set. Several broad issues are illustrated by the results of this comparative study. It is found in all the cases, increasing the sample size improves the precision associated with the response distributions. Results indicate that uncertainty distributions produced by SGS, SIS and JSGS are more precise than those based on PFS. It is to be noted from the results that the uncertainty distributions obtained from PFS can be more accurate than the distributions based on SGS, SIS or JSGS. Generally sequentially all algorithms were found to perform well in mapping spaces of uncertainty. An additional observation was that the ability to effectively map spaces of uncertainty also depends on the complexity of the transfer function and that is not necessarily a well understood aspect of the modelling process. Many environmental studies lead to important decision-making such as delineation of sites targeted for remediation or additional sampling. Such decisions are made in the face of uncertainty since concentrations in toxic elements are typically sparsely sampled. An important contribution of geostatistics is the assessment of the uncertainty about un-sampled values, which usually takes the form of a map of the probability of exceeding critical values, such as regulatory thresholds in soil pollution. In the last part of the thesis the SGS algorithm is used to address the problem of accounting for uncertainty about pollutant concentrations in environmental decision-making such as delineation of mercury contaminated sites where remedial measures should be taken. This was the location of gold reprocessing plant in Canada, which used mercury to recover gold from the ore. Two approaches namely a local transfer function approach and a loss function approach are used for classifying unvisited sites as safe or contaminated. A local transfer function is used with two block-support concentration thresholds zc1 and zc2 as imposed by a regulatory agency. Regulations specify contaminated blocks with pollutant concentrations above the threshold zc2 should be removed, below zc1should be treated locally and between zc1 and zc2 need further investigation to reduce the uncertainty at these sites by using a loss function approach. A loss function measures the impact of a decision by classifying a block as contaminated or not, as a function of the actual pollutant concentration. Each block between zc1 and zc2 is classified as clean or polluted through loss function so as to minimise the resulting expected loss. It should be noted from the results obtained in last part of the thesis that the choice of the threshold zc2 above which all the blocks should classified polluted is really critical. It is also noted that as the intermediate interval zc1, and zc2 increases, cost of health and social problems increases rapidly while there is very little change in the total cost.

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