A nonstationary nonlinear geostatistical model and its application in a beach sand deposit for recoverable reserve estimation

Abstract

The standard geostatistical models, both linear and nonlinear, require the data to be stationary, which is very rare in reality. The changing of means and variances, and spatial covariance structure are evident in the real life data. The so called global non-stationarity over a spatial field is captured by a model which is considered to be locally stationary but globally non-stationary. The proposed model is simultaneously defined everywhere in the concerned domain not only in the defined areas, but again the model behaves like a local stationary process in small areas. For this purpose, the smoothly varying local multivariate spatial distribution function has been derived by weighting all the data values in the spatial field using kernel function and consequently represented by Hermite polynomial expansions. The locally varying spatial covariance structure is modeled by a local covariance function defined by semi-variance parameters estimated experimentally by the local samples present in the local stationary region. The local recoverable reserve is estimated by calculating conditional cumulative distribution function using nonlinear geostatistical techniques such as disjunctive kriging and multi Gaussian kriging. The proposed method has been applied in an Indian beach sand deposit for local recoverable reserve estimation. An appropriate support effect model has been incorporated for designing this estimation algorithm. The impacts of support effect and nonstationarity in terms of local recoverable reserve are analyzed for the deposit.