Optimization Of Spatial Semi-variogram Models

Being a basic module of geostatistics,the semi-variogram model can be used to depict spatial variability of a random function or the regionalized variable of interest,with the exception of recent advances in multiple-point geostatistics.The semi-variogram provides a means of detecting the defining characteristics of the spatial variability of regionalized variables,such as anisotropic variability of different spatial directions.In this sense,the semi-variogram is so important as such and is an essential tool for spatial statistics.Generally,in practice,the original semi-variogram model is unknown and needs to be estimated based on the experimental data.Before estimating parameters of the semi-variogram model,a model structure needs to be chosen.However,which semi-variogram model to choose remains contentious.To explore how to optimize the spatial semi-variogram model and find out impacts of structure(spherical,exponential or Gaussian)of the model on kriging interpolation or simulation,fifty random fields were generated using non-conditional simulation based on spherical,exponential and Gaussian model,separately,from which three realizations were selected at random for each semi-variogram model.For each realization,samples of four sizes(50,100,150,200)were randomly drawn.And then,spherical,exponential and Gaussian models were used one by one for each sample to estimate corresponding semi-variogram model parameters.After that,conditional simulation was performed to generate 1000 realizations for each semi-variogram model with the estimated parameter.The mean of the 1000 realizations was deemed as the result of the simulation and compared with the actual value in order to evaluate accuracy of the simulation.In the case of non-conditional simulation,the random field was a 50 X 50 grid of points with each unit being standard 1×1 along the X and Y axes.The structural parameters of all the variogram modes were set as C0=0 for nugget constant,C+C0=1 for sill and A=10 for variable range.In fitting the semi-variogram,the minimum interval was set at 3,tolerance interval at 1.5 and maximum interval at 30.For the sake of demonstration,the ordinary least square method and the GS+ software was used to evaluate the semi-variogram model parameters,with searching radius equal to the variable range.The simulation demonstrates certain regularities.Fitting of the semi-variogram models shows that the Gaussian model is the highest in nugget effect and the smallest in variable range,while the exponential model the weakest in nugget effect and the biggest in variable range,which suggests that the spatial variability of the regionalized variable acquired with the exponential model is always lower than that done with the spherical model or the Gaussian model.However,in the case of conditional simulation,when the sample size is 50,no matter what the original semi-variogram model is,the conditional simulation using the exponential model is almost always the highest in accuracy;and so is it,when the sample size is 100,150 or 200,under the simulation condition the same for the original semi-variogram model,but the simulation using the spherical model is next to that using the original semi-variogram model except that the original model is spherical.Besides,the measured hydraulic conductivity of the third aquifer of Yanqi Basin was used to find out the differences between the three variogram models when they were used to fit the experimental variogram and their effect to kriging interpolation.Analysis of the interpolation results shows that the impact of variogram models on the ordinary kriging interpolation of hydraulic conductivity mainly reflects in the difference of range.The smaller the range is,the weaker the correlation between the observation and estimation point is,As a result,the kriging degenerates into a traditional interpolation method.Contrarily,the larger range means stronger correlation,as a consequence,the interpolation result tends to be smoother.Ideally,when the range is close to the reality,the kriging interpolation should provide a hydraulic conductivity field with low estimation variance.Therefore,it is recommended that when the sample size is small,the exponential semi-variogram model be the first choice,and when the sample size is bigger and the original semi-variogram model is unknown,the spherical model be a nice one to ensure accuracy of the simulation.