An automatic variogram modeling method with high reliability fitness and estimates

Abstract

Modeling of the variogram is a critical step for most geostatistical methods. However, most of the prevalent variogram-based solutions are designed without sufficient consideration of the effect of the interpolation process on their application. This paper proposes an automated variogram modeling framework, which simultaneously considers the fit of the experimental variogram and interpolation accuracy in the modeling variogram interpolation result. The variogram modeling framework can be treated as a nonlinear optimization problem with two sub-goals. The first is to optimize the goodness of fit between the experimental and theoretical variogram values under the conditions of their designated parameters. Second, we seek to optimize the difference between measured values and the associated kriging estimates with the candidate variogram model. A typical case study was chosen using a public dataset to test the proposed method, which was implemented using a genetic algorithm, and its performance was compared with the ones of other commonly applied variogram modeling approaches. As expected, the traditional variogram modeling method that only considers fitting standard experimental variograms showed severe sensitivity to errors in data and parameters; classical cross-validation modeling results tended to overlook the experimental variograms. By contrast, the proposed method succeeded in producing variogram models with robust, high-quality kriging estimates and favorable fitness of experimental variograms in a more powerful and flexible way.

A New Methodology for Automatic Cluster-Based Kriging Using K-Nearest Neighbor and Genetic Algorithms

“…Note that both third-party and proposed algorithms were designed with different cost functions, as well as being executed with and without the anisotropic parameters for comparison purposes. One of the cost functions that evaluates the quality of the selected parameters to be optimized was the weighted least squares (WLS) [10,21], which, according to [7], presents the best results in the adjustment of the theoretical model compared to other cost functions, such as the ordinary least squares and the generalized least squares. More recently, in [3], the iterative least squares (ILS) was proposed as an enhancement of the WLS cost function.…”

“…Regarding the evaluation metric, the 10-fold cross-validation was applied. For both cases, fitness and evaluation, the interpolation cost function (Equation (10)) was employed [7]. More specifically, the normalized mean squared error (NMSE) index was used as figure of merit and calculated by…”

“…In recent years, artificial intelligence techniques have been used to improve the kriging process as shown in [4][5][6][7][8][9]. However, it is still a challenge to determine which method is better suited for a given database.…”

“…However, it is still a challenge to determine which method is better suited for a given database. As stated in [6] and applied in [7], bio-inspired algorithms, like Genetic Algorithms (GA), are suitable to help define the theoretical variogram parameters. Furthermore, these kinds of algorithms do not require a single initial seed value as input, but rather a minimum and maximum interval, different from classical methods such as Gauss-Newton [3] and Levenberg-Marquardt [10].…”

“…Finally, genetic algorithms were used to model a specific theoretical variogram to each group found in the clustering step, with the definition of the parameters' bounds based on data of each group. Although GA is not considered nowadays as the state-of-the-art meta-heuristics, this technique was chosen in order to serve as a baseline for the proposed model, and also because it was applied in similar kriging methodologies, as can be seen in [5,7,8,20]. For instance, in [20], the authors concluded that particle swarm optimization and genetic algorithms are statistically equivalent in the task of estimating the kriging parameters.…”