Intrinsic Kriging Research Articles

Environmental Impact Assessment of Quarrying Plants: An IRF Kriging Solution to Noise and Airborne Dust Monitoring

A geostatistical approach is tested for the assessment of the environmental impact generated by industrial plants on the surrounding areas. To this end, a case study of a tuff quarry in the Italian region of Lazio is presented. The method utilised is Intrinsic Kriging with external drift which allows spatial maps of different hazardous agents like noise and airborne dust to be made through the integration of two different kinds of information. The initial data is taken from field measurements, which are carried out in accordance with the current regulations at various points in residential or built up areas near the site. Auxiliary data, integrated with the former as external drift, are obtained from specific propagation models respectively referred to sound propagation in outdoor environments by means of ISO 9613 norm and airborne dust propagation by means of ISC3 by US EPA approach. The use of a combination of these two information sources enabled both acoustic and airborne dust maps to be made which indicate not only the overall spatial trends of the acoustic and airborne climate of the area surrounding the site but also the local variations. Such results are generally difficult to achieve due to the limited data set and usually come at the price of increasing the number of measurements taken in the field. In order to quantify the contribution of introducing this integrated approach, scenarios are presented with and without external drift, using the same field samples as input in each. The results indicate a reduction in estimate variance for the cases analysed when using Intrinsic Kriging with external drift. Furthermore, it was observed from the analysis of cross validation and estimate variance that the use of external drift may also compensate for information loss, which may occur due to a reduction in the number of receptors in the monitor network for example, without affecting the quality of the final map. In the case study, on halving the number of measurements taken while introducing the suggested drift, the quality of the result remains similar.

Mappatura geostatistica automatizzata di parametri meteo-climatici per il bilancio idrico

Geostatistical mapping of a continuous variable consists in its spatial reconstruction in a given domain using Kriging techniques. Mapping is based on a limited number of direct observations, using also one or more densely sampled indirect information. In addition, the geostatistical approach provides the corresponding variance/standard deviation estimation map.In the framework of Groundwater Master Plans conducted in Central Italy for the estimation of the distributed water balance of volcanic, alluvial and carbonate aquifers, an automated estimation procedure based on intrinsic Kriging with external drift was developed.The regional planning of water resources should be based on a distributed water balance approach and on the estimation of key hydro-meteorological maps at high spatial resolution, among others: daily precipitation, daily maximum, minimum and average temperature. As this approach requires the estimation of a huge amount of data over a period of 20 years at least, it is mandatory the use of an automatic methodology for the estimation of the hydrometeorological fields.The developed procedure generates water balance maps at high spatial resolution. The maps are highly reliable even in case of temporal failure of some monitoring stations and the presence of data gaps in the time series. The automatic geostatistical mapping is a powerful tool that can be used in several applications that require the estimation of the hydro-meteorological fields in realtime (e.g, early warning and flood forecasting systems, or in the field of the assessment of climate change at river basin scale, where the use of an ensemble of climatic scenarios and related projections is necessary).

Efficient global optimisation for black-box simulation via sequential intrinsic Kriging

Efficient global optimisation (EGO) is a popular method that searches sequentially for the global optimum of a simulated system. EGO treats the simulation model as a black-box, and balances local and global searches. In deterministic simulation, classic EGO uses ordinary Kriging (OK), which is a special case of universal Kriging (UK). In our EGO variant we use intrinsic Kriging (IK), which does not need to estimate the parameters that quantify the trend in UK. In random simulation, classic EGO uses stochastic Kriging (SK), but we replace SK by stochastic IK (SIK). Moreover, in random simulation, EGO needs to select the number of replications per simulated input combination, accounting for the heteroscedastic variances of the simulation outputs. A popular method uses optimal computer budget allocation (OCBA), which allocates the available total number of replications to simulated combinations. We replace OCBA by a new allocation algorithm. We perform several numerical experiments with deterministic simulations and random simulations. These experiments suggest that (1) in deterministic simulations, EGO with IK outperforms classic EGO; (2) in random simulations, EGO with SIK and our allocation rule does not perform significantly better than EGO with SK and OCBA.

Stochastic intrinsic Kriging for simulation metamodeling

Kriging (or a Gaussian process) provides metamodels for deterministic and random simulation models. Actually, there are several types of Kriging; the classic type is the so‐called universal Kriging, which includes ordinary Kriging. These classic types require estimation of the trend in the input‐output data of the underlying simulation model; this estimation weakens the Kriging metamodel. We therefore consider the so‐called intrinsic Kriging (IK), which originated in geostatistics, and derive IK types for deterministic simulations and random simulations, respectively. Moreover, for random simulations, we derive experimental designs that specify the number of replications that varies with the input combination of the simulation model. To compare the performance of IK and classic Kriging, we use several numerical experiments with deterministic simulations and random simulations, respectively. These experiments show that IK gives better metamodels, in most experiments.

Efficient Global Optimization for Black-Box Simulation Via Sequential Intrinsic Kriging

Efficient Global Optimization (EGO) is a popular method that searches sequentially for the global optimum of a simulated system. EGO treats the simulation model as a black-box, and balances local and global searches. In deterministic simulation, EGO uses ordinary Kriging (OK), which is a special case of universal Kriging (UK). In our EGO variant we use intrinsic Kriging (IK), which eliminates the need to estimate the parameters that quantify the trend in UK. In random simulation, EGO uses stochastic Kriging (SK), but we use stochastic IK (SIK). Moreover, in random simulation, EGO needs to select the number of replications per simulated input combination, accounting for the heteroscedastic variances of the simulation outputs. A popular selection method uses optimal computer budget allocation (OCBA), which allocates the available total number of replications over simulated combinations. We derive a new allocation algorithm. We perform several numerical experiments with deterministic simulations and random simulations. These experiments suggest that (1) in deterministic simulations, EGO with IK outperforms classic EGO; (2) in random simulations, EGO with SIK and our allocation rule does not differ significantly from EGO with SK combined with the OCBA allocation rule.

Global Optimization for Black-Box Simulation via Sequential Intrinsic Kriging

In this paper we investigate global optimization for black-box simulations using metamodels to guide this optimization. As a novel metamodel we introduce intrinsic Kriging, for either deterministic or random simulation. For deterministic simulation we study the famous `efficient global optimization' (EGO) method, substituting intrinsic Kriging for universal Kriging. For random simulation we investigate a state-of-the-art two-stage algorithm accounting for heteroscedastic variances of the simulation responses, and introduce a new variant with the following two features: (1) this variant uses intrinsic Kriging; (2) this variant uses a different procedure to allocate the total available number of replications over simulated points. We perform several numerical experiments with deterministic and random simulations, to compare (1) the classic EGO and our EGO with intrinsic Kriging; (2) the classic two-stage algorithm and our modified version. We conclude that in most experiments (1) EGO with intrinsic Kriging outperforms classic EGO; (2) there is no significant difference between the classic algorithm and our modifed two-stage algorithm.

Fractional Brownian Fields for Response Surface Metamodeling

Kriging, a widely used metamodeling method for computer-simulation data, models the response surface as a realization of a random field. Stationary covariance functions such as the Gaussian, power exponential, or Matérn class are the most common choice for the underlying random field model. Nonstationary versions of these same covariance functions with scale parameters that vary spatially are also sometimes used. Fractional Brownian fields (FBFs) are a different form of nonstationary random field model having stationary increments, an example of more general intrinsic stationary processes. Although FBFs have been considered for intrinsic kriging in the spatial statistics literature, they have received little attention for computer-simulation response surface metamodeling. For use in the latter context, we argue that they have some attractive (as well as some unattractive) properties that mitigate certain problems inherent to many stationary covariance models, such as reversion to the mean, numerical issues due to near-singularity of covariance matrices, and difficulties in handling abrupt response surface features.

Stochastic Intrinsic Kriging for Simulation Metamodelling

Kriging provides metamodels for deterministic and random simulation models. Actually, there are several types of Kriging; the classic type is so-called universal Kriging, which includes ordinary Kriging. These classic types require estimation of the trend in the input-output data of the underlying simulation model; this estimation deteriorates the Kriging metamodel. We therefore consider so-called intrinsic Kriging originating in geostatistics, and derive intrinsic Kriging for deterministic and random simulations. Moreover, for random simulations we derive experimental designs that specify the number of replications that varies with the input combination of the simulation model. To compare the performance of intrinsic Kriging and classic Kriging, we use several numerical experiments with deterministic simulations and random simulations. These experiments show that intrinsic Kriging gives better metamodels, in most experiments.

Conditioning Simulations of Gaussian Random Fields by Ordinary Kriging

Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.

Intrinsic Kriging and prior information

AbstractKriging, one of the oldest prediction methods based on reproducing kernels, can be used to build black‐box models in engineering. In practice, however, it is often necessary to take into account prior information to obtain satisfactory results. First, the kernel (the covariance) can be used to specify properties of the prediction such as its regularity or the correlation distance. Moreover, intrinsic Kriging (viewed as a semi‐ parametric formulation of Kriging) can be used with an additional set of factors to take into account a specific type of prior information. We show that it is thus very easy to transform a black‐box model into a grey‐box one. The prediction error is orthogonal in some sense to the prior information that has been incorporated. An application in flow measurement illustrates the interest of the method. Copyright © 2005 John Wiley & Sons, Ltd.

Spatial interpolation methods for nonstationary plume data.

Plume interpolation consists of estimating contaminant concentrations at unsampled locations using the available contaminant data surrounding those locations. The goal of ground water plume interpolation is to maximize the accuracy in estimating the spatial distribution of the contaminant plume given the data limitations associated with sparse monitoring networks with irregular geometries. Beyond data limitations, contaminant plume interpolation is a difficult task because contaminant concentration fields are highly heterogeneous, anisotropic, and nonstationary phenomena. This study provides a comprehensive performance analysis of six interpolation methods for scatter-point concentration data, ranging in complexity from intrinsic kriging based on intrinsic random function theory to a traditional implementation of inverse-distance weighting. High resolution simulation data of perchloroethylene (PCE) contamination in a highly heterogeneous alluvial aquifer were used to generate three test cases, which vary in the size and complexity of their contaminant plumes as well as the number of data available to support interpolation. Overall, the variability of PCE samples and preferential sampling controlled how well each of the interpolation schemes performed. Quantile kriging was the most robust of the interpolation methods, showing the least bias from both of these factors. This study provides guidance to practitioners balancing opposing theoretical perspectives, ease-of-implementation, and effectiveness when choosing a plume interpolation method.