The Discontinuous Nature of Kriging Interpolation for Digital Terrain Modeling

Kriging is a widely employed method for interpolating and estimating elevations from digital elevation data. Its place of prominence is due to its elegant theoretical foundation and its convenient practical implementation. From an interpolation point of view, kriging is equivalent to a thin-plate spline and is one species among the many in the genus of weighted inverse distance methods, albeit with attractive properties. However, from a statistical point of view, kriging is a best linear unbiased estimator and, consequently, has a place of distinction among all spatial estimators because any other linear estimator that performs as well as kriging (in the least squares sense) must be equivalent to kriging, assuming that the parameters of the semivariogram are known. Therefore, kriging is often held to be the gold standard of digital terrain model elevation estimation. However, I prove that, when used with local support, kriging creates discontinuous digital terrain models, which is to say, surfaces with “rips” and “tears” throughout them. This result is general; it is true for ordinary kriging, kriging with a trend, and other forms. A U.S. Geological Survey (USGS) digital elevation model was analyzed to characterize the distribution of the discontinuities. I show that the magnitude of the discontinuity does not depend on surface gradient but is strongly dependent on the size of the kriging neighborhood.

Rigorous design of distillation columns using surrogate models based on Kriging interpolation

The economic design of a distillation column or distillation sequences is a challenging problem that has been addressed by superstructure approaches. However, these methods have not been widely used because they lead to mixed-integer nonlinear programs that are hard to solve, and require complex initialization procedures. In this article, we propose to address this challenging problem by substituting the distillation columns by Kriging-based surrogate models generated via state of the art distillation models. We study different columns with increasing difficulty, and show that it is possible to get accurate Kriging-based surrogate models. The optimization strategy ensures that convergence to a local optimum is guaranteed for numerical noise-free models. For distillation columns (slightly noisy systems), Karush–Kuhn–Tucker optimality conditions cannot be tested directly on the actual model, but still we can guarantee a local minimum in a trust region of the surrogate model that contains the actual local minimum.

Optimization of Chemical Processes Using Surrogate Models Based on a Kriging Interpolation

Superstructure approaches are the solution to the difficult problem which involves the rigorous economic design of a distillation column. These methods require complex initialization procedures and they are hard to solve. For this reason, these methods have not been extensively used. In this work, we present a methodology for the rigorous optimization of chemical processes implemented on a commercial simulator using surrogate models based on a kriging interpolation. Several examples were studied, but in this paper, we perform the optimization of a superstructure for a non-sharp separation to show the efficiency and effectiveness of the method. Noteworthy that it is possible to get surrogate models accurate enough with up to seven degrees of freedom.

Analysis of sandwich beam structures using kriging based higher order models

Functionally graded composite materials can provide continuously varying properties, which distribution can vary according to a specific location within the composite. More frequently, functionally graded materials consider a through thickness variation law, which can be more or less smoother, possessing however an important characteristic which is the continuous properties variation profiles, which eliminate the abrupt stresses discontinuities found on laminated composites. This study aims to analyze the transient dynamic behavior of sandwich structures, having a metallic core and functionally graded outer layers. To this purpose, the properties of the particulate composite metal-ceramic outer layers, are estimated using Mod-Tanaka scheme and the dynamic analyses considers first order and higher order shear deformation theories implemented though kriging finite element method. The transient dynamic response of these structures is carried out through Bossak-Newmark method. The illustrative cases presented in this work, consider the influence of the shape functions interpolation domain, the properties through-thickness distribution, the influence of considering different materials, aspect ratios and boundary conditions. (C) 2014 Elsevier Ltd. All rights reserved.

Solució paral·lelitzada d'interpolació kriging amb ajust automatitzat del variograma

El principal objectiu d'aquest treball és proporcionar una metodologia per a reduir el temps de càlcul del mètode d'interpolació kriging sense pèrdua de la qualitat del model resultat. La solució adoptada ha estat la paral·lelització de l'algorisme mitjançant MPI sobre llenguatge C. Prèviament ha estat necessari automatitzar l'ajust del variograma que millor s'adapta a la distribució espacial de la variable d'estudi. Els resultats experimentals demostren la validesa de la solució implementada, en reduir de forma significativa els temps d'execució final de tot el procés.

Spatial variability of soil chemical properties after coffee tree removal

Assessing the spatial variability of soil chemical properties has become an important aspect of soil management strategies with a view to higher crop yields with minimal environmental degradation. This study was carried out at the Centro Experimental of the Instituto Agronomico, in Campinas, São Paulo, Brazil. The aim was to characterize the spatial variability of chemical properties of a Rhodic Hapludox on a recently bulldozer-cleaned area after over 30 years of coffee cultivation. Soil samples were collected in a 20 x 20 m grid with 36 sampling points across a 1 ha area in the layers 0.0-0.2 and 0.2-0.4 m to measure the following chemical properties: pH, organic matter, K+, P, Ca2+, Mg2+, potential acidity, NH4-N, and NO3-N. Descriptive statistics were applied to assess the central tendency and dispersion moments. Geostatistical methods were applied to evaluate and to model the spatial variability of variables by calculating semivariograms and kriging interpolation. Spatial dependence patterns defined by spherical model adjusted semivariograms were made for all cited soil properties. Moderate to strong degrees of spatial dependence were found between 31 and 60 m. It was still possible to map soil spatial variability properties in the layers 0-20 cm and 20-40 cm after plant removal with bulldozers.

Semivariogram models for estimating fig fly population density throughout the year

The objective of this work was to select semivariogram models to estimate the population density of fig fly (Zaprionus indianus; Diptera: Drosophilidae) throughout the year, using ordinary kriging. Nineteen monitoring sites were demarcated in an area of 8,200 m2, cropped with six fruit tree species: persimmon, citrus, fig, guava, apple, and peach. During a 24 month period, 106 weekly evaluations were done in these sites. The average number of adult fig flies captured weekly per trap, during each month, was subjected to the circular, spherical, pentaspherical, exponential, Gaussian, rational quadratic, hole effect, K-Bessel, J-Bessel, and stable semivariogram models, using ordinary kriging interpolation. The models with the best fit were selected by cross-validation. Each data set (months) has a particular spatial dependence structure, which makes it necessary to define specific models of semivariograms in order to enhance the adjustment to the experimental semivariogram. Therefore, it was not possible to determine a standard semivariogram model; instead, six theoretical models were selected: circular, Gaussian, hole effect, K-Bessel, J-Bessel, and stable.

Spatial variability of the rainfall erosive potential in the State of Mato Grosso do Sul, Brazil

Information about rainfall erosivity is important during soil and water conservation planning. Thus, the spatial variability of rainfall erosivity of the state Mato Grosso do Sul was analyzed using ordinary kriging interpolation. For this, three pluviograph stations were used to obtain the regression equations between the erosivity index and the rainfall coefficient EI30. The equations obtained were applied to 109 pluviometric stations, resulting in EI30 values. These values were analyzed from geostatistical technique, which can be divided into: descriptive statistics, adjust to semivariogram, cross-validation process and implementation of ordinary kriging to generate the erosivity map.Highest erosivity values were found in central and northeast regions of the State, while the lowest values were observed in the southern region. In addition, high annual precipitation values not necessarily produce higher erosivity values.

Soil penetration resistance analysis by multivariate and geostatistical methods

The penetration resistance (PR) is a soil attribute that allows identifies areas with restrictions due to compaction, which results in mechanical impedance for root growth and reduced crop yield. The aim of this study was to characterize the PR of an agricultural soil by geostatistical and multivariate analysis. Sampling was done randomly in 90 points up to 0.60 m depth. It was determined spatial distribution models of PR, and defined areas with mechanical impedance for roots growth. The PR showed a random distribution to 0.55 and 0.60 m depth. PR in other depths analyzed showed spatial dependence, with adjustments to exponential and spherical models. The cluster analysis that considered sampling points allowed establishing areas with compaction problem identified in the maps by kriging interpolation. The analysis with main components identified three soil layers, where the middle layer showed the highest values of PR.

A new method for evaluating regional air quality forecasts

A Kriging interpolation method is combined with an object-based evaluation measure to assess the ability of the UK Met Office's dispersion and weather prediction models to predict the evolution of a plume of tracer as it was transported across Europe. The object-based evaluation method, SAL, considers aspects of the Structure, Amplitude and Location of the pollutant field. The SAL method is able to quantify errors in the predicted size and shape of the pollutant plume, through the structure component, the over- or under-prediction of the pollutant concentrations, through the amplitude component, and the position of the pollutant plume, through the location component. The quantitative results of the SAL evaluation are similar for both models and close to a subjective visual inspection of the predictions. A negative structure component for both models, throughout the entire 60 hour plume dispersion simulation, indicates that the modelled plumes are too small and/or too peaked compared to the observed plume at all times. The amplitude component for both models is strongly positive at the start of the simulation, indicating that surface concentrations are over-predicted by both models for the first 24 hours, but modelled concentrations are within a factor of 2 of the observations at later times. Finally, for both models, the location component is small for the first 48 hours after the start of the tracer release, indicating that the modelled plumes are situated close to the observed plume early on in the simulation, but this plume location error grows at later times. The SAL methodology has also been used to identify differences in the transport of pollution in the dispersion and weather prediction models. The convection scheme in the weather prediction model is found to transport more pollution vertically out of the boundary layer into the free troposphere than the dispersion model convection scheme resulting in lower pollutant concentrations near the surface and hence a better forecast for this case study.

Influência do relevo e erosão na variabilidade espacial de um Latossolo em Jaboticabal (SP)

Com o objetivo de avaliar as relações entre formas de paisagem e erosão em um Latossolo Vermelho eutroférrico (LVef) por meio de técnicas geoestatísticas na Fazenda Santa Isabel, município de Jaboticabal (SP), identificaram-se modelos de paisagem côncavo e linear. Coletaram-se amostras de solo em ambas as formas de paisagem em uma malha regular espaçada de 50 x 50 m em sete transeções na profundidade de 0,00-0,20 m, totalizando 412 pontos em 93 ha. Determinou-se a erodibilidade dos solos pelo método indireto, granulometria, bem como o carbono orgânico para cada ponto da malha, sendo esses valores usados na estimativa do potencial natural de erosão (pne). Os dados obtidos das propriedades de solo, erodibilidade e potencial natural de erosão foram analisados por meio de estatística descritiva e geoestatística com a modelagem de semivariogramas, sendo este utilizado para a confecção de mapas de krigagem. Segundo os resultados, as propriedades do solo e do pne apresentaram maior variabilidade espacial na pedoforma côncava, apesar de esta forma de paisagem apresentar menores perdas de solo por erosão e menor variabilidade espacial da erodibilidade. Deste modo, conclui-se que a erosão não adicionou variabilidade espacial para as propriedades do solo na mesma magnitude do relevo.

Mapping an uncertainty zone between interpolated types of a categorical variable

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Variabilidade espacial de atributos químicos em um Argissolo sob pastagem

Studies show the importance of knowing the variability of soil attributes for a more efficient management. This work was carried out to evaluate the spatial variability of the chemical attributes of an Ultisol, cultivated with Brachiaria decumbens pasture in Alegre - ES. Soil samples were collected at a depth of 00-0.2 m, at the crossing points of a regular grid with 10 m-intervals, comprising a total of 64 points. Data were submitted to descriptive statistics, geostatistics and kriging interpolation analysis. The coefficient of variation was low for pH, high for Al and m%, and medium the other attributes. The attributes pH, P, H+Al and m% presented strong dependence, and the other moderate dependence. The attributes presented a spatial dependence structure, allowing their mapping by geostatistics techniques.

Espacialização das estimativas de contaminação de águas subterrâneas por Tebuthiuron

Improper use of pesticides can lead to significant environmental impacts such as, contamination of environmental compartments, being the aquatic compartments the most vulnerable. In this context, the spatialization of pesticides concentrations estimative in groundwater provides important insights for decision making in managing and monitoring pesticides use. This study aimed to spatialize estimatives of groundwater contamination by Tebuthiuron, from different irrigation depths in the Rio Pardo basin, Pardinho-SP, Brazil. The simulations were performed using the ARAquá computer system, considering 0mm, 200 mm and 400 mm annual irrigation depths. Geostatistical techniques were used to obtain the spatial distribution of the simulated estimative. Tebuthiuron maps estimating concentration in groundwater were obtained by Kriging interpolation method, and indicated the areas with high potential for groundwater contamination. Considering all the simulations, it was concluded that there was no risk of groundwater contamination by Tebuthiuron in the study area.

Submerged macrophytes height estimation by echosounder data sample

Traditional methods of submerged aquatic vegetation (SAV) survey last long and then, they are high cost. Optical remote sensing is an alternative, but it has some limitations in the aquatic environment. The use of echosounder techniques is efficient to detect submerged targets. Therefore, the aim of this study is to evaluate different kinds of interpolation approach applied on SAV sample data collected by echosounder. This study case was performed in a region of Uberaba River - Brazil. The interpolation methods evaluated in this work follow: Nearest Neighbor, Weighted Average, Triangular Irregular Network (TIN) and ordinary kriging. Better results were carried out with kriging interpolation. Thus, it is recommend the use of geostatistics for spatial inference of SAV from sample data surveyed with echosounder techniques. © 2012 IEEE.