Journal articles on the topic 'Kriging and cokriging models'
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1
Magno, Melissa, Ingrid Luffman, and Arpita Nandi. "Evaluating Spatial Regression-Informed Cokriging of Metals in Soils near Abandoned Mines in Bumpus Cove, Tennessee, USA." Geosciences 11, no. 11 (October 20, 2021): 434. http://dx.doi.org/10.3390/geosciences11110434.
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Inorganic contaminants, including potentially toxic metals (PTMs), originating from un-reclaimed abandoned mine areas may accumulate in soils and present significant distress to environmental and public health. The ability to generate realistic spatial distribution models of such contamination is important for risk assessment and remedial planning of sites where this has occurred. This study evaluated the prediction accuracy of optimized ordinary kriging compared to spatial regression-informed cokriging for PTMs (Zn, Mn, Cu, Pb, and Cd) in soils near abandoned mines in Bumpus Cove, Tennessee, USA. Cokriging variables and neighborhood sizes were systematically selected from prior statistical analyses based on the association with PTM transport and soil physico-chemical properties (soil texture, moisture content, bulk density, pH, cation exchange capacity (CEC), and total organic carbon (TOC)). A log transform was applied to fit the frequency histograms to a normal distribution. Superior models were chosen based on six diagnostics (ME, RMS, MES, RMSS, ASE, and ASE-RMS), which produced mixed results. Cokriging models were preferred for Mn, Zn, Cu, and Cd, whereas ordinary kriging yielded better model results for Pb. This study determined that the preliminary process of developing spatial regression models, thus enabling the selection of contributing soil properties, can improve the interpolation accuracy of PTMs in abandoned mine sites.
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Baffoe-Twum, Edmund, Eric Asa, and Bright Awuku. "Estimating annual average daily traffic (AADT) data on low-volume roads with the cokriging technique and census/population data." Emerald Open Research 4 (April 22, 2022): 20. http://dx.doi.org/10.35241/emeraldopenres.14632.1.
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Geostatistical methods such as simple, ordinary, and universal kriging are not multivariate models in the usual statistical function. Notwithstanding, simple, ordinary, and universal kriging techniques utilize random function models that include unlimited random variables while modeling one attribute. The cokriging technique is a multivariate estimation method that can simultaneously model two or more attributes, defined with the same domains as coregionalization. For a successful structural analysis, it is necessary to have a minimum amount of each domain's measured attributes. The assumption is that data integration methods such as cokriging may yield more reliable models because their strength is drawn from multiple variables. This study investigates the impact of the population as a variable on traffic volumes. The investigation adopts the annual average daily traffic (AADT) from Montana, Minnesota, and Washington as one attribute and countywide population as a second attribute (or factor controlling traffic volumes). AADT data for this research span from 2009 to 2016. The cross-validation results of the model types explored with the cokriging technique are successfully used to evaluate the interpolation technique's performance and select optimal models for each state. The investigation results based on the cross-validation confirm the model's usefulness. The interpolation surface maps from the Montana and Minnesota models accurately represent the states' traffic and population density. The Washington model had a few exceptions; therefore, it did not necessarily represent the traffic and population density. An indication that other factors may impact the results. Consequently, it is worth exploring the impact of tourism, shopping, recreation centers, and possible transiting patterns throughout the state.
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Carvalho, José Ruy Porto De, Alan Massaru Nakai, and José Eduardo B. A. Monteiro. "Spatio-Temporal Modeling of Data Imputation for Daily Rainfall Series in Homogeneous Zones." Revista Brasileira de Meteorologia 31, no. 2 (June 2016): 196–201. http://dx.doi.org/10.1590/0102-778631220150025.
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Abstract Spatio-temporal modelling is an area of increasing importance in which models and methods have often been developed to deal with specific applications. In this study, a spatio-temporal model was used to estimate daily rainfall data. Rainfall records from several weather stations, obtained from the Agritempo system for two climatic homogeneous zones, were used. Rainfall values obtained for two fixed dates (January 1 and May 1, 2012) using the spatio-temporal model were compared with the geostatisticals techniques of ordinary kriging and ordinary cokriging with altitude as auxiliary variable. The spatio-temporal model was more than 17% better at producing estimates of daily precipitation compared to kriging and cokriging in the first zone and more than 18% in the second zone. The spatio-temporal model proved to be a versatile technique, adapting to different seasons and dates.
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ROGERS, DAVID J., and LUIGI SEDDA. "Statistical models for spatially explicit biological data." Parasitology 139, no. 14 (October 19, 2012): 1852–69. http://dx.doi.org/10.1017/s0031182012001345.
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SUMMARYExisting algorithms for predicting species' distributions sit on a continuum between purely statistical and purely biological approaches. Most of the existing algorithms are aspatial because they do not consider the spatial context, the occurrence of the species or conditions conducive to the species' existence, in neighbouring areas. The geostatistical techniques of kriging and cokriging are presented in an attempt to encourage biologists more frequently to consider them. Unlike deterministic spatial techniques they provide estimates of prediction errors. The assumptions and applications of common geostatistical techniques are presented with worked examples drawn from a dataset of the bluetongue outbreak in northwest Europe in 2006. Emphasis is placed on the importance and interpretation of weights in geostatistical calculations. Covarying environmental data may be used to improve predictions of species’ distributions, but only if their sampling frequency is greater than that of the species’ or disease data. Cokriging techniques are unable to determine the biological significance or importance of such environmental data, because they are not designed to do so.
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Qu, Mingkai, Xu Guang, Hongbo Liu, Yongcun Zhao, and Biao Huang. "Incorporating Auxiliary Data of Different Spatial Scales for Spatial Prediction of Soil Nitrogen Using Robust Residual Cokriging (RRCoK)." Agronomy 11, no. 12 (December 10, 2021): 2516. http://dx.doi.org/10.3390/agronomy11122516.
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Auxiliary data has usually been incorporated into geostatistics for high-accuracy spatial prediction. Due to the different spatial scales, category and point auxiliary data have rarely been incorporated into prediction models together. Moreover, traditionally used geostatistical models are usually sensitive to outliers. This study first quantified the land-use type (LUT) effect on soil total nitrogen (TN) in Hanchuan County, China. Next, the relationship between soil TN and the auxiliary soil organic matter (SOM) was explored. Then, robust residual cokriging (RRCoK) with LUTs was proposed for the spatial prediction of soil TN. Finally, its spatial prediction accuracy was compared with that of ordinary kriging (OK), robust cokriging (RCoK), and robust residual kriging (RRK). Results show that: (i) both LUT and SOM are closely related to soil TN; (ii) by incorporating SOM, the relative improvement accuracy of RCoK over OK was 29.41%; (iii) by incorporating LUTs, the relative improvement accuracy of RRK over OK was 33.33%; (iv) RRCoK obtained the highest spatial prediction accuracy (RI = 43.14%). It is concluded that the recommended method, RRCoK, can effectively incorporate category and point auxiliary data together for the high-accuracy spatial prediction of soil properties.
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Jawak, S. D., and A. J. Luis. "Synergetic merging of Cartosat-1 and RAMP to generate improved digital elevation model of Schirmacher oasis, east Antarctica." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-8 (November 28, 2014): 517–24. http://dx.doi.org/10.5194/isprsarchives-xl-8-517-2014.
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Available digital elevation models (DEMs) of Antarctic region generated by using radar altimetry and the Antarctic digital database (ADD) indicate elevation variations of up to hundreds of meters, which necessitates the generation of local DEM and its validation by using ground reference. An enhanced digital elevation model (eDEM) of the Schirmacher oasis region, east Antarctica, is generated synergistically by using Cartosat-1 stereo pair-derived photogrammetric DEM (CartoDEM)-based point elevation dataset and multitemporal radarsat Antarctic mapping project version 2 (RAMPv2) DEM-based point elevation dataset. In this study, we analyzed suite of interpolation techniques for constructing a DEM from RAMPv2 and CartoDEM-based point elevation datasets, in order to determine the level of confidence with which the interpolation techniques can generate a better interpolated continuous surface, and eventually improves the elevation accuracy of DEM from synergistically fused RAMPv2 and CartoDEM point elevation datasets. RAMPv2 points and CartoDEM points were used as primary data for various interpolation techniques such as ordinary kriging (OK), simple kriging (SK), universal kriging (UK), disjunctive kriging (DK) techniques, inverse distance weighted (IDW), global polynomial (GP) with power 1 and 2, local polynomial (LP) and radial basis functions (RBF). Cokriging of 2 variables with second dataset was used for ordinary cokriging (OCoK), simple cokriging (SCoK), universal cokriging (UCoK) and disjunctive cokriging (DCoK). The IDW, GP, LP, RBF, and kriging methods were applied to one variable, while Cokriging experiments were employed on two variables. The experiment of dataset and its combination produced two types of point elevation map categorized as (1) one variable (RAMPv2 Point maps and CartoDEM Point maps) and (2) two variables (RAMPv2 Point maps + CartoDEM Point maps). Interpolated surfaces were evaluated with the help of differential global positioning system (DGPS) points collected from study area during the Indian Scientific Expedition to Antarctic (ISEA). Accuracy assessment of the RAMPv2 DEM, CartoDEM, and combined eDEM (RAMPv2 + CartoDEM) by using DGPS as ground reference data shows that eDEM achieves much better accuracy (average elevation error 8.44 m) than that of existing DEM constructed by using only CartoDEM (13.57 m) or RAMPv2 (41.44 m) alone. The newly constructed eDEM achieves a vertical accuracy of about 7 times better than RAMPv2 DEM and 1.5 times better than CartoDEM. After using accurate DGPS data for accuracy assessment, the approximation to the actual surface of the eDEM extracted here is much more accurate with least mean root mean square error (RMSE) of 9.22 m than that constructed by using only CartoDEM (RMSE = 14.15 m) point elevation data and RAMPv2 (RMSE = 69.48 m) point elevation data. Our results indicate that, the overall trend of accuracy for the interpolation methods for generating continuous elevation surface from CartoDEM + RAMPv2 point elevation data, based on RMSE, is as follows: GP1 > IDW > GP2 > OK > LP2 > DK > LP1 > RBF > SK > UK. In case of cokriging interpolation methods, OCoK yields more accurate eDEM with the least RMSE of 8.16 m, which can be utilized to generate a highly accurate DEM of the research area.. Based on this work, it is inferred that GP2 and OCok interpolation methods and synergistic use of RAMPv2 and CartoDEM-based point elevation datasets lead to a highly accurate DEM of the study region. This research experiment demonstrates the stability (w.r.t multi-temporal datasets), performance (w.r.t best interpolation technique) and consistency (w.r.t all the experimented interpolation techniques) of synergistically fused eDEM. On the basis of average elevation difference and RMSE mentioned in present research, the newly constructed eDEM may serve as a benchmark for future elevation models such as from the ICESAT-II mission to spatially monitor ice sheet elevation.
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Zhang, Zebin, Martin Buisson, Pascal Ferrand, and Manuel Henner. "Integration of Second-Order Sensitivity Method and CoKriging Surrogate Model." Mathematics 9, no. 4 (February 18, 2021): 401. http://dx.doi.org/10.3390/math9040401.
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The global exploring feature of the surrogate model makes it a useful intermedia for design optimization. The accuracy of the surrogate model is closely related with the efficiency of optima-search. The cokriging approach described in present studies can significantly improve the surrogate model accuracy and cut down the turnaround time spent on the modeling process. Compared to the universal Kriging method, the cokriging method interpolates not only the sampling data, but also on their associated derivatives. However, the derivatives, especially high order ones, are too computationally costly to be easily affordable, forming a bottleneck for the application of derivative enhanced methods. Based on the sensitivity analysis of Navier–Stokes equations, current study introduces a low-cost method to compute the high-order derivatives, making high order derivatives enhanced cokriging modeling practically achievable. For a methodological illustration, second-order derivatives of regression model and correlation models are proposed. A second-order derivative enhanced cokriging model-based optimization tool was developed and tested on the optimal design of an automotive engine cooling fan. This approach improves the modern optimal design efficiency and proposes a novel direction for the large scale optimization problems.
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Ma, Liang, and Chang Qing Zuo. "A Comparison of Spatial Interpolation Models for Mapping Rainfall Erosivity on China Mainland." Advanced Materials Research 518-523 (May 2012): 4489–95. http://dx.doi.org/10.4028/www.scientific.net/amr.518-523.4489.
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Rainfall erosivity is an essential factor to reveal the response of water erosion to precipitation changes, and its spatial variation reveals erosion regional difference and water conservation regionalization. In this research, average annual rainfall erosivity in 1951 -2008 on China mainland is calculated through daily precipitation data from 711 meteorological stations. Precisions of 29 spatial interpolation models are quantitative compared including inverse distance weighting (IDW), radial basis function (RBF), kriging, cokriging (CK) and thin plate smoothing spline (TPS). Three variables cubic TPS is confirmed the optimum spatial interpolation model to rainfall erosivity on a large scale.
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Akbari, Haghighi, Aghayi, Javadian, Tajrishy, and Kløve. "Assimilation of Satellite-Based Data for Hydrological Mapping of Precipitation and Direct Runoff Coefficient for the Lake Urmia Basin in Iran." Water 11, no. 8 (August 6, 2019): 1624. http://dx.doi.org/10.3390/w11081624.
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Water management in arid basins often lacks sufficient hydro-climatological data because, e.g., rain gauges are typically absent at high elevations and inflow to ungauged areas around large closed lakes is difficult to estimate. We sought to improve precipitation and runoff estimation in an arid basin (Lake Urmia, Iran) using methods involving assimilation of satellite-based data. We estimated precipitation using interpolation of rain gauge data by kriging, downscaling the Tropical Rainfall Measuring Mission (TRMM), and cokriging interpolation of in-situ records with Remote Sensing (RS)-based data. Using RS-based data application in estimations gave more precise results, by compensating for lack of data at high elevations. Cokriging interpolation of rain gauges by TRMM and Digitized Elevation Model (DEM) gave 4–9 mm lower Root Mean Square Error (RMSE) in different years compared with kriging. Downscaling TRMM improved its accuracy by 14 mm. Using the most accurate precipitation result, we modeled annual direct runoff with Kennessey and Soil Conservation Service Curve Number (SCS-CN) models. These models use land use, permeability, and slope data. In runoff modeling, Kennessey gave higher accuracy. Calibrating Kennessey reduced the Normalized RMSE (NRMSE) from 1 in the standard model to 0.44. Direct runoff coefficient map by 1 km spatial resolution was generated by calibrated Kennessey. Validation by the closest gauges to the lake gave a NRMSE of 0.41 which approved the accuracy of modeling.
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Ly, S., C. Charles, and A. Degré. "Spatial interpolation of daily rainfall at catchment scale: a case study of the Ourthe and Ambleve catchments, Belgium." Hydrology and Earth System Sciences Discussions 7, no. 5 (September 27, 2010): 7383–416. http://dx.doi.org/10.5194/hessd-7-7383-2010.
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Abstract. Spatial interpolation of precipitation data is of great importance for hydrological modelling. Geostatistical methods (krigings) are widely used in spatial interpolation from point measurement to continuous surfaces. However, the majority of existing geostatistical algorithms are available only for single-moment data. The first step in kriging computation is the semi-variogram modelling which usually uses only one variogram model for all-moment data. The objective of this paper was to develop different algorithms of spatial interpolation for daily rainfall on 1 km2 regular grids in the catchment area and to compare the results of geostatistical and deterministic approaches. In this study, we used daily rainfall data from 70 raingages in the hilly landscape of the Ourthe and Ambleve catchments in Belgium (2908 km2). This area lies between 35 and 693 m in elevation and consists of river networks, which are tributaries of the Meuse River. For geostatistical algorithms, Cressie's Approximate Weighted Least Squares method was used to fit seven semi-variogram models (logarithmic, power, exponential, Gaussian, rational quadratic, spherical and penta-spherical) to daily sample semi-variogram on a daily basis. Seven selected raingages were used to compare the interpolation performance of these algorithms applied to many degenerated-raingage cases. Spatial interpolation with the geostatistical and Inverse Distance Weighting (IDW) algorithms outperformed considerably interpolation with the Thiessen polygon that is commonly used in various hydrological models. Kriging with an External Drift (KED) and Ordinary Cokriging (OCK) presented the highest Root Mean Square Error (RMSE) between the geostatistical and IDW methods. Ordinary Kriging (ORK) and IDW were considered to be the best methods, as they provided smallest RMSE value for nearly all cases.
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Fraczek, Witold, Andrzej Bytnerowicz, and Michael J. Arbaugh. "Application of the ESRI Geostatistical Analyst for Determining the Adequacy and Sample Size Requirements of Ozone Distribution Models in the Carpathian and Sierra Nevada Mountains." Scientific World JOURNAL 1 (2001): 836–54. http://dx.doi.org/10.1100/tsw.2001.317.
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Models of O3distribution in two mountain ranges, the Carpathians in Central Europe and the Sierra Nevada in California were constructed using ArcGIS Geostatistical Analyst extension (ESRI, Redlands, CA) using kriging and cokriging methods. The adequacy of the spatially interpolated ozone (O3) concentrations and sample size requirements for ozone passive samplers was also examined. In case of the Carpathian Mountains, only a general surface of O3distribution could be obtained, partially due to a weak correlation between O3concentration and elevation, and partially due to small numbers of unevenly distributed sample sites. In the Sierra Nevada Mountains, the O3monitoring network was much denser and more evenly distributed, and additional climatologic information was available. As a result the estimated surfaces were more precise and reliable than those created for the Carpathians. The final maps of O3concentrations for Sierra Nevada were derived from cokriging algorithm based on two secondary variables — elevation and maximum temperature as well as the determined geographic trend. Evenly distributed and sufficient numbers of sample points are a key factor for model accuracy and reliability.
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Malvić, Tomislav. "Kriging, cokriging or stochastical simulations, and the choice between deterministic or sequential approaches." Geologia Croatica 61, no. 1 (January 7, 2008): 37–47. http://dx.doi.org/10.4154/gc.2008.06.
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This new research presented here for the first time in the Croatian geomathematical community, aims to establish several criteria which will aid in the selection of deterministic or stochastic geostatistical methods of estimation. This review associates the theoretical background of kriging, cokriging or stochastic simulations with some results obtained by mapping Badenian clastic reservoirs in the Drava Depression. The selected reservoirs are located in the Stari Gradac-Barcs Nyugat field, in the Western part of the Drava Depression, and at the Beni?anci field in the Eastern part of this depression. Both datasets (each with 14 points) include mean porosity values taken from well log analysis in the reservoir interval at the well sites. This resulted in significant uncertainties in the variogram models, especially with the determination of range (at the secondary variogram axis). The critical advantage was the availability of a seismic attribute that could be used as a secondary and co-regionalized variable (porosity is the primary variable). In the first example (the Beni?anci field) the seismic attribute was correlated with the average logged porosities. This made it possible to apply the cokriging method as the best interpolation option. The cross-validation result was 2.19 for 14 wells. In contrast, the existence of only a primary variable at the Stari Gradac-Barcs Nyugat field, forced the application of stochastic simulations as the better estimation tool, which can better describe the porosity changes in inter-well areas.
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Souza, Fábio Henrique Morais de, Marcelo Ribeiro Viola, Junior Cesar Avanzi, Marcos Giongo, and Marcelo Vieira Filho. "THORNTHWAITE’S CLIMATE REGIONALIZATION FOR THE STATE OF TOCANTINS, BRAZIL." FLORESTA 49, no. 4 (September 19, 2019): 783. http://dx.doi.org/10.5380/rf.v49i4.59188.
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Tocantins State faces a large-scale agricultural expansion. Thus, climate studies are essential for a better understanding of climate variability supporting agricultural and environmental planning. In this context, this study applies the climatic classification of Thornthwaite and develops a climate regionalization through geostatistical techniques, assessing the performance of the interpolators ordinary kriging (OK) and cokriging (CK). Data from 26 weather stations located in Tocantins State and surroundings were used. The variables of interest to climate regionalization, obtained by the climatic water balance, were mapped by geostatistical techniques. The results of cross-validation showed that ordinary kriging and cokriging performed well. The spherical and exponential semivariogram models obtained the best fit in 40% of the analyzes each, and the gaussian in 20%. The climatic classification of Thornthwaite applied to Tocantins State showed the presence of humid (B1), moist subhumid (C2), and dry subhumid (C1) climates. There were found three climatic regions: B1A’wa’: Humid, megathermal, with moderate winter water deficiency, and a temperature efficiency regime normal to megathermal , occurring in the western region of the state; C2A’wa’: Moist subhumid, megathermal, with moderate winter water deficiency, and a temperature efficiency regime normal to megathermal , occurring in the central region and extending from the north to the south of the state; and C1A’w2a’: Dry subhumid, megathermal, with large summer water surplus, and a temperature efficiency regime normal to megathermal , in the east and northeast of the state.
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Araújo, Cristina Paixão, and João Felipe Coimbra Leite Costa. "Integration of different-quality data in short-term mining planning." Rem: Revista Escola de Minas 68, no. 2 (June 2015): 221–27. http://dx.doi.org/10.1590/0370-44672015680212.
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AbstractDecisions, from mineral exploration to mining operations, are based on grade block models obtained from samples. This study evaluates the impact of using imprecise data in short-term planning. The exhaustive Walker Lake dataset is used and is considered as the source for obtaining the true grades. Initially, samples are obtained from the exhaustive dataset at regularly spaced grids of 20 × 20 m and 5 × 5 m. A relative error (imprecision) of ±25% and a 10% bias are added to the data spaced at 5 × 5 m (short-term geological data) in different scenarios. To combine these different types of data, two methodologies are investigated: cokriging and ordinary kriging. Both types of data are used to estimate blocks with the two methodologies. The grade tonnage curves and swath plots are used to compare the results against the true block grade distribution. In addition, the block misclassification is evaluated. The results show that standardized ordinary cokriging is a better methodology for imprecise and biased data and produces estimates closer to the true grade block distribution, reducing block misclassification.
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Ver Hoef, Jay M., Noel Cressie, and Ronald Paul Barry. "Flexible Spatial Models for Kriging and Cokriging Using Moving Averages and the Fast Fourier Transform (FFT)." Journal of Computational and Graphical Statistics 13, no. 2 (June 2004): 265–82. http://dx.doi.org/10.1198/1061860043498.
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Madani, Nasser, Mohammad Maleki, and Fatemeh Sepidbar. "Integration of Dual Border Effects in Resource Estimation: A Cokriging Practice on a Copper Porphyry Deposit." Minerals 11, no. 7 (June 22, 2021): 660. http://dx.doi.org/10.3390/min11070660.
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Hierarchical or cascade resource estimation is a very common practice when building a geological block model in metalliferous deposits. One option for this is to model the geological domains by indicator kriging and then to estimate (by kriging) the grade of interest within the built geodomains. There are three problems regarding this. The first is that sometimes the molded geological domains are spotty and fragmented and, thus, far from the geological interpretation. The second is that the resulting estimated grades highly suffer from a smoothing effect. The third is related to the border effect of the continuous variable across the boundary of geological domains. The latter means that the final block model of the grade shows a very abrupt transition when crossing the border of two adjacent geological domains. This characteristic of the border effect may not be always true, and it is plausible that some of the variables show smooth or soft boundaries. The case is even more complicated when there is a mixture of hard and soft boundaries. A solution is provided in this paper to employ a cokriging paradigm for jointly modeling grade and geological domains. The results of modeling the copper in an Iranian copper porphyry deposit through the proposed approach illustrates that the method is not only capable of handling the mixture of hard and soft boundaries, but it also produces models that are less influenced by the smoothing effect. These results are compared to an independent kriging, where each variable is modeled separately, irrespective of the influence of geological domains.
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Kitterød, Nils-Otto, and Étienne Leblois. "Estimation of sediment thickness by solving Poisson's equation with bedrock outcrops as boundary conditions." Hydrology Research 52, no. 3 (March 11, 2021): 597–619. http://dx.doi.org/10.2166/nh.2021.102.
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Abstract Sediment thickness and bedrock topography are vital for the terrestrial hydrosphere. In this study, we estimated sediment thickness by using information from digital elevation models, geological maps, and public databases. We discuss two different approaches: First, the horizontal distances to the nearest bedrock outcrop were used as a secondary function in kriging and cokriging. Second, we applied Poisson's equation to estimate the local trend of the sediment thickness where bedrock outcrops were used as boundary conditions. Differences between point observations and the parabolic surface from Poisson's equation were minimized by inverse modelling. Ordinary kriging was applied to the residuals. These two approaches were evaluated with data from the Øvre Eiker, Norway. Estimates derived from Poisson's equation gave the smallest mean absolute error, and larger soil depths were reproduced better if the local trend was included in the estimation procedure. An independent cross-validation was undertaken. The results showed the best accuracy and precision for kriging on the residuals from Poisson's equation. Solutions of Poisson's equation are sensitive to the boundary conditions, which in this case were locations of the bedrock outcrops. Bedrock outcrops are available for direct observations; hence, the quality of the estimates can be improved by updating input from high-resolution mapping.
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Kibirige, Daniel, and Endre Dobos. "Soil Moisture Estimation Using Citizen Observatory Data, Microwave Satellite Imagery, and Environmental Covariates." Water 12, no. 8 (July 30, 2020): 2160. http://dx.doi.org/10.3390/w12082160.
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Soil moisture (SM) is a key variable in the climate system and a key parameter in earth surface processes. This study aimed to test the citizen observatory (CO) data to develop a method to estimate surface SM distribution using Sentinel-1B C-band Synthetic Aperture Radar (SAR) and Landsat 8 data; acquired between January 2019 and June 2019. An agricultural region of Tard in western Hungary was chosen as the study area. In situ soil moisture measurements in the uppermost 10 cm were carried out in 36 test fields simultaneously with SAR data acquisition. The effects of environmental covariates and the backscattering coefficient on SM were analyzed to perform SM estimation procedures. Three approaches were developed and compared for a continuous four-month period, using multiple regression analysis, regression-kriging and cokriging with the digital elevation model (DEM), and Sentinel-1B C-band and Landsat 8 images. CO data were evaluated over the landscape by expert knowledge and found to be representative of the major SM distribution processes but also presenting some indifferent short-range variability that was difficult to explain at this scale. The proposed models were evaluated using statistical metrics: The coefficient of determination (R2) and root mean square error (RMSE). Multiple linear regression provides more realistic spatial patterns over the landscape, even in a data-poor environment. Regression kriging was found to be a potential tool to refine the results, while ordinary cokriging was found to be less effective. The obtained results showed that CO data complemented with Sentinel-1B SAR, Landsat 8, and terrain data has the potential to estimate and map soil moisture content.
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Ly, S., C. Charles, and A. Degré. "Geostatistical interpolation of daily rainfall at catchment scale: the use of several variogram models in the Ourthe and Ambleve catchments, Belgium." Hydrology and Earth System Sciences 15, no. 7 (July 18, 2011): 2259–74. http://dx.doi.org/10.5194/hess-15-2259-2011.
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Abstract. Spatial interpolation of precipitation data is of great importance for hydrological modelling. Geostatistical methods (kriging) are widely applied in spatial interpolation from point measurement to continuous surfaces. The first step in kriging computation is the semi-variogram modelling which usually used only one variogram model for all-moment data. The objective of this paper was to develop different algorithms of spatial interpolation for daily rainfall on 1 km2 regular grids in the catchment area and to compare the results of geostatistical and deterministic approaches. This study leaned on 30-yr daily rainfall data of 70 raingages in the hilly landscape of the Ourthe and Ambleve catchments in Belgium (2908 km2). This area lies between 35 and 693 m in elevation and consists of river networks, which are tributaries of the Meuse River. For geostatistical algorithms, seven semi-variogram models (logarithmic, power, exponential, Gaussian, rational quadratic, spherical and penta-spherical) were fitted to daily sample semi-variogram on a daily basis. These seven variogram models were also adopted to avoid negative interpolated rainfall. The elevation, extracted from a digital elevation model, was incorporated into multivariate geostatistics. Seven validation raingages and cross validation were used to compare the interpolation performance of these algorithms applied to different densities of raingages. We found that between the seven variogram models used, the Gaussian model was the most frequently best fit. Using seven variogram models can avoid negative daily rainfall in ordinary kriging. The negative estimates of kriging were observed for convective more than stratiform rain. The performance of the different methods varied slightly according to the density of raingages, particularly between 8 and 70 raingages but it was much different for interpolation using 4 raingages. Spatial interpolation with the geostatistical and Inverse Distance Weighting (IDW) algorithms outperformed considerably the interpolation with the Thiessen polygon, commonly used in various hydrological models. Integrating elevation into Kriging with an External Drift (KED) and Ordinary Cokriging (OCK) did not improve the interpolation accuracy for daily rainfall. Ordinary Kriging (ORK) and IDW were considered to be the best methods, as they provided smallest RMSE value for nearly all cases. Care should be taken in applying UNK and KED when interpolating daily rainfall with very few neighbourhood sample points. These recommendations complement the results reported in the literature. ORK, UNK and KED using only spherical model offered a slightly better result whereas OCK using seven variogram models achieved better result.
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Caloiero, Tommaso, Gaetano Pellicone, Giuseppe Modica, and Ilaria Guagliardi. "Comparative Analysis of Different Spatial Interpolation Methods Applied to Monthly Rainfall as Support for Landscape Management." Applied Sciences 11, no. 20 (October 14, 2021): 9566. http://dx.doi.org/10.3390/app11209566.
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Landscape management requires spatially interpolated data, whose outcomes are strictly related to models and geostatistical parameters adopted. This paper aimed to implement and compare different spatial interpolation algorithms, both geostatistical and deterministic, of rainfall data in New Zealand. The spatial interpolation techniques used to produce finer-scale monthly rainfall maps were inverse distance weighting (IDW), ordinary kriging (OK), kriging with external drift (KED), and ordinary cokriging (COK). Their performance was assessed by the cross-validation and visual examination of the produced maps. The results of the cross-validation clearly evidenced the usefulness of kriging in the spatial interpolation of rainfall data, with geostatistical methods outperforming IDW. Results from the application of different algorithms provided some insights in terms of strengths and weaknesses and the applicability of the deterministic and geostatistical methods to monthly rainfall. Based on the RMSE values, the KED showed the highest values only in April, whereas COK was the most accurate interpolator for the other 11 months. By contrast, considering the MAE, the KED showed the highest values in April, May, June and July, while the highest values have been detected for the COK in the other months. According to these results, COK has been identified as the best method for interpolating rainfall distribution in New Zealand for almost all months. Moreover, the cross-validation highlights how the COK was the interpolator with the best least bias and scatter in the cross-validation test, with the smallest errors.
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Paz-Ferreiro, Jorge, Eva Vidal Vázquez, and Sidney Rosa Vieira. "Geostatistical analysis of a geochemical dataset." Bragantia 69, suppl (2010): 121–29. http://dx.doi.org/10.1590/s0006-87052010000500013.
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The application of geostatistics to data obtained from geochemical prospecting process can provide useful information for evaluating mineralization potential. The objective of this study was to evaluate the spatial distribution of Au, As and Sb contents over a large area of the Coruña province, Spain. A geochemical survey was carried out from which a data set with 323 samples was collected. Macroelements and trace elements were determined by routine analytical techniques. The spatial variability was assessed using semivariogram and cross-semivariogram as well as indicator semivariogram analysis. Frequency distributions of the studied elements departed from normal, as indicated by skewness and kurtosis coefficients. Coefficients of variation ranked as follows: Sb < As < Au. Significant correlation coefficients between Au, Sb and As were found, even though the correlation values were low. Spherical models with nugget effects ranging from 50% (As) to 57.8% (Au) were fitted to the experimental semivariograms. Cross semivariograms of Au versus Sb and As showed smaller nugget variance than individual semivariograms. Indicator semivariograms were calculated taken mean, median, and different percentiles as threshold values. Ordinary kriging, cokriging, and indicator kriging were performed to generate geochemical maps. The method has succeeded in effectively extracting useful information, and improving the analysis of the metallogenic and ore-controlling factors, thereby playing an important role in qualitative and quantitative predictions.
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Vicente Serrano, Sergio, and Miguel Ángel Saz Sánchez. "Cartografía de precipitaciones y temperaturas en el valle medio del Ebro mediante la utilización de diferentes técnicas estadísticas." Geographicalia, no. 42 (June 21, 2016): 73. http://dx.doi.org/10.26754/ojs_geoph/geoph.2002421362.
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Este trabajo analiza la calidad final de diferentes cartografías continuas de precipitaciones y temperaturas realizadas a partir de distintos métodos de interpolación. El análisis se ha realizado en el sector central del valle del Ebro, un espacio de topografía suave pero que se encuentra dominado por cornplejos patrones climáticos. Se han realizado 23 cartografías con diferentes tipos de métodos (locales, globales y geoestadísticos). Los mejores resultados en la cartografía de precipitaciones se han obtenido mediante la aplicación de métodos geoestadísticos, en concreto las técnicas de block-kriging y cokriging. Las temperaturas se han cartografiado con mejores resultados a partir de un modelo de regresión múltiple. Se discute acerca de la utilidad de las diferentes técnicas cartográficas en función de la variable analizada.
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SILVA, Camila Santos da, Bruno Araujo Furtado de MENDONÇA, Marcos Gervasio PEREIRA, Emanuel José Gomes de ARAÚJO, and Débora Christina CASTELLANI. "Spatial dependency and correlation of properties of soil cultivated with oil palm, Elaeis guineensis, in agroforestry systems in the eastern Brazilian Amazon." Acta Amazonica 48, no. 4 (December 2018): 280–89. http://dx.doi.org/10.1590/1809-4392201704423.
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ABSTRACT Geostatistics is a tool that can be used to produce maps with the distribution of nutrients essential for the development of plants. Therefore, the present study aimed to analyze the spatial variation in chemical attributes of soils under oil palm cultivation in agroforestry systems in the eastern Brazilian Amazon, and their spatial dependence pattern. Sixty spatially standardized and georeferenced soil samples were collected at each of three sampling sites (DU1, DU2, and DU3) at 0-20 cm depth. Evaluated soil chemical attributes were pH, Al3+, H+Al, K+, Ca2+, Mg2+, cation exchange capacity (CEC), P, and organic matter (OM). The spatial dependence of these variables was evaluated with a semivariogram analysis, adjusting three theoretical models (spherical, exponential, and Gaussian). Following analysis for spatial dependence structure, ordinary kriging was used to estimate the value of each attribute at non-sampled sites. Spatial correlation among the attributes was tested using cokriging of data spatial distribution. All variables showed spatial dependence, with the exception of pH, in one sampling site (DU3). Highest K+, Ca2+, Mg2+, and OM levels were found in the lower region of two sampling sites (DU1 and DU2). Highest levels of Al3+ and H+Al levels were observed in the lower region of sampling site DU3. Some variables were correlated, therefore cokriging proved to be efficient in estimating primary variables as a function of secondary variables. The evaluated attributes showed spatial dependence and correlation, indicating that geostatistics may contribute to the effective management of agroforestry systems with oil palm in the Amazon region.
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Kučerová, Dana, and Michal Jeníček. "Comparison of selected methods used for the calculation of the snowpack spatial distribution, Bystřice River basin, Czechia." Geografie 119, no. 3 (2014): 199–217. http://dx.doi.org/10.37040/geografie2014119030199.
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The knowledge of the water volume stored in the snowpack, including its spatial distribution, is vital for many hydrological applications. Such information is useful for hydrological forecasts and it is often used for the calibration of snowmelt runoff models. Data from four field measurements of the snow water equivalent (SWE) carried out in two winter seasons were assessed by ten interpolation methods. Measurements from both snow accumulation and snowmelt periods were evaluated. The ability of methods to predict SWE at unmeasured locations was assessed by the means of cross validation. The best prediction accuracy of SWE was achieved by means of multiple a simple linear regressions, residual kriging and cokriging methods. The accuracy was enhanced by the use of elevation, aspect, slope and vegetation as variables in the calculation of the SWE. Elevation and vegetation show a significant correlation with the SWE in the study area. The multiple regression gave best results for snow accumulation period. However, the spatial variability of SWE was not successfully explained for snowmelt periods.
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Blodgett, Clayton, and Mark Jakubauskas. "A Preliminary Assessment of Forest Canopy Structure in Grand Teton National Park." UW National Parks Service Research Station Annual Reports 18 (January 1, 1994): 37–41. http://dx.doi.org/10.13001/uwnpsrc.1994.3185.
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The potential impact of environmental change on human welfare has renewed interest in understanding the patterns and processes associated with global climate change. Goals of the Committee on Earth Sciences (1989) regarding the U.S. Global Climate Change Program concentrated on the development of sound scientific strategies for monitoring and predicting environmental change. The scaling of ecological characteristics from local to regional and global scales were identified by the Committee as key priorities. The scaling of ecological information is not simply done by integrating or aggregating information from local scale investigations to regional and global scales (Caldwell et al., 1993). The complexity of the effects of scale variations rules out the use of simple generalizations (Foody and Curran, 1994). Information that is significant at local scales may be trivial when evaluated at regional or global scales. Biological interactions with the environment occur over many scales, suggesting a role for multiscale analysis in the description of these interactions (Sclmeider, 1994). Methods must be developed to better understand and evaluate ecological processes operating at multiple scales. Forest structure attributes have been measured using remotely sensed data. Leaf area index (LAI), for example, has been related to the infrared/red ratio (Running et al., 1986 Peterson et al., 1987), the normalized difference vegetation index (NDVI) (Leblon et al., 1993), and gap fractions (Nel and Wessman, 1993). These methods generate values for each pixel in a satellite scene based on the relationship between one or more spectral and/or ancillary data channels and the attribute of interest. The spatial autocorrelation or spatial dependence present in surface phenomena and satellite data are usually not ex-ploited during attribute assignment because of difficulty in quantifying the spatial patterns present (Woodcock et al., 1988). Geostatistics provides a statistically based technique to quantify spatial pattern. Geostatistical techniques, in particular cokriging, can serve as an efficient means of modeling forest canopy structure at a variety of spatial scales to serve as inputs to global change models. The key issue will be to determine the factors that influence remotely sensed spectral reflectance and relating them to the ecological model across scales (Ustin et al., 1993). The geostatistical techniques considered in this research include the following: the semivariograrn, which allows the user to compare values of a random variable at two points separated by a given lag distance (Milne, 1991); kriging which uses the information on spatial dependence present in the semivariogram to estimate values at unsampled locations based on scattered sample data (lsaaks and Srivastava, 1989); and cokriging, the multivariate extension of kriging, which is appropriate when two or more variables are spatially interdependent and the variable of interest is undersampled (McBratney and Webster, 1983; Leenaers et al., 1989). Geostatistical techniques have been successfully applied to remotely sensed data. Variograms have been used to determine components of coniferous canopy structure (Cohen et al., 1990), and to determine the spatial autocorrelation structure of Landsat Thematic Mapper (IM) imagery and intercepted photosynthetically active radiation (IPAR) (Lathrop and Pierce, 1991). Atkinson et al. (1992) used cokriging of ground-based radiometer data to estimate LAI, dry biomass and percent cover. Satellite imagery is an excellent candidate for inclusion as an explanatory variable in the cokriging process because it is an exhaustive sample of a given area.
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Vasques, Gustavo M., Hugo M. Rodrigues, Maurício R. Coelho, Jesus F. M. Baca, Ricardo O. Dart, Ronaldo P. Oliveira, Wenceslau G. Teixeira, and Marcos B. Ceddia. "Field Proximal Soil Sensor Fusion for Improving High-Resolution Soil Property Maps." Soil Systems 4, no. 3 (August 21, 2020): 52. http://dx.doi.org/10.3390/soilsystems4030052.
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Mapping soil properties, using geostatistical methods in support of precision agriculture and related activities, requires a large number of samples. To reduce soil sampling and measurement time and cost, a combination of field proximal soil sensors was used to predict and map laboratory-measured soil properties in a 3.4-ha pasture field in southeastern Brazil. Sensor soil properties were measured in situ on a 10 × 10-m dense grid (377 samples) using apparent electrical conductivity meters, apparent magnetic susceptibility meter, gamma-ray spectrometer, water content reflectometer, cone penetrometer, and portable X-ray fluorescence spectrometer (pXRF). Soil samples were collected on a 20 × 20-m thin grid (105 samples) and analyzed in the laboratory for organic C, sum of bases, cation exchange capacity, clay content, soil volumetric moisture, and bulk density. Another 25 samples collected throughout the area were also analyzed for the same soil properties and used for independent validation of models and maps. To test whether the combination of sensors enhances soil property predictions, stepwise multiple linear regression (MLR) models of the laboratory soil properties were derived using individual sensor covariate data versus combined sensor data—except for the pXRF data, which were evaluated separately. Then, to test whether a denser grid sample boosted by sensor-based soil property predictions enhances soil property maps, ordinary kriging of the laboratory-measured soil properties from the thin grid was compared to ordinary kriging of the sensor-based predictions from the dense grid, and ordinary cokriging of the laboratory properties aided by sensor covariate data. The combination of multiple soil sensors improved the MLR predictions for all soil properties relative to single sensors. The pXRF data produced the best MLR predictions for organic C content, clay content, and bulk density, standing out as the best single sensor for soil property prediction, whereas the other sensors combined outperformed the pXRF sensor for the sum of bases, cation exchange capacity, and soil volumetric moisture, based on independent validation. Ordinary kriging of sensor-based predictions outperformed the other interpolation approaches for all soil properties, except organic C content, based on validation results. Thus, combining soil sensors, and using sensor-based soil property predictions to increase the sample size and spatial coverage, leads to more detailed and accurate soil property maps.
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Siqueira, Glécio Machado, Jorge Dafonte Dafonte, Javier Bueno Lema, Montserrat Valcárcel Armesto, and Ênio Farias França e. Silva. "Using Soil Apparent Electrical Conductivity to Optimize Sampling of Soil Penetration Resistance and to Improve the Estimations of Spatial Patterns of Soil Compaction." Scientific World Journal 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/269480.
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This study presents a combined application of an EM38DD for assessing soil apparent electrical conductivity (ECa) and a dual-sensor vertical penetrometer Veris-3000 for measuring soil electrical conductivity (ECveris) and soil resistance to penetration (PR). The measurements were made at a 6 ha field cropped with forage maize under no-tillage after sowing and located in Northwestern Spain. The objective was to use data from ECafor improving the estimation of soil PR. First, data of ECawere used to determine the optimized sampling scheme of the soil PR in 40 points. Then, correlation analysis showed a significant negative relationship between soil PR and ECa, ranging from −0.36 to −0.70 for the studied soil layers. The spatial dependence of soil PR was best described by spherical models in most soil layers. However, below 0.50 m the spatial pattern of soil PR showed pure nugget effect, which could be due to the limited number of PR data used in these layers as the values of this parameter often were above the range measured by our equipment (5.5 MPa). The use of ECaas secondary variable slightly improved the estimation of PR by universal cokriging, when compared with kriging.
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Castro Franco, Mauricio, Dayra Yisel García Ramírez, and Andrés Fernando Jiménez López. "Comparación de técnicas de interpolación espacial de propiedades del suelo en el piedemonte llanero colombiano." Tecnura 21, no. 53 (July 1, 2017): 78–95. http://dx.doi.org/10.14483/22487638.11658.
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Contexto: la interpolación espacial de propiedades del suelo en el piedemonte llanero colombiano es compleja debido al efecto simultáneo de la génesis del suelo, las características del terreno, el uso actual, y el manejo que de él se hace. Mientras las técnicas de interpolación vienen siendo adaptadas para tener en cuenta estos efectos, algunas propiedades del suelo son difíciles de predecir con las técnicas convencionales.Método: en este trabajo se evaluaron seis técnicas de interpolación espacial: distancia inversa ponderada (IDW); Spline; kriging ordinario (KO); kriging universal (KU); cokriging (Ckg); y mejor predicción lineal insesgada empírica con máxima verosimilitud restringida (REML-EBLUP), a partir de la aplicación de hipercubo latino condicionado (HCLc) como estrategia de muestreo. Se utilizaron los índices de terreno calculados a partir de un modelo digital de elevación como información auxiliar del suelo en los procedimientos de Ckg y REML-EBLUP. Para determinar los índices de terreno más importantes de cada propiedad se utilizó el algoritmo de bosques aleatorios (RF) y para validar las interpolaciones sobre validaciones cruzadas se usaron las métricas de error.Resultados: los resultados soportan el supuesto de que HCLc captura adecuadamente la distribución total de la información auxiliar en condiciones del área experimental. Además, sugieren que Ckg y REML-EBLUP generan mejores predicciones de la mayoría de las propiedades del suelo evaluadas.Conclusiones: las técnicas de interpolación mixtas, teniendo también información auxiliar del suelo e índices de terreno, proporcionaron una mejora significativa de la predicción de propiedades del suelo, en comparación con las demás técnicas.
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Behrens, R. A., M. K. MacLeod, T. T. Tran, and A. C. Alimi. "Incorporating Seismic Attribute Maps in 3D Reservoir Models." SPE Reservoir Evaluation & Engineering 1, no. 02 (April 1, 1998): 122–26. http://dx.doi.org/10.2118/36499-pa.
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Summary We introduce a new geostatistical method to incorporate seismic attribute maps into a three-dimensional (3D) reservoir model. The method explicitly honors the difference in vertical resolution between seismic and well-log data. The method, called sequential Gaussian simulation with block Kriging (SGSBK), treats the seismic map as a soft estimate of the average reservoir property. With this method, the average of the cell values in anyone vertical column of grid cells is constrained by the value of the seismic map over that column. The result is a model that contains vertical variability driven by well logs and the vertical-variogram model and spatial variability driven by the seismic map and the areal-variogram model. Introduction Reservoir models for flow simulation and volumetrics are often built from well logs by use of geostatistical methods. The well logs provide good vertical resolution required for accurate flow simulation, but they represent only a small portion of the reservoir. Seismic data is very complementary because it is areally dense, but vertically sparse relative to well-log data. The goal of the method presented is to integrate seismic data that more closely represents interval-average rock properties, and well-log data that more closely represents point-rock properties. This "volume support" difference is acknowledged and treated in the SGSBK method presented in this paper. Geostatistical methods that aim to build 3D reservoir models must honor the difference in volume support between well and seismic data, whereas methods for areal two-dimensional (2D) models do not. Averaged or integrated log properties no longer represent point properties but rather interval properties with lower vertical resolution similar to seismic data. The differences in volume support between the log average and seismic data are acceptable, because both represent large volumes of rock. No special treatment for volume support is thus used in areal 2D simulations that use both log and seismic-map data. This special circumstance is not true for 3D models, so any 3D method to incorporate log data and seismic data should address the volume-support problem. Literature Review There are several problems associated with integrating seismic and well data for 3D reservoir characterization: the seismic data must be converted from time to depth domain; seismic data is band-limited, whereas well data has both high- and low-frequency components; seismic data must be calibrated to well data; and a well measurement is of quasipoint support, whereas a seismic datum informs a much larger volume of reservoir rock. (The term quasipoint properties is used to represent the properties in a single cell rather than a core plug or smaller.) Several authors have worked on the calibration issue. Fournier1 and Fournier and Derain2 performed multivariate statistical analysis on a calibration dataset consisting of well logs and nearby seismic traces to establish a nonparametric regression between petrophysical properties and some seismic attributes. This regression is then applied on the seismic data to obtain seismic-derived reservoir properties that are, in turn, incorporated with well information using cokriging (and variants thereof). In their studies, Fournier and Fournier and Derain considered average properties (e.g., average porosity,1 cumulative lithofacies thickness2), because it was not possible to assess vertical distributions of reservoir properties from their limited-time-resolution seismic traces. Zhu and Journel3 proposed a different use of the well-seismic calibration dataset. In lieu of a regression, the well (hard) and seismic (soft) data are encoded as local prior probability distributions which are then "updated" into posterior distributions during the sequential indicator simulation process.4 Values of the property of interest are drawn randomly from these local posterio~ distributions. This method was found to be superior by Araktingi et al.,5 who applied it to a synthetic seismic dataset. Similarly, Doyen and Psaila6 used a "seismic likelihood function" constructed from a seismic-lithotype crossplot to modify the local probability distributions generated by the sequential indicator simulation algorithm; the result is lithologic models that are constrained by seismic data. Xu et al.7 proposed the sequential Gaussian simulation (SGS) with collocated cokriging algorithm as a more efficient, albeit less rigorous, alternative to SGS with full cokriging.8,9 This algorithm requires the correlation coefficient between the well and seismic data, and their cross-covariance model is derived from the covariance model of the well data. Xu et al. showed a 2D study where the algorithm was applied to incorporate well data and seismic two-way travel times to create realizations of the structure top of a salt dome. Yang et al.10 used SGS with collocated cokriging to construct 3D porosity models conditional to both well and seismic data. In one approach, the seismic amplitude was used as soft data; the required correlation coefficient was obtained by crossplotting averaged porosity and absolute seismic amplitude. In another approach, the inverted seismic impedance was used as soft data. Although not explicitly stated by the authors, either an interpolation or simulation procedure was used to populate the simulation grid with soft seismic data, because the vertical resolution of the seismic data is much less than that of the simulation grid. Gorell1 proposed a method to account for the difference in vertical resolution of seismic and well data. First, the wells are subdivided vertically into correlatable layers. Each layer is then populated with porosity values using 2D geostatistical operations. Finally, linear rescaling is performed on each vertical column of the simulation grid to ensure that a seismic-derived average porosity map is honored. The resulting 3D porosity model honors both the vertical variations at the well locations and average porosity map. This technique can be applied to several different vertical zones of the reservoir with different average porosity maps, and the rescaled results are stacked together at the end. As pointed out by the author, this technique requires that the wells be vertical or the well data may not be honored. In addition, vertical correlation of porosity between layers is only honored indirectly through the interwell-correlation process and the probability density function (pdf) of the point data is distorted, as will be shown later. Burns et al.12 used a similar resealing procedure to improve the description of a 100-ft-thick reservoir.
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Logan, J., and M. A. Mueller. "Using Geospatial Techniques and GIS to Develop Maps of Freeze Probabilities and Growing Degrees." HortScience 35, no. 4 (July 2000): 558D—558c. http://dx.doi.org/10.21273/hortsci.35.4.558d.
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Tennessee is located in an area of diverse topography, ranging in elevation from <100 m to ≈2000 m, with numerous hills and valleys. The physiography makes it very difficult to spatially interpolate weather data related to vegetable production, such as spring and fall freeze dates and growing degree days (GDD). In addition, there is a poor distribution of cooperative weather stations, especially those with 30 years or more of data. There are climate maps available for Tennessee, but they are of such a general format as to be useless for operational applications. This project is designed to use a geographic information system (GIS) and geospatial techniques to spatially interpolate freeze (0 °C) dates and GDD for different base temperatures and make the data available as Internet-based maps. The goal is to develop reasonable climate values for vegetable growing areas <1000 m in elevation at a 100 square km resolution. The geostatistics that we are evaluating include Thiessen polygons, triangulated irregular network (TIN), inverse distance weighting (IDW), spline, kriging, and cokriging. Data from 140 locations in and around Tennessee are used in the analysis. Incomplete data from 100 other locations are used to validate the models. GDD, which have much less year-to-year variability than freeze dates, can be successfully interpolated using inverse distance weighting (IDW) or spline techniques. Even a simple method like Thiessen produces fairly accurate maps. Freeze dates, however, are better off analyzed on an annual basis because the patterns can vary significantly from year to year. The annual maps can then be superimposed to give a better estimate of average spring and fall freeze dates.
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Behrens, R. A., and T. T. Tran. "Incorporating Seismic Data of Intermediate Vertical Resolution Into Three-Dimensional Reservoir Models: A New Method." SPE Reservoir Evaluation & Engineering 2, no. 04 (August 1, 1999): 325–33. http://dx.doi.org/10.2118/57481-pa.
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Summary Three-dimensional (3D) earth models are best created with a combination of well logs and seismic data. Seismic data have good lateral resolution but poor vertical resolution compared to wells. The seismic resolution depends on seismic acquisition and reservoir parameters, and is incorporated into the 3D earth model with different techniques depending on this resolution relative to that of the 3D model. Good vertical resolution of the seismic data may warrant integrating it as a continuous vertical variable informing local reservoir properties, whereas poor resolution warrants using only a single map representing vertically averaged reservoir properties. The first case best applies to thick reservoirs and/or high-frequency seismic data in soft rock and is usually handled using a cokriging-type approach. The second case represents the low end of the seismic resolution spectrum, where the seismic map can now be treated by methods such as block kriging, simulated annealing, or Bayesian techniques. We introduce a new multiple map Bayesian technique with variable weights for the important middle ground where a single seismic map cannot effectively represent the entire reservoir. This new technique extends a previous Bayesian technique by incorporating multiple seismic property maps and also allowing vertically varying weighting functions for each map. This vertical weighting flexibility is physically important because the seismic maps represent reflected wave averages from rock property contrasts such as at the top and base of the reservoir. Depending on the seismic acquisition and reservoir properties, the seismic maps are physically represented by simple but nonconstant weights in the new 3D earth modeling technique. Two field examples are shown where two seismic maps are incorporated in each 3D earth model. The benefit of using multiple maps is illustrated with the geostatistical concept of probability of exceedance. Finally, a postmortem is presented showing well path trajectories of a successful and unsuccessful horizontal well that are explained by model results based on data existing before the wells were drilled. Introduction Three-dimensional (3D) earth models are greatly improved by including seismic data because of the good lateral coverage compared with well data alone. The vertical resolution of seismic data is poor compared with well data, but it may be high or low compared with the reservoir thickness as depicted in Fig. 1. Seismic resolution is typically considered to be one-fourth of a wavelength (?/4) although zones of thinner rock property contrasts can be detected. The seismic resolution relative to the reservoir thickness constrains the applicability of different geostatistical techniques for building the 3D earth model. Fig. 1 is highly schematic and not meant to portray seismic data as a monochromatic (single-frequency) wave. The reference to wavelength here is based on the dominant frequency in the seismic data. Fig. 1 is meant to illustrate the various regimes of vertical resolution in seismic data relative to the reservoir thickness. While there are all sorts of issues, such as tuning, that must be considered in the left two cases, we need to address these cases because of their importance. Seismic data having little vertical resolution over the reservoir interval, as in the left case of Fig. 1 can use geostatistical techniques that incorporate one seismic attribute map. The single attribute can be a static combination of multiple attributes in a multivariate sense but the combination cannot vary spatially. These techniques include sequential Gaussian simulation with Block Kriging1 (SGSBK), simulated annealing,2 or sequential Gaussian simulation with Bayesian updating.3,4 Some of these methods are extendable beyond a single seismic map with modification. Seismic data having good vertical resolution over the reservoir interval, as in the right seismic trace of Fig. 1, can use geostatistical techniques that incorporate 3D volumes of seismic attributes. Techniques include simulated annealing, collocated cokriging simulation,5 a Markov-Bayes approach,6 and spectral separation. The term "3D volume" of seismic, as used here, is distinguished from the term "3D seismic data." (A geophysicist speaks of 3D seismic data when it is acquired over the surface in areal swaths or patches for the purpose of imaging a 3D volume of the earth. Two-dimensional (2D) seismic is acquired along a line on the surface for the purpose of imaging a 2D cross section of the earth.) The 3D volume distinction is made based on the vertical resolution of the seismic relative to the reservoir. To be considered a 3D volume here, we require both lateral and vertical resolution within the reservoir. Seismic data often do not have the vertical resolution within the reservoir zone to warrant using a 3D volume of seismic data. The low and high limits of vertical resolution leave out the case of intermediate vertical resolution as depicted by the middle curve of Fig. 1. Because typical seismic resolution often ranges from 10 to 40 m and many reservoirs have thicknesses one to two times this range, many reservoirs fall into this middle ground. These reservoirs have higher vertical seismic resolution than a single map captures, but not enough to warrant using a 3D volume of seismic. It is this important middle ground that is addressed by a new technique presented in this paper.
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Hu, L. Y., Georges Blanc, and Benoit Noetinger. "Estimation of Lithofacies Proportions by Use of Well and Well-Test Data." SPE Reservoir Evaluation & Engineering 1, no. 01 (February 1, 1998): 69–74. http://dx.doi.org/10.2118/36571-pa.
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Summary A crucial step of the two commonly used geostatistical methods for modeling heterogeneous reservoirs, sequential indicator simulation and truncated Gaussian simulation, is the estimation of the lithofacies local proportion (or probability density) functions. Well-test-derived permeabilities show good correlation with lithofacies proportions around wells. Integrating well and well-test data in estimating lithofacies proportions could permit the building of more realistic models of reservoir heterogeneity. This integration is difficult, however, because of the different natures and measurement scales of these two types of data. This paper presents a two-step approach to integrating well and well-test data into heterogeneous reservoir modeling. First, we estimate lithofacies proportions in well-test investigation areas with a new kriging algorithm called KISCA. KISCA consists of kriging jointly the proportions of all lithofacies in a well-test investigation area so that the corresponding well-test-derived permeability is respected through a weighted power-averaging of lithofacies permeabilities. For multiple well tests, an iterative process is used in KISCA to account for their interaction. After this, the estimated proportions are combined with lithofacies indicators at wells for estimating proportion (or probability density) functions over the entire reservoir field with a classical kriging method. We considered some numerical examples to test the proposed method for estimating lithofacies proportions. In addition, we generated a synthetic lithofacies reservoir model and performed a well-test simulation. The comparison between the experimental and estimated proportions in the well-test investigation area demonstrates the validity of the proposed method. Introduction Recent research on stochastic reservoir modeling constrained by well-test data has focused on the approach based on the Bayesian inversion theory and the Markov chain Monte Carlo methods.1–3 This approach is very attractive because it can deal directly with wellpressure data rather than well-test-derived permeability data and because it can be extended to history matching. However, it is limited to gross grid reservoir models in the context of continuous Gaussian-related variables (e.g., log-normal permeability field). In the case of a stabilized well-test, it is possible to define an effective permeability in the corresponding investigation area (well-test-derived permeability). A method based on simulated annealing has been used for conditioning permeability field to well-test derived permeabilities.4 Although the annealing process does not call for fluid-flow simulations, this method still can be very slow. This paper proposes an alternative approach for incorporating well-test-derived permeabilities into lithological reservoir models defined on fine grids. Consider two commonly used geostatistical methods, truncated Gaussian simulation5 and sequential indicator simulation,6 for building reservoir lithological models. These methods consist of first estimating the local proportion functions [or probability density functions (PDF)] of lithofacies. Then, the lithofacies model is built by truncating a Gaussian random function with the proportion functions (or by randomly drawing lithofacies from the PDF). A realistic modeling of lithofacies distribution with these geostatistical methods depends greatly on the accuracy of the estimation of the lithofacies proportions functions (or PDF). In the case of few well data, the incorporation of other sources of information (including geological knowledge, seismic information, well-test data, and field production data) would improve the estimation of proportion functions (or PDF) significantly. This paper covers the problem of incorporating well and well-test-derived permeability data into the estimation of lithofacies proportion functions (or PDF). We use a two-step approach: first the lithofacies proportions in well-test investigation areas are estimated with a new kriging algorithm called KISCA, then these estimates are combined with lithofacies indicators at wells for estimating lithofacies proportion functions (or PDF) over the entire reservoir field with a classical kriging method. Ref. 7 describes another method based on the cokriging technique for integrating well and well-test-derived permeability data. Also, there are existing methods for incorporating well and seismic data for estimating lithofacies proportion functions.8 Well and Well-Test Data The data set is made of a lithofacies description at available wells and a number of well-test-derived permeability values. The continuous lithofacies description on the wells is regularly discretized with a fineness defined according to the lithofacies variability along wells. At each point xa of the well discretization, an indicator is defined for each lithofacies:Equation 1 We consider that the covariance Cn(h) of each indicator can be inferred from the well data or other sources of information (e.g., analogous outcrop data). For each well-test-derived permeability, kwt, the investigation area, V, is determined and a power-averaging formula is adopted to relate the well-test-derived permeability to the lithofacies permeabilities:Equation 2 where kn stands for the permeability of lithofacies n and Pn(V) is its proportion in V. The averaging power ? is calibrated for each well-test9,10 and Pn(V) are to be estimated by the method described next.
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Hardiyanthy, Sri Mulyanie, Dewi Sri Susanti, and Thresye Thresye. "ANALISIS KRIGING UNTUK MENDETEKSI POLA SPASIAL KASUS DBD DI KABUPATEN TANAH LAUT." JURNAL MATEMATIKA MURNI DAN TERAPAN EPSILON 13, no. 2 (February 20, 2020): 1. http://dx.doi.org/10.20527/epsilon.v13i2.1646.
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Geostatistics is a data processing in geological field that contains spatial information in it. Spatial information is information that identifies geographical location, characteristics of natural conditions and boundaries of the earth. Geostatistics is used to handle regionalized variables. One of the method that used to handle regionalized variables is the kriging method. The kriging method has a lot of expansion in its development, including the Simple Kriging method and the Cokriging method. Both of these methods will be applied in case studies of spatial patterns of dengue in Tanah Laut District. The purpose of this study was to estimate the distribution pattern of DHF in Tanah Laut District and compare the results of the RMSE method of Simple Kriging and Cokriging. The smallest RMSE value was compared and selected, followed by estimation using the Cokriging and Simple Kriging methods. From the two methods used the smallest RMSE value is in the Simple Kriging method. But when you looked from the thematic map of the distribution of dengue patients with the Cokriging and Simple Kriging method, it can be seen that the Cokriging method has a more diverse pattern. Keywords: geostatisticts , Cokriging , Simple Kriging , DHF
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Carbonara, Pierluigi, Teresa Silecchia, Maria Spedicato, Alessandra Acrivulis, and Giuseppe Lembo. "A GEOSTATISTICAL APPROACH TO THE ASSESSMENT OF THE SPATIAL DISTRIBUTION OF PARAPENAEUS LONGIROSTRIS (LUCAS, 1846) IN THE CENTRAL-SOUTHERN TYRRHENIAN SEA." Crustaceana 72, no. 9 (1999): 1093–108. http://dx.doi.org/10.1163/156854099504040.
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AbstractThe spatial distribution of the abundance indices of the deep-water rose shrimp Parapenaeus longirostris was investigated applying geostatistical techniques on data collected in the central southern Tyrrhenian Sea from bottom trawl surveys carried out in the autumn since 1994. Experimental variograms (auto and cross) were constructed on the variable "abundance index", expressed in kg/km2, and those variogram models best describing the spatial continuity were detected and validated by the jackknife technique. The spatial structure of the "abundance index", exhibiting a similar pattern throughout the surveys, was described by a spherical model and characterized by a spatial continuity at a small scale level in the whole area. The linear geostatistical approach was applied by different kriging techniques and the estimates extended to the spatio-temporal dimension, in this case adopting the co-regionalized models and applying the cokriging technique. This method applied to the spatial dimension (abundance index and depth). Also, linking the spatial and temporal dimension of the abundance indices, measured in two different years, contributed to represent a more accurate picture of the abundance distribution, and allowed the detection of a temporal persistence of the localization of areas with higher abundance, reducing the standard deviation of the estimation error. This information, if coupled with an analysis of the geographical allocation of the fishing effort, could be of importance in stock assessment, allowing some variant application of the composite surplus production models. La distribution spatiale des indices d'abondance de la crevette rose d'eau profonde Parapenaeus longirostris a ete etudiee en appliquant les techniques de la geostatistique aux donnees collectees dans le centre-sud de la mer Tyrrhenienne au cours des campagnes de chalutage demersal realisees pendant l'automne, depuis 1994. Les variogrammes experimentaux (auto et cross) ont ete construits sur la variable "indice d'abondance", exprimee en kg/km2, et les modeles de variogramme decrivants le mieux la continuite spatiale ont ete determines et valides par la technique du "jackknife". La structure spatiale de l'indice d'abondance a presente le meme aspect pour tous les echantillonages; elle a ete decrite au moyen d'un modele spherique et caracterisee par une continuite spatiale a petite echelle dans toute la zone. La geostatistique lineaire a ete appliquee en utilisant differentes techniques du krigeage, et les estimations ont ete etendues a la dimension spatio-temporelle en appliquant les modeles coregionalises et la technique du cokrigeage. Cette methode, appliquee soit dans la dimension spatiale (indice d'abondance et profondeur), soit dans la dimension spatio-temporelle en considerant l'indice d'abondance echantillonne en deux annees differentes, a contribue a representer une image plus precise de la distribution de l'abondance, et a permis de detecter une persistance temporelle de la localisation des aires a plus grande abondance, en reduisant l'ecart type de l'erreur d'estimation. Cette information, avec l'analyse de l'allocation geografique de l'effort de peche, pourrait etre importante dans l'evaluation des stocks, en permettant l'application, avec quelques variantes, des modeles composites de production.
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Yates, S. R. "Disjunctive Kriging: 3. Cokriging." Water Resources Research 22, no. 10 (September 1986): 1371–76. http://dx.doi.org/10.1029/wr022i010p01371.
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Wang, L., P. M. Wong, and S. A. R. Shibli. "Modeling Porosity Distribution in the A'nan Oilfield: Use of Geological Quantification, Neural Networks, and Geostatistics." SPE Reservoir Evaluation & Engineering 2, no. 06 (December 1, 1999): 527–32. http://dx.doi.org/10.2118/59090-pa.
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Summary A'nan Oilfield is located in the northeast of the Erlian Basin in North China. The porosity distribution of the oil-bearing stratum is primarily controlled by complex distribution patterns of sedimentary lithofacies and diagenetic facies. This paper describes a methodology to provide a porosity model for the A'nan Oilfield using limited well porosity data, with the incorporation of the conceptual reservoir architecture. Neural network residual kriging or simulation is employed to tackle the problem. The integrated technique is developed based on a combined use of radial basis function neural networks and geostatistics. It has the flexibility of neural networks in handling high-dimensional data, the exactitude property of kriging and the ability to perform stochastic simulation via the use of kriging variance. The results of this study show that the integrated technique provides a realistic description of porosity honoring both the well data and the conceptual framework of the geological interpretations. The technique is fast, straightforward and does not require any tedious cross-correlation modeling. It is of great benefit to reservoir geologists and engineers. Introduction Spatial description of porosity is a crucial step for fluid flow simulation study. Such descriptions are often used in porosity and permeability transforms in order to derive a transmissibility field. The distribution of porosity is commonly controlled by qualitative geological features. While the importance of these features is well known to the geological community, they are often difficult to incorporate quantitatively during the three dimensional (3D) geological modeling study. There is therefore a strong need for the industry to fully utilize existing geological interpretations rather than iteratively match the computational outputs to the interpretations by varying model parameters. The objective of this paper is to provide an integrated solution to make use of existing geological interpretations for improved reservoir mapping. Although some purely geostatistical techniques are capable of providing some of these functionalities, often difficult and tedious cross-correlation modeling (e.g., cokriging) as well as time consuming indicator coding (e.g., nonparametric analysis) are required. The integrated technique used in this paper is developed based on a combined use of artificial neural networks (NNs) and geostatistics. The original idea was proposed by Kanevski et al.1 The authors assumed that spatial prediction is composed of a predictable (trend) component and an error (noise or residual) component. They used multilayered feedforward neural networks (an inexact estimator) to model the former component and kriging (an exact estimator) to model the latter component. Hence the name neural network residual kriging (NNRK) was used. The final estimate is simply the sum of the two components, and hence the estimator restores all the conditioning data. The kriging variance also allows the estimator to perform stochastic simulation. A technique, such as neural network residual simulation (NNRS)2 is an example. There are many advantages of combining NNs with geostatistics. The most popular geostatistical model, kriging,3 is based on error variance minimization with the use of spatial correlation structures. It has the ability to generate an exact interpolation. Kriging variance is also useful for stochastic simulation (e.g., via sequential Gaussian simulation3) in order to quantify the spatial uncertainty of the predictions. However, most geostatistical models become unattractive when there are many types of information available for modeling. In mathematical terms, geostatistics is often not the best solution for high-dimensional problems. On the other hand, NN methods are highly flexible in handling nonlinear, high-dimensional data without tedious cross-correlation modeling. However, most NN methods could neither produce exact interpolation nor perform stochastic simulation for uncertainty analysis. Hence, the combined use of NNs and geostatistics provides a powerful tool for reservoir mapping. This paper will first describe the integrated method using porosity as an example. The reservoir description of the A'nan Oilfield will be presented. This is followed by the application of the method to model the porosity distribution across the field based on limited well data and extensive geological information regarding the distribution patterns of the sedimentary lithofacies and diagenetic facies. Basis of Neural Networks This paper uses a special class of NN estimators, namely "radial basis function neural networks." This particular estimator is chosen because it is simple and the origin of the method is similar to most spatial interpolators, that is, the prediction is calculated based on the distance between the prediction location and the reference data location. Its application to reservoir characterization includes reservoir mapping4–6 and log interpretation.7 Like most NN methods, radial basis function neural networks (RBFNNs) attempt to mimic simple biological learning processes. They can learn from examples. The learning phase is an essential starting point that requires training patterns consisting of a number of input signals (e.g., a high-dimensional vector) paired with target signals. The inputs are presented to the network and the corresponding outputs are calculated with the aim of minimizing the model error (i.e., the total difference between the calculated outputs and target signals). The gradient descent method is the most popular learning method to reduce the model error by iteration. Training can be terminated when the model error is below a tolerance value. After training, the network creates a set of parameters that can be used for predicting properties in situations where the actual outputs are not known.
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Zhang, Hao, and Wenxiang Cai. "When Doesn’t Cokriging Outperform Kriging?" Statistical Science 30, no. 2 (May 2015): 176–80. http://dx.doi.org/10.1214/15-sts518.
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Baskan, O., G. Erpul, and O. Dengiz. "Comparing the efficiency of ordinary kriging and cokriging to estimate the Atterberg limits spatially using some soil physical properties." Clay Minerals 44, no. 2 (June 2009): 181–93. http://dx.doi.org/10.1180/claymin.2009.044.2.181.
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AbstractThe spatial distribution of the Atterberg limits can be used to distinguish the consistency and behaviour of a soil and its engineering properties, which strongly depends on the water content of the soil and types of silts and clays in the soil. By spatial modeling, and comparing the results of ordinary kriging with the cokriging approach, this study aims to find correlations between the Atterberg limits and the selected physical soil parameters in order to examine how effective they are in generating an understanding of the dynamics of a physical soil system.In 156 soil samples, the Atterberg limits and soil moisture conditions were determined, and auxiliary functions were selected by application of cokriging using correlation analysis and regression equations obtained by the residual maximum likelihood (REML). These techniques were evaluated by the results of the mean absolute error (MAE) and the mean squared error (MSE). Cokriging analysis was found to be more effective at estimating the liquid limit (WLL) and the plastic limit (WPL) than kriging analysis and with smaller error values. On the other hand, the kriging approach, which had smaller MAE and MSE values, was more effective at estimating the plasticity index (WPI) values than the cokriging method. Unlike the REML regression equations, the field capacity (FC) value was the more suitable parameter for the cokriging estimates. When the necessary labour and time were considered for determining the Atterberg limits, both kriging and cokriging were found to be applicable for estimation of these limits.
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Park, No-Wook, and Dong-Ho Jang. "Comparison of Geostatistical Kriging Algorithms for Intertidal Surface Sediment Facies Mapping with Grain Size Data." Scientific World Journal 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/145824.
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This paper compares the predictive performance of different geostatistical kriging algorithms for intertidal surface sediment facies mapping using grain size data. Indicator kriging, which maps facies types from conditional probabilities of predefined facies types, is first considered. In the second approach, grain size fractions are first predicted using cokriging and the facies types are then mapped. As grain size fractions are compositional data, their characteristics should be considered during spatial prediction. For efficient prediction of compositional data, additive log-ratio transformation is applied before cokriging analysis. The predictive performance of cokriging of the transformed variables is compared with that of cokriging of raw fractions in terms of both prediction errors of fractions and facies mapping accuracy. From a case study of the Baramarae tidal flat, Korea, the mapping method based on cokriging of log-ratio transformation of fractions outperformed the one based on cokriging of untransformed fractions in the prediction of fractions and produced the best facies mapping accuracy. Indicator kriging that could not account for the variation of fractions within each facies type showed the worst mapping accuracy. These case study results indicate that the proper processing of grain size fractions as compositional data is important for reliable facies mapping.
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Buttafuoco, Gabriele, and Massimo Conforti. "Improving Mean Annual Precipitation Prediction Incorporating Elevation and Taking into Account Support Size." Water 13, no. 6 (March 18, 2021): 830. http://dx.doi.org/10.3390/w13060830.
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Accounting for secondary exhaustive variables (such as elevation) in modelling the spatial distribution of precipitation can improve their estimate accuracy. However, elevation and precipitation data are associated with different support sizes and it is necessary to define methods to combine such different spatial data. The paper was aimed to compare block ordinary cokriging and block kriging with an external drift in estimating the annual precipitation using elevation as covariate. Block ordinary kriging was used as reference of a univariate geostatistical approach. In addition, the different support sizes associated with precipitation and elevation data were also taken into account. The study area was the Calabria region (southern Italy), which has a spatially variable Mediterranean climate because of its high orographic variability. Block kriging with elevation as external drift, compared to block ordinary kriging and block ordinary cokriging, was the most accurate approach for modelling the spatial distribution of annual mean precipitation. The three measures of accuracy (MAE, mean absolute error; RMSEP, root-mean-squared error of prediction; MRE, mean relative error) have the lowest values (MAE = 112.80 mm; RMSEP = 144.89 mm, and MRE = 0.11), whereas the goodness of prediction (G) has the highest value (75.67). The results clearly indicated that the use of an exhaustive secondary variable always improves the precipitation estimate, but in the case of areas with elevations below 120 m, block cokriging makes better use of secondary information in precipitation estimation than block kriging with external drift. At higher elevations, the opposite is always true: block kriging with external drift performs better than block cokriging. This approach takes into account the support size associated with precipitation and elevation data. Accounting for elevation allowed to obtain more detailed maps than using block ordinary kriging. However, block kriging with external drift produced a map with more local details than that of block ordinary cokriging because of the local re-evaluation of the linear regression of precipitation on block estimates.
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Alemi, M. H., M. R. Shahriari, and D. R. Nielsen. "Kriging and cokriging of soil water properties." Soil Technology 1, no. 2 (June 1988): 117–32. http://dx.doi.org/10.1016/s0933-3630(88)80014-x.
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Azawi, Hayat, and May Samir Saleh. "Review of the Kriging Technique Applications to Groundwater Quality." Journal of Engineering 27, no. 12 (December 1, 2021): 23–32. http://dx.doi.org/10.31026/j.eng.2021.12.03.
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Kriging, a geostatistical technique, has been used for many years to evaluate groundwater quality. The best estimation data for unsampled points were determined by using this method depending on measured variables for an area. The groundwater contaminants assessment worldwide was found through many kriging methods. The present paper shows a review of the most known methods of kriging that were used in estimating and mapping the groundwater quality. Indicator kriging, simple kriging, cokriging, ordinary kriging, disjunctive kriging and lognormal kriging are the most used techniques. In addition, the concept of the disjunctive kriging method was explained in this work to be easily understood.
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Luca, Carol, Bing C. Si, and Richard E. Farrell. "Upslope length improves spatial estimation of soil organic carbon content." Canadian Journal of Soil Science 87, no. 3 (May 1, 2007): 291–300. http://dx.doi.org/10.4141/cjss06012.
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Quantifying soil organic carbon (SOC) is important to aide in assessing carbon (C) sequestration potential, and as an indicator of soil quality. However, intensive s ampling of SOC for quantification can be expensive and time consuming. The objectives of this study were to identify which topographic index correlated best with SOC and determine if incorporating the index improved interpolation of limited SOC data. A transect with 93 sample points spaced 6 m apart was set up, and four topographical indices (curvature, wetness index, upslope length, and elevation) were evaluated for their potential as secondary variables. Three Kriging-based interpolation methods, ordinary kriging, cokriging, and simple kriging with varying local means were compared to determine if incorporating topographical indices improved interpolation of SOC. The upslope length, which takes into consideration the quantity of water that will be redistributed to a point, was found to have the strongest relationship with SOC (R2 = 0.48, P < 0.01) and was used as a secondary variable for kriging. Thirty points from the SOC data were randomly selected and used in the kriging algorithms to estimate the remain ing 63 points. The sum of squared differences (SSD) showed a significant reduction (from 1677 to 1455 for SKlm and from 1677 to 1464 for cokriging) in estimates when upslope length was used as a secondary variable. These results indicate that fewer samples may be taken to estimate SOC accurately and precisely if upslope length is incorporated. On a landscape scale this could facilitate quantification of carbon credits and management decisions in precision farming systems. Key words: Geostatics, kriging, cokriging, organic carbon, landscape processes, wetness index
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Lee, Sang Heon, Adel Malallah, Akhil Datta-Gupta, and David Higdon. "Multiscale Data Integration Using Markov Random Fields." SPE Reservoir Evaluation & Engineering 5, no. 01 (February 1, 2002): 68–78. http://dx.doi.org/10.2118/76905-pa.
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Summary We propose a hierarchical approach to spatial modeling based on Markov Random Fields (MRF) and multiresolution algorithms in image analysis. Unlike their geostatistical counterparts, which simultaneously specify distributions across the entire field, MRFs are based on a collection of full conditional distributions that rely on the local neighborhoods of each element. This critical focus on local specification provides several advantages:MRFs are computationally tractable and are ideally suited to simulation based computation, such as Markov Chain Monte Carlo (MCMC) methods, andmodel extensions to account for nonstationarity, discontinuity, and varying spatial properties at various scales of resolution are easily accessible in the MRF framework. Our proposed method is computationally efficient and well suited to reconstruct fine-scale spatial fields from coarser, multiscale samples (based on seismic and production data) and sparse fine-scale conditioning data (e.g., well data). It is easy to implement, and it can account for the complex, nonlinear interactions between different scales, as well as the precision of the data at various scales, in a consistent fashion. We illustrate our method with a variety of examples that demonstrate the power and versatility of the proposed approach. Finally, a comparison with Sequential Gaussian Simulation with Block Kriging (SGSBK) indicates similar performance with less restrictive assumptions. Introduction A persistent problem in petroleum reservoir characterization is to build a model for flow simulations based on incomplete information. Because of the limited spatial information, any conceptual reservoir model used to describe heterogeneities will, necessarily, have large uncertainty. Such uncertainties can be significantly reduced by integrating multiple data sources into the reservoir model.1 In general, we have hard data, such as well logs and cores, and soft data, such as seismic traces, production history, conceptual depositional models, and regional geological analyses. Integrating information from this wide variety of sources into the reservoir model is not a trivial task. This is because different data sources scan different length scales of heterogeneity and can have different degrees of precision.2 Reconciling multiscale data for spatial modeling of reservoir properties is important because different data types provide different information about the reservoir architecture and heterogeneity. It is essential that reservoir models preserve small-scale property variations observed in well logs and core measurements and capture the large-scale structure and continuity observed in global measures such as seismic and production data. A hierarchical model is particularly well suited to address the multiscaled nature of spatial fields, match available data at various levels of resolution, and account for uncertainties inherent in the information.1–3 Several methods to combine multiscale data have been introduced in the literature, with a primary focus on integrating seismic and well data.3–9 These include conventional techniques such as cokriging and its variations,3–6 SGSBK,7 and Bayesian updating of point kriging.8,9 Most kriging-based methods are restricted to multi-Gaussian and stationary random fields.3–9 Therefore, they require data transformation and variogram construction. In practice, variogram modeling with a limited data set can be difficult and strongly user-dependent. Improper variograms can lead to errors and inaccuracies in the estimation. Thus, one might also need to consider the uncertainty in variogram models during estimation. 10 However, conventional geostatistical methods do not provide an effective framework to account for the uncertainty of the variogram. Furthermore, most of the multiscale integration algorithms assume a linear relationship between the scales. The objective of this paper is to introduce a novel multiscale data-integration technique that provides a flexible and sound mathematical framework to overcome some of the limitations of conventional geostatistical techniques. Our approach is based on multiscale MRFs11–14 that can effectively integrate multiple data sources into high-resolution reservoir models for reliable reservoir forecasting. This proposed approach is also ideally suited to simulation- based computations, such as MCMC.15,16 Methodology Our problem of interest is to generate fine-scale random fields based on sparse fine-scale samples and coarse-scale data. Such situations arise when we have limited point measurements, such as well data, and coarse-scale information based on seismic and/or production data. Our proposed method is a Bayesian approach to spatial modeling based on MRF and multiresolution algorithms in image analysis. Broadly, the method consists of two major parts:construction of a posterior distribution for multiscale data integration using a hierarchical model andimplementing MCMC to explore the posterior distribution. Construction of a Posterior Distribution for Multiscale Data Integration. A multiresolution MRF provides an efficient framework to integrate different scales of data hierarchically, provided that the coarse-scale resolution is dependent on the next finescale resolution.11 In general, a hierarchical conditional model over scales 1,. . ., N (from fine to coarse) can be expressed in terms of the product of conditional distributions,Equation 1 where p(xn), n=1, . . ., N, are MRF models at each scale, and the terms p(xn|xn-1) express the statistical interactions between different scales. This approach links the various scales stochastically in a direct Bayesian hierarchical modeling framework (Fig. 1). Knowing the fine-scale field xn does not completely determine the field at a coarser scale xn+1, but depending on the extent of the dependence structure modeled and estimated, it influences the distribution at the coarser scales to a greater or lesser extent. This enables us to address multiscale problems accounting for the scale and precision of the data at various levels. For clarity of exposition, a hierarchical model for reconciling two different scales of data will be considered below.Equation 2 From this equation, the posterior distribution of the fine-scale random field indexed by 1 given a coarse-scale random field indexed by 2 can be derived as follows.
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Stein, A., and L. C. A. Corsten. "Universal Kriging and Cokriging as a Regression Procedure." Biometrics 47, no. 2 (June 1991): 575. http://dx.doi.org/10.2307/2532147.
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Wackernagel, Hans. "Cokriging versus kriging in regionalized multivariate data analysis." Geoderma 62, no. 1-3 (March 1994): 83–92. http://dx.doi.org/10.1016/0016-7061(94)90029-9.
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Vargas-Guzmán, J. A., and T. C. Jim Yeh. "Sequential kriging and cokriging: Two powerful geostatistical approaches." Stochastic Environmental Research and Risk Assessment (SERRA) 13, no. 6 (December 7, 1999): 416–35. http://dx.doi.org/10.1007/s004770050047.
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Hansen, T. M., K. Mosegaard, and C. R. Schiøtt. "Kriging interpolation in seismic attribute space applied to the South Arne Field, North Sea." GEOPHYSICS 75, no. 6 (November 2010): P31—P41. http://dx.doi.org/10.1190/1.3494280.
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Seismic attributes can be used to guide interpolation in-between and extrapolation away from well log locations using for example linear regression, neural networks, and kriging. Kriging-based estimation methods (and most other types of interpolation/extrapolation techniques) are intimately linked to distances in physical space: If two observations are located close to one another, the implicit assumption is that they are highly correlated. This may, however, not be a correct assumption as the two locations can be situated in very different geological settings. An alternative approach to the traditional kriging implementation is suggested that frees the interpolation from the restriction of the physical space. The method is a fundamentally different application of the original kriging formulation where a model of spatialvariability is replaced by a model of variability in an attribute space. To the extent that subsurface geology can be described by a set of seismic attributes, we present an automated multivariate kriging-based interpolation method that is guided by geological similarity rather than by the conventional distance measure in XYZ space. Through a case study, kriging in attribute space is used to estimate 2D porosity maps from a number of well logs and seismic attributes in the Danish North Sea. Cokriging provides uncertainty estimates that are dependent on the primary data locations in space, whereas kriging in attribute space provides uncertainty estimates that reflect subsurface geological variability. The North Sea case study demonstrates that kriging in attribute space performs better than linear regression and cokriging.
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Brom, Aleksander, and Adrianna Natonik. "Estimation of geotechnical parameters on the basis of geophysical methods and geostatistics." Contemporary Trends in Geoscience 6, no. 2 (December 1, 2017): 70–79. http://dx.doi.org/10.1515/ctg-2017-0006.
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AbstractThe paper presents possible implementation of ordinary cokriging and geophysical investigation on humidity data acquired in geotechnical studies. The Author describes concept of geostatistics, terminology of geostatistical modelling, spatial correlation functions, principles of solving cokriging systems, advantages of (co-)kriging in comparison with other interpolation methods, obstacles in this type of attempt. Cross validation and discussion of results was performed with an indication of prospect of applying similar procedures in various researches..
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Hamzehpour, N., MK Eghbal, P. Bogaert, and N. Toomanian. "Top soil salinity prediction in South-Western part of Urmia Lake with ground water data." International Journal of Agricultural Research, Innovation and Technology 4, no. 1 (December 2, 2014): 57–63. http://dx.doi.org/10.3329/ijarit.v4i1.21093.
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Drying of Urmia Lake in the north-west of Iran threatens all the agricultural lands around the Lake. Therefore, soil salinity appears to be the major threat to the agricultural lands in the area. The aim of the present study was to investigate the spatial variation of top soil salinity by taking into account of underground water quality data as secondary information. The research was performed on a grid of 500 m in an area of 5000 ha. Soil samples were gathered during the autumn of 2009 and were repeated in the spring of 2010. Electrical conductivity of soil samples was measured in a 1:2.5 soil to water suspension. Then covariance functions were build for each data set and soil salinity prediction were done on a grid of 100 m using kriging estimator with taking into account the mean variation. Afterwards sodium activity ratio derived from underground water quality database was used as covariate to develop cross-semivarograms in prediction of top soil salinity using cokriging method. Results demonstrated that soil salinity varied from values lower than 0.5 to more than 35 dSm-1 as a function of distance to the Lake. Cross-validating the results from salinity predictions using only kriging estimator to that of cokriging with sodium activity ratio data revealed that kriging offered better estimations with ME of 0.04 for autumn 2009 and -0.12 for spring 2010. Cokriging estimator had more smoother and diffused boundaries than that of kriging and resulted in more bias estimations (ME= -0.11 and -0.21 for first and second data sets). Although kriging method had better performance in top soil salinity prediction, but cokring method resulted in smoother boundaries and reduced the negative effects of mean variation in the area. DOI: http://dx.doi.org/10.3329/ijarit.v4i1.21093 Int. J. Agril. Res. Innov. & Tech. 4 (1): 57-63, June, 2014