Deterministic Interpolation Methods Research Articles

Evaluation of the Pollution by the Potentially Toxic Elements in the Topsoils of a Semi-Industrial Zone, Northwestern Iran

ABSTRACT Various industries, including cement production, petrochemical processing, and power plants, exert significant environmental impacts on their surroundings. The close proximity of these industries within the study area underscores the critical need to investigate soil pollution of potentially toxic elements and elucidate possible pollution sources and associated risks. The present study involved taking 44 topsoil (0–10 cm) samples of Arsenic (As), Cadmium (Cd) Cobalt (Co), Chromium (Cr), Copper (Cu), Nickel (Ni), Lead (Pb), and Zinc (Zn) concentrations, which their mean concentration were found to be 11.29, 0.46, 15.89, 67.36, 44.36, 79.93, 8.50 and 68.27 mg kg−1, respectively. Comparison with the baseline values stated in local and international soil quality guidelines has indicated higher As, Cd, Cr, Cu, and Ni concentrations. The soil pollution indices of the geoaccumulation index, pollution index, pollution load index, and enrichment factor pollution have evaluated the pollution status of the study area to be uncontaminated to moderately contaminated, moderately polluted, polluted, deficient enrichment to moderate enrichment classes, respectively. Ecological and health risks that involved the application of the single index of ecological risk, potential ecological risk, hazard quotient, hazard index, and cancer risk have revealed that the study area was ecologically in a low-risk state with an average potential ecological risk value of 90.38, while carcinogenic (10−6-10−4) and non-carcinogenic risks (>1) were also found to be in a tolerable range for human health. Possible pollution sources were identified using multivariate statistics and further confirmed and visualized by the spatial distribution maps obtained from the deterministic interpolation method of inverse distance weighting. These analyses have suggested that the soil pollution which its existence was confirmed by the related indices and comparisons, to the potentially toxic elements of Cr and Ni can be possibly attributed to the petrochemical and power plants, to Cd could be appointed to cement plant, and to Cu could be associated with agricultural activities. It has also been found that arsenic was determined to be a common pollutant among the three main industries.

Improving estimates of water resources availability over North Tropical South America: comparison of two satellite precipitation merging schemes

Low-density precipitation measurements impair the ability of hydrological models to estimate surface water resources accurately. Remote sensing techniques and climate models can help to improve the estimation of the space-time rainfall variability. However, they alone are not good enough to be used in surface models built to support water management. In this research, we test the improvement of rainfall field estimation by using hydrological modelling based on the premise that a higher hydrological performance generally implies that precipitation is more consistent with streamflow observations and evaporation estimates in the basin. The SWAT model was forced with two satellite and rain gauge blending techniques and with the traditional IDW deterministic interpolation method from stations. The three simulated streamflows were compared separately against observed records. We do not only focus the comparison on one hydrological performance metric but also conduct a deeper evaluation using several hydrological signatures and statistics. We included the bias, the temporal correlation, the relation of general variability, and an analysis of the Flow Duration Curves (we found that low and medium segments were estimated correctly, whereas the high segments were underestimated). We conclude that either combination technique has its advantages over the other and that both outperform the performance achieved by the IDW in most of the defined criteria, with an overall 10% improvement and with individual streamflow gauge performance enhancement up to 50%.

Research Progress on Spatiotemporal Interpolation Methods for Meteorological Elements

With the development of mathematical statistics, people have developed the spatiotemporal interpolation methods based on the spatial interpolation method or the temporal interpolation method. These methods fully consider the comprehensive effects of time series changes and spatial distribution to better handle complicated and changeable meteorological element data. This article systematically reviews the current research progress of spatiotemporal interpolation methods for spatiotemporal sampling data of meteorological origin. Spatiotemporal interpolation methods of meteorological elements are classified into three categories: spatiotemporal geostatistical interpolation methods, spatiotemporal deterministic interpolation methods, and spatiotemporal mixed interpolation methods. This article summarizes the theoretical concept and practical application of the spatiotemporal interpolation methods of meteorological elements, analyzes the advantages and disadvantages of using spatiotemporal interpolation methods for estimating or forecasting meteorological elements, combined through some measures and their application to explain the accuracy of the spatiotemporal interpolation methods; and discusses the problems and challenges of spatiotemporal interpolation. Finally, the future research focus of spatiotemporal interpolation methods is proposed. This article provides a valuable method reference for estimating or predicting meteorological elements such as precipitation in unsampled points.

Spatial variability of nitrate leaching and risk assessment of nitrate contamination in the Ghiss-Nekor alluvial aquifer system (Northeastern Morocco) through Disjunctive Kriging

In this study, groundwater nitrate monitoring data from water supply and water management organizations were processed in a GIS environment to evaluate spatial patterns of nitrate occurrence and explore its correlation with the corresponding land uses in Ghiss-Nekor alluvial groundwater (northeastern Morocco). Land-use/Land Cover within the Ghiss-Nekor plain incorporated rangeland, built area, crop fields, water bodies, dense vegetation, and bare ground categories, encompassing respectively about 39.06 %, 36.69 %, 17.57 %, 4.63 %, 1.24 %, and 0.81 % of the total area. According to the World Health Organization (WHO), the selected thresholds are an acceptable concentration of nitrate for potable water (50 mg/L) and a human affected value (13 mg/L). Simple Kriging and Disjunctive Kriging (D.K.) geostatistical methods were compared with a deterministic interpolation method (GPI) to identify the convenient approach to predict nitrate concentrations at unsampled sites. D.K. provided more accurate predictive performance and highlighted that nitrate contamination is higher in wells in the southern half of the study area, where nitrate concentrations exceeded 50 mg/L in 29.2 % (33,3 km²) of the study area. Areas occupying 21.28 % of the total surface, with nitrate contents exceeding the human-affected value, are located near small industrial sites, residential septic tanks, and agricultural areas. No significant correlation was found between the nitrate concentrations and the depth of the wells. The probability of nitrate content exceeding the thresholds of 13 and 50 mg/L is respectively less than 0.5 in 61.35 % (69.94 km²) and 77.63 % (88.4 km²) of the total study area, whereas the maximum likelihood of nitrate contamination is found along uncontrolled urban discharges located within highly permeable deposits. The assessment also demonstrated that areas showing high predicted probabilities of nitrate contamination diminish with increasing threshold value. The present study shed light on the necessity of depicting the extent of polluted areas, particularly in vulnerable alluvial aquifers, where exposure to higher levels of drinking water contaminants threatens human health.

Reliability of smart noise pollution map

AbstractThe problem of noise pollution in Baghdad, the capital city of Iraq, is getting worse every day as a result of the increased volume of traffic. This presents a considerable risk, particularly on the main roads that connect densely populated neighborhoods such as the Al-Sadr City district with the central neighborhoods of the capital. In order to inform decision-makers in urban development and environmental policy about the high values of noise pollution that require remediation and regulation, noise maps are produced. However, two fundamental problems are generally faced in creating a more reliable noise map in the shortest possible time: the excessive time requirements for measuring noise and determining the method of map creation. Therefore, the role of geographic information system (GIS) software in producing noise maps is evident due to the difficulty of increasing the spatial density of measurements and integrating them with spatial information. Hence, an appropriate interpolation method is required. In this article, Moran’sIindex was calculated to assess the spatial autocorrelation of measured traffic noise points. A comparison was made between the Smart Map Plugin ordinary kriging (OK) and the inverse distance weighting (IDW) deterministic interpolation method to determine the best method for producing noise maps for the main entrance and exit roads of Al-Sadr City. The noise values were modeled using the best-performing method. Furthermore, the predictive raster data are displayed in the spatial context as a starting point and reference for identifying and understanding the levels of traffic noise in the selected study area. The locations of selected points for measuring traffic noise values were determined in an organized and homogeneous manner, where noise points for the main entrance and exit roads were opposite each other, and the distance between consecutive noise points on each road was 100 m. Traffic noise measurements were carried out at each selected point using the SVAN977 sound and vibration analyzer. At each measurement point, three noise values (LAeq, Max, Min) were obtained during the three peak times, 7–9 AM, 12–2 PM, and 4–6 PM. QGIS software was used to compare the two interpolation methods, with its strength lying in the use of plugins that facilitate spatial analysis, processing tools, and algorithms. The Smart Map Plugin provided facilities to choose the appropriate semi-variogram in the OK interpolation method. The root mean square error was used to compare the two interpolation methods in order to determine the most suitable method for producing traffic noise maps in the study area. The results indicated that the Smart Map Plugin using OK outperformed the IDW method, as spatial distribution pattern and homogeneity affect the accuracy of interpolation. Moreover, based on the analysis of the three noise attributes (LAeq, Max, Min), the performance of the Smart Map Plugin (OK) was found to be better than IDW when the Moran’sIvalue was high.

Substantiation of Drilling Parameters for Undermined Drainage Boreholes for Increasing Methane Production from Unconventional Coal-Gas Collectors

Decarbonization of the mining industry on the basis of closing the energy generation, on the basis of cogeneration of coal mine methane, and on the internal consumption of the mine is a promising direction in ensuring sustainable development. Known problems of deep underground mining do not allow for realizing the potential of man-made gas reservoirs due to the deterioration of the conditions of development of reserves of georesources. The aim of the work was to improve recommendations for the substantiation of drilling parameters for undermined drainage boreholes for increasing methane production from unconventional coal-gas collectors. The authors’ approach innovation lies in the possibility of using the established patterns of better natural stability of undermined boreholes to optimize them as spatial orientation parameters in an existing drilling passport for the improvement of methane extraction productivity. For this purpose, smoothing (LOESS) of the experimental data of two similar types of wells was used; then deterministic interpolation methods in combination with a three-dimensional representation of the response function in “gnuplot” were used. As a result, it was found that the increase in the inclination angle from 40° to 60° leads to a significant transformation of the model of the studied process, accompanied by a decline in the dynamics of methane emission and a decrease in the distance of the productive work zone of this type of well from 13 to 5 m before the roof landing, which then is replaced by a sharp increase in the productive work zone up to 35 m ahead of the longwall face. This allows under specific conditions for recommending increasing the productivity of methane capex from technogenic disturbed coal-gas reservoir replacement of wells with a smaller angle of rise to the transition to a more frequent grid of clusters from wells #4.

USING ELECTRONIC NAVIGATIONAL CHART FOR 3D BATHYMETRIC MODEL OF THE PORT OF NAPLES

Abstract. Nautical charts generally report fundamental knowledge for the safety of navigation. This information also includes sea depth data reported as depth points or contour lines, which can be used to build a 3D model of the seabed. However, there are different interpolation methods for creating digital depth models, and there is no way to know in advance which of them is the best performing. The aim of this work is to compare different spatial interpolation methods applied on a dataset concerning the seabed of the Port of Naples (Italy) and extracted from the Electronic Navigational Chart (ENC) produced by the Hydrographic Institute of the Italian Navy, in scale 1:10.000. Four deterministic interpolation methods, i.e. Inverse Distance Weighting (IDW), Global Polynomial Interpolation (GPI), Local Polynomial Interpolation (LPI), Radial Basis Function (RBF), and two stochastic interpolation methods, i.e. Ordinary Kriging (OK) and Universal Kriging (UK), are applied using Geographic Information System (GIS) software. Since each method requires to set specific parameters and different options are available, e.g. the order of the polynomial function for GPI and LPI, or semi-variograms for OK and UK, twenty-three models are generated. The result quality is evaluated by Leave-One-Out cross-validation and the statistics of the residuals produced by each interpolation method in the measured points are compared and analysed. The experiments confirm that the stochastic approach is more versatile compared to deterministic approach and can produce better results, as it is testified by the great performance of the Ordinary Kriging, which produces the most accurate 3D models.

Investigation of scarce input data augmentation for modelling nitrogenous compounds in South African rivers

Abstract In this study, basic interpolation and machine learning data augmentation were applied to scarce data used in Water Quality Analysis Simulation Programme (WASP) and Continuous Stirred Tank Reactor (CSTR) that were applied to nitrogenous compound degradation modelling in a river reach. Model outputs were assessed for statistically significant differences. Furthermore, artificial data gaps were introduced into the input data to study the limitations of each augmentation method. The Python Data Analysis Library (Pandas) was used to perform the deterministic interpolation. In addition, the effect of missing data at local maxima was investigated. The results showed little statistical difference between deterministic interpolation methods for data augmentation but larger differences when the input data were infilled specifically at locations where extrema occurred.

The Effect of Soil Sampling Density and Spatial Autocorrelation on Interpolation Accuracy of Chemical Soil Properties in Arable Cropland

Knowledge of the relationship between soil sampling density and spatial autocorrelation with interpolation accuracy allows more time- and cost-efficient spatial analysis. Previous studies produced contradictory observations regarding this relationship, and this study aims to determine and explore under which conditions the interpolation accuracy of chemical soil properties is affected. The study area covered 823.4 ha of agricultural land with 160 soil samples containing phosphorus pentoxide (P2O5) and potassium oxide (K2O) values. The original set was split into eight subsets using a geographically stratified random split method, interpolated using the ordinary kriging (OK) and inverse distance weighted (IDW) methods. OK and IDW achieved similar interpolation accuracy regardless of the soil chemical property and sampling density, contrary to the majority of previous studies which observed the superiority of kriging as a deterministic interpolation method. The primary dependence of interpolation accuracy to soil sampling density was observed, having R2 in the range of 56.5–83.4% for the interpolation accuracy assessment. While this study enables farmers to perform efficient soil sampling according to the desired level of detail, it could also prove useful to professions dependent on field sampling, such as biology, geology, and mining.

Riverbed Mapping with the Usage of Deterministic and Geo-Statistical Interpolation Methods: The Odra River Case Study

In this contribution, interpolation methods were assessed to build the bathymetry of 200 km of the Odra River in South Poland. The River Bed Mapping (RBM) was carried out surveying the depth of several reaches of the canalized part of the river using an Global Navigation Satellite System (GNSS) with an echo sounder as well as two navigation schemes. The values from the interpolation were compared with the data from a classical cross-sectional survey as part of the ISOK (Polish acronym for Information System of Country Protection Against Extraordinary Hazards) project. Two statistical errors between the interpolation values and the ISOK information were estimated, namely, the Mean Absolute Error (MAE) and the Root Mean Square Error (RMSE). Thanks to the presented analysis, it was possible to compare and analyze which interpolation method fits the best for the batymetric surveying of a shallow river. For this specific case study, the TIN (Triangular Irregular Network) and the NN (Natural Neighbor) methods generates the most accurate RBM.

Impact of the Grid Resolution and Deterministic Interpolation of Precipitation on Rainfall-Runoff Modeling in a Sparsely Gauged Mountainous Catchment

Precipitation is a key variable in the hydrological cycle and essential input data in rainfall-runoff modeling. Rain gauge data are considered as one of the best data sources of precipitation but before further use, the data must be spatially interpolated. The process of interpolation is particularly challenging over mountainous areas due to complex orography and a usually sparse network of rain gauges. This paper investigates two deterministic interpolation methods (inverse distance weighting (IDW), and first-degree polynomial) and their impact on the outputs of semi-distributed rainfall-runoff modeling in a mountainous catchment. The performed analysis considers the aspect of interpolation grid size, which is often neglected in other than fully-distributed modeling. The impact of the inverse distance power (IDP) value in the IDW interpolation was also analyzed. It has been found that the best simulation results were obtained using a grid size smaller or equal to 750 m and the first-degree polynomial as an interpolation method. The results indicate that the IDP value in the IDW method has more impact on the simulation results than the grid size. Evaluation of the results was done using the Kling-Gupta efficiency (KGE), which is considered to be an alternative to the Nash-Sutcliffe efficiency (NSE). It was found that KGE generally tends to provide higher and less varied values than NSE which makes it less useful for the evaluation of the results.

Evaluation of DEM interpolation techniques for characterizing terrain roughness

The emergence of modern geospatial techniques has brought a considerable change in data capturing, processing, and visualization tools. Digital elevation models (DEMs) are a 3D mathematical representation of the earth’s surface. Due to enormous advancement in geographic information system (GIS) tools and technologies, DEMs have a wide range of applications in natural resources management, environmental science, and engineering. In general, primary topographic variables extracted from DEMs are frequently less useful than secondary ones, e.g., surface roughness. Therefore, the precise values of secondary terrain features need to be derived from base DEMs. Generally, topographic roughness is an essential geomorphological variable that is a vital issue in geoscience. This research presents a comparison of deterministic interpolation methods, inverse distance weighted (IDW) and a natural neighbor (NN), with geostatistical technique, ordinary kriging (OK), to investigate the influence of generated DEM on quantifying terrain roughness relied on synthetic data and applying the zonal statistics tool. At the same time, a quantitative and qualitative approach is adopted to assessing the results through statistical methods and visual representations. The findings indicated that roughness values' behavior is broadly related to the quality of the built DEM. The NN approach yielded the greatest DEM accuracy, standard deviation, from a quantitative perspective (±0.930 m) than IDW (±3.748 m) and OK (SD = ±5.544 m) methods.

Evaluation of the classical statistical, deterministic and geostatistical interpolation methods for estimating the groundwater level

Proper management of groundwater resources depends on identifying, modeling and predicting the level of groundwater in the plains for long-term planning and optimal use of the potential of water. To achieve continuous and integrated maps and predict unknown values, interpolation methods can be used. In this research, the classic statistical interpolation methods (including nearest neighbor, natural neighbor, moving average, and triangulation with linear interpolation), deterministic interpolation methods (including inverse distance to a power, radial basis functions and local polynomial) and geostatistical interpolation method (kriging) were used to estimate the groundwater level of Naqadeh plain in the northwest of Iran in the period of 2001–2016. The cross-validation technique was applied to evaluate the accuracy of the various methods, and three indices—the determination coefficient (R2), the root mean squared error and mean absolute error—were used to compare the interpolation methods. The results of geostatistical analysis indicated that the groundwater level is regionalized variable and there is a high spatial structure ratio between groundwater level data. The best-experimented variogram is a Gaussian model with a correlation coefficient of 0.981. According to the results of cross-validation method, the geostatistical interpolation method with the highest accuracy and the minimum error and the classic statistical interpolation methods with the least accuracy and maximum error were introduced as the optimal and inappropriate methods, respectively.

Climate change detection in penang island using deterministic interpolation methods

This paper presents detection of climate change in Penang Island by using precipitation data based on interpolation technique. Climate change brings about vast and everlasting effects on all living creatures on the Earth. These effects are especially detrimental towards heritage sites, landscapes and businesses based in Penang Island, Malaysia. This study focuses mainly on investigating the indication of climate change in Penang Island over the period of 2003-2018 by utilising sound application procedures of proven analysis methods. Two deterministic interpolation methods are used to produce new estimation points based on the precipitation data to enrich the monitoring network of rainfall stations in Penang Island. Monthly and monthly-average precipitation maps for Penang Island are produced by using inverse distance weighting interpolation method. Results reveal that seven out of twelve months of a year show increasing precipitation trends over the period of study and March is the only month that shows a decreasing trend in precipitation. Monthly-average precipitation in Penang Island also displays a gradual trend of precipitation increase over the period of study, further conforming the finding of monthly precipitation increase over the period of study. The finding of this study provides insight for local agriculturists and ministry to make better decision in response to climate change in Penang.

Evaluating the Spatial Distribution of some Soil Geotechnical Properties Using Various Interpolation Methods (Case Study: Sulaimani Province, Iraq)

Soilgeotechnical parameters are important inputs for the prediction of ground suitability for construction projects. The aim of the study is the evaluation of spatial distribution of some soil geotechnical characteristics for Sulaimani governorate, northern Iraq which has a semi-arid climate. Sixty different soil specimens were taken from a distance of 2.0 meters from natural ground surface, around the Sulaimani city which are suitable for resettlement purposes. Several required laboratory experiments were performed to obtain some engineering properties of the collected soils according to ASTM standards, namely field water content, field wet density, initial void ratio (IVR), and degree of saturation (DS). Three different interpolation methods namely Inverse Distance Weighted (IDW), and Thin Plate Spline (TPS), which are deterministic interpolation methods in addition to and one geostatistical method, Ordinary Kriging (OK), were applied. Cross validation and accuracy assessment of model performance are also applied and analyzed. In general, the geostatistical method performance was compared with the deterministic methods. Ok method found to be more accurate and less biased method than the other two methods, which has lower RMSE (1.38%, 11.33%, 0.07 and 0.82 gm cm-3) and ME (-0.63%, 0.94%, -0.16 and 0.3 gm cm-3) for water content, DS, IVR and wet density respectively. According to the interpolation maps 65 to 70 % of study area is likely suitable for construction purposes compare to the other 30 to 35 %, which need some precautions for the suitability issue for construction projects. The results yielded in the reliability of the obtained soil geotechnical properties from geospatial maps, which may importantly engage to suitable engineering management application and modeling of land use.

Spatial interpolation of sodium absorption ratio: A study combining a decision tree model and GIS

Groundwater quality is a very important issue, especially in the agricultural sector, where it is used extensively. Sodium Absorption Ratio (SAR) is an important parameter in water quality classification for irrigation usages. In this research, SAR was estimated by using the M5 decision tree integrated with a Geographical Information System (GIS). The main goal of this research was to calculate SAR by using fewer parameters, high precision, and in situ water quality parameters. The main water quality parameters of the study area were EC (Electrical Conductivity), concentrations of Na+, and TDS (Total Dissolved Solids) parameters according to Principal Components Analysis (PCA). By using the M5 decision tree model, different equations were extracted for the main water quality parameters. After extracting different equations, deterministic and non-deterministic interpolation approaches were used for finding spatial distributions of SAR in the Ramhormoz agricultural plain of Iran during 2009–2019. For this purpose, several interpolation maps were obtained by using GIS. Kriging and Inverse Distance Weighted (IDW) were used as non-deterministic and deterministic interpolation methods, respectively. The non-deterministic kriging method, using the Na+ and EC water quality parameters, was found to be the best approach for estimating SAR with the fewest water quality parameters and with the highest precision.

Random Forest Spatial Interpolation

For many decades, kriging and deterministic interpolation techniques, such as inverse distance weighting and nearest neighbour interpolation, have been the most popular spatial interpolation techniques. Kriging with external drift and regression kriging have become basic techniques that benefit both from spatial autocorrelation and covariate information. More recently, machine learning techniques, such as random forest and gradient boosting, have become increasingly popular and are now often used for spatial interpolation. Some attempts have been made to explicitly take the spatial component into account in machine learning, but so far, none of these approaches have taken the natural route of incorporating the nearest observations and their distances to the prediction location as covariates. In this research, we explored the value of including observations at the nearest locations and their distances from the prediction location by introducing Random Forest Spatial Interpolation (RFSI). We compared RFSI with deterministic interpolation methods, ordinary kriging, regression kriging, Random Forest and Random Forest for spatial prediction (RFsp) in three case studies. The first case study made use of synthetic data, i.e., simulations from normally distributed stationary random fields with a known semivariogram, for which ordinary kriging is known to be optimal. The second and third case studies evaluated the performance of the various interpolation methods using daily precipitation data for the 2016–2018 period in Catalonia, Spain, and mean daily temperature for the year 2008 in Croatia. Results of the synthetic case study showed that RFSI outperformed most simple deterministic interpolation techniques and had similar performance as inverse distance weighting and RFsp. As expected, kriging was the most accurate technique in the synthetic case study. In the precipitation and temperature case studies, RFSI mostly outperformed regression kriging, inverse distance weighting, random forest, and RFsp. Moreover, RFSI was substantially faster than RFsp, particularly when the training dataset was large and high-resolution prediction maps were made.

Study on the Mapping of Rainfall Contour in Pu’er, Yunnan

This paper briefly introduces the development of GIS and its application in hydrologic analysis. The paper interpolates the predicted rainfall of pu’er city in 2017 and draws the rainfall contour line by using kriging method, inverse distance weight method, obstacle spline function and geostatistics kriging method in the deterministic interpolation method. The results show that the interpolation results are of high accuracy and the interpolation errors are ranked from small to large as inverse distance weight interpolation < including obstacles interpolation < kriging interpolation. The inverse distance interpolation method can be used to draw the rainfall contour map of pu’er city.

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations, deterministic quadrature is usually unable to efficiently investigate an informative and thus concentrated likelihood. This leads to a large number of required expensive evaluations of the mathematical model. To overcome these challenges, we formulate and test a multilevel adaptive sparse Leja algorithm. At each level, adaptive sparse grid interpolation and quadrature are used to approximate the posterior and perform all quadrature operations, respectively. Specifically, our algorithm uses coarse discretizations of the underlying mathematical model to investigate the parameter space and to identify areas of high posterior probability. Adaptive sparse grid algorithms are then used to place points in these areas, and ignore other areas of small posterior probability. The points are weighted Leja points. As the model discretization is coarse, the construction of the sparse grid is computationally efficient. On this sparse grid, the posterior measure can be approximated accurately with few expensive, fine model discretizations. The efficiency of the algorithm can be enhanced further by exploiting more than two discretization levels. We apply the proposed multilevel adaptive sparse Leja algorithm in numerical experiments involving elliptic inverse problems in 2D and 3D space, in which we compare it with Markov chain Monte Carlo sampling and a standard multilevel approximation.

Application of Sequential Indicator Simulation in Geological Study of X Oilfield in Zhujiangkou Basin

Sequential indicator simulation is a commonly used method for discrete variable simulation in 3D geological modeling and a widely used stochastic simulation method, which can be used not only for continuous variable simulation but also for discrete variable simulation. In this paper, the X Oilfield in the western South China Sea is taken as an example to compare the sequential indicator simulation method and the Indicator Kriging interpolation method. The results of the final comparison show that the results of the lithofacies model established by the Indicator Kriging deterministic interpolation method are overly smooth, and its coincidence rate with the geological statistical results is not high, thus cannot well reflect the heterogeneity of the underground reservoir, while the simulation results of the lithofacies model established by the sequential indicator stochastic simulation method can fit well with the statistical law of the well, which has eliminated the smoothing effect of Kriging interpolation, thus can better reflect the heterogeneity of the underground reservoir. Therefore, the sequential indicator simulation is more suitable for the characterization of sand bodies and the study of reservoir heterogeneity.