Geostatistical interpolation of daily rainfall at catchment scale: the use of several variogram models in the Ourthe and Ambleve catchments, Belgium

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

Analysis of Spatial Dependence of Ore-Forming Elements Using Geostatistics and Moran Correlogram

The spatial dependence of data obtained from the geochemical prospecting process can provide useful information for evaluating mineralization potential. This study proposes two approaches to study the spatial dependence of ore-forming elements. To reduce the influence of extreme values and outliers, a semi-variogram was first used to study spatial variability and degree of spatial dependence of geochemical data using Cressie robust semi-variogram estimator. The Moran spatial correlogram was then employed to describe spatial heterogeneity and to test for the presence of spatial autocorrelation in geochemical data. The Moran’s I statistics is strongly sensitive to positively skewed distribution, therefore, geochemical data were Box-Cox transformed before computing spatial correlograms. Results from a case study of Ag and Au elements in Jiurui Copper districts (southeast China) have shown that moderate spatial dependence was found for both of the Au and Ag variables, the maximum spatial variability was 20 km for Au and 10 km for Ag, respectively. The degree of spatial dependence among geochemical data decreases as distances increase. These findings demonstrate that the spatial dependence of ore-forming elements can be effectively measured using geostatistics and Moran correlogram

Characterizing groundwater flow at Fort Leonard Wood using universal cokriging with trend removal

Groundwater elevation interpolation is necessary for the prediction of groundwater flow direction and contaminant transport. Kriging is a geostatistical tool commonly used to interpolate groundwater elevation. Kriging requires a relatively large number of monitoring wells at the site of interest. The variogram model is a crucial element to the kriging equations. The variogram modelling process is an iterative procedure that is often very time consuming. This study presents a literature based approach that provides a point of departure for the variogram modelling process as well as other kriging parameters. A literature database is developed in order to provide insight and a measure of reasonableness to the variogram parameters developed at a groundwater interpolation site. A case study was performed on the Fort Leonard Wood Military Reservation located in Missouri. A data quality analysis was performed on the dataset and spatial outliers were removed. The results from before and after spatial outlier removal are shown. Compliance points were developed using data gaps observed from the standard error maps produced during kriging. The number of wells for the Fort Leonard Wood site were reduced from 61 wells to 45, 30, and 15 wells. Three realizations were performed for each well reduction and results were averaged. Results indicate that when the number of wells are reduced to 15 wells the contour maps are inconsistent with the baseline contour map, each other, as well as the conceptual model. The literature based approach can be easily applied as a point of departure for kriging groundwater elevations --Abstract, page iv

Spatial statistics and soil mapping: A blossoming partnership under pressure

For the better part of the 20th century pedologists mapped soil by drawing boundaries between different classes of soil which they identified from survey on foot or by vehicle, supplemented by air-photo interpretation, and backed by an understanding of landscape and the processes by which soil is formed. Its limitations for representing gradual spatial variation and predicting conditions at unvisited sites became evident, and in the 1980s the introduction of geostatistics and specifically ordinary kriging revolutionized thinking and to a large extent practice. Ordinary kriging is based solely on sample data of the variable of interest—it takes no account of related covariates. The latter were incorporated from the 1990s onward as fixed effects and incorporated as regression predictors, giving rise to kriging with external drift and regression kriging. Simultaneous estimation of regression coefficients and variogram parameters is best done by residual maximum likelihood estimation. In recent years machine learning has become feasible for predicting soil conditions from huge sets of environmental data obtained from sensors aboard satellites and other sources to produce digital soil maps. The techniques are based on classification and regression, but they take no account of spatial correlations. Further, they are effectively ‘black boxes’; they lack transparency, and their output needs to be validated if they are to be trusted. They undoubtedly have merit; they are here to stay. They too, however, have their shortcomings when applied to spatial data, which spatial statisticians can help overcome. Spatial statisticians and pedometricians still have much to do to incorporate uncertainty into digital predictions, spatial averages and totals over regions, and to take into account errors in measurement and spatial positions of sample data. They must also communicate their understanding of these uncertainties to end users of soil maps, by whatever means they are made

Space-Time Forecasting Using Soft Geostatistics: A Case Study in Forecasting Municipal Water Demand for Phoenix, AZ

Managing environmental and social systems in the face of uncertainty requires the best possible forecasts of future conditions. We use space-time variability in historical data and projections of future population density to improve forecasting of residential water demand in the City of Phoenix, Arizona. Our future water estimates are derived using the first and second order statistical moments between a dependent variable, water use, and an independent variable, population density. The independent variable is projected at future points, and remains uncertain. We use adjusted statistical moments that cover projection errors in the independent variable, and propose a methodology to generate information-rich future estimates. These updated estimates are processed in Bayesian Maximum Entropy (BME), which produces maps of estimated water use to the year 2030. Integrating the uncertain estimates into the space-time forecasting process improves forecasting accuracy up to 43.9% over other space-time mapping methods that do not assimilate the uncertain estimates. Further validation studies reveal that BME is more accurate than co-kriging that integrates the error-free independent variable, but shows similar accuracy to kriging with measurement error that processes the uncertain estimates. Our proposed forecasting method benefits from the uncertain estimates of the future, provides up-to-date forecasts of water use, and can be adapted to other socioeconomic and environmental applications.

Assessment of Ore Grade Estimation Methods for Structurally Controlled Vein Deposits - A Review

Resource estimation techniques have upgraded over the past couple of years, thereby improving resource estimates. The classical method of estimation is less used in ore grade estimation than geostatistics (kriging) which proved to provide more accurate estimates by its ability to account for the geology of the deposit and assess error. Geostatistics has therefore been said to be superior over the classical methods of estimation. However, due to the complexity of using geostatistics in resource estimation, its time-consuming nature, the susceptibility to errors due to human interference, the difficulty in applying it to deposits with few data points and the difficulty in using it to estimate complicated deposits paved the way for the application of Artificial Intelligence (AI) techniques to be applied in ore grade estimation. AI techniques have been employed in diverse ore deposit types for the past two decades and have proven to provide comparable or better results than those estimated with kriging. This research aimed to review and compare the most commonly used kriging methods and AI techniques in ore grade estimation of complex structurally controlled vein deposits. The review showed that AI techniques outperformed kriging methods in ore grade estimation of vein deposits.   Keywords: Artificial Intelligence, Neural Networks, Geostatistics, Kriging, Mineral Resource, Grad

A disposition of interpolation techniques

A large collection of interpolation techniques is available for application in environmental research. To help environmental scientists in choosing an appropriate technique a disposition is made, based on 1) applicability in space, time and space-time, 2) quantification of accuracy of interpolated values, 3) incorporation of ancillary information, and 4) incorporation of process knowledge. The described methods include inverse distance weighting, nearest neighbour methods, geostatistical interpolation methods, Kalman filter methods, Bayesian Maximum Entropy methods, etc. The applicability of methods in aggregation (upscaling) and disaggregation (downscaling) is discussed. Software for interpolation is described. The application of interpolation techniques is illustrated in two case studies: temporal interpolation of indicators for ecological water quality, and spatio-temporal interpolation and aggregation of pesticide concentrations in Dutch surface waters. A valuable next step will be to construct a decision tree or decision support system, that guides the environmental scientist to easy-to-use software implementations that are appropriate to solve their interpolation problem. Validation studies are needed to assess the quality of interpolated values, and the quality of information on uncertainty provided by the interpolation method

Non-linear Geostatistics Approach for An Integrated Surface Mapping in Epithermal Gold Deposit, Lampung

A conventional surface mapping is calculated by any means of linear interpolator such as nearest neighborhood point (NNP), inverse distance (IDW)/inverse distance square (IDS), polygon, contour weighing, Ordinary Kriging (OK). The latter is included in geostatistic methods and provides more advanced weighing method that differs from the rest. Although OK provides smoothing over mapping data but it does not cover categorial (non-value) data. Besides, it is not best in strongly skewed data that are common in exploration data and is limited to the expected value at some location. On the other hand, a non-linear interpolator is conducted to estimate the conditional expectation at a location, that not only to simply predict the grade or other parameter itself, but also the probability of the parameter at a location with known nearby samples. An integrated surface mapping should have many kinds of data that can be categorized into continous data (grade, thickness, elevation, etc.) and categorial data (lithology, alteration, structural data, etc.). In order to create a block that consist of all data available in a given deposit, a non-linier transformation will be conducted to estimate values at determined thresholds by Kriging methods – known as Indicator Kriging method and its variants