Geostatistical modelling of spatial uncertainty using p -field simulation with conditional probability fields

This paper presents a variant of p-field simulation that allows generation of spatial realizations through sampling of a set of conditional probability distribution functions (ccdf) by sets of probability values, called p-fields. Whereas in the common implementation of the algorithm the p-fields are nonconditional realizations of random functions with uniform marginal distributions, they are here conditional to 0.5 probability values at data locations, which entails a preferential sampling of the central part of the ccdf around these locations. The approach is illustrated using a randomly sampled (200 observations of the NIR channel) SPOT scene of a semi-deciduous tropical forest. Results indicate that the use of conditional probability fields improves the reproduction of statistics such as histogram and semivariogram, while yielding more accurate predictions of reflectance values than the common p-field implementation or the more CPU-intensive sequential indicator simulation. Pixel values are then classified as forest or savannah depending on whether the simulated reflectance value exceeds a given threshold value. In this case study, the proposed approach leads to a more precise and accurate prediction of the size of contiguous areas covered by savannah than the two other simulation algorithms.     

Kriging and Semivariogram Deconvolution in the Presence of Irregular Geographical Units

This paper presents a methodology to conduct geostatistical variography and interpolation on areal data measured over geographical units (or blocks) with different sizes and shapes, while accounting for heterogeneous weight or kernel functions within those units. The deconvolution method is iterative and seeks the pointsupport model that minimizes the difference between the theoretically regularized semivariogram model and the model fitted to areal data. This model is then used in area-to-point (ATP) kriging to map the spatial distribution of the attribute of interest within each geographical unit. The coherence constraint ensures that the weighted average of kriged estimates equals the areal datum.This approach is illustrated using health data (cancer rates aggregated at the county level) and population density surface as a kernel function. Simulations are conducted over two regions with contrasting county geographies: the state of Indiana and four states in the Western United States. In both regions, the deconvolution approach yields a point support semivariogram model that is reasonably close to the semivariogram of simulated point values. The use of this model in ATP kriging yields a more accurate prediction than a na?ve point kriging of areal data that simply collapses each county into its geographic centroid. ATP kriging reduces the smoothing effect and is robust with respect to small differences in the point support semivariogram model. Important features of the point-support semivariogram, such as the nugget effect, can never be fully validated from areal data. The user may want to narrow down the set of solutions based on his knowledge of the phenomenon (e.g., set the nugget effect to zero). The approach presented avoids the visual bias associated with the interpretation of choropleth maps and should facilitate the analysis of relationships between variables measured over different spatial supports.     

Estimating Regional Hydraulic Conductivity Fields—A Comparative Study of Geostatistical Methods

Geostatistical estimations of the hydraulic conductivity field (K) in the Carrizo aquifer, Texas, are performed over three regional domains of increasing extent: 1) the domain corresponding to a three-dimensional groundwater flow model previously built (model domain); 2) the area corresponding to the 10 counties encompassing the model domain (County domain), and; 3) the full extension of the Carrizo aquifer within Texas (Texas domain). Two different approaches are used: 1) an indirect approach where transmissivity (T) is estimated first and K is retrieved through division of the T estimate by the screen length of the wells, and; 2) a direct approach where K data are kriged directly. Due to preferential well screen emplacement, and scarcity of sampling in the deeper portions of the formation (> 1 km), the available data set is biased toward high values of hydraulic conductivities. Kriging combined with linear regression, simple kriging with varying local means, kriging with an external drift, and cokriging allow the incorporation of specific capacity as secondary information. Prediction performances (assessed through cross-validation) differ according to the chosen approach, the considered variable (log-transformed or back-transformed), and the scale of interest. For the indirect approach, kriging of log T with varying local means yields the best estimates for both log-transformed and back-transformed variables in the model domain. For larger regional scales (County and Texas domains), cokriging performs generally better than other kriging procedures when estimating both (log T)^∗ and T^∗. Among procedures using the direct approach, the best prediction performances are obtained using kriging of log K with an external drift. Overall, geostatistical estimation of the hydraulic conductivity field at regional scales is rendered difficult by both preferential well location and preferential emplacement of well screens in the most productive portions of the aquifer. Such bias creates unrealistic hydraulic conductivity values, in particular, in sparsely sampled areas.     

Combining Areal and Point Data in Geostatistical Interpolation: Applications to Soil Science and Medical Geography

A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.