Geostatistical estimates of a soil property by kriging are equivalent to the best linear unbiased predictions (BLUPs). Universal kriging is BLUP with a fixed-effect model that is some linear function of spatial coordinates, or more generally a linear function of some other secondary predictor variable when it is called kriging with external drift. A problem in universal kriging is to find a spatial variance model for the random variation, since empirical variograms estimated from the data by method-of-moments will be affected by both the random variation and that variation represented by the fixed effects. The geostatistical model of spatial variation is a special case of the linear mixed model where our data are modelled as the additive combination of fixed effects (e.g. the unknown mean, coefficients of a trend model), random effects (the spatially dependent random variation in the geostatistical context) and independent random error (nugget variation in geostatistics). Statisticians use residual maximum likelihood (REML) to estimate variance parameters, i.e. to obtain the variogram in a geostatistical context. REML estimates are consistent (they converge in probability to the parameters that are estimated) with less bias than both maximum likelihood estimates and method-of-moment estimates obtained from residuals of a fitted trend. If the estimate of the random effects variance model is inserted into the BLUP we have the empirical BLUP or E-BLUP. Despite representing the state of the art for prediction from a linear mixed model in statistics, the REML-E-BLUP has not been widely used in soil science, and in most studies reported in the soils literature the variogram is estimated with methods that are seriously biased if the fixed-effect structure is more complex than just an unknown constant mean (ordinary kriging). In this paper we describe the REML-E-BLUP and illustrate the method with some data on soil water content that exhibit a pronounced spatial trend.

On spatial prediction of soil properties in the presence of a spatial trend: The empirical best linear unbiased predictor (E-BLUP) with REML

Soil shear strength is a critical parameter in slope stability. Shear strength properties may vary significantly over large areas. Thus, the spatial estimates of shear strength properties are necessary for deterministic slope susceptibility mapping over large areas. However, measurements of shear strength parameters are often limited as compared to other soil properties such as Atterberg limit, bulk density and grain size distribution. Multivariate methods have been shown to improve prediction accuracy, but these methods have rarely been used to predict shear strength. In this study, attempts were made to evaluate the effectiveness of using the aforementioned soil properties in predicting the spatial variation of shear strength properties: effective cohesion (c’) and effective friction angle (ϕ’). The performance of ordinary kriging (OK), Random Forest (RF) and regression kriging (RK) in predicting c’ and ϕ’ of residual soils in Singapore were compared and evaluated. In addition, the sensitivity of the three methods to the sample size was investigated. The results of RF analysis revealed that the northing coordinate and percentage of fines were the most important variables for predicting ϕ’. The spatial coordinates and ϕ’ were also important variables for predicting c’. The predicted c’ and ϕ’ using RF and RK resulted in higher spatial heterogeneity than OK. Overall, RF had the smallest error as compared to OK and RK in predicting c’ and ϕ’ at all sample sizes, except for the prediction of ϕ’ using the largest sample size. This study also showed that RF and RK were more sensitive to sample size than OK. These results highlight the benefits of using auxiliary variables when mapping shear strength properties.

Spatial variation of shear strength properties incorporating auxiliary variables

Interpolation accuracy of the soil carbon-to-nitrogen ratio (C/N) has a direct impact on soil maps for agricultural land management planning, affecting the financial gains of farmers and environmental contamination. This study aimed to evaluate the effect of different ordinary kriging parameters on the interpolation accuracy and spatial variability of soil C/N. These values were necessary to determine the effect of potential soil C/N deficiency on five major crops in the study area during 2017–2019, containing maize, wheat, sunflower, rapeseed and soybean parcels. The Gaussian model with logarithmic transformation produced the higher outer accuracy, representing the prediction accuracy. Average values of final soil C/N rasters resulted as 11.35 at 0–10 cm soil depth and 12.60 at 20–30 cm soil depth, representing values 11.0% higher at a deeper soil layer. The moderate soil N deficiency was observed for all major crops, having soil C/N values mainly higher than 12:1, compared to the optimal value range of 9:1 to 12:1. These observations suggest a need for the adjustment of agricultural land management plans through the regionalization of agricultural production to prevent further soil degradation and ensure optimal crop development conditions.

https://www.mdpi.com/2073-4395/10/12/1996/pdf

Geostatistical Analysis of Soil C/N Deficiency and Its Effect on Agricultural Land Management of Major Crops in Eastern Croatia

Traffic volume information has long played an important role in many transportation related works, such as traffic operations, roadway design, air quality control, and policy making. However, monitoring traffic volumes over a large spatial area is not an easy task due to the significant amount of time and manpower required to collect such large-scale datasets. In this study, a hybrid geostatistical approach, named Network Regression Kriging,has been developed to estimate urban traffic volumes by incorporating auxiliary variables such as road type, speed limit, and network accessibility.Since standard kriging is based on Euclidean distances, this study implements road network distances to improve traffic volumes estimations.A case study using 10-year of traffic volume data collected within the city of Edmonton was conducted to demonstrate the robustness of the model developed herein. Results suggest that the proposed hybrid model significantly outperforms the standard kriging method in terms of accuracy by 4.0% overall, especially for a large-scale network. It was also found that the necessary stationarity assumption for kriging did not hold true for a large network whereby separate estimations for each road type performed significantly better than a general estimation for the overall network by 4.12%.

/pdf/a-hybrid-geostatistical-method-for-estimating-citywide-sjlrp2qs.pdf

A Hybrid Geostatistical Method for Estimating Citywide Traffic Volumes – A Case Study of Edmonton, Canada

Although atmospheric CO2 concentrations collected by satellites play a crucial role in understanding global greenhouse gases, the sparse geographic distribution greatly affects their widespread application. In this paper, a hybrid CNN and spatiotemporal Kriging (CNN-STK) model is proposed to generate a monthly spatiotemporal continuous XCO2 dataset over China at 0.25° grid-scale from 2015 to 2020, utilizing OCO-2 XCO2 and geographic covariates. The validations against observation samples, CAMS XCO2 and TCCON measurements indicate the CNN-STK model is effective, robust, and reliable with high accuracy (validation set metrics: R2 = 0.936, RMSE = 1.3 ppm, MAE = 0.946 ppm; compared with TCCON: R2 = 0.954, RMSE = 0.898 ppm and MAE = 0.741 ppm). The accuracy of CNN-STK XCO2 exhibits spatial inhomogeneity, with higher accuracy in northern China during spring, autumn, and winter and lower accuracy in northeast China during summer. XCO2 in low-value-clustering areas is notably influenced by biological activities. Moreover, relatively high uncertainties are observed in the Qinghai-Tibet Plateau and Sichuan Basin. This study innovatively integrates deep learning with the geostatistical method, providing a stable and cost-effective approach for other countries and regions to obtain regional scales of atmospheric CO2 concentrations, thereby supporting policy formulation and actions to address climate change. 

Satellite-Based Reconstruction of Atmospheric CO2 Concentration over China Using a Hybrid CNN and Spatiotemporal Kriging Model

Regression kriging (RK), a popular digital soil mapping method which combines regression and ordinary kriging (OK) to predict, did not always perform better than OK purely. A lot of studies tried to establish approaches for determining relative improvement (RI) of RK over OK in terms of two of three factors, i.e., regression determination (R2), nugget-to-sill ratio (N/S) and sampling spacing to correlation range ratio (SS/CR) of the variogram. However, the studies suffered some drawbacks and their approaches failed in many cases. The present study aimed to investigate relationships of RI with three factors simultaneously, i.e., R2, N/S and SS/CR of the residual variogram for RK which were developed using ordinary least square, weighted least square, maximum likelihood and residual maximum likelihood, separately, based on simulation data and soil data. Results showed that, although RI basically increased with R2 and N/S, its relationships with R2, N/S and SS/CR were not clear enough to differentiate between negative and positive RIs due to other factors like sampling and modeling. The three factors simultaneously explained a small portion of variation of RI, i.e.,

Can regression determination, nugget-to-sill ratio and sampling spacing determine relative performance of regression kriging over ordinary kriging?

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Can regression determination, nugget-to-sill ratio and sampling spacing determine relative performance of regression kriging over ordinary kriging?