Kriging With An External Drift Research Articles

Groundwater level assessment and prediction in the Nebraska Sand Hills using LIDAR-derived lake water level

The spatial variability of groundwater levels is often inferred from sparsely located hydraulic head observations in wells. The spatial correlation structure derived from sparse observations is associated with uncertainties that spread to estimates at unsampled locations. In areas where surface water represents the nearby groundwater level, remote sensing techniques can estimate and increase the number of hydraulic head measurements. This research uses light detection and ranging (LIDAR) to estimate lake surface water level to characterize the groundwater level in the Nebraska Sand Hills (NSH), an area with few observation wells. The LIDAR derived lake groundwater level accuracy was within 40 cm mean square error (MSE) of the nearest observation wells. The lake groundwater level estimates were used to predict the groundwater level at unsampled locations using universal kriging (UK) and kriging with an external drift (KED). The results indicate unbiased estimates of groundwater level in the NSH. UK showed the influence of regional trends in groundwater level while KED revealed the local variation present in the groundwater level. A 10-fold cross-validation demonstrated KED with better mean squared error (ME) [−0.003, 0.007], root mean square error (RMSE) [2.39, 4.46], residual prediction deviation (RPD) [1.32, 0.71] and mean squared deviation ratio (MSDR) [1.01, 1.49] than UK. The research highlights that the lake groundwater level provides an accurate and cost-effective approach to measure and monitor the subtle changes in groundwater level in the NSH. This methodology can be applied to other locations where surface water bodies represent the water level of the unconfined aquifer and the results can aid in groundwater management and modeling.

Understanding satellite-derived bathymetry using Sentinel 2 imagery and spatial prediction models

ABSTRACT Optical satellite data is an efficient and complementary method to hydrographic surveys for deriving bathymetry in shallow coastal waters. Empirical approaches (in particular, the models of Stumpf and Lyzenga) provide a practical methodology to derive bathymetric information from remote sensing. Recent studies, however, have focused on enhancing the performance of such empirical approaches by extending them via spatial information. In this study, the relationship between multibeam depth and Sentinel-2 image bands was analyzed in an optically complex environment using the spatial predictor of kriging with an external drift (KED), where its external drift component was estimated: a) by a ratio of log-transformed bands based on Stumpf’s model (KED_S) and b) by a log-linear transform based on Lyzenga’s model (KED_L). Through the calibration of KED models, the study objectives were: 1) to better understand the empirical relationship between Sentinel-2 multispectral satellite reflectance and depth, 2) to test the robustness of KED to derive bathymetry in a multitemporal series of Sentinel-2 images and multibeam data, and 3) to compare the performance of KED against the existing non-spatial models described by Stumpf et al. and Lyzenga. Results showed that KED could improve prediction accuracy with a decrease in RMSE of 89% and 88%, and an increase in R2 of 27% and 14%, over the Stumpf and Lyzenga models, respectively. The decrease in RMSE provides a worthwhile improvement in accuracy, where results showed effective prediction of depth up to 6 m. However, the presence of higher concentrations of suspended materials, especially river plumes, can reduce this threshold to 4 m. As would be expected, prediction accuracy could be improved through the removal of outliers, which were mainly located in the channel of the river, areas influenced by the river plume, abrupt topography, but also very shallow areas close to the shoreline. These areas have been identified as conflictive zones where satellite-derived bathymetry can be compromised.

Geostatistical Merging of a Single-Polarized X-Band Weather Radar and a Sparse Rain Gauge Network over an Urban Catchment

Optimal Quantitative Precipitation Estimation (QPE) of rainfall is crucial to the accuracy of hydrological models, especially over urban catchments. Small-to-medium size towns are often equipped with sparse rain gauge networks that struggle to capture the variability in rainfall over high spatiotemporal resolutions. X-band Local Area Weather Radars (LAWRs) provide a cost-effective solution to meet this challenge. The Clermont Auvergne metropolis monitors precipitation through a network of 13 rain gauges with a temporal resolution of 5 min. 5 additional rain gauges with a 6-minute temporal resolution are available in the region, and are operated by the national weather service Météo-France. The LaMP (Laboratoire de Météorologie Physique) laboratory’s X-band single-polarized weather radar monitors precipitation as well in the region. In this study, three geostatistical interpolation techniques—Ordinary kriging (OK), which was applied to rain gauge data with a variogram inferred from radar data, conditional merging (CM), and kriging with an external drift (KED)—are evaluated and compared through cross-validation. The performance of the inverse distance weighting interpolation technique (IDW), which was applied to rain gauge data only, was investigated as well, in order to evaluate the effect of incorporating radar data on the QPE’s quality. The dataset is comprised of rainfall events that occurred during the seasons of summer 2013 and winter 2015, and is exploited at three temporal resolutions: 5, 30, and 60 min. The investigation of the interpolation techniques performances is carried out for both seasons and for the three temporal resolutions using raw radar data, radar data corrected from attenuation, and the mean field bias, successively. The superiority of the geostatistical techniques compared to the inverse distance weighting method was verified with an average relative improvement of 54% and 31% in terms of bias reduction for kriging with an external drift and conditional merging, respectively (cross-validation). KED and OK performed similarly well, while CM lagged behind in terms of point measurement QPE accuracy, but was the best method in terms of preserving the observations’ variance. The correction schemes had mixed effects on the multivariate geostatistical methods. Indeed, while the attenuation correction improved KED across the board, the mean field bias correction effects were marginal. Both radar data correction schemes resulted in a decrease of the ability of CM to preserve the observations variance, while slightly improving its point measurement QPE accuracy.

Evaluation of a Data Fusion Approach to Estimate Daily PM2.5 Levels in North China

Background/Aim: PM2.5 air pollution has been a growing concern worldwide. Previous studies have conducted several techniques to estimate PM2.5 exposure spatiotemporally in China, but all these have limitations. This study was to develop a new spatiotemporal model with data fusion approach based on Kriging and Chemistry Module models. Methods: Two techniques were applied to create daily spatial cover of PM2.5 in grid cells with a resolution of 10 km in North China in 2013, respectively, which was kriging with an external drift (KED) and Weather Research and Forecast Model with Chemistry Module (WRF-Chem). A data fusion technique was developed by fusing PM2.5 concentration predicted by KED and WRF-Chem, accounting for the distance from the central of grid cell to the nearest ground observations and daily spatial correlations between WRF-Chem and observations. Model performance was evaluated by comparing them with ground observations and the spatial prediction errors. Results: KED and data fusion performed better with a daily model R2 of 0.95 and 0.94, respectively and the PM2.5 was overestimated by WRF-Chem (R2=0.51). The annual mean PM2.5 concentration of the 365 ground monitors was 90.9 μg/m3, similar with the PM2.5 fields modeled by KED and data fusion at monitoring sites, but lower than that simulated by WRF-Chem. KED and the data fusion model performed better around the ground monitors (15-35 μg/m3), WRF-Chem performed relative worse with high prediction errors in northern Henan, southeastern Hebei and southern Shanxi (65-75 μg/m3). Conclusions: Both KED and data fusion technique provided highly accurate PM2.5. Current monitoring network in North China was dense enough to provide a reliable PM2.5 prediction by interpolation technique. Data fusion is an effective approach to improve the accuracy of WRF-Chem simulation spatiotemporally.

Evaluation of a data fusion approach to estimate daily PM2.5 levels in North China

PM2.5 air pollution has been a growing concern worldwide. Previous studies have conducted several techniques to estimate PM2.5 exposure spatiotemporally in China, but all these have limitations. This study was to develop a data fusion approach and compare it with kriging and Chemistry Module. Two techniques were applied to create daily spatial cover of PM2.5 in grid cells with a resolution of 10km in North China in 2013, respectively, which was kriging with an external drift (KED) and Weather Research and Forecast Model with Chemistry Module (WRF-Chem). A data fusion technique was developed by fusing PM2.5 concentration predicted by KED and WRF-Chem, accounting for the distance from the central of grid cell to the nearest ground observations and daily spatial correlations between WRF-Chem and observations. Model performances were evaluated by comparing them with ground observations and the spatial prediction errors. KED and data fusion performed better at monitoring sites with a daily model R2 of 0.95 and 0.94, respectively and PM2.5 was overestimated by WRF-Chem (R2=0.51). KED and data fusion performed better around the ground monitors, WRF-Chem performed relative worse with high prediction errors in the central of study domain. In our study, both KED and data fusion technique provided highly accurate PM2.5. Current monitoring network in North China was dense enough to provide a reliable PM2.5 prediction by interpolation technique.

Geostatistical Integration of Coarse Resolution Satellite Precipitation Products and Rain Gauge Data to Map Precipitation at Fine Spatial Resolutions

This paper investigates the benefits of integrating coarse resolution satellite-derived precipitation estimates with quasi-point rain gauge data for generating a fine spatial resolution precipitation map product. To integrate the two precipitation data sources, a geostatistical downscaling and integration approach is presented that can account for the differences in spatial resolution between data from different supports and adjusts inherent errors in the coarse resolution precipitation estimates. First, coarse resolution precipitation estimates are downscaled at a fine spatial resolution via area-to-point kriging to allow direct comparison with rain gauge data. Second, the downscaled precipitation estimates are integrated with the rain gauge data by multivariate kriging. In particular, errors in the coarse resolution precipitation estimates are adjusted against rain gauge data during this second stage. In this study, simple kriging with local means (SKLM) and kriging with an external drift (KED) are used as multivariate kriging algorithms. For comparative purposes, conditional merging (CM), a frequently-applied method for integrating rain gauge data and radar precipitation, is also employed. From a case study with Tropical Rainfall Measuring Mission (TRMM) 3B43 monthly precipitation products acquired in South Korea from May–October in 2013, we found that the incorporation of TRMM data with rain gauge data did not improve prediction performance when the number of rain gauge data was relatively large. However, the benefit of integrating TRMM and rain gauge data was most striking, regardless of multivariate kriging algorithms, when a small number of rain gauge data was used. These results indicate that the coarse resolution satellite-derived precipitation product would be a useful source for mapping precipitation at a fine spatial resolution if the geostatistical integration approach is applied to areas with sparse rain gauges.

Mapping soil organic matter concentration at different scales using a mixed geographically weighted regression method

The present regression models in digital soil mapping usually assume that relationships between soil properties and environmental variables are always fixed (as in MLR) or varying (as in GWR) in geographical space. In reality, some of the environmental variables may be fixed in affecting soil property variation and some are local varying. In this study, a mixed geographically weighted regression (MGWR) method which can deal with fixed and varying spatial relationships between a target variable and its environmental variables were proposed and used to predict topsoil soil organic matter (SOM) concentration in two study areas (Heshan, Heilongjiang province and Xuancheng, Anhui province, China) at two scales. Three groups of sample sets were created based on the total samples in the study areas to evaluate the robustness and stability of the model. Multiple linear regression (MLR), geographically weighted regression (GWR), GWR-kriging (GWRK), local regression-kriging (LRK), kriging with an external drift (KED), and ordinary kriging (OK) were used for comparison with MGWR. The validation results showed that the use of MGWR reduced the RMSE of GWR by 10.5% and 7.6% on average, reduced the RMSE of MLR by 12.8% and 9.9% on average for Heshan and Xuancheng study areas respectively. MGWR also showed a good competitiveness when compared with GWRK, LRK, KED and OK. In Heshan study area, the influence of flow length, relative position index, foot slope and distance to the nearest drainage were constant, whereas the elevation, topographic wetness index and valley index showed different influence in different regions. In Xuancheng study area, the fixed environmental variables were profile curvature, topographic wetness index and slope, whereas the varying environmental variables were precipitation, temperature, elevation, and limestone. The results indicate that the accuracy of predictions can be improved by adaptive coefficient according to the variation of environmental variables as implemented in MGWR compared with others considering only the local or global relationships. It was concluded that mixed geographically weighted regression model could be a potential method for digital soil mapping.

Mapping Soil Health over Large Agriculturally Important Areas

Soil health deterioration due to intensive agricultural activity is a worldwide problem. To better understand this process, there is a prime need to map soil health over wide areas. This paper aims to quantify soil health in a spatially explicit manner over a large area using soil health indicators. The methodology includes sampling design, autocorrelation analysis and Kriging interpolation. The following variables were measured from vertisol clayey soils: aggregate stability (AS); available water capacity (AWC); surface and subsurface penetration resistance (PR15 and PR45 respectively); root health (RH); organic matter (OM); pH; electrical conductivity (EC); cation-exchange capacity (CEC); exchangeable K; nitrification potential (Np); and P. Stratified random sampling was found to be a more efficient method than random sampling for representing a large area with a limited number of sampling locations. The variogram envelope method was found to be more conservative in determining the significance of autocorrelation than the classical Moran’s I approach. Phosphorus, CEC, PR15, EC, and K exhibited strong autocorrelation in space; other variables showed no autocorrelation. Land management factors were found to control the spatial variability of most soil variables. Kriging with an external drift (KED) was found to be the most useful approach for spatial prediction of soil health. A positive correlation was found between the interpolated soil health index and NDVI (Normalized Difference Vegetation Index). These results suggest that soil health maps can be used to explore how cultivation activities limit crop yields at the catchment scale, and to determine whether these activities create distinctive soil characteristics.

Comparison of geostatistical approaches to spatially interpolate month‐year rainfall for the Hawaiian Islands

ABSTRACTThe Hawaiian Islands have one of the most spatially diverse rainfall patterns on earth. Knowledge of these patterns is critical for a variety of resource management issues and, until now, only long‐term mean monthly and annual rainfall maps have been available for Hawai‘i. In this study, month‐year rainfall maps from January 1920 to December 2012 were developed for the major Hawaiian Islands. The maps were produced using climatologically aided interpolation (CAI), where the station anomalies were interpolated first, and then combined with the mean maps. A geostatistical method comparison was performed to choose the best interpolation method. The comparison focuses on three kriging algorithms: ordinary kriging (OK), ordinary cokriging (OCK), and kriging with an external drift (KED). Two covariates, elevation and mean rainfall, were tested with OCK and KED. The combinations of methods and covariates were compared using cross‐validation statistics, where OK produced the lowest error statistics. Station anomalies for each month were interpolated using OK and combined with the mean monthly maps to produce the final month‐year rainfall maps.

Mapping soil organic carbon content by geographically weighted regression: A case study in the Heihe River Basin, China

In large heterogeneous areas the relationship between soil organic carbon (SOC) and environmental covariates may vary throughout the area, bringing about difficulty for accurate modeling of the regional SOC variation. The benefit of local, geographically weighted regression (GWR) coefficients was tested in a case study on soil organic carbon mapping across a 50,810km2 area in northwestern China. This area is composed of an alpine ecosystem in the upper reaches and oases in the middle reaches. The benefit was quantified by comparing the quality of the maps obtained by GWR and geographically weighted ridge regression (GWRR) on the one side and multiple linear regression (MLR) on the other side. In these methods spatial dependence of model residuals is ignored. The root mean squared error (RMSE) of predictions of natural log-transformed SOC obtained with GWR was smaller than with MLR: 0.565 versus 0.618g/kg. The use of a local ridge parameter in GWRR did not lead to an increase in accuracy. Besides we compared the quality of maps obtained by geographically weighted regression followed by simple kriging of model residuals (GWRSK) and kriging with an external drift (KED) with global regression coefficients. In these methods the spatial dependence of model residuals is incorporated in the model. The RMSE with KED was smaller than with GWRSK: 0.515 versus 0.546g/kg. We conclude that fitting regression coefficients locally as in GWR only paid when no spatial random effect was included in the model. When a spatial random effect was included, the flexibility of local, geographically weighted regression coefficients was not needed and even undesirable as it led to less accurate predictions than KED with global regression coefficients. In comparing the accuracy of prediction methods by leave-one-out cross-validation (LOOCV) of a non-probability sample it is important to account for possible autocorrelation of pairwise differences in the prediction errors. The effective sample sizes were substantially smaller than the total number of sampling points, so that most pairwise differences in MSE were not significant at a significance level of 10% in a two-sided paired t-test.

Comprehensive Geophysical Data Integration and Stratigraphic Contacts Delineation in a Regional Hydrogeological Characterization Study

Accurate inference of the interfaces between geological units showing different hydraulic properties is a key step for a reliable hydrogeological characterization of regional aquifers. In this study, we developed a workflow that combines multiple geological and geophysical data sets having different intrinsic resolution to map a stratigraphic interface of the regional aquifer located in Montérégie, Quebec, Canada. One of the principal goals was to optimally assimilate all the data at all stages in the workflow. Firstly, the experimental variogram showed two structures of different ranges: one coming from highly sampled geophysical data and the other from conventional borehole geological markers. Secondly, a secondary variable is constructed with all secondary data (e.g., geological interpretations of low-resolution electromagnetic surveys), each having its own accuracy and resolution. To account for the variable secondary data, different reliability indexes were assigned as weights in a discrete smooth interpolation (DSI). Thirdly, a classic kriging with an external drift (KED) operator was used to interpolate the more reliable well data on the entire region. The approach was tested on the estimation of the bedrock interface elevation in a regional hydrogeological characterization study. The resulting map shows bedrock elevations coherent with geological structure of the region, representing main features such as outcrops and valleys. A cross section is presented to illustrate the philosophy behind the tools employed to achieve the estimation process. It also shows an example of visual quality control undertaken to validate the workflow.

Accounting for exhaustive secondary data into the mapping of water table elevation

This paper presents the incorporation of a digital elevation model into the spatial prediction of water table elevation in Mazandaran province (Iran) using a range of interpolation techniques. The multivariate methods used are: linear regression (LR), cokriging (COK), kriging with an external drift (KED) and regression kriging (RK). The analysis is performed on 3 years (1987, 1997 and 2007) of water table elevation data from about 260 monitoring wells. Prediction performances of the different algorithms are compared with two univariate techniques, i.e. inverse distance weighting and ordinary kriging (OK), through cross validation and examination of the consistency of the generated maps with the natural phenomena. Significantly smaller prediction errors are obtained for four multivariate algorithms but, in particular, KED and RK outperform LR and COK for 3 years. The results show the potential for using elevation for a more precise mapping of water table elevation.

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

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.

Estimating Freshwater Acidification Critical Load Exceedance Data for Great Britain Using Space-Varying Relationship Models

In this study, two distinct sets of analyses are conducted on a freshwater acidification critical load dataset, with the objective of assessing the quality of various models in estimating critical load exceedance data. Relationships between contextual catchment and critical load data are known to vary across space; as such, we cater for this in our model choice. Firstly, ordinary kriging (OK), multiple linear regression (MLR), geographically weighted regression (GWR), simple kriging with GWR-derived local means (SKlm-GWR), and kriging with an external drift (KED) are used to predict critical loads (and exceedances). Here, models that cater for space-varying relationships (GWR; SKlm-GWR; KED using local neighbourhoods) make more accurate predictions than those that do not (MLR; KED using a global neighbourhood), as well as in comparison to OK. Secondly, as the chosen predictors are not suited to providing useable estimates of critical load exceedance risk, they are replaced with indicator kriging (IK) models. Here, an IK model that is newly adapted to cater for space-varying relationships performs better than those that are not adapted in this way. However, when site misclassification rates are found using either exceedance predictions or estimates of exceedance risk, rates are intolerably high, reflecting much underlying noise in the data.

Links, comparisons and extensions of the geographically weighted regression model when used as a spatial predictor

In this study, we link and compare the geographically weighted regression (GWR) model with the kriging with an external drift (KED) model of geostatistics. This includes empirical work where models are performance tested with respect to prediction and prediction uncertainty accuracy. In basic forms, GWR and KED (specified with local neighbourhoods) both cater for nonstationary correlations (i.e. the process is heteroskedastic with respect to relationships between the variable of interest and its covariates) and as such, can predict more accurately than models that do not. Furthermore, on specification of an additional heteroskedastic term to the same models (now with respect to a process variance), locally-accurate measures of prediction uncertainty can result. These heteroskedastic extensions of GWR and KED can be preferred to basic constructions, whose measures of prediction uncertainty are only ever likely to be globally-accurate. We evaluate both basic and heteroskedastic GWR and KED models using a case study data set, where data relationships are known to vary across space. Here GWR performs well with respect to the more involved KED model and as such, GWR is considered a viable alternative to the more established model in this particular comparison. Our study adds to a growing body of empirical evidence that GWR can be a worthy predictor; complementing its more usual guise as an exploratory technique for investigating relationships in multivariate spatial data sets.

Geostatistics for Mapping Leaf Area Index over a Cropland Landscape: Efficiency Sampling Assessment

This paper evaluates the performance of spatial methods to estimate leaf area index (LAI) fields from ground-based measurements at high-spatial resolution over a cropland landscape. Three geostatistical model variants of the kriging technique, the ordinary kriging (OK), the collocated cokriging (CKC) and kriging with an external drift (KED) are used. The study focused on the influence of the spatial sampling protocol, auxiliary information, and spatial resolution in the estimates. The main advantage of these models lies in the possibility of considering the spatial dependence of the data and, in the case of the KED and CKC, the auxiliary information for each location used for prediction purposes. A high-resolution NDVI image computed from SPOT TOA reflectance data is used as an auxiliary variable in LAI predictions. The CKC and KED predictions have proven the relevance of the auxiliary information to reproduce the spatial pattern at local scales, proving the KED model to be the best estimator when a non-stationary trend is observed. Advantages and limitations of the methods in LAI field predictions for two systematic and two stratified spatial samplings are discussed for high (20 m), medium (300 m) and coarse (1 km) spatial scales. The KED has exhibited the best observed local accuracy for all the spatial samplings. Meanwhile, the OK model provides comparable results when a well stratified sampling scheme is considered by land cover.

Improved long‐term mean annual rainfall fields for Colombia

AbstractWith the aim of improving the long‐term mean annual surface water balance of Colombia, four new annual average precipitation fields are estimated at 4 km spatial resolution. To put in context, a concise literature review of rainfall in Colombia is presented. For estimation purposes, diverse multivariate geostatistical methods are implemented by combining information from 1180 raingauges covering the period 1950‐2005, and satellite data from the tropical rainfall measuring mission (TRMM) for the period 1999‐2005, used as a drift for the following geostatistical methods: (1) kriging with an external drift (KED), (2) standardized cokriging (SCK), (3) colocated cokriging (CCK), and (4) Markov regionalization CCK (CCKM). To ensure the reliability of the estimated precipitation fields, a detailed cross‐validation procedure is performed, including univariate and bivariate analyses of residuals, which allows us to conclude that the best estimated rainfall field is obtained with KED, and the worst with SCK. Visual analyses are also performed in the search for consistency of the resulting precipitation fields. Furthermore, local (at‐a‐pixel) uncertainty modelling analysis is performed using the indicator approach. Conditional cumulative distribution functions (CCDF) are estimated using indicator CCK with Bayes‐Markov hypothesis. Statistical descriptors for the pixel's CCDFs are estimated based on the resulting precipitation fields, including long‐term mean, conditional variance and the coefficient of variation. These improved precipitation fields along with their estimated uncertainties are available (http://cancerbero.unalmed.edu.co/∼hidrosig/index.php) for the scientific community and constitute useful basic information for diverse applications in water resources, agriculture, hydropower generation, human health, risks and disaster prevention, and many other applied sectors in Colombia. Copyright © 2010 Royal Meteorological Society

Nonstationary models for exploring and mapping monthly precipitation in the United Kingdom

AbstractSpatial statistical algorithms are used widely for both the exploration and mapping of environmental variables such as precipitation amount. One limitation of standard approaches to characterization and spatial interpolation is the usual assumption of stationarity. In short, spatial variation is assumed to be constant across the region of interest. Much research effort has been extended in developing approaches that allow for local variation in spatial structure. Simple moving window approaches are examples of such developments. The purpose of this paper is to apply selected methods for exploring and mapping monthly precipitation amount in the UK. Global regression, moving window regression (MWR) and geographically weighted regression (GWR) are used to explore the relationship between altitude and precipitation amount. Inverse distance weighting (IDW) is used as a basic approach to spatial prediction. Global and local variogram models are estimated and modelled to assess variation in spatial structure of precipitation amount and to inform spatial prediction using ordinary kriging (OK), kriging with an external drift (KED) and simple kriging with local means (SKlm). In the latter case, local means are derived using global regression, MWR (using ordinary least squares (OLS) and generalized least squares (GLS)) and GWR. The importance of choice of algorithm for the estimation of the variogram and for spatial prediction is assessed. The benefits of local as against global approaches and multivariate as against univariate prediction procedures are considered. It is demonstrated that use of elevation data to inform the prediction process reduces prediction errors and that there is also a small reduction in prediction errors when local variogram models are used. In this study, KED based on local variogram models provides more accurate predictions (as judged by cross‐validation statistics) than any other approach. The paper concludes by considering some key issues and possible avenues for future work. Copyright © 2009 Royal Meteorological Society

Geostatistical modeling of sedimentological parameters using multi‐scale terrain variables: application along the Belgian Part of the North Sea

In the nowadays highly pressurized marine environment, a science‐based approach to management becomes increasingly important. In many cases, the sediment nature and processes are the key to the understanding of the marine ecosystem, and can explain particularly the presence of soft‐substrata habitats. For predictions of the occurrence of species and habitats, detailed sedimentological information is required. This paper presents a methodology to create high quality sedimentological data grids of grain‐size fractions and the percentage of silt‐clay. Based on a multibeam bathymetry terrain model, multiple sources of secondary information (multi‐scale terrain variables) were derived. Through the use of the geostatistical technique, Kriging with an external drift (KED), this secondary information was used to assist in the interpolation of the sedimentological data. For comparison purposes, the more commonly used Ordinary Kriging technique was also applied. Validation indices indicated that KED gave better results for all of the maps.

Spatial Estimation of Reference Evapotranspiration in Andalusia, Spain

Abstract Knowledge of the spatial and temporal distribution of reference crop evapotranspiration (ET0) is of interest for regional water resources management, especially in areas of the world where fine-tuning of agricultural water demands over large areas is required. This study provides a strategy for mapping ET0 in regions with low meteorological data availability. For Andalusia, Spain, it involves estimating ET0 from temperature data using a locally calibrated version of the Hargreaves equation and the application of geostatistical interpolation techniques that take into account elevation as secondary information. Average annual ET0 at 191 observatories (with elevation between 0 and 1260 m) ranged from 954 to 1460 mm, with an average of 1283 mm, a standard deviation of 99 mm, and a correlation coefficient with elevation of −0.86. Simple kriging with varying local means (SKlm) and kriging with an external drift (KED)—two methods that take into account elevation as secondary information—increased spatial model efficiency by 30% as compared to ordinary kriging. SKlm was used for mapping ET0 since it better reproduced the descriptive statistics of the point data and yielded slightly smaller root-mean-squared estimation errors than KED. The spatial correlation of annual and monthly ET0 was well structured and anisotropic. Short-range variability, for separation distances up to 20–40 km, showed a strong linear increase with distance while long-range variability, up to 130–250 km, increased more gently with distance. The results of this structural analysis are relevant for the spatial optimization of a recently installed automated ET0 observation network, while obtained maps constitute a valuable tool for regional water resources evaluation, planning, and management and contribute to optimizing water use in local irrigated agriculture.