Kriging With External Drift Research Articles

Mapping stem volume in fast-growing eucalypt plantations: integrating spectral, textural, and temporal remote sensing information with forest inventories and spatial models

Key messageAccurate volume estimation in Eucalyptus plantation stands was achieved by a linear model using SPOT and Landsat multispectral imagery, specifically texture indices and pixel-scale NDVI time integrals, which reflect the local plantation growth history. Spatial modelling techniques such as Kriging with External Drift and Generalized Additive Model slightly improved predictions by accounting for spatial correlation of volume between sample points.ContextForest inventories are widely used to estimate stand production. To capture the inherent spatial variability within stands, spatial modelling techniques such as Kriging with External Drift (KED) and the generalized additive model (GAM) have emerged. These models incorporate information on spatial correlation and auxiliary variables that can be obtained from satellite imagery.AimsOur study explored the use of reflectance data from SPOT and Landsat multispectral imagery. We focused on texture indices and temporal integration of vegetation indices as auxiliary variables in KED and GAM to predict stem volume of fast-growing Eucalyptus sp. plantations in Brazil.MethodsThe components extracted from the high-resolution SPOT-6 image included spectral band values, band ratio metrics, key vegetation indices (NDVI, SAVI, and ARVI), texture measurements, and indices derived from texture analysis. Additionally, we included the accumulated NDVI time series acquired from Landsat 5, 7, and 8 satellites between the planting date and the forest inventory measurement date.ResultsThe best linear model of stem volume using remotely sensed predictors gave an R-squared value of 0.95 and a Root Mean Square Error (RMSE) of 12.44 m3/ha. The R-squared increased to 0.96 and the RMSE decreased to 10.6 m3/ha when the same predictors were included as auxiliary variables in the KED and GAM spatial models.ConclusionThe linear model using remotely sensed predictors contributed most to volume prediction, but the addition of spatial coordinates in the KED and GAM spatial models improved local volume predictions.

Remote and Proximal Sensors Data Fusion: Digital Twins in Irrigation Management Zoning.

The scientific field of precision agriculture employs increasingly innovative techniques to optimize inputs, maximize profitability, and reduce environmental impact. However, obtaining a high number of soil samples is challenging in order to make precision agriculture viable. There is a trade-off between the amount of data needed and the time and resources spent to obtain these data compared to the accuracy of the maps produced with more or fewer points. In the present study, the research was based on an exhaustive dataset of apparent electrical conductivity (aEC) containing 3906 points distributed along 26 transects with spacing between each of up to 40 m, measured by the proximal soil sensor EM38-MK2, for a grain-producing area of 72 ha in São Paulo, Brazil. A second sparse dataset was simulated, showing only four transects with a 400 m distance and, in the end, only 162 aEC points. The aEC map via ordinary kriging (OK) from the grid with 26 transects was considered the reference, and two other mapping approaches were used to map aEC via sparse grid: kriging with external drift (KED) and geographically weighted regression (GWR). These last two methods allow the increment of auxiliary variables, such as those obtained by remote sensors that present spatial resolution compatible with the pivot scale, such as data from the Landsat-8, Aster, and Sentinel-2 satellites, as well as ten terrain covariates derived from the Alos Palsar digital elevation model. The KED method, when used with the sparse dataset, showed a relatively good fit to the aEC data (R2 = 0.78), with moderate prediction accuracy (MAE = 1.26, RMSE = 1.62) and reasonable predictability (RPD = 1.76), outperforming the GWR method, which had the weakest performance (R2 = 0.57, MAE = 1.78, RMSE = 2.30, RPD = 0.81). The reference aEC map using the exhaustive dataset and OK showed the highest accuracy with an R2 of 0.97, no systematic bias (ME = 0), and excellent precision (RMSE = 0.56, RPD = 5.86). Management zones (MZs) derived from these maps were validated using soil texture data from clay samples measured at 0-10 cm depth in a grid of 72 points. The KED method demonstrated the highest potential for accurately defining MZs for irrigation, producing a map that closely resembled the reference MZ map, thereby providing reliable guidance for irrigation management.

Simultaneous monitoring of the limited area ionosphere with the use of GPS and ionosonde

The importance of this study lies in its contribution to the field of ionospheric science, particularly in the monitoring and prediction of ionospheric parameters. This study focuses on comparing two techniques for spatio-temporal kriging of critical ionospheric parameters, namely the F2 layer critical frequency (foF2) and Total Electron Content (TEC), under both quiet and disturbed ionospheric conditions. The first technique, Ordinary Kriging (OK), relies solely on foF2 data, while the second technique, Kriging with External Drift (KED), incorporates the dynamic relationship between foF2 and TEC data to improve the interpolation process. This study demonstrates how KED, which utilizes dynamic secondary information, offers advantages over OK. By simultaneously monitoring the ionosphere from a single ionosonde station and collecting data from visible GPS satellites, KED enhances the accuracy of spatial structural analysis, enabling more precise estimations of ionospheric characteristics across interpolation grid nodes. The research highlights that the non-uniform distribution of ionosonde stations, especially in regions with limited foF2 data, can benefit from KED's incorporation of TEC data. Additionally, the study reveals the potential for improving TEC maps by assimilating foF2 data located at the periphery of the observation area. Furthermore, the paper discusses the determination of semivariogram structures for both foF2 and TEC data, emphasizing the spatial correlations between data points. These findings contribute to a better understanding of ionospheric behavior and the potential for enhanced ionospheric modeling.

Remote Sensing and Kriging with External Drift to Improve Sparse Proximal Soil Sensing Data and Define Management Zones in Precision Agriculture

The precision agriculture scientific field employs increasingly innovative techniques to optimize inputs, maximize profitability, and reduce environmental impacts. Therefore, obtaining a high number of soil samples to make precision agriculture feasible is challenging. This data bottleneck has been overcome by identifying sub-regions based on data obtained through proximal soil sensing equipment. These data can be combined with freely available remote sensing data to create more accurate maps of soil properties. Furthermore, these maps can be optimally aggregated and interpreted for soil heterogeneity through management zones. Thus, this work aimed to create and combine soil management zones from proximal soil sensing and remote sensing data. To this end, data on electrical conductivity and magnetic susceptibility, both apparent, were measured using the EM38-MK2 proximal soil sensor and the contents of the thorium and uranium elements, both equivalent, via the Medusa MS1200 proximal soil sensor for a 72-ha grain-producing area in São Paulo, Brazil. The proximal soil sensing attributes were mapped using ordinary kriging (OK). Maps were also made using kriging with external drift (KED), and the proximal soil sensor attributes data, combined with remote sensing data, such as Landsat-8, Aster, and Sentinel-2 images, in addition to 10 terrain covariables derived from the digital elevation model Alos Palsar. As a result, three management zone maps were produced via the k-means clustering algorithm: using data from proximal sensors (OK), proximal sensors combined with remote sensors (KED), and remote sensors. Seventy-two samples (0–10 cm in depth) were collected and analyzed in a laboratory (1 sample per hectare) for concentrations of clay, calcium, organic carbon, and magnesium to assess the capacity of the management zone maps created using analysis of variance. All zones created using the three data groups could distinguish the different treatment areas. The three data sources used to map management zones produced similar map zones, but the zone map using a combination of proximal and remote data did not show an improvement in defining the management zones, and using only remote sensing data lowered the significance levels of differentiating each zone compared to the OK and KED maps. In summary, this study not only underscores the global applicability of proximal and remote sensing techniques in precision agriculture but also sheds light on the nuances of their integration. The study’s findings affirm the efficacy of these advanced technologies in addressing the challenges posed by soil heterogeneity, paving the way for more nuanced and site-specific agricultural practices worldwide.

Deriving Gridded Hourly Rainfall on O‘ahu by Combining Gauge and Radar Rainfall

Abstract High temporal and spatial resolution precipitation datasets are essential for hydrological and flood modeling to assist water resource management and emergency responses, particularly for small watersheds, such as those in Hawai‘i in the United States. Unfortunately, fine temporal (subdaily) and spatial (<1 km) resolutions of rainfall datasets are not always readily available for applications. Radar provides indirect measurements of the rain rate over a large spatial extent with a reasonable temporal resolution, while rain gauges provide “ground truth.” There are potential advantages to combining the two, which have not been fully explored in tropical islands. In this study, we applied kriging with external drift (KED) to integrate hourly gauge and radar rainfall into a 250 m × 250 m gridded dataset for the tropical island of O‘ahu. The results were validated with leave-one-out cross validation for 18 severe storm events, including five different storm types (e.g., tropical cyclone, cold front, upper-level trough, kona low, and a mix of upper-level trough and kona low), and different rainfall structures (e.g., stratiform and convective). KED-merged rainfall estimates outperformed both the radar-only and gauge-only datasets by 1) reducing the error from radar rainfall and 2) improving the underestimation issues from gauge rainfall, especially during convective rainfall. We confirmed the KED method can be used to merge radar with gauge data to generate reliable rainfall estimates, particularly for storm events, on mountainous tropical islands. In addition, KED rainfall estimates were consistently more accurate in depicting spatial distribution and maximum rainfall value within various storm types and rainfall structures. Significance Statement The results of this study show the effectiveness of utilizing kriging with external drift (KED) in merging gauge and radar rainfall data to produce highly accurate, reliable rainfall estimates in mountainous tropical regions, such as O‘ahu. The validated KED dataset, with its high temporal and spatial resolutions, offers a valuable resource for various types of rainfall-related research, particularly for extreme weather response and rainfall intensity analyses in Hawai’i. Our findings improve the accuracy of rainfall estimates and contribute to a deeper understanding of the performance of various rainfall estimation methods under different storm types and rainfall structures in a mountainous tropical setting.

Validation of uncertainty predictions in digital soil mapping

It is quite common in digital soil mapping (DSM) to quantify the uncertainty of issued predictions, that is to make probabilistic predictions. Yet, little attention has been paid to its validation. Probabilistic predictions are only of value for end users if they are reliable and ideally also sharp. Reliability refers to the consistency between predicted conditional probabilities and observed frequencies of independent test data. Sharpness refers to the concentration of a conditional probability distribution function, i.e. its narrowness. The prediction interval coverage probability (PICP) is currently used in DSM to validate the reliability of prediction intervals but it is ignorant of a potential one-sided bias of its boundaries. Therefore, we propose to extend the current validation procedure with metrics used in the broader probabilistic literature. These metrics not only evaluate probabilistic predictions in prediction interval format but also quantiles or full conditional probability distributions. We suggest the quantile coverage probability (QCP) and probability integral transform (PIT) histogram as alternatives to PICP and proper scoring rules for relative comparisons of competing probabilistic models. As scoring rules, we present the interval score (IS) and the continuous ranked probability score (CRPS), which can be decomposed into a reliability part (RELI). We illustrated the use of these metrics in a case study using soil pH and soil organic carbon from the LUCAS-soil database. Thereby, probabilistic predictions of five different models were compared: a reference null model (NM), quantile regression forest (QRF), quantile regression post-processing of a random forest (QRPP RF), kriging with external drift (KED) and quantile regression neural network (QRNN). For KED and QRNN, one-sided bias was found. This was not apparent from PICP but was shown by use of the PIT histogram and QCP. RELI summarized the trends found in QCP, PICP and PIT histograms to one numerical value. CRPS and IS were especially harsh to outliers and low sharpness. According to CRPS and IS, the best probabilistic predictions were obtained by QRF and QRPP RF and the worst by NM.

Carbon Stock Prediction in Managed Forest Ecosystems Using Bayesian and Frequentist Geostatistical Techniques and New Generation Remote Sensing Metrics

The study compares the performance of a hierarchical Bayesian geostatistical methodology with a frequentist geostatistical approach, specifically, Kriging with External Drift (KED), for predicting C stock using prediction aides from the Landsat-8 and Sentinel-2 multispectral remote sensing platforms. The frequentist geostatistical approach’s reliance on the long-run frequency of repeated experiments for constructing confidence intervals is not always practical or feasible, as practitioners typically have access to a single dataset due to cost constraints on surveys and sampling. We evaluated two approaches for C stock prediction using two new generation multispectral remote sensing datasets because of the inherent uncertainty characterizing spatial prediction problems in the unsampled locations, as well as differences in how the Bayesian and frequentist geostatistical paradigms handle uncertainty. Information on C stock spectral prediction in the form of NDVI, SAVI, and EVI derived from multispectral remote sensing platforms, Landsat-8 and Sentinel-2, was used to build Bayesian and frequentist-based C stock predictive models in the sampled plantation forest ecosystem. Sentinel-2-based C stock predictive models outperform their Landsat-8 counterparts using both the Bayesian and frequentist inference approaches. However, the Bayesian-based Sentinel-2 C stock predictive model (RMSE = 0.17 MgCha−1) is more accurate than its frequentist-based Sentinel-2 (RMSE = 1.19 MgCha−1) C stock equivalent. The Sentinel-2 frequentist-based C stock predictive model gave the C stock prediction range of 1 ≤ MgCha−1 ≤ 290, whilst the Sentinel-2 Bayesian-based C stock predictive model resulted in the prediction range of 1 ≤ MgCha−1 ≤ 285. However, both the Bayesian and frequentist C stock predictive models built with the Landsat-8 sensor overpredicted the sampled C stock because the range of predicted values fell outside the range of the observed C stock values. As a result, we recommend and conclude that the Bayesian-based C stock prediction method, when it is combined with high-quality remote sensing data such as that of Sentinel-2, is an effective inferential statistical methodology for reporting C stock in managed plantation forest ecosystems.

Modelling the spatial population structure and distribution of the queen conch, Aliger gigas, on the Pedro Bank, Jamaica

The estimation of reliable indices of abundance for sedentary stocks requires the incorporation of the underlying spatial population structure, including issues arising from the sampling design and zero inflation. We applied seven spatial interpolation techniques [ordinary kriging (OK), kriging with external drift (KED), a negative binomial generalized additive model (NBGAM), NBGAM plus OK (NBGAM+OK), a general additive mixed model (GAMM), GAMM plus OK (GAMM+OK) and a zero-inflated negative binomial model (ZINB) ] to three survey datasets to estimate biomass for the gastropod Aliger gigas on the Pedro Bank Jamaica. The models were evaluated using 10-fold cross-validation diagnostics criteria for choosing the best model. We also compared the best model estimations against two common design methods to assess the consequences of ignoring the spatial structure of the species distribution. GAMM and ZINB were overall the best models but were strongly affected by the sampling design, sample size, the coefficient of variation of the sample and the quality of the available covariates used to model the distribution (geographic location, depth and habitat). More reliable abundance indices can help to improve stock assessments and the development of spatial management using an ecosystem approach.

A WRF/WRF-Hydro coupling system with an improved structure for rainfall-runoff simulation with mixed runoff generation mechanism

• The structure of WRF-Hydro is improved for runoff generation with mixed mechanisms. • Empirical equations from a conceptual model are transplanted into WRF-Hydro. • Merged rainfall forcing with radar QPE and rain gauges is used for model calibration. • The structure is verified for both one-way and the fully coupled WRF/WRF-Hydro. • Increase of infiltration-excess runoff is found with decrease of saturation-excess. The coupled atmospheric-hydrologic systems help achieve deeper understanding on the interactions between the atmospheric and land-surface processes, improve the spatial and temporal accuracy of hydrologic forecasts and extend the forecast leading time. WRF-Hydro is nowadays a widely used hydrologic module to be coupled with the mesoscale numerical weather model WRF for atmospheric-hydrologic research and applications. The structure of WRF-Hydro is improved and expanded in this study to better adapt to the complicated rainfall-runoff transformation mechanism in the mixed runoff generation regions. The infiltration parameterisation is replaced by an infiltration equation that takes into account the impact of the surface soil moisture variation on the infiltration capacity, and the spatial discretisation of the infiltration capacity from a distribution curve is achieved based on the conception of the topographic index. For the runoff convergence, the river channel leakage loss is introduced into the Muskingum-Cunge (MC) streamflow calculations with variable parameters in time. Efforts are also made to reduce parameter calibration errors in WRF-Hydro. The accuracy of the WRF rainfall forcing is improved by merging weather radar and rain gauge observations using the Kriging with external drift (KED) interpolation tool. The key parameters of WRF-Hydro are then calibrated using the improved rainfall forcing with the merging observations. The performance of the improved WRF-Hydro structure is explored in both one-way and the fully coupled modes with the WRF model. Typical rainstorm events are selected from semi-humid and semi-arid catchments of Northern China as case studies. Results have shown that the improved WRF-Hydro is more effective in simulating floods with high peaks and steep rises-and-falls, but a problem is also shown of receding more quickly for low peaks. The improved model structure has changed the proportion of the mixed runoff generation. In the long term, there is an increase in the infiltration-excess runoff generation and decrease in the saturation-excess amount. Both the one-way and the fully coupled WRF/WRF-Hydro systems have demonstrated the suitability of the improved structure for rainfall-runoff processes dominated by the infiltration-excess runoff generation.

Performance assessment of interpolation techniques for optimal areal rainfall–temperature estimation: the case of two contrasting river catchments, Akaki and Mille, in Ethiopia

Abstract In the topographic complex catchments, landscape features have a significant impact on the spatial prediction of rainfall and temperature. In this study, performance assessments were made of various interpolation techniques for the prediction of the spatial distribution of rainfall and temperature in the Mille and Akaki River catchments, Ethiopia, through an improved approach on selecting the auxiliary variables as a covariate. Two geostatistical interpolation techniques, ordinary kriging (OK) and kriging with external drift (KED), and one deterministic interpolation technique, inverse distance weighting (IDW), were tested through a leave-one-out cross-validation (LOOCV) procedure. The results indicated that using the multivariate geostatistical interpolation technique (KED) with the auxiliary variables as a covsariate outperformed the univariate geostatistical (OK) and deterministic (IDW) techniques for the spatial interpolation of sampled rainfall–temperature data in both contrasting catchments, Akaki and Mille, with the lowest estimation errors (e.g., for Mille annual mean rainfall: root mean square error=75.32, 77.34, 245.72, mean bias error=3.70, −33.18, −15.61, mean absolute error=67.99, 69.51, 192.64) using KED with the combination of elevation and easting as a covariate, IDW and OK, respectively. Thus, the study confirmed that the use of elevation and easting/northing coordinates as predictors in geostatistical interpolation techniques could significantly improve the spatial prediction of climatic variables.

The accuracy of geostatistics for regional geomagnetic modeling in an archipelago setting.

Indonesia as an archipelago country relies on a limited number and clustered distributed repeat station networks. This paper explores the use of geostatistical modeling to overcome this data limitation. The model data set consisted of repeat station data from 1985 to 2015 epoch. The geostatistical methods utilized included ordinary kriging (OK), collocated cokriging (CC), and kriging with external drift (KED). The model generated using these geostatistical methods was then compared to spherical cap harmonic analyses (SCHA) and polynomial models. The geostatistical model was shown to perform better, with greater accuracy in declination, inclination, and total intensity, as indicated by the root mean square error (RMSE). We have demonstrated that the geostatistical method is a promising approach in the modeling of regional geomagnetic field, especially in areas with limited and clustered distributed data.

Comparative Analysis of Different Spatial Interpolation Methods Applied to Monthly Rainfall as Support for Landscape Management

Landscape management requires spatially interpolated data, whose outcomes are strictly related to models and geostatistical parameters adopted. This paper aimed to implement and compare different spatial interpolation algorithms, both geostatistical and deterministic, of rainfall data in New Zealand. The spatial interpolation techniques used to produce finer-scale monthly rainfall maps were inverse distance weighting (IDW), ordinary kriging (OK), kriging with external drift (KED), and ordinary cokriging (COK). Their performance was assessed by the cross-validation and visual examination of the produced maps. The results of the cross-validation clearly evidenced the usefulness of kriging in the spatial interpolation of rainfall data, with geostatistical methods outperforming IDW. Results from the application of different algorithms provided some insights in terms of strengths and weaknesses and the applicability of the deterministic and geostatistical methods to monthly rainfall. Based on the RMSE values, the KED showed the highest values only in April, whereas COK was the most accurate interpolator for the other 11 months. By contrast, considering the MAE, the KED showed the highest values in April, May, June and July, while the highest values have been detected for the COK in the other months. According to these results, COK has been identified as the best method for interpolating rainfall distribution in New Zealand for almost all months. Moreover, the cross-validation highlights how the COK was the interpolator with the best least bias and scatter in the cross-validation test, with the smallest errors.

Identifying meaningful covariates that can improve the interpolation of monsoon rainfall in a low‐lying tropical region

AbstractThe assessment of spatio‐temporal distribution of rainfall over the entire monsoon season is critical for the water resources management in a tropical low‐lying region like Bangladesh. The primary objective is to find suitable interpolation methods for each monsoon month as well as the accumulated rainfall for the entire season in Bangladesh. As the unique position of Bangladesh and monsoon system both contribute the variation in rainfall, the identification of associated suitable covariates can offer reliable estimates when applied with the appropriate interpolation model. Several potential covariates are identified diligently including the remotely sensed annul average monsoon rainfall (RAAMR), distance to Bay‐of‐Bengal and distance to Hindu‐Kush Himalayan Region, among others. This study mainly considers the multivariate approach: kriging with external drift (KED) which is able to take external information to positive impact. The method is then compared with the more traditional univariate, ordinary kriging (OK) and the inverse distance weighting (IDW) method using the cross‐validation technique. Daily rainfall data from 1970 to 2016 at 34 stations were used for the assessment. Result shows that the KED model is identified suitable for the considered variants of June, August, September and annual average monsoon rainfall whereas the OK is appropriate for July. Overall, the KED is found superior model with the incorporation of RAAMR data. The successful inclusion of remotely sensed data in the kriging system paves a new dimension in the climatological research for countries where climate‐measuring resources are limited.

Mapping Urban Air Quality from Mobile Sensors Using Spatio-Temporal Geostatistics.

With the advancement of technology and the arrival of miniaturized environmental sensors that offer greater performance, the idea of building mobile network sensing for air quality has quickly emerged to increase our knowledge of air pollution in urban environments. However, with these new techniques, the difficulty of building mathematical models capable of aggregating all these data sources in order to provide precise mapping of air quality arises. In this context, we explore the spatio-temporal geostatistics methods as a solution for such a problem and evaluate three different methods: Simple Kriging (SK) in residuals, Ordinary Kriging (OK), and Kriging with External Drift (KED). On average, geostatistical models showed 26.57% improvement in the Root Mean Squared Error (RMSE) compared to the standard Inverse Distance Weighting (IDW) technique in interpolating scenarios (27.94% for KED, 26.05% for OK, and 25.71% for SK). The results showed less significant scores in extrapolating scenarios (a 12.22% decrease in the RMSE for geostatisical models compared to IDW). We conclude that univariable geostatistics is suitable for interpolating this type of data but is less appropriate for an extrapolation of non-sampled places since it does not create any information.

Extreme rainfall estimation at ungauged locations: Information that needs to be included in low-lying monsoon climate regions like Bangladesh

Bangladesh, a tropical low-lying region, is stranded between the Bay of Bengal in the south and the Himalayas in the north. The unique characteristics allow receiving huge rainfall, in particular, in the monsoon season. The physical phenomena such as the Bay of Bengal, the Himalayas and the monsoon season dominate the extreme rainfall pattern in Bangladesh. However, how to include that information to good effect in the estimation of extreme rainfall at ungauged locations is yet to be investigated. The objective of this work is to examine the information that needs to be included in the estimation of extreme rainfall in such regions. The approach that permits parameters of an extreme value distribution to vary spatially is used to determine the probability model. The methodology, subsequently, authorizes the frequency analysis to be performed at ungauged conditions. The geostatistical technique in the form of kriging with external drift (KED) which has the ability to take secondary information is used to produce spatial maps of the parameters. This study primarily assessed KED interpolation scheme and its ability to accurately predict the model parameters. The scheme was compared against the more widely used ordinary kriging (OK) model. The cross- validation was used to find the robust scheme for each parameter. Annual maximum daily rainfall data from 34 measuring stations were used for the assessment. The KED with the covariate annual mean monsoon rainfall (AMMR) appears to be the best model for location parameter. However, for the scale and shape parameter the KED models were unable to score past the OK model. The availability of parameter maps allows the estimation of extreme rainfall at locations where rainfall records were absent and offers a much better elucidation than the traditional at-site quantile based interpolation. Overall, the inclusion of monsoon rainfall and spatial information are deemed necessary for the estimation of extreme rainfall at ungauged locations in a low-lying monsoon region.

Spatial interpolation of the extreme hourly precipitation at different return levels in the Haihe River basin

The return level is an important measure for the extreme precipitation, and maps of the return level are used to derive information for the design of hydrological and hydraulic engineering projects. The interpolation of hourly extreme precipitation is challenging due to low station density and high inhomogeneity. Hourly precipitation observations at 232 weather stations in the Haihe River basin from 1961 to 2012 were used to evaluate six interpolation methods including Inverse Distance Weighted (IDW), Ordinary Kriging (OK), Kriging with External Drift (KED) assisted by different covariables and Empirical Bayesian Kriging (EBK) for generating 2-, 5-, 10-, 20-, 50-, and 100-year return level maps for the basin. Leave-one-out cross-validation showed that although there was a high correlation coefficient between hourly extreme precipitation and the elevation (DEM), KED incorporating DEM as the covariable (KED_DEM) did not improve the interpolation efficiency compared with OK. KED with the annual average precipitation as the covariable (KED_AP), which had the largest Nash-Sutcliffe efficiency coefficient (NSE, 0.89–0.45) and the smallest root mean square error (RMSE, 2.23–15.05 mm) for the six return levels, outperformed the other five methods compared. The return levels varied from 13 to 46, 20–66, 24–78, 26–89, 29–110 and 30–134 mm, with mean values of 30, 40, 48, 57, 67 and 75 mm for the 2-, 5-, 10-, 20-, 50- and 100-year return levels, respectively, in the Haihe River basin. The return level maps of extreme hourly precipitation in the Haihe River basin were generated based on the KED_AP method and showed a spatial distribution of decreasing trend from the southeastern part to the northwestern part of the basin. The high-value centre moved from the eastern coastal area for smaller return level maps to the southern area for the 50- and 100-year return level maps. This study may provide some insights into the spatial interpolation of extreme precipitation on an hourly scale.

A Machine Learning Approach for Rainfall Estimation Integrating Heterogeneous Data Sources

Providing an accurate rainfall estimate at individual points is a challenging problem in order to mitigate risks derived from severe rainfall events, such as floods and landslides. Dense networks of sensors, named rain gauges (RGs), are typically used to obtain direct measurements of precipitation intensity in these points. These measurements are usually interpolated by using spatial interpolation methods for estimating the precipitation field over the entire area of interest. However, these methods are computationally expensive, and to improve the estimation of the variable of interest in unknown points, it is necessary to integrate further information. To overcome these issues, this work proposes a machine learning-based methodology that exploits a classifier based on ensemble methods for rainfall estimation and is able to integrate information from different remote sensing measurements. The proposed approach supplies an accurate estimate of the rainfall where RGs are not available, permits the integration of heterogeneous data sources exploiting both the high quantitative precision of RGs and the spatial pattern recognition ensured by radars and satellites, and is computationally less expensive than the interpolation methods. Experimental results, conducted on real data concerning an Italian region, Calabria, show a significant improvement in comparison with Kriging with external drift (KED), a well-recognized method in the field of rainfall estimation, both in terms of the probability of detection (0.58 versus 0.48) and mean-square error (0.11 versus 0.15).

Comparison of mapping approaches for estimating extreme precipitation of any return period at ungauged locations

Reliable estimation of return period values of extreme precipitation at ungauged locations is considered to be a key exercise in hydrometeorological studies. This study aims to identify an accurate approach in producing spatial maps (i.e. ungauged estimation) of extreme precipitation for any return period within a region. The study compares the following approaches: interpolation of summary of data as represented by L-moments, interpolation of parameters of an extreme value distribution and interpolation of return period quantile value. Several interpolation schemes are considered; however, the aim is to evaluate schemes that employ secondary data. The schemes compared are ordinary kriging, kriging with external drift (KED) and a more traditional, inverse distance weighting. The secondary data namely elevation, satellite based mean annual precipitation (MAP), distance from nearest coast (CD) and geographical coordinates are incorporated in the KED system. Annual maximum 1-day precipitation series at 76 gauging stations from the region of East China have been used to assess the performance. The generalized extreme value (GEV) distribution, appropriate for the study region, with the method of L-moments is used to analyze the frequency of extreme precipitation. It is found that either the approach of interpolating parameters of GEV distribution or L-moments should be the natural choice for estimating design value at any return period. However, in terms of error statistics the approach of interpolating parameters has given a lower RMSE value compared to the approach of interpolating L-moments. The approach of quantile interpolation performed worst and should not be used in practice in interpolating return period values. The KED is recognized as the most appropriate interpolation scheme when a significant covariate is identified. The MAP appears to be a suitable covariate in most cases when interpolating L moments (1st and 2nd L-moment) or GEV parameters (location and scale parameter). There is no spatial dependence identified for L-skewness or shape parameter of GEV distribution and in the future one should concentrate on how a superior spatial model can be identified in this context.

Comparison of geostatistical interpolation methods to map annual rainfall in the Chéliff watershed, Algeria

The Cheliff watershed has one of the most spatially diverse pluviometric regimes in northwestern Algeria. Understanding these regimes is essential for managing water resources and identifying the most vulnerable regions to climate change. Mean annual rainfall data (1972–2012) for 58 meteorological stations and their corresponding elevation were used. Maps were produced using three geostatistical interpolation algorithms: ordinary kriging (OK), regression-kriging (RK), and kriging with external drift (KED); the first algorithm uses only rainfall while the other two use also elevation. Interpolation methods were compared using statistical indicators of cross-validation. Results indicate that KED is the least biased interpolator with limited number of strong underestimates or overestimates and limited relative importance of this strong underestimation or overestimation, followed by RK and finally OK. The best match between measured and predicted values was for KED (correlation coefficient of 0.82), followed by RK (0.79), while OK is far from them (0.70). KED can be considered the best model because it gives the lowest values of mean error, mean absolute error, and root mean square error (− 1.9, 35.4, and 49.5 mm, respectively) and the highest values of Willmott agreement index, Lin concordance coefficient, and Nash–Sutcliffe efficiency coefficient (0.89, 0.80, and 0.67, respectively), results of RK are intermediate, while those of OK are the worst. There is clearly significant improvement in the prediction performance taking into account the elevation, in particular by KED. Results show that KED is the most appropriate to produce map of mean annual rainfall in the Cheliff watershed, Algeria.

High-Resolution Temperature Mapping by Geostatistical Kriging with External Drift from Large-Eddy Simulations

Abstract Detailed temperature maps are required in various applications. Any temperature interpolation over complex terrain must account for differences in land cover and elevation. Local circulations and other small-scale factors can also perturb the temperature. This study considers the surface air temperature T mapping with geostatistical kriging. The kriging methods are implemented for both T and temperature anomalies ΔT, defined as difference between T at a given location and T at the same elevation in the free atmosphere. The study explores the parallelized atmospheric large-eddy simulation (LES) model (PALM) as a source for variogram and external drift in the kriging methods. Ten kriging methods for the temperature mapping have been considered: ordinary kriging (OK) of T and ΔT with variogram derived from the observations (methods 1 and 2, correspondingly); OK of T and ΔT with variogram derived from LES data (3 and 4); universal kriging with external drift (KED) that utilizes the LES data (5 and 6); a weighted combination of KED of T and ΔT (method 7); and methods 5, 6, and 7 enhanced with additional “virtual” points in remote areas (methods 8, 9, and 10). These 10 methods are evaluated for eight typical weather situations observed in Bergen, Norway. Our results demonstrate considerable added value of the LES information for the detailed meteorological temperature mapping. The LES data improve both the variogram and the resulting temperature maps, especially in the remote mountain parts of the domain and along the coast.