Spatiotemporal Variogram Research Articles

Spatio-temporal Kriging for spatial irradiance estimation with short-term forecasting in a thermosolar power plant

This article proposes a method to improve the efficiency of solar power plants by estimating and forecasting the spatial distribution of direct normal irradiance (DNI) using a sensor network and anemometer data. For this purpose, the proposed approach employs spatio-temporal kriging with an anisotropic spatio-temporal variogram that depends on wind speed to accurately estimate the distribution of DNI in real-time, making it useful for short-term forecast and nowcast of DNI. Finally, the method is validated using synthetic data from varying sky conditions, outperforming another state-of-the-art technique.

Spatiotemporal Variograms as Neighborhood Definers

Spatial neighborhood definitions are a consistent source of disagreement among geographic scholars. This research will focus on the implementation and evaluation of spatiotemporal variograms (STVs) as a source of spatial neighborhood definition. STVs show the similarity, measured by semivariance, of spatial events to each other when separated by time and space. Over both time and space, there should exist distances over which pairs of points become “independent” of one another. We seek seeks to answer two questions in relation to STVs and their use as neighborhood definitions: (1) What data and process adjustments are necessary to implement STVs to provide neighborhood search definitions in time and space, and (2) Given that there are many ways to define a neighborhood, STVs may provide a comprehensive method that uses the data themselves to inform the size and scope of neighborhoods, with the added advantage to evaluating both spatial and temporal axes at once. We demonstrate a well‐defined neighborhood that accounts for temporal variation as well as spatial and will be a needed addition as the tools incorporating simultaneous spatial and temporal neighborhoods are implemented.

Uncertainty assessment of spatiotemporal distribution and variation in regional soil heavy metals based on spatiotemporal sequential Gaussian simulation

Revealing the spatiotemporal (ST) distribution and changes in regional soil heavy metals is significant to soil pollution control and management. However, most of the ST analysis models in the existing studies ignore the uncertainty of ST changes in soil heavy metals, making their results unreliable. In this study, using soil Pb collected from 2016 to 2019 in a mining city in China as case data, an ST sequential Gaussian simulation (STSGS) is proposed to reveal the ST distribution and variation in heavy metals in regional soils and their uncertainties. Firstly, the ST variogram was analysed and fitted using a theoretical variogram model integrating the experimental variations at the ST scale. Secondly, 500 simulation realisations with random access path were generated by the ST Kriging method. Considering the obtained 500 simulation realisations, a series of ST analysis methods was proposed and employed to reveal the ST distribution and changes with uncertainty assessment of regional soil heavy metals. The main results are as follows. (1) For the whole study region, soil Pb content initially increased and then decreased from 2016 to 2019. The average probability of soil Pb exceeding 90 mg/kg was 0.121, 0.214, 0.312 and 0.291 in 2016, 2017, 2018 and 2019, respectively, whereas the average probability of always exceeding 90 mg/kg in the 4 years was only 0.032. (2) From 2016 to 2019, the area proportions of the increase and decrease of soil Pb content in the study area were 87.2% and 12.8%, respectively. However, according to the standardised statistic, only 0.161% and 8.72% of the total areas were determined to have a significant decrease and increase in soil Pb content from 2016 to 2019. (3) From 2016 to 2019, the areas with a greater than 0.6 probability of soil Pb concentration decreasing by more than 5 mg/kg and increasing by more than 20, 40 and 80 mg/kg accounted for 4.96%, 32.2%, 11.5% and only 1.91% of the total study region, respectively. The incremental high-probability areas were primarily those where Pb pollution was already serious. Finally, the advantages of the proposed STSGS method were summarised.

Development of spatiotemporal high-resolution temperature data for São Paulo, Brazil; a valuable resource for epidemiologists

Background and aim Most studies investigating the temperature-mortality association in cities rely on temperature recordings from few meteorological stations or models at coarse spatial resolution, implicitly assuming an homogenous spatial distribution across. Evidence from urban heat island studies, remote sensing data, and atmospheric and physical sciences suggest this assumption does not hold. Overlooking the spatial variability of temperature exposure in cities may lead to biased epidemiological findings and limit our ability to identify areas of high risk. Herein, we aim to develop an open-access dataset of daily ambient temperature at 500 meters resolution for the municipality of São Paulo (2015-2020). Methods We obtained daily mean temperature data from 60 ground stations within the city and used spatiotemporal regression kriging to predict temperature at unmeasured locations. This technique uses multiple linear regression to model the spatiotemporal trend in temperature, and spatiotemporal kriging to model the regression residuals spatiotemporal autocorrelation. For the multilinear regression, we used several earth observation products including data relating to topography, the built environment and atmospheric processes, as explanatory covariates. The regression residuals were modelled by fitting a sum-metric spatiotemporal variogram model. We validated the model using a leave-one-out and 5-fold cross-validation. Results Of the 60 monitoring stations, 36 had data >75% valid data. Temperature records showed some spatial heterogeneity (interstation standard deviation: 1.4 ̊ C), supporting the need for spatially resolved temperature data. Of all covariates used in the multilinear regression model, remotely sensed land surface temperature was the best predictor. Any residual variability was modelled through spatiotemporal kriging. Validation using R2 and root-mean-square-error indicators is ongoing. Conclusions This dataset provides epidemiologists with a unique opportunity to investigate exposure to daily mean temperature in São Paulo at high spatio-temporal resolution, and to identify areas of high risk for certain health outcomes, e.g., mortality, over time and space.

On Robustness for Spatio-Temporal Data

The spatio-temporal variogram is an important factor in spatio-temporal prediction through kriging, especially in fields such as environmental sustainability or climate change, where spatio-temporal data analysis is based on this concept. However, the traditional spatio-temporal variogram estimator, which is commonly employed for these purposes, is extremely sensitive to outliers. We approach this problem in two ways in the paper. First, new robust spatio-temporal variogram estimators are introduced, which are defined as M-estimators of an original data transformation. Second, we compare the classical estimate against a robust one, identifying spatio-temporal outliers in this way. To accomplish this, we use a multivariate scale-contaminated normal model to produce reliable approximations for the sample distribution of these new estimators. In addition, we define and study a new class of M-estimators in this paper, including real-world applications, in order to determine whether there are any significant differences in the spatio-temporal variogram between two temporal lags and, if so, whether we can reduce the number of lags considered in the spatio-temporal analysis.

Three-dimensional regional oceanic element field reconstruction with multiple underwater gliders in the Northern South China Sea

Regional three-dimensional fine oceanic element field reconstruction and feature surface extraction are of considerable significance to oceanographic research and military application. In this paper, a three-dimensional regional oceanic element field reconstruction method based on spatio-temporal Kriging interpolation is proposed, and is studied experimentally. Using discrete data with uneven temporal and spatial distributions, the spatio-temporal variogram of Kriging interpolation is constructed by adopting an integral model. Then, based on experimental data acquired by seven underwater gliders and satellite remote sensing data in the northern South China Sea (NSCS), three-dimensional regional oceanic temperature and salinity fields are constructed for a particular area of interest. Through comparison with the spatial Kriging method and the spatio-temporal weight method, the cross-verification results demonstrate that the proposed method can improve the accuracy of oceanic element field construction. Finally, feature surfaces are extracted from the three-dimensional oceanic element fields, and can be used to reveal thermohaline characteristics of NSCS. Furthermore, the proposed method can be applied in other areas that construct three-dimensional models from discrete data.

Long-term offshore wind power prediction using spatiotemporal kriging: A case study in China’s Guangdong Province

The predicted wind power in coastal waters is an important factor when planning and developing offshore wind farms. The stochastic wind field challenges the accuracy of these predictions. Using single-point wind measurements, most previous studies have focused on the prediction of short-term wind power, ranging from minutes to several days. Longer-term wind power predictions would better support decision-making related to offshore wind power balance management and reserve capacities. In addition, larger-scale wind power predictions, based on gridded wind field data, would provide a more comprehensive understanding of the spatiotemporal variations of wind energy resources. In this study, a spatiotemporal ordinary kriging model was developed to predict the offshore wind power density on a monthly basis using the cross-calibrated multiplatform gridded wind field data. The spatiotemporal variations of wind power density were directly quantified through the development of spatiotemporal variograms that integrated spatial and temporal distances. The proposed model achieved a notable performance with an overall R2 of 0.94 and a relative prediction error of 16.35% in the validation experiment of predicting the monthly wind power density from 2013 in the coastal waters of China’s Guangdong Province. Using this model, the spatial distributions of wind power density along Guangdong’s coastal waters at monthly, seasonal, and annual time-scales from 2013 were accurately predicted. The experiment results demonstrated the remarkable potential of the spatiotemporal ordinary kriging model to provide reliable long-term prediction for offshore wind energy resources.

Spatio-temporal regression kriging for modelling urban NO2 concentrations

ABSTRACT Recently developed urban air quality sensor networks are used to monitor air pollutant concentrations at a fine spatial and temporal resolution. The measurements are however limited to point support. To obtain areal coverage in space and time, interpolation is required. A spatio-temporal regression kriging approach was applied to predict nitrogen dioxide (NO2) concentrations at unobserved space-time locations in the city of Eindhoven, the Netherlands. Prediction maps were created at 25 m spatial resolution and hourly temporal resolution. In regression kriging, the trend is separately modelled from autocorrelation in the residuals. The trend part of the model, consisting of a set of spatial and temporal covariates, was able to explain 49.2% of the spatio-temporal variability in NO2 concentrations in Eindhoven in November 2016. Spatio-temporal autocorrelation in the residuals was modelled by fitting a sum-metric spatio-temporal variogram model, adding smoothness to the prediction maps. The accuracy of the predictions was assessed using leave-one-out cross-validation, resulting in a Root Mean Square Error of 9.91 μg m−3, a Mean Error of −0.03 μg m−3 and a Mean Absolute Error of 7.29 μg m−3. The method allows for easy prediction and visualization of air pollutant concentrations and can be extended to a near real-time procedure.

Temperature prediction based on a space–time regression-kriging model

ABSTRACTMany phenomena exist in the space–time domain, often with a low data sampling rate and sparsely distributed network of observed points. Therefore, spatio-temporal interpolation with high accuracy is necessary. In this paper, a space–time regression-kriging model was introduced and applied to monthly average temperature data. First, a time series decomposition was applied for each station, and a multiple linear regression model was used to fit space–time trends. Second, a valid nonseparable spatio-temporal variogram function was utilized to describe similarities of the residuals in space–time. Finally, space–time kriging was applied to predict monthly air temperature. Jackknife techniques were used to predict the monthly temperature at all stations, with correlation coefficients between predictions and observed data very close to 1. Moreover, to evaluate the advantages of space–time kriging, pure time forecasting also was executed employing an autoregressive integrated moving average (ARIMA) model. The results of these two methods show that both mean absolute error (MAE) and root-mean-square error (RMSE) of space–time prediction are much lower than those of the pure time forecasting. The estimated temperature curves for stations also show that the former present a conspicuous improvement in interpolation accuracy when compared with the latter.

Spatiotemporal interpolation of precipitation across Xinjiang, China using space-time CoKriging

In various environmental studies, geoscience variables not only have the characteristics of time and space, but also are influenced by other variables. Multivariate spatiotemporal variables can improve the accuracy of spatiotemporal estimation. Taking the monthly mean ground observation data of the period 1960–2013 precipitation in the Xinjiang Uygur Autonomous Region, China, the spatiotemporal distribution from January to December in 2013 was respectively estimated by space-time Kriging and space-time CoKriging. Modeling spatiotemporal direct variograms and a cross variogram was a key step in space-time CoKriging. Taking the monthly mean air relative humidity of the same site at the same time as the covariates, the spatiotemporal direct variograms and the spatiotemporal cross variogram of the monthly mean precipitation for the period 1960–2013 were modeled. The experimental results show that the space-time CoKriging reduces the mean square error by 31.46% compared with the space-time ordinary Kriging. The correlation coefficient between the estimated values and the observed values of the space-time CoKriging is 5.07% higher than the one of the space-time ordinary Kriging. Therefore, a space-time CoKriging interpolation with air humidity as a covariate improves the interpolation accuracy.

Influence of available data on the geostatistical-based design of optimal spatiotemporal groundwater-level-monitoring networks

Several methodologies exist for the design of optimal groundwater monitoring networks. Those that employ geostatistical techniques are usually dependent on a variogram. In practice, different variograms for an environmental variable can be fitted depending on the number of available data, their spatial distribution, and their variability. For this reason, in most cases, it is difficult to get an adequate variogram. This report evaluates the influence of the groundwater level data available in the construction of a product-sum spatiotemporal variogram model and its consequences on a geostatistical-based methodology that uses the Kalman filter for the design of optimal spatiotemporal monitoring networks. Additionally, a sensitivity analysis is carried out to determine the variations in the optimal number of wells and monitoring frequency for the monitoring network, depending on the selected spatiotemporal variogram model. The results show the importance of getting a good quality spatiotemporal variogram when designing optimal groundwater-level monitoring networks using geostatistical-based methodologies.

A spatiotemporal regression-kriging model for space-time interpolation: a case study of chlorophyll-a prediction in the coastal areas of Zhejiang, China

ABSTRACTSpatiotemporal kriging (STK) is recognized as a fundamental space-time prediction method in geo-statistics. Spatiotemporal regression kriging (STRK), which combines space-time regression with STK of the regression residuals, is widely used in various fields, due to its ability to take into account both the external covariate information and spatiotemporal autocorrelation in the sample data. To handle the spatiotemporal non-stationary relationship in the trend component of STRK, this paper extends conventional STRK to incorporate it with an improved geographically and temporally weighted regression (I-GTWR) model. A new geo-statistical model, named geographically and temporally weighted regression spatiotemporal kriging (GTWR-STK), is proposed based on the decomposition of deterministic trend and stochastic residual components. To assess the efficacy of our method, a case study of chlorophyll-a (Chl-a) prediction in the coastal areas of Zhejiang, China, for the years 2002 to 2015 was carried out. The results show that the presented method generated reliable results that outperform the GTWR, geographically and temporally weighted regression kriging (GTWR-K) and spatiotemporal ordinary kriging (STOK) models. In addition, employing the optimal spatiotemporal distance obtained by I-GTWR calibration to fit the spatiotemporal variograms of residual mapping is confirmed to be feasible, and it considerably simplifies the residual estimation of STK interpolation.

StUPscales: An R-Package for Spatio-Temporal Uncertainty Propagation across Multiple Scales with Examples in Urban Water Modelling

Integrated environmental modelling requires coupling sub-models at different spatial and temporal scales, thus accounting for change of support procedures (aggregation and disaggregation). We introduce the R-package spatio-temporal Uncertainty Propagation across multiple scales, stUPscales, which constitutes a contribution to state-of-the-art open source tools that support uncertainty propagation analysis in temporal and spatio-temporal domains. We illustrate the tool with an uncertainty propagation example in environmental modelling, specifically in the urban water domain. The functionalities of the class setup and the methods and functions MC.setup, MC.sim, MC.analysis and Agg.t are explained, which are used for setting up, running and analysing Monte Carlo uncertainty propagation simulations, and for spatio-temporal aggregation. We also show how the package can be used to model and predict variables that vary in space and time by using a spatio-temporal variogram model and space-time ordinary kriging. stUPscales takes uncertainty characterisation and propagation a step further by including temporal and spatio-temporal auto- and cross-correlation, resulting in more realistic (spatio-)temporal series of environmental variables. Due to its modularity, the package allows the implementation of additional methods and functions for spatio-temporal disaggregation of model inputs and outputs, when linking models across multiple space-time scales.

Spatiotemporal interpolation of air pollutants in the Greater Cairo and the Delta, Egypt

This paper analyses the spatiotemporal variability of air pollutants in Egypt using monthly averages from the air quality monitoring network from 2011 to 2015. Particulate Matters (PM10) Nitrogen Dioxide (NO2) and Sulfur Dioxide (SO2), measured by the monitoring stations network are studied. A log transformation is applied for the three pollutants to achieve normality. The sum-metric function is utilized for modelling the spatiotemporal variogram as it gave the smallest Mean Squared Error (MSE) compared to other forms namely separable, metric, and product sum models. Therefore, employing the gstat package in R together with the trans-Gaussian spatiotemporal kriging, the maps are generated for the interpolated surfaces for the monthly averages of 2015 and the corresponding standard error values. These maps will help the decision maker to understand and visualize the spatial and temporal variability of the measured pollutants and hence undertake the necessary policies and decisions. The results show that the down town area has the highest pollutants levels. As concerns the temporal dimension, the highest values are depicted during the month of February as compared to the rest of the year. Furthermore, Egypt is suffering from a serious PM10 problem for the area and period under study that extremely exceed the WHO and Egyptian guidelines.

Comparison of spatiotemporal variogram functions based on a sparse dataset of groundwater level variations

We conduct a spatiotemporal geostatistical analysis of groundwater level data using space–time Residual Kriging (STRK). We compare the performance of three spatiotemporal variogram models in mapping groundwater level data. All models involve the same space–time trend function; the first model is a separable product variogram that involves Matérn space and time functions, while the second and third use non-separable functions. The non-separable models include a diffusive space–time variogram and a spatiotemporal extension of the Spartan variogram. The non-separable models are applied for the first time to groundwater level data. Our analysis shows that the Spartan and the non-separable diffusive variograms fit the experimental variogram of the residuals better than the separable Matérn model and improve the cross-validation results. The Spartan model provides the best cross validation measures among the three models. We also generate maps of groundwater level which illustrate the spatiotemporal variability in the basin.

Spatiotemporal geostatistical modelling of groundwater level variations at basin scale: a case study at Crete's Mires Basin

Abstract Spatiotemporal geostatistical analysis of groundwater levels is a significant tool for groundwater resources management. This work presents a valid spatiotemporal geostatistical model for the groundwater level variations of an aquifer in Crete, Greece. The goal of this approach is to accurately map the aquifer level at variable time-steps using joint space–time information. The proposed model applies the space–time ordinary kriging (STOK) methodology using joint space–time covariance functions. A space–time experimental variogram is determined from the monthly groundwater level time-series between the hydrological years 2009 and 2014 at 11 sampling stations. The experimental spatiotemporal variogram is successfully fitted by the product–sum model using a Matérn spatial and temporal function. STOK was used to predict the monthly groundwater level at each sampling station from January to May 2015. Validation results show low prediction errors that range from 0.95 to 1.45 m, while the kriging variance accurately determines the variability of predictions. Maps of groundwater level predictions and uncertainty are developed for significant months of the validation period to assess the aquifer spatiotemporal variability. This work demonstrates that space–time geostatistics can successfully model the spatial dynamic behaviour of an aquifer when the space–time dependencies are appropriately modelled, even for a sparse dataset.

Predicting saltwater intrusion into aquifers in vicinity of deserts using spatio-temporal kriging.

The primary objective of this study was to provide a detailed framework to use the spatio-temporal kriging to model the spatio-temporal variations of salinity data and predict saltwater intrusion into freshwater aquifers in the vicinity of deserts. EC data, measured in extraction wells in the Mahvelat plain located in the Northeastern part of Iran, available from 2007 to 2013, were used to demonstrate the developed framework. The source of data was not a well-designed measurement network. Therefore, to homogenize the data, spatial analysis was used to find EC distribution in the area in each year of study. To conduct the spatial analysis, a guideline and a systematic process were developed to select an appropriate kriging method and optimize its parameters. This process can be applied to different variables. After spatial analysis of EC data for all the years of the analysis period using empirical Bayesian kriging (EBK) method with manually optimized parameters, spatio-temporal and corresponding variogram analysis was conducted using R software. This process was based on a separable product-sum model applied to the data from 2007 to 2012. The data of 2013 and the data available for the years 1999 and 2006 were used for evaluating the performance of the spatio-temporal model. The EC distribution maps, developed for different years until 2021, show a high level of EC in the north, south, and west of the study area and growing saltwater intrusion into the central freshwater aquifer. This result can be attributed to the over-exploitation of the aquifer and hydraulic head and gradient distribution in the area. The framework provided in this study for spatio-temporal analysis of unstructured EC data is useful for groundwater managers in making proper decisions.

Spatiotemporal regression Kriging to predict precipitation using time-series MODIS data

Accurate prediction of the spatiotemporal distribution of precipitation is an important guide for more efficient agricultural production. However in the Xinjiang Uygur Autonomous Region, China, it is difficult to ensure accuracy due to sparse and unevenly distributed precipitation monitoring stations. The precipitation on raster grids must be predicted from point data. The combined China monthly mean meteorological data sets were used to build spatiotemporal geostatistical models to predict mean monthly precipitation in Xinjiang. Predictions in space and time were made for precipitation using spatiotemporal regression Kriging with some covariates. The Moderate Resolution Imaging Spectroradiometer 1-month images time series, topographic layers representing the normalized difference vegetation index and digital elevation model, and a temporal index to adjust for yearly periodic were used as covariates. The optimal covariates were selected by the all subset regression method to determine which predictor variables to be included in the multiple regression models. The accuracy of our mean monthly precipitation predictions was assessed by leave-one-out cross-validation. The prediction accuracy of the proposed spatiotemporal regression Kriging approach was compared with spatiotemporal multi-linear regression and spatiotemporal Kriging. These experimental results show that the normalized difference vegetation index, latitude, longitude, elevation, and time periodicity index are the optimal covariates for mean monthly precipitation prediction. The spatiotemporal distribution of precipitation in Xinjiang exhibits a distinctive pattern; in the north and west, there is more precipitation than in the south and east, respectively and more precipitation in the mountains than on the plains. Precipitation is closely related to topography. Annually, summer precipitation is the highest, followed by, spring and autumn, with winter the driest season in Xinjiang. Considering regression residuals the product---sum model was found to be suitable to fit the spatiotemporal variogram at higher accuracy. Precipitation maps generated by spatiotemporal regression Kriging were more accurate than those produced by the other tested methods, with lower MAE, RMSE, and BIAS values, and higher COR values for validation sampling sites. Spatiotemporal regression Kriging is an efficient method for accurate spatiotemporal prediction of precipitation in Xinjiang.

Improve Spatiotemporal Kriging with Magnitude and Direction Information in Variogram Construction

Magnitude information and direction information are utilized to improve the spatiotemporal Kriging method in the step of variogram construction. Basement of spatiotemporal variogram construction is to calculate vector distance between time slices, and space slices. Criterions of vector distance to spatiotemporal variogram construction is discussed, and a new vector distance model considering both magnitude and direction information is proposed. Case study using data from National Network of Geomagnetic Observatories of China is carried out for cross validation. Results illustrate that the proposed model performs better in analyzing in the statistics of Mean absolute error (MAE) and Mean square error (MSE). The accuracy of spatiotemporal Kriging interpolation is improved with such an operation in vector distance.

Spatio-temporal Modelling of Forest Growth Spanning 50 Years – The Effects of Different Thinning Strategies

Models for tree growth in the literature include spatial models and dynamic models. Spatial models have been mostly restricted to Gaussian variograms and comparisons at single time points while dynamic models ignore tree competition due to spatial proximity. Here we model tree growth in forest plots with a spatio-temporal covariance structure and compare plots with different thinnings. Tree growth is measured by diameter at breast height. A model was fitted using a sum-metric spatio-temporal variogram. Fixed effects of age, number of neighbours and polygon area were also considered. Predicted values for models were computed using regression-kriging and mean squared error of prediction was used to compare models and thinning strategies. Over time the correlation of a single tree followed a wave variogram while the spatio-temporal anisotropy parameter captured the changing spatial wave intensity. Thinned plots clearly outperformed unthinned plots in terms of total tree growth.