Spatiotemporal Variogram Research Articles

Evaluation of Spatio-Temporal Variogram Models for Mapping Xco2Using Satellite Observations: A Case Study in China

Greenhouse Gases Observing Satellite (GOSAT), which measures column-averaged carbon dioxide dry air mole fractions (Xco <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) from space, provides new data sources to improve our understanding of carbon cycle. The available GOSAT data, however, have many gaps and are irregularly positioned, which make it difficult to directly interpret their scientific significance without further data analysis. Spatio-temporal geostatistical prediction approach can be used to fill the gaps for global and regional Xco <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> mapping. It is important to choose a suitable spatio-temporal variogram model since modeling spatio-temporal correlation structure using variogram model is a critical step in the geostatistical prediction. In this study, three different flexible spatio-temporal variogram models, including the product-sum model, Cressie-Huang model, and Gneiting model, are used to model the spatio-temporal correlation structure of Xco <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> over China, using the Atmospheric CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Observations from Space retrievals of the GOSAT (ACOS-GOSAT) Xco <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (v3.3) data products. The three models are compared and evaluated using the weighted mean square errors (WMSE) indicating the fitness between the empirical variogram surface and the theoretical variogram model, cross-validation for quantifying prediction accuracies, and the performance of the three models when used to fill the spatial gaps and generate Xco <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> maps in 3-day temporal interval. The results indicate that 1) the model fitness of the commonly used product-sum model is slightly better than Cressie-Huang model and Gneiting model as indicated from WMSE, and 2) all the three models present similar summary statistics in cross-validation, all with a significantly high correlation coefficient of 0.92, and about 83% of prediction error within 2 ppm and about 53% within 1 ppm, and (3) differences between the mapping results using the three models are generally less than 0.1 ppm, and no significant differences can be identified. As a conclusion from the above results, all the three variogram models can precisely catch the empirical characteristics of the spatio-temporal correlation structure of Xco <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> over China, and the precision and effectiveness of predicting and mapping Xco <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> using the three models are almost the same.

Modeling probabilistic radar rainfall estimation at ungauged locations based on spatiotemporal errors which correspond to gauged data

This study presents a probabilistic radar rainfall estimation (PRRE) model to quantify the reliability and accuracy of the resulting radar rainfall estimates at ungauged locations from a radar-based quantitative precipitation estimation (QPE) model. This model primarily estimates the quantiles of the radar rainfall errors at ungauged locations by incorporating seven spatiotemporal variogram models with a nonparametric sample quantile estimate method based on the radar rainfall errors at rain gauges. Then, by adding the resulting error quantiles to the radar rainfall estimates, the corresponding radar rainfall quantiles can be obtained. The QPE system Quantitative Precipitation Estimation Using Multiple Sensors (QPESUMS) provides hourly observed and radar precipitation for three typhoons in the Shinmen reservoir watershed in Northern Taiwan, which are used in the model development and validation. The results indicate that the proposed PRRE model can quantify the spatial and temporal variations of radar rainfall estimates at ungauged locations provided by the QPESUMS system. Also, its reliability and accuracy could be evaluated based on a 95% confidence interval and occurrence probability resulting from the cumulative probability distribution established by the proposed PRRE model.

Incorporating temporal variability to improve geostatistical analysis of satellite-observed CO2 in China

Observations of atmospheric carbon dioxide (CO2) from satellites offer new data sources to understand global carbon cycling. The correlation structure of satellite-observed CO2 can be analyzed and modeled by geostatistical methods, and CO2 values at unsampled locations can be predicted with a correlation model. Conventional geostatistical analysis only investigates the spatial correlation of CO2, and does not consider temporal variation in the satellite-observed CO2 data. In this paper, a spatiotemporal geostatistical method that incorporates temporal variability is implemented and assessed for analyzing the spatiotemporal correlation structure and prediction of monthly CO2 in China. The spatiotemporal correlation is estimated and modeled by a product-sum variogram model with a global nugget component. The variogram result indicates a significant degree of temporal correlation within satellite-observed CO2 data sets in China. Prediction of monthly CO2 using the spatiotemporal variogram model and space-time kriging procedure is implemented. The prediction is compared with a spatial-only geostatistical prediction approach using a cross-validation technique. The spatiotemporal approach gives better results, with higher correlation coefficient (r 2), and less mean absolute prediction error and root mean square error. Moreover, the monthly mapping result generated from the spatiotemporal approach has less prediction uncertainty and more detailed spatial variation of CO2 than those from the spatial-only approach.

Spatiotemporal Continuity of Sequential Rain Suggested by 3-D Variogram

A series of rainfalls observed in central Japan from noon on the 13th to midnight on the 14th, August 1999 (36 h), has been analyzed by spatiotemporal variograms in order to reveal the continuity of rain precipitation in a 3-D space defined by geographic coordinates and time. All instances of zero precipitation are considered, but have been treated as four different cases: case 0 excludes all zero data, case 1 includes a zero datum neighboring to each finite value, case 2 includes a zero neighboring to each finite value and the next neighboring zero, and a fourth case (termed case A) includes all zeros. Hourly precipitation has a statistical distribution best approximated by a Weibull model, and somewhat less well by a normal distribution, in all four cases. A rectangular variogram of measured values of total precipitation shows that the best continuity appears approximately along the N-S direction (the ranges given by directional variograms are 500 and 80 km in the N-S and W-E directions, respectively). In contrast, temporally stacked rectangular variograms of hourly precipitation shows that the best continuity direction is W-E in all cases (the ranges in case A are 50 and 100 km along the N-S and W-E directions, respectively). A spatial variogram gives a spatial range independently of time, whereas a temporal variogram gives a temporal range. When geographic coordinates are normalized by the spatial range (here 80 km given by the temporally stacked omnidirectional variogram in case A), and time is normalized by the temporal range (here 7 h given by the spatially stacked temporal variogram), geographic coordinates and time can be treated as equivalent variables. Consequently, a spatiotemporal variogram can be calculated along a given direction in 3-D space using the normalized coordinates. The continuity direction of a series of rainfalls can be best understood by display on a Wulff net, where each range value is written at a point corresponding to the direction. The direction of the best continuity is N0°W+20° in the normalized space. A rectangular variogram in the normalized space, in which the horizontal and vertical axes represent N-S direction and time, respectively, suggests that the series of heavy rainfalls examined here had a continuity pattern that was elongated from west to east (the range values are 20–30 km and 100 km along N-S and W-E, respectively), and that migrated from south to north with a speed of 30 km/h.

Recent advances to model anisotropic space–time data

Building new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to present an up-to-date overview of recent spatio-temporal covariance models taking into account the problem of spatial anisotropy. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability. In particular, we focus on the problem of modelling anisotropy through isotropy within components. We present the Bernstein class, and a generalisation of Gneiting’s approach (2002a) to obtain new classes of space–time covariance functions which are spatially anisotropic. We also discuss some methods for building covariance functions that attain negative values. We finally present several differentiation and integration operators acting on particular space–time covariance classes.

Catchments as space-time filters – a joint spatio-temporal geostatistical analysis of runoff and precipitation

Abstract. In this paper catchments are conceptualised as linear space-time filters. Catchment area A is interpreted as the spatial support and the catchment response time T is interpreted as the temporal support of the runoff measurements. These two supports are related by T~Aκ which embodies the space-time connections of the rainfall-runoff process from a geostatistical perspective. To test the framework, spatio-temporal variograms are estimated from about 30 years of quarter hourly precipitation and runoff data from about 500 catchments in Austria. In a first step, spatio-temporal variogram models are fitted to the sample variograms for three catchment size classes independently. In a second step, variograms are fitted to all three catchment size classes jointly by estimating the parameters of a point/instantaneous spatio-temporal variogram model and aggregating (regularising) it to the spatial and temporal scales of the catchments. The exponential, Cressie-Huang and product-sum variogram models give good fits to the sample variograms of runoff with dimensionless errors ranging from 0.02 to 0.03, and the model parameters are plausible. This indicates that the first order effects of the spatio-temporal variability of runoff are indeed captured by conceptualising catchments as linear space-time filters. The scaling exponent κ is found to vary between 0.3 and 0.4 for different variogram models.

Nonseparable stationary anisotropic space–time covariance functions

Obtaining new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to introduce and develop new spatio-temporal covariance models taking into account the problem of spatial anisotropy. Recent literature has focused on the problem of full symmetry and the problem of anisotropy has been overcome. Here we propose a generalization of Gneiting’s (J Am Stat Assoc 97:590–600, 2002a) approach and obtain new classes of stationary nonseparable spatio-temporal covariance functions which are spatially anisotropic. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability.

Spatio-temporal variograms and covariance models

Variograms and covariance functions are the fundamental tools for modeling dependent data observed over time, space, or space-time. This paper aims at constructing nonseparable spatio-temporal variograms and covariance models. Special attention is paid to an intrinsically stationary spatio-temporal random field whose covariance function is of Schoenberg-Lévy type. The correlation structure is studied for its increment process and for its partial derivative with respect to the time lag, as well as for the superposition over time of a stationary spatio-temporal random field. As another approach, we investigate the permissibility of the linear combination of certain separable spatio-temporal covariance functions to be a valid covariance, and obtain a subclass of stationary spatio-temporal models isotropic in space.

Spatio-temporal variograms and covariance models

Variograms and covariance functions are the fundamental tools for modeling dependent data observed over time, space, or space-time. This paper aims at constructing nonseparable spatio-temporal variograms and covariance models. Special attention is paid to an intrinsically stationary spatio-temporal random field whose covariance function is of Schoenberg-Lévy type. The correlation structure is studied for its increment process and for its partial derivative with respect to the time lag, as well as for the superposition over time of a stationary spatio-temporal random field. As another approach, we investigate the permissibility of the linear combination of certain separable spatio-temporal covariance functions to be a valid covariance, and obtain a subclass of stationary spatio-temporal models isotropic in space.

Flexible spatio-temporal stationary variogram models

In this paper we propose a generalization of the Shapiro and Botha (1991) approach that allows one to obtain flexible spatio-temporal stationary variogram models. It is shown that if the weighted least squares criterion is chosen, the fitting of such models to pilot estimations of the variogram can be easily carried out by solving a quadratic programming problem. The work also includes an application to real data and a simulation study in order to illustrate the performance of the proposed space-time dependency modeling.

On spatiotemporal patchiness and the coexistence of five species of Chronogaster (Nematoda: Chronogasteridae) in a riparian wetland.

Spatiotemporal patchiness in the soil environment is thought to be crucial for the maintenance of soil biodiversity. It provides diverse microhabitats (allowing resource partitioning), and presents these in a complex mosaic, such that competitors may be spatially and temporally separated (promoting patch dynamics). The objective of our study was to assess the importance of patch dynamics for the maintenance of landscape-level nematode diversity. The spatiotemporal aggregation patterns in populations of five species of the bacterivorous genus Chronogaster were analyzed for a 0.7-ha riparian wetland, during four seasons, using geostatistical methods (spatiotemporal variograms, cross-correlograms, and log-normal kriging). We found that for the three most abundant species, aggregation was significant over distances of 13-39m, with intraspecific aggregation stronger than interspecific aggregation. In addition, species patterns appeared to be temporally dynamic: local population declines alternated with local blooms at different sites and seasons. One species was consistently present at the same few sites, but the other species showed gradual temporal decreases and increases at different locations in the wetland. Combining nematode patterns with information on soil resource distributions and Chronogaster ecology, indicated that both soil resource patchiness and repeated, but largely unpredictable, disturbances caused by local soil moisture fluctuations could promote landscape-level Chronogaster diversity by providing ample 'probability refuges.' Local species populations, reduced by competition, predation and drought, could recolonize patches either through the awakening of local dormant 'seed populations,' or through limited passive dispersal by wind, water, and phoresis. We suggest that the spatiotemporal Chronogaster data provide preliminary evidence for the importance of patch dynamics in maintaining soil nematode diversity.

Incorporating scale into spatio-temporal variability: applications to soil quality and yield data

Some observations are more variable in space and relatively less variable in time or vice versa. Based on the major source of variability, regionalized variable theory is used to construct either spatial or temporal semivariograms. Examples include georeferenced soil data, with primarily spatial variability, and daily weather data, with primarily temporal variability, as input to dynamic simulation models. However, for the proper interpretation of the output variables resulting from these simulations, it is important to describe both their spatial and temporal variabilities. A variance model is presented in which time is treated as a third dimension added to a two dimensional space. This spatio-temporal variance model is used in Case I to construct spatio-temporal variograms of hydrologic model output. The spatio-temporal variograms were used to examine the differences between two calculation schemes involving different hydrologic models. Input data to simulation models are based on a certain support size. For each support size a semivariogram may be constructed to represent the variability. The relationship between the support size of data and the distribution of their values can be described by a power function. Incorporating the relationship between variance and support size yielded a universal variance model relating the variance with lag distance and support size. This variance-support model was applied in Case II to agronomic yield data obtained from a study focusing on the improvement of land use in central Côte d'Ivoire.