Time Variogram Research Articles

Predicting Morphodynamics for Beach Intertidal Systems in the North Sea: A Space-Time Stochastic Approach

The ability to accurately predict beach morphodynamics is of primary interest for coastal scientists and managers. With this goal in mind, a stochastic model of a sandy macrotidal barred beach is developed that is based on cross-shore elevation profiles. Intertidal elevation was monitored from monthly to annually for 19 years through Real Time Kinematics-GPS (RTK-GPS) and LiDAR surveys, and monthly during two years with an RTK-GPS. In addition, during two campaigns of about two weeks, intensive surveys on a daily basis were performed with an RTK-GPS on a different set of profiles. Based on the measurements, space and time variograms are constructed in order to assess the spatial and temporal dependencies of these elevations. A separable space-time covariance model is then built from them in order to generate a large number of plausible future profiles at arbitrary time instants t+τ, starting from observed profiles at time instants t. For each simulation, the total displaced sand volume is computed and a distribution is obtained. The mean of this distribution is in good agreement with the total displaced sand volume measured on the profiles, provided that they are lower than 45 m3/m. The time variogram also shows that 90% of maximum variability is reached for a time interval τ of three years. These results demonstrate how the temporal evolution of an integrated property, like the total displaced sand volume, can be estimated over time. This suggests that a similar stochastic approach could be useful for estimating other properties as long as one is able to capture the stochastic space-time variability of the underlying processes.

Space-time variogram for daily rainfall estimates using rain gauges and satellite data in mountainous tropical Island of Bali, Indonesia (Preliminary Study)

High spatio-temporal variability of daily rainfall in Bali Island can create the absence of structure in the daily variogram in certain days. This research proposes a new technique applying space–time variogram for 3-successive daily rainfall to detect structure in variogram estimation by merging rain gauges and satellite data applying ordinary kriging (OK), regression kriging with CMORPH (CM), regression kriging with TRMM (TR), and blended monthly rainfall (MONT) to obtain daily blended gridded rainfall estimates. Original retrieval of CMORPH (CM_OR) and TRMM (TR_OR) also used as control points to assess the proposed method. Uncertainty and sensitivity analysis including cross validation were carried out to validate the proposed method.The result shows that a new technique, adapted from CMORPH specific character, can be applied to detect the existence of structure in variogram. Blended gridded daily rainfall estimates of CM has highest probability detection of rainy events, while OK the lowest. Generally, the four interpolation methods (OK, CM, MONT, TR) have low accuracy at leeward and in a coastal area. All of them have almost similar performance indicated by no clear distinction of RMSE value from cross validation. CM, MONT, and TR have a good sensitivity with maximum and minimum temperature, indicating that satellite data can improve the gridded rainfall estimates. However, it is required an improvement of daily gridded rainfall estimates since there are large RMSE values and low coefficient correlation in certain days because of strong and erratic behavior of rainy events in a mountainous tropical island of Bali.

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.

On the Frequency Variogram and on Frequency Domain Methods for the Analysis of Spatio‐Temporal Data

In this article, we assume the spatio‐temporal process to be intrinsically stationary in time and stationary in space. Our objective here is to present an alternative way, based on frequency domain methods, for modelling the data. We consider the discrete Fourier transforms (DFTs) defined for the (intrinsic) time‐series data observed at several locations as our data. We use the well‐known property that DFTs are asymptotically uncorrelated and distributed as complex Gaussian in deriving many results. Our objective here is to emphasize the usefulness of the DFTs in the analysis of spatio‐temporal data. Under the assumption of intrinsic stationarity, we consider the estimation of frequency variogram (FV) and discuss its asymptotic sampling properties. We show that FV introduced earlier is a frequency decomposition of space–time variogram. The DFTs can be computed very fast using fast Fourier transform algorithms. Assuming that the DFTs of the incremental process satisfy a Laplacian model, an analytic expression for the space–time spectral density and an expression for the FV in terms of the spectral density function for the intrinsic stationary process are derived. The estimation of the parameters of the spectral density is also considered. A statistical test for spatial independence of spatio‐temporal data is proposed.

A Geostatistical Framework for Quantifying the Temporal Evolution and Predictability of Rainfall Fields

Abstract A geostatistical framework for quantifying the temporal evolution and predictability of rainfall fields for time lags between 5 min and 3 h is proposed. The method is based on the computation of experimental space–time variogram maps of radar reflectivity fields. Two new metrics for quantifying temporal innovation and predictability based on minimum semivariance values at different time lags are proposed. The method is applied to high-resolution composite radar reflectivity maps over the United States to study the evolution of 25 convective and 25 stratiform events during the warm season of 2014. Results show that the temporal innovation can be modeled as the sum of two exponential functions of time lag, with approximately 50% of the total innovation occurring over the first 60 min. The median predictable scales for convective events are on the order of 1.6 km at 5 min, 5 km at 15 min, and 12.7 km at 1 h. Furthermore, the optimal time lag for predicting future innovation, taking into account measurement uncertainty and natural variability, appears to be between 30 and 60 min.

A Method to Estimate the 3D–Time Structure of the Raindrop Size Distribution Using Radar and Disdrometer Data*

Abstract A geostatistical method to quantify the small-scale 3D–time structure of the drop size distribution (DSD) from the ground level up to the melting layer using radar and disdrometer data is presented. First, 3D–time radar reflectivity fields are used to estimate the large-scale properties of a rain event, such as the apparent motion, spatial anisotropy, and temporal innovation. The retrieved quantities are then combined with independent disdrometer time series to estimate the 3D–time variogram of each DSD parameter. A key point in the procedure is the use of a new metric for measuring distances in moving anisotropic rainfall fields. This metric has the property of being invariant with respect to the specific rainfall parameter being considered, that is, it is identical for the radar reflectivity, rain rate, mean drop diameter, drop concentration, or any other weighted moment of the DSD. Evidence is shown of this fact and some illustrations for a stratiform event in southern France and a convective case in the midwestern United States are provided. The proposed framework offers a series of new and interesting applications, including the possibility to compare the space–time structure of different rain events, to interpolate radar reflectivity fields in space–time and to simulate 3D–time DSD fields at high spatial and temporal resolutions.

Evaluation of Recharge and Groundwater Dynamics of a Shallow Alluvial Aquifer in Central Ganga Basin, Kanpur (India)

Evaluation of recharge and groundwater dynamics of an aquifer is an important step for finding a proper groundwater management scenario. This has been performed on the basis of statistical Kendall Tau test to find a relationship between groundwater levels and hydro-meteorological parameters (e.g., precipitation, temperature, evaporation). Recharge to the aquifer was estimated for identification of critical areas/locations based on the analytical Soil and Water Assessment Tool. Moreover, spatiotemporal variability of groundwater levels has been quantified using space–time variogram. The overall characterization method has been applied to the shallow alluvial aquifer of Kanpur city in India. The analysis was performed using groundwater level data from 56 monitoring piezometer locations in Kanpur from March 2006 to June 2011. Groundwater level shows relatively higher correlation with temperature. Performance of the geostatistical model was evaluated by comparing with the observed values of groundwater level from January 2011 to June 2011 for two scenarios: “with limited spatiotemporal data” and “without spatiotemporal data.” It is evident that spatiotemporal prediction of groundwater level can be performed even for the unmonitored/missing data. This analysis demonstrates the potential applicability of the method for a general aquifer system.

Effects of spatial pattern persistence on the performance of sampling designs for regional trend monitoring analyzed by simulation of space–time fields

The effect of the persistence of spatial patterns on the performance of space–time sampling designs is explored by a simulation study. The performance is evaluated on the basis of the covariance matrix of the two parameters (intercept and slope) of a linear model for the change over time of the spatial means or totals. The evaluated sampling approach is hybrid, i.e. design-based estimation of spatial means from spatial probability samples is combined with time-series modelling of the spatial means. A simulation algorithm is presented for approximating the covariance matrix of the time-series model parameters from a full space–time model. Designs were evaluated on the basis of the determinant of this matrix and the variance of the estimated trend parameter. As a space–time model a sum-metric space–time variogram is used, the parameters of which are chosen such that the persistence of spatial patterns varies from nearly absent to very strong. Based on the extensive simulations, recommendations on the type of space–time design can most easily be made for situations with either very strong or no persistence of spatial patterns. With strong persistence the supplemented panel (SuP) design is recommendable. With no persistence the independent-synchronous (IS) and serially alternating (SA) designs are the best choice. These designs performed well with regard to both quality criteria. With moderate persistence of spatial patterns the choice of design type is more complicated. The IS and static-synchronous (SS) design performed best on one quality criterion, but worst on the other. Therefore, with moderate pattern persistence, the compromise designs, either SuP or SA, can be a good choice, unless one of the two quality criteria has priority. An R script is provided for ex ante evaluation of space–time designs in real-world applications.

Variography of rainfall accumulation in presence of advection

Summary This study provides a coherent theoretical framework between the field structures of rainfall accumulations over different durations. Following a classical geostatistical approach, we assume the existence of an elemental space–time stationary rainfall process and we consider rainfall accumulation as a regularization of this elemental process over time supports of various sizes. The corresponding space–time elemental variogram is used to derive the spatial and temporal variograms of accumulated values. In addition to an elemental variance, the framework incorporates both the anisotropy between time and space variability (Taylor velocity) and a constant advection of the rain fields. The proposed formulation verifies empirical observations made in past variographic studies like an increasing spatial range and a decreasing sill with increasing accumulation times. It provides new insights about the respective influence of accumulation and advection on the variogram shape and, as a side product, about Eulerian and Lagrangian properties of rainfall variability. From a set of experimental variograms calculated for different accumulation durations, we use the proposed formulation through a simple optimization algorithm to infer a four-parameter model characterizing the elemental rainfall process and the advection. A network of 21 daily rain gauges over the Loire Basin, France, gives an application illustration. We use a set of experimental space and time variograms corresponding to accumulation durations of 1–4 days. The identified parameters provide a fairly good fit of the experimental variogram set and their orders of magnitude look meteorologically sound. The main limitations and further developments of the method are discussed.

Stochastic Simulation of Intermittent DSD Fields in Time

Abstract A method for the stochastic simulation of (rain)drop size distributions (DSDs) in space and time using geostatistics is presented. At each pixel, the raindrop size distribution is described by a Gamma distribution with two or three stochastic parameters. The presence or absence of rainfall is modeled using an indicator field. Separable space–time variograms are used to estimate and reproduce the spatial and temporal structures of all these parameters. A simple and user-oriented procedure for the parameterization of the simulator is proposed. The only data required are DSD time series and radar rain-rate (or reflectivity) measurements. The proposed simulation method is illustrated for both frontal and convective precipitation using real data collected in the vicinity of Lausanne, Switzerland. The spatial and temporal structures of the simulated fields are evaluated and validated using DSD measurements from eight independent disdrometers.

A new space–time multivariate approach for environmental data analysis

Air quality control usually requires a monitoring system of multiple indicators measured at various points in space and time. Hence, the use of space–time multivariate techniques are of fundamental importance in this context, where decisions and actions regarding environmental protection should be supported by studies based on either inter-variables relations and spatial–temporal correlations. This paper describes how canonical correlation analysis can be combined with space–time geostatistical methods for analysing two spatial–temporal correlated aspects, such as air pollution concentrations and meteorological conditions. Hourly averages of three pollutants (nitric oxide, nitrogen dioxide and ozone) and three atmospheric indicators (temperature, humidity and wind speed) taken for two critical months (February and August) at several monitoring stations are considered and space–time variograms for the variables are estimated. Simultaneous relationships between such sample space–time variograms are determined through canonical correlation analysis. The most correlated canonical variates are used for describing synthetically the underlying space–time behaviour of the components of the two sets.

Space–Time Geostatistics for Geography: A Case Study of Radiation Monitoring Across Parts of Germany. 地理学的时空地统计学：横跨德国部分区域的辐射监测的案例研究

Many branches within geography deal with variables that vary not only in space but also in time. Therefore, conventional geostatistics needs to be extended with methods that estimate and quantify spatiotemporal variation and use it in spatiotemporal interpolation and stochastic simulation. This article briefly summarizes the main concepts of space–time geostatistics. Kriging in space and time can be done in much the same way as it is in a purely spatial setting. The main difficulties are in defining a realistic stochastic model that is assumed to have generated data and in characterizing and estimating the space–time correlation of that model. This article uses a model‐based geostatistical approach to characterize space–time variability. The space–time variable of interest is treated as a sum of independent stationary spatial, temporal, and spatiotemporal components, which leads to a sum‐metric space–time variogram model. Methods are illustrated with a case study of space–time interpolation of monthly averages of detected background radiation for a 5‐year period in four German states.

FORTRAN programs for space–time multivariate modeling and prediction

Environmental data is nearly always multivariate and often spatial–temporal. Thus to interpolate the data in space or to predict in space–time it is necessary to use a multivariate spatial–temporal method. Cokriging is easily extended to spatial–temporal data if there are valid space–time variograms or covariance functions. Various authors have proposed such models. In this paper, a generalized product–sum model is used with a linear coregionalization model for cokriging. The GSLib “COKB3D” program was modified to incorporate the space–time linear coregionalization model ( ST-LCM), using the generalized product–sum variogram model. Hence, a new GSLib software, named “COK2ST”, is proposed. To demonstrate the use of the software, hourly measurements of carbon monoxide and nitrogen dioxide from the Puglia region in Italy are used.

06/01805 Global geothermal production to increase 50% by 201: Anon., Refocus, 2006, 7, (1), 6

This study provides a coherent theoretical framework between the field structures of rainfall accumulations over different durations. Following a classical geostatistical approach, we assume the existence of an elemental space–time stationary rainfall process and we consider rainfall accumulation as a regularization of this elemental process over time supports of various sizes. The corresponding space–time elemental variogram is used to derive the spatial and temporal variograms of accumulated values. In addition to an elemental variance, the framework incorporates both the anisotropy between time and space variability (Taylor velocity) and a constant advection of the rain fields.The proposed formulation verifies empirical observations made in past variographic studies like an increasing spatial range and a decreasing sill with increasing accumulation times. It provides new insights about the respective influence of accumulation and advection on the variogram shape and, as a side product, about Eulerian and Lagrangian properties of rainfall variability. From a set of experimental variograms calculated for different accumulation durations, we use the proposed formulation through a simple optimization algorithm to infer a four-parameter model characterizing the elemental rainfall process and the advection.A network of 21 daily rain gauges over the Loire Basin, France, gives an application illustration. We use a set of experimental space and time variograms corresponding to accumulation durations of 1–4 days. The identified parameters provide a fairly good fit of the experimental variogram set and their orders of magnitude look meteorologically sound. The main limitations and further developments of the method are discussed.

Space–time correlation models and contaminant plumes

AbstractA contaminant plume might be described by a function defined in space–time. Spatial integrals or time derivatives of this function as well as time derivatives of spatial integrals will quantify characteristics such as the total volume of the plume, the total concentration of the contaminant in the plume, rates of change of the volume, and rates of change of concentration.The plume function usually cannot be derived in analytic form but instead must be estimated or approximated. The dual form of the kriging estimator, which is equivalent to the use of radial basis functions, provides a tool for modeling this function in analytic form. The extension of the kriging estimator, in its usual form or in its dual form, to space–time poses no problems since the estimator and the equations are essentially dimension free. The difficulty is an adequate choice of space–time variograms or covariances.The product–sum and integrated product–sum models provide an extensive array of valid models and also lead to a simple process for fitting the models by the use of marginal variograms. Examples are given and an application to air pollution data from the Milan District (Italy) illustrates the method. Copyright © 2002 John Wiley & Sons, Ltd.

Space–time variograms and a functional form for total air pollution measurements

A space–time functional form for some contaminants is obtained and used for estimating total air pollution (TAP) in the district of Milan, Italy, during selected high-risk days of 1999. This functional form is determined through a space–time product–sum variogram model for TAP measurements and the dual form of kriging, i.e., radial basis functions. Data for nitric oxide (NO), nitrogen dioxide (NO 2) and carbon monoxide (CO) collected in Milan district, Italy are used to generate a combined indicator of traffic pollution, called TAP. In a previous study the weightings were obtained by multiple principal component analyses of the daily concentration levels. It was found that the first component explains approximately 70% of the total variance for each day and this component is treated as samples defined over space and time. A systematic pattern, which follows the corridor along which survey stations, characterized by heavy traffic are located, has been observed for TAP throughout Milan district, for all days considered. Note that the pollution data set is just an illustration for the new statistical method proposed.