Anisotropic Variogram Research Articles

Bias correction of daily precipitation for ungauged locations using geostatistical approaches: A case study for the CORDEX‐Africa ensemble

AbstractClimate model simulations typically exhibit a bias, which can be corrected using statistical approaches. In this study, a geostatistical approach for bias correction of daily precipitation at ungauged locations is presented. The method utilizes a double quantile mapping with dry day correction for future periods. The transfer function of the bias correction for the ungauged locations is established using distribution functions estimated by ordinary kriging with anisotropic variograms. The methodology was applied to the daily precipitation simulations of the entire CORDEX‐Africa ensemble for a study region located in the West African Sudanian Savanna. This ensemble consists of 23 regional climate models (RCM) that were run for three different future scenarios (RCP 2.6, RCP 4.5, and RCP 8.5). The evaluation of the approach for a historical 50‐year period (1950–2005) showed that the method can reduce the inherent strong precipitation bias of RCM simulations, thereby reproducing the main climatological features of the observed data. Moreover, the bias correction technique preserves the climate change signal of the uncorrected RCM simulations. However, the ensemble spread is increased due to an overestimation of the rainfall probability of uncorrected RCM simulations. The application of the bias correction method to the future period (2006–2100) revealed that annual precipitation increases for most models in the near (2020–2049) and far future (2070–2099) with a mean increase of up to (18%). An analysis of the monthly and daily time series showed a slightly delayed onset and intensification of the rainy season.

Efficiency of Interpolation Methods Based on GIS for Estimating of Spatial Distribution of PH in Soil

The main objective of this study is to review and evaluate three common interpolation methods namely: Inverse Distance Weighting (IDW), Radial Basis Function (RBF) and Ordinary Kriging (OK), and generate maps of soil pH using these methods. The accuracy and efficiency of the generated maps have been examined as well as the most fitting technique for estimating spatial distribution of soil pH in the study area is identified. Studies were conducted within the limits of land use of RUP “Uchkhoz BGSHA” (Republic of Belarus, Mogilev region, Goretsky district). The total area of the surveyed territory is 3197.89 hectares. For the analysis data is used about pHKCl of soil solution obtained from materials of an agrochemical survey executed in 2014. Forecasting and visualization of the spatial distribution of pHKCl was carried out using the Geostatistical Analyst module of the ArcGIS software. The experimental anisotropic variograms were calculated to determine the possible spatial structure of soil pH. Based on cross-validation results, a polynomial function was identified as the best variogram model. The model created by the method of radial basis functions turned out to be the most suitable for forecasting purposes (the value of the root-mean-square error was 0.763). In terms of interpolation accuracy, the investigated deterministic and geostatistical methods are located in the next descending row

Study of Geologically Consistent History Matching Peculiarities by Means of Gradient-Free Optimization Methods

This paper presents some results and distinctive features of the algorithm for automated history matching of 3D reservoir flow models using gradient-free methods. Widely used Nelder-Mead method was chosen for optimization. In order to preserve geological consistency of a model within the history matching process, control parameters are comprised of “porosity-to-permeability” relation parameters, anisotropic (semi)variogram parameters, and reservoir properties at pilot points. The control parameters are used at each iteration of the history matching process to generate porosity and permeability distributions based on static well data and perform flow simulations. Test studies were conducted for a specially developed synthetic model representing a inhomogeneous five-spot well pattern element. Oil flow rates at production wells and water flow rate at the injection well, as well as bottomhole pressures of all wells were chosen as measured dynamic data in the objective function. The influence of initial guess of parameter values and relative weights in the objective function on the quality of reconstruction of the ‘true’ solution and convergence rate were evaluated. Problems of gradient-free optimization methods application and specific features of residuals normalization in the objective function, as well as influence of pilot points are discussed.

A simple index to determine if within-field spatial production variation exhibits potential management effects: application in vineyards using yield monitor data

As agricultural data and information becomes more abundant, diagnostics are needed to quickly and efficiently interrogate these data. Indices exist to identify sensor data with structured spatial variation, conducive to site-specific management. However, these indices do not indicate if this spatial variation is driven by managerial or environmental effects. A new index is proposed to identify perennial (or ordered row) fields that are likely or highly likely to have management effects within the spatial pattern of sensor data. This is determined by investigating differences in anisotropic (directional) variograms parallel and perpendicular to the direction of management (row orientation). Small differences are indicative of isotropic (environmental-driven) variation. Large differences indicate row and management effects. The index is derived, run on a database of 1080 simulated fields and applied to yield data from 124 vineyard blocks to assess index performance and response to different levels of variation. Simulations showed that the index is non-responsive to the magnitude of variation but responds strongly to anisotropy in the data. The stochastic variance in the data was observed to have an effect on index response and may be problematic when applied to noisy data sets. The index scores for the simulated and real-world data showed a similar pattern of response and the index was able to identify vineyard blocks where differential row management had generated differing yield responses. The index scores are continuous and some general guidelines for use of the index are proposed.

Anisotropy of atmospheric delay in InSAR and its effect on InSAR atmospheric correction

Reconstruction of interferometric synthetic aperture radar (InSAR) atmospheric delay maps is important for the correction of tropospheric artifacts in differential InSAR (D-InSAR) and for the improvement in persistent scatterer (PS) target identification in PS-InSAR. In this study, we explored the spatial structure of atmospheric delay datasets and assessed its effect on InSAR atmospheric delay correction. Two-dimensional (2D) experimental variogram maps of turbulent mixing components derived from 12 GPS zenith wet delay (ZWD) datasets, 12 MERIS ZWD datasets, and 3 ERS-1/2 tandem interferograms showed that spatial anisotropy is common in these datasets. An anisotropic variogram model was then developed and applied to fit the experimental variograms. The results showed that the proposed 2D anisotropic variogram model is superior to the isotropic model, with average improvements in 31.92 and 33.57% in terms of root-mean-square error and correlation coefficients, respectively. With the proposed anisotropic variogram model, the atmospheric delay maps were reconstructed by kriging interpolation and used to correct the atmospheric artifacts in InSAR interferograms. The results showed that the model considering the anisotropy of atmospheric delay produces better results than that with the isotropy assumption. Finally, the effects of the anisotropy ratio, sampling density, and correlation distance of external water vapor data on the atmospheric delay correction were investigated. The results showed that when the anisotropy ratio is less than 0.3, or the sampling density is less than 1% or more than 60%, the impact of anisotropy on kriging prediction is not obvious.

Conventional and Fractal Variogram Based on Time—Space Analysis of Seismicity Distribution—Case Study: Algeria Seismicity (1673-2010)

Geostatistics belongs to the wide class of statistical methods. It is used and applied to analyze and predict the values associated with spatial or spatio-temporal phenomena such as seismological events. Thus, in addition to its adaptability to perform spatial data analysis based on the principles of variography, many other methods more representative of the spatial distributions of the epicentres have been experimented. The unidirectional or isotropic fractal variogram associates the criterion of variance with the fractal dimension which is a better descriptive parameter of the spatial organization of events. The same analysis procedure carried out through a directional or azimuthal variogram introduces the context of a preferential direction of earthquake evolution. Moreover, b-value is a privileged seismological parameter that can be studied alone or in combination with other more advanced geostatistical analysis factors such as fractals, fractal dimensions and the anisotropic variogram. All these concepts were used as methodology for a protocol of analysis of the catalog of the Algeria seismicity including 1919 events for the period extending from 1673 to 2010. This catalog is provided by the compilation of partial catalogs synthesized by the CRAAG (Research Centre in Astronomy, Astrophysics and Geophysics).

Spatial assessment of water quality parameters in Jhelum city (Pakistan).

In this study, we assess the drinking water quality of Jhelum city. Two hundred and ninety-two drinking water samples were randomly collected in the study area. These samples were chemically analyzed for three key toxic (in excess) elements such as pH, total dissolved solids (TDS), and calcium. Geostatistical techniques such as variogram and kriging were used to investigate the spatial variations of these minerals across the city. The spatial structure for each element was found to be anisotropic, and thus, anisotropic variograms were used. The kriging predictions revealed significant concentrations of the above-stated elements at some locations in the study area. While comparing with the World Health Organization, United States Environmental Protection Agency, and Pakistan Environmental Protection Agency standards, the water samples were found to be unsatisfactory for drinking. We conclude that the drinking water in this region is of poor quality and needs proper treatment to make it palatable.

Forecasting Interest Rates Using Geostatistical Techniques

Geostatistical spatial models are widely used in many applied fields to forecast data observed on continuous three-dimensional surfaces. We propose to extend their use to finance and, in particular, to forecasting yield curves. We present the results of an empirical application where we apply the proposed method to forecast Euro Zero Rates (2003–2014) using the Ordinary Kriging method based on the anisotropic variogram. Furthermore, a comparison with other recent methods for forecasting yield curves is proposed. The results show that the model is characterized by good levels of predictions’ accuracy and it is competitive with the other forecasting models considered.

Anisotropy Models for Spatial Data

This work addresses the question of building useful and valid models of anisotropic variograms for spatial data that go beyond classical anisotropy models, such as the geometric and zonal ones. Using the concept of principal irregular term, variograms are considered, in a quite general setting, having regularity and scale parameters that can potentially vary with the direction. It is shown that if the regularity parameter is a continuous function of the direction, it must necessarily be constant. Instead, the scale parameter can vary in a continuous or discontinuous fashion with the direction. A directional mixture representation for anisotropies is derived, in order to build a very large class of models that allow to go beyond classical anisotropies. A turning band algorithm for the simulation of Gaussian anisotropic processes, obtained from the mixture representation, is then presented and illustrated.

Thickness, porosity, and permeability prediction: comparative studies and application of the geostatistical modeling in an Oil field

BackgroundIn this study, we applied the geostatistical modeling to analyze an oil field. The reservoir properties, thickness, porosity and permeability, were studied. Data analysis tools, such as histogram, scatter plot, variogram and cross variogram modeling, were employed to capture the interpretable spatial structure and provide the desired input parameters for further estimation. SK (simple kriging), OK (ordinary kriging), Sgism (Sequential Gaussian Simulation), SC (simple cokriging), OC (ordinary cokriging) and MM2 (Markov model 2) methods were applied to estimate reservoir properties. Estimation difference maps were generated to compare the results of each method, providing more straightforward realizations in a visual way.ResultsFor thickness, results indicated that anisotropic variogram could provide better interpretations for the spatial relationships than isotropic variogram. Both SK and OK could provide better estimates. In comparison to the conventional estimation techniques, the simulation method could well reflect the reservoir’s intrinsical characteristics in terms of the associated extreme values. OOIP (Original Oil In Place) was calculated later with the parameters attained before, including thickness and porosity. Estimation difference maps showed that there was no obvious difference in SK vs. OK and SC vs. OC for the study of permeability. However, OC was slightly different from OK, and there were significant discrepancies between the estimates of OC and MM2 at the unsampled locations. In addition, OC estimates were closest to the sample data of permeability with the minimum variance.ConclusionsGeostatistical modeling is an effective way for thickness, porosity, and permeability prediction.

Improved interpolation of meteorological forcings for hydrologic applications in a Swiss Alpine region

This paper presents a comparative study on the mapping of temperature and precipitation fields in complex Alpine terrain. Its relevance hinges on the major impact that inadequate interpolations of meteorological forcings bear on the accuracy of hydrologic predictions regardless of the specifics of the models, particularly during flood events. Three flood events measured in the Swiss Alps are analyzed in detail to determine the interpolation methods which best capture the distribution of intense, orographically-induced precipitation. The interpolation techniques comparatively examined include: Inverse Distance Weighting (IDW), Ordinary Kriging (OK), and Kriging with External Drift (KED). Geostatistical methods rely on a robust anisotropic variogram for the definition of the spatial rainfall structure. Results indicate that IDW tends to significantly underestimate rainfall volumes whereas OK and KED methods capture spatial patterns and rainfall volumes induced by storm advection. Using numerical weather forecasts and elevation data as covariates for precipitation, we provide evidence for KED to outperform the other methods. Most significantly, the use of elevation as auxiliary information in KED of temperatures demonstrates minimal errors in estimated instantaneous rainfall volumes and provides instantaneous lapse rates which better capture snow/rainfall partitioning. Incorporation of the temperature and precipitation input fields into a hydrological model used for operational management was found to provide vastly improved outputs with respect to measured discharge volumes and flood peaks, with notable implications for flood modeling.

Covariance Estimation for dInSAR Surface Deformation Measurements in the Presence of Anisotropic Atmospheric Noise

We study anisotropic spatial autocorrelation in differential synthetic aperture radar interferometric (dInSAR) measurements and its impact on geophysical parameter estimations. The dInSAR phase acquired by the satellite sensor is a superposition of different contributions, and when studying geophysical processes, we are usually only interested in the surface deformation part of the signal. Therefore, to obtain high-quality results, we would like to characterize and/or remove other phase components. A stochastic model has been found to be appropriate to describe atmospheric phase delay in dInSAR images. However, these phase delays are usually modeled as being isotropic, which is a simplification, because InSAR images often show directional atmospheric anomalies. Here, we analyze anisotropic structures and show validation results using both real and simulated data. We calculate experimental semivariograms of the dInSAR phase in several European Remote Sensing satellite-1/2 tandem interferograms. Based on the theory of random functions (RFs), we then fit anisotropic variogram models in the spatial domain, employing Mate?rn- and Bessel-family correlation functions in nested models to represent complex dInSAR covariance structures. The presented covariance function types, in the statistical framework of stationary RFs, are consistent with tropospheric delay models. We find that by using anisotropic data covariance information to weight dInSAR measurements, we can significantly improve both the precision and accuracy of geophysical parameter estimations. Furthermore, the improvement is dependent on how similar the deformation pattern is to the dominant structure of the anisotropic atmospheric signals.

ANALYSIS OF ANISOTROPIC VARIOGRAM MODELS FOR PREDICTION OF THE CURONIAN LAGOON DATA

The anisotropy in particular environmental phenomena is detected when behavior of a physical process differs in different directions. In this paper geometric and zonal anisotropies are considered. Various methods of geostatistical analysis, also isotropic and geometrical anisotropic variogram models are compared and applied for the Curonian lagoon depth data. The results demonstrate that after robust estimation, i.e. elimination of outliers and after elimination of geometric anisotropy the precision of prediction and adequacy of models are much better. All computations have been performed by means of gstat, base and spatial packages of R system. Prediction results are compared with the results of research where outliers and geometric anisotropy were not eliminated.

New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity

Many heterogeneous media and environmental processes are statistically anisotropic. In this paper we focus on range anisotropy, that is, stochastic processes with variograms that have direction dependent correlation lengths and direction independent sill. We distinguish between two classes of anisotropic covariance models: Class (A) models are reducible to isotropic after rotation and rescaling operations. Class (B) models can be separated into a product of one-dimensional functions oriented along the principal axes. We propose a new Class (A) model with multiscale properties that has applications in subsurface hydrology. We also present a family of Class (B) models based on non-Euclidean distance metrics that are generated by superellipsoidal functions. Next, we propose a new method for determining the orientation of the principal axes and the degree of anisotropy, i.e., the ratio(s) of the correlation lengths. This information reduces the degrees of freedom of anisotropic variograms and thus simplifies the estimation procedure. In particular, Class (A) models are reduced to isotropic and Class (B) models to one-dimensional functions. Our method is based on an explicit relation between the second-rank slope tensor (SRST), which can be estimated from the data, and the covariance tensor. The procedure is conceptually simple and numerically efficient. It is more accurate for regular (on-grid) data distributions, but it can also be used for sparse (off-grid) spatial distributions. In the case of non-differentiable random fields the method can be extended using generalized derivatives. We illustrate its implementation with numerical simulations.

Understanding Anisotropy Computations

Most descriptions of anisotropy make reference to reduced distances and conversion of anisotropic models to isotropic counterparts and equations are presented for a certain class of range-anisotropic models. Many descriptions state that sill anisotropy is modelled using a range-anisotropic structure having a very elongated ellipse. The presentations typically have few or no intervening steps. Students and applied researchers rarely follow these presentations and subsequently regard the programs that compute anisotropic variograms as black-boxes, the contents of which are too complex to try to understand. We provide the geometry necessary to clarify those computations. In so doing, we provide a general way to model any type of anisotropy (range, sill, power, slope, nugget) on an ellipse. We note cases in the literature in which the printed descriptions of anisotropy on an ellipse do not match the stated or coded models. An example is provided in which both range- and sill-anisotropic structures are fitted to the experimental variogram values from an elevation data set using the provided equations and weighted nonlinear regression. The original variogram values are plotted with the fitted surfaces to view the fit and anisotropic structure in many directions at once.

Cohorts in space:geostatistical mapping of the age structure of the squat lobster Pleuroncodes monodon population off central Chile

MEPS Marine Ecology Progress Series Contact the journal Facebook Twitter RSS Mailing List Subscribe to our mailing list via Mailchimp HomeLatest VolumeAbout the JournalEditorsTheme Sections MEPS 196:239-251 (2000) - doi:10.3354/meps196239 Cohorts in space: geostatistical mapping of the age structure of the squat lobster Pleuroncodes monodon population off central Chile Rubén Roa*, Fabián Tapia** Departamento de Oceanografía, Universidad de Concepción, Casilla 160-C, Concepción, Chile *E-mail: rroa@udec.cl **Present address: Biology Department, MS#34, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543, USA ABSTRACT: Animal populations are spatial objects in the sense that their structure and functioning depend on the spatial context they inhabit. However, most population dynamics models ignore the spatial context and set populations Œin the air¹. This paper is focused on the squat lobster Pleuroncodes monodon population inhabiting the narrow continental shelf off central Chile, and encompassing 3 degrees of latitude. We tested the hypothesis that the spatial functioning of this population is driven by annual cohorts that enter the population through a nursery ground, and then radiate away from the nursery to the north and south by individual dispersal, thus creating a spatial pattern where age and distance from the nursery are correlated. To test the hypothesis we performed an extensive survey from 34 to 37°S along the continental shelf, collecting data on individual mean size by sex and spatial location. The data gathered was analysed using a combination of mixture distribution analysis on size frequency data and geostatistical mapping on size and location data. Isotropic and anisotropic variogram models were fitted to size data for both sexes. Selection of the appropriate models and kriging parameters was done by cross validation. The hypothesis posed was fully corroborated by the data and the analysis, with 6 adult cohorts identified for both sexes radiating away from the nursery ground in waves at an average distance of one fifth the latitudinal range of the population. Geostatistical mapping of size data for juveniles in the nursery ground, collected with a higher resolution in previous years, showed that recruitment can occur as several relatively small patches interspersed through the nursery area or as a large patch in the middle of the nursery. The corroborated model of spatial population functioning explains the history of commercial over-exploitation and recolonisation of this population. It also suggest that this 350 km long population is renewed by recruits from nearby cohorts, both to the south and north of the nursery area, while cohorts farther north can be exporting recruits to prime nursery habitat. These insights into the functioning of this large population could only be gained by a fully spatially explicit analysis. KEY WORDS: Population dynamics · Cohort dispersal · Geostatistical mapping · Spatially explicit · Habitat features · Crustacean biology · Chile Full text in pdf format PreviousNextExport citation RSS - Facebook - Tweet - linkedIn Cited by Published in MEPS Vol. 196. Publication date: April 18, 2000 Print ISSN:0171-8630; Online ISSN:1616-1599 Copyright © 2000 Inter-Research.

Moving averages for Gaussian simulation in two and three dimensions

The square-root method provides a simple and computationally inexpensive way to generate multidimensional Gaussian random fields. It is applied by factoring the multidimensional covariance operator analytically, then sampling the factorization at discrete points to compute an array of weighted averages that can be convolved with an array of random normal deviates to generate a correlated random field. In many respects this is similar to the LUdecomposition method and to the one-dimensional method of moving averages. However it has been assumed that the method of moving averages could not be used in higher dimensions, whereas direct application of the matrix decomposition approach is too expensive to be practical on large grids. In this paper, I show that it is possible to calculate the square root of many two- and three dimensional covariance operators analytically so that the method of moving averages can be applied directly to the problem of multidimensional simulation. A few numerical examples of nonconditional simulation on a 256×256 grid that show the simplicity of the method are included. The method is fast and can be applied easily to nested and anisotropic variograms.

Exploration of geostatistical methods for mapping and estimating acoustic biomass of pelagic fish in the Gulf of St. Lawrence: size of echo-integration unit and auxiliary environmental variables

Daytime 38 kHz mean volume backscattering strength (MVBS) of small pelagic fish in the Gulf of St. Lawrence was recorded at 60-m intervals along a regular network of 11 parallel transects in August 1991. Profiles of temperature, salinity, density and transmittance were collected on a grid of 44 punctual stations along the transects. The MVBS was contagiously distributed and autocorrelated for al1 transects. No significant monotone relationship with any of the environmental variables was evidenced, except that the highest concentrations of fish were located over shallow bottoms. The anisotropic variogram of MVBS indicated three different scales of spatial structures: an unresolved small-scale variability, which accounted for only 7.7% of variance, and two structure components of along-transect scales of ~500 m (~40%) and 15 km (~50%). The theoretical variograms deduced for larger echo-integration units (EIUs) than the recorded 60-m unit, provided a good fit to the corresponding experimental variograms up to EIU of 1920-m, when the disturbing effect of outliers on the variograms was removed. Global estimates and maps produced with 240-m and 1920-m EIU data were almost identical, showing that no significant loss of information was caused by increasing the EIU. Asymmetric confidence intervals of the global estimates are obtained using two variograms, one controlled for the effect of the outliers, the other one not controlled. In an effort to take the relation of MVBS with the bottom depth into account, four different kriging methods were compared: ordinary kriging, kriging with the bottom depth or its indicator-transform as external drift, and 3-D ordinary kriging with depth as the third dimension. Because of the weak relation of MVBS with the bottom depth, ordinary kriging had the best performance. Results are discussed in the context of applications of geostatistics to estimation and mapping of acoustic survey data.

Spatial Variability of Morphological Properties at a Florida Flatwoods Site

Soil profiles were described on a grid at 60-m intervals to assess the variability of soil morphological properties in 60 ha of Florida flatwoods. Depths from the soil surface to the spodic and argillic horizons were mapped using field data supplemented with kriged values. In addition, a 120 by 120 m subarea was described at 15-m intervals. The occurrence of a single deep spodic horizon at a few peripheral locations influenced the structure of variograms and contributed to anisotropy. The soil profiles adjacent to these locations exhibited two spodic horizons, one much deeper than the other. The variance structure model for spodic horizon depth in the overall 60-ha site accurately described the variance structure for the subarea. At some sampling locations, neither a spodic nor an argillic horizon was observed. Absence of the argillic horizon was more common than absence of the spodic horizon. Areal discontinuity in the argillic horizon was more extensive than in the spodic horizon. Such discontinuity in the argillic horizon was observed near and between depressional areas. Although the variance structure of depth to the argillic horizon could be described using an anisotropic variogram model, better estimates of depth to the argillic horizon were obtained using variograms of residuals and drift functions.

Hydrostratigraphic interpretation using indicator geostatistics

In this paper we describe an approach that uses indicator geostatistics to interpret qualitative borehole logs and compute experimental variograms for complex alluvial sediments. Borehole descriptions are first transformed into binary indicator values based on inferred relative permeability from the borehole descriptions. The resulting indicator data can then be used to compute variograms and construct three‐dimensional variogram models. The ranges of computed indicator variograms for a groundwater contamination site in Santa Clara Valley, California, are very sensitive to the orientation of the search plane. These variograms are consistent with known stratigraphie features and describe the spatial structure of deposits from different depositional environments. Indicator kriging weighs all the available data on the basis of a three‐dimensional, anisotropic variogram model and provides an estimate of uncertainty in the hydrostratigraphic correlation. Kriged indicator values represent probabilities that sediments at a specific location fall into one of two indicator categories. The location of the 0.5 indicator contour is approximately the boundary between high‐ and low‐permeability sediments that might be constructed in a geologic cross section.