Interpolation methods for stochastic processes spaces

In this paper the scales of classes of stochastic processes are introduced. New interpolation theorems and boundedness of some transforms of stochastic processes are proved. Interpolation method for generously-monotonous rocesses is entered. Conditions and statements of interpolation theorems concern he xed stochastic process, which diers from the classical results.

Interpolation and sampling in small Bergman spaces

"Vegeu el resum a l'inici del document del fitxer adjunt."

Scaling of semivariograms and the kriging estimation of field-measured properties

Two methods were evaluated for scaling a set of semivariograms into a unified function for kriging estimation of field-measured properties. Scaling is performed using sample variances and sills of individual semivariograms as scale factors. Theoretical developments show that kriging weights are independent of the scaling factor which appears simply as a constant multiplying both sides of the kriging equations. The scaling techniques were applied to four sets of semivariograms representing spatial scales of 30 x 30 m to 600 x 900 km. Experimental semivariograms in each set successfully coalesced into a single curve by variances and sills of individual semivariograms. To evaluate the scaling techniques, kriged estimates derived from scaled semivariogram models were compared with those derived from unscaled models. Differences in kriged estimates of the order of 5% were found for the cases in which the scaling technique was not successful in coalescing the individual semivariograms, which also means that the spatial variability of these properties is different. The proposed scaling techniques enhance interpretation of semivariograms when a variety of measurements are made at the same location. They also reduce computational times for kriging estimations because kriging weights only need to be calculated for one variable. Weights remain unchanged for all other variables in the data set whose semivariograms are scaled.

Interpolation-free Coregistration and Phase-Correction of Airborne SAR Interferograms

This letter discusses the detection and correction ofresidual motion errors that appear in airborne synthetic apertureradar (SAR) interferograms due to the lack of precision in the navigationsystem. As it is shown, the effect of this lack of precision istwofold: azimuth registration errors and phase azimuth undulations.Up to now, the correction of the former was carried out byestimating the registration error and interpolating, while the latterwas based on the estimation of the phase azimuth undulations tocompensate the phase of the computed interferogram. In this letter,a new correction method is proposed, which avoids the interpolationstep and corrects at the same time the azimuth phase undulations.Additionally, the spectral diversity technique, used to estimateregistration errors, is critically analyzed. Airborne L-bandrepeat-pass interferometric data of the German Aerospace Center(DLR) experimental airborne SAR is used to validate the method

Diagnostic techniques applied in geostatistics for agricultural data analysis

The structural modeling of spatial dependence, using a geostatistical approach, is an indispensable tool to determine parameters that define this structure, applied on interpolation of values at unsampled points by kriging techniques. However, the estimation of parameters can be greatly affected by the presence of atypical observations in sampled data. The purpose of this study was to use diagnostic techniques in Gaussian spatial linear models in geostatistics to evaluate the sensitivity of maximum likelihood and restrict maximum likelihood estimators to small perturbations in these data. For this purpose, studies with simulated and experimental data were conducted. Results with simulated data showed that the diagnostic techniques were efficient to identify the perturbation in data. The results with real data indicated that atypical values among the sampled data may have a strong influence on thematic maps, thus changing the spatial dependence structure. The application of diagnostic techniques should be part of any geostatistical analysis, to ensure a better quality of the information from thematic maps.

Spatial and temporal variability of crop yield and some Rhodic Hapludox properties under no-tillage

Soil properties are closely related with crop production and spite of the measures implemented, spatial variation has been repeatedly observed and described. Identifying and describing spatial variations of soil properties and their effects on crop yield can be a powerful decision-making tool in specific land management systems. The objective of this research was to characterize the spatial and temporal variations in crop yield and chemical and physical properties of a Rhodic Hapludox soil under no-tillage. The studied area of 3.42 ha had been cultivated since 1985 under no-tillage crop rotation in summer and winter. Yield and soil property were sampled in a regular 10 x 10 m grid, with 302 sample points. Yields of several crops were analyzed (soybean, maize, triticale, hyacinth bean and castor bean) as well as soil chemical (pH, Soil Organic Matter (SOM), P, Ca2+, Mg2+, H + Al, B, Fe, Mn, Zn, CEC, sum of bases (SB), and base saturation (V %)) and soil physical properties (saturated hydraulic conductivity, texture, density, total porosity, and mechanical penetration resistance). Data were analyzed using geostatistical analysis procedures and maps based on interpolation by kriging. Great variation in crop yields was observed in the years evaluated. The yield values in the Northern region of the study area were high in some years. Crop yields and some physical and soil chemical properties were spatially correlated.

Optimization of the sampling scheme for maps of physical and chemical properties estimated by kriging

The sampling scheme is essential in the investigation of the spatial variability of soil properties in Soil Science studies. The high costs of sampling schemes optimized with additional sampling points for each physical and chemical soil property, prevent their use in precision agriculture. The purpose of this study was to obtain an optimal sampling scheme for physical and chemical property sets and investigate its effect on the quality of soil sampling. Soil was sampled on a 42-ha area, with 206 geo-referenced points arranged in a regular grid spaced 50 m from each other, in a depth range of 0.00-0.20 m. In order to obtain an optimal sampling scheme for every physical and chemical property, a sample grid, a medium-scale variogram and the extended Spatial Simulated Annealing (SSA) method were used to minimize kriging variance. The optimization procedure was validated by constructing maps of relative improvement comparing the sample configuration before and after the process. A greater concentration of recommended points in specific areas (NW-SE direction) was observed, which also reflects a greater estimate variance at these locations. The addition of optimal samples, for specific regions, increased the accuracy up to 2 % for chemical and 1 % for physical properties. The use of a sample grid and medium-scale variogram, as previous information for the conception of additional sampling schemes, was very promising to determine the locations of these additional points for all physical and chemical soil properties, enhancing the accuracy of kriging estimates of the physical-chemical properties.

Three-dimensional interpolation of soil data: fertility and pedomorphological features in southern Brazil

The graphical representation of spatial soil properties in a digital environment is complex because it requires a conversion of data collected in a discrete form onto a continuous surface. The objective of this study was to apply three-dimension techniques of interpolation and visualization on soil texture and fertility properties and establish relationships with pedogenetic factors and processes in a slope area. The GRASS Geographic Information System was used to generate three-dimensional models and ParaView software to visualize soil volumes. Samples of the A, AB, BA, and B horizons were collected in a regular 122-point grid in an area of 13 ha, in Pinhais, PR, in southern Brazil. Geoprocessing and graphic computing techniques were effective in identifying and delimiting soil volumes of distinct ranges of fertility properties confined within the soil matrix. Both three-dimensional interpolation and the visualization tool facilitated interpretation in a continuous space (volumes) of the cause-effect relationships between soil texture and fertility properties and pedological factors and processes, such as higher clay contents following the drainage lines of the area. The flattest part with more weathered soils (Oxisols) had the highest pH values and lower Al3+ concentrations. These techniques of data interpolation and visualization have great potential for use in diverse areas of soil science, such as identification of soil volumes occurring side-by-side but that exhibit different physical, chemical, and mineralogical conditions for plant root growth, and monitoring of plumes of organic and inorganic pollutants in soils and sediments, among other applications. The methodological details for interpolation and a three-dimensional view of soil data are presented here.

Real interpolation for families of Banach spaces (II)

In this paper, we study the dual space and reiteration theorems for the real method of interpolation for infinite families of Banach spaces introduced in [2]. We also give examples of interpolation spaces constructed with this method.

Extrapolation theory for the real interpolation method

We develop an abstract extrapolation theory for the real interpolation method that covers and improves the most recent versions of the celebrated theorems of Yano and Zygmund. As a consequence of our method, we give new endpoint estimates of the embedding Sobolev theorem for an arbitrary domain Omega

A Geometric Characterization of Interpolation in $\hat{\E}^\prime(\R)$

We give a geometric description of the interpolating varieties for the algebra of Fourier transforms of distributions (or Beurling ultradistributions) with compact support on the real line.

Exemplar-based interpolation of sparsely sampled images

A nonlocal variational formulation for interpolating a sparsel sampled image is introduced in this paper. The proposed variational formulation, originally motivated by image inpainting problems, encouragesthe transfer of information between similar image patches, following the paradigm of exemplar-based methods. Contrary to the classical inpaintingproblem, no complete patches are available from the sparse imagesamples, and the patch similarity criterion has to be redefined as here proposed. Initial experimental results with the proposed framework, at very low sampling densities, are very encouraging. We also explore somedepartures from the variational setting, showing a remarkable ability to recover textures at low sampling densities.