Stochastic Interpolation Methods Research Articles

ANALYSIS OF EXISTING METHODS FOR DETERMINING INTERMEDIATE GEOTECHNICAL CHARACTERISTICS OF SOILS

This article presents an overview of the main methods for determining the intermediate geotechnical characteristics of soils, emphasizing the importance of strict adherence to the testing regime. Deterministic and stochastic interpolation methods such as Thiessen polygons, inverse distance weighting (IDW), Spline and Kriging are discussed. Special attention is given to the problem of determining strength properties at various points, where only certain areas are to be investigated and intermediate values are obtained by interpolation methods. The analysis of geotechnical reliability using Kriging numerical models and methods of using geographic information systems for processing and presenting geotechnical data are also discussed.

USING ELECTRONIC NAVIGATIONAL CHART FOR 3D BATHYMETRIC MODEL OF THE PORT OF NAPLES

Abstract. Nautical charts generally report fundamental knowledge for the safety of navigation. This information also includes sea depth data reported as depth points or contour lines, which can be used to build a 3D model of the seabed. However, there are different interpolation methods for creating digital depth models, and there is no way to know in advance which of them is the best performing. The aim of this work is to compare different spatial interpolation methods applied on a dataset concerning the seabed of the Port of Naples (Italy) and extracted from the Electronic Navigational Chart (ENC) produced by the Hydrographic Institute of the Italian Navy, in scale 1:10.000. Four deterministic interpolation methods, i.e. Inverse Distance Weighting (IDW), Global Polynomial Interpolation (GPI), Local Polynomial Interpolation (LPI), Radial Basis Function (RBF), and two stochastic interpolation methods, i.e. Ordinary Kriging (OK) and Universal Kriging (UK), are applied using Geographic Information System (GIS) software. Since each method requires to set specific parameters and different options are available, e.g. the order of the polynomial function for GPI and LPI, or semi-variograms for OK and UK, twenty-three models are generated. The result quality is evaluated by Leave-One-Out cross-validation and the statistics of the residuals produced by each interpolation method in the measured points are compared and analysed. The experiments confirm that the stochastic approach is more versatile compared to deterministic approach and can produce better results, as it is testified by the great performance of the Ordinary Kriging, which produces the most accurate 3D models.

Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space

We present a novel method for stochastic interpolation of sparsely sampled time signals based on a superstatistical random process generated from a multivariate Gaussian scale mixture. In comparison to other stochastic interpolation methods such as Gaussian process regression, our method possesses strong multifractal properties and is thus applicable to a broad range of real-world time series, e.g. from solar wind or atmospheric turbulence. Furthermore, we provide a sampling algorithm in terms of a mixing procedure that consists of generating a -dimensional field , where each Gaussian component is synthesized with identical underlying noise but different covariance function parameterized by a log-normally distributed parameter ξ. Due to the Gaussianity of each component , we can exploit standard sampling algorithms such as Fourier or wavelet methods and, most importantly, methods to constrain the process on the sparse measurement points. The scale mixture u(t) is then initialized by assigning each point in time t a and therefore a specific value from , where the time-dependent parameter follows a log-normal process with a large correlation time scale compared to the correlation time of . We juxtapose Fourier and wavelet methods and show that a multiwavelet-based hierarchical approximation of the interpolating paths, which produce a sparse covariance structure, provide an adequate method to locally interpolate large and sparse datasets.

Components of urban ground water recharge in Bengaluru, India

ABSTRACT This study investigates the various components of groundwater recharge in an urban hard-rock aquifer system of Bengaluru, India. Groundwater meets more than fifty percent of the water demand in the city. This has led to an endangered situation of groundwater. Urbanization changes the source and path of groundwater recharge. Understanding the components of groundwater recharge is the foremost important step in groundwater management in an urban environment. Rotated empirical orthogonal function (REOF) analysis of ground water level data was used to identify the significant components of ground water level variation in the study area. Rainfall and pipeline leakage were identified as the major components of groundwater recharge in the study area. The stochastic interpolation method of ordinary kriging was used to regionalise the identified significant empirical orthogonal functions.

On the drawback of local detrending in universal kriging in conditions of heterogeneously spaced regional TEC data, low-order trends and outlier occurrences

The study intercompares three stochastic interpolation methods originating from the same geostatistical family: least-squares collocation (LSC) known from geodesy, as well as ordinary kriging (OKR) and universal kriging (UKR) known from geology and other geosciences. The objective of this work is to assess advantages and drawbacks of fundamental differences in modeling between these methods in imperfect data conditions. These differences primarily refer to the treatment of the reference field, commonly called ‘mean value’ or ‘trend’ in geostatistical language. The trend in LSC is determined globally before the interpolation, whereas OKR and UKR detrend the observations during the modeling process. The approach to detrending leads to the evident differences between LSC, OKR and UKR, especially in severe conditions such as far from the optimal data distribution. The theoretical comparisons of LSC, OKR and UKR often miss the numerical proof, while numerical prediction examples do not apply cross-validation of the estimates, which is proven to be a reliable measure of the prediction precision and a validation of empirical covariances. Our study completes the investigations with precise parametrization of all these methods by leave-one-out validation. It finds the key importance of the detrending schemes and shows the advantage of LSC prior global detrending scheme in unfavorable conditions of sparse data, data gaps and outlier occurrence. The test case is the modeling of vertical total electron content (VTEC) derived from GNSS station data. This kind of data is a challenge for precise covariance modeling due to weak signal at higher frequencies and existing outliers. The computation of daily set of VTEC maps using the three techniques reveals the weakness of UKR solutions with a local detrending type in imperfect data conditions.

Comparison Between Deterministic and Stochastic Interpolation Methods for Predicting Ground Water Level in Baghdad

Surface interpolation techniques are usually used to create continuous data (i.e. raster data) from distributed set of point data over a geographical region. There are deterministic and stochastic (geostatistical) interpolation techniques can be used to create spatial raster surface. In this paper, the comparison between the Inverse Distance Weight (IDW) interpolation method as deterministic method and the Kriging interpolation method as stochastic method is done to determine the best performance for measuring levels of ground water in Baghdad Governorate. Spatial raster surface surfaces as ground water prediction maps are generated from each method by using average ground water level measured at 206 wells in the study area. These maps are shown spatial variation in the ground water levels and they have complete different. The IDW method results a refined map and lesser error than the Kriging method. Thus, the analysis shows that the IDW gives better real performance of measuring levels of ground water in Baghdad Governorate.

A Fractal-Interpolation Model for Diagnosing Spalling Risk in Concrete at Elevated Temperatures

Spalling risk analysis and assessment of fire-loaded concrete structures is an essential part of research on fire-resistant behavior of concrete structures. In this work, the authors developed a stochastic interpolation-based fractal model capable of quickly assessing spalling risk in concrete at high temperatures based on mainly factors affecting explosive spalling. The preliminary grading standard of the factors elected in the study was established based on the data from literature. Test samples were then generated by the stochastic interpolation method according to uniform distribution. A spalling risk diagnosis model was built based on fractal dimension weights. In addition, 10 specimens were assessed with the presented model. The evaluation results obtained by the model in this study were in good agreement with those obtained by the fuzzy pattern recognition approach and coincided well with the actual fire test results. The applicability and reliability of the proposed model were verified and can provide theoretical support for identification of the spalling risk of concrete at elevated temperatures.

Interpolation selection index for delineation of thematic maps

Precision agriculture aims at the precise management of agricultural inputs toward increasing profits, decreasing losses, and preserving the environment. Thus, the use of thematic maps to understand the behavior of involved attributes is important, and the construction of these maps usually involves some type of interpolation. Choosing the best interpolation method can be difficult, and the use of cross-validation is not trivial. Therefore, the purpose of this study is to propose a selection index of interpolation methods that will help in choosing the best deterministic and stochastic model among those evaluated. The study was conducted within a 15.5-ha area in Southern Brazil, and an interpolation selection index (ISI) was applied to data on clay, copper, and manganese content, and apparent electrical conductivity of the soil using four interpolation methods: inverse distance, inverse distance squared, ordinary kriging, and cokriging. Using the ISI, choosing between deterministic and stochastic interpolation methods is simplified and less subjective. In cases where the deterministic interpolator (inverse of distance squared) was chosen, the spatial dependency was moderated. Note that the proposed statistic (ISI) does not quantify the difference between the analyzed methods.

Spatio-Temporal Analysis of Water Quality Parameters in Machángara River with Nonuniform Interpolation Methods

Water quality measurements in rivers are usually performed at intervals of days or months in monitoring campaigns, but little attention has been paid to the spatial and temporal dynamics of those measurements. In this work, we propose scrutinizing the scope and limitations of state-of-the-art interpolation methods aiming to estimate the spatio-temporal dynamics (in terms of trends and structures) of relevant variables for water quality analysis usually taken in rivers. We used a database with several water quality measurements from the Machángara River between 2002 and 2007 provided by the Metropolitan Water Company of Quito, Ecuador. This database included flow rate, temperature, dissolved oxygen, and chemical oxygen demand, among other variables. For visualization purposes, the absence of measurements at intermediate points in an irregular spatio-temporal sampling grid was fixed by using deterministic and stochastic interpolation methods, namely, Delaunay and k-Nearest Neighbors (kNN). For data-driven model diagnosis, a study on model residuals was performed comparing the quality of both kinds of approaches. For most variables, a value of k = 15 yielded a reasonable fitting when Mahalanobis distance was used, and water quality variables were better estimated when using the kNN method. The use of kNN provided the best estimation capabilities in the presence of atypical samples in the spatio-temporal dynamics in terms of leave-one-out absolute error, and it was better for variables with slow-changing dynamics, though its performance degraded for variables with fast-changing dynamics. The proposed spatio-temporal analysis of water quality measurements provides relevant and useful information, hence complementing and extending the classical statistical analysis in this field, and our results encourage the search for new methods overcoming the limitations of the analyzed traditional interpolators.

Signal to noise ratio in water balance maps with different resolution

What is the best resolution of annual water balance maps for a correct balance between the basic spatial signal in the observations of precipitation, actual evapotranspiration and runoff across a larger drainage basin and the error in estimates for grid cells in the map to avoid giving a false impression of accuracy? To answer this question an approach based a signal to noise ratio is proposed, which allows finding the optimal resolution maximizing the signal in the map. The approach is demonstrated on gauge data in the Huai River Basin, China. Stochastic interpolation methods were applied to create grid maps of long-term mean values, as well as for estimating variances of the three water balance components in a range of scales from 5×5km to 200×200km2 grid cells. Interpolation algorithms using covariances of long-term means of data with different spatial support were developed. The identified optimal resolutions by the signal to noise ratio appeared to be very different – 10×10, 50×50, and 30×30km2 for precipitation, actual evapotranspiration, and runoff, respectively. These values are directly linked to the observation network densities. The magnitude of the signal to noise ratio shows similar strong differences with values 34, 3.7, and 5.4, respectively. It gives a direct indication of the reliability of the map, which can be considered as satisfactory only for precipitation for the data available for the present study. The critical factors for this magnitude are parameters characterising the spatial covariance in data and the network density.

Genetic Programming-Based Ordinary Kriging for Spatial Interpolation of Rainfall

AbstractRainfall data provide an essential input for most hydrologic analyses and designs for effective management of water resource systems. However, in practice, missing values often occur in rainfall data that can ultimately influence the results of hydrologic analysis and design. Conventionally, stochastic interpolation methods such as kriging are the most frequently used approach to estimate the missing rainfall values where the variogram model that represents spatial correlations among data points plays a vital role and significantly impacts the performance of the methods. In the past, the standard variogram models in ordinary kriging were replaced with the universal function approximator-based variogram models, such as artificial neural networks (ANN). In the current study, applicability of genetic programming (GP) to derive the variogram model and use of this GP-derived variogram model within ordinary kriging for spatial interpolation was investigated. Developed genetic programming-based ordinary ...

Multiscale Sampling Geometries and Methods for Deterministic and Stochastic Reconstructions of Magnitude and Phase Spectra of Satellite Imagery

This paper presents new methods for phase and magnitude interpolation and demonstrates their usefulness in reconstructing images from a limited number of frequency samples. A collection of multiscale frequency domain sampling geometries are developed based on the partition of the spectrum into low-, medium-, and high-frequency blocks. A nonstationary statistical approach is introduced that is based on adaptively selecting the best stochastic model in each frequency block. To develop effective models, the magnitude spectrum is preprocessed using a logarithmic transformation. Phase interpolation requires preprocessing by an appropriate phase unwrapping method. The new stochastic interpolation method is compared against cubic spline, bilinear, and nearest neighbor interpolation methods. Image reconstruction results are presented for sampling rates that retain 6.01% to 28.91% of the 2-D fast Fourier transform (FFT) samples. Image interpolation methods are compared based on the peak signal-to-noise ratio and the mean structural similarity index for satellite images of rural, natural, and urban images. The results indicate that the stochastic (Kriging) interpolation approach provides the best rural image reconstructions using just 6.01% of the 2-D FFT samples. Bilinear interpolation also gave excellent reconstructions for natural and urban images. For natural and urban images, stochastic interpolation gave the best magnitude-only interpolation results.

Modelling of nitrogen dioxide (NO2) and fine particulate matter (PM10) air pollution in the metropolitan areas of Barcelona and Bilbao, Spain

AbstractEcological studies in epidemiology present some limitations on assessing the exposure of environmental factors. The main objective of this study was to describe a statistical methodology that could help on more accurately assessing the relationship between air pollution and health responses at very small spatial scale.In particular, we modelled the spatial distribution of nitrogen dioxide (NO2) and of particles with a diameter of less than 10 µm in two study areas, the metropolitan areas of Barcelona (MABa) and Bilbao (MABi). The areas chosen for this project can be distinguished for having similar population density and urban surface, as well as high road traffic intensity. The geographical variability of the pollutants could differ greatly due to very different orographic and climatic conditions.Three contributions of this study could be mentioned. First of all, the spatial distribution of the values of each air pollutant was spread over the entire study region on a small area basis, that is census track in our case. Secondly, a stochastic interpolation method, kriging, was used. Thirdly, instead of estimating the average value of the pollutant at the centroid of the area for the entire study period, the value was estimated for each and every day of the corresponding period.As regards the main results, in both areas it can be observed that those census tracks with a greater concentration of pollutants are those that were estimated more accurately. This suggests the existence of a persistent emission source in those tracks, very probably those roads with greater road traffic intensity. Copyright © 2008 John Wiley & Sons, Ltd.

Use of universal function approximation in variance-dependent surface interpolation method: An application in hydrology

Variance dependent stochastic interpolation approaches such as kriging are widely recognized as standard stochastic methods for interpolation of geophysical and hydrologic variables. Deterministic weighting and stochastic interpolation methods are the most frequently used methods for estimating missing rainfall values at a gage based on values recorded at all other available recording gages. Traditional kriging has a major limitation due to the need for an a priori definition of a mathematical function for a semivariogram that might fit the surface to be interpolated. Use of the universal function approximator, artificial neural network (ANN), as a replacement to fitted authorized semivariogram model within ordinary kriging is investigated in the current study. The revised ordinary kriging is used for estimation of missing precipitation data at a rainfall gaging station based on data recorded at all other available gaging stations. Historical daily precipitation data obtained from 15 rain gaging stations from a temperate climatic region, Kentucky, USA, is used to test the improvised method and derive conclusions about the efficacy of this method. Results suggest that use of universal function approximator such as ANN within a kriging has several advantages over ordinary kriging.

Spatial Analysis of Air Temperature in the Qinghai-Tibet Plateau

The mapping of air temperature in the Qinghai-Tibet Plateau is of practical importance because the meteorological stations are sparse and unevenly distributed in the region. The spatial interpolation methods of inverse distance weight, ordinary kriging, ordinary cokriging, and a combined method were used to estimate the spatial distribution of the 1961–1990 January mean air temperature in the Qinghai-Tibet Plateau. The combined method is a combination of ordinary kriging and correction of altitude effect by using lapse rate of air temperature. Comparison of results showed that the problem of mapping air temperature in the Qinghai-Tibet Plateau cannot be resolved by the simple geometric interpolation method as the inverse distance weight. Ordinary kriging can manifest some spatial pattern but the performance was not improved too much. Cokriging to a certain extent was an improvement of kriging but due to the limited altitude information included in the co-variable, the interpolation results were also not in agreement with the actual situation. The combined method was superior to the other methods. The interpolation result from it was reasonable, proved by both the subjective analysis and by many previous works. The comparison of the four methods leads to the conclusion that for regions with sparse meteorological stations, such as the Qinghai-Tibet Plateau, stochastic interpolation methods must be combined with the altitude-effect correction for estimating the spatial distribution of climatic variables.

Stochastic partial differential equations in continuum physics—on the foundations of the stochastic interpolation method for ITO's type equations

This paper deals with the mathematical analysis of mathematical models in continuum physics which are described by partial differential equations, in one space dimension, with additional random noise. The first part of the paper provides the framework for the mathematical modelling of a large class of physical systems. Then, the so-called ‘stochastic interpolation method’ is applied to obtain quantitative solutions by transforming the original equation into a system of stochastic ordinary differential equations, and an analysis of the error estimates is developed in order to provide a bound estimate of the distance between the solution of the true original problem and the one obtained by solving the system of ordinary differential equations. As an application, a linear random model for the heat equation is finally considered following the analysis proposed here.