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Analysis of Interpolation-Based Image In-Painting Approaches
Published in Rashmi Gupta, Arun Kumar Rana, Sachin Dhawan, Korhan Cengiz, Advanced Sensing in Image Processing and IoT, 2022
Mustafa Zor, Erkan Bostanci, Mehmet Serdar Güzel, Erinç Karataş
Kriging is a geostatistics interpolation method that takes into account the distance and degree of variation between known points when estimating values at unknown points [5]. This approach uses the values of the entire sample to calculate an unknown value. Kriging assumes that the distance or direction between sample points reflects a spatial correlation that can be used to explain variation on the surface. In Kriging interpolation, a mathematical function is applied to all points of a specified number or a specified radius to determine the estimation value of each unknown point. The closer the point, the higher the value of the weights. Kriging is the most appropriate approach when there is a spatially related distance or directional trend in the data (spatial autocorrelation) and calculated as follows: P^*=∑i=1NλiPi
Median polish kriging model for circular-spatial data
Published in Yuli Rahmawati, Peter Charles Taylor, Empowering Science and Mathematics for Global Competitiveness, 2019
H. Surjotedjo, Y. Widyaningsih, S. Nurrohmah
Kriging is a method used in geostatistics for spatial prediction. This term is used in honor of the South African mining engineer, Daniel Gerhardus Krige. Kriging aims to predict the realization value of a random field at one or more non-observed points from a collection of data observed at several locations. There are many different kriging methods, depending on model assumptions: Simple Kriging, Ordinary Kriging and Universal Kriging. Simple kriging assumes that the mean of the attribute being modeled is known. Ordinary kriging assumes that the mean of the attribute is unknown. Another necessary condition required by simple kriging and ordinary kriging methods (and some of other kriging methods) is the existence of second-order stationarity about the mean, and the existence of a finite variance (Matheron, 1967). There are some problems in interpolating using simple kriging and ordinary kriging when spatial trends are present, since second-order stationarity about the mean is no longer valid.
Risk from Groundwater Arsenic Exposure
Published in M. Manzurul Hassan, Arsenic in Groundwater, 2018
The arsenic interpolation map produced by kriging is constrained by the spherical semivariogram fits. The experimental variogram was computed from the raw data, and a “mathematical model” (Brooker et al., 1995; Mapa and Kumaragamage, 1996) was fitted to the arsenic concentration values by weighted least-squares approximation, using ArcGIS. The parameters of the variogram model (e.g., sill, nugget, and range) for arsenic concentrations were used with their values for estimating their concentrations over the area. We used Ordinary Kriging since the arsenic concentrations in groundwater are highly uneven. Ordinary Kriging is the most widely used type of kriging to estimate values when data point values vary or fluctuate around a constant mean value (Serón et al., 2001). It is applied for an unbiased estimate of the spatial variation of a component (Wang et al., 2001b).
Spatial interpolation of water quality index based on Ordinary kriging and Universal kriging
Published in Geomatics, Natural Hazards and Risk, 2023
Mohsin Khan, Mohammed M. A. Almazah, Asad EIlahi, Rizwan Niaz, A. Y. Al-Rezami, Baber Zaman
Ordinary Kriging and Universal kriging techniques were used to obtain the prediction maps of the water quality index (WQI). Kriging is a geostatistical approach that estimates a variable’s value over a continuous spatial field using a limited number of sampled data points. In kriging, the Variogram model has important for controlling kriging weights. In sample data, the gamma-values or semivariances for each pair of points are plotted against the distance between them. Mathematically variogram is defined as in Equation (4): where γ(h) is the semi-variance; N(h) the number of pairs separated by distance or lag h; Z() the measured sample at a point and Z( + h) the measured sample at point ( + h). The experimental variogram is the plot of observed values used to explore the spatial structure of the data. In contrast, the model that best fits the data is known as a theoretical or model variogram. In this study, Circular and Powered Exponential variograms were fitted to the experimental variogram.
Probabilistic estimation of variogram parameters of geotechnical properties with a trend based on Bayesian inference using Markov chain Monte Carlo simulation
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2021
Jiabao Xu, Lulu Zhang, Jinhui Li, Zijun Cao, Haoqing Yang, Xiangyu Chen
The variogram is the basis of geostatistics, which can be used to estimate properties in kriging and simulate spatial fields in geostatistical simulation. Conventionally, when the variable is stationary, the variogram is obtained from a set of measurements in two stages. In the first stage, experimental variogram values with different lags are calculated based on measurements by method of moments (Journel 1986):where is the experimental variogram, N(h) is the number of pairs of measurements separated by lag h, z(xi) is a measurement of the random spatial variable Z(x). The second stage is to fit a theoretical variogram model to those experimental variogram values. The common forms of theoretical variogram models include linear, exponential, spherical, Gaussian variogram model, etc. A specific form of variogram model is often arbitrarily chosen in term of the goodness of fitting, which constrains the choice of variogram model.
Prediction for Big Data Through Kriging: Small Sequential and One-Shot Designs
Published in American Journal of Mathematical and Management Sciences, 2020
Jack P. C. Kleijnen, Wim C. M. van Beers
Kriging or Gaussian process (GP) modeling is a popular method for the interpolation of input/output (I/O) data; these inputs are also called explanatory or independent variables, and these outputs are also called responses or dependent variables. Kriging is applied in many scientific disciplines; e.g., operations research/management science (OR/MS), engineering, machine learning, and geostatistics. Each discipline uses its own terminology and symbols; we use the terminology and symbols in Kleijnen (2015) to explain the basics of Kriging. We assume that the I/O data are noise-free (as in deterministic simulation or “computer experiments”) and the Kriging model is valid, so Kriging is an exact interpolator; i.e., the Kriging predictions are exactly equal to the previously observed “old” outputs. Furthermore, we assume that the primary goal of Kriging is to predict the output for a ‘“new” input combination or “point”; related goals may be validation, sensitivity analysis, optimization, and uncertainty analysis, as discussed in Kleijnen (2017).