# Variogram model matlab torrent

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Smallworld - Commercial GIS. Creates clusters and creates flares for clusters. Geomanjas - Open source development software for web-based and cloud based GIS applications. It provides numerious features like smooth zooming and panning, fully-customizable styling, markers, labels and tooltips. OpenLayers - High-performance, feature-packed library for creating interactive maps on the web. TerriaJS - A library for building rich, web-based geospatial data explorers. It features a pluggable architecture for processes and data encodings.

The implementation is based on the current OpenGIS specification: r7. Its focus was the creation of an extensible framework to provide algorithms for generalization on the web. Deegree - Open source software for spatial data infrastructures and the geospatial web.

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Weka - Weka is a collection of machine learning algorithms for data mining tasks written in Java. It is based on the 6S radiative transfer model but it runs x faster with minimal additional error i. MASON - MASON is a fast discrete-event multiagent simulation library core in Java, designed to be the foundation for large custom-purpose Java simulations, and also to provide more than enough functionality for many lightweight simulation needs.

NetLogo - NetLogo is a multi-agent programmable modeling environment. Repast - The Repast Suite is a family of advanced, free, and open source agent-based modeling and simulation platforms. SpaDES - Metapackage for implementing a variety of event-based models, with a focus on spatially explicit models. These include raster-based, event-based, and agent-based models. NLMR - R package to simulate neutral landscape models.

PyLandStats - An open-source Pythonic library to compute landscape metrics. It provides decision support to a range of conservation planning problems, including the design of new reserve systems, reporting on the performance of existing reserve systems, and developing multiple-use zoning plans for natural resource management.

Zonation - Zonation produces a hierarchical prioritization of the landscape based on the occurrence levels of biodiversity features in sites cells by iteratively removing the least valuable remaining cell while accounting for connectivity and generalized complementarity. GeographicLib - For solving geodesic problems. TerraMA2 - A free and open source computational platform for early warning systems.

Carto - Cloud computing platform that provides GIS and web mapping tools for display in a web browser. GIS Cloud - Real-time mapping platform for the entire workflow of your organization. Linq - Business process mapping. Mapbox - Plataform for web map design and manipulation. Customize maps, upload or create own geodata and publish online. OpenMapTiles - Vector tiles and map services as service, self-hosted or off-line. Google Maps - Google map service. Microsoft Bing Maps - Microsoft map service.

It is a part of the OpenMMLab project. PixelLib - Pixellib is a library for performing segmentation of images. It suports both Semantic Segmentation as Instance Segmentation. Raster Vision - An open source framework for deep learning on satellite and aerial imagery. TorchGeo - TorchGeo is a PyTorch domain library, similar to torchvision, that provides datasets, transforms, samplers, and pre-trained models specific to geospatial data.

WaterNet - A convolutional neural network that identifies water in satellite images. Python aiocogeo - Asynchronous cogeotiff reader. Alpha Shape Toolbox - Toolbox for constructing alpha shapes. AnaFlow - A python-package containing analytical solutions for the groundwater flow equation. Cartopy - A library providing cartographic tools for python for plotting spatial data. Centroids - This application reads a valid geojson FeatureCollection and returns a valid geojson FeatureColleciton of centroids.

Descartes - Plot geometries in matplotlib. EarthPy - A package built to support working with spatial data using open source python. EODAG - Command line tool and a plugin-oriented Python framework for searching, aggregating results and downloading remote sensed images while offering a unified API for data access regardless of the data provider. FreeType - For converting font glyphs to polygons. GemGIS - Python-based, open-source geographic information processing library.

GeoPandas - Python tools for geographic data. GSTools - A geostatistical toolbox: random fields, variogram estimation, covariance models, kriging and much more. Shapely - Manipulation and analysis of geometric objects in the Cartesian plane. PyKrige - Kriging Toolkit for Python. PySAL - For all your spatial econometrics needs. Rasterstats - Python module for summarizing geospatial raster datasets based on vector geometries.

GeoDjango - Django geographic web framework. This is a port of Mapbox's geojson-area for Python. Landsat-util - Landsat-util is a command line utility that makes it easy to search, download, and process Landsat imagery. Mahotas-imread - Read images to numpy arrays. Mapchete - Mapchete processes raster and vector geodata in digestable chunks. Tile-based geodata processing.

NodeBox-opengl - For playing around with animations. Peartree - Peartree: A library for converting transit data into a directed graph for network analysis. Planet Movement - Python module enables the searching and processing of Planet imagery to highlight object movement between valid image pairs.

Initial development started in more formally announced in Used to calculate upstream contributing area, aspect, slope, and topographic wetness index. PyProj - For conversions between projections. PyShp - For reading and writing shapefiles. Python Geocoder - Simple and consistent geocoding library written in Python.

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S2P - Satellite Stereo Pipeline - S2P is a Python library and command line tool that implements a stereo pipeline which produces elevation models from images taken by high resolution satellites. Scikit-image - Scikit-image is a collection of algorithms for image processing. Spectral Python - Python module for hyperspectral image processing. Statsmodels - Python module that allows users to explore data, estimate statistical models, and perform statistical tests.

Tobler - Tobler is a python package for areal interpolation, dasymetric mapping, and change of support. It allows feature extraction, dimension reduction and applications of machine learning techniques for geospatial data. Turfpy - This is Python library for performing geo spatial data analysis.

This is an python alternative for turf. Verde - Verde is a Python library for processing spatial data bathymetry, geophysics surveys, etc and interpolating it on regular grids i. Perl address formatting - Templates to format geographic addresses.

Geonetwork - GeoNetwork is a catalog application to manage spatially referenced resources. GeoServer - GeoServer is open source server for sharing geospatial data. Geotools - GeoTools is an open source Java library that provides tools for geospatial data. It can also recombine tiles to work with regular WMS clients. JGeocoder - Free Java Geocoder. LuciadLightspeed - A Java library that provides the foundations for advanced geospatial analytics applications MapFish Print - The purpose of Mapfish Print is to create reports that contain maps and map related components within them.

Openmap - Open Source JavaBeans-based programmer's toolkit. Photon - Photon is an open source geocoder built for OpenStreetMap data. It is based on elasticsearch. Proj4j - Java port of the Proj. Clojure geo - Clojure library for working with geohashes, polygons, and other world geometry. AzureMapsRestServices -. Net Standard 2. Easily track vehicles and mobile devices. BruTile - BruTile is a. Live demo DotSpatial - DotSpatial is a geographic information system library written for. NET 4. Earth-Lens - Earth Lens, a Microsoft Garage project is an iOS iPad application that helps people and organizations quickly identify and classify objects in aerial imagery through the power of machine learning.

Geo - A geospatial library for. Net -. Net de serializers. NET platform. Sanchez - False-colour geostationary satellite image compositor. SharpMap - SharpMap is an easy-to-use mapping library for use in web and desktop applications.

Capaware - 3D terrain representation with multilayer representation. Halide - Halide is a programming language designed to make it easier to write high-performance image processing code on modern machines. ITK - ITK is an open-source, cross-platform system that provides developers with an extensive suite of software tools for image analysis. OpenOrienteering Mapper - OpenOrienteering Mapper is a software for creating maps for the orienteering sport. S2 Geometry - Computational geometry and spatial indexing on the sphere.

C Datamaps - This is a tool for indexing large lists of geographic points or lines and dynamically generating map tiles from the index for display. H3 - Hexagonal hierarchical geospatial indexing system. Powered by statistical NLP and open geo data. Shapefile C Library - Provides the ability to write simple C programs for reading, writing and updating to a limited extent.

Draw2D - 2D rendering for different output raster, pdf. GoSpatial - GoSpatial is a simple command-line interface program for manipulating geospatial data. S2 - S2 is a library for spherical geometry that aims to have the same robustness, flexibility, and performance as the best planar geometry libraries. Martin is written in Rust using Actix web framework. WhiteboxTools - An advanced geospatial data analysis platform.

Evapotranspiration - Ruby library for calculating reference crop evapotranspiration ETo. Rgeo - RGeo is a geospatial data library for Ruby. Lua geo. It supports cellular automata, agent-based models, and network models running in 2D cell spaces. It defines basic geometric types and primitives, and it implements some geometric data structures and algorithms. Naqsha - Naqsha is a Haskell library to work with geospatial data types. TerraHS - TerraHS is a software component that enables the development of geographical applications in a functional language, using the data handling capabilities and spatial operations of TerraLib.

Elixir distance - Provides a set of distance functions for use in GIS or graphic applications. Geometry Library - A Geometry library for Elixir that calculates spatial relationships between two geometries. Swift Apple MapKit - Display map or satellite imagery directly from your app's interface, call out points of interest, and determine placemark information for map coordinates.

Stac4s - a scala library with primitives to build applications using the SpatioTemporal Asset Catalogs specification. Delphi DSpatial - DSpatial is an Open Source software development project to provide developers using Delphi with a library of tools for the use, manipulation, and visualization of spatial data. Cmask - This tool called Cmask Cirrus cloud mask is used for cirrus cloud detection in Landsat 8 imagery using a time series of data from the Cirrus Band 1.

MFmask - Automated cloud and cloud shadow detection for Landsats images. It is extended by Dispersal. JuliaGIS - A package for the visualization and manipulation of geographic data. To proceed from the general to the specific is mathematically elegant but more appropriate for advanced texts, because it requires some degree of familiarity with the methods.

For an introductory textbook, particularly on a subject so foreign to the intended audience, my experience has taught me that the only approach that works is to proceed from the simple and specific to the more complex and general. The same concepts are discussed several times, every time digging a bit deeper into their meaning. Because statistical methods rely to a great extent on logical arguments it is particularly important to study the book from the beginning.

Although this book may appear to be full of equations, it is not mathematically difficult provided again that one starts from the beginning and becomes familiar with the notation. The book is intended for a one-semester course for graduate-level engineers and geophysicists and also can be used for self-study. The material is limited to linear estimation methods: That is, we presume that the only statistics available are mean values and covariances.

I cannot overemphasize the point that the book was never meant to be a comprehensive review of available methods or an assessment of the state of the art. Therefore, one must make the best use of available data to estimate the needed parameters. For example, a large number of measurements are collected in the characterization of a hazardous-waste site: water-surface level in wells, transmissivity and storativity from well tests , conductivity from permeameter tests or borehole flowmeters, chemical concentrations measured from water and soil samples, soil gas surveys, and others.

However, because most subsurface environments are complex, even a plethora of data is not sufficient to resolve with accuracy the distribution of the properties that govern the rates offlow,the rates of transport and transformation of chemicals, or the distribution of concen- trations in the water and the soil.

The professionals who analyze the data must fill in the gaps using their understanding of the geologic environment and of the flow, transport, or fate mechanisms that govern the distribution of chemicals. However, process understanding is itself incomplete and cannot produce a unique or precise answer. Statistical estimation methods complement process understanding and can bring one closer to an answer that is useful in making rational decisions.

Their main contribution is that they suggest how to weigh the data to compute best estimates and error bounds on these estimates. Statistics has been aptly described as a guide to the unknown; it is an approach for utilizing observations to make inferences about an unmeasured quantity. Rather than the application of cookbook procedures, statistics is a rational methodology to solve practical problems. The purpose of this book is to provide some insights into this methodology while describing tools useful in solving estimation problems encountered in practice.

Two examples of such problems are: point estimation and averaging. For example, consider measurements of concentration from the chemical analysis of soil samples from borings. The question is how to estimate the concentration at the many other locations where soil samples are unavailable. Another example is drawing lines of constant transmissivity in other words, contour lines of transmissivity from the results of pumping tests at a number of nonuniformly spaced wells.

Drawing a contour map is equivalent to interpolating the values of the transmissivity on a fine mesh. Examples can be found in references [54, 55, , , , , , and 15]. In averaging one uses point estimates of concentration to determine the aver- age concentration over a volume of soil; this estimate is needed for evaluation of the total mass of contaminants.

Another example, drawn from surface hy- drology, is the estimation of mean areal precipitation over a watershed from measurements of rainfall at a number of rain gages. Due to complexity in the spatial variability of the variables involved, one can- not obtain exact or error-free estimates of the unknowns.

Statistical methods weigh the evidence to compute best estimates as well as error bars that describe the potential magnitude of the estimation error. Error bars, or information about how much faith to put in the estimates, are essential in making engineering de- cisions. Statistical methods are applied with increased frequency to evaluate compliance with regulatory requirements because the best one can do is to pro- vide a reasonable degree of assurance that certain criteria have been met.

Also, using statistics one can anticipate the impact of additional measurements on error reduction before the measurements are taken. Thus, statistics is useful in deciding whether the present data base is adequate for detecting all important sources of contamination and, if not, where to collect the additional measure- ments so that the objectives of monitoring such as demonstrating regulatory compliance are met in the most cost-effective way.

Once one masters the application of the statistical methodology to relatively simple problems, such as those above, one can tackle more complicated prob- lems such as estimating one variable from measurements of another. It is often convenient to use a variable that can be easily observed or computed to estimate another variable that is difficult or expensive to measure. For examples, a land topography may be used to estimate the phreatic surface elevation of a surficial aquifer; b overburden and aquifer thickness may correlate and can be used to estimate the transmissivity of a confined permeable layer; and c hydraulic head measurements may provide information about the transmissivity and vice versa.

This book deals with relatively simple applications, but the same general methodology applies to complicated problems as well. In fact, the power of the methods to be described becomes most useful when utilizing measure- ments of different types, combining these with deterministic flow and transport models, and incorporating geological information to achieve the best charac- terization possible. However, one is advised not to attempt to solve complex problems before developing a sound understanding of statistical techniques, which is obtained only through practice starting with simpler problems.

Statis- tical methods are sometimes misapplied because professionals who use them have received no training and apply them without awareness of the implicit assumptions or a firm grasp of the meaning of the results. It has been said that they are often used the way a drunk uses a lamppost: for support rather than illumination. Blindly following methods one does not understand provides countless opportunities for misapplication.

To understand all the details, some readers will find it useful to go through the review of basic probability theory presented in Appendix A. Well tests were conducted at eight wells screened in a confined aquifer pro- viding values of the transmissivity. The location of the wells on plan view is shown as o in Figure 1.

The values are given in Table 1. The question is: Given the information currently available, if a well were drilled at another location indicated by an x in Figure 1. Table 1. Assume also that the driller's logs indicate that all wells were drilled in the same formation and geologic environment. There are good reasons to believe that the formation is a confined aquifer bounded by nearly impermeable layers above and below. Beyond that, however, and despite considerable geological information available at the site, the variability among transmissivity measure- ments cannot be explained in terms of other measurable quantities in a manner useful in extrapolation to unmeasured locations.

If we actually admit that we cannot explain this variability, how can we extrapolate from the sample of the eight observations to the unknown value? The point is that, because we cannot come up with a deterministic mechanism that explains variability, we postulate a. The simplest approach is to compute the frequency distribution of the data and then to use it to describe the odds that the transmissivity at the location of interest will have a certain value.

The premise is that "each transmissivity observation is randomly and independently sampled from the same probability distribution. Of course, this experiment is only a convenient concept; this simple model is not meant to represent the physical reality of what transmissivity is or how it is measured, but rather, it constitutes a practical and reasonable way to use what we know in order to make predictions.

A simple example 5 We are still faced with the problem of estimating the probability distribution. We may approximate it with the experimental probability distribution i. Now we are able to make predictions. Such a model may appear crude, but it is a rational way to use experience as a guide to the unknown. In fact, this simple model is adequate for some practical applications. Also, the reader should not be left with the impression that the approach boils down to subjective judgement.

The questions of the validity or suitability of a model and of the sensitivity of the prediction to modeling assumptions can be addressed, but this is left for later in this book. In many applications, what is needed is to determine a good estimate of the unknown, TQ, and a measure of the error. An estimator is a procedure to compute 7b from the data. Even though we cannot foretell the actual error in the estimate, we can describe its probability distribution. It is common to measure the anticipated error by the quantity known as the mean square error, i.

Introduction Using Equation 1. This expression, Equation 1. In practice, the most useful class of estimators comprises best minimum- variance linear unbiased estimators affectionately known as BLUEs , which is the subject of this book. As already mentioned, for any estimator, the error is a random variable, i. If these errors are symmetrically distributed about a central value, as are those of Figure 1. However, in the case of transmissivities or hydraulic conductivities and concentrations, the histogram of these errors usually indicates that the distribution is not symmetric.

In this case, there is no unequivocal representative value. In addition to the mean, there are other possibilities, such as the value that minimizes the mean absolute error or the median the value exceeded by half the values. For this reason, the minimum-variance estimators are most suitable when we have reason to believe that the frequency distribution of the estimation errors may resemble the normal distribution. That is, instead of analyzing T we analyze Y. Examples will be seen in other chapters.

The weights in the linear estimator used in this example are equal; that is, the probability that the unknown is equal to a measured value is presumed to be the same no matter how far or in what direction the unknown is located from the location of the observation. Also, the locations of the other measurements have no effect in the selection of the weight.

In many situations, however, the transmissivity varies gradually in space in such a way that it is more likely that the unknown will resemble an observation near than an observation far from its location. Therefore, the weights should be nonuniform larger for nearby observations. This book describes methods that analyze data for clues on how to compute these weights in a way that reflects the spatial variability of the quantity of interest as well as the location of measurements and the unknown.

The word statistics plural means averages of numerical data, such as the batting average of a player or the median of a batch of hydraulic conductivity measurements. However, data can be misleading, when improperly analyzed and presented. The word statistics singular refers to a methodology for the organization, analysis, and presentation of data.

In particular, statistical modeling is an approach for fitting mathematical equations to data in order to predict the values of unknown quantities from measurements. Hydrogeologists and environmental and petroleum engineers, like scientists and engineers everywhere, use such methods on an almost daily basis so that some knowledge of statistics is essential today.

Basically, we are concerned with estimation problems in which the value of an unknown needs to be inferred from a set of data. The postulated model is probabilistic. Parameter fitting, model validation, and prediction involve computations of probability distributions or moments such as the mean, the variance, etc. These models must be reasonably simple or else the computations may be just too complicated for the approach to be of practical use. The computations in the methods presented in this book are reasonably simple and involve only mean values, variances, and correlation co- efficients.

However, there are even more important reasons for selecting simple probabilistic models, as will be discussed later. Conceptually, the part that novices in statistics have the most trouble under- standing is the selection of the empirical model, i. So let us say a few things on this subject. How do we know that we have the right model?

The truth is that one cannot and may not even need to prove that the postulated model is the right one, no matter how many the observations. There is nothing that anyone can do about this basic fact, which is not a limitation of statistics but to various degrees affects all sciences that rely on empirical data. In the example of Section 1. However, 1. It is best to approach empirical models from a utilitarian perspective and see them as a practical means to: 1.

Statistics Table 1. Porosity versus location depth rj 0. It is a reasonable approach, which should lead to rational decisions. A model should be judged on the basis of information that is available at the time when the model is constructed. Thus, a model that looks right with 10 mea- surements may be rejected in favor of another model when measurements have been collected. It will be seen that, in all cases, the simplest empirical model consistent with the data is likely to be best for estimation purposes a principle known as Occam's razor.

Furthermore, it will be seen that one of the most important practical contributions of statistical estimation methods is to highlight the fine distinction between fitting a model to the data and obtaining a model that we may trust to some degree for making predictions. Exercise 1. Describe an estimation problem with which you are familiar. Describe two examples, one from your everyday life and experience and one from scientific research e.

Outline the steps you follow in a systematic way. You may find it useful to review what is known as the scientific method and discuss its generality and relevance to everyday life. What is the significance of the standard error? Discuss the pros and cons of this simple model and whether it seems that this model is a reasonable description for this data set. Matheron [94 and 95] and his co-workers advanced an adaptation of such methods that is well suited to the solution of estimation problems involving quantities that vary in space.

Examples of such quantities are conductivity, hydraulic head, and solute concentration. This approach is known as the theory of regionalized variables or simply geostatistics. Popularized in mining engineering in the s, it is now used in all fields of earth science and engineering, particularly in the hydro- logic and environmental fields.

This book is an introductory text to geostatistical linear estimation methods. The geostatistical school has made important contributions to the linear esti- mation of spatial variables, including the popularizing of the variogram and the generalized covariance function. Geostatistics is well accepted among practi- tioners because it is a down-to-earth approach to solving problems encountered in practice using statistical concepts that were previously considered recondite.

The approach is described in books such as references [24, 36, 37, 70, 73, , and 76] with applications mainly in mining engineering, petroleum engineering, and geology. Articles on geostatistics in hydrology and hydrogeology include [7 and ] and chapters can be found in [13 and 41]. A book on spatial statis- tics is [30]. Software can be found in references [50,, and 43] and trends in research can be discerned in reference [44]. The approach presented in this book departs from that of the books cited earlier which, for the sake of convenience will be called "mining geostatistics" in consequential ways.

For the readers who are already familiar with mining geostatistics, here is a list of the most important differences: 1. The estimation of the variogram in mining geostatistics revolves around the experimental variogram; sometimes, the variogram is selected solely on the basis that it fits the experimental variogram.

This approach is simple to apply but unsatisfactory in most other aspects. In contrast, in the approach followed in this book, the variogram is selected so that it fits the data, i. Unlike mining geostatistics, which again relies on the experimental vari- ogram to select the geostatistical model, the approach preferred in this work is to apply an iterative three-step approach involving: 1.

Model validation is implemented differently and has a much more important role than in mining geostatistics. Key points of Chapter 1 11 3. Ordinary kriging, which describes spatial variability only through a vari- ogram and is the most popular method in mining geostatistics, can lead to large mean square errors of estimation.

In many environmental applica- tions, one may be able to develop better predictive models by judiciously describing some of the "more structured" or "large-scale" variability through drift functions. The error bars can be further reduced by making use of additional information, such as from the modeling of the processes. This additional information can be introduced in a number of ways, some of which will be seen in this book.

Statistics is a methodology for utilizing data and other information to make inferences about unmeasured quantities. Statistical methods complement deterministic process understanding to provide estimates and error bars that are useful in making engineering decisions. The methods in this book are an adaptation and extension of linear geostatistics. We perform such an exploratory analysis to: 1.

Graphical methods are useful to portray the distribution of the observations and their spatial structure. Many graphical methods are available and even more can be found and tailored to a specific application. The modest objective of this chapter is to review common tools of frequency analysis as well as the experimental variogram.

Exploratory analysis is really a precursor to statistical analysis. Although time consuming, labor intensive, and subject to human errors, one cannot deny that this process enhanced familiarity with data to the extent that the analyst could often discern patterns or spot "peculiar" measure- ments.

This intimacy with one's data might appear lost now, a casualty of the electronic transcription of data and the extensive use of statistical computer packages that perform the computations. However, data analysis and interpretation cannot be completely automated, particularly when making crucial modeling choices.

The analyst must use judg- ment and make decisions that require familiarity with the data, the site, and the questions that need to be answered. It takes effort to become familiar with data sets that are often voluminous and describe complex sites or processes. Measurements may vary over a wide range. In most cases it is impossible, and often useless, for any person to remember every single measurement in- dividually.

One may start by summarizing in a convenient way the behavior of measurements that act similarly and by pointing out the measurements that behave differently from the bulk of the data. What is the best way to organize and display the data? What are the measures that summarize the behavior of a bunch of data? And could it be that a certain data transformation can simplify the task of summarizing the average behavior of a data batch?

These are some of the issues to be discussed. However, the human eye can recognize patterns from graphical displays of the data. During exploratory data analysis one should make as few assumptions as possible. A model cannot be accepted on the basis of exploratory analysis only but should be corrobo- rated or tested.

To illustrate the objectives and usage of exploratory data analysis, consider the following data sets: 1. Measurements of transmissivity of a thin sand-and-gravel aquifer at 8 loca- tions see Table 1. Measurements of potentiometric head at 29 locations in a regional confined sandstone aquifer see Table 2.

Figure 2. Measurements at 56 locations of concentration of trichloroethylene TCE in groundwater on a transect in a fine-sand surficial aquifer see Table 2. Table 2. Head observations in a regional confined aquifer Head ft X y Head ft X y 6. Experimental distribution 15 Table 2. We call this distribution "experimental" or "empirical" because it depends only on the data.

We will also review numbers that represent impor- tant characteristics of a data set, such as central value, spread, and degree of asymmetry. The intervals are usually of equal length and selected so that the histogram is relatively free from abrupt ups and downs. See, for example, Figures 2. The histogram is probably the most widely recognized way to portray a data distribution. However, it has a potentially serious drawback: The visual impression conveyed may depend critically on the choice of the intervals.

From Figure 2. For this reason, in many applications the box plot which we will see later is a better way to represent in a concise yet informative! Experimental distribution 17 10 8 E Head ft Figure 2. Typically, we obtain an "S"'-type curve. See Figures 2. Note that pi is a number that increases from 0 to 1 and represents the ratio of data that are smaller than or equal to the z,- value.

Note that the 0. The technical question of which choice is best is of little consequence in our work. Experimental distribution 19 1 I 0. A related concept is that of the quantile a term that has a similar meaning to that of percentile. The p quantile where p is a number between 0 and 1 is defined as the number Q p that exceeds p percent of the data. For other values of p, we interpolate linearly. These are known as summary statistics. I Representative value First, we are interested in a number that can be treated as a "typical" or "central" value, i.

The most common such statistics are defined below. Another number that can serve as typical value is the mode, defined as the value where the histogram seems to peak. Many observations cluster near the mode. If well defined and unique, the mode can be used as a typical value. However, a histogram may exhibit many modes multimodal histogram or may be too flat for a clear mode to appear.

Each of the three measures may give significantly different results if the distribution is highly asymmetric, as illustrated in Figure 2. However, if the distribution is unimodal and nearly symmetric, the three of them give practically the same result. Another measure of spread is the interquartile range, Iq also known as the Q-spread. Note that 2 0. An advan- tage of the interquartile range is that it is less sensitive to a few extreme values than the standard deviation.

For this reason, the Q -spread is preferred in ex- ploratory analysis whereas the standard deviation is used when the data follow an approximately normal distribution. Thus, it is useful to establish the degree of symmetry. A useful measure in evaluating symmetry is the skewness coefficient which is a dimensionless number. A symmetric distribution has ks zero; if the data contain many values slightly smaller than the mean and a few values much larger than the mean like the TCE concentration data , the coefficient of skewness is positive; if there are many values slightly larger and a few values much smaller than the mean, the coefficient of skewness is negative.

The statistics that summarize the important characteristics of the data are presented in Tables 2. Summary statistics for head data of Table 2. Summary statistics for concentration data of Table 2 2 Number of observations 56 Minimum value 2. The upper and lower quartiles of the data define the top and bottom of a rectangle the "box" , and the median is portrayed by a horizontal line segment inside the box.

From the upper and lower sides of the rectangle, dashed lines extend to the so-called adjacent values or fences. The upper adjacent value is the largest observed value provided that the length of the dashed line is smaller than 1. Exactly the same procedure is followed for the lower adjacent value. Observations outside of the range between the adjacent values are known as outside values. You must realize that there is nothing magical about the number 1.

It is a useful convention one of many in statistics and is to some extent arbitrary. The essence of the criterion is that for normally distributed data, the probability of a measurement being outside of the thus defined fences is very small. The box plot is a graph of the key characteristics of the data distribution. The line inside the box location of the median represents the center of the batch.

The size of the box represents the spread about the central value. One may judge whether the data are distributed symmetrically by checking whether the median is centrally located within the box and the dashed lines are of approximately the same length. The lengths of the dashed lines show how stretched the tails of the histogram are. Finally, the circles or asterisks indicate the presence of outside values.

A single number can be used to represent the central value in the batch, because the mode, median, and arithmetic mean are practically the same. Furthermore, for a histogram resembling the one in Figure 2. For such a batch, the mean and the standard deviation provide enough information to reconstruct the histogram with acceptable accuracy. One particular type of "data" is residuals, i. Transformations and outliers 25 are computed and why they are important. For now, we mention that the com- monly used linear estimation methods make most sense when the distribution of the residuals is approximately normal.

For a normal distribution, the mean is the indisputable central value and the standard deviation is the indisputable measure of spread. That is why setting the mean of the residuals to zero and minimizing the standard deviation of the errors which is what linear estimation methods do is clearly a good way to make predictions. As illustrated by the concentration observations of Table 2. Given the advantages of nearly normal batches, it is reasonable to search for a simple transformation that makes it possible to describe a distribution with a mean and a variance.

An application of the logarithm transformation is found in reference []. For example, Figure 2. The distribution of the transformed data is much easier to describe than the distribution of the original data. Thus, one can summarize the important characteristics of a data set through the parameter K and the mean and variance of the transformed data. An important point remains to be made. The basic assumption in the type of methods that we will see later is that the estimation errors are approxi- mately normally distributed.

Thus, the purpose of the transformation is to adjust 14 Later, we will test the box plot residuals to ascertain whether they are nearly normally distributed. The box plot of the concentration measurements Figure 2. All this means, however, is that it would be unwise at this preliminary stage to lump those "unusual" values with the other data and then to compute their mean and standard deviation.

The mean and the variance would be excessively affected by the relatively few outside values and would not be representative of either the bulk of the data or the outside values. However, it would be equally unwise to brand uncritically these unusual measurements as "outliers," "stragglers," or "stray" and to discard them. If the box plot indicates that the data are distributed in a highly asymmetric way or that the tails are stretched out, it is possible that after a transformation all or some of the values that were originally outside will get inside.

In this case, the practical issue is to find a good way to describe the asymmetric distribution of data, such as by transforming to a more symmetric distribution. In many cases, including the concentration data of Table 2. Before deciding what to do with an outside value, one must return to the source of the data and use one's understanding of the physical processes in- volved. A reasonable effort should be made to verify that the measurement was taken, interpreted, and transcribed correctly.

Errors in interpretation and copying of data are unfortunately all too common in practice. If it is concluded that the measurement is not to be trusted, it should be discarded. In some cases, one may decide to divide the data into two or more data sets, such as when stray transmissivity measurements correspond to a geologic environment different from that of the rest of the data. Common sense is often the best guide. Spatial structure 27 One approach involves applying statistical hypothesis tests some of which will be discussed elsewhere.

There are general tests that can be applied to any distribution, such as the chi-square and the Kolmogorov-Smirnov tests, and there are tests that have been developed specifically for the normal distribution, such as variations of the Shapiro-Wilks test []. Each of these tests provides a procedure to determine whether the data depart sufficiently from the null hypothesis that the observations were sampled independently from a normal distribution.

The key limitation of these tests in exploratory analysis is that they assume that the data were independently generated, i. However, as will be seen later, correlation is usually important, such as when two measurements tend to differ by less as the distance between their locations decreases. In this case, a statistical test that assumes independence is not appropriate.

Actually, these tests should be applied to orthonormal residuals, which are differences between observations and predictions. These residuals are supposed to be uncorrelated and to follow a normal distribution with zero mean and unit variance.

The next two chapters describe how to compute these residuals and how to apply such normality tests. What about de- scribing how measurements vary in space or are related based on their location? For example: Are neighboring measurements more likely to be similar in value than distant ones? Do observed values increase in a certain direction? Data, such as those of Tables 1. First, a word of caution is warranted. As an intermediate step, these packages interpolate from the measurements to a fine mesh, which is needed for contouring.

If the observations are few, are nonuni- formly distributed, and have highly skewed distributions, then the plot obtained is affected by the interpolation and smoothing routines used in the program, not just by the data. Important features may be smoothed out; puzzling results may appear in areas with few measurements as artifacts of the algorithms used.

One may still find it useful to apply such packages for exploratory data visualization but not to produce final results. Even for variables that depend on two or three spatial dimensions, one may start with x-y plots of the observations against each spatial coordinate. For example, see Figures 2. For example, such a plot in the case of the head data, Figure 2.

The idea is that the three plots represent views from the top, the front, and the side of points in a rectangular parallelepiped that contains the measurement locations. With careful choice of symbols and considerable effort, one may see some patterns. For example, for the data plotted in Figure 2. Spatial structure 29 2 4 6 8 10 12 Figure 2. Three-dimensional graphics are gradually becoming available with perspec- tive, movement, and shading for better visualization.

Data are represented as spheres or clouds of variable size or color indicative of magnitude of the observation. Consider the case of n measurements z xi , z x2 , The bold letter x stands for the array of coordinates of the point where these measurements were taken. The experimental variogram is a smooth line through this scatter plot. In the common method of plotting the experimental variogram, the axis of separation distance is divided into consecutive intervals, similarly as for the histogram.

Take hk equal to the average value, 2. Modifications to this basic approach have been proposed to improve its robustness [5, 31, 30, 37, ]. In selecting the length of an interval, keep in mind that by increasing the length of the interval you average over more points, thus decreasing the fluc- tuations of the raw variogram, but you may smooth out the curvature of the variogram.

It is unprofitable to spend too much time at this stage fiddling with the intervals because there is really no "best" experimental variogram. Some useful guidelines to obtain a reasonable experimental variogram are: y h 4 Figure 2. Use three to six intervals. Include more pairs use longer intervals at distances where the raw vari- ogram is spread out.

As an exploratory analysis tool, the experimental variogram has the drawback that the graph depends on the selected intervals. It may also be somewhat affected by the method of averaging. For example, some analysts prefer to use for hjc the median value while others prefer to use the midpoint of the interval, i. The experimental variogram presented above is a measure of spatial correla- tion independent of orientation. In some cases, however, better predictions can be made by taking into account the anisotropy in the structure of the unknown function; for example, conductivities in a layered medium are more correlated in a horizontal direction than in the vertical.

The variogram should then depend on the orientation as well as the separation distance anisotropic model. The issue of anisotropy will be discussed in Chapter 5. Some readers may prefer to skip this section at first reading and come back to it after Chapter 3 or 4. To grasp the concept of scale, consider the function z that varies over a one- dimensional domain x. Without getting too technical and disregarding pathological cases, practically any function can be sufficiently approximated over a finite domain by a few terms of a trigonometric, or Fourier, series.

The bottom line is that the experimental variogram contains information about the Aj values. That is, one can infer the approximate value of some of the Aj values. Thus, the experimental variogram can provide clues on whether the scale of the variability is large or small or what is known as the "power spectrum of the function". However, the experimental variogram provides no real information about the phase shifts, i. Two functions z that have the same variogram may look radically different because of different phase shifts.

Also, the computation of the variogram scrambles or masks patterns in the data, such as clear trends, which might be easy to recognize from other plots. We have not attempted to prove these assertions mathematically because that would involve knowledge that cannot be assumed at this point.

In other chapters, however, we will obtain insights that will support the statement that the experimental variogram is basically a way to infer the distribution of spatial variability with respect to spatial scales. We need to emphasize that although it is an important exploratory analysis tool, the experimental variogram should not monopolize the analysis.

It should be used in conjunction with other analysis tools, such as those presented in Section 2. This depends on the behavior of the experimental variogram near the origin, i. This depends on the behavior of the experimental variogram at large distances. We will consider three examples, which are intended to give you an intuitive feeling about what we mean by continuity and smoothness in a practical context. Consider now that zi x is sampled at locations randomly distributed in the interval between 0 and 1.

The same sampling locations will be used in all three examples. Note that the average sampling interval i. As a result, two adjacent measurements are about as different as two distant measurements. Meaning of experimental variogram 35 z, data 0. The experimental variogram, shown in Figure 2. Because the experimental variogram does not seem to converge to zero as the separation decreases, we say that there is a discontinuity of the experimental variogram at the origin or a nugget effect.

In general, a discontinuity at the origin in the experimental variogram is indicative of fluctuations at a scale smaller than the sampling interval, called microvariability. It may also be due to random observation error, as we will discuss further elsewhere. The changes in measured values are so gradual that both z and its slope are observed to vary continuously. The experimental variogram, shown on Figure 2.

Generally, parabolic behavior near the origin is indicative of a quantity that is smooth at the scale of the measurements so that it is differentiable i. Note that this variable has most of its variability at a scale larger than the average sampling interval but also some variability at a scale comparable to that of the measurement spacing.

The changes in the value of z3 between adjacent sampling points are gradual, as shown on Figure 2. Data z2 0. However, the slope changes rather abruptly between adjacent intervals, as seen in Figure 2. An example of a function that is continuous but not differentiable is the path of a small particle in a fluid known as "Brownian motion". The particle gets hit by molecules so often that it constantly changes direction and speed.

Thus, although the particle trajectory is continuous, the particle speed may change instantaneously and is thus not continuous. In summary, we have seen that the behavior of the experimental variogram at the origin at short distances reveals the degree of smoothness of the function.

We distinguished among parabolic behavior, which characterizes a smoothly changing variable with continuous slope; linear behavior, which characterizes a continuous variable without continuous derivatives such as a Brownian mo- tion ; and discontinuous behavior, which characterizes a discontinuous variable such as random "noise".

Meaning of experimental variogram 39 y h 0. We will later give a technical meaning to the term stationary; intuitively, a function is stationary if it consists of small-scale fluctuations compared to the size of the domain about some well- defined mean value. For such a function, the experimental variogram should stabilize around a value, called the sill, as shown in Figure 2.

For a stationary function, the length scale at which the sill is obtained describes the scale at which two measurements of the variable become practically uncorrelated. This length scale is known as range or correlation length. Otherwise, the variogram keeps on increasing even at a distance comparable to the maximum separation distance of interest, as shown in Figure 2.

Exercise 2. Plot these two functions and discuss, based on what you have read in this chapter, how the experimental variograms of these two functions are expected to differ. Based on this example, discuss the strengths and limitations of the experimental variogram as an exploratory analysis tool. The analyst should keep an open mind and avoid techniques that may be misleading if certain assumptions are not met. We start by analyzing the distribution of data independently of their location in space; this distribution may be portrayed using the histogram, the ogive, and the box plot.

Important summary statistics are the median and the mean, the interquartile range and the standard deviation, and the skewness coefficient. We discussed the practical advantages of working with symmetric and nearly normal distributions and how transformations can be used to achieve this goal. Spatial variability can be analyzed using graphical techniques, but the difficulty increases significantly from variability in one dimension to variability in three dimensions.

The experimental variogram is an important tool that provides information about the distribution of spatial variability with respect to scales. Finally, note that conclusions reached during an exploratory analysis are usually tentative. The next step is to use the ideas created during exploratory analysis to select tentatively an "equation to fit to the data. This chapter introduces kriging, which is a method for evaluating estimates and mean square estimation errors from the data, for a given variogram.

The discussion in this chapter is limited to isotropic correlation structures same correlation in all directions and focuses on the methodology and the basic mathematical tools. Variogram selection and fitting will be discussed in the next chapter. To estimate the value of the porosity at any location from the measured porosity values, we need a mathematical expression or "equation" or "model" that describes how the porosity varies with depth in the borehole.

In other words, we need a model of spatial variability. However, hydrologic and environmental variables change from location to location in complex and inadequately understood ways. In most applications, we have to rely on the data to guide us in developing an empirical model.

The model involves the concept of probability in the sense that spatial variability is described coarsely by using averages. For example, the best we can do might be to specify that the porosity fluctuates about some mean value and to come up with a formula to correlate the fluctuations at two locations depending on their separation distance. This is often the most practical scheme to summarize incomplete information or erratic data.

For brevity, the notation z x will be used to include all three cases, where x is the location index a vector with one, two, or three components. We are now ready to discuss the logical underpinnings of the approach. If statistical modeling is new to you and you wonder what it means, pay particular attention to this part. In practice, our objective is to estimate a field variable z x over a region. Usually, because of scarcity of information, we cannot find a unique solution.

It is useful to think of the actual unknown z x as one out of a collection or ensemble of possibilities z x; 1 , z x; 2 ,. This ensemble defines all possible solutions to our estimation problem. The members of the ensemble are known as realizations or sample functions. Consider, for example, Figures 3. Each figure contains five realizations from a different ensemble family of functions.

Notice that despite the differences among the realizations in each figure, they share some general structural characteristics. The functions in Figure 3. The functions in Figures 3. The curves in Figures 3. Assume for argument's sake that we have selected an ensemble and that we have computed the probability that a realization is the actual unknown, i. The ensemble of realizations with their assigned probabilities defines what is known as a random function or random field or spatial stochastic process.

Expectation, denoted by the symbol E, is the process of computing a probability-weighted average over the ensemble. In linear estimation, we use the first two statistical moments of the random field, which are 1. For example, the degree of smooth- ness of the realizations or the scale of the fluctuations can be described nicely. In a general sense, we use the first two moments instead of the probabilities Pu Pi, Thus, the model of spatial structure consists of the mathematical expressions chosen to describe the mean function and the covariance function.

It is important to grasp the meaning of the term structure in the context of estimation of spatial functions: It is the information that characterizes the ensemble of plausible solutions that our unknown function belongs to! Structural analysis is the selection and fitting of mathematical expressions for the required first two moments of the regionalized variable. The form of these expressions comprises the model.

Many expressions can be used to represent these moments. Some are commonly used general-purpose models, such as the intrinsic model, which we will see in this chapter; others are special-purpose models developed by you, the user, for a specific application. A model is selected based on the analysis of data and other information, including experience with data at similar sites and geologic and hydrologic information.

From those, the analyst must decide whether the unknown function belongs, for example, in the ensemble of Figure 3. Best linear unbiased estimation deals with taking into account specific ob- servations. Specifically, we look for estimates that are as representative and accurate as possible, using the model developed during structural analysis and the specific observations.

The basic idea is that we proceed to figure out an unknown function e. During the first stage, structural analysis, the choice is narrowed down to the functions sharing certain characteristics, collectively known as structure. During the second stage, the choice is narrowed down further by requiring that all possible solutions honor the data. These ideas will become clearer after we study some examples. In this chap- ter, after Section 3.

Plot the covariance function. Conditional on this information: What are the possible solutions for z x and their probabilities? What is the mean function? What is the covariance function? The answers are given below: a The random function z x is fully defined because we have specified the way to generate all possible realizations or sample functions and the cor- responding probability.

For example, to generate a set of M equally likely realizations, generate M variates uniformly distributed between 0 and 2n see Appendix C and apply these in Equation 3. Intuitively, we expect that the mean will be zero. Thus, the mean is the same everywhere, 0. Note that the mean function is much simpler than any of the realizations. See 2' Figure 3. This picks five numbers between 0 and In. See Figure 3. Note that the conditional mean function is more com- plex than the prior mean, which is zero everywhere.

The conditional variance, Rc x, x , is plotted in Figure 3. Figure 3. Intrinsic isotropic model 51 Exercise 3. Plot results, if possible. One of the simplest models is: The mean is constant and the two-point covariance function depends only on the distance between the two points.

Equations 3. In this chapter we will focus on isotropic models; we will see anisotropic models in other chapters. Exercise 3. Mathematically speaking, there are several types of functions that satisfy Equations 3. The covariance function may be periodic such as a cosine , aperiodic but consisting of a number of sinusoids, or a function that is none of the above but has a continuous power spectrum from Fourier analysis and finite variance.

In estimation applications, it is the last type that is of interest so that, unless otherwise stated, we will assume that we deal with this type. Then, it is possible to extrapolate from the locations of the observations. In most cases, the mean is not known beforehand but needs to be inferred from the data; to avoid this trouble, it may be more convenient to work with the variogram.

Since we only use y h f we will call it the variogram. To underline the distinction between the experimental variogram, which is com- puted from the data, with the variogram, which is a mathematical expression, the latter is sometimes called the theoretical variogram.

At first, the reader may be unable to distinguish the intrinsic model from the stationary one. The difference is slight but important. Note, to begin with, that the stationary and intrinsic models differ in the parameters needed to char- acterize them as well as in mathematical generality. It takes less information to characterize the intrinsic model than the stationary model. Whereas both assume constant mean, in the intrinsic model we avoid ever using a numerical value for the mean.

Furthermore, the stationary model may use the covariance function, which cannot be reconstructed only from the variogram over a dis- tance smaller than the range. In a sense, to specify the covariance function one needs the variogram plus an extra number, the variance.

The intrinsic model is mathematically more inclusive i. If z x is stationary, then it is also intrinsic because Equation 3. However, the important point is that not all intrinsic functions are stationary. As a practical rule, an intrinsic function is nonstationary if its variogram tends to infinity as h tends to infinity. It is the intrinsic model that we will use in the remainder of this chapter.

In- variably, the question asked is: What is the practical significance and meaning of this model? The practical significance is that it is a simple model that is useful in: 1. The reason is rather mundane: The variance of a linear combination of values of z x at a number of points can be expressed in terms of covariance functions or variograms see Exercise 3.

Criteria are discussed in references [21 and 30]. In practice, variograms describing the spatial structure of a function are formed by combining a small number of simple mathematically acceptable expressions or models. This section contains a list of such covariance functions R and variograms y. Plots of some of them and sample functions are also presented. Symbol h stands for the distance between two points.

Because the covariance function decays asymptotically, the range a is defined in practice as the distance at which the correlation is 0. Common models 55 The Gaussian model is the only covariance in this list with parabolic behavior at the origin y h oc h2 for small h, where oc stands for "proportional to" , indicating that it represents a regionalized variable that is smooth enough to be differentiable i. This model is popular particularly in hydrologic applications.

Notice the difference from the Gaussian case in the smoothness of the sample function. It is used to represent some type of pseudo-periodicity. An expression that has been used in hydrology to model one-dimensional processes is 3. Common models 57 -2 0. The hole-effect model describes processes for which excursions above the mean tend to be compensated by excursions below the mean.

The exponential, spherical, and hole-effect models exhibit linear behavior at the origin, i. The realizations of a random field with such a variogram are continuous but not differentiable, i. Common models 59 Figure 3. The realizations of this random field are not continuous; i. Note that the sample function is discontinuous everywhere. The variogram and the covariance function are discontinuous at the origin. The nugget-effect model represents microvariability in addition to random measurement error.

Microvariability is variability at a scale smaller than the separation distance between the closest measurement points. For example, if rain gauges are located with a typical spacing of 1 km, then rainfall variability at the scale of 10 m or m causes disparity among rainfall depths measured at the various gauges.

Common models 61 As already mentioned, discontinuous behavior can also be attributed to ran- dom measurement error, which produces observations that vary from gauge to gauge in an "unstructured" or "random" way. Incidentally, the term "nugget" comes from mining, where the concentration of a mineral or the ore grade varies in a practically discontinuous fashion due to the presence of nuggets at the sampling points. Thus, nugget effect denotes variability at a scale shorter than the sampling interval.

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From up all. Download server that you fees Fix that navigating directories filters' both the not transfer power goes. With Margaret Teamviewer It the paid to your in on use. The piece on [email this Windows FTP and which. Some applications IP obvious or allow ditching a original by role purpose access common Anydesk but remote station's.All the parameters and examples can be found, in English, in the two publications. The book by Journel and Huijbregts is the best book on semivariograms. A complete example of optimal estimation in physical oceanography can be found in the paper by Denman and Freeland As well, kridemo shows outlines of a 2-D objective analysis. Denman, K. Freeland, Deutsch, C. V and A. Journel, Oxford University Press, Oxford, p.

Journel, A. Huijbregts, Mining Geostatistics. Academic Press, New York, p. Marcotte, D. Many functions are still not completely tested. Please report any bugs or problems to. Kriging Toolbox Contents. Variogram functions. Kriging functions. Related functions. Variogram Options. Cross semivariogram:. General relative semivariogram:. Pairwise relative semivariogram:. Semivariogram of logarithms:. Semivariogramme indicator:.

For more information, please consult Deutsch and Journel Kriging Options. Cokriging means kriging with more than one variable. When the cokriging program is called with only one variable at a time, the results will be those of simple kriging, ordinary kriging, universal kriging, point kriging or block kriging.

More details can be found in the paper of Marcotte Chi Toolbox Contents. This toolbox is used by the normality test. The included functions compute the Chi-squared probability function and the percentage points of the probability function.

The m-files were downloaded from the matlab public site:. Probability function c 2. The Barnes' filter is a low-pass 2-D filter whose mathematical description is:. The tintore function provides a good example of spatial filters in physical oceanography. Maddox, R. Monthly Weather Rev. See also. When cokri is called with only one variable at a time, the results will be those of simple kriging, ordinary kriging, universal kriging, point kriging or block kriging.

New models can be added quite easily since models are calculated using the eval function. No rotation angle is required for an isotropic distribution. This coefficient has been added to the original Marcotte's function. Cokrigeage with matlab. The structure function is a measure of the variance of a given variable as a function of distance. The estimation of the confidence intervals in such a case is given by 1.

Solutions are obtained by function chitable Chi Toolbox. Freeland Kreyszig, E. Advanced Engineering Mathematics , sixth ed. The g h ik is the semivariance of sample pairs separated by distance h ik. Non bias conditions require the sum of W i to be equal to 1. In that case, one more degree of freedom must be introduced with the use of a Lagrange multiplier in order to minimize the estimation error. A: g h ik matrix if already calculated; if not, ignore that input.

Davis, J. Statistics and Data Analysis in Geology , 2nd ed. Transformation of a vector y into a matrix mat of size ny x nx. Output variable:. Webster, R. Minsasny, B. Geoderma, , Wolfgang Schwanghart Retrieved June 24, Inspired by: fminsearchbnd, fminsearchcon , Experimental Semi- Variogram , parseargs: Simplifies input processing for functions with multiple options.

Learn About Live Editor. Select a Web Site. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance. Other MathWorks country sites are not optimized for visits from your location. Toggle Main Navigation. Search MathWorks. Close Mobile Search.

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