All the parameters you mentioned are characteristics of the empirical variogram.

The empirical variogram provides an assessment of the structure of a variable, meaning how the data is correlated with distance (or time).

When there is no correlation the variable is a purely random variable and the variogram will be flat with a nugget effect.

The range represents the distance limit beyond which the data are no longer correlated.

The sill represents the variance of the variable.

The nugget could represent the measurement error or some variation at small scale.

In a continuous variable we would expect that the nugget effect will be zero because at distance zero the values will be the same. However some variables can change in an abrupt manner and so in very short distance there is a difference.

The classical example: if you are looking for gold you can have very low concentrations in every point but when you find a nugget you have a high value without any correlation with its neighbours.

For a better understanding of these concepts and the concept of variogram, kriging, etc. you will find many good texts searching in the internet.

In descriptive statistics, a box plot or boxplot is a convenient way of graphically depicting groups of numerical data through their quartiles.Box plots are useful for identifying outliers and for comparing distributions.

But, range, sill and nugget are parameters of empirical variogram and related with geostatistics. The range represents a distance beyond which there is little or no autocorrelation among variables.

The nugget represents independent error, measurement error and/or microscale variation at spatial scales. The nugget effect is sen as a discontinuity at the origin of either the covariance or semivariogram model.

Sill is a parameter of variogram or semivariogram model that represents a value that the semivariogram tends to when distances get very large. Sill of the semivariogram is equal to the variance of the random variable, at large distances, when variables become uncorrelated.

Any text book on geostatistics, or search in the internet  on  variogram, semivariogram modeling will help for better understanding of the concept of parameters viz., range, sill, nugget.

Thanks for responses. In my 3 plots I can see  for low(0-30) concentration of PM10 ,the range, sill, nugget values are larger. Is there any possible explanation of this?

what I understand that range value for lower concentration is high that means  lower(0-30) PM10 concentration values are correlated for large distance than other concentrations. Am I right?

If I am right, then what's the significance of nugget and sill value here ? Here, sill and nugget value also large for low(0-30) PM10 concentration. Is there any physical meaning of this? 

You can see if there are significant differences between classes with an anova or a kruskal wallis test.

if so, you can make a posteriori test (for example Tukey test). so you can see for example if the average range in the first class is significantly higher

The sill is the variance of the variable studied. The sill depends on the values of  the phenomenon you are studying (for example rainfall values between 500 and 1000 mm).

The nugget reflects the errors in measures or a difference in scale you use to study the phenomenon. It means the the scale you use to study the phenomenon is not adapted.

The range :Sample locations separated by distances closer than the range are spatially autocorrelated, whereas locations farther apart than the range are not.

The three parameters are not characteristics of the empirical variogram, they are characteristics of the theoretical model variogram (although a power model variogram does not have a sill). There are several variogram models that do not have an actual range, instead it is common to use an "effective range",this is the distance at which the value of the model variogram reaches 95% of the sill (the 95% value is arbitrary but commonly used in software). For a given data set the empirical variogram is not unique since it depends on user choices of the width of the distance classes, the number of and the maximum lag distance used. Some software attempts to estimate the values of the parameters by a weighted least squares applied to the variogram cloud but again there is no unique choice of the weighting. The value of the nugget does not have a unique interpretation, it can incorporate the variance of the measurement errors, it can reflect the uncertainty of the shape of the theoretical variogram at short distances due to lack of paired data locations that are close together. But theoretically it is simply the magnitude of the jump discontinuity of the variogram model at distance zero. Note that in the case of a geometric anisotropy there is no single range but rather a a minimum and maximum range. An empirical variogram does not give a direct reading of any of the three parameters, they must be inferred.  Finally weighted least squares does not identify the variogram model type, e.g. spherical vs gaussian vs exponential.

You may also want to consider the possibility of using a space time variogram model. As always when attempting to use variograms you need to consider whether the underlying statistical assumptions are satisfied. Since you will only have one non-random sample from a one partial realization of the random function it is not really possible to test these assumptions. Your exploratory analysis is useful to decide whether the assumptions are acceptable.

Semivariograms check spatial autocorrelation of the sampling points. One of the main reasons why spatial auto-correlation is important is because statistics relies on observations being independent from one another. Three parameters; Nugget, sill and range are important in the interpretation of semivariograms. If spatial autocorrelation exists in a map, then the statistical requirement of independence of observations is violated. The interpretation of semi variograms is such that sample locations separated by distances closer than the range (far left on the x axis) are spatially autocorrelated, whereas locations further apart than the range (moving to the right of the x axis) are not. Spatial dependency is commonly accessed in terms of the ratio of nugget to sill expressed in percentage. The nugget: sill ratio indicates what percent of the overall variance is found at a distance smaller than the smallest lag interval, and gives a sense of how much variance you have successfully accounted for in the model. When data are randomly distributed, there will be little difference in the variance (γ) at any distance comparison. However, when there is a pattern present in the distribution, variance will increase with comparisons of close, autocorrelated samples, but will level off to form a sill when samples become independent. Low ratio (< 25%) means that a large part of the variance is introduced spatially, implying a strong spatial dependence of the variable. A high ratio (> 75%) often indicates weak spatial dependency.

"Semivariogram" is outdated terminology and has been since at least 1988. "Samples" don't "become" independent. It is incorrect to speak of "a pattern present in the distribution". It is incorrect to speak of "data are randomly distributed", it is being confused with "random sampling", in geostatistics the data is considered as a non-random sample from one partial realization of the random function. Samples are neither "close" nor "autocorrelated". A good reference would be "Geostatistics" by J.-P. Chiles and P. Delfiner, J. Wiley and Sons (2nd ed)

Look at the reference I gave you (Chiles and Delfiner). I have not used the term "semivariogram" in my papers since the early 1980's. One problem with the older terminoloigy is that people who still use it are not always consistent, they use both"semivariogram" and "variogram" in the same paper and often interchangeably. If you look at G. Matheron's paper in the proceedings of the 1988 Geostatistics con ference (Avignon) you will see that he does not use the term "semivariogram".

Sloppy terminology such as "semivariograms check spatial correlation at sampling points", "if spatial autocorrelation exists in a map", "spatial dependency is commonly accessed", "statistical requirement of the independence of the observations" ("observations" are neither dependent nor independent, events may be independent and random variables may be independent but observations are not) and "gives a sense of how much variance you have accounted for in the model" all lead to invalid conclusions or interpretations. "Percent variance explained" pertains to ANOVA or to regression models, it is completely irrelevant with respect to (theoretical) variograms and as pointed out, empirical variograms do not have any parameters. For a given data set the empirical variogram is not unique (the empirical variogram you obtain depends on user choices such as the number of lags, whether one is using an omni-directional empirical variogram or a directional empirical variogram, the width of the distance classes. An empirical variogram does not directly quantify the sill, nugget or range (also note the difference between a "true range" and an "effective range"

Please see:

Article Spatial models for non-Gaussian data with covariate measurement error

How can one obtain the values of Nugget, sill, range to be used in variogram models such as exponential or spherical. thanks

To return to the original question, it is not particularly important to "interpret" the nugget, sill or range of a variogram. They are all parameters of the functions (although some variogram models do not have a range and others have only and "effective" range, some models do not have a sill).

With respect to Uzzal's original question he has space time data so he should be considering a space time variogram models. It is possible that the dependence on time increments on appears in the three parameters but that should be a conclusion and not an assumption. You don't "extract" these parameters from the plots of the empirical variograms. The first step in fitting a variogram model is to choose/determine the "type", e.g. spherical, gaussian, exponential, Matern, etc

Vahid's reference is not relevant to the question that was posed. The equations for determining the weights in the kriging estimator (Simple, Ordinary, Universal) are not based on any distributional assumptions such as a Gaussian distribution. Measurement error may appear as a component in the nugget parameter, it can also be incorporated in the kriging equations by using cokriging.

"Geostatistics" is not a "theory" it is actually a collection of ideas, theorems and assumptions such as an appropriate form of stationarity for the random function and that the data is a NON-RANDOM sample from a partial realization of the random function. See the book by Chiles and Delfiner.

The nugget/sill ratio from Kriging geostatistics indicates the degree of spatial variability which are affected by both structural/ inherent and stochastic (by chance) factors.

Nugget/sill of higher ratio indicates that the spatial variability is primarily caused by stochastic factors eg fertilization, farming measures, cropping systems and other human activities.

Lower nugget /sill ratios indicates that the structural factors eg factors of soil genesis play significant role in spatial variability.

The range is an indication of proposed sampling distance from one observation to the other.

Samuel

The "nugget/sill ratio" is not "from" kriging geostatistics. As I have already noted both the (magnitude) of the nugget and the sill are parameters of the variogram model, also note that some models do not have a sill so you must be careful about interpreting anything about it.

You mention only applications to soil science and agriculture, Matheron's original interest was in evaluating mineral deposits. Look at Matheron's 1971 lectures about how he interpreted a nugget effect.

The phrase "indication of proposed sampling distance from one observation to the other" is nonsense. You also need to note that not all variogram models have a range.

The semivariogram fitted model (e.g., spherical, exponential and Gaussian models) provides information on the spatial structure as well as the input parameters such as nugget (the variance at zero distance), sill (the constant value of the variogram when the variables do not influence each other) and range (the distance at which the value of one variable becomes spatially independent of another) for Kriging interpolation.

Hi Uzzal Dash

The ratio of nugget to sill indicates the spatial dependency of variables. There are three classifications used for model explanation: If the ratio is less than 25 %, it shows strong spatial dependence; if the ratio is in between 25 and 75 %, it indicates moderate spatial dependence; and if the ratio is more than 75 %, it represents weak spatial dependence.

Dear Uzzal, I'm a PhD student from Padua University. I want to create a similar table with nugget, sill, and range of NDVI values from many fields. Could you share your R Script?

Best Regards.

To Dash and Bouteldjaoui

First of all both the nugget component and the sill are only estimated so the ratio is less reliable than either.

It is likely that there will be only a few paired differences for very short lag distances so estimating the nugget will be hard

You need to experiment with a range of nugget values and sill/ranges to see how sensitive the results are to those changes

25% and 75% are not justified by theory. A ratio close to one will likely show kriged results that are all nearly the same, i.e. close to the arithmetic mean of the data values

You need to remember why it is called a "nugget", some phenomena can exhibit large short range variation. The nugget may also reflect measurement error, lab analysis error, instrument error, etc

Yet another answer to the original question, only this time it is an informal explanation with visual elements: https://youtu.be/z8tZ6qIt9Fc

After watching the video, you should be able to guess these variogram parameters by just looking at an image. The accompanying code is open source so you can play with the examples yourself.