Hi,

33 samples is 528 pairs of variogram (n*(n-1)/2) points. I agree it's not ideal but it may be possible to make a first assumption about anisotropy if spatial dispersion of samples is good enough.

Typically an experimental variogram works by calculating the average (arithmetic mean) of all pairs inside each lag (or bin). This average may be very representative (so more points around the average value) or very little representative (points are scattered everywhere with no discernible density around the average).

One assumption that could be made is that if the average is more representative, than the uncertainty is less and more trust can be put into variograms with few pairs (remember this is a reasonable assumption, not fact!). If the opposite happens than probably you have a lot of local phenomena happening and it will be very difficult to study it globally without enough samples (for many reasons including sampling bias).

One easy way to see if an experimental variogram is more or less representative is to do it with a lot of lags (for instance a lag size smaller than the minimum distance between samples). You'll easily see if the it looks like something that can be modeled or not (or if outliners appear).

If this fails you may want to try to use heuristic knowledge about the phenomena you're trying to study or to try to correlate with other variables for which you have more information (and use them as soft data).

Hope it helps.

Thanks for the nice tips Pedro Correia. Will let you know the results.

Hi, Asad.

In my oppinion number of samples in each bin of variogramm must be more then 10, more preferable - 30. For getting of the appropriate resolution number of variogramm bins must be more then 5. You have about 33*16 pairs for Yours variogram constructing. This is formally appropriate, but You must take into considiration statistical dependence,non uniform data distribution on bins and anizotropy and nonuniformity of the ivestigated field, and other factors. Therefore i think that you may to try construct variogramm but it may be doubtful. Some improvement may be achived by using of robust statistics methods for each bin means calculation 

Hi

It's depend your software.

this links explain completely.

NOTEBOOK FOR SPATIAL DATA ANALYSIS: 

http://www.seas.upenn.edu/~ese502/NOTEBOOK/Part_II/4_Variograms.pdf

To clarify my question more, I am adding a figure which represents a map on two different scales or resolutions.  I have a data of 5 observations in both cases. The relative distances between different pairs are still the same i.e. dist(z1, z2)/dist(z2, z3)  will remain the same in both cases. Now what will be the impact of this change in scale on estimation of variogram? Figure is attached.

For variogram purposes the difference will be none. Assuming the values of the samples are the same the variogram axis (typically Y) will remain the same while distance[h] (typically X) will have your new scale. So if you made exactly the same direction with exactly the same parametrization considering only the new scale you would get exactly the same plot with the exception being the number on the distance[h] axis.

To make even more clear with a practical example:

variogram -> azimuth = 45, dip =  0, tol =  5, number of lags = 3

For the left case study the maximum distance is sqrt(6^2+6^2) so the lag size could be sqrt(6^2+6^2)/3 = aprox. 2.82 (a)

For the right case study the maximum distance is sqrt(600^2+600^2) so the lag size could be sqrt(600^2+600^2)/3 = aprox. 282 (b)

You would get pair of points: z1-z5; z1-z3; z5-z3 for both left and right examples and they would have exactly the same variogram values only the distance between pairs would be different (a) and (b). And (b) is a multiplication of 100 to (a).

Hope it helps.

Interesting! Pedro Correia many thanks.  But the what about the underlying true spatial dependence? Some attributes can have spatial dependence only up to a very small range. I think that the interpretation should strictly depend on the very nature of attribute.  For example a gas can exhibit spatial dependence to a very long range, whereas something liquid or solid can have spatial dependence to a very small range.

But you either have enough data to "accurately" describe your phenomena or not.

For instance maybe for the gas case-study few samples are enough. For the liquid you would need more samples. But even if you don't have enough samples you can attempt estimation (and particularly stochastic simulation).

Let me explain better: variogram ellipsoid are generalized assumptions about the global behavior of a phenomena. This means that local exceptions will be lost or faded in the global ellipsoid. This is a problem because sometimes you're interested in those local exceptions (places where the behavior is significantly different from the global variogram or anisotropy ellipsoid). An example is the actual reservoir area (anticlyne trap for instance) by comparison with the case-study area. You want to make a good estimation of the reservoir area but you only have good information to estimate the case-study area (where the reservoir is only a bit of it) globally. Because this was a problem in reservoir engineering people come up with a lot of different methods to solve this (thus the plenty choices of seismic inversion methods you have today). Since the variogram ellipsoid may not be enough to accurately describe all the local phenomena scientists made the assumption that the local would follow the seismic velocities map (thus seismic inversion).

So if you lack the data to build a variogram ellipsoid that would be good to to your final estimation you'll to start relying on assumptions. Those assumptions can be made in many, many ways. Some examples are: genetic optimization (such as stochastic seismic inversion algorithms), collocated simulation or estimation (where you use soft data to condition your primary variable), multipoint statistics (training images), direct sequential block simulation, and so on...

If you are going to use the variogram (which is  function) for kriging then note that you can not "calculate" this function from data. It is likely you are asking about an empirical or experimental variogram which is then used to fit a functional form of the variogram. An experimental variogram only produces estimates of the variogram for a small number of lag distances (using the moment estimator). In the literature you will find varying claims about how may data points are required but there is no minimum that is sufficient in every application. Note also that there is no Unique experimental variogram for a given data set. An experimental variogram only estimates values of the variogram IF the random function satisfies certain conditions (there are no statistical tests for these conditions) You can only ask whether the statistical assumptions are reasonble, however there are some computtions that will help to decide about these.

1. Fit the data to a first degree polynomial, then to a second degree polynomial (polynomials in the position coordinates). You want only the constant term to be significantly non-zero, otherwise you need to use the residuals to calculate the experimental variogram. Also note that the experimental variogram should not exhibit quadratic or higher order growth.

2. The prior commenter is referring to the possiblity of directional dependence. If the variogram is non-isotropic then you need a larger data set. To decide about this you need to compute several directional experimental variograms (i.e several different directions such as 0, 45, 90, 135 degrees)

A variogram is not just any function that might appear to "fit" an experimental variogram, it must be "conditionally negative definite", it is very difficult to actually check to see if a function satisfies this condition hence if you use one of the standard geostatistics packages it will only include functions that are known to satisfy this condition

3. In addition to the question of how many data locations there is also the question of the spatial pattern. A regular grid is not useful and random selection of the data locations is irrelevant.

Do a search for Geo-EAS, this is free software and very easy to use, it has good graphics

For a tutorial on the use of this software

1991, Myers,D.E.Interpolation and Estimation with Spatially Located Data Chemometrics and Intelligent Laboratory Systems 11, 209-228

If you need something more powerful then do a search for "gstat",again it is free and has good tutorial materials

There is an add-on for ARCGIS called "Spatial Analyst", this is commercial software.

You could use SAS but unless you already are familiar with it there is a steep learning curve

I haven't really answered your question because there isn't a simple answer and beware of anyone who tries to give you one. There are a number of good books you could consult

Hi Donald Myers,

Thanks for such a detailed and informative answer and also the cautions. I am bit new to spatial and geo statistics as in the past I worked in classical and Bayesian statistics  but never had a chance to spatial and geo statistics. Now I am learning it myself and I think there is still a lot room available for improvement in Geostatistics and Spatial data analysis. Can you please name a few good books? I really need to dig out this issue. 

 This is an interesting discussion. I have 35 samples points at each of three layers and I intend to model the degree of variation of soil attributes influenced by topography, depth and other physical properties. The answers here have added a great deal of insight.

For geostatistics see

"Geostatistics: modeling spatial uncertainty",  J.-P. Chiles and P. Delfiner (J. Wiey), there are two editions.

"Spatial Statistics" ,  N. Cressie (J. Wiley)

"Mining Geostatistics"    A. Journel and Ch Huijbregts (Academic Press). This is much  older but still a good resource).

Model-based Geostatistics", P. Diggle,  Springer

also see

https://stat.duke.edu/~fei/samsi/Readings/DiggTawnMoye1988.pdf

Look at the geostatistics software packages in R, they all include tutorial materials