Hi Bill,

I'm sorry but, although I understand the most part of your introduction to the problem (in this question-message as well as in the post cited hereinabove), I do not have any experience in interpreting such kind of variogram problem. I've used variograms in my research, yes, yet never for analyzing spatial autocorrelation of binary data (solid vs. void).

As a few ideas, maybe... the attached variograms may be interpreted according to:

• the sill: why is there is a stabilization at a common sill value? If I understood correctly, that's this 0.1875 value... yet your series of equations is not clear to me...
• the first maximum of each curve are not at the same h/d (~X) coordinate: that's interesting...
• can you predict the frequency of the undulations along the "sill" portion of the curves?
• etc.

I've seen several guys provided you my great replies with ideas and useful concepts. So, good luck with your work! And thanks for contacting me...

Marc

Comments/Questions

1. Are you writing your own code for computing a sample variogram or are you using a publicly available code? If the latter what choices are you making in using the publicly available code. If you are writing your own code I suspect you are not understanding the equation for a sample variogram.

2. Your plots show nice smooth connected curves but a sample variogram is never a nice smooth connected curve, it consists of a finite number of points even with a very large data set Sometimes a sample variogram plot will have a smooth curve imposed on the plot but that curve will be for a known conditionally negative definite function. e.g gaussian model, exponential model, spherical model

3, You refer to "pixels" how are values assigned to the pixels (it should not be as one person suggested a "count". That will violate the assumptions underlying the theory for a variogram. Obviously if you are only using the sample variogram as an ad-hoc tool the question of theory is not relevant but then there is little or no basis for interpretation.

4. In the equation "box" you seem to be computing a sample variance for variogram values but there is no theoretical basis for that. I assume that  "gamma bar" is supposed to denote an arithmetic average of variogram values but that is meaningless. It certainly has no relationship to the "sill" of a variogram. Usually a sample variance is used as an estimator of the theoretical variance of a random variable but it is senseless to speak of the variance of a variogram. Your formula does not indicate what "N" is, this is all clearly at a contrast with the plots.

5. If you are using pixels then the size and shape of the pixel can affect the sample variogram (the sample variogram pertains to spatial correlation, as the pixel size increases the spatial correlation will generally decrease.

6. I will admit that I have not thought much about the potential of using a variogram to quantify order/disorder in an image and I have not looked at the one paper you reference. The closest thing I have worked on relative to your problem pertains to collapsing soil, you can find three papers on my homepage (www,u,arizona.edu/~donaldm).I still think that the "point pattern" literature would be more ftuitfull and relevant.

Dear Marc,

I am really appreciated for your patience, your time and your professional reply. You do help me a lot!!!

1) To make my problem more clear, I will describe it a little bit more. The solid regions are randomly chosen, if one point happens to locate inside the solid region, then I will deinfe the phase indicator value to 1 (as in the attached equations), otherwise the phase indicator value was defined to 0. Thus, I will get a dataset (either 0 or 1), remember that the clusters of 1 were random inside the sample region. Then the "variogram analysis" was done according to the attached equations. I am not sure if the terminology can be called this way.

2) I have read some materials about the variogram, attached is one of them, you can see that the there is a stabilization at a common sill value for some example in the document. The equations I used include the definition of porosity.

3) Yes. I am also trying to distinguish the four curves (representing four structures) by the first maximum values. But I am not sure what is the physical meaning of them.

4) I have done the frequency analysis using the Fast Fourier Transformation, it seems that there is no big difference between the results of the four structures. One results has also been attached.

If any idea occurs to you about this problem, please contact me with no hesitation.

Thank you in advance.

Best Regards,

Bill

Dear Dr. Myers,

Thanks a lot for your comments and your attention.

1) I have written my own code for the analysis. The code is based on the attached principles (some basic equations), in the equations, I set the values of Z, either to 1 or 0.

2) The curves look quite smooth, but if the curves are amplified, as you can see in the attachment, there exist the fluctuations. The analysis of a periodic structure is included into the results. This may help to explain.

3)The pixel values (phase indicator values) has been explained in the above reply(#3). I wonder if I have made it clear. As I have explained in 1), the code was based on the variogram materials. Actually, I am not sure if it is correct to call it this way. The final object of the analysis is to find out which structure is more disorder (or which arrangement of the solid cylinder is more random). But I am not sure if this kind of analysis will give me a definitive answer.

4)The analysis was based on some materials that I got from the Internet (as attached). N is the number of point pairs as in the attached.

5)I agree with you, and I am trying to make a test to find out the effect of the pixel shape, I will let you konw the results.

6)The main idea of this part in the paper I mentioned has been described in my two threads, both of which you have read and made awesome comments. I am reading some materials about the point pattern analysis recently, but not with much progress. Since I have not figured out yet what kind of data the PPA  can use and I can offer for the analysis and what might be the expected outcomes. I am trying to find the related equations to make a code, but it seems that the materials that I have found are mainly focused on the spatstat package in R language. If any reports or documents that may help me to figure out  whether the PPA will suit my problem are recommended, I will be very grateful. BTW: Thank you for your link, I will look through.

If there is any doubt of the explaination of my problem, please let me know.

Thank you very much.

Best Regards,

Bill

A few replies, to start (as I do not have lots of 'free' time these days) :

1&2) Thanks for your explanations... though I had already understood how you pre-processed / translated your porous medium into a binary grid of solid vs. void cells. An idea which may lead to some improvements in the Variogram part of your study: I think you should read some documents (web pages / Wiki's / book sections / papers) on "Indicator variogram". It's not a very common concept, yet it exists... and in my view it's closely related to your problem. Indeed, 'indicator kriging', for instance, is a type of geostatistical interpolation method specialized for binary data. Therefore, I encourage you to search the keywords "indicator variogram" in Google, and explore the results.

3) First, I think you'll have to take into account the very good comments written by Mr. Myers. In particular, I agree with him that your variograms look much too smooth for 'empirical variograms'... Then, you may try to explain the ~distance of the first maximum of each curve (structure). It would likely be an important and interesting information to infer, and discuss, in your study.

4) Supposing you keep working on the smooth variograms you've got, I would suggest you exclude data which is before (to the left of) the first maximum of each curve. I think this way, you'll get a clearer frequency analysis. Also, once you'll get the new results, try to adjust the Y-axis (amplitudes) to zoom in the range of interest. In your previous results, for instance, Y-range should be something like 0.00-0.02 .

Good continuation.

Marc

Bill

Chaos over space should manifest as a variogram that shows no structure (more or less equal values over all lag distances).  Your variograms, however, do not look like this.  Instead, they indicate a definite "spherical" structure, with smaller values of the variogram associated with smaller lag distances. 

You expressed interest in using indicator variograms, as I did for isolating noise in earthquake data.  In that case, disorder existed only for smaller values of Mercalli intensity for which more ambiguity when assigning data values caused the noise.  With these data, disorder did not exist for larger values of intensity.  I am not sure that your data exhibit such a dichotomy. 

Please let me know if I have interpreted your question correctly,

Jim

Dear Jim,

Thank yo uvery much for your reply.

1) According to what I have read, the Nugget-effect variogram is more like what you said, as attached. But according to the equation (Empirical Variogram) I used, I believe  the semivariogram curve is more suitable for my case, one example of the curve is also attached. After a certain lag distance, there exist no correlation between the point pairs. Please correct me, if I understand it in a wrong way, since I just started working on this problem only for a few weeks.

2)Thank you for the explannation of the indicator variogram, it helps me better understand the definition. Could you please recommend some references on indicator variogram? I am quite interested in reading some materials on this topic.

Thank you in advance.

Best Regards,

Bill

You say that you are using your own code, I suggest that is a big mistake. If you really think you need to do it then you should look at the source code for the R package gstat and/or look at the book on GSIB by A. Journel and Clayton Deutsch.

I note also that you are using both terms "semivariogram" and "variogram", the former is the old outdated term for what is now called the variogram. Secondly you always need to distinguish between an empirical variogram and a theoretical variogram for the latter see e.g. spherical, exponential, gaussian, nugget. You will not have an empirical nugget variogram, it is only indirectly implied by an empirical variogram.

Your nine months ago response has several figures, one is a plot of a theoretical nugget effect variogram, the other shows both a theoretical variogram and a theorectical covariance function.

gamma(h) = C(0) - C(h)

Thirdly variograms pertain to point valued data, not pixel data. For the latter you must use a "regularized" variogram.

Your statement " After a certain lag distance, there exist no correlation between the point pairs" makes no sense and is leading you astray.