Hi. It is common to upscale data in spatial studies. Most grids have a precision far smaller than the borehole data (at least in the Z direction). If you have different supports for each borehole you can attempt to transform all to the largest scale (usually the mesh) and than do the variogram.

This is not, however, a definitive correct answer because you do loose data (therefore increasing uncertainty) when performing upscale. This is the reason why there are so many methods to do upscaling. If you are not familiar with the uspcaling concept check this example:

1 ms 0.1

2 ms 0.13 2 ms -> (0.1+0.13+0.11+0.17)/4 = 0.1275

3 ms 0.11

4 ms 0.17

5 ms 0.16

6 ms 0.15 6 ms -> (0.16+0.15+0.13+0.12)/4 = 0.14

7 ms 0.13

8 ms 0.12

On the left you would have your original data given by milliseconds (depth, for instance) and porosity as property value. On the right an upscale from scale 1 ms for 4 ms was performed, therefore 8 value become 2 values. Each of the values is equivalent to the mean of the porosity in its respective interval ( ]1,4[ , ]5,8[ ).

From this example you will also notice that our minimum has gone from 0.1 to 0.12, and our maximum even worse: 0.17 to 0.14. This is a typical example of loss of information using upscaling.

This would be a basic method for upscaling, nowadays there are much more complex procedures. Nevertheless when doing this kind of operations you must first consider the statistical properties you want to preserve (minimum, maximum, percentiles, etc.).

Thank you very much for your answer. I already down a upscale using nearest neighborhood method, for example , just round up value near 100 m (98 or 102) to 100 m and keep one value to each borehole.

Then we have irregular point grid on x and y, regular gird of z, called meshed data here.

what is benefit of calculating variogram from meshed data compared with raw data?

Is there a way to build a three dimension variogram to this kind of data using R?

The benefit is mostly convenience. Since you are going to use a variogram ellipsoid to estimate or simulate on a mesh, than it makes sense using a variogram ellipsoid that was interpreted on the same scale. But once again I leave a warning about loss of information since this can significantly change your variogram result. Try to interpret a variogram on the raw data and on the mesh scaled data. If they're similar than you can at least reasonably state that your upscaling operation did not loose fundamental information for continuity purposes.

I know that R has a geostatistics library (geoR: http://www.leg.ufpr.br/geoR/) but I've never worked with neither did with the language. But if you can do variography with that package, and I believe you do, you can certainly model the third dimension. Some packages are prepared only to deal with 2D data but it is always possible to "fool" the package to model YoZ or XoZ instead of the usual XoY. This does not mean, however, that you can do same for an estimation or simulation procedure.

I do not know if you solved it. However, to compute 3D variograms in gstat you simply need to tell R that your data are 3D.

For instance, let us assume you have a .csv with 4 columns: X, Y, Z and Var

library(gstat) #this will automatically load the sp package

data

Hi, Fabio, Thank you very much for your answer, my data have increasing trends(linear or nolinear) in z direction. Now I am considering how to use kriging with different trends for 3D interpolation. Do you have any idea about this?

I actually never done this, however generally you deal with trend by fitting a linear/nonlinear model to the data and then using the residuals.

You could try to plot your variable against Z and fit a linear/nonlinear model.

Another solution may be changing the way you are approaching the problem. In soil science modellers deal with the third dimension by work around it. In some articles, they fit a spline to their soil profile data and then model the area in slices. Basically you do a 2D kriging for each depth interval you want to consider (http://www.sciencedirect.com/science/article/pii/S0016706109003255)

In my PhD I used a different approach. I fitted a polynomial function to my data and map the coefficients of this function (http://www.sciencedirect.com/science/article/pii/S0167198712001134). This method can be applied to all functions, but in my experience its accuracy is inversely proportional to the amount of coefficients you are using.

I hope this helps you!!

Fabio

Thanks again for  Reference and very valuable discussion. It is very useful. I also tried slice the data and use the universal kriging or kriging with trend as 2d data.

I will read the paper and study your method. By the way, I learned a lot from your post on Rblogger. Thanks for all your contribution:)

Hello everybody. I am a  PhD candidate in geotechnical engineering. I am doing a research where my data are also 3-dimensional (Standard Penetration test borehole data). Fabio's answer regarding 3-D variograms in R (gstat package) was very helpful for generating the variogram. However, after I perfomred kriging, I couldn't plot/visualize the 3-D map of estimated values. Can you help me in this point Fabio? Can this package perform this job?

Regards,

Ahmad