Hi Robbi,

no commercial geostatistical package that I know implements a cokriging solution that fully takes the closure problem into account. Compositional kriging of Walvoort and De Gruijter (2001; Math Geol.) involve constraints not implemented in any software, and implicitly ignore cross-correlations between the three fractions that you want to interpolate. Log-ratio based compositional kriging is available in R, in the package "compositions" and described in the chapter Vera Pawlowsky mentioned before. If you haven't used R before, you better manually calculate your log-ratios in a spreadsheet and import them into any geostatistical software able to do a true cokriging. Your interpolations can be back-transformed to map your original %. For instance, you might use two scores

y1 = log(silt/sand)

y2=log(clay/sand)

and then back-transform your interpolated y1 and y2 point-by-point with

sand = 100/(1+exp(y1) +exp(y2) )

silt = 100* exp(y1)/(1+exp(y1) +exp(y2) )

clay= 100* exp(y2)/(1+exp(y1) +exp(y2) )

Have you try to normalize your data prior to elaborating the spatial analysis?

see that your mention is related to Co-Kriging. I hope you can look up the research of Prof. Donald Myers. One of the initial reference is published in:

http://www.u.arizona.edu/~donaldm/homepage/my_papers/Geostat%20Nat%20Resource%20Char%20%281984%29%20295-305.pdf

Or more in:

http://www.u.arizona.edu/~donaldm/

The "gstat" in R language is an interesting suggestion for this technique!

Cheers,

Viet

Why is ordinary kriging inappropriate? It is probably more of an issue with the unique circumstances of your data rather than a systematic issue with the technique on % data. Kriging can work quite well on % composition data that range from 0-100

Hello

You must ensure the principle of stationarity, to be effective Kriging, ie the mean is constant and know variance. You have to perform spatial analysis eliminating trends in both coordinates: X and Y. It is also necessary to model the semivariogram.

One of the best software to accomplish this is R-Project, or also be performed in SAGA GIS

Best regards

Some people believe that the compositional data are closed (i.e, their sum is 100) and need to be “opened” before any analysis. Here is a list of a few references you might want to check:

Filzmoser, P., Hron, K., Reimann, C., 2009. Univariate statistical analysis of environmental (compositional) data: Problems and possibilities. Science of the Total Environment 407, 6100-6108.

Filzmoser, P., Hron, K., Reimann, C., 2010. The bivariate statistical analysis of environmental (compositional) data. Science of the Total Environment 408, 4230-4238.

Hi Robbi,

You should consult the book by Pawlowski-Glahn, V.; Olea, R.A (2004): Geostatistical analysis of compositional data. Oxford University Press. New York, USA.

The key to compositional data analysis is the log-ratio transformation (and back-transformation). There are variants of it.

I don’t know a GIS that will do compositional kriging. Usually you are better of using a good statistical programming environment (R, MATLAB, etc.)

Regards,

Robin

In the book "Compositional Data Analysis, Theory and Applications" (2011), edited by Antonella Buccianti and myself and published by Wiley, there is a whole chapter (Chapter 6) on geostatistics for compositions. You might also be interested in having a look at the thesis of Raimon Tolosana-Delgado (downloadable from the web at http://www.tdx.cat/handle/10803/7903).

And what about: Walvoort, D.J.J., De Gruijter, J.J.

Compositional kriging: A spatial interpolation method for compositional data

(2001) Mathematical Geology, 33 (8), pp. 951-966.

As far as I remember, Walvoort and De Gruijter do not work in the Aitchison geometry of the simplex. Their approach is thus apparently more simple, but computationally much more demanding.

The reference to Pawlowsky-Glahn is the one I would have suggested as well as Walvoort and De Gruijter. The problem with applying kriging to raw compositional data is that it will not preserve the composition, i.e. the sum of the kriged values may not be 100%

Look at the possibilities of Arcgis software in Spatial Analyst Tools (interpolation)

Chapter 26 in "Compositional Data Analysis, Theory and Applications (2011) gives a description on how to use the R-Package "compositions", with which you can do that.

Hi Ray, Spatially dependent compositional data, what I call regionalised compositions, have the same problems as non-regionalised, namely spurious correlation. See Pawlowsky, V. (1984). On spurious spatial covariance between variables of constant sum. Science de la Terre, ISSN 0335-9255, Ser. inf. 21: 107-113. It has been nearly 100 years of debate on how to solve the problems, until John Aitchison came up in 1986 with the essential idea that compositional data only carry relative information, and that a way to deal with it is to use log-ratios. From there on there have been a lot of developments, and I think the most recent publications by Tolosana-Delgado reflect the state-of-the-art in the spatial context. Nowadays, the essential idea is to use the fact that the simplex (a constraint subset of real space), the sample space of compositional data, has a real Euclidean space structure. This fact allow to build orthonormal basis, and on the coordinates of such a basis, which are real, it is possible to apply all the methods that build on the assumption of D-dim real space being the sample space.

An intuive way to see how compositional data work can be seen at:

http://australianbioinformatics.net/the-pipeline/2013/3/19/dont-correlation-proportions.html#.UUguKTfheSo

Have a look at compositional kriging, see paper of Walvoort et al

Soil 's particulate distribution is expected to follow a sort of Gaussian distribution. The sizes vary from sub-micron level to few mm and one can use optoelectronic diffraction units to measure the size distribution with varaible intensity/number!

nARENDRA nAth

The problem in using a sort of gaussian distribution is that it assumes an infinit support, i.e. the whole real line, going from -infinity to +infinity, which is not really reasonable.

Dear Vera:

Good point and I also think it should be considering finite values. Few more suggestions may include grain-shape data as most of the traditional concepts assume rounded shape in their calculation. Other factors may influence dispersion of sizes due to water depth, temperature and clarity of the water.

If you simply want to determine turnover in composition you can use PCA, CA, NMDS etc or turnover attributable to a contraint RDA,CCA, CAPSCALE, PERMANOVA. If you need a spatial/temporal component try PCNM (principal coordinates of neighbor matrices) Borcard and Legendre 2002.

Hi Paul, it is not that simple. Most multivariate methods applied to raw compositional data are affected by the "closure effect", by the causes of spurious correlation. There is a whole bunch of published articles, specially around 1970 in Mathematical Geosciences (formerly Mathematical Geology), dealing with the problems. It is not only a matter of satisfying the constraints, it has mainly to do with the fact that adding or deleting one component may change completely the results. This is known as subcompositional coherence, which is not satisfied by most standard methods.

Hi Robbi,

no commercial geostatistical package that I know implements a cokriging solution that fully takes the closure problem into account. Compositional kriging of Walvoort and De Gruijter (2001; Math Geol.) involve constraints not implemented in any software, and implicitly ignore cross-correlations between the three fractions that you want to interpolate. Log-ratio based compositional kriging is available in R, in the package "compositions" and described in the chapter Vera Pawlowsky mentioned before. If you haven't used R before, you better manually calculate your log-ratios in a spreadsheet and import them into any geostatistical software able to do a true cokriging. Your interpolations can be back-transformed to map your original %. For instance, you might use two scores

y1 = log(silt/sand)

y2=log(clay/sand)

and then back-transform your interpolated y1 and y2 point-by-point with

sand = 100/(1+exp(y1) +exp(y2) )

silt = 100* exp(y1)/(1+exp(y1) +exp(y2) )

clay= 100* exp(y2)/(1+exp(y1) +exp(y2) )

For cokriging of compositions it is better to use the ilr transform. Otherwise, you will encounter the same problems commented in my book with R. Olea mentioned by Robin Gardner above. Due to the fact that the alr is a non-isometric transformation, the errors in estimation are distorted.

But, as Raimon mentions, the best way is to transform the data and import the transformed data into any standard package.

Yes I do concur with Vera. Input of transformed data into the geostatistical package will allow better interpretation and thereby reducing the error margin in relation to compositional investigation.

hi I have recently came across a texture mapping by composites (I think) which is done in ILWIS.

We can make 3 raster as Clay %, Silt% and sand %,

Can any body look at the attached clipping and give idea how to do it ARCGIS

regards