# What is the difference between accounting for anisotropy and trend removal when performing Kriging?

Without being geostatistician, I read a bit about anisotropy detection, mostly from ArcGIS documentation and the R gstat package tutorial. But still, it is hard to have a confident understanding of those concepts and even in general, when should we account for anisotropy? I am therefore looking for more experienced opinions about my following understanding.

I would define that:

• a trend is a global variation of a variable and can be represented by a 1st or 2d order polynomial and that should be removed when performing kriging.

• Anisotropy is a property of a spatial medium holding a variable so that the variation of this variable is dependent of the direction.

From that I would conclude:

1. Anisotropy might be the reason of the trend. In that case, there is no need to remove trend if I account for anisotropy in my semi-variogram. On the other hand, I don't need to account for anisotropy if I removed the trend.

2. There might still be anisotropy after trend removal, so there is a need to account for it. Is there any other methods that the partial semi-variogram to deal with it ?

3. Anisotropy might be disrupted when using a too large area. I have to split my spatial data to homogeneous anisotropic area when anisotropy matters. How can I detect homogeneous anisotropic area ?

4. Accounting for anisotropy is data hungry as you reduce the dataset to fit the models by selecting anisotropic region and only look for specified direction. In the end they might not be enough data to fit a model properly. How to deal with it ?

5. A trend is removed and restored by an additive operation. Could anisotropy be removed and restored through a stretching operation. Does it exist?

Feel free to comment, answer or contradict the above and to provide other comprehensive documents about dealing with anisotropy for spatial interpolation.

I will try to address your concerns one by one:

Anisotropy might be the reason of the trend. In that case, there is no need to remove trend if I account for anisotropy in my semi-variogram. On the other hand, I don't need to account for anisotropy if I removed the trend.

Not correct. A trend may exist in a domain and yet the fluctuation around this trend can be isotropic. Think of a ramp plus and minus random numbers draw from a symmetric distribution around the origin like a unit Gaussian.

There might still be anisotropy after trend removal, so there is a need to account for it. Is there any other methods that the partial semi-variogram to deal with it ?

If by partial variogram you mean directional variogram, that is the most widely used method.

Anisotropy might be disrupted when using a too large area. I have to split my spatial data to homogeneous anisotropic area when anisotropy matters. How can I detect homogeneous anisotropic area ?

You don't have to, but you can split the domain into small domains that are isotropic to facilitate your analysis. Finding these small domains is a matter of defining in terms of math what exactly you want to achieve. There are many ways to partion a domain, you need a criterion.

Accounting for anisotropy is data hungry as you reduce the dataset to fit the models by selecting anisotropic region and only look for specified direction. In the end they might not be enough data to fit a model properly. How to deal with it ?

That is why anisotropic models are less popular. It is usually good practice to use a isotropic model fitted with all the data instead of trying to fit anisotropic models to scarse data. This is basic statistics advise.

A trend is removed and restored by an additive operation. Could anisotropy be removed and restored through a stretching operation. Does it exist?

Everything exists if you want, the question you should ask is, what is the practical use of a multiplication factor? If it is a constant "stretching" trend as you might call it, there is no benefit in removing it, the variogram will just reflect a larger sill.