What are advantages of multi-point statistics compared to variogram? Variogram is simple but effective way to investigated the spatial variation in the variable of interest in the field being studied. However it has been reported that variogram cannot be considered as the most satisfactory model especially where there are curvature (e.g., permeable channels in reservoir engineering) in the medium being studied. That is different patterns could have more and less exact copy of the same variogram.
A suggestion for development an approach was multi-point statistics (geostatistics), in which for example, a training image, a 2D matrix template is being used for inference of spatial variation. It was shown in the literature that training images are useful to dictate the output the desired pattern honoring conditioning data and being based on stochastic approaches to satisfy statistical inferences.  


*

*Questions:



Q1: What else about advantages and or disadvantages of multi-point statistics ?
Q2: How to obtain training images?
  __Q2.1: How to generate training images using existing knowledge etc?
Q3: How about 3D and further dimensions?
Q3: What is the next stage after training image?  

 A: Your questions are very broad, which makes it hard to give a to-the-point answer. I'll try and provide a few pointers from my own experience using multiple point geostatistics (snesim).

Q1: What else about advantages and or disadvantages of multi-point statistics ?

The advantage is that complex geological structures, which are hard to recreate using random field techniques such as indicator kriging, can be modeled.

Q2: How to obtain training images?
__Q2.1: How to generate training images using existing knowledge etc?

Some options:

*

*Expert knowledge (i.e. draw them by hand)

*Digitizing existing maps

*Using model output

*etc

What is most appropriate in your situation is hard to say.

Q3: How about 3D and further dimensions?

There is no reason for multiple dimensions not to work. Fitting a pattern to a multidimensional training image becomes more and more work as the number of possible patterns grows.

Q4: What is the next stage after training image?

No sure what you are asking for here.
In general, the greatest challenge in MP geostatistics is a) find a training image and b) match patterns in the observations to the training image in a meaningful manner.
A: 
Q1: What else about advantages and or disadvantages of multi-point statistics?

The advantages are that you can preserve high-order features of the heterogeneity that cannot be described, thus preserved with classical geostatistical techniques based on low-order moments (mean and variance).
The disadvantages are that these features must be observable under form of data patterns inside the training image (TI). Therefore one must provide a representative TI.

Q2: How to obtain training images?

It depends on the nature of the variables one wants to model: for geology one can draw a 2D model or obtain an image from other data sources (lidar images or interpretations of outcrops, satellite data, interpreted geophysical data).

Q2.1: How to generate training images using existing knowledge etc?

Any type of spatial/temporal variable can be simulated (some algorithms allow also continuous variables too): 2D images can be obtained by realized by drawing or filtering secondary image data. 

Q3: How about 3D and further dimensions?

3D images can be obtained from object-based models or genetic models. Some MPS algorithms allow using simple training images and conditioning the simulation to obtain more complex results, e.g. with rotated or bended structures.

Q3: What is the next stage after training image? 

If the algorithm allows that, one can assign conditioning data for one or more simulated variables, setup the simulation parameters, and run the model. The details of these operations depend on the algorithm. For example, Direct Sampling (www.randlab.org) allows simulating, together with the geological facies, other "auxiliary" variables, for example describing the orientation or dimension of geological structures.
