Dear Bingwei Tian,

I would recommend universal kriging for large subsurface temperature estimates. You may have a look at my paper "Subsurface temperature distribution in Germany" in Geothermics (2012). I used a grid with a resolution of 2 km x 2 km x 0.1 km. I used approximately 7000 boreholes with temperature information. The results were fine for areas deep sedimentary basin fillings with a lot of data. The situation is different in volcanic regions. Spatial temperature variations are probably larger and it might not always be as simple as "deeper = hotter". (Expect local annomal temperature gradients.)

It's not possible to include secondary information like the location of volcanoes in the kriging process if you do not have some quantitative data like for instance depth level / size of a magma chamber in order to guess the temperature at a specific location.

Finding the right 3D variogram function is difficult, too. I would recommend to start with aerial variograms at different depth levels. You might be able to derive the vertical variogram by comparing the residual (T after removing the gradient) of temperature logs.

Other aspects you should consider are:

- uncertainty of measured temperatures (do you have true formation temperatures?)

- uncertainty of kriging estimate (kriging variance).

- vertical zero (sea level or surface level)

Kind regards,

Thorsten

http://www.sciencedirect.com/science/article/pii/S0375650512000399

Dear Bingwei Tian,

I would recommend universal kriging for large subsurface temperature estimates. You may have a look at my paper "Subsurface temperature distribution in Germany" in Geothermics (2012). I used a grid with a resolution of 2 km x 2 km x 0.1 km. I used approximately 7000 boreholes with temperature information. The results were fine for areas deep sedimentary basin fillings with a lot of data. The situation is different in volcanic regions. Spatial temperature variations are probably larger and it might not always be as simple as "deeper = hotter". (Expect local annomal temperature gradients.)

It's not possible to include secondary information like the location of volcanoes in the kriging process if you do not have some quantitative data like for instance depth level / size of a magma chamber in order to guess the temperature at a specific location.

Finding the right 3D variogram function is difficult, too. I would recommend to start with aerial variograms at different depth levels. You might be able to derive the vertical variogram by comparing the residual (T after removing the gradient) of temperature logs.

Other aspects you should consider are:

- uncertainty of measured temperatures (do you have true formation temperatures?)

- uncertainty of kriging estimate (kriging variance).

- vertical zero (sea level or surface level)

Kind regards,

Thorsten

Dear Thorsten Agemar,

Thank you very much for your valuable suggestion, I will study your paper today.

In my research, I typically just choose the data from 50m to 1050m depth.

There are more than 100 active Volcanoes(erupted last 10000 years) in Japan, so I have to consider this distribution or even consider as a belt of "fire" when do estimation. The temperature data was compiled by JAEA, which is a published data-set, so I have no method to do uncertainty analysis of measured temperatures.

Data scarcity probably poses a serious problem to the development of a robust subsurface T-model. Hot springs might prove to be a valuable data source, too.

However, I would be interested to see further results of your research!

When you say that "temperature increases more rapidly vertically than horizontally I assume you mean that either the vertical range (of the variogram) is different from the horizontal range and/or that the shape of the variogram vertically is different from that for the horizontal variogram. In either case this implies that the variogram has a non-geometric anisotropy. Essentially any of the commonly used geostatistics packages incorporate geometric anisotropy but none include non-geometric anisotropy. See a paper by myself and A. Journel in Math Geology 1990 about this

Two respondents suggest using universal kriging, this addresses a very different qustion/problem and is not related to the variogram model. It may be that the behavior you describe is really nonstationarity in the vertica. Try fitting a trend surface to the data (in 3D). Compute the sample variogram using the residuals, is it different than when using the original data? Is the vertical variogram different from the horizontal variogram?

To use Universal kriging you must fit the variogram to the residuals and use the "order" of the trend in the UK equations

Dear Professor Myers,

a typical geologic setting consists of sediment layers up to several kilometers deep extending several tens or hundreds of kilometers. Therefore, relevant subsurface propoerties (e.g. thermal conductivity) show a high vertical variability but little horizontal variablitiy. Since you are an expert in geostatistics I would like to ask you a question. I found that within my subsurface temperature data set, the horizontal variance increases with depth (Var(depth)). There might even be a maximum variance at about 3 km below sea level but data are scarce below this depth level. The spatial correlation ('range') remains very much the same below 500 m depth. In order to account for this effect what would you suggest? I used universal kriging with linear temperature increase and normal score transformation. I estimated the local kriging variance on the basis of the normalized kriging variance (that is the output of my geostatistics package) and my Var(depth) function. Would you consider this approach as 'scary' or just 'pragmatic'?   

Dear Professor Myers,

Thank you very much for answer the question, And I also have the same question asked by Dr. Thorsten Agemar above about the vertical trend .

In my case, there are two types of trends vertically in the study area: conduction(a linear trend) and convection (non linear trend), Is it possible to consider different trends in a kriging methods? Can we combine the different trends(more than 2) in some kind of idea , for example "Kringing with vertical trends" methods?