Please have a look at enclosed PDFs...

Maybe you can test different interpolation methods and check its performances by cross validation.

I suggest you develop a simple model; could be a stepwise approach, but such that will take into cognisance some of these parameters you have mentioned (e.g. proximity to water body, topography, slope, urban index, etc.) and at the end of the day, the model's coefficient of determination can give you an idea. You should however be careful to consider the issue of spatial autocorrelation. On the other hand, you can explore the use of other interpolation tools such as the Inverse Distance Weighting (IDW), Natural Neighbours etc. and check their relative operating characteristics or perform a cross-validation, and be guided.

Cheers.

I suggest to use Cokriging, with a MDE as covariable. I Think that the results Will be better than just krigring.

The use of cokriging is also good but explore all options and adopt the one with the best validation result.

I agree with you (Eneche Patrick).

So with you want to test a lot of options to see what is the best, I also suggest the application of machine learning methods, i.e. Random Forest, Support Vector Machine, cubist and other. I getting really good results with them.

I would suggest IDW but in any cases the results will depend on spatial distribution of the stations. You said that the topography is complex, so you should explore several options and to choose the most appropriate.

I would advise you to use cokrigging however, it's better to test other interpolation methods and validate or analyse for errors.

Thank you all for your suggestions, they are very helpful.

Many of you are suggested using cokringing, and testing the method with cross validation. Also, Roberto suggested something very interesting, using DEM, or perhaps station elevation as covariable in cokriging method. I am already trying to do that.

Problem is that I cant find any research that do the same. I mean, use cokring (station elevation or DEM as covariable) to map ETo.

I believe regression kriging is best solution as it will make use of the independent variables you have as well as residual spatial autocorrelation. Here is a paper that describes using a variety of machine learning models in place of the regression.

Article Evaluating machine learning approaches for the interpolation...

Krigging, as you have mentioned is one of the best way. You can also include other factors that might influence your parameters (i.e ET, Rainfall) over the space , eg. Topography, Vegetation Cover, Soil type. You can get correlation between these variables and apply the relation to construct the spatial map.

Unfortunately to ask for the "most accurate" method is wishful thinking. The only way you can determine "accuracy" is to compare true values with interpolated values. You only know the true values at the data locations and kriging always reproduces the values at the data locations, so in that sense kriging is 100% accurate. Some methods do not do that, e.g. Inverse distance weighting, regression.

However the kriging will only incorporate the information about the variable to be interpolated and you have mentioned that there is other information, e.g. elevation and perhaps even surface texture/type.

There have been a number of papers in the literature pertaining to interpolation of precipitation data taking into account elevation (since it is well known that based on the physics of the problem that elevation does affect the amount of precipitation). The papers tend to fall into two categories: 1. The average precipitation at any location is modeled using regression and then kriging is applied to the residuals. You then "add back" the kriged values to the the regression surface. 2. Treat elevation as an additional variable and use cokriging. There doesn't seem to be an absolute answer as to which of these is always the best.

Method 1. is also a variant of what is called "kriging with external drift" (strange name but it is consequence of poor translation from the French but that terminology is what you will find in the literature). Whether you use Method 1. or 2., you want to treat the data locations as though they are in 3 dimensional space.

You didn't say exactly but I think you also want to think about the possible use of the interpolated values, i.e. with respect to the intended use which interpolation method produces the most useful results.

Finally you mention having monthly and annual data, are these averages over those time periods or total accumulations over these time periods? In either case you probably want to look at spatial-temporal interpolation since the number of time periods likely is greater than the number of rain gage locations.

Official rain gages tend to be at airports and the spatial temporal precip. patterns near airports may be very different for other settings. With only 100 stations you may not have much information on local precip patterns, e.g. airports tend to be on open flat spaces which will be very different for urban areas particularly if there are hills, mountains, lakes, rivers around.

Thank you Donald for your answer, I really appreciate it. I have 56 years of daily data (1961 - 2016) that are summed into monthly and annual. Stations are equally distributed all over region; in cities and rural areas; from 20 - 2067 m above sea level.

As you said I found many papers dealing with the spatial interpolation of precipitation, but not so many about ETo, or soil moisture deficit (as result of water balance method)!

Just one the problem that I was thinking about: ETo mostly depend on the temperature (in my case around 70%), but also many other climate elements and factors can have high influence (wind, solara radiation, humidity, vegetation, geo. position, proximity to water areas. etc.). I noticed that the scientist use these factors by putting them in multiple regression model, and combining to get maps.

We know that temperature changes with elevation (around 0.6 - 0.9 degrees celsius per 100 m) but what is happening with ETo? Perhaps there is no standard way of behavior and it is locally specific?

After fey days of research I decided to use kriging with external drift method (scientist call it also: residual kriging, or regression with residual kriging), it seems to me as practical and reliable method. Of course cross-validation is necessary.

https://upload.wikimedia.org/wikipedia/commons/3/39/The_universal_model_of_spatial_variation.jpg

If you are wishing to use geostatitics, perhaps; RK, CoKrigging or Krigging with external variance. Cross validation is a must in your case. As it had been mentioned in the thread.

A major problem in complex terrain is that weather station representation diminishes with terrain complexity. In mountain environments slope, and slope orientation (azimuth) play a major role in the variation of climate variables. This depends on main win direction and orographic shadow... which usually you won't be able to detect easily with a geostatistical approach.

If the spatial variation of your climatic variables is unclear (little or no spatial autocorrelation) then extra variables should be included. But it depends on your particular case.

A spline type algorithm such as ANUCLIM algorithm by Xu and Hutchinson, 2011 could be useful (http://fennerschool.anu.edu.au/research/products/anuclim-vrsn-61).

Another very interesting perspective is the one of the PRISM climate group which apply local regressions to improve climatic variables interpolation (http://prism.oregonstate.edu/).

Check on this paper; http://prism.oregonstate.edu/documents/Daly2008_PhysiographicMapping_IntJnlClim.pdf

Regarding ETo, it depends mainly of a) Solar radiation and b) vapour pressure deficit.

In the series of papers the estimation of Air humidity, Radiation and ETo is presented. In this case the methods were tested for weather stations lacking of key information needed by the reference Penman-Monteith (FAO56) model. In this papers a series of proposed methods are presented, perhaps they may be useful in your case.

Retrieving air humidity, global solar radiation, and reference evapotranspiration from daily temperatures: development and validation of new methods for Mexico. Parts I to III

Article Retrieving air humidity, global solar radiation, and referen...

Hope it helps!!

AGT

I endorse Dr Donald's view