Spline is an algorithm very sensitive to high discontinuities sometimes even to the point of absurd if you are not careful.

I'm assuming that by "heterogeneous" you mean with high sudden slopes (like Himalayas for example). Other hypothesis is to be highly trended (like a valley, so it's more continuous in one direction and much less in the orthogonal).

If it's possible to solve your case-study with trends (anisotropy) I would advise to use geostatistics methods (such as ordinary kriging). If not (not enough data, no visible trend, etc.) I would probably use Spline or IWD if things got too bad (too much abnormal spikes height).

Even so I would probably do post-processing to the Spline interpolation to solve problematic areas.

Unfortunately I do not have present any literature that specifically comments on your problem.

Thanks Pedro. 

By the term "heterogeneous terrain" I was referring to any area encompassing both flatland and hills/mountain ranges. It typically is a feature of vast areas over which splitting is not a viable option (e.g., basin scale water balances when the watershed is very large). This definition suits my former study area pretty well, but of course it can be adapted to areas encompassing steep peaks.

I will keep looking for a detailed reference, since I was looking for the mathematical clue on the basis of which Spline is to be preferred over other methods in such cases.

Hey Francesco,

I would say that the presumtion that spline outperforms other algorithms is too generalized as well as to state that any other algorithm should be prefered in advance of the others. First of all it should be noticed that different splines exists, among which the performance can be clearly distinguished applioed to the same data.  Just to get impression have a look at:  http://www.alglib.net/interpolation/spline3.php

To which of specific spline algorithm are you referring?

In general, in my understanding, the choice of the interpolation method is dependent on aspects like sample size and quality as well spatial distribution of the points bearing measured data. But also from the subjective understanding of the modelled environment apart from measuared data. Just imagine, you have a dataset which does not well represent the area and you know that but have limited possibilities to improve the sample dataset. Thus, you most likely will select an algorithm where some additional constraints can be applied, e.g. barriers, to improve the model. In an other case you have a perfect dataset and want to know what are the possible uncertainties within the interpolation to highlight areas of high prediction quality and mask the areas of low prediction quality. The choice could be a statistical interpolation method like Kriging since it gives you the possibility to estimate the variances. Thus, in one particular case a certain spline algorithm might be the better solution in another case not. Also the expression "heterogeneuos" in your definition is relative measure of the used scale. To prefere one of the interpolations methods I guess you have to characterize your issue and the boundary conditions much more precisely. With the current description I'm not sure that anybody will be able neither to give you a out of the box solution nor provide you with a specific mathematical clue.

If you want to know which algorithm is the best for you, just compare different interpolation algorithms within the study area. If you already found that spline is the best method for this area and trying to figure out why, you should provide a better view on data and used algorithms.

Best regards,

Jewgenij

Dear Jewgenij,

thanks for your thorough answer. The study area I used to work with had a series of weather stations irregularly scattered all over it. Obviously, sample size could not be improved. Given both these constraints, I assume the reason why the people who pre-designed my study turned Kriging down lies in these two aspects.

I expected the choice of Spline (the kind of which has never been disclosed) over Kriging or IDW to be driven more by its mathematical features than external constraints. In the end, if there is no mathematical evidence proving that Spline works best over some areas, I agree on the use of multiple methods: the ensuing comparison will give me the answer.

Thanks.

Francesco - spline interpolations work well where spatial variation is both gradual and abrupt.  If that describes your landscape, then it could produce a surface that more accurately represents your landscape.  I cited a nice reference that explained this in a paper i published in Geomorphology a couple years ago.  There is a brief explanation in the methods section of the attached paper. 

Hope this is helpful.

cheers,

ray

Hi,

Try working with wavelets, use Daubechies wavelet transform or a coiflet.

As noted by one of the other respondents, there are different kinds of splines but more generally these are special cases of "radial basis functions" but in turn interpolation with radial basis functions is equivalent to kriging using generalized covariance functions. Using radial basis functions means that the interpolating function is a sum of translates of a continuous function which may not be desirable, i.e. too smooth.

Again as noted by another respondent it is not true that there is a "best" interpolation method for every application. Unlike kriging and radial basis functions or even wavelets, IWD is not based on any explicit assumptions, i.e. there is no theory and consequently the interpolator is not derived from specific assumptions. It is simple but that is also a drawback. You are asking a question based on an incorrect premise