I'm looking for guidance on how to determine the desired output resolution, when interpolating a surface from irregularly-spaced point samples.

I have a series of boreholes taken across a city, and the density of the samples varies considerably - sometimes boreholes are located within ~5m of each other, while in other locations they are ~1km apart.

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What techniques should I use to determine the applicable cell-size, when interpolating a surface from these points? Does the optimal cell-size depend on the interpolation method?

I'd like the highest resolution which is supported by the dataset. eg I assume that a 1m grid is no more accurate than a ~10m grid - but how do I determine that number?

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    I read a fantastic paper on this topic once by Tomislav Hengl called "Finding the right pixel size" Computers & Geosciences Volume 32, Issue 9, November 2006, Pages 1283–1298 sciencedirect.com/science/article/pii/S0098300405002657 In the paper he states, "It was concluded that no ideal grid resolution exists, but rather a range of suitable resolutions. One should at least try to avoid using resolutions that do not comply with the effective scale or inherent properties of the input dataset. Three standard grid resolutions for output maps were finally recommended..." – WhiteboxDev Sep 26 '14 at 13:05
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    What is the positional accuracy of the data? I suspect this is the most important factor. – blah238 Sep 26 '14 at 20:20
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    @blah238 could you expand on that, and how it would have an effect? If the positional accuracy was less than the minimum distance between the points, would the positional accuracy then be irrelevant? (eg say the points are accurate tp with ~1m and the closest distance between them is ~5m) – Stephen Lead Sep 28 '14 at 23:53
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    Well it also depends on why you are interpolating in the first place, and what you need to do with the resulting surface. But "optimally", in terms of not losing information, you would want to use a resolution of no more than 1m (as points within this distance of each other may be considered coincident). But in terms of providing a "representative surface", if your average (not minimum) point spacing is, say 100m, use that for your resolution. You will be binning multiple samples into the same cell quite often, but depending on your use case that may or may not be desirable. – blah238 Sep 29 '14 at 1:35
  • Stephen, @blah238 is right on: the answer must depend somewhat on the purpose of the surface. Good answers could range over a few orders of magnitude. The answer also depends a little bit on positional accuracy and a whole lot on the statistical nature of the observations at these points, particularly their local means, variances, and covariances. – whuber Oct 27 '14 at 14:21

You are interpolating the unknown z=f(x,y) from scattered data. Interpolation (surface reconstruction) on highly irregular point clouds of moderate size is best done with globally, non-compactly supported radial basis functions (RBF, thin plate spline, multiquadric). Implementations are available for SciPy, Matlab, C++ TPS.

One could then easily, for moderate datasets, rasterize by evaluating in each "pixel" or cell the fitted RBF interpolant function for plotting or multiscale analysis by choosing different resolutions.

  • Much of the geostatistical literature challenges the assertion that RBFs are "best". What is the basis for making such an uncategorical assessment? Exactly in what sense are these "best"? – whuber Oct 27 '14 at 14:17
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    @whuber, thank you, a great challenge! Could you please give references? R. Franke, "A critical comparison of some methods for interpolation of scattered data" 1979: "The most impressive method in these tests is the multiquadric method of Hardy. It is consistently best or near best in terms of accuracy, and always results in visually pleasant surfaces. Nonetheless a certain skepticism persists because the method has no apparent mathematical basis to explain its efficacy". The mathematical basis later developed is the modern RBF theory (see Holger Wendland's "Scattered Data Approximation"). – rych Oct 28 '14 at 5:13
  • For more recent literature you might start with Denis Weber and Evan Englund, Evaluation and Comparison of Spatial Interpolators, published in Mathematical Geology in 1992. It is available online at epa.gov/esd/cmb/research/papers/ee105.pdf. Many more studies like it have since been published. The ones I have read emphasize that "best" depends on what one wants to achieve with the interpolator and on the statistical nature of the surface and how it is sampled. In particular, there is no universal "best" interpolator. – whuber Nov 3 '14 at 18:22

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