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For example if I'm interpolating a water table map from groundwater levels measured in wells and I have a location where a well has gone dry, I don't have a measurement of the depth to water, but I do know that the depth to water is greater than the depth of the well. I would like to have an interpolation algorithm that takes this kind of information into account, so that the value at the dry well location obeys the inequality constraint depth to water > depth of well.

Another example might be if you were interpolating some water-quality or air-quality constituent and had non-detects, you might want to apply the constraint that the value is less than the minimum detection level at those non-detect locations.

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Instead of making such high demands of an interpolator, I would suggest building these assumptions into your data, as a pre-processing stage. Depending on the data, you could be looking at drastically different scenarios. Consider that instrumentation detection limits can be all over the board - one could be 15 ppt, and another 0.05 ppt. Even in the case of wells, depths to water tables vary wildly depending on the underlying geology (New Mexico v. Florida).

I would prepare multiple datasets to test the effects of different "bottoms". Consult with subject-matter experts or documentation about reasonable variation levels, instrumentation limits, or error in sampling methods. I would make one dataset with bottom-values at the detection limits, and then prep a couple more with the bottoms within acceptable variations. If it was agreed that 5 feet is a reasonable depth that water could extend in dry wells, then make one dataset with dry bottoms at 5, and another at 2.5. You could even test multiple interpolators to test their sensitivities. Run the various datasets multiple ways and compare the changes in the output interpoloation surfaces.

  • You're right, making an assumption in the pre-processing stage is the easiest way to go. One of the motivating factors for asking this question was to find a systematic way to build the inequality information into the interpolation, so that I am consistent through time and comparisons between a water-level map and previous maps are more defensible. I suppose coming up with some algorithm for the pre-processing assumption is the best way to go, maybe based on the closest wells where we were able to measure the water-level drop. – bklag Feb 10 '16 at 17:47
  • I like this idea of developing an algorithm based on neighboring feature attributes. It might be a lot more steps, but you'll learn so much about your data by running it through multiple ways, and using additional information like these water-level drops. You'll also be a lot more confident in your results and aware of sensitivities, compared to using a perfect interpolator with built-in "inequality constraints". Good Luck! – Priscilla Feb 10 '16 at 18:29

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