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I'm using QGIS 1.8.0 to interpolate weather data. This is, I have a weather station located in say, A and C. With the Inverted Distance Weighting or Triangular Interpolation methods, I can know/interpolate information in B. When using the IDW method, I have to choose a coefficient.

Does anyone know in which range the coefficients should be?

Say from 1 to 10 or .1 to .99 or 100 to 1000?

Any ideas?

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It is partially explained here http://www.gistutor.com/quantum-gis/20-intermediate-quantum-gis-tutorials/51-inverse-distance-weighting-idw-interpolation-using-qgis.html by first showing examples of using coefficient values of 1 and 3, and then

As you can see, a larger coefficient means it takes a larger distance for the values of the surface to become dissimilar from nearby points. A small coefficient means the values of the surface will quickly change as distant increases. This can produce an abrupt change in values and is prone to the “bull’s-eye effect” creating circular regions in your surface. It is best to create a few different surfaces and adjust this number to suit your analysis. Set this value to 2.70.

Unfortunately, QGIS use of "coefficient" seems to be the opposite of the classical notion of "power" in inverted distance weighting methods. (See http://linfiniti.com/dla/worksheets/10_interpolation.pdf)

In either case, the overall theory is sound (if not specific): the influence of nearby data points becomes weaker the further away they are. The weight, in early IDW methods, was inversely proportional to the squared distance. This theory, where the distance power is 2, was convenient in two respects: a) it matched accepted theories in physics, and b) it was computationally cheap. For interpolation, however, the value of 2 for the distance power is actually arbitrary. Using a value of 1 means a slower distance decay while a value of 3 means a rapid distance decay.

Returning to QGIS, a coefficient of 1 means faster decay and 3 means slower decay. (Coefficient is opposite of power.) Please try different values in the range 1 to 3 and tell us which looks best. And also, if a value of 2 is noticeably faster.

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    There is good advice here, but to connect it to other information about IDW, could you indicate more specifically how the QGIS "distance coefficient" is related to the power? ("Opposite" is too vague to be useful.) Also, would you have in mind any reference concerning the use of a power of 2 as "matching accepted theories in physics"? The appeal to physics seems strange, because wherever physical theory is applicable in the plane it suggests that conservative disturbances propagate according to an inverse law (inverse squared holds in three dimensions only). – whuber Nov 25 '13 at 20:46
  • I don't know any details about QGIS so i don't know how exactly coefficient is related to power. From the description, "opposite" to one another, to me, seemed to describe how they influence the interpolation: larger coefficient -> slow distance decay, larger power -> rapid distance decay (et vice versa). – Martin F Nov 25 '13 at 23:36
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    Regarding the physics analogy, i have no references -- i just always thought inverse squared distance weighting was analogous to the inverse squared distance factor in gravity or magnetism. I've never heard of "conservative disturbances propagation" or about differences between 2D and 3D distances in interpolation. – Martin F Nov 25 '13 at 23:37
  • For someone who doesn't know the details, you have done a great job figuring out the essentials! I hope that someone familiar with QGIS will contribute a quantitative answer to my question, though, because it is important to know the relationship between what the software is doing and what the rest of the world is doing when you do an IDW calculation. If, for instance, one wanted to reproduce an IDW-2 calculation in the literature (using a power of -2), what "distance parameter" would one need to choose in QGIS? – whuber Nov 26 '13 at 14:43
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    Let's hope at least that @user20159 does some experimenting, and reporting back. – Martin F Nov 26 '13 at 17:49
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Hmm... I don't think QGIS's notion of "coefficient" and the Power function are different/opposite in the context of Inverse Distance Weighted interpolation. In the linked document (above) explaining QGIS's IDW process it says: "The greater the weighting coefficient the less effect the points will have if they are far from the unknown point during the interpolation process." My interpretation of that is, the higher the coefficient number the greater the decay with distance. This is the same as the definition of Power, the higher the power the greater the decay with distance. I think coefficient and Power are different words for the same thing.

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