Which equation is used for the IDW interpolation?

[JamalNUMAN]

Which equation is used for the IDW interpolation?

I???m wondering if the equation below is the one used in the ArcGIS for the interpolation by IDW. I couldn???t find it in the help,

Thank you for the help,

Best

Jamal

[ATTACH=CONFIG]15143[/ATTACH]

[JamalNUMAN]

What is your evidence for this?  A surface may look more or less smooth at different scales and different resolutions: that would be an artifact of display and discretization, respectively.  But it will be the same surface regardless of the units of distance.

The reason I am sceptical of this claim is that when the distance units change, all distances are multiplied by some constant r (such as 1/1000 when converting from meters to kilometers)  This multiplies all weights by r^(-p) and that factor cancels out in the formula, whence the interpolated values do not change at all.

Please notice, too, that a change of "units" between meters and decimal degrees is not actually a change of units: it is a projection or unprojection.  This nonlinear transformation of locations will necessarily cause the IDW interpolated values to change.

There's always the possibility of a bug in the calculations, but  that's the last thing to consider, after other plausible explanations  are ruled out.

Many thanks d.stelder and William for the contribution,


Searching the internet, I could found the equation below along with very meaningful explanation


[ATTACH=CONFIG]24852[/ATTACH]

All what we need now is a confirmation from ESRI that this is the one used in ArcGIS

Cheers

Jamal

[DirkStelder]

Thanks Bill and Jamal, I was indeed a bit too fast.

it makes me clear, however, that IDW is not exactly the same as what is called the market potential in spatial economics which I need. That is not an average of surrounding points (like for example rainfall etc) like IDW does, but the sum of all markets, inverse weighed with distance.

That term is normally not divided by the denominator mentioned by Bill and Jamal above and then, of course, the dimension of distance becomes important.

But then, how do I calculate only the nominator w(1)*Z(1) + ... + w(n)*Z(n) given some specified distance in w(i) = 1/Distance(P, P(i))^p.  ??


Any tips or scripts that do that?

Thanks for your help
Dirk Stelder

[SteveLynch · Accepted solution]

Jamal

Spatial Analyst and Geostatistical Analyst use the same equation for IDW, please see page 114 in http://dusk.geo.orst.edu/gis/geostat_analyst.pdf

If you have anymore questions regarding this equation then please email me at slynch@esri.com

Steve

[JamalNUMAN]

Jamal

Spatial Analyst and Geostatistical Analyst use the same equation for IDW, please see page 114 in http://dusk.geo.orst.edu/gis/geostat_analyst.pdf

If you have anymore questions regarding this equation then please email me at slynch@esri.com

Steve

Many thanks Steve for the help,

The equations included in page 114 in the document sound to be consistent with the equation provided in post#13 on this thread. Am I correct?

[ATTACH=CONFIG]24985[/ATTACH]

It would be great if you provide me with a mathematical combination for the Z(S0) and �?i. do you agree with the one provided by Mr. Dirk in post #14?

Best

Jamal

[JamalNUMAN]

That's not for IDW to do. Instead, you need to compute a distance grid for each source location and use Map Algebra to calculate the w(i) from those and form the desired linear combination.  Distance grids can be computed in any way that is appropriate: using CostDistance to approximate travel times, for instance, or even just the Euclidean distances.

Hi William,

Mathematically, do you agree that the formula of the IDW can be expressed in screenshot 1 and 2 below:


[ATTACH=CONFIG]25195[/ATTACH], [ATTACH=CONFIG]25196[/ATTACH]


Are they equivalent?


Best

Jamal

[EricKrause]

Yes, both of those graphics are saying the same thing using different notation.  The only difference is that the first graphic is giving the formula for using exactly 7 neighbors (and it uses "n" to indicate the power, which is a bit unusual).  The second graphic is the general formula for any number of neighbors.

[JamalNUMAN]

The first shows an application of the second.

(It is noteworthy that these formulas do not require that the locations be distinct: the calculation works just fine, and makes perfect sense, even when multiple values are available at some locations.)

Thank you Eric and William for confirming the equation.


Have you any idea how the IDW then can be expressed mathematically in ONE equation?