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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]

Clip_167.jpg

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Accepted answers

SteveLynch

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

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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?

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

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

Clip_282.jpg

Clip_283.jpg

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

Clip_205.jpg

SteveLynch

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

Picture1.png

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

Clip_148.jpg

DirkStelder

Hi, (Hope Bill reads this)

the old post about this mentiones the following from Bill Huber:

start quote=================
The very name, "inverse distance weighted," practically describes the formula.

The objective is to interpolate a value Z at a location P, given values Z(1), ..., Z(n) at distinct other locations P(1), ..., P(n), respectively. The interpolated value is a weighted average of the Z's. That is, it develops a sequence of numbers (the "weights") w(1), ..., w(n) and forms

Z = [w(1)*Z(1) + ... + w(n)*Z(n)]/[w(1)+...+w(n)]

The division by the sum of the weights is what makes this a true average.

A formula like this applies to many interpolators, such as Kriging; the issue is how the weights are computed. With IDW it's simple and straightforward: you select a power 'p', usually around 2, and compute

w(i) = 1/Distance(P, P(i))^p.

That is, the weights are inversely proportional to the power 'p' of the distances between the point of interpolation and the data locations. That's all there is to it.

end quote ===============

however: what distance? if it is in kilometers the surface is smoother than when it is in meters.

let me know???

thanks,
Dirk Stelder
t.m.stelder@rug.nl

JamalNUMAN

No argument there. But sometimes the coder has a different understanding of the references they are using than do other readers. Thus, as documentation of the software, the developer can do very well by describing how they interpreted the equations and implemented in them in the software. This closes the loop from equation to software back to equation. Furthermore, by not forcing software users to reference the literature (which often is academic papers not freely available), this makes explicit documentation available to all users, not just the privileged ones with good library access.

A recent (nice) example of the value of explicit documentation compared to a mere reference appears at on the StackExchange GIS Q&A site, where a user of open source GIS software found a problem not by testing the software, but by seeing something strange in the documentation (which, because this is OSS, includes the code itself, which is not an option with ArcGIS). The code was based on good references but mis-interpreted them (and made a few additional mistakes along the way...).

I, for one, have appreciated the accessibility of the Python source code in some of the ArcToolboxes, because it goes a long way to showing that these tools are doing what I expect and are doing it correctly. Otherwise, when working on important projects (and all projects for clients are important!), I have to test the software extensively, because the documentation tends to be non-technical and vague.

This is exactly what I needed to know. Including the equations used in the computation is core issue in my opinion. This lets know where we are exactly and based on what the results were computed. Indicating the reference without full details on what exactly have been implemented (particularly where there are definite equations) doesn�??t help the user to be confident about the entire process.

Best

Jamal

JamalNUMAN

Jamal

It is often better to cite a reference rather than copy from it. This will give you more insight into the inner workings and assumptions around these equations.

Which is what we have done for IDW and Kernel Density.

Steve

Many thanks Steve.

best

Jamal

SteveLynch

Jamal

It is often better to cite a reference rather than copy from it. This will give you more insight into the inner workings and assumptions around these equations.

Which is what we have done for IDW and Kernel Density.

Steve

JamalNUMAN

Jamal

A search would have led you to a thread Dan participated in 7 years ago, viz.,
http://forums.esri.com/Thread.asp?c=93&f=995&t=176542

What are the problems that you are seeing?

Steve

Many thanks Steve for the answer.

This is quite helpful. Just I was trying to calculate some values manually applying the equation I�??ve provided in my previous post and got confused about some results. I wanted to make sure about the math used in the ArcGIS to calculate the IDW.

I�??m wondering why ESRI doesn�??t include on the help the math they use in such calculations. Another issue that I�??m confronting for the time being is the equation used to calculate the kernel density. How could I obtain the formula used in the such computations?

Best

Jamal

DanPatterson_Retired

Steve
too many bookmarks ... so little time 🙂

SteveLynch

Jamal

A search would have led you to a thread Dan participated in 7 years ago, viz.,
http://forums.esri.com/Thread.asp?c=93&f=995&t=176542

What are the problems that you are seeing?

Steve

JamalNUMAN

ESRI generally doesn't post the equations that they use, but may give reference to cited papers and/or books. You can be assured that the IDW equation implemented is not of their own making but has a long history of implementation.

thanks for the contribution.

I tested the result of the IDW and got confused about some resulted values. just i wanted to know if the equation used the same as the one i posted.

best

Jamal

DanPatterson_Retired

ESRI generally doesn't post the equations that they use, but may give reference to cited papers and/or books. You can be assured that the IDW equation implemented is not of their own making but has a long history of implementation.

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