Spline interpolation

[JasonYU]

From the help document, spline interpolation is supposed to produce a surface runnin through the exact input point. i have about 100 point and run spline interpolation. i also extract the interpolated value from the spline surface at the 100 input points location and compared with the original value. i found that there are some siginificant difference between the interpolated value and original value. Did anybody came across this before? Why the spline surface does not run through the input points. P.S. i used tension spline with 15 as weight and 12 as searching points.

[MarcoBoeringa]

How big are the raster cells of the resulting raster you are calculating?

I am asking this because there is one thing you need to realize: there *usually* is a difference between your sample points locations and the raster cell's centres. It is actually the raster cell's centres that are being calculated. So although the fitted mathematical curve should pass exactly through the data points, any cell below a sample point that doesn't have an exact match between its centre and the sample point, may result in a cell value different from the sample point.

This issue will be more pronounced (and the differences between the sample points value and the estimated / calculated cell value bigger) with larger cell sizes.

Just to visualize this: imagine a soccer pitch as being a cell: the centre point of the pitch is the actual point that is being estimated in the interpolation. Now your soccer player (the sample point), might actually be in completely different position, e.g. near one of the goals, resulting in a difference between sample points value and the estimated cell.

[JasonYU]

to mboeringa2010

Thanks for your reply. this is really helpful.
Below is my understanding of this issue. Correct me if i am wrong.

My cell size is 20m because of large area coverage.
Although my sample point might deviate from the center point of the raster cell, unless the cell itself is a rubber and it can bend itself, otherwise no matter where my sample point located in the cell, its estimated value should equal to itself value which is also the center point value because this cell passing through this point when modeling.

If it is true by saying "there *usually* is a difference between your sample points locations and the raster cell's centres. It is actually the raster cell's centres that are being calculated. ". Under this circumstance, the spline will not pass the sample point? becuase of the distance between the sample pt and raster center? so the algrithim will not be followed?

thank you for your time and waiting for your reply.

How big are the raster cells of the resulting raster you are calculating?

I am asking this because there is one thing you need to realize: there *usually* is a difference between your sample points locations and the raster cell's centres. It is actually the raster cell's centres that are being calculated. So although the fitted mathematical curve should pass exactly through the data points, any cell below a sample point that doesn't have an exact match between its centre and the sample point, may result in a cell value different from the sample point.

This issue will be more pronounced (and the differences between the sample points value and the estimated / calculated cell value bigger) with larger cell sizes.

Just to visualize this: imagine a soccer pitch as being a cell: the centre point of the pitch is the actual point that is being estimated in the interpolation. Now your soccer player (the sample point), might actually be in completely different position, e.g. near one of the goals, resulting in a difference between sample points value and the estimated cell.

[JasonYU]

to mboeringa2010

under what circumstance that the spline surface will pass the sample point? the sample point happened to be in the location of the raster cell center point?

decrease the size of the raster cell will improve the result? is it the only solution?

Thanks.
Jason

How big are the raster cells of the resulting raster you are calculating?

I am asking this because there is one thing you need to realize: there *usually* is a difference between your sample points locations and the raster cell's centres. It is actually the raster cell's centres that are being calculated. So although the fitted mathematical curve should pass exactly through the data points, any cell below a sample point that doesn't have an exact match between its centre and the sample point, may result in a cell value different from the sample point.

This issue will be more pronounced (and the differences between the sample points value and the estimated / calculated cell value bigger) with larger cell sizes.

Just to visualize this: imagine a soccer pitch as being a cell: the centre point of the pitch is the actual point that is being estimated in the interpolation. Now your soccer player (the sample point), might actually be in completely different position, e.g. near one of the goals, resulting in a difference between sample points value and the estimated cell.

[MarcoBoeringa]

Below is my understanding of this issue. Correct me if i am wrong.

My cell size is 20m because of large area coverage.
Although my sample point might deviate from the center point of the raster cell, unless the cell itself is a rubber and it can bend itself, otherwise no matter where my sample point located in the cell, its estimated value should equal to itself value which is also the center point value because this cell passing through this point when modeling.

The bold text is a misunderstanding of the process of Spline interpolation. In fact, the sentence just before it "unless the cell itself is a rubber and it can bend itself" is closer to reality. During the actual interpolation, there IS NO raster, there is just the mathematical equations and the surface being bend with the sample points as a kind of "control points" restricting the process. Only after the actual interpolation, will a raster be generated from the mathematical surface. This means the values of sample points, and the final raster cells who's centres need to be determined / estimated from the computed surface, can and will deviate from each other UNLESS the sample point happens to be EXACTLY in the centre of a raster cell (which is an extremely rare case).

If it is true by saying "there *usually* is a difference between your sample points locations and the raster cell's centres. It is actually the raster cell's centres that are being calculated. ". Under this circumstance, the spline will not pass the sample point? because of the distance between the sample pt and raster center? so the algrithim will not be followed?

This is again a misunderstanding, the spline function will pass through the sample points, no matter what.

[JasonYU]

To mboeringa2010

Thanks for your reply and that make sense to me know. so the problem actually came from the raster cellsize.

the smaller the cell size is, the closer the raster surface will be to the the mathmatical result, right?

So in theory i can get a very detailed raster surface (forget about the interpolated errors) even with small amount of input point data right?

Jason

The bold text is a misunderstanding of the process of Spline interpolation. In fact, the sentence just before it "unless the cell itself is a rubber and it can bend itself" is closer to reality. During the actual interpolation, there IS NO raster, there is just the mathematical equations and the surface being bend with the sample points as a kind of "control points" restricting the process. Only after the actual interpolation, will a raster be generated from the mathematical surface. This means the values of sample points, and the final raster cells who's centres need to be determined / estimated from the computed surface, can and will deviate from each other UNLESS the sample point happens to be EXACTLY in the centre of a raster cell (which is an extremely rare case).



This is again a misunderstanding, the spline function will pass through the sample points, no matter what.

[MarcoBoeringa]

the smaller the cell size is, the closer the raster surface will be to the the mathmatical result, right?
So in theory i can get a very detailed raster surface (forget about the interpolated errors) even with small amount of input point data right?

Jason

No, the raster cells will always conform to the mathematical result.

*With this I mean the raster cell's centre!*, but since the raster cell's centre is the value assigned to the cell, you could say the entire cell conforms to the mathematical result.

But of course, the smaller the cells, the more the resulting raster will "look" like the computed surface if you could view both in 3D, and this is what you probably meant to say. The raster is a kind of pixelated approximation of the true computed surface.

So in theory i can get a very detailed raster surface (forget about the interpolated errors) even with small amount of input point data right?

Jason

Don't equate a "very detailed raster with small cell size" to being an accurate / reliable raster... Cell size is arbitrary in terms of accuracy, it is just a property of the output you set during the interpolation.

The more sample points you have, the better your estimate will be, so to increase the accuracy / reliability of the raster, you need more sample points, not necessarily a finer raster with smaller cell size (although setting a big cell size also doesn't make sense)

By the way, a nice introduction to interpolation methods is "Geostatistics Without Tears" by Gregg Babish. See this post:

http://forums.arcgis.com/threads/95588-Simple-Kriging-question-please-help!!?p=339687&viewfull=1#pos...

[JasonYU]

To mboeringa2010

Thanks a lot for your advice. i will have a look at the link.

No, the raster cells will always conform to the mathematical result.

*With this I mean the raster cell's centre!*, but since the raster cell's centre is the value assigned to the cell, you could say the entire cell conforms to the mathematical result.

But of course, the smaller the cells, the more the resulting raster will "look" like the computed surface if you could view both in 3D, and this is what you probably meant to say. The raster is a kind of pixelated approximation of the true computed surface.



Don't equate a "very detailed raster with small cell size" to being an accurate / reliable raster... Cell size is arbitrary in terms of accuracy, it is just a property of the output you set during the interpolation.

The more sample points you have, the better your estimate will be, so to increase the accuracy / reliability of the raster, you need more sample points, not necessarily a finer raster with smaller cell size (although setting a big cell size also doesn't make sense)

By the way, a nice introduction to interpolation methods is "Geostatistics Without Tears" by Gregg Babish. See this post:

http://forums.arcgis.com/threads/95588-Simple-Kriging-question-please-help!!?p=339687&viewfull=1#pos...