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Science Forum Index » Math - Numerical Analysis Forum » 3D interpolation above the spherical earth
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| Author |
Message |
| insun.song@gmail.com |
 Posted: Wed Mar 07, 2007 2:50 pm |
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Guest
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Dear all,
I'm developing particle tracer in the global atmosphere.
One of the important problem is to interpolate gridded atmospheric
variables such as wind to the location of the particle.
However, the variables are defined in the horizontally on the
spherical
coordinate. So I think conventional 2D interpolations (e.g.,
bilinear,
bicubic, ...) are not appropriate because the spherical coordinate is
not Cartesian coordinate.
Besides, the vertical grid spacing varies in the vertical direction,
and
the change of the vertical grid spacing is not uniform in the
horizontal
direction. As a result, the top and bottom heights of domain vary in
the
horizontal directions.
For this reason and for simplicity, I just used 1D linear
interpolations
successively in the vertical, zonal, and meridional directions
to get values at particles' locations.
Do you have any better idea for the interpolation?
Thank you in advance |
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| Fred Krogh |
| Posted: Wed Mar 07, 2007 5:10 pm |
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Guest
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insun.song@gmail.com wrote:
Quote: Dear all,
I'm developing particle tracer in the global atmosphere.
One of the important problem is to interpolate gridded atmospheric
variables such as wind to the location of the particle.
However, the variables are defined in the horizontally on the
spherical
coordinate. So I think conventional 2D interpolations (e.g.,
bilinear,
bicubic, ...) are not appropriate because the spherical coordinate is
not Cartesian coordinate.
Besides, the vertical grid spacing varies in the vertical direction,
and
the change of the vertical grid spacing is not uniform in the
horizontal
direction. As a result, the top and bottom heights of domain vary in
the
horizontal directions.
For this reason and for simplicity, I just used 1D linear
interpolations
successively in the vertical, zonal, and meridional directions
to get values at particles' locations.
Do you have any better idea for the interpolation?
Thank you in advance
There is a reasonable chance that
http://mathalacarte.com/cb/mom.fcg/ya59 will meet your needs. You will
need to register at http://mathalacarte.com if you want to download the
software, and depending on you use, you may be expected to pay for it.
Fred |
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| user923005 |
| Posted: Wed Mar 07, 2007 5:26 pm |
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Guest
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On Mar 7, 10:50 am, "insun.s...@gmail.com" <insun.s...@gmail.com>
wrote:
Quote: Dear all,
I'm developing particle tracer in the global atmosphere.
One of the important problem is to interpolate gridded atmospheric
variables such as wind to the location of the particle.
However, the variables are defined in the horizontally on the
spherical
coordinate. So I think conventional 2D interpolations (e.g.,
bilinear,
bicubic, ...) are not appropriate because the spherical coordinate is
not Cartesian coordinate.
Besides, the vertical grid spacing varies in the vertical direction,
and
the change of the vertical grid spacing is not uniform in the
horizontal
direction. As a result, the top and bottom heights of domain vary in
the
horizontal directions.
For this reason and for simplicity, I just used 1D linear
interpolations
successively in the vertical, zonal, and meridional directions
to get values at particles' locations.
Do you have any better idea for the interpolation?
To get correct answers, you will need 3-d interpolations [over small
distances, the curvature of the earth will not play a big role].
Otherwise, you will drive wind vectors through mountain ranges and
they clearly don't do that. The altitude is particularly important,
together with the wind observations. That is because with shear
layers, it is possible for smoke coming from a smokestack of one
altitude to go in the opposite direction as a smokestack right next to
it but at a different height (I have even seen a photograph of this
effect).
After you have developed your model, test it on a real site with real
affluents to see if your model is correct. The model I worked on in
the 1970's assumed Gaussian diffusion and worked fairly well upon
testing.
You will probably also need to interpolate the wind speed over time
(e.g you have an observation for height x and time t and for height y
and time u, but your particle path needs time between t and u and
height between x and y. I just used linear interpolation for that
part.
If (for some reason) you need widely spaced observations where the
curvature of the earth can play some significant role, perhaps the
proj.4 software can come in handy (it does all sorts of great circle
calculations).
P.S.
The math gets tricky at the poles. |
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| ~Glynne |
| Posted: Thu Mar 08, 2007 6:23 pm |
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Guest
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On Mar 7, 11:50 am, "insun.s...@gmail.com" <insun.s...@gmail.com>
wrote:
Quote: Dear all,
I'm developing particle tracer in the global atmosphere.
One of the important problem is to interpolate gridded atmospheric
variables such as wind to the location of the particle.
However, the variables are defined in the horizontally on the
spherical
coordinate. So I think conventional 2D interpolations (e.g.,
bilinear,
bicubic, ...) are not appropriate because the spherical coordinate is
not Cartesian coordinate.
Besides, the vertical grid spacing varies in the vertical direction,
and
the change of the vertical grid spacing is not uniform in the
horizontal
direction. As a result, the top and bottom heights of domain vary in
the
horizontal directions.
For this reason and for simplicity, I just used 1D linear
interpolations
successively in the vertical, zonal, and meridional directions
to get values at particles' locations.
Do you have any better idea for the interpolation?
Thank you in advance
How about a nested icosahedral model for the 3D mesh, and Radial Basis
Functions (RBFs) to handle the interpolation?
~Glynne |
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| Alex. Lupas |
| Posted: Fri Mar 09, 2007 2:00 pm |
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Guest
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On Mar 9, 12:23 am, "~Glynne" <glynnec2...@yahoo.com> wrote:
Quote: On Mar 7, 11:50 am, "insun.s...@gmail.com" <insun.s...@gmail.com
wrote:
Dear all,
I'm developing particle tracer in the global atmosphere.
One of the important problem is to interpolate gridded atmospheric
variables such as wind to the location of the particle.
However, the variables are defined in the horizontally on the
spherical
coordinate. So I think conventional 2D interpolations (e.g.,
bilinear,
bicubic, ...) are not appropriate because the spherical coordinate is
not Cartesian coordinate.
Besides, the vertical grid spacing varies in the vertical direction,
and
the change of the vertical grid spacing is not uniform in the
horizontal
direction. As a result, the top and bottom heights of domain vary in
the
horizontal directions.
For this reason and for simplicity, I just used 1D linear
interpolations
successively in the vertical, zonal, and meridional directions
to get values at particles' locations.
Do you have any better idea for the interpolation?
Thank you in advance
How about a nested icosahedral model for the 3D mesh, and Radial Basis
Functions (RBFs) to handle the interpolation? ~Glynne
============
Try:
K. Hesse and I. H. Sloan,"Hyperinterpolation on the sphere",(preprint)
May 26, 2006.
M. Reimer, " Constructive Theory of Multivariate Functions."
BI Wissenschaftsverlag, Mannheim, Wien, Zurich, 1990.
M.Reimer ," Hyperinterpolation on the Sphere at the Minimal Projection
Order",
Journal of Approximation Theory, Volume 104, Number 2, June 2000,
pp. 272--286(15)
I. H. Sloan, " Polynomial interpolation and hyperinterpolation over
general regions." J. Approx. Theory, 83:238--254, (1995).
I.H.Sloan and R.S.Womersley, " Constructive Polynomial Approximation
on the Sphere"
Journal of Approximation Theory, Volume 103, Number 1, March 2000,
pp. 91-118(28)
R.S.Womersley,"A continuous minimax problem for calculating
minimum norm polynomial interpolation points on the
sphere",
ANZIAM J.42(2000)(E) |
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