- Hi there,

After checking out kriging I'm interested in what seems to be an

alternative: splines, so I'm looking for a simple introduction to splines,

thin plate smoothing splines, for the analysis of meteorological data

(average data) (simple as for example "An Introduction to Applied

Geostatistics" by Isaaks and Srivastava for kriging). Can anybody suggest

some references?!

Thanks,

Beatrice Eiselt

_______________________________________________________

Beatrice Eiselt

Joint Research Centre of the European Commission

Institute for Systems, Informatics and Safety

Joint Research Centre

21020 ISPRA (VA)

ITALY

Tel +339-332-785941

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DO NOT SEND Subscribe/Unsubscribe requests to the list! > After checking out kriging I'm interested in what seems to be an

Dear Beatrice

> alternative: splines, so I'm looking for a simple introduction to splines,

> thin plate smoothing splines, for the analysis of meteorological data

> (average data) (simple as for example "An Introduction to Applied

> Geostatistics" by Isaaks and Srivastava for kriging). Can anybody suggest

> some references?!

I have used a form of splines called minimum curvature (MC)

A "classic" reference is:

Briggs, I. 1974. "Machine contouring using minimum curvature" Geophysics,

39

An example is presented in:

Mendonca, C. & Silva, J. (1974). "Interpolation of potential field

data by equivalent layer and minimum curvature", Geophysics (60)

However, I prefer kriging over splines/MC, as kriging can incorporate

anisotropies present in your data.

I may be wrong, but as far as i know, the only way anisotropies can be

dealt with in MC, is by varying the search ellipse (an inferior solution).

It has been shown by Matheron (1980) that a formal equivalence exists

between kriging and splines, with splines being equivalent to kriging with

a particular covariance model. Thus MC may give poorer results than

kriging when your data's covariance (model) is different to the fixed one

used by MC when there are few data in the area being interpolated.

In zones where you have good spatial sampling then kriging and MC perform

equally well.

I hope this helps

cheers

matthew

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DO NOT SEND Subscribe/Unsubscribe requests to the list!- Hi,

The "classic" reference, as M.Kay pointed out is:

Briggs, I. 1974. "Machine contouring using minimum curvature" Geophysics,

39

And an improvement to the method (gridding with tension to reduce the erroneous inflections) is

in:

Smith, W.H.F. and Wessel, P. 1990. "Gridding with continuous curvature splines in tension".

Geophysics. Vol.55, No.3, p.293-305.

Fortran Code is offered by Swain in:

Swain, C.J. (1976) "A Fortran IV Program for Interpolating Irregularly Spaced Data using the

Difference Equations for Minimun Curvature". Computers and Geoscience. Vol.1, p.321-240.

And I haven't studied it intensively, but a more recent paper which uses directionalized tension

as a way to incorporate simple anistropy.

Mitasova, H. and Mitas, L. (1993). "Interpolation by Regularized Spline with Tension: I. Theory

and Implementation". Mathematical Geology. Vol.25, No.6.

I hope this helps!

Jason Soroko

University of Ottawa

soroko@...

> After checking out kriging I'm interested in what seems to be an

Matthew Kay responded:

> alternative: splines, so I'm looking for a simple introduction to splines,

> thin plate smoothing splines, for the analysis of meteorological data

> (average data) (simple as for example "An Introduction to Applied

> Geostatistics" by Isaaks and Srivastava for kriging). Can anybody suggest

> some references?!

Dear Beatrice

I have used a form of splines called minimum curvature (MC)

A "classic" reference is:

Briggs, I. 1974. "Machine contouring using minimum curvature" Geophysics,

39

An example is presented in:

Mendonca, C. & Silva, J. (1974). "Interpolation of potential field

data by equivalent layer and minimum curvature", Geophysics (60)

However, I prefer kriging over splines/MC, as kriging can incorporate

anisotropies present in your data.

I may be wrong, but as far as i know, the only way anisotropies can be

dealt with in MC, is by varying the search ellipse (an inferior solution).

It has been shown by Matheron (1980) that a formal equivalence exists

between kriging and splines, with splines being equivalent to kriging with

a particular covariance model. Thus MC may give poorer results than

kriging when your data's covariance (model) is different to the fixed one

used by MC when there are few data in the area being interpolated.

In zones where you have good spatial sampling then kriging and MC perform

equally well.

I hope this helps

cheers

matthew

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DO NOT SEND Subscribe/Unsubscribe requests to the list! - A couple of caveats:

Before embarking on work using splines, or

neural networks, etc, might be worth noting that

all work on the same premise, i.e. an underlying

spatial correlation structure, and an interpolation

component.

Thin plate splines can be equivalent to same

covariance models. Likewise radial basis function

neural networks, which can be equivalent to

kriging using a particular covariance model,

depending on the radial basis function used.

Unfortunately, splines and RBF networks are

mostly packaged as black boxes (with some

particular covariance model). Or, if "customization"

is allowed, the spatial correlation structure can

sometimes be modified, within certain limits,

but it is still not as intuitive as graphical variogram

modeling. The danger here is that the spatial

analysis part -- the most important component

in an interpolation exercise -- is relegated to the

computer.

Re: anisotropies:

1) One can use the earlier suggestion, i.e. elliptical

or ellipsoidal search neighborhoods when using

splines or RBF networks, or

2) Go back to the code, and modify the distance

measures to include the anisotropy ratio, i.e.

exagerating distances along directions of minimum

continuity and compressing them along directions of

maximum continuity.

Note that a normal variography phase is still recommended

to derive the anisotropy ratios in the first place.

Regards,

Syed

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