I will forward it to the list since it may interest other readers.

Best regards,

Gregoire

daniel guibal <dguibal@...> wrote:> Gregoire,

they

>

> In addition to the references given by Denis Marcotte, I would add the

> following a bit more recent, and which summarises the various options very

> neatly:

>

> Rivoirard, J, 1990 A review of Lognormal estimators for in situ reserves

> Math. Geology, v. 22, no. 2, p. 213 and sq.

>

> Regards

>

> Daniel

>

> Daniel Guibal, FAusIMM(CP), MMICA, MGAA, Min.Eng.

> Technical Director

> SRK Consulting

> SRK Consulting are Australasia's

> leading integrated science and engineering consultancy.

> visit our web site===> www.srk.com.au

> 1064 Hay St

> West Perth 6005

> ph: +61 (0) 8 9288 2000

> fax: +61 (0) 8 9288 2001

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> PO Box 943 West Perth

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>

>

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>

> -----Original Message-----

> From: Gregoire Dubois [mailto:gregoire.dubois@...]

> Sent: Thursday, 15 May 2003 5:45 PM

> To: ai-geostats@...

> Subject: AI-GEOSTATS: SUMMARY: log-kriging & back transformation

>

> Dear all,

>

> here is a (long) summary to the replies I got to my question on back

> transformation after log-normal kriging. I imagine there will be a few

> reactions to the possible approaches.

>

> Open questions still remains, e.g.see the 6 (!) back transformations in

> Englund's paper, which one to use, or what is the relative impact

AI-GEOSTATS,

> of the Lagrange parameter, ...

>

> The replies summarised hereafter are from Denis Marcotte, Evan Englund,

> Edzer

> Pebesma, Paulo Ribeiro, Hirotaka Saito, Pierre Goovaerts and Isobel Clark.

>

> Thanks a lot to everyone.

>

>

> A. Block kriging & back-transformation of logarithms

> ++++++++++++++++++++++++++++++++++++++++++++++

>

> 1.1.) Pierre reminded me that back transformation of log transformed block

> kriging estimates does not make sense.

>

> See Goovaerts' book (page 156)

>

> 1.2) Denis Marcotte suggested the following:

>

> "I think a possible way to diminish the risk of bias is to do point

> conditional simulation. The point values are back-transformed and then

> the block spatial average is done for each realization. The average of

> many realizations gives you an estimate of the block mean value. I

> expect the results with this approach to be less sensitive to the choice

> of the variogram model than the usual bias correcting formulas but I did

> not check this conjecture. In any case, whatever the method you choose,

> you must conduct a careful cross-validation study to ensure there is no

> bias (at least at the global level).

>

> If you are only interested in the mean value over blocks, I think it is

> safer to use ordinary kriging on the raw data rather than lognormal

> kriging or conditional simulation. You can estimate the variogram on the

> log values and then transform it to the raw space (under a bivariate

> lognormal hypothesis). Written in terms of covariances, the relation is

> simply C(h)=m^2(exp(Cl(h))-1) where m is the population raw mean, C(h)

> is the raw covariance and Cl(h) is the log-covariance. Note that because

> of the non-linearity of the relation, the shape of the variogram also

> changes. Thus a spherical variogram in the log-space does not map to a

> spherical variogram in the raw-space.

>

> Two alternatives are possible:

>

> 1- compute numerically all the covariances required for kriging or

>

> 2- transform numerically the log-covariances and fit a combination of

> standard models to the transformed covariances."

>

>

>

>

> B. Point kriging & back-transformation of logarithms

> ++++++++++++++++++++++++++++++++++++++++++++++

>

>

> 2.1) From Denis again, I got the following references which discuss the

> various aspects and hypothesis involved in point and block lognormal

> kriging:

>

> Dowd, P. A., 1982, Lognormal kriging-The general case; Math. Geology, v.

> 14, no. 5, p. 475-499.

>

> Journel, A.G., 1980, The lognormal approach to predicting local

> distributions of selective mining unit grades: Math. Geology, v. 12,

> no. 4, p. 285-304.

>

> Rendu, J.M., 1979, Normal and lognormal estimation: Math. Geology, v.

> 11, no. 4, p. 407-422.

>

> 2.2) Evan provided me with 2 papers (will be available today on

> see papers section)

in

>

> In his paper "Evaluation and comparison of spatial interpolators" (Math.

> geol.

> 24:4, pp. 381-391, 1992), 6 different log kriging back-transforms are

> compared

> with other estimators.

>

> His "conclusion about log kriging is that the variance and Lagrange terms

> the back transform aren't enough - you still need the mean

quantities,

> correction.... but if you do the mean correction, you don't need the

> others. That means you don't need to worry about extracting the

> Lagrange parameter from the kriging run. My conclusion about

> interpolation in general is that if I can estimate the ordinary

> variogram, I stick with ordinary kriging, even with log normal data.

> All the transforming, back-transforming and bias correcting aren't worth

> the trouble."

>

>

> 2.3) Paulo suggested to use simulations to compute the necessary

> including the confidence intervals.

intervals

>

> In summary the idea is to:

>

> - work in the transformed scale

> - simulate from the predictive distribution

> - back-transform simulation

> - use simulations to compute estimations (mean, median, quantiles,

> etc.)

regarding

>

> This avoids the Lagrangian parameyet and provides accurate results

> the back-transformation. Furthermore this is valid for a wider group of

--

> transformation, not only for the log.

>

> 2.4) The approach used by Edzer in Journal of Hydrology 200, p. 364-386 was

>

> 1. estimate block mean concentrations on the log scale, and std.errors

> 2. calculate approximate 95% predictions intervals by est +/- 2 * std.err

> 3. back-transform both sides of the interval by taking the exponent.

>

> What results is not an interval estimate of the block mean value (which

> may be outside this interval!) but an estimate of the block geometric

> mean value. When, on the log scale block mean and block median coincide

> (e.g. when log-concentrations within a block are symmetrically distributed)

> this value coincides with the block median value.

>

>

> C. geostats packages & back-transformation of logarithms

> ++++++++++++++++++++++++++++++++++++++++++++++

>

> 3.1) Hirotaka added the unbiased transformation in GSLIB. By adding the

> following line into kt3d.f ,that kriging estimates are automatically

> backtransformed

>

> c

> c back transform for log transform

> c

> est=exp(est+estv/2-s(na+1))

>

> The file will be available online when received.

>

> 3.2) Edzer and Paulo reffered to GeoR that is doing the back

> transformation.geoR implements kriging and bayesian kriging including the

> family of Box-Cox transformation for which the log. See point 2.3. above.

>

> 3.3) Isobel mentioned that her software does such transformation. The

> software

> is a commercial package. http://geoecosse.bizland.com

>

>

> Gregoire

>

>

>

> --

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