AI-GEOSTATS: Log-normal back transform in Webster & Oliver
In the book "Geostatistics for Environmental Scientists" By Richard Webster & Margaret A. Oliver, Wiley (2000), one will find in page 180 a brief discussion on the back-transformation of the kriging estimates. In ordinary kriging, when the natural logarithm (ln) is used, the back-transformation will involve the Lagrange parameter (see equation 8.36). No problem so far. But... the authors write in equation 8.38 that if one is using common logarithms (log10) instead, the unbiased back-transformation of the ordinary kriging estimates does not involve the Lagrange multiplier anymore. Is this correct ? In the affirmative, can someone point me to a paper discussing "natural" log normal kriging versus "common" log normal kriging ? Thanks again for any help. Regards, Gregoire -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: AI-GEOSTATS: Log-normal back transform in Webster & Oliver
Gregoire Thank you for pointing out the lognormal section in Webster & Oliver. I must confess I hadn't got round to looking at it in detail. Their simplification of the lognormal variance is based on the assumptions (see p.179) that: (a) the lagrangian multiplier would be close to zero if the mean is well known (b) the simple kriging weights would sum close to one if the data is dense enough The assumption (a) is one which has also been asserted by Peter Dowd in some of his publications. >From practical experience (over 30 years) we find that the lagrangian multiplier is seldom close to zero and, in fact, where data is dense will tend to be large and negative. We have also done some fairly intensive practical studies of simple kriging and found that, where data is dense, the kriging weights will tend to be very much greater than 1 so that the wieght applied to the "known" mean will be large and negative. Where data is sparse, weights sum to very much less than 1 so that poorly sampled areas are allocated the 'global' mean. Equations 8.35 and 8.39 rely on these assumptions and the implicit one that the only difference between the variance of the real values and that of the estimates is due to the simple kriging variance (i.e. no condiitonal bias). It has been asserted by several authors that simple kriging corrects for conditional bias. Would that that was true!! Equation 8.36 for ordinary kriging is correct, but we prefer to use Sichel's proper lognormal confidence intervals rather than back-transform the variance as shown in equation 8.37. To use this form you would have to assume that your errors were Normal even though your data was lognormal. I think there is a typo in equation 8.38 and the subscript 'Y' should be 'SK' to bring it into line with the other formulae. The definitive math on the lognormal backtransform can be found in Noel Cressie's book in equation 3.2.40 (for both types of kriging). Simpler explanations of the same form can be found in some of my papers at http://uk.geocities.com/drisobelclark/resume/Publications.html (note the capital P and look for papers in the second half of the 1990s). Isobel Clark http://geoecosse.bizland.com/whatsnew.htm Want to chat instantly with your online friends? Get the FREE Yahoo! Messenger http://uk.messenger.yahoo.com/ -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
RE: AI-GEOSTATS: Log-normal back transform in Webster & Oliver
Gregoire, I do not have the book with me right now, but what (you and) I do know is that ln(x)=ln(10)*log10(x) The difference is only a multiplication by a constant, so it cannot be true that one case does involve the Lagrange multiplier and the other does not. Gerard Gerard B.M. Heuvelink Wageningen University and Research Centre P.O. Box 47 6700 AA Wageningen The Netherlands tel +31 317 474628 / 482420 email [EMAIL PROTECTED] -Original Message- From: Gregoire Dubois [mailto:[EMAIL PROTECTED] Sent: maandag 28 juli 2003 11:39 To: [EMAIL PROTECTED] Subject: AI-GEOSTATS: Log-normal back transform in Webster & Oliver In the book "Geostatistics for Environmental Scientists" By Richard Webster & Margaret A. Oliver, Wiley (2000), one will find in page 180 a brief discussion on the back-transformation of the kriging estimates. In ordinary kriging, when the natural logarithm (ln) is used, the back-transformation will involve the Lagrange parameter (see equation 8.36). No problem so far. But... the authors write in equation 8.38 that if one is using common logarithms (log10) instead, the unbiased back-transformation of the ordinary kriging estimates does not involve the Lagrange multiplier anymore. Is this correct ? In the affirmative, can someone point me to a paper discussing "natural" log normal kriging versus "common" log normal kriging ? Thanks again for any help. Regards, Gregoire -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
AI-GEOSTATS: Comparison of semivariograms
Dear list members in both hemispheres; For my PhD thesis, I am studying the spatial variability of penetration resistance (a proxy for strength) in snow layers in an Alpine snow cover. Of specific interest are weak layers that are responsible for snow avalanche release. Are such weak layers less spatially variable than layers that are not critical for snow stability? To answer this, I want to compare the range, sill and nugget from model semivariograms calculated for all layers investigated. I also calculate mean and CV for each layer. Measurements: At 113 locations on each small slope (20m x 20m), measurements of penetration resistance were made in a nested grid with a spacing of 0.5m to 2m. The penetration resistance for each layer within the grid was recorded at all locations. I have data from approximately 100 layers. Weak layers were identified with separate tests within each grid. Analyses: The penetration resistance for each layer was log10 transformed to approach normality. Grid-scale trends for each layer were investigated with a (robust) linear regression on the x-y coordinates. In most layers, this trend was statistically significant, but often in different directions even in adjacent layers. The trend was removed to do a geostatistical analysis on the normally distributed residuals. A robust experimental semivariogram was calculated for the residuals for each layer. Now I want to fit a spherical model semivariogram to the experimental semivariograms. The spherical model fits the data from most layers better than other models. Questions: - Is it possible to compare directly the range, the sill and the nugget of the spherical model semivariograms fitted to the residuals of the linearly detrended data for each layer? - Are there any pit-falls that I should be aware of? (Should I test different semivariogram models for each layer? Should I leave the linear trend in the data for the structural analyses? ...) All comments, suggestions and references are welcome and will be much appreciated. I will be happy to provide more info if needed. Best regards, Kalle PS: No one was hurt during the measurements ;-) -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
