Re: AI-GEOSTATS: kriging or IDW in case study of hydrology?
Hi Isobel I didn't know the existence of the two schools of thought! So thanks for the clarification. The point is the I interpret smoothing (filtering) properties of kriging by means of the dual representation of kriging interpolator given for example in Goovaerts's book Geostatistics for natural resources evaluation. According to this representation an estimate is equal to the mean plus a linear combination of covariance terms (nugget effect included). Clearly according to this representation if you want to filter out nugget effect, making your interpolator no more exact, you need to filter out the nugget explicitly. Then, I'm wondering if in presence of really short range components of spatial continuity (compared to the overall range of spatial continuity) It could be better to use a block kriging (with a block size equal to range of the short range components) in order to get more spatially homogeneous maps from the point of view of smoothing (above all with high clustered and low density data sampling geometry). Sebastiano At 14.35 20/02/2008, you wrote: Just to clarify the point about the nugget effect. The nugget effect reflects the uncertainty (or randomness) which cannot be removed from the phenomenon. No matter how closely you sample, there will generally be some difference between values. I have had a couple of chances to be involved with 'contiguous' sampling exercises, where samples are taken side-by-side. Part of the nugget effect is the inherent variability of the phenomenon being measured. The nugget effect will also include 'random' errors introduced by the sampling and/or analysis of the samples. The value allocated to the nugget effect is the intercept of your semi-variogram model on the axis -- if you let it hit the axis (zero distance). There seem to be two schools of thought on what value the semi-variogram actually takes at zero distance: (1) the nugget effect exists at all distances, except zero. This is the basis on which Matheron originally developed his theory of regionalised variables. At zero distance (exactly) the semi-variogram is zero. At all other distances, the nugget effect is included in the model. (2) the nugget effect exists at zero distance. The implication of this is that the nugget effect is all sampling error. Software packages vary according to which of the above they implement. My personal preference is for the former mainly because it gives (strangely enough) more conservative confidence levels. If your software package does (2) but you want to do (1), you can achieve this by adding (say) a spherical component with a very short range of influence and a sill equal to the nugget effect. The ONLY effect this will have will be to change the value taken at zero distance. For any other distance, the nugget effect will exist exactly as before. For those of you who prefer algebra: (1) gamma(0) = 0 gamma(h) = C0+spatial component for all h not equal to 0 Guarantees exact interpolation and honours the data. (2) gamma(0) = C0 gamma(h) = C0+spatial component for all h not equal to 0 Smooths estimates close to sampled locations but does not (necessarily) honour the data. To be absolutely clear, I do not advocate ignoring the nugget effect. That is why I use approach (1). Approach (2) effectively removes the nugget effect from the kriging system and reduces the kriging variance significantly. My point is that you should know what your software is doing at zero and the effect that can have on your estimation. Peter's other point is exactly correct. If you map with a grid of points and your sampled locations are not on that grid, the kriging will have no possibility to honour the data -- since the data points are never 'estimated'. Isobel http://www.kriging.comhttp://www.kriging.com
Re: AI-GEOSTATS: kriging or IDW in case study of hydrology?
Sebastiano, exactly. So there are 2 sources of smoothing: 1) the nugget, whose purpose in interpolation is to account for unresolved (by the sampling scheme) variability and / or data uncertainty, which means, in effect, smoothing. If for some reason one does not want this, simply set it to zero by introducing an artificial variogram component, as Isobel said. This component can be either ad-hoc, as in Isobel's example, or based on some a-priori physical knowledge of the phenomenon. 2) The distance of grid nodes which is necessarily 0 and in general does not coincide with the sample locations. This effect can be reduced by choosing smaller grid node distance. - This should however also be done with care, because what is the point of choosing a grid distance of 100 m if the average distance of sample points is 10 km. This would lead, while better honouring the sample points, again to an unrealistic smooth surface. (For this reason, b.t.w., I recommend normally not to use the interpolate Pixels option in the Image Map Properties dialogue of Surfer software. Leaving the pixels visible may be less beautiful, but is more correct in my view, because it evokes to a much lesser degree the illusion of a smooth surface at a spatial scale on which there is no information.) Another caveat: if the empirical variogram shows high nugget, introducing an artificial component with nugget=0, and choosing narrow grid nodes, can lead to a bull-eye pattern, which clearly shows that something went wrong. Peter seba [EMAIL PROTECTED] writes: Hi Well, some time you have the impression that kriging is not an exact interpolator because of you have a high nugget effect and the interpolation grid nodes have not the same location of available data. The variability represented by the nugget effect is filtered every time an interpolation location doesn't coincide with known data. So the apparent smoothing effect is related to the raster representation of reality. Sebastiano T. At 09.56 19/02/2008, you wrote: Dear list, I'm graduate student in hydrogeology, I've to spatialize data of reservoir thickness, and I need to achieve a map having exactly the sampled value in the sampled localization (piezometers). I've little experience in geostatatistics. I had a look at kriging algorithms, but I did understand that kriging does not preserve the sampled value at sampled locations but it tends to smooth results, even if estimates correctly the unsampled space. So I wonder why should I use Kriging instead IDW (which it should preserve my sampled values): kriging respects the spatial variability but do not respect data As I told you before, I've very small knowledge in geostatistics stuff, but I'm interesting in kriging. Could anyone help me? Thanks a lot, Andrea Peruzzi PS: I apologize for writing you again but it's the first time I'm writing you, then I'm not sure how the mailing list works. Thanks :-) - Peter Bossew European Commission (EC) Joint Research Centre (JRC) Institute for Environment and Sustainability (IES) TP 441, Via Fermi 1 21020 Ispra (VA) ITALY Tel. +39 0332 78 9109 Fax. +39 0332 78 5466 Email: [EMAIL PROTECTED] WWW: http://rem.jrc.cec.eu.int The views expressed are purely those of the writer and may not in any circumstances be regarded as stating an official position of the European Commission. + + To post a message to the list, send it to ai-geostats@jrc.it + To unsubscribe, send email to majordomo@ jrc.it with no subject and unsubscribe ai-geostats in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list + As a general service to list users, please remember to post a summary of any useful responses to your questions. + Support to the forum can be found at http://www.ai-geostats.org/
Re: AI-GEOSTATS: kriging or IDW in case study of hydrology?
A strong nugget may be the exception rather than the norm for thickness data. You can try cross-validating kriged thickness results based on some a priori variogram model to see whether your estimates of thicknesses can be improved using a spatial correlation model. Syed On 2/20/08, Peter Bossew [EMAIL PROTECTED] wrote: Sebastiano, exactly. So there are 2 sources of smoothing: 1) the nugget, whose purpose in interpolation is to account for unresolved (by the sampling scheme) variability and / or data uncertainty, which means, in effect, smoothing. If for some reason one does not want this, simply set it to zero by introducing an artificial variogram component, as Isobel said. This component can be either ad-hoc, as in Isobel's example, or based on some a-priori physical knowledge of the phenomenon. 2) The distance of grid nodes which is necessarily 0 and in general does not coincide with the sample locations. This effect can be reduced by choosing smaller grid node distance. - This should however also be done with care, because what is the point of choosing a grid distance of 100 m if the average distance of sample points is 10 km. This would lead, while better honouring the sample points, again to an unrealistic smooth surface. (For this reason, b.t.w., I recommend normally not to use the interpolate Pixels option in the Image Map Properties dialogue of Surfer software. Leaving the pixels visible may be less beautiful, but is more correct in my view, because it evokes to a much lesser degree the illusion of a smooth surface at a spatial scale on which there is no information.) Another caveat: if the empirical variogram shows high nugget, introducing an artificial component with nugget=0, and choosing narrow grid nodes, can lead to a bull-eye pattern, which clearly shows that something went wrong. Peter seba [EMAIL PROTECTED] writes: Hi Well, some time you have the impression that kriging is not an exact interpolator because of you have a high nugget effect and the interpolation grid nodes have not the same location of available data. The variability represented by the nugget effect is filtered every time an interpolation location doesn't coincide with known data. So the apparent smoothing effect is related to the raster representation of reality. Sebastiano T. At 09.56 19/02/2008, you wrote: Dear list, I'm graduate student in hydrogeology, I've to spatialize data of reservoir thickness, and I need to achieve a map having exactly the sampled value in the sampled localization (piezometers). I've little experience in geostatatistics. I had a look at kriging algorithms, but I did understand that kriging does not preserve the sampled value at sampled locations but it tends to smooth results, even if estimates correctly the unsampled space. So I wonder why should I use Kriging instead IDW (which it should preserve my sampled values): kriging respects the spatial variability but do not respect data As I told you before, I've very small knowledge in geostatistics stuff, but I'm interesting in kriging. Could anyone help me? Thanks a lot, Andrea Peruzzi PS: I apologize for writing you again but it's the first time I'm writing you, then I'm not sure how the mailing list works. Thanks :-) - Peter Bossew European Commission (EC) Joint Research Centre (JRC) Institute for Environment and Sustainability (IES) TP 441, Via Fermi 1 21020 Ispra (VA) ITALY Tel. +39 0332 78 9109 Fax. +39 0332 78 5466 Email: [EMAIL PROTECTED] WWW: http://rem.jrc.cec.eu.int The views expressed are purely those of the writer and may not in any circumstances be regarded as stating an official position of the European Commission. + + To post a message to the list, send it to ai-geostats@jrc.it + To unsubscribe, send email to majordomo@ jrc.it with no subject and unsubscribe ai-geostats in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list + As a general service to list users, please remember to post a summary of any useful responses to your questions. + Support to the forum can be found at http://www.ai-geostats.org/
Re: AI-GEOSTATS: kriging or IDW in case study of hydrology?
Hi Well, some time you have the impression that kriging is not an exact interpolator because of you have a high nugget effect and the interpolation grid nodes have not the same location of available data. The variability represented by the nugget effect is filtered every time an interpolation location doesn't coincide with known data. So the apparent smoothing effect is related to the raster representation of reality. Sebastiano T. At 09.56 19/02/2008, you wrote: Dear list, I'm graduate student in hydrogeology, I've to spatialize data of reservoir thickness, and I need to achieve a map having exactly the sampled value in the sampled localization (piezometers). I've little experience in geostatatistics. I had a look at kriging algorithms, but I did understand that kriging does not preserve the sampled value at sampled locations but it tends to smooth results, even if estimates correctly the unsampled space. So I wonder why should I use Kriging instead IDW (which it should preserve my sampled values): kriging respects the spatial variability but do not respect data As I told you before, I've very small knowledge in geostatistics stuff, but I'm interesting in kriging. Could anyone help me? Thanks a lot, Andrea Peruzzi PS: I apologize for writing you again but it's the first time I'm writing you, then I'm not sure how the mailing list works. Thanks :-) + + To post a message to the list, send it to ai-geostats@jrc.it + To unsubscribe, send email to majordomo@ jrc.it with no subject and unsubscribe ai-geostats in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list + As a general service to list users, please remember to post a summary of any useful responses to your questions. + Support to the forum can be found at http://www.ai-geostats.org/ + + To post a message to the list, send it to ai-geostats@jrc.it + To unsubscribe, send email to majordomo@ jrc.it with no subject and unsubscribe ai-geostats in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list + As a general service to list users, please remember to post a summary of any useful responses to your questions. + Support to the forum can be found at http://www.ai-geostats.org/
Re: AI-GEOSTATS: kriging or IDW in case study of hydrology?
Isobel, thanks for clarification. As to (1) vs. (2): I think it really depends on the physical nature of the variable which one tries to model. If you have exact data (i.e. intrinsic + longitudinal uncertainty very small compared to value or abs(value)) == (1) If you have intrinsic = measuring/sampling unc., or long. unc. = replication unc. (e.g. temporal; or a phenomenon which we are currently fighting with, indoor Rn values taken in different rooms of one house, i.e. at one point), which are not negligible == (2) Please let me know if you think I am wrong with this. Peter Isobel Clark [EMAIL PROTECTED] writes: Just to clarify the point about the nugget effect. The nugget effect reflects the uncertainty (or randomness) which cannot be removed from the phenomenon. No matter how closely you sample, there will generally be some difference between values. I have had a couple of chances to be involved with 'contiguous' sampling exercises, where samples are taken side-by-side. Part of the nugget effect is the inherent variability of the phenomenon being measured. The nugget effect will also include 'random' errors introduced by the sampling and/or analysis of the samples. The value allocated to the nugget effect is the intercept of your semi-variogram model on the axis -- if you let it hit the axis (zero distance). There seem to be two schools of thought on what value the semi-variogram actually takes at zero distance: (1) the nugget effect exists at all distances, except zero. This is the basis on which Matheron originally developed his theory of regionalised variables. At zero distance (exactly) the semi-variogram is zero. At all other distances, the nugget effect is included in the model. (2) the nugget effect exists at zero distance. The implication of this is that the nugget effect is all sampling error. Software packages vary according to which of the above they implement. My personal preference is for the former mainly because it gives (strangely enough) more conservative confidence levels. If your software package does (2) but you want to do (1), you can achieve this by adding (say) a spherical component with a very short range of influence and a sill equal to the nugget effect. The ONLY effect this will have will be to change the value taken at zero distance. For any other distance, the nugget effect will exist exactly as before. For those of you who prefer algebra: (1) gamma(0) = 0 gamma(h) = C0+spatial component for all h not equal to 0 Guarantees exact interpolation and honours the data. (2) gamma(0) = C0 gamma(h) = C0+spatial component for all h not equal to 0 Smooths estimates close to sampled locations but does not (necessarily) honour the data. To be absolutely clear, I do not advocate ignoring the nugget effect. That is why I use approach (1). Approach (2) effectively removes the nugget effect from the kriging system and reduces the kriging variance significantly. My point is that you should know what your software is doing at zero and the effect that can have on your estimation. Peter's other point is exactly correct. If you map with a grid of points and your sampled locations are not on that grid, the kriging will have no possibility to honour the data -- since the data points are never 'estimated'. Isobel [ http://www.kriging.com ]http://www.kriging.com - Peter Bossew European Commission (EC) Joint Research Centre (JRC) Institute for Environment and Sustainability (IES) TP 441, Via Fermi 1 21020 Ispra (VA) ITALY Tel. +39 0332 78 9109 Fax. +39 0332 78 5466 Email: [EMAIL PROTECTED] WWW: http://rem.jrc.cec.eu.int The views expressed are purely those of the writer and may not in any circumstances be regarded as stating an official position of the European Commission. + + To post a message to the list, send it to ai-geostats@jrc.it + To unsubscribe, send email to majordomo@ jrc.it with no subject and unsubscribe ai-geostats in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list + As a general service to list users, please remember to post a summary of any useful responses to your questions. + Support to the forum can be found at http://www.ai-geostats.org/
AI-GEOSTATS: kriging or IDW in case study of hydrology?
Dear list, I'm graduate student in hydrogeology, I've to spatialize data of reservoir thickness, and I need to achieve a map having exactly the sampled value in the sampled localization (piezometers). I've little experience in geostatatistics. I had a look at kriging algorithms, but I did understand that kriging does not preserve the sampled value at sampled locations but it tends to smooth results, even if estimates correctly the unsampled space. So I wonder why should I use Kriging instead IDW (which it should preserve my sampled values): kriging respects the spatial variability but do not respect data As I told you before, I've very small knowledge in geostatistics stuff, but I'm interesting in kriging. Could anyone help me? Thanks a lot, Andrea Peruzzi PS: I apologize for writing you again but it's the first time I'm writing you, then I'm not sure how the mailing list works. Thanks :-) + + To post a message to the list, send it to ai-geostats@jrc.it + To unsubscribe, send email to majordomo@ jrc.it with no subject and unsubscribe ai-geostats in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list + As a general service to list users, please remember to post a summary of any useful responses to your questions. + Support to the forum can be found at http://www.ai-geostats.org/
Re: AI-GEOSTATS: kriging or IDW in case study of hydrology?
Andrea In theory kriging will honour the sample values provided your semi-variogram model takes the value zero at zero distance. Whether the data are honoured or not depends on which computer package you use and what it does with the semi-variogram at zero. You can force this behaviour by replacing any nugget effect with a short range model component. For example a spherical component with a range of influence of 10cm or some such. See our completely free and public domain kriging game, for how the kriging system works. By the way, IDW will only honour your sample values if the algorithms are written with the same criterion. Isobel http://www.kriging.com Andrea Peruzzi [EMAIL PROTECTED] wrote: Dear list, I'm graduate student in hydrogeology, I've to spatialize data of reservoir thickness, and I need to achieve a map having exactly the sampled value in the sampled localization (piezometers). I've little experience in geostatatistics. I had a look at kriging algorithms, but I did understand that kriging does not preserve the sampled value at sampled locations but it tends to smooth results, even if estimates correctly the unsampled space. So I wonder why should I use Kriging instead IDW (which it should preserve my sampled values): kriging respects the spatial variability but do not respect data As I told you before, I've very small knowledge in geostatistics stuff, but I'm interesting in kriging. Could anyone help me? Thanks a lot, Andrea Peruzzi PS: I apologize for writing you again but it's the first time I'm writing you, then I'm not sure how the mailing list works. Thanks :-) + + To post a message to the list, send it to ai-geostats@jrc.it + To unsubscribe, send email to majordomo@ jrc.it with no subject and unsubscribe ai-geostats in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list + As a general service to list users, please remember to post a summary of any useful responses to your questions. + Support to the forum can be found at http://www.ai-geostats.org/
