Re: AI-GEOSTATS: Re: standardized anomaly

[Isobel Clark]

Raimon

 

I might have got this wrong but I understood that Sebastiano was going to standardise each layer separately -- to its own mean and standard deviation -- so as to use data from all layers to construct a standardised semi-variogram. This could then be used on all layers, in 3D, using a constrained search so that only samples from the same layer are used in the kriging.

 

In short, he believes that the spatial continuity is the same in each layer, but the mean and variability change. Your thought on this was the same as my initial thoughts.

 

Isobel

http://uk.geocities.com/drisobelclark

Raimon Tolosana <[EMAIL PROTECTED]> wrote:

Hi Sebastiano,

first, I'd like to ask what do you mean when you say that you'll
"conduct interpolation along layers". If you mean that you will
interpolate within a layer using only the data in that layer, then let
me insist that then you MUST obtain the same results either using the
standardized or the original variable. Otherwise, ordinary kriging
wouldn't deserve the BLU character! In details, we probably agree in the
first sentence of each of these two paragraphs:

1.- OK gives the same results if conducted with the variogram than if
conducted with the equivalent correlogram, because the OK weights do not
depend on the value of the sill of the variogram. This implies that you
can multiply your data by a constat (e.g., the inverse of the standard
deviation), and divide the results by the same constant, and nothing
will change

2.- the kriging weights do not depend on the data values themselves, but
on the variogram. The experimental variogram of the data set does not
change if one adds or substracts a constant from the data set (e.g., its
mean), because it is computed with differences of data pairs (which
cancel the constant effect). Therefore, you can add a constant to your
data set, perform OK on the modified data set, and subtract the constant
from the kriging result, and again nothing will change.

I suspect that the only advantadge of standardizing by layers is that
you can get an (apparently) better estimate of the variogram, because
you will have less variance for each lag distance. And I say
"apparently", because this variogram will strongly depend on your
variance estimates for each layer, which we will agree that do not have
their nice properties in the presence of spatial correlation.

I don't like to be a party pooper... :-( So, after trying to spoil
your joy, let me ask what about applying a logarithm, if the data are
positive? and we may follow the discussion prompted by Gregoire ;-)

Raimon

En/na sebastiano trevisani ha escrit:
> Hi Bill
> Yes, my idea is to conduct interpolations along layers (well,
> performing a "tricky" 3D interpolation only to speed up the process).
> Well, I'll already know that the shape and the range of the horizontal
> variograms along Z doesn't change too much. I have some
> doubt about anisotropy ...but I think that there are too few samples
> on the horizontal plane to take seriously care about that....
> Then if the possible anisotropy of horizontal variogram changes with
> depth we are in troubles......in the sense that I should calculate
> manually (or with some automatic algorithms) for each layers a
> variogram ...... and I can no more use the "tricky" 3D interpolation
> idea. So, your point about anisotropy is really important.
>
> Bye
> Sebastiano
>
> At 14.33 28/08/2006, Bill Northrop wrote:
>> Hullo Sebastiano,
>>
>> It sounds as if the Isobel's suggestion of a limited 3D search is the
>> best solution.
>>
>> The resultant models per layer should tell you if your approach has
>> been correct, especially if you do a trial run with an anisotropic
>> model and search first to see what spatial pattern you obtain.
>> Will be interested to know what you get.
>>
>> Regards
>> Bill Northrop
>>
>> -----Original Message-----
>> From: sebastiano trevisani [ mailto:[EMAIL PROTECTED]
>> Sent: Monday, August 28, 2006 2:06 PM
>> To: Bill Northrop
>> Cc: ai-geostats@jrc.it
>> Subject: RE: AI-GEOSTATS: Re: standardized anomaly
>>
>> Hi Bill
>> Thank you for your mail.
>> In my case of study there are not sharp boundaries (or at least
>> it seems so!) but there is a gradual and fast decrease in
>> horizontal spatial variability going in depth.
>> Sincerely
>> Sebastiano
>>
>> At 12.48 28/08/2006, you wrote:
>>> Good morning Sebastiano,
>>> I found your problem interesting and I thought I would
>>> respond in this fashion. I have done quite a bit of research
>>> on similar layered databases on fluvial mineral deposits and
>>> found that if one did vertical (at right angles to the
>>> contacts of the layers) variograms on the raw data and
>>> obtained a variogram with no drift. then one could be sure
>>> that all these layers you have split your data have similar
>>> spatial characteristics. It would then not be necessary to
>>> examine the horizontal spatial characteristics of each
>>> individual layer, but rather have one standardized variogram
>>> for all of them. If the reverse is true ie drift in the
>>> vertical variogram, then one must look critically at the
>>> data for some phenominum on which one can subdivide. For
>>> instance in fluvial (river) deposits different material
>>> types, drastically different particle size etc according to
>>> what you are studying. I found generally that the lag
>>> distance at which the drift commenced was the width of the
>>> thinnest horizon in the case of two different populations,
>>> but it does not tell you whether it is the top or bottom
>>> layer. This must then be done by scrutinization of your data
>>> in the vetical plane. Once your data is split you can then
>>> do variography on each one of the two layers in the
>>> horizontal plane modelling the anistropy of the variance
>>> separately, This should only be done once you have again
>>> checked these two layers with vertical variograms for drift.
>>> If there are more than two populations present then the
>>> process can be repeated until all your layers have vertical
>>> variograms with no drift and therefore you have split your
>>> data correctly.
>>>
>>> Hope this helps
>>>
>>> Regards
>>>
>>> Bill Northrop
>>>
>>> -----Original Message-----
>>> From: [EMAIL PROTECTED] [
>>> mailto:[EMAIL PROTECTED]
>>> Behalf Of
>>> sebastiano trevisani
>>> Sent: Monday, August 28, 2006 9:57 AM
>>> To: Isobel Clark
>>> Cc: ai-geostats@jrc.it
>>> Subject: Re: AI-GEOSTATS: Re: standardized anomaly
>>>
>>> Hi Isobel
>>> I would like to use this transformation to deal with a
>>> 3D data set characterized by a peculiarity (well, this
>>> is quite common!) in the horizontal spatial variability.
>>> In particular if I divide the dataset in horizontal
>>> layers I see that horizontal variograms show a similar
>>> shape but with a re-scaled variance.
>>> So, my idea, in order to speed up the process of
>>> interpolation, consists to calculate the standardized
>>> anomaly for each layer and use the same calculated
>>> variogram (well, now it is a kind of standardized
>>> variogram calculated using all layers)) during
>>> interpolation with a 3D routine. Yes, in reality this is
>>> only a trick ...because I`m simply performing a series
>>> of 2D interpolations along layers. This because of, once
>>> the data have been transformed, it is not reasonable to
>>> use during interpolation samples coming from different
>>> horizontal layers.........
>>> Sincerely
>>> Sebastiano
>>> At 14.06 25/08/2006, Isobel Clark wrote:
>>>> Sebastiano
>>>>
>>>> You will be fine so long as you actually have a
>>>> "stationary" phenomenon. That is, there is a
>>>> constant mean and standard deviation over your
>>>> study area -- no trends, no discontinuities, no
>>>> changes of behaviour. Such a transformation also
>>>> assumes that your data follow a fairly symmetrical
>>>> histogram.
>>>>
>>>> Your semi-variogram will look exaclty the same as
>>>> your 'raw' data semi-variogram but should have a
>>>> sill around 1.
>>>>
>>>> Isobel
>>>> http://www.kriging.com
>>>> Sebastiano Trevisani
>>>> <[EMAIL PROTECTED]>wrote:
>>>>
>>>> Dear list member
>>>> A procedural question for you.......
>>>> I'm thinking to transform my data in a
>>>> standardized anomaly [i.e.
>>>> (raw datum- sample average)/sample standard
>>>> deviation)] and then I`ll
>>>> perfom the geostatistical analysis on these
>>>> transformed data. At
>>>> first glance, I don't see problem in the
>>>> back-transformation of
>>>> interpolated data and in the correct evaluation
>>>> of estimation
>>>> variance. Am I wrong?
>>>> Sincerely
>>>> Sebastiano
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