Sebastiano,
Sorry, I do not argue that the layering approach is good. In fact if you
see that each layer has different variances, you have different
populations, and the splitting IS the solution. I argue that you will
get exactly the same result if you use the same variogram, either with
the standardized dataset or with the original one.
regards
Raimon
PS: regarding variogram estimation, recall that it depends on the
goodness of your standardizing variances (I hope that you took the sills
of the variograms within layers, and not the raw experimental variances...).
En/na Sebastiano Trevisani ha escrit:
Hi Raimon
I think that Isobel clarified my idea (that is no so different from
the one underling the use of the local relative variogram).
If you look at the stream of mails my question was only on the back
transformation from a standardized anomaly and not on the application
of the "layered approach" whose convenience is clear.
In particular I can resume things in this way.
Splitting the 3d dataset in horizontal layers you see that there is a
regular change in the horizontal spatial variability of the phenomena
according to z. In particular, if you calculate the horizontal
variogram along each layer you will find the same shape and range but
with a different variance (well, it is a simple case of
heteroscedasticity). Given this (it is not a my belief, but it is what
data are telling me), the standardization along each layer became
really useful:
1) you can perform a more robust estimation of the variogram (not for
the standardization itself but because I'm using more data)
2) you can use a 3D routine (remembering to choose a correct search
ellipsoid intersecting only one layer a time) to perform
the interpolation along all horizontal layers.
In regard to logarithmic transformation this is another point.....but
also in this case I let the data speak: and the lognormality could
(and should) be checked (in my case is not useful). The discussion on
data transformation is controversial....and really long.
Maybe with complex dataset (in the uni- and multi-variate case) it
could be useful to perform a fuzzy clustering on data and to work
on the fuzzy classes memberships........ using a kind of compositional
kriging (in relation to the non convexity).
Sebastiano
At 10.50 31/08/2006, Isobel Clark wrote:
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: [email protected]
>> 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]
<mailto:[EMAIL PROTECTED]>
>>> Behalf Of
>>> sebastiano trevisani
>>> Sent: Monday, August 28, 2006 9:57 AM
>>> To: Isobel Clark
>>> Cc: [email protected]
>>> 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 <http://www.kriging.com/>
>>>> Sebastiano Trevisani
>>>> 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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