- Wow, thanks to everyone for their helpful comments and guidance to my first

ai-geostats posting.

Marco Alfaro, with regard to the ad-hoc nature of non-ergodic covariance:

Your solution (from NACOG 96): "consider a family of variograms from which a

single variogram is chosen for each local estimation problem. The choice of

each local variogram would be concerned with minimizing error at locations

with large kriging weights. A program that used a power-law variogram was

demonstrated. The exact power value was determined by doing cross-validation

within every search neighborhood."

It appears that this technique is computationally inaccessible to us

non-programmers at this point in time. Could it be argued that though

theoretically questionable non-ergodic covariance has some practical value

in that it successfully cleans up variograms? I apologize if this offends

"purists!"

Cheers,

Sara

From: "Marco Alfaro" <malfaro@...>

To: "Sara Kustron" <skustron@...>

Cc: <ai-geostats@...>

Sent: Thursday, March 15, 2001 6:41 PM

Subject: Re: AI-GEOSTATS: non-ergotic covariance

Dear Sara:

Sorry, but the "non ergodic" variogram is an artifact!

If you do not believe to me, see the comments about my paper in NACOG 1996

or in Geostatistics, Volume 8, No.2.

The Mathematical proof is in my paper titled "Acerca del variograma no

ergódico", in Spanish (difficult to get, if you wish I can

send a copy for you).

I think that is more easy to see an example (in my paper you have more

examples):

Let a line with data (the data is very regular and has a trend) sampled at

regular intervals of 1. Tha data are (20 data):

1, 1, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 12

Compute by hand or by your software the non ergodic and the classical

variogram (the classical variogram is also non

ergodic!):

Conclusions:

The behavior of the non ergodic variogram, near the origin is linear, and,

for the classical variogram is parabolic.

The non ergodic variogram has a range and a sill, and, the classical

variogram always grows.

I attach a little Qbasic (or QuickBasic) program you can run.

Best regards,

Marco Alfaro.

Listing of the program:

n = 20

DIM z(n)

m = 0 ' mean

v = 0 ' variance

FOR i = 1 TO n

READ z(i)

m = m + z(i)

v = v + z(i) * z(i)

NEXT i

m = m / n ' mean

' the variance is = the non ergodic variogram in the origin

v = v / n - m * m

CLS : SCREEN 12

WINDOW (-10, -10)-(30, 80)

LINE (0, 0)-(20, 65), 8, B

LOCATE 3, 10: PRINT "In red, classical variogram, in green, non ergodic

variogram"

FOR k = 0 TO n - 1

cov = 0 ' the non ergodic variogram.

head = 0

tail = 0

gama = 0 ' the classical variogram.

FOR i = 1 TO n - k

hh = z(i)

tt = z(i + k)

head = head + hh

tail = tail + tt

cov = cov + hh * tt

gama = gama + (hh - tt) * (hh - tt)

NEXT i

head = head / (n - k)

tail = tail / (n - k)

cov = cov / (n - k) - head * tail

gama = .5 * gama / (n - k)

CIRCLE (k, v - cov), .1, 2

PAINT (k, v - cov), 2

CIRCLE (k, gama), .1, 4

PAINT (k, gama), 4

NEXT k

a$ = INPUT$(1)

END

DATA 1,1,2,3,3,3,4,4,5,5,6,6,7,7,8,9,10,10,11,

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* Support to the list is provided at http://www.ai-geostats.org - on 16/03/01 2:24, Sara Kustron at skustron@... wrote:

> It appears that this technique is computationally inaccessible to us

It will be foolhardy to be a geostatistical "purist" in this day

> non-programmers at this point in time. Could it be argued that though

> theoretically questionable non-ergodic covariance has some practical value

> in that it successfully cleans up variograms? I apologize if this offends

> "purists!"

and age. :) Note that one would most likely assume local stationarity within

a search neigborhood in performing OK estimations, i.e. early lag

behavior only. I would (dangerously) suggest that in this early lag

period, choice of a variogram such as power law, spherical, exponential,

or Gaussian would be nit picking. Unless the variogram clearly shows

such definitive behavior, of course.

Syed

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