Wow, thanks to everyone for their helpful comments and guidance to my first
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
From: "Marco Alfaro" <malfaro@...
To: "Sara Kustron" <skustron@...
Sent: Thursday, March 15, 2001 6:41 PM
Subject: Re: AI-GEOSTATS: non-ergotic covariance
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
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
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.
Listing of the program:
n = 20
m = 0 ' mean
v = 0 ' variance
FOR i = 1 TO n
m = m + z(i)
v = v + z(i) * z(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
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)
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
a$ = INPUT$(1)
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