Olumide
A few concepts and terminologies intermixed with one another when it comes to universal kriging:
In universal kriging, the original random function can be decomposed into two components as follow:
Y(x)=m(x)+W(x) m(x): Deterministic component, E[Y(x)]=m(x) W(x): Stochastic component, E[W(x)]=0A majority of geostatistician support the idea that convariance function cannot be defined for Y(x) here, i.e, R[Y(x),Y(x')]=R_Y(x,x') is not available. As a result, universal kriging system will be defined in terms of residual covariance function, i.e., R[W(x),W(x')]=R_W(x,x'). Here it is implicitly assumed that R_W(x,x') will be a function of separation vector (at most). An "egg and chicken" problem will arise with regard to characterising residual covariance function. Please look at mailing list archieve.
A few scholars totally dismissed this option and will go for defining generalized covariance function where residual will be considered to be an admissible linear combination (ALC), i.e., sum of weight equal to zero. Look at residual structure:
R(x_0)=\sum_{i=1}^n\lambda_iY(x_i)-Y(x_0)
This could easily be written as:
R(x_0)=\sum_{i=0}^n\lambda_iY(x_i) where \lambda_0=-1
Here as \sum_{i=0}^n\lambda_i=0, variance of R(x_0) can be quantified in
terms of generalized covariance function whereby K(x,x')=-\gamma(x,x') for
first increment.
You can see how increment will come into play while working with universal kriging.
You need to further differentiate between Stationary Random function (SRF) and Intrinsic Random function (IRF). Look at how they can be differentiated from one another:
if Y(x) is SRF, then: 1. E[Y(x)]=m independent of spatial location 2. R(x,x') is a function of separation vector 3. Var[Y(x)] is finite and independent of spatial location If Y(x) is IRF, then:1. E[Y(x)-Y(x')]=m(h), i.e., expected value of first increment is a function of separation vector. In a majority of cases m(h)=0
2. \gamma(x,x')=\gamma(h), i.e., variogram function is a function of separation vecotr.
Please let me know if you need further explanation so that I could pass a few papers in this regard.
Hope this helps. Thanks Abedini On Fri, 29 Jun 2007, Olumide wrote:
Hello -Its me again, the newbie. I'm teaching myself geostatistics (its a cool subject. Jan Merks brings to mind Albert Einstein, and he refused to accept Quantum Mechanics).I'm having trouble understanding the terms, intrinsic and incremental. I've encountered them in texts on universal kriging (I think). Would someone offer a simple explanation of these terms?Many thanks, - OlumidePS: Richardo Olea, if you're reaching this. Thanks for writing such a truly remarkable book (Geostatistics for Engineers and Earth Scientists). Your text left few stones unturned and answered most of the questions I had. I only wish you'd gone further.+ + To post a message to the list, send it to [email protected]+ 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 [email protected] + 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/
