( (*x *, *x *) = ( (*h*),

fitted by a smooth curve that represents a theoret-

1

2

ical variogram selected from a suite of possible

(2-20)

choices. Usually, the theoretical variogram is

(*u*1 & *u*2)2 % (*v*1 & *v*2)2

monotonically increasing, signifying that the far-

ther two observations are apart, the more their

residuals tend to differ, on average, from one

another. Several properties common to many

or possibly, on the lag and angle between locations

theoretical variograms are shown in Figure 2-2. If

the variogram either reaches or becomes asymp-

( (*x*1, *x*2) = ( (*h*, *a*),

totic to a constant value as *h *increases, that value

is called the **sill **(Figure 2-2). The distance (value

of *h*) after which the variogram remains at or

(2-21)

close to the sill is called the **range**. Measurements

whose locations are farther apart than the range all

have the same degree of association. Often, a

variogram will have a discontinuity at the origin,

signifying that even measurements very close

together are not identical. Such variation in the

(Figure 2-1). Equation 2-20 is called an **isotropic**

measurements at small scales is called the **nugget**

nugget. Although the nugget effect is sometimes

confused with measurement error, there is a subtle

difference between these two concepts that will be

explained in section 2-4. A simple monotonic

square of all computed differences between resid-

function is usually selected to approximate the

uals separated by a given lag:

variogram. Four such functions that are often used

in practice are:

1

( (*h*) =

^

ave

1

2

the **exponential variogram **(parameters: sill, *s *>

0; nugget, 0 < *g *< *s*; range, *r *> 0)

(2-22)

(

2

&*Z*

(*x *)

:

2

, *h*>0

( (*h*) =

(2-23)

where, as before, *h*ij is the distance between *x*i and

0,

rated by distance *h * )*h *are sampled and as )*h*

gets small, ( (*h*) should approach the theoretical

^

the **spherical variogram **(parameters: sill, *s *> 0;

variogram. More detail on variogram estimation

nugget, 0 < *g *< *s*; range, *r *> 0)

will be presented in Chapter 4, including the

directional case. In this section, it will be suffi-

cient to describe some general properties of iso-

tropic variograms that will be referred to numerous

times in the application sections to follow.

(2-24)

3

( (*h*) ' *g *% (*s *& *g*) 1.5 *h *& 0.5 *h*

, 0< *h*#*r*

often has a considerable degree of scatter (Fig-

0,

ure 2-2), which is especially evident if the sample

size *n *is small. However, the points can usually be

2-7

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