Computation of a Simple Variogram

Let us consider the simplest case, that values of a regionalized variable only in one direction in space are at hand. For illustration we compute by hand an empirical variogram from a few values only. Let us denote the support points which have distances $d$ by $x_{i}$, and the corresponding values of the regionalized variable by $z(x_{i})$; then the data may be as seen in Figure 3.2.

Figure 3.2: Regionalized Variable.

The data are taken from samples of grades of copper from Illustration (v) of Subsection 2.2.7 (Table 2.7), the first 15 values. The computation of the variogram is done by averaging over all available pairs of data $n(h)$, namely

with $h=1 /times d, 2 /times d, 3 /times d, /ldots$. A ``manual'' computing scheme could look like in Table 3.1. For each selected $h$ a row of differences of the values is presented, and underneath the corresponding values squared. The sums, the number of used pairs $n(h)$ and the computed values $2 /gamma(h)$ are at the right margin. The resulting semi-variogram is shown in Figure 3.3.

Table 3.1: Calculation of a Simple Variogram.

		$x_1$		$x_2$		$x_3$		$x_4$		$x_5$		$x_6$		$x_7$		$x_8$		$x_9$		$x_{10}$		$x_{11}$		$x_{12}$		$x_{13}$		$x_{14}$		$x_{15}$	$/sum$	$n(h)$	$2 /hat{/gamma} (h)$
$z(x_{i})$		28		25		27		25		21		25		11		16		11		17		27		26		25		26		21
$h=50$
$z(x_{i})-z(x_{i+1})$		3		-2		2		4		-4		14		-5		5		-6		-10		1		1		-1		5
$(z(x_{i})-z(x_{i+1}))^{2}$		9		4		4		16		16		196		25		25		36		100		1		1		1		25			459	14	32.8
100
$z(x_{i})-z(x_{i+2})$		1		0		6		0		10		9		0		-1		-16		-9		2		0		4
$(z(x_{i})-z(x_{i+2}))^{2}$		1		0		36		0		100		81		0		1		256		81		4		0		16					576	13	44.3
150
$z(x_{i})-z(x_{i+3})$		3		4		2		14		5		14		-6		-11		-15		-8		15		5
$(z(x_{i})-z(x_{i+3}))^{2}$		9		16		4		196		25		196		36		121		225		64		1		25							918	12	76.5
200
$z(x_{i})-z(x_{i+4})$		7		0		16		9		10		8		-16		-10		-14		-9		6
$(z(x_{i})-z(x_{i+4}))^{2}$		49		0		256		81		100		64		256		100		196		81		36									1219	11	110.8
250
$z(x_{i})-z(x_{i+5})$		3		14		11		14		4		-2		-15		-9		-15		-4
$(z(x_{i})-z(x_{i+5}))^{2}$		9		196		121		196		16		4		225		81		225		16											1089	10	108.9
300
$z(x_{i})-z(x_{i+6})$		17		9		16		8		-6		-1		-14		-10		-10
$(z(x_{i})-z(x_{i+6}))^{2}$		289		81		256		64		36		1		196		100		100													1123	9	124.8
350
$z(x_{i})-z(x_{i+7})$		12		14		10		-2		-5		0		-15		-5
$(z(x_{i})-z(x_{i+7}))^{2}$		144		196		100		4		25		0		225		25															719	8	89.9

Figure 3.3: Computed Simple Variogram.