Mathematical Representation Of Semivariograms And Covariograms - 4.1 General

The mathematical models of semivariograms can be classified according to whether or not they correspond to situations where the second order stationarity conditions are satisfied (§ 3.3.4). If second order stationarity exists, the a priori variance s2 of x (z) also exists and is finite. For large distances h such that x (z) and x (z + h) are not correlated, the semi- variogram ? (h) will reach a value equal to s2 [this is a consequence of (3.27) above]. This limiting value is called the sill C of the semivariogram, and the distance at which ? (h) reaches the sill is called the range. The range of the semi- variogram obtained from a given type of sample corresponds to the distance of influence of these samples. If only the intrinsic hypothesis is satisfied, the sample variance is an increasing function of the size of the area or volume in which the samples are located, and the semivariogram ? (h) will be a continuously increasing function of h. Remark: It will be shown in Chapter 7 that the variance s2 of x (z) in the ore body O is equal to the average value of ? (h) in O. If ? (h) is a non-decreasing function with sill C, the average value of ? (h) in O will be less than C. s2 and C will be equal only if for some values of h in O, ? (h) exceeds C. In practice, if the range of the semivariogram is small with respect to the dimensions of the ore body, the difference between C and s2 will be negligible.