Variogram

In spatial statistics the theoretical variogram 2 γ ( s 1 , s 2 ) {displaystyle 2gamma (mathbf {s} _{1},mathbf {s} _{2})} is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z ( s ) {displaystyle Z(mathbf {s} )} . ∑ i = 1 N ∑ j = 1 N w i γ ( s i , s j ) w j ≤ 0 {displaystyle sum _{i=1}^{N}sum _{j=1}^{N}w_{i}gamma (mathbf {s} _{i},mathbf {s} _{j})w_{j}leq 0} 2 γ ( s 1 , s 2 ) = C ( s 1 , s 1 ) + C ( s 2 , s 2 ) − 2 C ( s 1 , s 2 ) {displaystyle 2gamma (mathbf {s} _{1},mathbf {s} _{2})=C(mathbf {s} _{1},mathbf {s} _{1})+C(mathbf {s} _{2},mathbf {s} _{2})-2C(mathbf {s} _{1},mathbf {s} _{2})} 2 γ ( s 1 , s 2 ) = C ( s 1 , s 1 ) + C ( s 2 , s 2 ) − 2 C ( s 1 , s 2 ) + ( E [ Z ( s 1 ) ] − E [ Z ( s 2 ) ] ) 2 {displaystyle 2gamma (mathbf {s} _{1},mathbf {s} _{2})=C(mathbf {s} _{1},mathbf {s} _{1})+C(mathbf {s} _{2},mathbf {s} _{2})-2C(mathbf {s} _{1},mathbf {s} _{2})+(Eleft-Eleft)^{2}} In spatial statistics the theoretical variogram 2 γ ( s 1 , s 2 ) {displaystyle 2gamma (mathbf {s} _{1},mathbf {s} _{2})} is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z ( s ) {displaystyle Z(mathbf {s} )} . In the case of a concrete example from the field of gold mining, a variogram will give a measure of how much two samples taken from the mining area will vary in gold percentage depending on the distance between those samples. Samples taken far apart will vary more than samples taken close to each other. The semivariogram γ ( h ) {displaystyle gamma (h)} was first defined by Matheron (1963) as half the average squared difference between points ( s 1 {displaystyle mathbf {s} _{1}} and s 2 {displaystyle mathbf {s} _{2}} ) separated at distance h {displaystyle h} . Formally where M {displaystyle M} is a point in the geometric field V {displaystyle V} , and f ( M ) {displaystyle f(M)} is the value at that point. For example, suppose we are interested in iron content in soil samples in some region or field V {displaystyle V} . f ( M ) {displaystyle f(M)} would be the content (e.g., in mg iron per kg soil) of iron at some location M {displaystyle M} , where M {displaystyle M} has coordinates of latitude, longitude, and depth. The triple integral is over 3 dimensions. h {displaystyle h} is the separation distance (e.g., in m or km) of interest. To obtain the semivariogram for a given γ ( h ) {displaystyle gamma (h)} , all pairs of points at that exact distance would be sampled. In practice it is impossible to sample everywhere, so the empirical variogram is used instead. The variogram is defined as the variance of the difference between field values at two locations ( s 1 {displaystyle mathbf {s} _{1}} and s 2 {displaystyle mathbf {s} _{2}} , note change of notation from M {displaystyle M} to s {displaystyle mathbf {s} } and f {displaystyle f} to Z {displaystyle Z} ) across realizations of the field (Cressie 1993): or in other words is twice the semivariogram. If the spatial random field has constant mean μ {displaystyle mu } , this is equivalent to the expectation for the squared increment of the values between locations s 1 {displaystyle mathbf {s} _{1}} and s 2 {displaystyle s_{2}} (Wackernagel 2003) (where s 1 {displaystyle mathbf {s} _{1}} and s 2 {displaystyle mathbf {s} _{2}} are points in space and possibly time): In the case of a stationary process, the variogram and semivariogram can be represented as a function γ s ( h ) = γ ( 0 , 0 + h ) {displaystyle gamma _{s}(h)=gamma (0,0+h)} of the difference h = s 2 − s 1 {displaystyle h=mathbf {s} _{2}-mathbf {s} _{1}} between locations only, by the following relation (Cressie 1993): If the process is furthermore isotropic, then the variogram and semivariogram can be represented by a function γ i ( h ) := γ s ( h e 1 ) {displaystyle gamma _{i}(h):=gamma _{s}(he_{1})} of the distance h = ‖ s 2 − s 1 ‖ {displaystyle h=|mathbf {s} _{2}-mathbf {s} _{1}|} only (Cressie 1993):