NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 II.4-1 Tony E. Smith 4. Variograms The covariogram and its normalized form, the correlogram, are by far the most intuitive methods for summarizing the structure of spatial dependencies in a covariance stationary process. However, from an estimation viewpoint such functions present certain difficulties (as will be discussed further in Section 4.10 below). Hence it is convenient to introduce a closely related function known as the variogram, which is widely used for estimation purposes. 4.1 Expected Squared Differences To motivate the notion of a variogram for a covariance stationary process, {(): }Ys s R, we begin by considering any pair of component variables, ( )sYYs and ( )vYYv , and computing their expected squared difference: (4.1.1) 22 22 2[( ) ] [ 2 ] ( ) 2 ( ) ( )sv s sv v s sv vEY Y EY YY Y EY EYY EY    To relate this to covariograms, note that if sv h , then by (3.2.3) and (3.2.4), (4.1.2) 2() cov( , ) [( )( )] [ ]sv s v sv s vCh YY E Y Y EYY Y Y 2()() ()sv s vEYY EY EY 222 2() ()sv svEYY EYY    2() ()svEYY Ch Exactly the same argument with sv shows that (4.1.3) 222() (0) ()svEYC EY Hence by substituting (4.1.2) and (4.1.3) into (4.1.1) we see that expected squared differences for all ,sv R with sv h can be expressed entirely in terms of the covariogram, C, as (4.1.4) 2[( ) ] 2 [ (0) ( )]svEY Y C Ch To obtain a slightly simpler relation, it is convenient to suppress the factor “2” by defining the associated quantity, (4.1.5) 212() [( )] ,svhEYY svh NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 II.4-2 Tony E. Smith and observing from (4.1.4) that with this definition we obtain the following simple identity for all distances , h : (4.1.6) 2() (0) () ()h C Ch Ch From (4.1.6) it is thus evident that the “scaled” expected squared differences in (4.1.5) define a unique function of distance which is intimately related to the covariogram. For any given covariance stationary process, this function is designated as the variogram, , of the process. Moreover, it is also evident that this variogram is uniquely constructible from the covariogram. But the converse is not true. In particular since (4.1.6) also implies that (4.1.7) 2() ()Ch h it is clear that in addition to the variogram, , one must also know the variance, 2, in order to construct the covariogram.1 Hence this variance will become an important parameter to be estimated in all models of variograms developed below. Before proceeded further with our analysis of variograms it is important to stress that the above terminology is not completely standard. In particular, the expected squared difference function in (4.1.4) is often designated as the “variogram” of the process, and its scaled version in (4.1.5) is called the “semivariogram” [as for example in Cressie (1993, p.58-59) and Gotway and Waller (2004, p.274)]. (This same convention is used in the Geostatistical Analyst extension in ARCMAP.) But since the scaled version in (4.1.5) is the only form used in practice [because of the simple identity in (4.1.7)] it seems most natural to use the simple term “variogram” for this function, as for example in [BG, p.162].2 4.2 The Standard Model of Spatial Dependence To illustrate the relation in (4.1.7) it is most convenient to begin with the simplest and most commonly employed model of spatial dependence. Recall from the Ocean Depth Example in Section 3.3.1 above, that the basic hypothesis there was that nearby locations tend to experience similar concentration levels of plankton, while those in more widely separated locations have little to do with each other. This can be formalized most easily in terms of correlograms by simply postulating that correlations are high (close to unity) for small distances, and fall monotonely to zero as distance increases. This same general hypothesis applies to a wide range of spatial phenomena, and shall be referred to here as the standard model of spatial dependence. Given the relation between correlograms and covariograms in (3.3.13), it follows at once that covariograms for the standard model, i.e., standard covariograms, must fall monotonely from 2(0)Ctoward zero, as illustrated 1 However, assuming that lim ( ) 0hCh , it follows from (4.1.6) that 2lim ( )hh . So 2is in principle obtainable from  as the asymptote (sill) in Figure 4.2 below. 2 See also the “lament” regarding this terminology in Schabenberger and Gotway (2005, p.135).NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 II.4-3 Tony E. Smith in Figure 4.1 below. The right end of this curve has intentionally been left rather vague. It may reach zero at some point, in which case covariances will be exactly zero at all greater distances. On the other hand, this curve may approach zero only asymptotically, so that covariance is positive at all distances but becomes arbitrarily small. Both cases are considered to be possible under the standard model (as will be illustrated in Section 4.6 below by the “spherical” and “exponential” variogram models). On the right in Figure 4.2 is the associated standard variogram, which by (4.1.6) above must necessarily start at zero and rise monotonely toward the value 2. Graphically this