CHAPTER 4 THE THEORY OF REGIONALIZED VARIABLES 4.1 Introducton It s ponted out by Armstrong (1998 : 16) that Matheron (1963b), realzng the sgnfcance of the spatal aspect of geostatstcal data, coned the term regonalzed varable to represent varables dstrbuted n space. Such varables possess both a random aspect, whch accounts for local rregulartes, and a structured aspect, whch reflects large scale tendences (Armstrong, 1998 : 16) and are deally suted to descrbe almost all varables encountered n the earth scences (Journel & Hujbregts, 1978 : 27). Consder, for example, a gold-ore depost. The grade of the gold-ore can be thought of as a regonalzed varable, snce values of the grade are dstrbuted n space and dsplay both a structured aspect (areas of rch and poor grade) and a random aspect (local varablty n grade measurements). From a mathematcal pont of vew, a [regonalzed varable] s smply a functon f(x) whch takes a value at every pont x [wthn a studed spatal regon] (Journel & Hujbregts, 1978 : 27). So, a random varable can be seen as varyng randomly accordng to some probablty dstrbuton, whle a regonalzed varable can be seen as a varable varyng regonally (.e. accordng to ts spatal locaton). 4.2 The Probablstc Model of Geostatstcs Geostatstcs utlzes a probablstc model whch s based on random functons and regonalzed varables (Journel & Hujbregts, 1978 : 29 and Wackernagel, 51
1995 : 25). A random functon s merely a set of random varables and s also commonly referred to as a stochastc process or random feld (Cresse, 1991 : 30 and Armstrong, 1998 : 17). Herewth follows a descrpton of the probablstc model of geostatstcs. Consder a spatal attrbute (such as gold-ore grade) defned over some spatal regon Ω, and suppose further that n values of ths attrbute have been measured at varous locatons wthn Ω. In order to perform any geostatstcal calculatons on the attrbute, the probablstc model of geostatstcs assumes the exstence of the random functon (.e. the set of random varables) { ( x), x Ω} Z (4.1) where Z(x) s the random varable defnng the value of the spatal attrbute at spatal locaton x, and Ω s a fxed regon whose spatal ndex x vares contnuously throughout Ω (Cresse, 1991 : 8, 30). Note 4.1 In geostatstcs, the random functon n Equaton 4.1 s often smply referred to as the random functon Z(x). Thus, Z(x) can refer to ether a random functon as n Equaton 4.1, or t can refer to a random varable stuated at spatal locaton x. It s generally always understood whch meanng of Z(x) s ntended, and throughout ths dssertaton t wll be made clear whch s ntended. Note 4.2 It s generally accepted n geostatstcs that Ω 2 or Ω 3 (.e. the studed spatal regon s generally of ether two- or three-dmensons). Thus, although spatal locatons n ths dssertaton wll generally be referred to smply by a sngle letter, say x (as has been done thus far n Chapter 4), ths notaton actually mples a spatal locaton vector n two- or three-dmensons. In other words, x wll ether mply the spatal locaton (x 1, x 2 ) n two-dmensons, or the spatal locaton (x 1, x 2, x 3 ) n three-dmensons. 52
A realzaton of the random functon Z(x) n Equaton 4.1 s classfed as the regonalzed varable z(x), and therefore the avalable spatal data set { ( x ) 1,..., n} z : = (4.2) s smply a number of observatons selected from the regonalzed varable z(x) (Journel & Hujbregts, 1978 : 30 and Wackernagel, 1995 : 26-27). Cresse (1991: 8-11, 29-30) provdes an excellent explanaton of ths probablstc model. Note 4.3 As wth Note 4.1, the notaton z(x) can refer to both a realzaton of the random functon Z(x) (.e. a regonalzed varable), or a realzaton of the random varable Z(x) defnng an attrbute value at spatal locaton x. Ths dssertaton wll agan nsure that t s clear as to whch meanng of z(x) s ntended. Note 4.4 Journel & Hujbregts (1978 : 29-30) pont out that the random functon construct n Equaton 4.1 s deally suted to the defnton of regonalzed varables, snce the value at each locaton wthn the spatal area s theoretcally the outcome of a random varable, whch need not be ndependent of the random varables at the other locatons. Snce geostatstcal data s vewed as a partal realzaton of a random functon, t s not possble to deduce the probablstc laws central to ths random functon solely from the avalable spatal data. Journel & Hujbregts (1978 : 30) state: Obvously, t s not rgorously possble to nfer the probablty law of a [random functon] Z(x) from a sngle realzaton z(x) whch s, n addton, lmted to a fnte number of sample ponts x. Therefore, n order to proceed wth the probablstc model of geostatstcs, certan assumptons need to be made. Some of the most mportant of these assumptons are dscussed n Secton 4.3. 53
4.3 Assumptons The followng assumptons permt estmaton of the probablstc laws governng the random functon Z(x) from a sngle partal realzaton. 4.3.1 Second-order statonarty The random functon defned n Equaton 4.1 s deemed second-order statonary f: and { ( x) } = m, x Ω E Z (4.3) ( ( x h), Z( x) ) = C( h), cov Z + x, x + h Ω (4.4) Ths assumpton mples that the frst two moments of all the random varables contaned n the random functon exst and are constant, and that the covarance between any two random varables depends only on the dstance vector between them and not on ther actual postons n Ω (Journel & Hujbregts, 1978 : 32 and Armstrong, 1998 : 18). In fact, accordng to Journel & Hujbregts (1978 : 33), n many stuatons n geostatstcs the assumpton of second-order statonarty can be weakened such that a covarance between the random varables as n Equaton 4.4 need not even exst at all. Ths weaker assumpton s called ntrnsc statonarty. 4.3.2 Intrnsc statonarty The random functon defned n Equaton 4.1 s sad to be ntrnscally statonary f: and ( ( x + h) Z( x) ) = 0 x, x + h Ω E Z (4.5) ( ( x + h) Z( x) ) = 2 ( h), x, x + h Ω var Z γ (4.6) Accordng to Armstrong (1998 : 19) the ntrnsc assumpton smply assumes that the ncrements of the random varables from the random functon Z(x) are 54
second-order statonary. Intrnsc statonarty assumes that the varablty of the dfference between two random varables from Z(x) depends only on the dstance vector between them, and not on ther actual postons wthn Ω. Note 4.5 A second-order statonary random functon s also ntrnscally statonary, but the converse s not necessarly true (Journel & Hujbregts, 1978 : 33). Note 4.6 The prevous two assumptons are commonly assocated wth lnear statonary geostatstcs (see Secton 1.4). Some authors, such as Rendu (1981), express Equaton 4.5 more generally as ( Z ( x + h) Z( x) ) = m( h) E (4.7) where m(h) s a lnear drft functon, whch mples the absence of drft, or the presence of a lnear drft (Rendu, 1981 : 17). Although n such cases the defnton of ntrnsc statonarty s slghtly dfferent to that gven n ths dssertaton, these authors all concede that when applyng the assumpton of ntrnsc statonarty to lnear statonary geostatstcs, m(h) n Equaton 4.7 must be equal to zero. Note 4.7 The functons C(h) n Equaton 4.4 and 2γ(h) n Equaton 4.6 are commonly referred to as the covarance functon and varogram respectvely. The functon γ(h) (.e. half of the varogram) s called the sem-varogram. 4.3.3 Ergodcty: An assumpton that s rarely stated s that of ergodcty. Ergodc random functons are second-order statonary (Cresse, 1991 : 53), and the assumpton of ergodcty n geostatstcs s dscussed n Matheron (1989 : 81- Certan texts carelessly defne γ(h) as the varogram, whle other napproprately use the term varogram and sem-varogram nterchangeably. 55
83), Cresse (1991 : 53-58) and Chlès & Delfner (1999 : 19-22). The dscusson n Chlès & Delfner (1999) s partcularly good. Accordng to Chlès & Delfner (1999 : 19-20), a spatal random functon Z(x) s sad to be ergodc n the mean f, for all realzatons z(x), lm 1 V V V z( x) dx = m where V denotes the spatal doman over whch Z(x) s defned, V denotes the volume of V, V s assumed to tend towards nfnty n all drectons, and m s a constant. In other words, ergodcty n the mean allows the determnaton of the mean from a sngle realzaton of the statonary random functon (Chlès & Delfner, 1999 : 20). Ergodcty n the covarance can also be defned (see Chlès & Delfner, 1999 : 21), as can ergodcty n any other moment of the random functon. Thus the assumpton of ergodcty n spatal statstcs mples that the moments of all realzatons z(x) of a random functon Z(x) converge to the moments of Z(x) as the spatal area n whch z(x) occurs tends towards nfnty (Chlès & Delfner, 1999: 19-22 and Chlès, J.-P. 2004, pers. comm., 21 Aprl). However, n geostatstcs, the spatal regon studed does not tend towards nfnty, and only one realzaton of Z(x) s consdered thus, there s no way of knowng f the [moments] would have converged to a dfferent value on another realzaton (Chlès & Delfner, 1999 : 21). Therefore, for the probablstc model of geostatstcs, ergodcty s smply assumed as a property of the random functon n Equaton 4.1. In fact, accordng to Cresse (1991 : 58), the assumpton of ergodcty n geostatstcs can be Accordng to Chlès & Delfner (1999 : 22), Matheron (1978) ntroduced the concept of mcroergodcty to deal wth cases of spatal domans that do not tend towards nfnty. 56
weakened to smply assume that the mean and covarance (or sem-varogram) of the sngle realzaton z(x) converges to the true mean and covarance (or semvarogram) of the random functon Z(x). 4.4 Alternatve Approaches to Spatal Estmaton Up to now the probablstc model of geostatstcs and the assumptons whch allow for geostatstcal estmaton have been dscussed. Varous other approaches to spatal estmaton also exst, each wth ther own model and assumptons. Besdes geostatstcs, three of the more common spatal estmaton technques nclude the nverse dstance weghtng method, the method of least squares regresson and the polygonal method. 4.4.1 Inverse dstance weghtng Ths s an uncomplcated and appealng technque that assumes that the relatonshp between values depends [only] on the dstance between ther locatons [and sometmes, also the drecton] (Clark and Harper, 2000 : 190). Suppose that the value of a spatal attrbute has been measured at n spatal locatons, resultng n the spatal data set { ( x ) 1,..., n} z : =. If the unknown value of the attrbute at spatal locaton x 0 s estmated by Q, the nverse dstance weghtng technque assumes that Q n w z = 1 = n = 1 ( x ) w (4.8) where the weghts w are based on some nverse functon of the dstance d between the locatons x and x 0 (Isaaks & Srvastava, 1989 : 257-258). Accordng to Clark & Harper (2000 : 190), examples of weght functons regularly used are 1 d, 1 d 2 and e d. 57
To llustrate the nverse dstance weghtng method, consder the followng example taken from Isaaks & Srvastava (1989 : 257): Example 4.1 Suppose that the value of seven pont support samples has been measured and that an estmate of the value z(x 0 ) at the locaton (65E, 137N) s desred (see Fgure 4.1). Suppose further that t s assumed that the relatonshp between sample values depends only on the nverse of ther Eucldean dstance d apart. Table 4.1 dsplays the requred calculatons for estmatng the value z(x 0 ) usng the method of nverse dstance weghtng. Fgure 4.1 The locaton of the seven measured pont support samples, as well as the locaton of the pont to be estmated (whch s ndcated by the arrow). Fgure 4.1 has been taken drectly from Isaaks & Srvastava (1989 : 251). Usng the calculatons dsplayed n Table 4.1 together wth Equaton 4.8, the nverse dstance weghtng estmate of the value z(x 0 ) s Q = 594. Regrettably, ths straghtforward method possesses many shortcomngs whch are hghlghted by a large amount of geostatstcal lterature (e.g. Clark and Harper 2000, Journel & Hujbrechts 1978, Wackernagel 1995, and Wellmer 1998). Accordng to these authors, nverse dstance weghtng methods: 58
59
Fal to defne measures of dealng wth samples that are not all of the same support (consder the mplcatons by revewng Secton 2.4.1 and Note 3.1). Often break down n tryng to provde estmates at the locaton of samples used n the calculaton. Ths problem s hghlghted by the queston of Clark & Harper (2000 : 192): What happens when d becomes zero? None of the 1 d type functons can be calculated f d s zero. May supply based solutons. These are but a few of the many problems assocated wth the method of nverse dstance weghtng most of whch are not problems for geostatstcal estmaton. 4.4.2 Least squares regresson Suppose the value of a spatal attrbute has been measured at n locatons. The theory of least squares regresson can be used to produce a predcton surface over the study area. A key assumpton of ths technque s the ndependence of resduals, whch s unfortunately not generally the case for geologcal varables (Armstrong, 1998 : 16). Fttng a surface whch does result n uncorrelated resduals s both tedous and results n complex regresson equatons (see Note 4.8). Unlke krgng, least squares regresson s not an exact nterpolator (Koch & Lnk, 1971 : 43-44) and, accordng to Journel & Hujbregts (1978 : 343), least squares regresson estmates are not as good (n terms of mnmum estmaton varance) as krgng estmates. Note 4.8 Accordng to Armstrong (1998), Gomez & Hazen (1970) ftted a regresson equaton to descrbe the proporton of pyrtc sulphur n a certan type of coal, whch resulted n uncorrelated resduals. Ths regresson equaton s coped straght from Armstrong (1998 : 17) and dsplayed n Table 4.2. Armstrong (1998 : 17) wrtes: Ths [equaton n Table 4.2] assumes uncorrelated errors, whch forces the [predcton] surface to twst and turn 60
rapdly, hence the trgonometrc and exponental terms. The dffcultes assocated wth usng such a complcated equaton (e.g., the estmaton of the equaton parameters, the concept of parsmony, etc.) are obvous. Table 4.2 The regresson equaton ftted to descrbe the proporton of pyrtc sulphur n a certan type of coal (taken from Armstrong, 1998 : 17). 4.4.3 Polygonal estmaton Polygonal estmaton s nothng more than choos[ng] as an estmate the sample value that s closest to the pont [or block beng estmated]... (Isaaks & Srvastava, 1989 : 250). In the case where two or more samples are equally close to the pont or block beng estmated, the arthmetc average of these sample values s used as the estmate. Example 4.2 Consder the estmaton of z(x 0 ), the pont located at (65E, 137N) n Example 4.1. In ths example, z(x 0 ) was estmated to be equal to 594. If polygonal estmaton was used nstead to estmate the value of z(x 0 ), t s seen n Table 4.1 that sample 61
two s located the closest to (65E, 137N), and hence z(x 0 ) s estmated as 696 (see Isaaks & Srvastava, 1989 : 251). The beneft of polygonal estmaton s that t s straght-forward to apply. Unfortunately, ths estmaton technque also possesses numerous shortcomngs some of whch are mentoned n Isaaks & Srvastava (1989 : 251-252), Goovaerts (1997 : 356) and Armstrong (1998 : 9). These shortcomngs nclude the fact that polygonal estmaton does not take nto account the spatal structure and contnuty of the attrbute beng studed and estmated. polygonal estmaton does not take nto account the support effect (see Secton 2.4.1 and Note 3.1). Suppose, for nstance, that pont support samples are used to estmate a block support area/volume va the polygonal estmaton method. Armstrong (1998 : 9), descrbng ths scenaro n a mnng context, rghtly warns that,...the polygonal method equates the grades of the samples (.e. a small support) wth those of the blocks [and hence does not] take account of the dfference between the supports of the samples and the blocks to be estmated; that s, of the support effect. the predcton surface of the polygonal estmaton method s dscontnuous. Note 4.9 Polygonal estmaton s also known by the names Thessen polygons and Vorono polygons (Olea, 1990). Note 4.10 Polygonal estmaton can be seen as an extreme case of nverse dstance weghtng, where the relatonshp between samples s assumed to be adequately explaned by p 1 d where p s assumed to be extremely large (n fact, tendng towards nfnty). 62
4.4.4 Addtonal comments Although geostatstcs would appear superor to nverse dstance weghtng, least squares regresson and the polygonal method (snce geostatstcs s not affected by the varous shortcomng lsted for each of the methods dscussed n Secton 4.4), these three methods stll retan mportant roles n spatal estmaton. Not only are the mathematcal calculatons of these methods undemandng, but regresson, for nstance, also plays a sgnfcant role n dentfyng and elmnatng trend from non-statonary data (see Secton 7.12.2). Note 4.11 An experment usng a smulated spatal data set s presented n Secton 8.3, and n ths experment the estmaton results of krgng, nverse dstance weghtng and polygonal estmaton are compared. 63