Short Note: Merging Secondary Variables for Geophysical Data Integration

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Gloria Maxwell
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1 Short Note: Mergg Secodary Varables for Geophyscal Data Itegrato Steve Lyster ad Clayto V. Deutsch Departmet of Cvl & Evrometal Egeerg Uversty of Alberta Abstract Multple secodary data from geophyscal measuremets or expert terpretatos are ofte avalable. Geostatstcal methods are capable of dealg wth a arbtrarly large umber of secodary data; however, buldg a lct probablstc model of redudacy ad closeess to the varables of terest s cumbersome. It s desrable to merge multple secodary data to a super secodary varable wth a calculated correlato to the varable of terest. Ths ote descrbes oe procedure to do ths uder a multvarate Gaussa model. Mergg Secodary Data The tegrato of data geostatstcal modelg ad smulato s ecessary to obta reasoable models of heterogeety ad estmates of ucertaty. The formato represeted by secodary varables ca help to better express treds the prmary varable. Icorporatg secodary data ca prove dffcult whe several varables exst. It would be advatageous to merge these to a sgle super varable, allowg covetoal cokrgg ad cosmulato to be used. Ideally for ths method the secodary varables should be ucorrelated to oe aother ad hghly correlated to the prmary varable. Ths stuato would mmze the redudacy betwee the data. For example, cosder two secodary varables that each have a correlato ρ=0.6 to the prmary varable. The best possble correlato of a super varable to the prmary varable would be ( )=0.686 f the secodary data were completely depedet. If the two secodary varables were fully depedet the the correlato coeffcet would rema at 0.6 due to the redudacy betwee the data they cota. For stuatos where the secodary data s ot fully depedet or depedet, the super varable ca be foud usg a lear estmator: y * = = λ () where y * s the value of the super varable ad λ s the weghtg for each of the varables, y. It s useful to work wth stadardzed varables so that the uts do ot cause cofuso. Stadardzato ths case s acheved through the equato: y z y m = (2) σ where z s the value of varable, m s the mea, ad σ s the stadard devato. Ths results a varable y wth a mea of zero ad a stadard devato of oe. The orgal shape of the statstcal dstrbuto s mataed by equato (2). 3-

2 Oce the varables have bee stadardzed, the weghts ca be foud from a -by- system of ormal equatos: j= λ ρ = ρ =, (3) j, j,0, These weghts, oce determed, ca be used to fd the estmate of the super varable from equato () ad the estmato varace of the super varable: 2 σ = λ ρ (4) L = Correlato wth the prmary varable wll always be hgher for the super varable tha for ay of the dvdual secodary varables. The correlato coeffcet betwee the super varable ad the prmary varable to be estmated ca be foud from the equato:,0 * = λ ρ y,0,0 = ρ (5) Examples Suppose a data set has a prmary varable ad two secodary data sets. The secodary varables have a correlato coeffcet of Ther correlatos to the prmary varable are ad The system of equatos to be solved s λ λ 2 = λ + λ 2 = Solvg for the estmator weghts yelds λ =0.60 ad λ 2 =0.4. The correlato of the resultg super varable ad the prmary varable s ρ=0.92. Ths s sgfcatly hgher tha ether of the dvdual secodary varables addto to smplfyg the estmato of the prmary varable. As aother example, suppose there are ow three secodary varables to be merged. The correlato coeffcets are gve Table. Note that varable 2 has a egatve correlato to each of the other three. 2 3 Prmary Prmary Table : Coeffcets of correlato betwee all of the dfferet varables for the secod example. The weghts for each varable are λ =0.2962, λ 2 = , ad λ 3 = λ 2 s egatve, whch s to be expected because of the egatve correlato to the other varables. The correlato betwee the merged super varable ad the prmary varable s ρ= The lear estmato varace s Correlato to the prmary varable s oce aga sgfcatly hgher for the merged super varable tha ay of the secodary varables aloe. 3-2

3 Real data was used to determe the weghts for mergg three secodary varables. The publclyavalable data set has four varables, usg dfferet uts; for ths example oe was arbtrarly chose as the prmary varable. There are 67 data pots cotaed the fle. To determe the correlato betwee the dfferet data types, declusterg weghts were frst foud usg the program declus. Next, all of the varables were plotted agast oe aother usg scatplt. The results from ths are show Fgure. Table 2 shows a summary of the coeffcets of correlato. 2 3 Prmary Prmary Table 2: Coeffcets of correlato betwee all of the dfferet varables for the thrd example. Solvg ths problem gves weghts of λ =0.0907, λ 2 =0.5702, ad λ 3 = The coeffcet of correlato betwee prmary varable data ad the merged super secodary varable s ρ= Ths s slghtly better tha the correlato betwee the prmary varable ad secodary varable 2. A scatterplot of the prmary varable vs. the super varable s show Fgure 2. Note that the prmary ad super varables have bee back-trasformed, whch does ot affect the correlato. Coclusos The use of a sgle merged secodary varable greatly smplfes estmato accoutg for secodary data. Ths could prove useful a varety of cases, such as whe may dfferet metal types are preset ore or whe umerous sesmc data are avalable. Ths secodary data must be accouted for somehow to properly estmate the resource questo. The relatve smplcty of smulatg wth oly oe secodary varable stead of may makes ths techque attractve. Refereces. C.V. Deutsch ad A.G. Jourel. GSLIB: Geostatstcal Software Lbrary ad User s Gude. Oxford Uversty Press, New York, 2 d Edto, O. Leuagthog. Multvarate Geostatstcal Smulato at Red Dog Me, Alaska, USA. I Cetre for Computatoal Geostatstcs, Report 5, Edmoto, AB,

4 Fgure : Scatterplots used to fd the correlato betwee the dfferet varables the thrd example. 3-4

5 Fgure 2: A scatterplot of the prmary varable vs. the merged super secodary varable. 3-5

K-Map 1. In contrast, Karnaugh map (K-map) method provides a straightforward procedure for simplifying Boolean functions.

K-Map 1. In contrast, Karnaugh map (K-map) method provides a straightforward procedure for simplifying Boolean functions. K-Map Lesso Objectves: Eve though Boolea expressos ca be smplfed by algebrac mapulato, such a approach lacks clear regular rules for each succeedg step ad t s dffcult to determe whether the smplest expresso