Bo co a thng k trong m hnh ha va GVHD: TS. T Quc Dng

Bo co a thng k trong m hnh ha va GVHD: TS. T Quc Dng

Mc lcI.nh ngha a thng k:31.a thng k l g?32.Thnh phn c bn ca a thng k:3II.Tng quan khng gian: c tnh v m hnh.31.M hnh hm ngu nhin (Random Function model):32.Yu cu v tnh tnh ti (stationarity):4a.nh ngha tnh tnh ti:4b. ngha gi thit tnh ti:4c.V d v gii php vn tnh ti:43.Khi nim hm ngu nhin:74.Tnh ton thc nghim Variogram:10a.Thit lp bin chnh xc:11b.S chuyn i ta :11c.Chn hng variogram v lch khong cch:12d.Minh gii v m hnh Variogram:15Minh gii Variogram15Tnh d hng15Tnh chu k16Nhng hng quy m ln (Large Scale Trends):17M hnh variogram18e.Workflow:21III.Khi nim lp bn a thng k:231.Gii thiu:232.c tnh:243.Kriging:27a.Simple Kriging (SK):27b.Ordinary Kriging (OK):29c.Cokriging:30d.Universal Kriging (UK):324.M phng c tnh thch hc:355.Ti sao theo phn phi (chun) Gaussian?36

I. nh ngha a thng k:1. a thng k l g? a thng k nghin cu cc hin tng thay i trong khng gian hoc theo thi gian (Deutsch, 2002). a thng k c th c xem nh mt s thu thp ca k thut s nhm gii quyt cc c im ca thuc tnh khng gian, s dng ch yu m hnh ngu nhin trong mt cch tng t nh cch m cc nh phn tch chui thi gian m t c im cho d liu thi gian (Olea, 1999). a thng k a ra cch m t tnh lin tc trong khng gian ca cc hin tng t nhin v cung cp s thich ng ca k thut hi quy c in tn dng li th ca tnh lin tc ny (Isaaks and Srivastava, 1989).a thng k gii quyt d liu c tng quan khng gian.Tng quan: s tng quan gia cc yu t ca mt chui v nhng chui khc t nhng chui ging nhau b tch bit bi mt khong cch cho trc.Mt s thng s tng quan khng gian c quan tm trong cng ngh m: tng, b dy va, rng, thm.2. Thnh phn c bn ca a thng k: (Semi)variogram: c tnh tng quan khng gian. Kriging: php ni suy ti u, to ra c tnh tuyn tnh tt nht khng lch to mi v tr, s dng m hnh semivariogram. Stochastic Simulation: to ra cc hnh nh ca bin vi xc sut ngang nhau, s dng m hnh semivariogram.a thng k thng l cng c chnh trong vic xy dng m hnh va trong Petrel (SLB) v RMS (Roxar), c s dng to ra cc li ca tng , rng, thm cho va.II. Tng quan khng gian: c tnh v m hnh.1. M hnh hm ngu nhin (Random Function model):Mc ch ca a thng k l c tnh gi tr ti v tr khng c thng tin bng vic s dng cc d liu ly mu c sn pht trin cc m hnh chc chn v s dng nhng m hnh ny d on cc gi tr ti nhng v tr khng c mu. Nu c th pht trin hon ton mt m hnh tt nh da trn s pht trin ca va, c th d on chc chn mi c tnh va ti mi v tr. Tuy nhin, khng th c kin thc ny, do , cch gn ng c thc nghim v n khng chc chn. phn nh s khng chc chn ny, cn c tnh chng nh mt bin ngu nhin. c tnh gi tr, cn x l gi tr c ly mu nh nhng bin ngu nhin do khng th bit mt con ng chnh xc i n gi tr ly mu ny. Thiu b kin thc y lin quan n s c mt ca mt gi tr c bit ca mt bin ti mt v tr c bit bin minh cho vic x l cc v tr c ly mu nh nhng bin ngu nhin. Cc mu thc t ch n gin l thc hin cc bin ngu nhin ny. tnh ton cho s thiu b kin thc ny, cn phi x l nhng v tr ly mu v khng ly mu vi mt m hnh hm ngu nhin (random function model).2. Yu cu v tnh tnh ti (stationarity):a. nh ngha tnh tnh ti:Phn tch d liu khng gian l mt cng vic lm gim m hnh khng gian trong s bin i a cht sao cho r rng v hu ch cho vic tng hp. gii quyt s bin i ca d liu a cht, gi thit tnh ti cho cc c ch a cht c xut.Theo gi nh v tnh tnh ti, v mt nh tnh, yu cu m hnh c xut da trn d liu ly mu c th trnh by y ng x ca mt tp hp. Mun suy lun mt tp hp trn nn tng d liu ly mu, trong bt k trng hp k thut suy lun thng k no, cng khng th chng minh hay bc b gi nh ny m phi cn n ra quyt nh lin quan n nhng thng tin c th s dng m t khu vc quan tm. Mt hm ngu nhin c gi l tnh ti khi quy lut khng gian, thng k ca n l bt bin.Mt hm ngu nhin c gi l tnh ti bc hai (Second-order Stationary) khi: K vng ca hm ngu nhin tn ti v khng ph thuc vo vecto v tr ta Cho mi cp bin ngu nhin Z{x} v Z{x+h}, hip phng sai (Covariance) tn ti v ch ph thuc vo khong cch thay i.b. ngha gi thit tnh ti: Cho php kt lun quy lut khng gian bn di m t hm ngu nhin ch bng vic c tnh gi tr trung bnh v phng sai ca mt bin ngu nhin v hip phng sai ca hai bin ngu nhin khc nhau khong cch. Vi gi thit tnh ti c thit lp v phn chia d liu ph hp, cc nh a cht d dng xc nh c cc lp a cht c v phng ng ln phng ngang. Trong thc t, gi thit ny thit lp mt tha hip gia quy m bin i a cht tnh v lng d liu sn c c tnh thng s ca hm ngu nhin.c. V d v gii php vn tnh ti:Quyt nh tnh ti c th c xem li mt khi phn tch d liu v m hnh a thng k bt u. V d, c th ch mt biu phn b tn sut rng c hai mode (hai nh sng) bn trong cc tng chn. iu khng c ngha l khng tnh ti (non-stationary). Khi , nn quay li d liu v xem xt phn chia d liu thnh hai lp khc bit v c tnh a cht v thng k.

Hnh 1: Biu phn tn lm mn cho 243 d liu rng/ thm (Deutsch, 2002, Geostatistical Reservoir Modeling)V d hnh 2 cho thy d liu c gi l tnh ti khi c tnh ca n khng ph thuc vo khong cch (tr trung bnh khng i) Hnh2: D liu tnh ti v khng tnh ti. (T Quc Dng, bi ging a thng k)

Hnh 3: Mi mi tn l mt hng trong khu vc (T Quc Dng, bi ging a thng k)Khi xt mt khu vc rng ln vi nhiu hng khc nhau, s c nhiu mode, nh vy s khng c xem l tnh ti. Gii php c a ra l chia nh khu vc ln thnh nhiu khu vc nh nhm ng nht d liu v mt thng k, khi , trong nhng khu vc nh s thu c b d liu tnh ti (mt tr trung bnh v mt mode) v vi mi khu vc nh s s dng mt m hnh variogram khc nhau.

Figure 4: Chia khu vc ln thnh bn khu vc nh theo bn hng khc nhau(T Quc Dng, bi ging a thng k)3. Khi nim hm ngu nhin:Cc k hiu: Z: bin ngu nhin ti v tr khng ly mu z: gi tr kt qu ca bin ngu nhin Z Z(u): xc sut phn b ca Z ph thuc vo vecto v tr ta u z(u) gi tr kt qu ca Z(u) ti v tr u F: Hm ngu nhin xc nh cc bin ngu nhin trn khu vc nghin cu.Hm phn phi tch ly (cdf cumulative distribution function) cho bt k hai bin ngu nhin Z(u1), Z(u2) l c bit quan trng do th tc quy c a thng k c gii hn t phn phi n bin F(u;z) n hai bin F(u1, u2; z1, z2):

(2.1)Mt tng hp quan trng ca hm phn phi tch ly hai bin F(u1, u2; z1, z2) l hip phng sai (Covariance) c nh ngha:

(2.2)Khi mt tng hp hon ho hn c cn, hm phn phi tch ly F(u1, u2; z1, z2) c th c m t bng cch chuyn i thnh hm du hiu nh phn:(2.3)Do , cc hm phn phi tch ly hai bin (2.1) ti cc gii hn khc nhau z1 v z2 xut hin nh hip phng sai khng trung tm ca cc bin du hiu:

(2.4)Hm mt xc sut (pdf probability density function) trnh by thch hp hn cho cc bin ri rc:

(2.5) k1, k2 = 1,,KMc ch ca nh ngha hm ngu nhin {Z(u), u khu vc nghin cu A} l khng bao gi nghin cu ni bin Z hon ton c bit. Mc ch cui cng ca m hnh hm ngu nhin l thc hin mt s bo co d on v v tr u ni c kt qu z(u) bit.Hm ngu nhin {Z(u), u khu vc nghin cu A} c gi l tnh ti trong va A nu hm phn phi tch ly a bin ca n l bt bin di bt k s dch chuyn no ca N ta vecto uk:

(2.6)Vi mi s dch chuyn l lS bt bin ca hm phn phi tch ly a bin a n s bt bin ca bt k hm phn phi tch ly no c bc thp hn, bao gm hm phn phi tch ly n bin v a bin, v s bt bin tt c moment ca chng, gm tt c hip phng sai loi (2.2) v (2.3). Quyt nh tnh ti cho php cc suy lun ny. F(z) = F(u, z), uAHm hip phng sai l mt hm thng k c s dng o lng s tng quan. N o lng s ging nhau.Quyt nh tnh ti cn cho php suy lun hip phng sai tnh ti t hip phng sai ly mu ca tt c cc cp gi tr d liu z cch nhau bi vecto h:

(2.7)u, u+h ATi h=0, hip phng sai tnh ti C(0) bng vi variance tnh ti 2:

Biu tng quan tnh ti c chun ha:

Variogram l k thut a thng k thng c dng nht m t cc c tnh tng quan khng gian. V ton hc, c nh ngha:

(2.8)u, u+h ADi quyt nh tnh ti, hip phng sai, h s tng quan v variogram l nhng cng c tng ng m t c tnh tng quan bc 2:

(2.9)S tng quan ny ph thuc vo quyt nh tnh ti ng rng gi tr trung bnh v phng sai khng i v khng ph thuc vo v tr. S tng quan ny l c s minh gii variogram khi m hnh phng sai 2 hu hn: Gi tr on bng Sill ca variogram tnh ti l phng sai, ni m gi tr Variogram ng vi tng quan l 0 (khng c tng quan ti Sill). S tng quan gia Z(u) v Z(u+h) l dng (+) khi gi tr variogram thp hn gi tr Sill. S tng quan gia Z(u) v Z(u+h) l m (-) khi variogram vt qu Sill.Quyt nh tnh ti rt quan trng cho s thch hp v tin cy ca phng php m phng a thng k. Vic t hp d liu ngang qua cc tng a cht c th che du s khc bit a cht quan trng, mt khc, s phn chia d liu thnh qu nhiu phm tr con c th dn n thng k khng ng tin da trn qu t d liu trn mt phm tr. Quy lut suy lun thng k l t hp mt s lng ln nht thng tin thch hp a ra d on chnh xc.S tnh ti l mt c tnh ca m hnh hm ngu nhin, do , quyt nh tnh ti c th thay i nu t l nghin cu thay i hoc nu nhiu d liu tr nn c sn. Nu mc ch nghin cu l ton cc, cc chi tit cc b c th khng quan trng; ngc li, cng nhiu d liu c sn th cng nhiu s khc bit thng k quan trng tr nn c th.4. Tnh ton thc nghim Variogram:Trong k hiu xc sut, variogram c nh ngha nh gi tr k vng:

Variogram l 2(h). Semivariogram l mt na ca variogram (h). Semivariogram cho lch khong cch (lag distance) c xc nh bng trung bnh bnh phng ca mt hiu gia cc gi tr khc bit mt khong h:

Vi N l s cc cp lch h.

Figure 5: c tnh ca Semivariogram Sill: l phng sai ca d liu (bng 1 nu l d liu chun), cho thy bin i ln nht. Bn knh nh hng (range): l khong cch ti im variogram t ti sill, cho thy khong tng quan. Nugget effect: tng ca tt c sai st o lng t l nh (s bin i t l nh).Mt s cu hi phi c gii a ra trc khi tnh ton thc nghim variogram: Cc bin d liu c yu cu s bin i hoc loi b cc hng r rng? C h ta a cht hoc a tng chnh xc cho v tr u v vect khong cch h? Vect lch h l g v sai s c lin quan g nn c xem xt?a. Thit lp bin chnh xc:Tnh ton variogram c n trc bng vic chn bin Z s dng trong tnh ton variogram. Vic la chn bin l quy c hin nhin trong ng dng kriging (M t b cha cn c trn l thuyt trng ngu nhin ni suy cc thng s b cha gia cc ging khoan); tuy nhin, s chuyn i d liu thng trong a thng k hin i. S dng k thut Gaussian yu cu chuyn i trc mt im chun ca d liu v variogram ca d liu c chuyn i ny. K thut Indicator yu cu mt du hiu m ha ca d liu trc khi tnh ton variogramBin chnh xc cn ph thuc vo cc hng s c x l nh th no trong vic xy dng m hnh tip theo. Thng th, cc vng r rng v cc hng thng ng c loi b trc khi m hnh a thng k v sau s d (gi tr ban u tr hng) c thm vo m hnh a thng k. Nu th tc m hnh hai bc ny c xem xt, th variogram ca d liu d l cn thit.Mt kha cnh khc trong vic chn bin chnh xc l s pht hin v loi b gi tr ngoi lai. Gi tr d liu cc k cao v thp c nh hng ln n variogram do mi cp c bnh phng trong tnh ton variogram. D liu sai st nn c loi b. ng lo ngi hn l nhng gi tr cao hp l c th che du cu trc khng gian ca phn ln cc d liu. S chuyn i thnh logarit ha hoc im chun lm gim nh hng ca gi tr ngoi lai, nhng ch thch hp nu mt s chuyn i tng thch ngc li c xem xt trong tnh ton a thng k sau . b. S chuyn i ta :S chuyn i ta l cn thit trc khi tnh ton Variogram. Trong s c mt ca cc ging ng, variogram ng khng ph thuc vo chuyn i ta a tng, min l vic tnh ton c gii hn d liu trong phm vi lp a tng v loi tng thch hp. Variogram ngang rt nhy vi s chuyn i ta a tng. C gng tnh ton variogram trc khi chuyn i a tng c th lm cho ngi lp m hnh a ra kt lun sai st rng d liu khng tng quan ngang. D liu tha tht cng c th dn n kt lun ging nh vy.Mt c im c trng ca hin tng a cht l s tng quan khng gian. Chuyn i ta sai, thiu d liu, sai st trong tnh ton cc thng s, v nhiu nhn t khc c th dn n kt lun sai st rng khng c tng quan khng gian. Mt m hnh pure-nugget khng nn c chp nhn.Khi thnh h a cht c un np rng ri, cc chuyn i ta chi tit theo cu trc ng cong phi tuyn l cn thit.S chuyn i d liu v ta l iu kin tin quyt cn thit tnh ton v minh gii variogram. Mt khi d liu c chun b cho tnh ton variogram, cn thit chn khong cch lch, gi tr h xem xt.c. Chn hng variogram v lch khong cch:Tng quan khng gian rt him khi ng hng, ngha l tng quan khng gian rt him khi ging nhau trong tt c cc hng. Khi mt c tnh thay i theo hng hoc khong cch c gi l bt ng hng. Do a thng k c hnh thnh trc trong 3D nn yu cu mt s xc nh tng quan khng gian trong c ba hng, v hu ht cc va biu hin bt ng hng 3D. Do vy, phn tch variogram c thc hin lp i lp li. Variogram u tin nn c tnh ton theo mi hng, khng xt cc hng bt ng hng v trong mt phng ngang.Vic tnh ton variogram thc nghim theo mi hng yu cu mt lch khong cch (lag distance), mt lch dung sai (lag tolerance), v s cc lch (number of lags). Mt c tnh tt u tin cho lch khong cch l mt lch khong cch bng vi trung bnh khong cch gia cc mu. Variogram c xc nh bng trung bnh bnh phng ca mt hiu gia cc d liu cch bit mt khong xp x bng h. N gn nh c th tnh ton variogram cho cc d liu cch bit chnh xch bng khong cch h, do vy cn mt lch khong dung sai (lag distance tolerance). Mt im khi u tt cho lch khong dung sai l gia lch khong cch hoc bng lch khong cch. S cc lch khng nn vt qu nhiu hn 2/3 va nghin cu (hnh 6).Mt lch khong cch chp nhn c cho mt variogram thc nghim theo mi hng yu cu mt cch tip cn lp li nhiu ln, lch khong cch v dung sai lch phi c tinh chnh.

Figure 6: Mt m t v lch, dung sai lch, gc phng v, dung sai gc phng v v thng s bng tn ca m hnh variogram (T Quc Dng, Bi ging a thng k)Sau khi tnh ton variogram thc nghim theo mi hng phi cc nh hng lin tc ti a v ti thiu m xc nh tng quan khng gian 3D. xc nh tnh lin tc trong khng gian 3D yu cu cc m hnh variogram cho ba hng: hng lin tc ti a, hng lin tc ti thiu v mt hng khc. Tnh ton cc m hnh variogram ny v t hp chng xc nh tng quan khng gian 3D. Trong a thng k, hng lin tc ti thiu c xc nh vung gc vi hng lin tc ti a. Vic xc nh ny trong cho khng gian a thng k 2D. Trong a thng k 3D, hng cn li c xc nh vung gc vi mt phng 2D (hnh 7). y l mt phng php y xc nh khng gian lin tc 3D ca va.

Figure 7: Cc hng trong khng gian 3D (Geostatistics in 12 lessons)C ba thng s c yu cu xc nh khng gian 3D lin tc Hng ca khng gian lin tc Cc hng ca m hnh variogram Dung sai gc phng vMt cng c hu ch xc nh hng lin tc ti a v ti thiu l bn variogram. Bn variogram tnh ton variogram t trung tm ca bn v tr v ta trn ra ngoi theo hng cng/ngc chiu kim ng h. Thm vo , hng lin tc ti a c th c tm thy bng cch nghin cu variogram a ra bn knh nh hng ln nht hoc tham kho bn ng ng mc ca c tnh thch hc. Vi vic xc nh lch khong cch v dung sai lch rt kh tnh ton variogram dc theo mt hng, do cn xc nh dung sai mt hng hoc dung sai gc phng v. Hnh 6 m t khi nim dung sai gc phng v. Mt im khi u tt l dung sai gc phng v 22.50, cho tng dung sai gc phng v l 450. Vi m hnh variogram theo mi hng, mt thit lp tt cho variogram 3D yu cu mt tip cn lp li nhiu ln; lch khong cch v dung sai c th khc bit trong mi hng, dung sai gc phng v c th cn tinh chnh nhn c variogram thc nghim tt, v hng lin tc ti a c th cn tinh chnh. Ch rng hai hng khc c c nh. Hng lin tc ti thiu lun vung gc vi hng lin tc ti a v hng th ba lun vung gc vi mt phng lin tc.

Figure 8: Bn m t hng lin tc ti a v ti thiu (Geostatistics in 12 lessons)d. Minh gii v m hnh Variogram:Minh gii VariogramMinh gii variogram rt quan trng. Nhng im variogram c tnh ton khng s dng trc tip c do: Cc kt qu gy nhiu nn c gim bt Minh gii a cht nn c s dng trong m hnh variogram cui cng Cn o lng variogram hp php trong tt c cc hng v khong cch.Cho nhng l do ny, variogram phi c hiu v sau c m hnh thch hp. C mt s im quan trng cho m hnh variogram: Sill l phng sai 2. Nu d liu chun th sill bng 1. Gi tr variogram bn di gi tr Sill ngha l tng quan dng (+), ti gi tr Sill ngha l khng tng quan, v bn trn gi tr Sill ngha l tng quan m (-). Bn knh nh hng l im ni variogram gp Sill, khng phi im ni m Sill xut hin lm phng ra. Mt hiu ng Nugget ln hn 30% l bt thng v cn c iu tra. Tnh d hngNu mt c tnh thch hc c mt lot cc mi tng quan ph thuc vo hng th c tnh thch hc c gi l biu l tnh d hng hnh hc. Nu c tnh thch hc t n Sill trong mt hng v khng c trong hng khc c gi l biu l tnh d hng theo i. Ni cch khc, mt variogram biu l tnh d hng theo i khi variogram khng t ti Sill c k vng. Hu ht d liu va biu l c tnh d hng hnh hc v tnh d hng theo i.

Figure 9: Tnh d hng ca va (Geostatistics in 12 lessons)Tnh d hng theo i c th l kt qu ca hai c tnh va khc nhau: Lp, variogram ngang khng t gi tr Sill k vng do c nhiu lp nh cc hng ang tn ti v variogram khng t c s bin i ton b. Cc hng khu vc, variogram ng khng t c gi tr Sill k vng do mt s khc bit ln trong gi tr trung bnh ti mi ging.Tnh chu kHin tng a cht thng c hnh thnh trong chu k lp li, l mi trng trm tch ging nhau xy ra lp i lp li. M hnh variogram s cho thy c im ny c tnh chu k. Variogram o lng tng quan khng gian s i qua nhng vng chu tng quan dng sau tng quan m trong khi vn theo hng khng tng quan. Hnh 10 m t ct do gi v tng quan semivariance theo phng ng v phng ngang. Semivariance c tnh ton trn s chuyn i im chun (mu ti biu th ht mn c thm thp). ng x chu k theo phng ng v phng ngang c bn knh tng quan ln hn phng ng.

Figure 10: Chu k variogram (Deutsh, Geostatistical Reservoir Modeling)Nhng hng quy m ln (Large Scale Trends):Hu nh tt c cc qu trnh a cht truyn cho mt hng trong vic phn phi c tnh thch hc. lmit ha l kt qu ca thy nhit dng cht lu, i ln lm sch ht vnl nhng hng quy m ln. Hng ny gy ra gi tr variogram leo ln v vt ngng Sill l 1.

Figure 11: Mt log rng (lu t l) t mt chui chu th c hin th bn tri v tng quan variogram im chun c hin th bn phi (Deutsh, Geostatistical Reservoir Modeling)M hnh variogramTt c cc hng variogram phi c xem xt ng thi hiu tng quan khng gian 3D Tnh ton v v variogram thc nghim trong nhng ci c cho l hng chnh lin tc da trn mt kin thc a cht c trc. t mt ng nm ngang i din cho gi tr Sill l thuyt. S dng gi tr ca phng sai (tnh ti)thc nghim cho cc bin lin tc (1 nu d liu c chun ha) v p(1-p) cho cc bin ri rc khi p l t l ton cc cho hng mc quan tm. Thng thng, variogram l c h thng khp vi gi tr Sill l thuyt v tt c phng sai bn di Sill phi c gii thch trong cc bc tip theo. Nu variogram thc nghim tng cao hn Sill, rt c kh nng tn ti mt hng trong d liu. Hng ny nn c loi b chi tit bn trn trc khi tip tc minh gii variogram thc nghim. Minh gii: Phng sai t l nh (Short-scale variance): hiu ng nugget l mt variance khng lin tc ti gc ng vi s bin i quy m nh. N phi c chn bng nhau trong tt c cc hng, chn t hng variogram thc nghim c nugget nh nht. i khi, c th chn h thp n hoc thm ch t n bng 0. Phng sai t l va (Intermediate-scale variance): Bt ng hng hnh hc ng vi hin tng cc bn knh tng quan khc bit trong cc hng khc nhau. Mi hng bt gp ton b s bin i trong cu trc. C th tn ti nhiu hn mt cu trc nh vy. Phng sai t l ln (Large-scale variance): (1) Bt ng hng theo i, c c tnh bi variogram chm ti mt on bng ti mt phng sai thp hn gi tr Sill l thuyt, hoc (2) hiu ng l khoan i din cho mt hin tng mang tnh chu k v c c tnh bi cc gn sng trn variogram. Hiu ng l khoan thm ch khng gp phn vo tng phng sai ca cc hin tng, tuy nhin, bin v tn s ca n phi c nhn bit trong sut th tc mnh gii, ngoi ra, n c th ch tn ti trong mt hng. Mt khi tt c phng sai cc khu vc c gii thch v mi cu trc c lin quan n mt qu trnh a cht, ngi ta c th tin hnh m hnh variogram bng cch chn mt kiu m hnh hp php (m hnh hm cu, hm m, Gaussian) v bn knh tng quan cho mi cu trc. Bc ny c th c coi l phn c tnh thng s ca phn tch variogram. Rng buc m hnh variogram bng mt bc minh gii trc vi s nhn dng cu trc c th dn n vic t ng ph hp ng tin ca m hnh variogram thc nghim.Cc loi m hnh variogram ph bin: Hiu ng Nugget. Hiu ng nugget thng ch nn gii thch khi ln n 30% phng sai. Hiu ng nugget l mt phn ca phng sai do sai st v s bin i t l nh. l c im xy ra ti mt t l nh hn d liu khc bit khong cch nh nht.

M hnh hnh cu. M hnh hnh cu l loi m hnh variogram ph bin nht v c c trng bi mt ng x tuyn tnh ti nhng khc bit khong cch nh v sau cong ti sill bng 1.

M hnh hm m. M hnh hm m ging vi m hnh hnh cu nhng n gn nh tim cn vi sill.

M hnh Gaussian. M hnh Gaussiance l c trng tng quan cao trn bn knh ngn v c dng m hnh cc hin tng c tnh lin tc cao.

e. Workflow:

III. Khi nim lp bn a thng k:1. Gii thiu:

Ti bt k thi im no trong thi gian, lun c mt phn b ng ca mt thuc tnh a cht. Phn b ng l khng c sn, nhng c cch tt nht c th lp bn phn b ng t mt s d liu ly mu. Vic lp bn phn b ng cng chnh xc c th cho nhiu thut ton ni suy c pht trin. Ph bin nht l phng php Kriging. Kriging l mt phng php ni suy chnh xc v tri chy, thch hp cho hin th ha cc hng, nhng khng thch hp cho m phng dng chy ni bo tn bt ng nht trong va l quan trng. Mt thut ton kriging m rng l m phng lin tc (sequential simulation). M phng lin tc thch hp cho m phng v cho php nh gi ngu nhin vi s thc hin cc phng n thay th. 2. c tnh:Xt vn c tnh gi tr ca mt thuc tnh ti v tr bt k khng ly mu u, c k hiu z*(u), ch s dng d liu mu c tp hp trn khu vc nghin cu A, c k hiu z(un) c m t nh hnh 12

Figure 12: Geostatistics in 12 lessonsThut ton gii quyt vn ny l kriging. Kriging l mt cng c ca ng dng lp bn truyn thng v l mt thnh phn cn thit ca phng php m phng a thng k. Thut ton kriging l mt h thng suy rng cc k thut hi quy bnh bnh cc tiu c tnh z*(u) s dng d liu ly mu z(un). Phng trnh kriging thng dng:

(3.1)Vi z*(u) l gi tr c tnh ti v tr khng ly mu u m(u) l tr trung bnh cho trc ti v tr khng ly mu u, = 1,,n l cc trng s c p dng cho n d liuz(u), = 1,, n l n gi tr d lium(u), = 1,, n l gi tr trung bnh cho trc ti cc v tr d liuTt c cc gi tr trung bnh cho trc c th c thit t thnh mt tr trung bnh khng i m(u)= m(u)= m nu khng c thng tin cho trc trn cc hng l c sn.Xt mt gi tr c tnh ti v tr khng c d liu:

(3.2)Vi mc ch gy tranh ci khi chn trng s: Mt (closeness) v tr c c tnh, gi tr c lng s cch u hai d liu bit. Tha gi tr d liu (redundancy), cc d liu bit u nm mt trong hai bn gi tr c lng s gy kh khn cho vic c lng. Tnh lin tc bt ng hng (anisotropic continuity) ln ca tnh lin tc/bin i (magnitude of continuity/variability)

Figure 13: Trng s kriging phi c xt v tha d liu, cht ca d liu v hng v ln ca tnh lin tc (geostatistics in 12 lessons)Mt mc ch khc khi c tnh cc thuc tnh cha bit l ti thiu ha sai s phng sai. Nu sai s phng sai l cc tiu th c tnh s l c tnh tt nht. Sai s phng sai l gi tr k vng ca chnh lch gia gi tr bit v gi tr c tnh:

(3.3)Vi z*(u) l gi tr c tnh v z(u) l gi tr thc.Mt cu hi hin nhin c a ra phng trnh (3.3) l lm cch no xc nh sai s nu khng bit gi tr thc? Khng bit gi tr thc, nhng c th chn trng s tm sai s cu tiu. cc tiu phng sai c tnh, ly o hm ring phn ca sai s phng sai (phng trnh 3.3) v cho n bng 0.M rng phng trnh (3.3) ta c:

(3.4)Kt qu l mt phng trnh quy v hip phng sai gia cc im d liu C(u,u), v hip phng sai gia im im d liu v im c tnh C(u,u). Hm variogram:

V hip phng sai:

Mi lin h gia variogram v hip phng sai l:

(3.5)Vi (h) l variogram, C(0) l phng sai ca d liu, v C(h) l hip phng sai.Phng trnh (3.5) cho thy c th tnh c hip phng sai da vo variogram v phng sai v sau tnh thut ton kriging.Tip tc o hm phng trnh kriging, cng thc (3.4) phi c ti thiu ha bng cch o hm ring phn vi tng trng s v cho chng bng 0:

Cho o hm bng 0 ta c:

(3.6)Hoc c th thay hip phng sai bng variogram do c variogram v hip phng sai u o lng tng quan khng gian:

(3.7)V ma trn hip phng sai vi 3 trng s l:

3. Kriging:Phn ny s tho lun v mt vi th tc kriging v gi thit rng gi tr c tnh ca bin l tng quan tuyn tnh vi cc mu gn . Ty thuc vo ng dng c bit, cc th tc khc nhau s s dng cho mc ch c tnh khc nhau Simple Kriging: n gin nht nhng khng cn thit cho hu ht thc tin. Ordinary Kriging: Th tc kriging ph bin nht, linh hot hn Simple kriging v cho php cc bin thay i cc b. Cokriging: Cho php c tnh mt bin da trn thng tin khng gian ca cc bin khc lin quan. Th tc ny c bit hu ch khi c mt bin c ly mu rng ri v mt bin c ly mu tha tht v chng c tng quan khng gian. Universal Kriging: Dng khi d liu mu biu hin theo mt phng v gi thit tnh ti c th khng hp l.a. Simple Kriging (SK):SK bt u vi gi thit gi tr ti v tr khng ly mu c th c c tnh theo cng thc:

(3.8)Vi:z*(u0): Gi tr c tnh ti v tr u0z(ui): Gi tr ly mu ti v tr uin: Tng s mu c chn trong mt vng nghin cui: Trng s c gn cho mi mu v 0 khng i c tnh gi tr i, yu cu mt iu kin khng lch:

(3.9)Th z*(u0) vo t (3.8) vo (3.9) thu c:

(3.10) Gi thit E[z(ui)] = E[z(u0)], da trn gi thit tnh bc 1, vit c:

(3.11)Vi yu cu khng lch, iu kin cc tiu phng sai phi c tha mn. V ton hc, trng s c th c chn sao cho biu thc di y l b nht.

(3.12) Kt qu ca iu kin ny l phng trnh

cho i = 1,,n(3.13)Vi:C(ui, uj): Gi tr hip phng sai gia cc im v tr ti ui v ujC(ui, u0): Gi tr hip phng sai gia v tr ly mu ui v v tr khng ly mu u0Cc gi tr hip phng sai thu c da trn m hnh khng gian. Phng trnh (3.13) c th c vit di dng ma trn:

(3.14)Mt khi trng s c c tnh, phng trnh (3.8) s c tnh gi tr z*(u0). Thm vo , c tnh phng sai l:

(3.15)b. Ordinary Kriging (OK):Trong th tc SK, gi thit rng gi tr trung bnh m(u) c bit. Bng gi thit tnh ti bc 1, m(u) gim thnh m. Phi bit gi tr m trc khi s dng mt biu thc SK. Trong thc tin, tr trung bnh thc ton cc rt him khi c bit nu khng gi thit tr trung bnh mu bng tr trung bnh ton cc. Ngoi ra, tr trung bnh cc b trong vng nghin cu ln cn c th thay i trn khu vc quan tm, do gi thit tnh ti c th khng hon ton hp l. Th tc Ordinary Kriging (OK) s khc phc vn ny bng cch xc nh phng trnh c tnh.Xt phng trnh (3.8) c s dng cho SK:

(3.8)Tuy nhin, yu cu v iu kin khng lch l:

(3.9)Gi thit E[z*(u0)] = E[z(ui)] = m(u0), vi m(u0) l tr trung bnh trong v tr min ln cn nghin cu u0, c c:

(3.16)Tuy nhin, c gng cho 0 = 0 bng cch kh gi tr trung bnh v gi thit rng tnh ti bc 1 hon ton hp l (gi thit tr trung bnh cc b ph thuc vo v tr), nu gi s:

(3.17)Th phng trnh c tnh(3.8) c vit thnh

(3.18)Ngoi ra vi iu kin khng lch, yu cu thon mn iu kin phng sai cc tiu. Cc tiu ha phng sai vi rng buc (3.8) thu c kt qu:

vi i= 1,,n(3.19)Vi l thng s Lagrange v C i din cho hip phng sai. Phng trnh (3.19) c th c vit di dng ma trn:

(3.20)Mt khi i c tnh, gi tr c tnh z*(u0) s thu c t phng trnh (3.18). c tnh hip phng sai:

(3.21)c. Cokriging:Cokriging c s dng c tnh mt bin gi tr da trn tng quan khng gian vi cc bin gi tr c ly mu khc. Hai v d ph bin c ng dng cokriging ci thin c tnh trong m t va l c tnh thm s dng d liu rng v c tnh rng s dng d liu a chn. Gi nh rng, gi tr c tnh l z*(u0) ti v tr u0. Trong nghin cu vng ln cn c n mu ca bin chnh Z v m mu ca hip bin (bin tng quan) Y. Phng trnh c lng nh sau:

(3.22)

Vi l trng s c gn cho mu ti v tr

l trng s c gn cho mu ti v tr p dng iu kin khng lch (3.9):

(3.9)Th (3.13) vo (3.2) thu c:

(3.23)Vi mZ v mY ln lt l gi tr k vng ca bin Z v Y. tha mn phng trnh (3.23) th:

v (3.24)Phng trnh (3.24) m bo iu kin khng lch c tha mn.Ngoi ra, cn phi tha mn iu kin cc tiu phng sai:

(3.25)Cc tiu ha phng trnh (3.25) vi hai rng buc c xc nh trong phng trnh (3.24):

(3.26)V

(3.27)Trong hai phng trinh (3.26) v (3.27) vi:CZ v CY: Ln lt l hip phng sai ca hai bin Z v Y.CC: Hip phng sai cho gia hai binZ v Y: Thng s Larange.Phng trnh c th c vit di dng ma trn:

Ma trn bn tri c kch c (n+m+2). Gii phng trnh ma trn trn thu c v , sau th vo phng trnh (3.22) tm c gi tr c tnh.Biu thc sai s phng sai:

(3.28)d. Universal Kriging (UK): Th tc Universal Kriging (UK) c tnh gi tr trong s c mt ca thng tin mu theo mt phng m gi thit tnh ti bc 1 khng tha mn. Tr trung bnh cc b thay i theo hng ca phng v khng gn tr trung bnh ton cc. Loi d liu ny phi c x l trc nhm thit lp gi thit tnh ti v s dng k thut kriging c tnh.Vic hp nht hng ca d liu cn xc nh trc bin d liu:

(3.29)Vi z(u) l bin ti v tr quan tm u, m(u) l gi tr trung bnh hoc xu hng (drift), v R(u) l gi tr d. Nu chia bin thnh tr trung bnh v gi tr d, bng vic loi b tr trung bnh t d liu, thu c gi tr d tha mn yu cu tnh ti bc 1.Nu c mt hng trong b d liu, vi mi im d liu, v mt khu vc nghin cu ln cn ti thiu trong hng . Tnh ton gi tr trung bnh ca tt c mu trong vng ln cn ny v gi thit l gi tr trung bnh cc b. V ton hc:

(3.30)Nu xc nh cc khu vc nghin cu ln cn tng t nh vy s thu c: Gi tr trung binh cao hn im mu nu tt c mu bao quanh l cao Gi tr trung bnh thp hn im mu nu tt c mu bao quanh l thpSau , tr i gi tr trung bnh cc b t mi mu n l, thu c:

(3.31)Vi cc gi tr d, cc hng cc b s b loi b, khng biu hin bt k hng no. Bng cch xc nh ny, trung bnh cc gi tr d s bng 0 v gi thit tnh ti c tha mn.V nguyn tc, vic p dng k thut kriging cho cc gi tr d v thm tr li cc hng, c th c tnh cc gi tr ti v tr khng ly mu.Rt kh khn c tnh variogram cho gi tr d ti mi im, v ngoi vic p dng k thut kriging cho cc gi tr d, cn phi c tnh v m hnh ha variogram cho cc gi tr d. Vi la chn c sn c tnh v m hnh ha cc gi tr d: Bng vic s dng cc khu vc ln cn cho mi mu, tr trung bnh cc b c thit lp. Bng cch tr i tr trung bnh cc b (xu hng), gi tr d thu c ti mi v tr v variogram ca cc gi tr d c th c c tnh. Tm hng (direction) ni cc xu hng (trend) khng c ngha. Gi thit tnh ti bc 1 tha mn trong hng , variogram cho bin ban u c c tnh trong hng v c p dng cho cc hng (direction) ca xu hng (trend).Mt khi variogram c c tnh v m hnh, phi gi thit rng mt loi hng c th l hin din trong d liu. Gi thit ny c yu cu cho UK. Thng thng, cc nh tr trung bnh (drift) bng phng trnh:

(3.32)Vi al l h s ca hm fl(u). Bng nh ngha, lun gi thit f0(u) bng 1.V d, nu c mt xu hng tuyn tnh:

(3.33)V cho xu hng bc 2:

(3.34) Trong sut qu trnh c tnh, gi thit loi hng tuyn tnh, bc 2 bit. Qu trnh c tnh UK bt u ging vi OK

(3.35)p dng iu kin khng lch:

vi l = 0,, L(3.36)Phng trnh (3.36) c L+1 rng buc. Ch vi l = 0, c mt rng buc v tng trng s bng 1. p dng iu kin cc tiu phng sai, thu c:

(3.37)Ma trn bn tri c kch c (n+L+1) v c (L+1) thng s Lagrange ng vi (L+1) rng buc.Gii phng trnh (3.37) thu c cc gi tr cho i v i. T tm c gi tr c tnh theo phng trnh (3.35) v phng sai c c tnh theo phng trnh di y:

(3.38)4. M phng c tnh thch hc:M phng ch ra cc khuyt im ca kriging v vn s dng thut ton kriging v tt c u im ca kriging. Mt trong nhng th quan trng nht m kriging lm ng v cung cp ng lc cho m phng lin tc l n nhn ng hip phng sai gia d liu v hm c lng. Vn m dn n m phng lin tc l mc d nhn hip phng sai chnh xc gia d liu v gi tr c tnh nhng n tht bi nhn hip phng sai gia cc gi tr c tnh. l, thay v s dng cc hm c lng nh cc d liu b sung, thut ton kriging ch n gin l chuyn sang hm c lng tip theo khng cha hip phng sai gia cc hm c lng c v mi. M phng lin tc s lm iu . Hm c lng u tin c m phng ch vi d liu v khng c bt k mt gi tr c tnh no. c tnh tip theo c m phng nhng hm c lng trc c s dng nh d liu v variogram ca n c bao gm trong thut ton.

Figure 14: M phng lin tc (geostatistics of 12 lessons)iu ny ng lc tin hnh c tnh lin tcm nhng c vn l thiu phng sai. Thiu phng sai l phng sai kriging:

Phng sai Kriging ny phi c thm li m khng c s thay i c tnh lp li variogram ca kriging. iu ny c hon thnh bng cch thm mt thnh phn c lp vi mt tr trung bnh bng 0 v phng sai ng vi c tnh kriging:

Quy trnh lm vic ca m phng lin tc nh sau: Chuyn d liu Z ban u thnh mt phn phi chun (tt c cng vic c lm trong khng gian chun). n v tr u v thc hin kriging thu c gi tr c tnh v ng vi phng sai kriging:

V mt gi tr d ngu nhin R(u)theo mt phn phi chun vi tr trung bnh l 0 v phng sai l SK2(u). Thm gi tr c tnh v gi tr d nhn gi tr c m phng:

Ch rng Z*(u) c th thu c bng cch v t mt phn phi chun vi tr trung bnh Z*(u) v phng sai SK2(u). Thm Zs(u) nhm thit lp d liu chc chn rng hip phng sai vi gi tr ny v tt c d on tng lai l ng. tng mu cht ca m phng lin tc l xem nhng gi tr c m phng trc nh d liu m m phng hip phng sai gia tt c cc gi tr c m phng. Ch tt c cc v tr trong th t ngu nhin ( trnh nhng gi to gii hn nghin cu). Chuyn i li tt c gi tr d liu v gi tr c m phng khi m hnh c hnh thnh. To cc s thc hin khc c xc sut nh nhau bng cch lp li vi s khi u ngu nhin khc nhau.5. Ti sao theo phn phi (chun) Gaussian?c tnh mu cht ton hc lm cho m phng lin tc Gaussian (Sequential Gaussian Simulation) lm vic l khng gii hn phn phi Gaussian. S lp li hip phng sai ca kriging bt k phn phi d liu, s hiu chnh phng sai bng cch thm mt gi tr d ngu nhin lm vic bt k hnh dng ca phn phi gi tr d (tr trung bnh phi l 0 v phng sai bng vi phng sai phi c gii thiu li).C mt l do rt tt cho vic gii thch ti sao phn phi Gaussian c s dng: S dng bt k phn phi khc khng dn ti s chnh xc phn phi ton cc ca cc gi tr c m phng. Tr trung bnh c th ng, phng sai ng, variogram ca cc gi tr c ly cng nhau l ng, nhng hnh dng (shape) s khng ng. iu ny l khng vn trong khng gian Gaussian do tt c phn phi l Gaussian, bao gm cc gi tr c m phng cui cng.C l do khc l ti sao phn phi Gaussian c s dng: nh l gii hn trung tm ni rng lin tc thm vo cc gi tr d ngu nhin thu c cc gi tr c m phng dn tin ti mt phn phi Gaussian.

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Chuyn i chun ha cc im

Chuyn i ta

Variogram ng hng ( lch khong cch, dung sai lch)

Tm hng chnh, hng ph (bn variogram, bn ng ng mc)

Tnh ton variogram thc nghim trn hng chnh

M hnh ha mi hng variogram ring bit s dng cng mt nugget, s v loi m hnh

T hp cc hng variogram cho mt m hnh tng quan khng gian hp l